Session KA: Bio-Fluid Dynamics: Cells and Capsules

Simulation of Red Blood Cell Interaction with the Endothelial Cell Surface

It is hypothesized that the stress field on the EC surface felt through hydrodynamic interaction at the extracellular layer, known as the glycocalyx, is the pathophysiological link between the hemodynamics and the cell function. Simulations revealing the general stress distribution on the EC surface, and in particular the mechanical interactions of red blood cells (RBC's) with the EC's will be presented. The current focus is to investigate the drag force and the bending moment on the core proteins in the EC glycocalyx. The glycocalyx has been modeled as a quasiperiodic array of cylinders (Weinbaum et al., PNAS \textbf{100}, 1988-7995, 2003). The height and diameter of the cylinders were assumed to be 150 nm and 6 nm, respectively, and the gap between cylinders was 8 nm. Weinbaum et al. computed the average velocity profile by treating the glycocalyx as a porous medium. The focus of the work to be presented is on the effects upon the EC by close encounters with RBC's over a long period of time. We will present results for the flow in and above a model glycocalyx caused by the motion of a nearby surface. The flow was computed using the lattice Boltzmann method (LBM). The results of the LBM for the mean flow and the bending moment and drag on a model protein fiber will be compared with the predictions obtained from the model of Weinbaum.   

Noninvasive Visualization of Human Capillary Vessel Blood Flow

Human blood flows are highly susceptible to physical and health conditions. Hence quantitative evaluation of Blood flow is a useful parameter in the physical check up of individuals. However, the most convenient method is taking a blood sample, which can only examine ex vivo Blood condition. We turn our attention to the observation of the capillary loops of blood vessels in the finger skin nail fold, in which blood flow can be easily visualized without using complicated specialized tools other than capillaroscopy. We modified both the spatial and temporal resolution in capillaroscopy. A deep-focus high magnification zoom lens and a high speed video camera of 1000 fps allowed us to observe the motion of red blood cells, white blood cells and plasmas. Quantitative analysis of blood flow allowed us to observe the motion of red blood cells in capillary vessels with a diameter of about 10 micro meters. We discuss the quantitative evaluation of blood flow velocity in artery capillary vessels. We also conducted shape analysis of the capillary vessel, by using the level set method. By analyzing the obtained level set function, quantitative evaluation of the capillary blood shape, such as characteristic diameters and curvatures, are carried out.   

The effects of non-Newtonian viscosity on the deformation of red blood cells in a shear flow

The analyses of the effects of non-Newtonian viscosity on the membrane of red blood cells (RBCs) suspended in a shear flow are presented. The specific objective is to investigate the mechanical deformation on the surfaces of an ellipsoidal particle model. The hydrodynamic stresses and other forces on the surface of the particle are used to determine the cell deformation. We extended previous works, which were based on the Newtonian fluid models, to the non-Newtonian case, and focus on imposed shear rate values between 1 and 100 per second. Two viscosity models are investigated, which respectively correspond to a normal person and a patient with cerebrovascular accident (CVA). The results are compared with those obtained assuming a Newtonian model. We observed that the orientation of the cell influences the deformation and the imposed shear rate drives the local shear rate distribution along the particle surface. The integral particle deformation for the non-Newtonian models in the given shear rate regime is higher than that for the Newtonian reference model. Finally, the deformation of the cell surface decreases as the dissipation ratio increases.   

An Experimental Study of a Giant Vesicle in a Simple Shear Flow

Deformation and motion of lipid bilayer vesicles with the diameter of 10-50$\mu $m (giant vesicle, GV) in a simple-shear-flow have been observed using phase contrast microscopy. We developed a simple-shear-flow apparatus, which consists of two cylinders with the diameter of 50mm separated by a narrow gap of 0.5mm. A linear shear is created in the gap and GVs prepared by the gentle hydration method are transferred there. Their behaviors in the flow are observed with microscope from the direction of the axis of the cylinders. In our observation, GVs are deformed to steady ellipsoidal shapes and show constant orientations of \textit{$\theta $}, which is the angle between the major axis and the flow direction. It is also observed that \textit{$\theta $} becomes smaller with decrease of swelling ratio \textit{$\tau $ }, which indicates the degree of deflation. Our experimental result shows good agreement with those of the previous theory [Keller and Skalak, \textit{J. Fluid. Mech}. \textbf{120}, 27-47 (1982)] and numerical simulations.   

The dynamics of a deflated giant Vesicle in a simple shear flow

Giant vesicle (GV) is an artificial capsule which is composed of lipid bilayer and which has the size of several tens microns. In the present study, 3-D numerical simulation for the dynamics of a GV in a simple shear flow was conducted. Immersed-Boundary method was used to simulate a deformation of GV. We expressed a GV model with the fluidity of membrane taken into account and with its volume and surface area kept constant. At first, we calculated the deflated GV as an equilibrium shape. And a long stick-like shape called a prolate shape was obtained. Then we investigated the dynamics of a prolate GV in a simple shear flow for various values of viscosity inside the GV (\textit{$\mu $}$_{in})$ and swelling ratio, Sw. Sw denotes the degree of deflation of a GV. Depending on \textit{$\mu $}$_{in }$, GV showed two kinds of motions. When \textit{$\mu $}$_{in}$ is not large enough, a GV settled down a steady shape with its major axis at a certain angle \textit{$\theta $}. This motion is called tank-treading motion. We investigated the angle \textit{$\theta $} for various values of Sw. And our results are in good agreement with the results by Kraus (1996). Then we investigated the relationship between the angle \textit{$\theta $} and \textit{$\mu $}$_{in}$. As the value of \textit{$\mu $}$_{in}$ becomes larger, the angle \textit{$\theta $} becomes smaller. When \textit{$\mu $}$_{in}$ exceeded the threshold value, a GV started tumbling its major axis in clockwise direction. The transition between a prolate shape and a disc shape was observed during tumbling motion.   

Hydrodynamic interaction of two capsules in simple shear flow

We present a numerical model of the hydrodynamic interactions between two capsules freely suspended in a simple shear flow (SSF). The capsules are identical and consist of a liquid droplet enclosed by a thin hyper-elastic membrane. Such particles can be used in applications where encapsulation of living cells or of active agents in a protecting membrane is necessary. We assume a Stokes flow and use a boundary integral method to represent the fluid motion of the internal and suspending liquids. An isolated capsule subjected to SSF will interact with the two liquids until equilibrium is reached between the in-plane elastic stress and the viscous traction exerted on the membrane. The membrane may undergo very large deformations, thus making the problem non-linear. Monitoring the stress level in the membrane is important to predict burst. When two capsules interact in SSF, they eventually overlap and pass each other. During that process, the membranes are submitted to extra strain/stress which may lead to unexpected break-up. Pairwise interactions also cause an irreversible cross-flow trajectory shift, showing the self-diffusivity of the capsules. The comparison with a pair of droplets shows that the membranes have a strong effect on short range interactions.   

Effect of membrane constitutive equation on the recovery of capsules from large deformations

The recovery of capsules after large deformations can be used to calculate its material properties. We focus our attention on the influence of varying the membrane constitutive model and the initial geometry of the capsule on the recovery process. An axisymmetric computational model based on the boundary element method (BEM) is used to simulate the recovery of capsules from small and large deformations. Comparison is made between capsules having: (1) constant cortical (surface) tension [CCT], (2) two-dimensional Hooke's law [H], (3) Mooney-Rivlin law [MR] and (4) Evans and Skalak [ES] membrane models. At small initial deformations similar behavior is observed for all models and appears independent of initial geometry. The recovery process is more sensitive to initial conditions for large deformations due to the non-linear behavior of the elastic membranes. The difference in the local strain distribution caused by variations in the initial geometry significantly affects the membrane stress field at large deformations, and thus the recovery process.   

\textsc{Numerical simulations of cell interactions under shear flows in complex geometries}

The receptor-mediated leukocyte adhesion and rolling on endothelium under shear flows are of crucial importance in governing a range of cell functions: inflammatory response, lymphocyte homing, and sickle cell vascular occlusion. In vivo, an endothelium-lined blood vessel lumen has a non-flat irregular complex geometry presented to blood flows, and adherent leukocytes can lead to further geometry complexity. This geometry factor can have a prominent impact on the mechanics and hemodynamics of cell interactions and adhesions in high endothelial venules, non-uniform capillaries and post-capillary expansions to name a few. In this work, a ghost-cell immerse boundary/front tracking method is presented to examine the physiological role of the blood vessel geometry in microcirculation. Motions of deformable blood cells are computed via a multiphase front tracking method. Boundary conditions for arbitrary geometries are enforced through a high-order ghost cell immersed boundary method. The current method is validated and used to explore the potential roles of vessel geometry in modulating hemodynamics and kinetics of 2d/3d cell interactions, in particular leukocyte adhesion and accumulation.   

Numerical evaluation of stress contribution by model red blood cells in shear flow

Contributions to the stress of a dilute suspension of biconcave disks in simple shear flow are reported. The disks are a geometric but nondeformable model of red blood cells (RBCs), with biconcave disk geometry. Motion of disks and an equal density suspending liquid is computed using a lattice-Boltzmann equation technique. The disks, unless oriented with the normal to their ``flat'' side oriented along the vorticity direction of the shear flow, are found to tumble in a motion which becomes periodic after an initial transient. The stress contribution of the RBCs thus undergoes a periodic variation, and we report results of the instantaneous and averaged stress contributions of these particles in the dilute limit where computations of a single body motion suffice. The symmetric first moment of the surface force distribution, or stresslet is the primary contribution at low Reynolds number, and its shear and normal stress components are determined. The role of weak inertia, requiring integration over the volume of the fluid to capture the Reynolds stress contribution resulting from velocity fluctuations induced by the model RBC, will be discussed.   

Hydrodynamic interaction among blood cells in microcirculation

Particulate nature of blood plays an important role in many hemodynamic events in small vessels. One example is the Fahraeus-Lindqvist effect which arises due to the flow-induced deformation and lateral migration of red blood cells away from the vessel wall. The lateral migration creates a region of cell- free layer which has a reduced local viscosity, and thus a pronounced effect on the blood rheology and many physiological events. The formation of the cell-free layer also plays an important role in the wall-bounded motion (margination) and vascular adhesion of white blood cells, which are critical steps in the body's immune response. To explore hydrodynamic interactions among various blood cells, under normal and disease conditions, we are developing 2D/3D numerical simulations of multiple deformable cells using front tracking/immersed boundary method. In this talk, we will describe some numerical results on the effect of neighboring particles on the lateral migration, and the development of the cell-free layer. We will also explore the effect of flowing red blood cells on the wall-bounded rolling motion and adhesion of white blood cells, as well as the effect of the white blood cells on the dispersion of the red blood cells.   

Three-dimensional numerical simulation of cell deformation

Blood is a multiphase suspension of various deformable cells. The particulate nature of blood is absent in large blood vessels making a numerical/theoretical analysis somewhat easier. The analysis is also simplified for the flow through small capillaries, where blood cells flow in an ordered, `single-file' fashion. The main difficulty arises for the vessels of $\sim$10--500 micron diameter, where the cells move in a 'multi-file' fashion. The Casson fluid model, used to describe blood flow in such vessels, often fails to elucidate many microrheological events. In order to perform accurate and detailed numerical simulations of blood flow at microscales, we are developing 3D simulation techniques for multiple deformable cells using immersed boundary method. In this method, the cells are modeled as capsules, that is, liquid drops surrounded by elastic membranes. The model allows us to include various constitutive laws for the cell membrane, as well as the rheological properties of the liquid inside the cell. It also allows inclusion of the cell nucleus, as in case of a white blood cell or a neonatal red blood cell. In this talk we will describe the numerical techniques, and then explore the deformation dynamics of a nucleated/non-nucleated cell in a shear flow.   

Session KB: Minisymposium: Fluid Transport in Nanotubes and Nanochannels

Carbon Nanotube-Based Devices and the Study of Fluid Transport through Them

Experimental data pertaining to liquid transport through carbon nanotubes with diameters ranging from a few to hundreds of nanometers is briefly reviewed. A hybrid fabrication technique of carbon nanotube-based devices is described. The fabrication technique combines dielectrophoretic positioning of nanotubes and photolithography. The devices facilitate the introduction and control of fluid flow through the carbon nanotubes. Preliminary experimental results pertaining to capillary filling of, condensation in, evaporation from, particle flow into, and ionic current transmission through the nanotubes are discussed and compared with theoretical predictions. Although many of the observations indicate that the liquids behave classically, a few observations are still puzzling and await explanation. Finally, electron microscopy of controlled liquid flow is proposed as a new paradigm in fluid physics.   

Nanohydrodynamics within the electrical double layer

Charge transport in nanochanels is investigated using molecular dynamics. We focus in particular on the microscopic origin of the widely used ``Zeta potential.'' We show that the definition of this quantity not only involves the electrostatic nature of the interface, but is also intrinsincaly related to the dynamics of the solvent at the solid surface, providing new perspectives to control this quantity. We show in particular that the dynamics of the electric double layer (EDL) is very much dependent on the wettability of the charged surface on which the EDL develops. For a wetting surface, the dynamics, characterized by the so-called Zeta potential, is mainly controlled by the electric properties of the surface, and our work provides a clear interpretation for the traditionally introduced immobile Stern layer. In contrast, for non-wetting surfaces the immobile layer disappears and the Zeta potential deduced from electrokinetic effects is considerably amplified by the existence of a slippage at the solid substrate. The existence of strongly amplified electro-osmotic effects is accordingly demonstrated. Simulation results are shown to be in excellent agreement with predictions taking into account the slippage of the fluid the solid surface. The amplification effect is accordingly controlled by the ratio between the slip length (of the fluid at the solid surface), and the debye length. Such effects open the possibility of strongly enhanced electro-osmotic and electrophoretic effects in microchanels.   

Carbon nanotubes as molecular conduits: flow of water, protons, ions, and nucleic acids

The transport of water, protons, ions, and nucleic acids through carbon nanotubes was studied with all-atom molecular dynamics simulations. Water is found to fill even narrow pores of sub-nanometer diameter, but the filling is sensitive to the strength of attractive pore-water interactions and local electric fields. Motions of water through nanotubes is fast on a molecular scale. Protons were also found to move rapidly along one-dimensionally ordered water chains inside nanotubes. The transport of nucleic acids through nanotube membranes is dominated by polymer conformational dynamics during entry, and hydrophobic attachment to the pore walls during exit.   

Multiscale Simulations of Carbon Nanotubes and Liquids

We present molecular dynamics and hybrid continuum/atomistic simulations of carbon nanotubes in liquid environments with an emphasis on aqueous solutions. We emphasize computational issues such as interaction potentials and coupling techniques and their influence on the simulated physics. We present results from simulations of water flows inside and outside doped and pure carbon nanotubes and discuss their implications for experimental studies.   

Molecular Dynamics Study of Phase Change of Water inside a Single-Walled Carbon Nanotube

The phase change of liquid water to ice crystal inside a single-walled carbon nanotube (SWNT) was studied with molecular dynamics simulations. Water molecules were modeled with SPC/E potential and carbon-carbon interaction was expressed by Brenner potential. The carbon-water interaction was expressed with the Lennard-Jones function with the quadrupole interaction term. An SWNT with liquid water inside was initially kept in equilibrium at 300 K. Then, the carbon atoms were cooled at the constant cooling rate. The liquid to solid phase change for various cooling rates in a SWNT with various chiralities were examined. With certain cooling rate for a fixed chirality SWNT of (10, 10), the phase change was observed in the temperature range of 200K-220K. For sufficient slow cooling rate, the structure of ice crystal was a hollow octagonal tube. Similar simulations for several host SWNTs with different chiralities such as (8, 8), (9, 9), (11, 11) and (14, 3) were examined. It turned out that for thinner SWNTs the ice crystal favored the hollow tube structure such as pentagonal and hexagonal tubes depending on the diameter. On the other hand, for thicker nanotube the ice tube larger than octagonal structure was not obtained. Depending on the diameter of SWNT, the chiral water tube structure was also observed.   

Multiphase fluids confined in carbon nanotubes

The dynamics of liquid attoliter volumes contained in carbon nanotubes is investigated theoretically and experimentally. The experiments employ electron microscopy to visualize multiphase fluids in real time with spatial resolution approaching 1nm. The hydrophilic nanochannels studied include hydrothermally synthesized, CVD and commercially produced carbon nanotubes with inner diameters in the range 5-300 nm and wall thickness ranging from 1 nm to 40 nm. Dynamic phenomena are presented for aqueous fluids contained in closed-end nanotubes, and pure water condensing inside open-end carbon nanotubes. Some examples are given on filling nanotube channels with fluids impregnated with solid particles. A theoretical model formulated using a continuum approach, combines temperature-dependent mass diffusion with intermolecular (Lennard-Jones) interactions in the fluid bulk, as well as in the vicinity of the carbon walls. Several axisymmetric cases are considered, and comparisons between theoretical predictions and experimental data are performed. The current study shows the potential of using nanotube channels for understanding fluid behavior at the nanoscale.   

Session KC: Microfluidics: Numerical Studies

Simulation of $\lambda$-phage DNA in microchannels using a coarse-grained MD method

In this work we present Dissipative Particle Dynamics (\textsc{dpd}) simulations of polymers subject to the Marko-Siggia wormlike chain (\textsc{wlc}) spring law. We demonstrate the advantages of Lowe's \textsc{dpd} method, which simulates high Schmidt numbers for the solvent, and contrast it with the velocity-Verlet scheme. Shear flow results for the wormlike chain (\textsc{wlc}) simulating single \textsc{dna} molecules compare well with average extensions from experiments, irrespective of the number of beads. However, coarse-graining with more than a few beads degrades the agreement of the autocorrelation of the extension.   

Ion Separation using a Y-Junction Carbon Nanotube

Using molecular dynamics simulations, we show that a Y-junction carbon nanotube can be used to separate potassium and chloride ions from a KCl solution. The system consists of a KCl solution chamber connected to an (8,8) carbon nanotube, which acts as the stem. Two carbon nanotube branches of sizes (5,5) and (6,6) are connected to the (8,8) nanotube forming the Y-junction. Uncharged (5,5) and (6,6) carbon nanotubes show close to zero occupancy for transport of potassium and chloride ions. By functionalizing a (5,5) carbon nanotube with a negative charge, we show that we can selectively transport potassium ions. Similarly, by functionalizing a (6,6) carbon nanotube with a positive charge, we can selectively transport chloride ions. By performing molecular dynamics simulations on the entire system comprising the two branches, stem and the KCl solution chamber, we show that perfect ion separation is observed when (5,5) and (6,6) nanotubes are charged with $\sigma_{w,(5,5)}=-0.181$ C/m$^2$ and $\sigma_{w,(6,6)}=+0.143$ C/m$^2$, respectively, whereas for the system with $\sigma_{w,(5,5)}=-0.168$ C/m$^2$ and $\sigma_{w,(6,6)}=+0.131$ C/m$^2$ the separation is not perfect because of the formation of ion pairs. We discuss the formation and control of ion pairing, which is a common phenomenon in confined nanochannels.   

Efficient solutions of the nonlinear Boltzmann equation for low-speed applications

We show that efficient Monte Carlo solution methods for the nonlinear Boltzmann equation for low-speed applications can be constructed by expressing the single-particle distribution function as the sum of an equilibrium distribution and a deviational term. By considering the deviation from equilibrium when evaluating the collision integral, one can avoid simulating a large number of physically occuring collisions with no net effect and thus achieve a high degree of variance reduction. As the degree of deviation from equilibrium decreases, the degree of variance reduction increases, leading to a velocity signal to noise ratio that remains approximately constant and thus a computational cost which is essentially independent of the Mach number ($\textrm{Ma}$). These features are in sharp contrast to current particle-based simulation techniques (e.g. DSMC) in which statistical sampling leads to computational cost that is proportional to $\textrm{Ma}^{-2}$, making calculations at small Mach numbers very expensive. The present formulation can be incorporated into both direct numerical methods as well as particle-based methods. These approaches are validated by comparing results with analytical and direct simulation Monte Carlo (DSMC) solutions of the Boltzmann equation.   

High-speed microfluidic differential manometer for cellular-scale hydrodynamics

We propose a broadly applicable high-speed microfluidic approach for measuring dynamical pressure drop variations along a micron size channel and illustrate the technique by presenting the first measurements of the additional pressure drop produced at the scale of individual flowing cells. The influence of drug-modified mechanical properties of the cell membrane is shown. Finally, single hemolysis events during flow are recorded simultaneously with the critical pressure drop for the rupture of the membrane. This scale-independent measurement approach can be applied to any dynamical process or event that changes the hydrodynamic resistance of micro- or nanochannels.   

A hybrid continuum-atomistic simulation of heat transfer in micro flow

The heat transfer problem in a micro/nano flow is studied based on domain decomposition hybrid method. This method uses an atomistic description in one part of the domain and a continuum description in other place. Two solutions are matched in a coupling region which is necessary to ensure their consistency including the temperature and heat flux. In the coupling region, the statistical results from the atomistic simulation provides the boundary conditions for continuum energy equation, and the particle velocities are rescaled to account for the energy transfer from continuum domain to to particle domain. Simulation results for steady and unsteady heat transfer in a channel flow will be shown. The effect of rough wall on the heat transfer will also be discussed.   

Molecular Dynamics Investigation of Ionic Flow and Separation by Carbon Nanotube Electrodes

We report on molecular simulation studies of the ionic flow in the presence of charged carbon nanotubes. Our domain contains three species; viz. positively charged sodium ions, negatively charged chlorine ions and neutral water; and a pair of single-walled carbon nanotube electrodes. One of the nanotube is positively charged and the other is negatively charged. The system of 1024 atoms is initially allowed to equilibrate from an FCC crystal structure for the solution. The nanotubes are tethered and the carbon atoms are assumed to vibrate as in a One-Dimensional Harmonic Oscillator (ODHO) about their mean positions. The sodium ions travel towards the negatively charged carbon nanotube and the chlorine ions likewise flow towards the positively charged nanotube. The simulation uses a Lennard-Jones soft sphere potential model and coulombic potential for interaction between the charges. In addition to the ion transport mechanism, the hydrophobic character of carbon nanotubes is clearly evident from the simulated flow.   

Temperature gradient driven transport of water inside a carbon nanotube

With growing need for micro-nanoscale manipulators and transporters, liquid systems confined in small geometries are of a great interest. An extreme case would be single walled carbon nanotubes (SWNTs), where the liquid motion is confined in quasi-one-dimensional geometry. In this work, by means of molecular dynamics simulations, we consider the transport of a water cluster which consists of a few hundred molecules in an SWNT with a diameter of about 1.4 nm. Especially, the influence of the non-equilibrium thermal boundary condition on the water cluster is investigated by imposing a longitudinal temperature gradient to the SWNT. Water molecules are modeled with the SPC/E model whereas the carbon-carbon and carbon-water interactions are expressed using the Brenner potential and a simple Lennard-Jones potential, respectively. It is demonstrated that the water cluster is transported with the temperature gradient at an average velocity that is proportional to the temperature gradient. The trend exhibits good correlation with the temperature dependence of the overall potential energy between water and carbon molecules. Together with the comparative case of a water cluster adsorbed on the outer-wall of an SWNT, the molecular dynamics of the transport phenomenon will be discussed.   

Numerical Solutions and Structures of Double Quantum Jet Solving by an Upwind Scheme

The solutions of a double quantum jet are analyzed by solving the quantum fluid dynamical formulation (QFD) of the Schr\"{o}dinger equation. The QFD equations are obtained by expressing the Schr\"{o}dinger wave function as $\varphi =\rho ^{1/2}\exp (iS/\hbar )$and $\vec {u}=(u,v)$. In QFD, $Q=-\rho ^{-1/2}\Delta \rho ^{1/2}$ is called as quantum potential. An upwind method is developed to solve the QFD equations. The method use a third-order upwind method to discrete convection terms and the central finite difference method to discrete the quantum potential. A fourth-order Runge-Kutta method is used for time marching. Two cases, one-dimensional free particle with external potential and two-dimensional free particle with external potential, are presented to illustrate the accuracy of the QFD solver. The computational results are compared well with the results obtained by solving the Schr\"{o}dinger equation. Finally, the QFD solver is applied to solve the solutions of a double quantum jet and to investigate its structures. First, a mathematical formulation is derived to describe the double quantum jet. The jet has the probability density equals 2 and the velocity equals 2 at the inlet of the jet. Then, the solutions are computed by the QFD solver. The structures of the solutions are affected by the strength of the quantum potential. The interesting phenomena of quantum clustering are found.   

Session KD: Liquid Breakup and Coalescence III

Unexpected Breakup Dynamics of Compound Jets

Understanding the breakup dynamics of a compound or a two-fluid jet is of great importance in applications such as micro-/nano-encapsulation and emulsion formation. Breakup dynamics of compound jets are studied computationally using finite element analysis. For single fluid jets, it has been known for over a century that a jet is unstable (stable) to small amplitude perturbations if the wavelength of the perturbation $\lambda $ is larger (smaller) than the unperturbed circumference of the jet. Response of compound jets is quite similar to that of single-fluid jets if $\lambda $ is larger (smaller) than the unperturbed circumference of the outer surface of the jet. However, it is shown that an unexpected oscillatory instability results if $\lambda $ is larger than the unperturbed circumference of the inner surface of the jet but smaller than that of the outer surface of the jet when the ratio of the interfacial tension of the outer interface to that of the inner interface is much larger than unity. Pressure fields, streamlines, and energies are interrogated to elucidate the physics of the instability.   

Multiple coalescence and pinch off at a fluid interface

We investigate numerically and experimentally the dynamics of the coalescence of a drop coming in contact with a horizontal interface. We focus on cases where the drop repeatedly coalesces and pinches off, forming a sequence of progressively smaller drops. We determine the regime in which such a cascade can occur and describe for the first time the details of the mechanism behind multiple coalescence. Viscous damping of capillary waves is found to be crucial in determining whether pinch off will occur or not, despite the fact that only a small fraction of the available energy is dissipated by viscous effects. When pinch off does occur, we also characterize the following bouncing of the residual drop on the oscillating interface.   

Coalescence of Spreading Droplets on a Wettable Substrate

We investigate experimentally the coalescence dynamics of two slowly spreading droplets on a highly wettable substrate. Upon contact, surface tension drives a rapid motion perpendicular to the line of centers that joins the drops and lowers the total surface area. The coalescence behavior is characterized by the time-dependent width $w_m$ of the growing meniscus bridge between the two drops. We find that the growth rate of $w_m$ is always viscously dominated and at early times exhibits power-law-like behavior wherein $w_m\sim t^{0.7\pm0.1}$. Moreover, the growth rate is highly sensitive to both the radii and heights of the droplets at contact. This feature differs significantly from the behavior of freely suspended droplets, in which the coalescence growth rate depends only weakly on the droplet size. We present scaling arguments that accord with the experimental observations.   

Cascade of coalescences for droplets near liquid interfaces

An experimental study on the coalescence of liquid droplets in a system composed of two immiscible liquids is carried out. When a liquid droplet is deposited on the interface separating both liquids it floats momentarily before coalescing with the bottom layer. In some cases the droplet does not coalesce entirely. A smaller satellite droplet is pinched off and partial coalescence may start again. This results in a cascade of successive coalescence. The coalescence cascade is found to be characterized by the liquid viscosities, the liquid densities and the initial radius of the released droplet. The fine study of these parameters yields some interesting information about multiple coalescence. The evolution of the radii of successive droplets is studied and found to be independent of the liquid viscosities. On the other hand the number of steps in a coalescence cascade is shown to be a function of the viscosities and the initial droplet radius. Several arguments are provided that give rise to the existence of a critical as well as a maximum droplet radius.   

Important Parameters in Drop Coalescence at Planar Surfaces

An experimental study has been performed to establish the principal elements that govern drop coalescence. The study consisted of placing drops of various sizes and physical properties on a planar interface with the aid of a high speed digital camera. The experimental portion of the project was aimed at capturing the time of coalescence and the size of the secondary drop that formed after coalescence had finished. Results of the experiments showed clear patterns with respect to inertial and viscous terms. Dimensional analysis indicated that \textit{Oh} had a strong influence on the behavior of drop coalescence. The ratio of secondary drop radius to primary drop radius was calculated to be approximately constant when Oh was much smaller than unity. However, as Oh approached unity from the lower bound, the value of r$_{i}$ decayed. No secondary drop was observed when Oh was greater than unity. Normalized coalescence times confirmed this trend by being properly scaled with inertial time scales for small Oh and preferring viscous time scales when Oh was greater than unity. During the coalescence of a drop with a planar interface, a hole is generated in a microscopic film that separates the drop from the interface. The experiment captured two separate processes, film rupture and the closing of the hole. During the film rupture, the hole radius demonstrated a power law time dependence. The dimensionless drop rupture radii and times fit onto a single master curve and were independent of their physical properties during the opening.   

Binary Drop Coalescence in Liquid/Liquid Systems

Drop pairs of water/glycerin mixture were injected horizontally into silicone oil and, due to gravitational effects, traveled on downward trajectories before colliding. Flow visualization and PIV measurements were obtained in index-matched fluids to characterize coalescence and rebounding behavior. Planar PIV was used to examine large-scale drop motion. In a dual field measurement, stereo PIV and planar long distance microscope PIV were used for resolving larger and smaller scale motion respectively. Experiments were performed for Weber numbers [We] in the range of 1-50. Higher We caused stronger drop deformation and enhanced interface instability, leading to film rupture. By adjusting the initial separation distance and drop volume, trajectory angles could be controlled somewhat. Steeper collision angles encouraged rebounding as opposed to coalescence. Velocity and vorticity fields of the impact zone will be discussed in relation to coalescence and rebounding behavior for several cases   

Drop coalescence and film rupture with van der Waals forces

During the collision of two drops the thin film that forms between the drops as they approach each other plays an important role. Viscous forces limit the rate at which the film thins, so the film pressure rises, which tends to cause the drops to rebound. For coalescence to occur the film must become thin enough for the small-scale, attractive van der Waals forces to overcome the high pressure. Two methods to compute van der Waals forces during drop coalescence have been developed. In one method the van der Waals potential between the two drops is computed directly to obtain the force. In the other method a disjoining pressure is applied to the film between the drops. A comparison of the results using the two methods is presented. For validation numerical simulations of the rupture of a fluid film using the disjoining pressure method are compared to the lubrication theory of Vaynblat {\it et al.} ({\it Phys. Fluids}, 2001). The effect of van der Waals forces on drop collision and film rupture is described.   

Boundary Integral Simulations of Drop Coalescence

We report on boundary integral simulations of the flow-induced coalescence of a pair of equal size drops undergoing a head-on collision in an axisymmetric extensional flow. For the small capillary numbers that are relevant to the coalescence of drops in the 10-50 micron diameter range, a direct comparison can be made between the theory and experimental data, including both head-on and glancing collisions. In the latter case, the time-dependent force due to rotation of the drop pair can be mimicked via head-on collisions with a time-dependent velocity gradient This analogy allows us to study the mechanism for coalescence in cases where it is observed experimentally to occur during the latter half of a collisions, after the hydrodynamic force has begun to pull the drops apart.   

A finite-element phase-field method for simulating interfacial dynamics in complex fluids

We present a novel and efficient finite-element method for treating interfacial problems involving rheologically complex fluids. Two key ingredients of the method are a phase-field representation of the interface and an adaptive meshing scheme that allows fine interfacial resolution at manageable computational cost. In the phase-field framework, the interface is seen as a thin layer across which material properties change rapidly but continuously. Thus, a set of governing equations are derived that hold for both fluids across the interface. This circumvents the cumbersome task of interface tracking. The surface tension emerges from the mixing energy at the interface, and the energy-based formalism easily incorporates complex rheology. The challenge of the method lies in resolving the interfacial layer on a fixed Eulerian grid. This is handled by adaptive meshing on a unstructured grid using the phase- field as the criterion for local refinement and coarsening. We will present several simulations on drop deformation, retraction, coalescence and breakup for Newtonian and viscoelastic liquids and nematic liquid crystals. While some of these serve as validations of our new method, the results also reveal novel physics governing the interplay between interfacial dynamics and bulk rheology.   

Effects of inertia on the rheology of a dilute emulsion of drops in steady shear

Effects of inertia on the rheology of a dilute Newtonian emulsion are investigated using DNS. The drop shape and flow are computed by solving the Navier-Stokes equation in two phases using Front-tracking method at nonzero Reynolds numbers of 0.1 and 1.0. Effective stresses are computed using Batchelor's formulation, where the interfacial stress is obtained from the simulated drop shape and the perturbation stress from the velocity field. At low Reynolds number, the simulation is compared successfully with various analytical results and experimental measurements. At higher inertia deformation is enhanced and the tilt angle of the drop becomes larger than forty-five degree. The inertial morphology directly affects interfacial stresses. The first and the second interfacial normal stress differences are found to change sign due to the change in drop orientation. The interfacial shear stress is enhanced by inertia and decreases with capillary number at lower inertia but increases at higher inertia. The total excess stresses including perturbation stress contribution shows similar patterns.   

Windswept droplets

A small droplet impacting a glass window usually remains stuck on the pane. How can we expel it? One possible solution consists in coating the glass surface with a hydrophobic layer. Another solution is to blow it off. We explore this last solution (partly combined with the first one). The droplet starts moving when the wind exceeds a threshold velocity, depending essentially on the surface wettability and the drop size. Above this threshold, the drift speed of the droplet results from a balance between aerodynamic drag and viscous dissipation near the contact lines. The results for different experimental conditions collapse on a master curve, once the wind speed is rescaled as a Weber number and the droplet velocity as a capillary number. While small droplets remain almost spherical caps, larger ones are strongly deformed and take the shape of a sausage, perpendicular to the wind direction. We finally determine the conditions in which satellite droplets are left at the rear of the moving drop, an issue crucial for blow drying processes.   

Session KE: Multiphase Flows: General II

On a multi-phase modeling framework for sediment transport

Understanding sediment transport in the heterogeneous environment, including river, beach, estuary, shelf and submarine canyon, is crucial to the preservation and restoration of coastal ecosystem. A multi-phase modeling framework is developed in order to study sediment transport driven by a variety forcing (current, tide, wave and gravity-driven flow) with a range of sediment characteristics. As an example, constitutive relation for intergranular interaction (particle stress) based on kinetic theory of granular flow is adopted in a two-phase model and is shown to be capable of modeling wave-induced sheet flow transport in the sandy-beach environments. Recently, to further model typical fine sediment transport processes of long timescales (e.g., tidal), spatial inhomogeneity, and multiple sediment classes, the two-phase model is rationally simplified. The simplified model is much efficient yet robust to retain essential mechanisms of fluid-sediment and intergranular interactions. Adopting rheological closure based on viscous suspension, preliminary results indicate that the model captures field observed lutocline behavior of fluid mud under tidal flow and wave-supported gravity-driven fluid mud on the continental shelf.   

Sediment dynamics over rippled beds in oscillatory flow: Experiments

The University of Maryland Oscillatory Sediment Flume (UMOSF) is an experimental facility built to investigate sediment transport mechanics within an oscillatory turbulent boundary layer is over a mobile sediment bed. The range of sediment size and density as well as the flow oscillation amplitude and period is selected in the current work to study flows which generate rippled bed forms. The measurement technique utilizes a simultaneous two-phase PIV method to examine fluid-particle interactions, focusing on the suspension mechanisms and to obtain statistics to describe the two-way coupling. Specifically, measurements will focus on the upslope face, crest and recirculation zone of the ripple, where previous simulations for steay flow\footnote{Chang \& Scotti, \textit{J. Turbulence}, \textbf{4}(019), pp.1-21, (2003).} have shown the strongest regions of suspension, injection into the boundary region, and mixing with the outer flow to occur. Results of these experiments are closely coordinated with ongoing numerical simulations, and discussed in a companion presentation.   

Sediment dynamics over rippled beds in oscillatory flow: Numerical results

The dynamic motion of solid particles is an important phenomena in a wide range of applications such as: coastal erosion, sand dune motion or fluidized bed. In these applications, an important issue is the effect of the wall-shape on the dynamical behavior of the dispersed phase and on the sediment deposition. In this study, Direct Numerical Simulations of steady fluid flow over a ripple have been coupled with Lagrangian tracking of discrete solid particles. The forces acting on the particles are reduced to the drag and the lift force induced by the fluid a flow. The particles are initially randomly distributed in the computational domain. On the bottom boundary, a saltating model is introduced to account for particle-wall interaction. Both steady and oscillatory conditions are considered, and the results are compare with ongoing experimental results discussed in a companion presentation.   

Modeling Sediment Transport under Waves

Modeling studies of the flux of sediment at the sea bed under energetic waves are presented. The transport of sediment is crucial to predicting many coastal engineering processes, such as erosion around structures and prediction of beach profiles. We model a two-phase system containing water and sediment particles approximated as a mixture having variable density and viscosity that depends on the local sediment concentration. We use a control volume approach on a three-dimensional staggered grid to solve the equations numerically. The expression for the stress-induced diffusion coefficient developed by Nir {\&} Acrivos for sediment flow is used and the Richardson {\&} Zaki relationship is used to the calculate sediment settling velocity as a function of concentration. The turbulent dynamics of an initially stationary densely packed sand layer are examined and model results are compared with experimental data collected in two lab experiments. The model also does a reasonable job of predicting concentration profiles and suspension properties across the bottom boundary layer.   

Concentration distribution in gravity driven mixing of two fluids in a tilted tube

The concentration distribution in the mixing zone of interpenetrating light and heavy fluids in a tilted tube is studied by laser induced fluorescence as a function of the tilt angle $\theta$ from vertical. At low $\theta$, the flow is turbulent, resulting in efficient mixing across the tube. With increasing $\theta$, a concentration difference appears across the tube section due to the transverse component of gravity. At large $\theta$, this segregation is efficient enough for the concentration contrast at the fronts to become equal to the global density difference between the two original fluids. At still larger tilt angles, there is no mixing between fluids but a stable parallel counterflow controlled by viscous dissipation in the bulk of the fluids. In the two first regimes, the local concentration contrast $\delta \rho$ at the interpenetration fronts is shown to be directly related to the front velocity through $V_f \propto \delta \rho/\rho$. This confirms that these regimes correspond to a local balance between inertia and gravity at the front.   

Effect of inertia and gravity on the turbulence in a suspension

A theoretical model is presented for the effect of particle inertia and gravity on the turbulence in a homogeneous suspension. It is an extension of the one-fluid model developed by L'vov, Ooms and Pomyalov (2003), in which the effect of gravity was not considered. In the extended model the particles are assumed to settle in the fluid under the influence of gravity due to the fact, that their density is larger than the fluid density. The generation of turbulence by the settling particles is described, special attention being paid to the turbulence intensity and spectra. A comparison is made with DNS calculations and experimental data. Also a sensitivity study is carried out to investigate at which conditions the gravity effect becomes important. With the model it is possible to calculate the significance of the two-way coupling effect as function of the relevant dimensionless groups. Also an explanation in physical terms is given.   

Solidification and the Effects of Internal Heat Generation

Solutions using the integral method are presented for solid-liquid phase change in materials that generate internal heat. This problem is solved for cylindrical, spherical, plane wall and semi-infinite slab geometries. The analysis assumes a temperature profile in the solid phase and constant temperature boundary conditions on the exposed surfaces. We derive differential equations governing the solidification thickness for the geometries as functions of the Stefan number and the internal heat generation (IHG). For cylindrical, spherical, and plane wall geometries, the solidification layer obtains a steady-state value which is related to the inverse of the square root of the IHG. The solutions to the semi-infinite slab geometry problem show that when the surface is cooled to below the freezing point, a solidification layer forms along the edge and begins to grow until it reaches a maximum, then begins remelt. The problem has application to diverse fields such as nuclear energy, materials processing, geophysical fluids, and bioengineering.   

Numerical Simulations of Solidification in a Convecting Supercooled Melt

We present a 2-D phase-field model with convection induced by a flow field applied to freezing into a supercooled melt of pure substance, nickle. Four-fold anisotropy is introduced to the interfacial energy. Renormalization group theory is applied to the phase-field model with convection to produce an efficient computational procedure for treating multiscales in both time and space. Numerical procedures and details of numerical parameters employed are provided, and convergence of the numerical method is demonstrated by conducting grid-function convergence tests. Dendrite structures, temperature fields, pressure fields, streamlines and velocity vector fields are presented at several different times during the dendrite growth process. Comparisons of dendrites and temperature fields with and without convection indicate that the flow field has a significant effect on the growth rate of the dendrites; in particular, it inhibits growth. In addition, the flow field influences the dendritic structural morphologies and thickness of the interface. Moreover, the dendrites behave as a solid body in the flow leading to stagnation points and other interesting flow features.   

Simulations of densely-packed cloth motion in water

Fluid-structure simulations of densely-packed immersed fabric model the clothes washing process. We have modified the Immersed Boundary Method (Peskin 1977) to handle the known but complex geometry of the washing machine and agitator as well as the unknown cloth structure immersed in the fluid. Extending the technique to three-dimensions has required improved computational efficiency and causes geometric singularities when cloth that is not sufficiently extensible bends in two directions. We present some preliminary comparisons to primarily two-dimensional experiments in the dilute cloth limit. Computational difficulties caused by cloth permeability and bending stiffness will be discussed.   

Numerical method for interaction among fluid, multi-particle and complex structures

We propose a numerical method for dealing with interactions among fluid, multiple particles and complex structures. This method is based on the level set method, the CIP method and DEM. In the formulation, the structures are represented on a grid by using the level set method. The interactions of particles and structures are calculated by a method based on the discrete element method. The method can treat the interaction among fluid, multi-particle and complex structures robustly.   

Session KF: Interfacial and Thin Film Instabilities III

Nonlinear Stability Analysis of a Two-Layer Thin Liquid Film: Dewetting and Autophobicity

The nonlinear analysis of a two-layer thin liquid film on a solid substrate is performed. Weakly nonlinear stability analysis of nonlinear evolution equations for the two interfaces reveals that coupling of van der Waals interactions in the layers can lead to an autophobic behavior of the film, similar to spinodal decomposition. Numerical simulations of the strongly nonlinear evolution equations confirm this conclusion. The effect of both soluble and insoluble surfactants on the film stability is also studied. It is shown that the presence of surfactants can lead to an osillatory instability of a two-layer film that manifests itself as dewetting waves.   

Instability of an interface, with large viscosity-contrast, under tangentially oscillatory motion

We shall present here an instability which initiates a pattern formation on the interface between two viscous fluids, with very strong viscosity contrast, subjected to tangential oscillatory motion at a moderate frequency. We carried out experiments and theoretical studies. Experimental results showed that the first selected wavelength, which is far from the capillary one, is controlled by the oscillation amplitude. A quasi static model is elaborated, predicting the instability threshold and the wavelength dependence on the amplitude, in agreement with the experiments. A detailed analysis reveals that the origin of the instability is not a simple Kelvin Helmholtz type, because the pressure perturbation distribution is different. This difference lies in the contribution of the streaming effect.   

Longwave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect. The case of a finite Biot number

We investigate the long-wave Marangoni instability in a binary-liquid layer with a deformable interface in the limit of a finite Biot number $B$ and a specified heat flux at the solid substrate and in the presence of the Soret effect. In the fundamental case (a) of both finite Galileo and Lewis numbers, $G$ and $L$, respectively, and a large inverse capillary number $S$, both monotonic and oscillatory instabilities are present. The monotonic instability takes place with the critical Marangoni number $M_{mon}=48\,L\,\chi^{-1}$, where $\chi$ is the Soret (separation) number when $-1<\chi<0$. When $(1+\chi)/\chi >0$, this instability emerges if $L [Preview Abstract]

Strongly nonlinear interfacial-surfactant instability and diffusion

The nonlinear stages of the recently uncovered instability due to insoluble surfactant at the interface between two fluids in a creeping plane Couette flow are investigated for the case when one of the fluids is a thin film and the other is semi- infinite in the cross-flow direction. Numerical simulation of strongly nonlinear longwave evolution equations which couple the film thickness and the surfactant concentration, assuming the latter sufficiently small, reveals that the instability saturation is only possible when the surfactant diffusion exceeds a threshold strength whose value depends on the interfacial shear rate and other parameters. The disturbance of surfactant concentration never remains small, so the evolution never can be completely described by weakly nonlinear equations. The evolution time scale appears to grow indefinitely as the interfacial shear goes to zero and/or the surfactant diffusion strength increases.   

Monolayer phase coarsening using oscillatory flow

The co-existing phase domains of monolayers commonly observed via microscope are examined on flowing systems. Recent evidence shows that co-existing phase domains have profound effects on monolayer response to bulk flow. The present flow geometry consists of an open-top rectangular cavity in which the flow is driven by the periodic oscillation of the floor in its own plane. The oscillation of the floor dilates and compresses any film at the gas/liquid interface while still maintaining an essentially flat interface. A range of flow conditions (oscillation frequency and amplitude) is chosen so that the flow remains essentially two-dimensional. Measurements at the interface, initially covered by an insoluble monolayer (vitamin K$_{1}$ or stearic acid), are made using a Brewster angle microscope system with a pulsed laser. Various phenomena such as fragmentation (breaking up of co-existing domains into finer ones) had previously been observed in sheared monolayer flows. In this new flow regime, we have seen dramatic coarsening of the domains. Interesting relaxation behavior at short and long time scales will also be discussed.   

Starbursts and Wispy Drops : Surfactants Spreading on Gel Substrates

We report a phase diagram for a novel instability seen in drops of nonionic surfactant solution (Triton X-305) spreading on viscoelastic agar gel substrate . This system allows us to examine the effect of varying the effective fluidity/stiffness of aqueous substrates. The morphology is strongly affected by the substrate fluidity, ranging from spreading starbursts of arms on weak gels, to wispy drops on intermediate strength gels, to circular drops on stiff gels. We analyze the dynamics of spreading in the starburst phase, where the arm length grows as t $^{3/4 }$at early times, independent of the gel strength and surfactant concentration. The number of arms is proportional to the surfactant concentration and inversely proportional to the gel strength. Ongoing work is exploring the effects of changing the drop volume.   

Domain Relaxation in Polymer Langmuir Layers

We report on an experimental, theoretical and computational study of a molecularly thin polymer Langmuir layer on the surface of a subfluid. When stretched (by a transient stagnation flow), the monolayer takes the form of a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape relaxes to the minimum energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is thin. We model these experiments by taking previous descriptions of the full hydrodynamics (primarily those of Stone \& McConnell and Lubensky \& Goldstein ), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. The problem has a boundary integral formulation which allows us to numerically simulate the tether relaxation; comparison with the experiments allows us to estimate the line tension in the system.   

Dynamics of a reactive falling film

We study the dynamics of a falling film in the presence of a first-order (exothermic or endothermic) chemical reaction. The heat released or absorbed by the reaction alters the surface tension, which in turn affects the evolution of the film, which in turn affects the rate of reaction and therefore the heat released by the reaction (feedback). Our analysis is based on an integral-boundary-layer approximation of the equations of motion, energy and concentration and associated free-surface boundary conditions. The heat/mass transport P\'eclet numbers are taken sufficiently large so that to take into account the convective terms of the heat/mass transport equations. Perticular emphasis is given to permanent-form traveling solitary waves. We show that the solitary waves can be dispersive and the size of dispersion depends on the size of the Prandtl and Schmidt numbers while its sign can change from positive to negative leading to negative-hump solitary waves. For large dispersion and for a sufficiently large region of Reynolds numbers, the liquid layer can be excited in the form of nondissipative solitary pulses which close to criticality assume the form of Kortweg-de Vries solitons.   

Dynamics of a falling film in the presence of surfactants

We investigate the dynamics of a falling film in the presence of surfactants. As a first step we consider insoluble surfactants thus ignoring diffusion from the bulk and desorption to the gas phase. We utilize an integral-boundary-layer approximation of the momentum and concentration equations and free-surface boundary conditions. We construct bifurcation diagrams for single-hump solitary wave solutions and we show that for vanishing Marangoni numbers the surfactants concentration becomes singular. The singularity appears at the front stagnation point of a solitary pulse due to accumulation of surfactants.   

Flow down an inclined plane with soluble surfactant

We study the flow of a thin film down an inclined plane in the presence of dilute concentrations of soluble surfactant. Lubrication theory and cross-sectional averaging are used to derive a coupled set of two-dimensional (2-D) evolution equations for the film thickness and surfactant surface and bulk concentrations in the limit of rapid vertical diffusion. These equations are closed by a linear equation of state and parameterized by bulk and surface Peclet numbers, and dimensionless solubility and sorption kinetics parameters; the contact-line singularity is relieved via use of a thin precursor layer. The results of our transient growth analysis and transient numerical simulations of the nonlinear 2-D equations reveal the presence of a fingering instability that targets the thickened advancing ridge where the film adjusts onto the precursor layer. Although these fingering phenomena are present in the surfactant-free case, the presence of surfactant enhances the instability over an intermediate range of solubilities.   

Marangoni instability of thin films on horizontal and inclined substrates

Nonlinear evolution of the Marangoni instability of thin liquid films is studied via direct integration of the thin film equation to identify the states selected dynamically by the instability. On a horizontal substrate the instability is subcritical and proceeds to rupture. On slightly inclined substrates rupture may occur depending on parameter values and initial conditions. In other cases the instability evolves into arrays of solitary waves with both periodic and nonperiodic time-dependence. The results extend earlier work (U. Thiele and E. Knobloch, Physica D 190, 213, 2004) into the dynamical regime.   

Density Effects on Immiscible Interface Breakup and Drop Formation Process

The interface dynamics of a single immiscible interface at varying density ratios with the viscosity ratio order of one in the presence of vortical flow is examined through Front-Tracking/Finite difference method to solve unsteady Navier-Stokes equations for both the disperse and continuous phase flow. It is observed that as the density ratio of both phases decreases the larger density difference prohibits the formation and growth of surface waves and results in the formation of a longer column, which persists for a longer time until first breakup occurs. In contrast, the numerical simulations show that the change of density ratio is accompanied by the relative small variation in the detached volume of column. In this work, we also show that there exist the upper and lower limits of density ratio. Moreover, the small density ratio effects on the viscosity ratio are investigated. Although the density ratio is responsible for the interface deformation and breakup process when the viscosity of dispersed phase is greater than that of the continuous phase, the small density ratio effects on the dynamics is not observed for the case that the dispersed phase is less viscous than the continuous phase.   

Session KG: Vortex Dynamics III

Extension of the formation time concept to general planforms and trajectories

We propose a generalization of the concept of formation time for flapping wings and appendages, that overcomes the drawbacks of Strouhal number and paves the way to the development of a unified theory for the unsteady aerodynamics of flapping propulsion. We show experimental evidence of the universality of the generalized formation time concept coming from results on flapping flat plates, shape optimization of flapping wings, and trajectory optimization for the caudal fin of an arrtificial fish.   

LES investigation of aircraft wake two-vortex system in low level atmospheric turbulence

The numerical simulation of realistic vortical aircraft wakes constitutes a challenging task, as the problem is of large size, the vortex cores are small, and the Reynolds number is very high. In particular, the vortex cores grow very little during the lifetime of the vortices. Large-eddy simulation (LES) at ``essentially'' infinite Reynolds number is here used: a Fourier-based pseudo-spectral method in ``quasi- Euler'' mode ($\nu$ set to zero and use of a high order $k^{16}$ hyperviscosity subgrid-scale model, which ensure negligible viscous core growth). The LES grid is also fine enough so as to properly capture relevant dynamics in the core region. A realistic case is simulated: circulation $\Gamma_0=400\,{\rm m} ^2/{\rm s}$, spacing $b_0=50\,{\rm m}$, core radius $r_c=2.5\, {\rm m}$. We use a $L^3$ computational box of $256^3$ grid points and with $L=200\,{\rm m}$. The background atmospheric turbulence is here at low level (dissipation $\epsilon = 0.0001\, {\rm m}^2/{\rm s}^3$) and was obtained as a pre-simulation. The vortex system engulfs the ambient turbulence, the non-linear interactions then amplify it and this eventually leads to a vortex pair where the surrounding turbulence is independent of the atmospheric background: the low level turbulence thus acts as a seed to create a ``turbulent vortex pair'' that then lives and decays on its own. We also observe significant axial velocities in the core region. Those results are further being used for simulations of LIDAR return signal.   

Stochastic analysis of wing-tip vortex wandering in turbulent free streams.

Instantaneous measurements of the local velocity vector and streamwise vorticity were performed in the tip vortex of a finite wing with a NACA-0012 profile and a rectangular tip at Re = 240000 and angle of attack of 5$^{o}$. Results are reported on six transverse planes downstream of the wing, in an unobstructed free stream with a turbulence intensity of 0.3{\%}, as well in grid-generated turbulence with intensities of 2.5{\%} and 5.0{\%}. It was found that, although vortex formation and mean trajectory were unaffected by the turbulence, time-averaged velocity statistics in the vortex core were strongly influenced by random lateral motion of the vortex, which increased with increasing turbulence intensity. Analysis of velocity signals identified instances when the vortex axis was on the same horizontal or vertical plane as the probe tip, which provided an estimate of the probability density function of the vortex axis position, found to be approximately Gaussian. Joint PDF of the measured velocity components perpendicular to the vortex axis indicated that the instantaneous peak tangential velocity decayed with increasing streamwise distance.   

Corotating Vortex Formation, Merger, and Modification

Experimental results from a combined wind tunnel and tow tank study on the evolution, interaction, and merger of two corotating trailing vortices are presented. In the current study, NACA 0012 airfoils positioned at positive and negative angles of attack are used to generate a corotating vortex pair. Measurements with a seven-hole probe are used to extract 3-D velocity and pressure fields in the tunnel while PIV is used in the tow tank to measure 2-D velocity fields in the vortex wake. The semi-span length is varied in the experiments to investigate the effects of initial separation on vortex formation and merger. Experimental information of the vortex motion is compared with theoretical predictions. Comparison shows that at larger initial separations, the motion of the vortex pair is well estimated from the 2-D analysis, while at smaller separations, the motion of the vortex pair cannot be well predicted from the results of the line vortex assumption. Effects in forcing vortex breakup using a virtual winglet from a plasma actuator are presented. At sufficient power inputs, the virtual winglet can substantially modify the vortex circulation magnitude and distribution. Comparisons of information confidence and data quality are made between the two measurement techniques and facilities. Finally, comparisons of differences in the vortex roll-up and evolution due to initial separation changes are discussed and contrasted to that of a single tip vortex and corotating vortex pair on a wing with flap.   

2D PIV of a pitching airfoil

Two dimensional particle image velocimetry (PIV) experiments were performed at the midspan of a sinusoidally pitching NACA 0020 airfoil. Measurements were taken for a range of Strouhal numbers up to 0.4 and at various Reynolds numbers. In addition, the velocity field around the airfoil moving in quiescent flow was obtained. The flow is measured in a phase-averaged sense. Far field measurements of the surrounding flow field are presented in order to investigate the generation and propagation of the wake of the pitching motion. Near field measurements are also presented at the leading edge of the airfoil in order to resolve and investigate the behavior of the dynamics stall vortex and the shear layer.   

Simulation of a 2D flow past a flexible fibre tethered at its center point: vortex shedding

Vortex shedding from an object immersed in a flowing fluid is an important and interesting topic and has been extensively studied experimentally, analytically and computationally. Most of the work focused on vortex shedding from a rigid body; for instance, a circular cylinder [1], a sphere [2] or an inclined flat plate [3]. Here we report our simulation of vortex shedding from the two free ends of a flexible fibre with its center point tethered (otherwise unrestricted) in a two-dimensional flowing viscous incompressible fluid by the immersed boundary method [4]. The motivation of our work is a laboratory experiment reported in [5]. The Reynold numbers range from $2000$ to $40,000$ in the experiment and the authors focused on drag reduction caused by self-similar bending of the fibre. Our work concentrates on the vortex shedding at lower Reynolds numbers ($12.5-375$), investigating the influences of inflow speed, fibre length and fibre bending rigidity on the vortex shedding. \vskip 2mm {\underline {References}} \small \noindent [1] C.H.K. Williamson and R. Govardhan, {\it Annu. Rev. Fluid Mech.} {\bf 36}, 413 (2004). \noindent [2] S. Lee, {\it Computers \& Fluids} {\bf 29}, 639 (2000). \noindent [3] T. Sarpkaya, {\it J. Fluid Mech.} {\bf 68}, 109 (1975) \noindent [4] C.S. Peskin, {\it Acta Numerica} {\bf 11}, 479 (2002). \noindent [5] S. Alben, M. Shelley, and J. Zhang, {\it Nature} {\bf 420}, 479 (2002).   

Vortex Shedding Dynamics in the Wake of Slender Cones at Low Reynolds Numbers

Since the original work of Gaster, the K\'{a}rm\'{a}n vortex shedding from slender cones placed normal to an oncoming uniform flow has been thought to lead to a series of stationary cells along the cone span, separated by zones of dislocations, with constant shedding frequency within each cell. Experiments on two cones with taper ratios of 3.2 10$^{-3}$ and 6.7 10$^{-3}$ are reported for local Reynolds numbers ranging between 40 and 180. By visualizing the plan view of the wake with hydrogen bubbles and determining local ``instantaneous'' frequencies, wave lengths and shedding angles from a digital movie, it is shown that shedding cells do appear but consistently move towards the thin end of the cone. An attempt is made to correlate this spanwise cell velocity with the speed of ``hole-solitons'' in the spanwise Ginzburg-Landau equation. Finally, it is shown why the data processing employed in previous studies can make moving cells look stationary.   

Free-stream shear effects on vortex shedding from step-cylinders.

In a recent experimental study we examined the complex process of vortex shedding from cylinders with a stepwise change in diameter inserted in uniform streams. The present work examines vortex shedding from a step-cylinder with a diameter ratio near 2 in uniformly sheared flow generated by a curved screen in a water channel. The Reynolds number, based on the centreline velocity and the large cylinder diameter, was in the range 268 to 622. The experimental techniques include laser Doppler velocimetry and flow visualization by electrolytic precipitation. Compared to the uniform flow case, vortices in the shear flow were generally less well-defined. In both the uniform and shear flows, spectra identified spanwise cells of constant frequency, including a distinct cell near the step, but the number of cells was larger in the shear flow case. The orientation of the cylinder axis relative to the shear direction affected the spanwise length of the near-step cell, the frequency difference between this cell and the adjacent cell behind the large cylinder, and the inclination of vortices behind both cylinders. Wavelet analysis showed that the vortex shedding frequency changed constantly with time near cell boundaries and within the near-step cell.   

Instability of a junction vortex

The flow field in the region where a moving wall, started from rest, slides under a stationary one, produces an interesting flow phenomena with relatively simple generation geometry. Experiments show that if the wall speed is high enough a vortex forms close to the junction of the moving wall with the stationary one. Vortex formation was observed for the range of Reynolds number $5 \times 10^2 \rightarrow 5 \times 10^5$, where the length scale is the distance the wall has moved from rest. The data reveals that in the absence of an apparatus length scale, the vortical structure appears to scale in a self- similar fashion that is consistent with the impulse provided by the moving wall. Over this Reynolds number range the vortical structure, which is initially laminar, begins to transition at $16 \times 10^3$ and appears to be turbulent by $40 \times 10^3$. The transitional regime is marked by the appearance of an instability wave on the perimeter of the vortical structure. The instability mechanism appears to be centrifugal in nature. The formation and non-linear growth of these structures and their ingestion into the primary vortex core is what causes the eventual breakdown of the primary vortex.   

Fluidic-Driven Ducted Heat Ejector

Unsteady, small-scale fluid mechanics and heat transport processes within a high-aspect ratio ducted heat ejector are investigated experimentally. The ducted heat ejector exploits the flow that is induced within the channel by the motion of a vibrating reed to cool the inner surfaces of the duct walls and thereby transport heat across its boundaries to cool electronic hardware by direct contact. This cooling approach is particularly attractive for low-power, densely-packed electronic hardware where heat is removed by direct conduction through the duct walls. The time harmonic motion of the reed results in the regular shedding of nominally two-dimensional counter-rotating vortical structures and induces a net flow through the duct. The flow characteristics are investigated using high-resolution particle image velocimetry (PIV). Of particular interest is the effect of the induced, small-scale motions and enhanced mixing on heat transfer across the duct boundaries which is comparable to conventional time-invariant channel flows at higher Reynolds numbers.   

Starting jets of finite width and formation time of vortex dipoles

Evolution of a two-dimensional flow induced by a jet ejected from a nozzle of finite size is studied experimentally. Vortex dipole forms at the front of the developing flow and then moves forward with constant speed. Trailing jet establishes behind the dipole. The dynamics of the flow is discussed on the basis of detailed measurements of vorticity and velocity fields which are obtained using particle image velocimetry. It is found that within the range of control parameters used in our experiments the dipoles never separate from the jet which is in contrast to the behavior of vortex rings reported previously by other authors. However, the formation time for the dipoles can be introduced such that after the formation the dipoles start moving away from the nozzle. Their dynamics after the formation is characterized by a reduced flux of vorticity from the jet. A value of the ratio of the speed of propagation of the dipole to the mean velocity of the jet is found to be 0.5 for later times of the evolution of the flow. A theoretical model is offered to predict quantitatively the initial propagation of the dipole as well as its steady-state regime.   

Merging of unequal strength co-rotating vortices

We present numerical simulation results on the 2D merging of a pair of co-rotating vortices and examine the effect of varying the circulation ratio between the two vortices. In a first step we present the results obtained for a pair of equal strength vortices ($\Gamma_2/\Gamma_1 = 1$) at a Reynolds number of $\mathit{Re} = \Gamma_1/\nu = 530$. A detailed quantitative analysis is made and shown to be in very good agreement with the experimental measurements of Cerretelli \& Williamson (2003, JFM, {\bf 475}, 41--77) for that same case. The results for vortex pairs with unequal strengths, i.e.\ with $\Gamma_2/\Gamma_1 < 1$, also at $\mathit{Re} = 530$, are presented next. The decomposition of the vorticity field into its symmetric and anti-symmetric components, as done by Cerretelli \& Williamson, shows that the structure of the anti-symmetric vorticity, in the convective merging phase, completely changes when $\Gamma_2/\Gamma_1$ reaches a value of about 0.8. Above that value, both vortices undergo deformation and contribute to the apparition of anti-symmetric vorticity. They also both move towards the center of rotation while they merge. Below that value, only the weaker vortex undergoes significant deformation and thus contributes to the apparition of anti-symmetric vorticity. Furthermore, the center of that weak vortex remains almost stationary (in the proper rotating frame of reference). The stronger vortex follows the opposite behavior with only a little deformation but with a more significant relative displacement.   

Session KH: Experimental Techniques: Particle-Based Anemometry

Energy dissipation rate measurements with PIV-PTV in a Partially Stirred Reactor (PaSR)

The work presented addresses some of the questions surrounding the experimental determination of the mean energy dissipation rate. The approach is based on a hybrid PIV-PTV (particle-tracking velocimetry) method. The flow investigated is in a Partially Stirred Reactor (PaSR) in which fluid is injected through 16 opposed jets that issue from top/bottom planes. PIV is first used to detect a locally averaged mean velocity of particle groups. A refined PTV algorithm is then used, with a resolution of $\sim $1 pixel, to determine the instantaneous, spatially resolved velocity field. Second-order structure functions are calculated as a function of spatial separation. Their small-scale limit provides estimates for the mean energy dissipation rate that are in good agreement with ones obtained from the inertial range of scales or very large scales (the last two are not affected by resolution). Three different methods allow for a proper determination of the dissipation rate, which is found to be 20 {\%} greater than that inferred from PIV alone. Our results suggest an improved methodology for the estimation of instantaneous values and high-order, small-scale, velocity-field statistics.   

Development of a PIV based technique for measurements of instantaneous pressure distributions

A non-intrusive PIV based method for measuring the instantaneous pressure distribution over a sample area simultaneously with the velocity field has been developed and tested. This method utilizes four-exposure PIV measurements to measure the distribution of material acceleration, and then integrating it to obtain the pressure. Two cameras are used, one for recording images 1 and 3, and the other for recording images 2 and 4. Validation tests of the principles of the technique using synthetic images of rotating and stagnation point flows show that the standard deviation of the measured pressure from the exact value can be kept within 1.0{\%}. A key to the success of this method is precision matching of images recorded by the two cameras. Images of stationary particles recorded at the same time are used for generating a map of local deformations for matching the two filed of views. Application of local image deformation correction to velocity vectors measured by the two cameras reduces the error due to misalignment and image distortion to about 0.01 pixels. An efficient and accurate acceleration integration algorithm has also been developed. This method has been used for measuring the instantaneous pressure distribution in a cavity shear flow, and samples will be presented.   

Experimental investigation of low Mach number flow past a rectangular cavity using dual-camera Cinematographic PIV system

Flow past cavity has been of interest due to its geometrical simplicity and complex flow characteristics. A dual-camera Cinematographic Particle Image Velocimetry (CPIV) system has been developed to study low Mach number flow over a rectangular cavity. This system consists of two high-repetition rate Nd:YAG lasers and two high-speed CMOS cameras registered to have sub-pixel alignment errors. A rectangular cavity with a length-to-depth ratio of 2 was mounted in the test section of a recirculating water tunnel providing free-stream flow speeds between 5$\sim $26 m/s. Consecutive CPIV images with a spatial resolution of 1632 x 800 pixels and 20 $\mu $s time delay were obtained at frame rate of 1.5 KHz. Time traces of surface pressures at the bottom of the cavity are acquired simultaneously by using flush-mounted dynamic pressure transducers. The temporal evolution of velocity and vortical fields reveals the time-dependence of the mixing and mass transport between the shear layer and the cavity. The simultaneous velocity and pressure measurements also show the unsteady interaction between vortical structures and the trailing edge of the cavity under resonating and non-resonating conditions. [Sponsored by National Science Foundation Grant: CTM 0203140]   

Cinematographic 3-D PIV of a Turbulent Jet

The structure of a fully developed turbulent jet at a Reynolds number of 5000 is investigated with cinematographic (1 kHz) stereoscopic PIV in a plane normal to the jet axis (i.e., ``end view''). The temporal resolution is sufficiently high that Taylor’s hypothesis can be used to enable the computation of velocity gradients in the axial direction. Furthermore, the resolution (in space and time) is approximately three Kolmogorov scales, and is therefore sufficient to resolve the structure of the dissipation field. The technique enables computation of all terms of the velocity gradient tensor in a plane, at kilohertz rates, and therefore at each point in the plane we can compute the complete vorticity vector, strain rate tensor and kinetic energy dissipation. We use the data to investigate the time-evolution of the dissipation field and its relationship to the vorticity and strain rate fields. The data can alternatively be used to form y-z-t volumes (with x the axial direction). These pseudo-volumes show that the vorticity field is dominated by tube-like structures and the dissipation structures are often sheet-like or more nondescript “blobs”. The spatial relationship among the dissipation, strain rate and vorticity will be discussed, as well as a statistical analysis of these quantities.   

WITHDRAWN: Use of Variable Thresholds in Post-Correlation PIV Outlier Correction

While PIV has successfully proven to be a valuable velocity measurement technique, it has still been susceptible to producing outliers due to a variety of reasons. Many methods of correcting these post-correlation outliers have been proposed with varying degrees of success. This paper proposes a method of correction using automated variable thresholds to identify and replace outliers. This method varies the threshold depending on the velocity around the node of interest, and utilizing statistical properties such as averages, and standard deviations to identify the local threshold value. This allows the threshold to better accommodate the changes in the velocity profile of a specific region. This method will be applied to simulated velocity profiles containing outliers, and the results will be compared with a fixed threshold method.   

Improving the spatial resolution of particle image velocimetry by means of weighting window functions

Due to its high robustness, correlation-based particle image velocimetry (PIV) has become the prime choice for processing image-based flow measurements in fluid dynamics experiments. The algorithm-inherent averaging process over certain image subspaces limits the achievable spatial resolution. By utilizing appropriate weighting window functions in an iterative refinement loop, the spatial resolution can be improved significantly. We will prove the concept by investigating the achievable stability and spatial resolution given by the frequency response function for different windowing functions. An algorithm that implements the windowing concept will be introduced yielding a densely sampled estimate of the velocity field. The ability of the entire procedure to improve the spatial resolution is shown using a simulated flow field with high velocity gradients.   

Random error due to Brownian motion in particle image velocimetry

In particle image velocimetry (PIV) experiments, the Brownian motion of seed particles results in the addition of a random error component to the measured velocity. As the number of seed particles within each control volume increases, this random error component will decrease. Previous researchers have assumed that this random error decreases proportionally with the square root of the number of seed particles in the measurement volume as the central limit theorem predicts. However, this conclusion is based on the assumption that each seed particle contributes equally to the measured velocity, which is not the case in PIV. This assumption is even further weakened in micro-PIV experiments, where the contribution of individual particles is dependent on their separation from the image plane, due to volume rather than sheet illumination. In the present work, a computer simulation of volume-illuminated and sheet-illuminated particle-image interrogation for flowfields with a Brownian motion component was performed for various experimental conditions to determine the proper relationship between seed particle density and random error due to Brownian motion. It was found that the actual random error is greater than that predicted by application of the central limit theorem. To assist in the design and analysis of PIV experiments, an empirical formula for estimating error in micro and standard PIV experiments with Brownian motion is presented.   

A novel 3D micron-scale DPTV (Defocused Particle Tracking Velocimetry) and its applications in microfluidic devices

The rapid advancements in micro/nano biotechnology demand quantitative tools for characterizing microfluidic flows in lab-on-a-chip applications, validation of computational results for fully 3D flows in complex micro-devices, and efficient observation of cellular dynamics in 3D. We present a novel 3D micron-scale DPTV (defocused particle tracking velocimetry) that is capable of mapping out 3D Lagrangian, as well as 3D Eulerian velocity flow fields at sub-micron resolution and with one camera. The main part of the imaging system is an epi-fluorescent microscope (Olympus IX 51), and the seeding particles are fluorescent particles with diameter range 300nm - 10um. A software package has been developed for identifying (x,y,z,t) coordinates of the particles using the defocused images. Using the imaging system, we successfully mapped the pressure driven flow fields in microfluidic channels. In particular, we measured the Laglangian flow fields in a microfluidic channel with a herring bone pattern at the bottom, the later is used to enhance fluid mixing in lateral directions. The 3D particle tracks revealed the flow structure that has only been seen in numerical computation. This work is supported by the National Science Foundation (CTS - 0514443), the Nanobiotechnology Center at Cornell, and The New York State Center for Life Science Enterprise.   

3D-flow measurements in micro channel and pipe with high time resolution using micro digital-holographic particle-tracking velocimetry

A micro digital-holographic particle-tracking velocimetry (micro-DHPTV) method for high time-resolution flow field measurement in a micro-channel was developed by Satake et al. (2005). The system consists of an objective lens, a high-speed camera and a single high-frequency double pulsed laser. Particle positions in a three-dimensional field can be reconstructed by a computer-generated hologram. The time evolution of a three-dimensional water flow in a semicircular micro-channel of 100-$\mu$m width and 40-$\mu$m depth and in a circular micro-pipe of 100-$\mu$m inner diameter are obtained successfully using this micro-DHPTV system. The three- dimensional measurement volume of the system is 410 $\mu$m $\times$ 100 $\mu$m $\times$ 40 $\mu$m and is irradiated by one laser beam with the time resolution of 100 $\mu$sec and a reputation rate of 1 kHz. Consequently, 130 velocity vectors in the semicircular micro-channel can be obtained instantaneously. Satake, S., Kunugi, T., Sato, K., Ito T., Taniguchi, J., ``Three- dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,'' To appear in Optical Review, 2005.   

Measurements of 3-D Flows with a Digital Holographic Microscope

Rising interests in micro-scale dynamics, such as turbulence in a near wall region or flow around a microorganism require measurements at compatible scales. A Digital Holographic Microscope (DHM) records magnified in-line holographic images and the 3D volumes are reconstructed numerically. This method offers inherent advantages over both conventional microscopy and lens-less in-line holography. DHM extends the depth of field of a conventional microscope by two orders of magnitude, to about 1000 time the target object diameter. It also reduces the depth-of-focus to less than ten particle diameters, two orders of magnitude lower than lens-less holography. For example, using segmentation and volume averages, one can detect displacements of 2$\mu $m particles in the depth direction at a resolution of about 10 $\mu $m. A single digital hologram can detect 5000 -- 10,000 particles. Examples of implementation of this method include near-wall velocity measurements of the channel flow at 0$<$y$^{+}<$60 (Re$_{h}$=60,000), as well as swimming behavior of a nauplius. Errors and techniques for determining velocity field and particle distributions are discussed.   

Measurement of high-speed water column inside a Steam Injector using Dynamic PIV

The Steam Injector is the superior system to pump the fluid without rotating machine. Because the water column is surrounded by the saturated steam, very high heat transfer is also expected with direct condensation. The inside of the Steam Injector is very complicated. To improve the efficiency of the Steam Injector, the water column behavior inside the Injector is visualized using the Dynamic PIV system. Dynamic PIV system consists of the high-speed camera and lasers. In this study, 384x180 pixel resolution with 30,000fps camera is used to visualize the flow. For the illumination CW green laser with 300mW is applied. To view inside the Injector, relay lens system is set at the Injector wall. Very high speed water column during the starting up of Steam Injector had been clearly visualized with 30,000fps. The wave velocity on the water column had been analyzed using PIV technique. The instability of the water column is also detected.   

Comparison of Data from Three PIV Configurations for a Supersonic Jet in Transonic Crossflow

Particle image velocimetry (PIV) data have been acquired using three different configurations in the far-field of the interaction of a transverse supersonic jet with a transonic crossflow. The configurations included two-dimensional PIV in the centerline streamwise plane at two overlapping stations, as well as stereoscopic PIV in both the same streamwise plane and the crossplane. The streamwise data show the downstream evolution of the interaction whereas the crossplane data directly reveal its vortex structure. The measurement planes intersect at a common line, allowing a comparison of those mean velocity components and turbulent stresses common to all configurations. All data from the streamwise plane agree to within their estimated uncertainties, but data from the crossplane exhibit reduced velocity and turbulent stress magnitudes by a small but significant degree. Additionally, the vertical positions of the peak velocities are slightly nearer the wall for the crossplane configuration. This comparison suggests that routine methods of uncertainty quantification for data used in the validation of computational models may not fully capture the error sources of an experiment.   

Session KJ: Granular Media: Surface and Tumbling Flows

Surface Mobility of Horizontally Vibrated Granular Layers as a Function of Depth

Stimulated by studies of avalanches where the critical slope angle is a function of layer depth [1], we investigate horizontally vibrated layers of various thickness, using acceleration to simulate the effects of gravity. The rectangular cell is 20 cm long in the direction of motion, and 8 cm transverse to that direction, containing polydisperse polystyrene particles of diameter 0.7-1.2 mm, 1-20 particles deep. We measure the RMS velocity of the mobilized surface particles in the frame of reference of the oscillating box, as a function of non-dimensional acceleration and layer depth. We find a depth-dependent threshold acceleration for surface mobility. The mobility also varies with time, due possibly to structural re-arrangement of the particles. The observations are compared to numerical simulations of the same phenomena using soft particle forces with friction, and to earlier experimental studies [2]. \newline \newline [1] O. Pouliquen, Phys. Fluids 11, 542 (1999). \newline [2] G. Metcalfe et al., Phys. Rev. E 61, 031302 (2002).   

Sensitivity of Granular Hopper Flows to Boundary Conditions

Granular hoppers provide an excellent experimental system for studying continuum descriptions of granular materials in part because there exist the well-established Jenike radial solutions. (Moreea, S. B. M. and Nedderman, R. M., Chem. Eng. Sci., 51, 3931--3942 (1996)) In the case of a perfectly symmetrical right hopper there are multiple constitutive relations that lead to the Jenike solutions. When asymmetry is introduced by tilting the hopper with respect to gravity, these constitutive relations lead to different predictions of how the granular flow will circulate. (Gremaud, P.A., Matthews, J.V. and Schaeffer, D. G., SIAM J. Appl. Math., 64, 583--600 (2003) and personnal communications) This experimental study finds that the Matsuoka-Nakai constitutive relation most accurately describes granular flow, but only for moderate tilt angles and only for a specific range of wall roughnesses. In more extreme cases, the velocity field is found to change substantially. We believe our results may allow for discrimination between different soil mechanics descriptions of dense granular flows.   

Surface Flow of Sloshing Granular Materials

We studied the fluidity of the flowing layer of grains in a tumbler through gentle sideways sloshing, and how the flow of large and small particles differs. Monodisperse beads were rotated in a long partially filled drum while also being gently vibrated sinusoidal below one g along the axial direction. As a result of the motion of the drum the beads slosh slightly back and forth also with sinusoidal motion. We compared the phase difference between the motion of the drum and the particles for big and small particles. It was determined that the phase difference is directly proportional to the frequency of vibration with a proportionality constant that increased with fluidity. The proportionality constant is a measure of the fluidity of the grains, and was shown to scale with the logarithm of rotation rate and the inverse square root of the bead diameter.   

Granular Flow in a Tumbler Under Variable g-Levels

The Froude number $\omega^2 r/g$, where $\omega$ is the rotational speed, $r$ the radius of the tumbler and $g$ the gravitational acceleration, is frequently used to characterize a granular flow. Although $g$ appears in the Froude number, little is understood about how its variation affects the nature of granular flow. Experiments were performed with $0.5mm$ glass beads in a half-full, quasi-two dimensional 45mm radius tumbler at high $g$-levels. The tumbler was mounted in a large centrifuge to provide high g-levels. At a particular tumbler rotational speed, the dynamic angle of repose decreases as the $g$-level increases from 1$g$ to 25$g$. However, the data at all g-levels collapses so that the angle of repose is independent of the $g$-level when plottet as a function of the Froude number. Furthermore, the shape of the surface of the flowing layer depends only on the Froude number, not directly on $g$. Thus, the Froude number appears to characterize the nature of the flowing layer in a tumbler when both $\omega$ and $g$ are varied.   

Core Precession and Erosion in a Tumbler Under Variable g-Levels

The precession and erosion of a core of granular material in a rotating tumbler that is more than half full provides a measure of the slow granular motion that occurs beneath the flowing surface layer. Since the effect of gravity on the subsurface flow has not been explored, experiments were performed in a~63{\%} to 83{\%} full tumbler apparatus mounted in a large centrifuge that can provide very high g-levels. Two colors of 0.5 mm glass beads were filled side by side to mark a vertical line in the 45mm radius quasi-two dimensional tumbler. The rotation of the core with respect to the tumbler and size of the core was monitored over 250 tumbler revolutions at accelerations between 1g and 12g. The degree of core precession increases with the g-level, while the core erosion depends less on g-level. The flowing layer thickness is essentially independent of the g-level for identical Froude numbers, suggesting that the shear rate in the flowing layer must increase with increasing g-level.   

Endwall Flow Effects in a Tumbler

The flow of granular material in rotating tumblers is limited to the thin flowing layer at the free surface. Particle tracking velocimetry was used to measure the surface velocity for 1 mm and 2 mm glass particles and sand in cylindrical tumblers of various diameters, lengths, and rotation rates. Friction at the end walls slows the streamwise surface velocity, yet material just in from the end wall flows faster than that in the center of the tumbler. Although the axial velocity in the center of a long tumbler is negligible, a non-negligible axial velocity exists near the end walls of cylindrical tumblers. The axial flow and increase in streamwise velocity are likely a result of the conservation of mass of particles passing through the flowing layer near the endwall. Increasing the end wall friction slows particles adjacent to the end wall, further enhancing the axial flow near the end wall. Decreasing the total axial length of the cylinder to quasi-two-dimensional causes the axial flow regions near the endwalls to merge and generates an even higher streamwise velocity than for three-dimensional tumblers.   

Granular Surface Flows in Three Dimensional Tumblers

Many granular flows are confined to thin layers of rapid surface flow. Therefore, a complete understanding of surface flows is the key to an accurate representation of the entire flow. Experiments were conducted measuring the surface flow in three-dimensional tumblers: cylindrical tumblers with lengths of 17.5 cm and diameters of 7.0 cm, 10.0 cm, 14.0 cm and 17.0 cm, a double-cone tumbler with a maximum diameter of 13.8 cm, and a spherical tumbler with a diameter of 13.6 cm. Surface velocity measurements for 1 mm and 2 mm glass particles were obtained using particle tracking velocimetry. Results indicate that the streamwise surface velocity at the midpoint of the flowing layer is a linear function of local flowing layer length, regardless of tumbler shape, particle size, rotation rate, and fill fraction. In addition, the axial surface velocity at the midpoint of the flowing layer is negligible. These results are key for the development of three-dimensional models of granular flows. [Supported by NSF and DOE.]   

Friction and the Dynamic Angle of Repose of a Granular Material

The angle of repose is one most common measures of the flow-ability of granular materials. Particle properties influencing the dynamic angle of repose of a granular material rotated in a cylinder are studied using particle dynamics simulations. The influence of each parameter in the force models of this commonly used simulation technique are evaluated through the use of a factorial set of designed simulation experiments. The friction coefficient is the most important parameter affecting the angle of repose. Bidisperse mixtures of particles with different friction coefficients are also studied. The angle of repose of the mixture depends on the concentration of the mixture, consistent with recent experiments, and also on how the friction coefficient for contacts between dissimilar particles is defined.   

Transverse Instability of Granular Avalanches

Recent experiments with dry and underwater avalanches revealed a surprising new phenomenon: transverse instability and fingering of avalanche front propagating on erodible granular substrate. In order to describe this phenomenon we applied order parameter model of partially fluidized granular flows which was successfully tested on downhill and triangular avalanches in thin granular layers. In the framework of our model we obtained a family of ``solitary'' front solutions with the velocity and the height of the front determined by the depth of erodible substrate, the inclination angle, and front’s total mass. We have found that the front exhibits transverse instability in the certain range of substrate depth and inclination angles, in good agreement with the experiment. The primary machanism of the transverse instability is related to the dependence of the front velocity on the mass of granular material it carries, and is not triggered by the granular size segregation   

New insight on the understanding of long runout avalanches: geometric lubrication

The unexpected long-runout landslides have been a controversial subject of discussion. In order to provide a new insight of this phenomena, we investigate the apparent reduction of friction resulting from the presence of small beads. Results obtained by means of a 2-D soft particle numerical simulation are presented. The numerical experiments consider an avalanche of two size disks, originally placed over an inclined plane. The friction coefficient for the particle-particle and wall-particle contacts is held fixed and is equal to 0.5. The granular mass is allowed to evolve with time, until it comes back to rest. The position of the center of mass is located, such that the runout length could be measured, L/H. Many simulations were performed keeping the area of the mass constant, varying only the percentage of small disks. The results show that the runout length increases with the percentage of small beads, reaching a maximum for approximately 25\% of small disks. These results indicate that the apparent friction coefficient is reduced and affected by the percentage area of small particles. Additionally, the formation of a layer of small disks at the base of the avalanche was observed. This layer is identified as the source of ``lubrication.''   

Bidisperse granular flows on inclined rough plane

Experiments on bidiperse dry granular flows on a rough inclined plane were performed in order to investigate its rheology. Flows, created by a localised input of granular matter onto the plane, propagate and spread laterally, being unconfined by the experimental set-up. Because of size segregation, small beads are found in a layer at the bottom of the flow and larger ones at the free surface, at the borders and at the front. These lateral and vertical inhomogeneous repartitions lead to two effects: the surround effect and the interface effect. The surround effect is due to the large beads at the front and borders of the flow. It can be interpreted considering the relative frictions of the two types of beads with the rough plane. We show that a maximum of the friction exists for a particular relative roughness. Depending on the values of relative frictions, obtained for monodisperse flow, the surround effect can lead to a narrowing of the bidisperse flow, an increase of the deposit thickness. The interface effect deals with the interaction between the layers of large and small beads. Depending of the size ratio between the beads, the large beads can be ``trapped'' in the deposit of small ones, or they can increase the velocity gradient in the layer of the small beads, by an entrainment phenomenon. The combination of these two effects results in the different behavior observed for bidispersed granular deposits, compared to those observed for monodisperse flows.   

Failure of Granular Slopes under Vibration

We report experimental measurements of the stability of a granular slope under external vibration. A 3D layer of glass beads is inclined and subjected to sinusoidal horizontal acceleration that continuously increases in amplitude. Video imaging of the upper surface of the slope provides sensitive detection of the onset of motion. For a wide range of inclination angles, even above the dynamic angle of repose, initial failure is transient: flow allows the material to find a stronger static configuration which then requires a higher acceleration to initiate the next failure. Measurements show a clear dependence of the acceleration at failure on the rate at which the acceleration increases, revealing microscopic strengthening before the onset of detectable bead motion.   

Session KK: Non-Newtonian Flows II

Multi-scale simulations of shear-dominated flows of rigid rod dispersions

The Doi-Hess theory coupled with an anisotropic Marrucci-Greco distortional elasticity potential provides a multi-scale description of the flowing nematic liquid crystalline polymers (LCPs). We have developed numerical simulation methods for the model equations for structure formation of LCPs in confied, planar Couette cells. In this talk we will provide some computational results from our numerical simulations. This includes the steady state structure in weak shear flow, structure transtions in the space determined by the Ericksen number and the Deborah number, the out-of-plane attractors and the chaos phenomena.   

WITHDRAWN: Taylor limit of equilibration for mass transport in transversely bounded rectilinear flows

The multimode diffusion approximation for solute dispersion in transversely bounded shear flows owes its origin to the formal method of eigenmode expansion. It is put forth upon the premise that a quasi-steady condition termed the Taylor limit of equilibration exists in the course of time when equilibrium estimates of the residual terms of the concentration distribution can be realistically made contigent to the evolution of their primary counterparts. By applying the Green's function for the diffusion equation, this paper provides a qualifying account for the establishment of the Taylor limit. A method of successive approximation is derived for the determination of the principal mode coefficient functions with the inclusion of bulk reaction and longitudinal diffusion. The resulting equations governing the evolution of these coeficient functions are truncated to conform to the multimode diffusion type. Examples are given to illustrate the attainment of a convergent solution.   

Changes of K\'{a}rm\'{a}n vortex shedding from a cylinder due to weak fluid elasticity

Experiments on vortex shedding from a cylinder placed in uniform flows of dilute polymer solutions are reported for Reynolds numbers from 50 to 150. The fluids used were aqueous solutions of polyethylene oxide (PEO) and rheological characterization showed them to have a constant viscosity over a wide range of shear rates. Using the Zimm model relaxation time the Deborah numbers calculated for the cylinder wake are O(10$^{-3})$. Parallel vortex shedding was induced with a combination of end-cylinders and end-plates. The resulting nominally two-dimensional cylinder wake was investigated using LDA, PIV, hydrogen bubble visualizations and hot film anemometry. The characteristics of the von K\'{a}rm\'{a}n instability - the critical Reynolds number, maximum perturbation amplitudes, etc. - are presented as a function of PEO concentration. It is shown that even small amounts of polymers, corresponding to low Deborah numbers, have a significant stabilizing effect which is only counteracted by shear-thinning at higher concentrations.   

Nonlinear instabilities in parallel shear flows of viscoelastic fluids

Newtonian fluids are known to exhibit turbulent behaviour at large enough Reynolds numbers. Recently, it has been discovered that flows of visco-elastic fluids in simple geometries become chaotic at arbitrary low Reynolds numbers (the so-called ``elastic turbulence''). When elastic stresses become large enough, laminar flows lose their stability and become turbulent. However, a little is known about the exact nature of this instability. Model calculations reveal that for some geometries the basic flow can become linearly unstable, while for the others it stays linearly stable for any value of the elastic stresses. Here we present a non-linear mechanism of the flow instability: independently of the presence or absence of the linear instability, the finite-amplitude disturbances can result in flow destabilization. We calculate the onset of this transition for plane Couette and plane Poiseuille flows and show that its sub-critical nature leads to the chaotic regime very close to the onset. We discuss briefly the role of these finite-amplitude solutions in sustaining visco-elastic turbulence.   

An exponential mapping for the conformation tensor for flow of viscoelastic fluids; application in turbulent channels

The conformation tensor, a quantity that describes the internal microstructure of polymer molecules, is usually being used as the primary variable in viscoelastic flow calculations. Its main property is that is a positive definite, second order, tensor. Unless special care is taken, the conformation tensor may lose this property resulting to instabilities during the calculations and finally either to break-up of the simulations or to non-physical results. This situation is greatly intensified under turbulent flow conditions. In order to resolve these problems we have expressed the conformation tensor, `c', as the exponential of another tensor `a', c=exp(a), and we solve for `a' instead of `c'. By construction, the positive definite property of `c' is always preserved since its eigenvalues are the exponential of the eigenvalues of `a'. The method is illustrated for viscoelastic turbulent channel flow. Direct Numerical Simulations are being performed using spectral spatial approximations and a stabilizing artificial diffusion term in the viscoelastic constitutive model. That term is needed to smooth the solution to be resolvable with the mesh size used due to the very fine scales that are being created in chaotic flow fields. The FENE-P constitutive model is used to represent the effect of polymer molecules in solution. We will offer a comparison of the results, for exactly the same flow, viscoelastic and numerical parameters, using the old and the new formulation of the constitutive model in terms of the conformation tensor and the exponential tensor, respectively.   

Statistical closure for homogeneous turbulent shear flow of a dilute polymer solution

Dilute polymer solutions exhibit macroscopic behaviors that distinguish them from ordinary Newtonian fluids. For example, minute concentrations of polymers (parts per million on a weight basis) can lead to impressive reductions in the drag on solid surfaces (by up to 80\%). Numerical simulations of viscoelastic flows are generally based on an evolution equation for the conformation tensor of the polymer, $C_{ij}=\langle r_i r_j\rangle$, where $\mathbf{r}$ is the separation vector between the ends of the molecule and the angle brackets indicate an average over the Brownian configuration space of the molecule. Direct numerical simulations (DNS) of viscoelastic turbulence are able to reproduce the key phenomenology found in experiments; however, they are limited to modest values of the Reynolds number. An alternative approach is to seek a closed equation for the average configuration tensor that could be coupled to a Reynolds averaged Navier Stokes (RANS) solver. We propose a set of effective equations of motion for the mean conformation tensor rooted in the analysis of Lagrangian stochastic models with independent correlation times for velocity rate-of-strain and rate-of-rotation tensors. The proposed closure is compared with numerical simulations of Gaussian stochastic flows and DNS of homogeneous turbulent shear flows.   

PIV and LIF measurements of a turbulent boundary layer with injected drag-reducing polymers at high Reynolds numbers

The injection of aqueous solutions of large molecular weight polymers into the near wall region of turbulent boundary layers (TBL's) has been known to produce significant reductions in friction drag. The goal of this study has been to make unique experimental measurements that illuminate the behavior of TBL's modified by slot injected polymers, and to assist with predictive code development efforts by producing experimental data sets at scales emulating prototype applications. For the present experiments, the model is a hydrodynamically smooth flat plate that measures approximately 13 m in length, and is 3 m wide. Free stream water speeds as high as 20 m/s were investigated, resulting in length-based Reynolds numbers above 200 million, and boundary layer thickness of $\sim $10 cm. Polyethylene oxide (PEO) based polymers were chosen for this study. We discuss near wall (y $<$ 0.2 cm) velocity measurements of the TBL produced by particle image velocimetry and polymer concentration measurements produced by non-simultaneous laser induced fluorescence. [Sponsored by DARPA]   

The Mechanism of Polymer Drag Reduction derived from Numerical Simulations

This talk revisits the mechanism of polymer drag reduction proposed by Dubief {\it et al.} ({\it J. Fluid Mech.}, {\bf 514}, pp 271-280, 2004) derived from the observation of coherent transfers of energy between polymers and velocity fluctuations. We present further proofs of this mechanism using Lagrangian tracers to represent polymer molecules in drag reduced flows as well as controlled numerical experiments to isolate various phenomena involved in the mechanism. Turbulence reduction, impacting mainly near-wall vortices, and increase, occurring in high-speed streaks very near the wall, are found to result from the dynamics of highly stretched polymers. We will discuss various scenarios to explain the occurrence of Maximum Drag Reduction based on our mechanism.   

Long Chain Polymers and Bubbly Drag Reduction in Taylor-Couette Flow

Small amounts of long chain polymers have been shown to dramatically reduce the drag of some turbulent flows. This effect is examined in a Taylor Couette apparatus ($Re=1.4\cdot 10^6$) instrumented to measure torque on the inner cylinder. Particular attention is paid to changes in drag reduction over time as a result of polymer degradation, and light scattering measurements are presented to quantify the change in polymer characteristics. Results are compared to drag reduction by bubble injection in the same apparatus and both methods are also examined in case of rough walls. The polymer used is polyacrylamide with mean molecular weights ranging from $8 \cdot 10^5$ to $1.8 \cdot 10^6$ Daltons. Concentrations range of 0.5 to 100 parts per million by mass.   

Stretching of dumbbells around the Kolmogorov scale in a turbulent shear flow

We present numerical studies of kinematic stretching of Hookean dumbbells in a turbulent Navier-Stokes flow with a linear mean profile, $\langle u_x\rangle=Sy$. The simulations combine Brownian dynamics of the dumbbells with a high-resolution pseudospectral calculation of the simple shear flow. Scales below the viscous Kolmogorov scale, at which most of the dumbbell dynamics is present, are well resolved. The variation of the constant shear rate $S$ causes a change of the velocity fluctuations on all scales and thus of the intensity of local stretching rate of the advecting flow. The latter is measured by the maximum Lyapunov exponent $\lambda_1$ and scales as $\lambda_1\sim \epsilon^{1/2}\sim S^{3/2}$. As suggested by de Gennes and Tabor, turbulence is found to stop the stretching of Hookean dumbbells when the full nonlinear velocity differences with respect to the bead positions are taken. The growth of anisotropy of stretching with increasing shear rate is confirmed by the joint statistics of the extension $R$ with the azimuthal angle $\phi$ and the polar angle $\theta$, respectively.   

\textbf{Initiation vs. Sustenance of Active Polymer-Turbulence Interactions}.

Lumley (1973) theorized that polymer molecules are passively stretched only by strain-dominated turbulent eddies with time scales below polymer relaxation time. Tabor {\&} DeGenne (1986) added the argument that \textit{active} turbulence-polymer energy exchange occurs only at strain-dominated eddies where polymer elastic and turbulent kinetic energies are comparable. The Lumley-Tabor-DeGenne (LTG) arguments suggest an interruption to the energy cascade with an effective increase in small-scale cutoff scale. With these concepts in mind, we analyzed the evolution of polymer-laden homogeneous shear turbulence---from an initial period of passive stretch, through initiation of active polymer-turbulence interaction, to quasi equilibrium---using DNS with accurate numerical implementation of the FENE-P model. We find at equilibrium, contrary to the LTG arguments, that polymer-turbulence energy exchange is concentrated at the largest eddies. Evolution backwards in time, however, leads to a state more consistent with LTG phenomenology, suggesting relevance to the model at the point of transition from passive to active polymer in quasi Newtonian turbulence. LTG phenomenology rapidly loses its relevance, however, as turbulence evolves towards an equilibrium state of active polymer-turbulence interactions.   

Session KL: General Fluid Dynamics II

Two-dimensional sails in a uniform potential flow

We consider a spatially two-dimensional potential flow model for the shape of and lift generated by a single or a pair of inextensible membranes or `sails' in a uniform stream. The problem is solved numerically via the boundary integral method. Of interest are (i) the `luffing' of a sail, which occurs when it is nearly aligned with the oncoming stream, (ii) the interaction of a pair of sails that share a common near-field, and (iii) possible optimal distributions of chord between two sails and relative orientations that can maximize the total lift generated.   

Cavitation inception on rudder models with smooth and scalloped leading edges

We present experimental results, based on water channel testing, comparing the lift and drag characteristics of rudder models with a smooth and scalloped leading edge at velocities which induce cavitation at the low pressure areas of the foils. We also compare the point of cavitation inception. We have found that leading edge cavitation occurs sooner in the scalloped leading edge model, but never progresses over the entire leading edge as it does with the smooth leading edge model.   

Exact solutions of the Navier-Stokes equations having steady vortex structures

We present two classes of exact solutions of the Navier-Stokes equations, which describe steady vortex structures with two-dimensional symmetry in an infinite fluid. The first is a class of similarity solutions obtained by conformal mapping of the Burgers vortex sheet to produce wavy sheets, stars, flowers and other vorticity patterns. The second is a class of non-similarity solutions obtained by continuation and mapping of the classical solution to steady advection-diffusion around a finite circular absorber in a two-dimensional potential flow, resulting in more complicated vortex structures that we describe as avenues, fishbones, wheels, eyes and butterflies. These solutions exhibit a transition from `clouds' to `wakes' of vorticity in the transverse flow with increasing Reynolds number. Our solutions provide useful test cases for numerical simulations, and some may be observable in experiments, although we expect instabilities at high Reynolds number. For example, vortex avenues may be related to counter-rotating vortex pairs in transverse jets, and they may provide a practical means to extend jets from dilution holes, fuel injectors, and smokestacks into crossflows.   

The Interplay of Solvation Forces with Lubrication Forces in Thin Gaps

We report on the motion of a planar sphere approaching a smooth planar surface in a Lennard--Jones fluid using molecular dynamics simulations to understand the effect of solvation forces on the hydrodynamic drag experienced by the sphere. The hydrodynamic theory is found to be well representative of the force experienced by the sphere, even as close as 5--6 molecular lengths from the surface. Close to the surface, the solvation force is observed to be prominent in two regards, in the presence of molecular--length oscillations in the total force, and in some cases being also the predominant contributor to the total force on the sphere. The density distribution of the fluid particles close to the surface is characterized by strong layering, indicative of a breakdown of the continuum approximation in the hydrodynamic theory. Studies employing a range of sphere sizes and approach velocities indicate that the effect of the solvation force is most pronounced for slow approach velocities and for large spheres. Additionally, for analyzing experimental total force data, we suggest a phenomenological approach to decompose the total force on the sphere into a static component (``solvation force'') and a dynamic component (``lubrication force'',) which should be of interest in sedimentation processes and Atomic Force Microscopy.   

Mode-Locking of Passive Scaler Transport to Chaotic Flows in a Disk

We study transport of a passive scaler in a bounded 2-dimensional, chaotic flow, specifically a disk with flow generated by tangential boundary motion over finite arcs of the disk, the RAM flow.\footnote{Metcalfe et al, to appear AIChE J (2005).} We calculate a spectral solution to the advection-diffusion equation, such that for a fixed Pecl{\'e}t number the full spectrum of eigensolutions---the so-called ``strange'' eigenmodes for the spatial distribution of the scaler\footnote{Pierrehumbert Chaos {\bf 10} 61 (2000).} and their decay rates---can be obtained for the entire space of controlling flow parameters with just a few additional matrix multiplies. Over this parameter space the dominant strange eigenmode symmetry locks to the wavenumber of the chaotic flow. For low values of stretching the eigenmode is symmetric. As stretching increases, the locked intervals grow, showing Arnol'd tongues and a transition to asymmetry of the scaler distribution. Preliminary experiments using an infrared camera to image temperature in a thin fluid layer show similar patterns and decay rates.   

Residence time of a buoyant ball in a hydraulic jump

We study experimentally the residence time required for a buoyant ball to cross upstream of a hydraulic jump. The experimental results indicate that the the distribution of residence times is exponential, and therefore history-independent. Based on this insight, we formulate a model in which an attempt at crossing the hydraulic jump is made at regular time intervals $\tau$ and with a constant probability of success $p$, use phenomenological theory of turbulence to obtain an expression for $\tau$, and ascertain that $p$ depends on a single dimensionless variable, which we identify. Last, we show that the predictions of this model are in good accord by the experimental data over a broad range of Froude numbers, ball sizes, and ball densities.   

Bounds on the enstrophy growth rate for solutions of the 3-d Navier-Stokes equations

It is still an open problem whether smooth solutions to the 3-d Navier-Stokes equations lose regularity in finite time. But it is known that if enstrophy ($\|\mathbf{\omega}\|^2$) remains finite, the solution is regular. The growth rate of enstrophy can be estimated from the Navier-Stokes equations by Sobelev inequalities. In general form, $d\|\mathbf{\omega}\|^2/dt \le c(\|\mathbf{\omega}\|^2)^\alpha$, where $c$ is a constant. In 2d, the exponent $\alpha$ is 2 and leads to regularity. However, $\alpha=3$ in 3d, which shows only finite-time regularity of the solutions. In these types of estimates, incompressibility is not used. We formulate the search for the maximal enstrophy growth rate as a variational problem and include incompessibility as a constraint. The variational problem is solved numerically by a gradient-flow type algorithm. Our preliminary results show that $\alpha\approx 1.75$, which hints that solutions of the 3-d Navier-Stokes equations are regular for all time.   

Vorticity Dynamics in Soap Films

Fast flowing soap films offer an experimental realization of two-dimensional flow that allows studies of fluid-structure interaction and vortex shedding mechanisms. The thickness of a gravity driven soap film can undergo significant variations in experiments but most modeling work in the past has focused on incompressible results that correspond to constant thickness films. We discuss a viscous compressible model of soap film flow that is equivalent to the Navier-Stokes equations with a film thickness dependent viscosity. A new vorticity transport equation for compressible soap film flow suggests effects unique to soap films and not predictible by a constant density incompressible theory. New exact solutions for the soap film model will be presented. Numerical simulations are shown to illustrate vortex shedding in soap film experiments.   

Drag Reduction Method using Combination of Hydrophobic and Hydrophilic Coatings

A new drag reduction method for a moving model in water is presented. This method applies a flow control using a combination of hydrophobic and hydrophilic coatings on the model surface. The flow is passively controlled by changing a chemical property on a model surface. As a preliminary result, a sphere with 2 mm in diameter is used as a basic model. The sphere is dropped in a 1 m height water tube, which has 100 mm in diameter. A flash lamp with 10 ms interval is used to capture the sphere motion at the terminal velocity. The drag coefficients, Cd, of different surface coatings are compared. Hydrophobic coating on the sphere increases drag with Cd of 0.49, while non-hydrophobic coated one shows Cd of 0.44. A sphere with hydrophilic coating gives Cd of 0.42. This tells that the hydrophilic coating on a sphere reduces drag instead of applying hydrophobic coating. In the final version, the time interval of dropping motion will be included. Besides a sphere model, other basic shapes, such as flat plate and cone, will be investigated. For a flat plate and a cone model, a combination of hydrophobic and hydrophilic coatings will be separately applied on a model surface to discuss the efficiency of a new drag reduction method.   

Fronts in high-temperature laminar jets

This paper addresses the slender laminar flow resulting from the discharge of a low-Mach-number hot gas jet of radius $a$ and moderately large Reynolds number $R_j$ into a cold atmosphere of the same gas. We give the boundary-layer solution for plane and round jets with very small values of the ambient-to-jet temperature ratio $\varepsilon$ accounting for the temperature dependence of the viscosity and conductivity typical of real gases. It is seen that the leading-order description of the jet in the limit $\varepsilon \rightarrow 0$ exhibits a front-like structure, including a neatly defined separating boundary at which heat conduction and viscous shear stresses vanish in the first approximation. Separate analyses are given for the jet discharging into a stagnant atmosphere, when the jet boundary is a conductive front, and for the jet discharging into a coflowing stream, when the jet boundary appears as a contact surface. We provide in particular the numerical description of the jet development region corresponding to axial distances of order $R_j a$ for buoyant and non-buoyant jets, as well as the self-similar solutions that emerge both in the near field and in the far field. In all cases considered, the comparisons with the numerical integrations of the boundary-layer problem for moderately small values of $\varepsilon$ indicate that these front descriptions give excellent predictions for the temperature and velocity fields in the near-axis region.   

Session KM: Materials Processing

Elastohydrodynamics of Step and Flash Imprint Lithography

We study theoretically and numerically the elastohydrodynamics of template filling and deformation in Step and Flash Imprint Lithography (SFIL). This is a photolithography process in which the photoresist is compressed in its liquid monomer form between a silicon wafer and a quartz template with desired features. The monomer is cured into the template pattern by flashing UV light through the quartz template, instead of using traditional optical systems. Features as small as 20 nm can be produced with SFIL. Surprisingly, the lubrication pressures in the filling process can be large enough to cause distortions in the template that are comparable to the feature size and hence reduce the fidelity of the imprinting process. An elastohydrodynamic simulation is developed combining lubrication theory with capillary forces for the fluid flow and thin plate theory for the template deformation to understand the dynamics of the process and how to mitigate the undesirable deformation. Imprint time, template deformation and possible contact of the template with the wafer are presented as a function of number of drops, their placement and imprinting speed.   

Multiscale modeling of interfacial flow in particle-solidification front dynamics

Particle-solidification front interactions are important in many applications, such as metal-matrix composite manufacture, frost heaving in soils and cryopreservation. The typical length scale of the particles and the solidification fronts are of the order of microns. However, the force of interaction between the particle and the front typically arises when the gap between them is of the order of tens of nanometers. Thus, a multiscale approach is necessary to analyze particle-front interactions. Solving the Navier-Stokes equations to simulate the dynamics by including the nano-scale gap between the particle and the front would be impossible. Therefore, the microscale dynamics is solved using a level-set based Eulerian technique, while an embedded model is developed for solution in the nano-scale (but continuum) gap region. The embedded model takes the form of a lubrication equation with disjoining pressure acting as a body force and is coupled to the outer solution. A particle is pushed by the front when the disjoining pressure is balanced by the viscous drag. The results obtained show that this balance can only occur when the thermal conductivity ratio of the particle to the melt is less than 1.0. The velocity of the front at which the particle pushing/engulfment transition occurs is predicted. In addition, this novel method allows for an in-depth analysis of the flow physics that cause particle pushing/engulfment.   

A `Coating' Operating-Window for a Metal-Casting Flow

Planar-flow single roll melt spinning is a promising technology for the next generation of continuous casting machines. In the process, a planar nozzle is held close to a rotating metal wheel and liquid metal is forced through the nozzle into the gap region between the nozzle and wheel where a puddle, constrained by surface tension, is formed. A solidification front grows along the wheel as it translates, forming a solid ribbon ($\sim $100 $\mu $m thick) which is continually ejected from the puddle ($\sim $ 10 m/s). An operating window can be predicted based on the possible curvatures of the upstream meniscus, similar to the approach taken for some liquid film coating flows. While the derivation of the window mirrors that of the coating-flow literature, the flow regimes are different. Apart from the presence of solidification, the pressure losses in casting flows are predominantly inertia dominated, while viscous effects are usually dominant in coating flows. A range of accessible product thicknesses can be predicted when casting open to atmosphere. This predicted thickness range is compared with well-established empirical windows. The predicted window also indicates considerable benefits (extended operability) from applying a pressure difference between the upstream and downstream menisci, as will be discussed.   

Fluid Mechanical Properties of Silkworm Fibroin Solutions

The aqueous solution behavior of silk fibroin is of interest due to the assembly and processing of this protein related to the spinning of protein fibers that exhibit remarkable mechanical properties. To gain insight into the origins of this functional feature, it is desired to determine how the protein behaves under a range of solution conditions. Pure fibroin at different concentrations in water was studied for surface tension, as a measure of surfactancy. In addition, shear induced changes on these solutions in terms of structure and morphology was also determined. Fibroin solutions exhibited shear rate-sensitive viscosity changes and precipitated at a critical shear rate where a dramatic increase of 75-150\% of the initial value was observed along with a decrease in viscosity. In surface tension measurements, critical micelle concentrations were in the range of 3-4\% w/v. The influence of additional factors, such as sericin protein, divalent and monovalent cations, and pH on the solution behavior in relation to structural and morphological features will also be described.   

Numerical Modeling of Hollow Fiber Drawing

Hollow optical fibers, which are widely used in medicine and in diagnostics, power delivery and communications, are typically manufactured by drawing a specially fabricated hollow preform down to a fiber in a conventional fiber-drawing tower. In this work, a numerical model based on the mass, momentum and energy equations is developed to investigate the drawing process. The axisymmetric flow of air in the central cavity, as well as the flow of glass and aiding purge gas, are considered. The complex computational domains are converted to cylindrical regions by using coordinate transformations. The two neck-down profiles, which are the inner and outer surfaces of the hollow fiber, are generated by using an iterative scheme. The optical thick approximation, as well as the zone model, are applied to calculate the radiative transport within the glass and the Boussinesq approximations are used for the buoyancy effects. The results obtained show that it is possible to predict the geometry of the final hollow fiber and to provide feasible combinations of parameters for successful hollow fiber drawing. The validation of the model is carried out by comparing the predictions with the results for solid-core fiber drawing and with available experimental and numerical results for hollow fibers. It is shown that the results from the model are consistent with the physical trends and agree well with the results in the literature.   

Effect of Curtet Number Variation on Dispersion of SWNTs in Epoxy Composites Prepared by a Continuous Impingement Mixing Process

Dispersion of nanoparticles is a key issue in preparing nanocomposites. Preparation of polymer nanocomposites is normally done by batch processing, with nanoparticles synthesized in-situ by a chemical reaction, which leads to a good dispersion. However, the in-situ synthesis technique is not readily applicable to the dispersion of single wall carbon nanotubes (SWNTs). A novel approach is presented here to improve the dispersion of SWNTs in polymers to enhance the structural, electrical and thermal properties that uses a continuous, high output impingement mixing process. In particular, we report on the dispersion and properties of composites of SWNTs in a Shell EPON-862/W system. The primary mechanism of dispersion is a high-speed jet immersed in a secondary stream confined by a constant area duct. The degree of dispersion is governed by the Curtet number (C$_{t}$) calculated using the diameter ratio and the velocity ratio. It was found that dispersion was affected by a critical value for C$_{t}$ of 0.75. Poor performance above the critical Curtet number is attributed to a reduction in residence time of fluid within the mixer. A series of composites with various SWNT loading were prepared at various C$_{t}$. The effect of C$_{t}$ on the degree of dispersion was evaluated by scanning electron microscopy and electrical conductivity measurements. Percolation curves were also obtained.   

Solvent-Free Thermal Spraying of Polymer Particles

During thermal spray deposition, jets of high temperature and high velocity gases are used to melt and accelerate particles towards the surface to be coated. Upon impact at the surface, multiple droplets deform, cool and consolidate to form a coating. A 3D model of particle impact and deformation on flat and rough surfaces has been developed for thermally sprayed polymer and metal particles. Fluid flow and particle deformation were predicted by the Volume of Fluid Method using Flow-3D software. A comparison between polymer and metal splatting demonstrates how the large physical property differences between these materials affect their flow behavior under similar thermal spray conditions. The higher viscosity of molten polymers leads to lower Reynolds numbers and less deformation, and lower thermal conductivity of polymers leads to higher Biot numbers and large temperature gradients in the polymer particles. Temperature gradients in a particle lead to a ``fried egg'' shaped splat characteristic of experimental observations of thermally sprayed polymer particles. The effect of roughness on the mechanics of splatting and final splat shapes was explored through the use of several prototypical rough surfaces, e.g. steps and grooves.   

Controlling Release Kinetics of PLG Microspheres Using a Manufacturing Technique

Controlled drug delivery offers numerous advantages compared with conventional free dosage forms, in particular: improved efficacy and patient compliance. Emulsification is a widely used technique to entrap drugs in biodegradable microspheres for controlled drug delivery. The size of the formed microspheres has a significant influence on drug release kinetics. Despite the advantages of controlled drug delivery, previous attempts to achieve predetermined release rates have seen limited success. This study develops a tool to tailor desired release kinetics by combining microsphere batches of specified mean diameter and size distribution. A fluid mechanics based correlation that predicts the average size of Poly(Lactide-co-Glycolide) [PLG] microspheres from the manufacturing technique, is constructed and validated by comparison with experimental results. The microspheres produced are accurately represented by the Rosin-Rammler mathematical distribution function. A mathematical model is formulated that incorporates the microsphere distribution function to predict the release kinetics from mono-dispersed and poly-dispersed populations. Through this mathematical model, different release kinetics can be achieved by combining different sized populations in different ratios. The resulting design tool should prove useful for the pharmaceutical industry to achieve designer release kinetics.   

Boltzmann Monte-Carlo simulations of a suspension of non-spherical particles in a parallel-wall channel

The evolution of a dilute suspension of axisymmetric particles confined between two parallel planar walls is investigated under creeping-flow conditions. The suspension undergoes a shear flow that results from the relative motion of the walls. The hydrodynamic interactions are accurately evaluated using our Cartesian-representation algorithm. In the absence of interparticle interactions the suspended particles undergo periodic motions, similar to Jefferey orbits in free space. However, the periods in the confined system are not identical. Due to the associated phase shifts a stationary state is reached at long times. Finite-concentration effects are included via a Boltzmann Monte-Carlo method. The state of the system is described by an ensemble of periodic particle trajectories, which are characterized by the vertical position and orientation of the particles crossing the horizontal plane. The ensemble is updated by computing a large number of binary collisions. Implications of our simulations for particle separation in microchannels are examined.   

Flow-induced instability of mushy layer permeability

The coupling of an external shear flow with the permeability of a solidifying mushy layer is investigated experimentally and theoretically. We grow a mushy layer from a trans-eutectic aqueous ammonium chloride solution from the base of a laboratory flume. The growth rate is constant and a laminar shear flow is applied. We find a threshold speed above which a spatiotemporal variation of the permeability of the layer appears with a planform wherein the long axis is transverse to the flow direction. Upon removal of the flow, the material returns to a uniform state. The growth of the pattern compares favorably with an analytical and numerical stability analysis which incorporates dissolution of the solid matrix.   

Three dimensional Solutocapillary Convection in Spherical Shells

Nonlinear, time-dependent, three-dimensional, variable viscosity, infinite Schmidt number solutocapillary convection in spherical shells is computed by a finite-volume method. The shell contains a solute and a solvent, and the inner boundary is impermeable and stress free. The solvent evaporates at the outer surface into a water-solvent environment with a prescribed mass transfer coefficient. Convection is driven by surface tension dependence on the solvent concentration $C$. A time-dependent diffusive state characterized by concentration $C_{d}(r,t)$ and a receding outer surface $r_{2d}(t)$ is possible and is a function of the mass transfer Biot number, a partition coefficient, and ambient solvent concentration $C_{\infty}$. It loses stability at critical values of the Marangoni number and degree of surface harmonic. In the limit of small Capillary number $Ca\to0$ the outer radius deviation from sphericity $\delta(\theta,\phi,t)$ is $O(Ca)$ and $r_{2}(\theta,\phi,t)$ is given by $r_{2d}(t)$ in the $O(1)$ convection. We compute supercritical motions and companion $\delta(\theta,\phi,t)$ in this moving boundary problem subject to random initial conditions and compare nonlinear results with those from linear theory, axisymmetric calculations and available experiments.   

Session KN: Geophysical Flows III

Bumps, witches and bouncing beams: lab investigations of internal waves

There is a great, ongoing effort to better understand the processes surrounding internal wave generation, propagation and dissipation in the oceans. To contribute to this effort, we are in the process of establishing a state-of-the-art experimental facility. The facility, based around the digital schlieren method, is designed to investigate both linear and non-linear phenomena in a laboratory setting. We here report the latest experimental results concerning tidal conversion by typical topographic features, such as a Gaussian bump and a knife-edge. The quantitative results compare very well with theoretical predictions developed from the classic work of Bell and Hurley, and more recent analysis of subcritical topography by Balmforth et al. In addition, we present more details on recently published results concerning the nonlinear generation of second-harmonic wavebeams at reflecting boundaries.   

2D PIV Measurements of flow between a pair of model buildings with varying geometries

The study of flow within a pair of three dimensional model buildings assumes paramount importance in developing an understanding of the mechanisms involved in the transport of pollutants within urban areas. For this work, 2D PIV measurements have been performed to add insight into urban flow physics. The experiments were performed in a 7.9 m long boundary layer wind tunnel facility with a cross section of 0.61 m x 0.91 m and at a free stream velocity of $\sim $7 m s-1. Two sets of experiments have been considered. In the first set of experiments, the spacing between two cubes was varied. For these highly idealized experiments, the results correspond well with the results in the literature for the wake interference and skimming flow regimes, but discrepancies have been found corresponding to the isolated roughness flow regime. The second set of experiments considers flow between two buildings with variable upwind and downwind building heights and separation distances. These experiments indicate that the mean and turbulence flow characteristics associated with even these very simple types of building configurations can vary considerably and thus significantly modify pollutant transport.   

Atmospheric Flow through Urban Street Canyons

Flow and turbulence through a network of urban street canyons (streets located within large buildings) were studied during two large-scale field experiments: the Mock Urban Setting Test (MUST-2000) at the US Army Dugway Proving Grounds and the Joint-Urban 2003 field experiment in Oklahoma City. Instrumented towers and tethersondes deployed by the authors and several other groups were analyzed in the framework of flow regimes corresponding to each of the sites (``isolated roughness'' at Dugway and ``skimming flow'' at OKC). The results show that the flow patterns are highly sensitive to the approach angle for angles greater than about 5 deg, and that when the flow is normal to the building cluster the canyons are dominated by recirculating flow. The production of turbulence is highest near the ground and near the top of the buildings, and the variations of turbulent shear stresses could be scaled using local similarity variables. The mean flow in the roughness and inertial layers were compared with available theoretical formulations, and the flow in MUST was also studied using numerical simulations.   

Stratified Flow over Rough Topography

Linear theory and Long's model for stratified flow over topography both assume free-slip lower boundary conditions and so neglect the possibility of boundary-layer separation either between successive hill crests or in the lee of a range of hills. Here we report upon laboratory experiments that focus upon periodic finite-amplitude hills which are representative of the Earth's major mountain ranges as well as the fractured crevasses of the ocean floor. The topographic shapes are selected to encompass varying degrees of roughness, from smoothly-varying sinusoidal hills to steeper triangular and rectangular hills. For low flow speeds, U (and hence low values of the excitation frequency), the internal wave frequencies are consistent with those predicted by linear theory. However, when the excitation frequency exceeds the buoyancy frequency, N, internal waves are still excited and their frequencies are found to be an approximately constant fraction of N. In all experiments the wave amplitudes are much smaller than predicted because, through boundary-layer separation, fluid is trapped in the valleys between hills effectively reducing the peak-to-peak hill height, H. This is true even if NH/U is moderately less than 1. For rough topographies, turbulent structures emerge even at low towing speeds and waves are generated by the nonlinear interactions between the flow, lee waves and turbulence far in the lee.   

Intrusions on a density interface

An accurate theoretical prediction of the speed of an intrusion propagating along the interface between two uniform layers has defied analysis for the past 25 years. Theories by Holyer \& Huppert (1980) and, more recently, Sutherland, Kyba \& Flynn (2004) give only approximate agreement with experiments. We describe an experimental and numerical study of an intrusion and show that, except when the density of the intrusion is the depth-weighted mean of the layer densities, the interface ahead of the intrusion is displaced vertically. We predict this vertical displacement, which takes the form of an upstream-propagating long wave, and use the predicted value to determine the intrusion speed. For the case when the interface is undisturbed the intrusion propagation speed is a minimum. We develop an energy argument that describes the observed variation of the intrusion speed from this minimum speed as a function of the intrusion and layer densities and the ratio of the layer depths. We also show that if, and only if, the layer depths are equal, the speed of the intrusion is \emph {independent} of the density of the intrusion.   

Internal Wave Transmission Across the Equatorial Undercurrent

We examine the propagation of internal waves from the surface mixed region of the equatorial Pacific Ocean through the equatorial undercurrent. Thus we are able to assess the spectrum of waves capable of penetrating to the deep ocean and energising the deep equatorial countercurrents. We show that heuristics based on ray theory are not sufficient to make this assessment. A numerical code is developed to solve the Taylor-Goldstein equation for arbitrary buoyancy frequency and background flow profiles. In addition to numerical integration of the governing equation, the code also uses a initial condition driver that finds a unique causal wave-like solution and thus determines the transmitted and reflected wave amplitudes. From these we determine the transmission coefficient defined to be the ratio of transmitted to incident pseudoenergy. Using equatorial ocean density and background flow speed observations, we develop characteristic analytic basic state profiles. Applying the code, the transmission coefficient is calculated for given incident initial internal wave frequencies $\omega$ and horizontal wavenumbers $k$. For a range of $\omega$ and $k$ for which waves do not encounter a critical level and the Doppler-shifted frequency is less than the local buoyancy frequency at all depths (and hence for waves that one might expect on the basis of ray theory to transmit perfectly) we find wave transmission is largest for incident waves that are close to being harmonic with vertical modes in the duct.   

Internal Gravity Waves in a Dipolar Wind.

An experimental study on the interaction of the internal wave field generated by oscillating cylinders in a stratified fluid with a pancake dipole is presented. The experiments are carried out in a salt-stratified water tank with constant Brunt-V\"{a}is\"{a}l\"{a} frequency. When the wave and the dipole propagate horizontally in opposite directions (counterpropagating case), the phase line of the gravity wave beam steepens towards the vertical as it enters the dipolar field and it may even reach a turning point where the wave is reflected. When the dipole and the wave propagate in the same direction (copropagating case), the wave beam is bent towards the horizontal and may be absorbed by the dipole. In that case, the waves encounter a critical layer, and momentum is transferred to the dipole. Three-dimensional effects of the dipolar velocity field on the propagating internal waves induce focusing and refraction of a wave beam, that in ocean flows may lead to wave breaking.   

Effects of particle inertia on gravity current flows

Gravity currents are buoyancy-driven flows well known to be one of the main sediment transport mechanism into deep sea. A new mathematical model for particulate gravity currents based on the well-accepted formalism of two-phase flow is introduced. The model includes settling and particle inertia effects to $O(\tau V_s + \tau^2 + V_s^2$) (tau represents the dimensionless particle inertia and $V_s$ the dimensionless settling rate). By means of highly resolved simulations the effect of particle inertia on front velocity, ambient fluid entrainment, flow structure and deposition patterns is addressed. The simulations are performed with a de-aliased pseudo-spectral code which allows accurate representation of length and time scales present in the flow. The results show a strong variation of the flow structure caused by particles migration from regions with strong vorticity and accumulation into regions of high shear. These effects contribute to the variation of the current front velocity with varying particle size, even in the case of same net buoyancy. The analysis of the effect of particle inertia on deposition patterns and ambient fluid entrainment is under way.   

Mixing efficiency in lock release gravity currents

We consider numerically and experimentally mixing in Boussinesq lock release gravity currents. We show quantitatively that mixing is strongly dependent on the lower boundary condition. As the fluid in the current slumps and flows under gravity, Kelvin-Helmholtz billows grow at the current head, entrain ambient fluid as they are swept backwards, and then fall down over the current tail, interacting strongly with a thin layer of dense fluid which remains at the lower boundary. For flows with free-slip lower boundaries, the billows remain largely two- dimensional. Conversely, for flows with no-slip lower boundaries, the current front develops lobes and clefts as ambient fluid is engulfed horizontally and overrun vertically, thus leading inevitably both to convection, and also to strong three-dimensionality. Using the APE framework of Winters et al. (1995), we quantify the time-dependent mixing associated with the engulfed fluid, the overrun fluid, and the billows as they develop streamwise secondary instabilities which trigger turbulence. We also compare the cumulative mixing efficiency with laboratory experiments.   

Interaction of finite volume gravity currents with a two-layer stratified interface

An experimental study of two-dimensional gravity currents impinging on a stratified interface in a two-layer stratified environment has been conducted. The gravity currents are created by the release of a finite volume of dense fluid along a 6$^{o}$ inclined boundary. The effect of the stratified interface on the entrainment and mixing processes is quantified by the use of Planar Laser Induced Fluorescence (PLIF) and Particle Image Velocimetry (PIV). The instantaneous velocity and vorticity fields are quantified and averages are computed over 0.2 seconds. For both experimental techniques, the laser sheet is positioned at mid-span and extends in the streamwise direction. This allows for the measurements to be centered on the impact region between the gravity current and the stratified interface. We have previously determined the entrainment rate and studied the internal structure of a gravity current created by a continuous source with similar experimental conditions (for the slope and the ambient stratification) as in the current study. A thorough comparison of the two cases is provided.   

Front velocity of lock exchange gravity currents on a slope

We present an investigation of lock exchange gravity currents in sloping channels. Two-dimensional Navier-Stokes simulations show the existence of two phases for the flow. During the initial phase, the front velocity is seen to be constant. Its magnitude depends on the slope angle and reaches a maximum near 40 degrees. This quasisteady initial phase gives way to a second phase of higher, unsteady front velocity. This second phase is dominated by the accelerating motion of the two stratified fluid layers past each other. We develop a simple model that predicts the time of the transition between the two phases. Experimental observations are presented that support the numerical findings.   

Mixing induced in a dense current flowing down a sloping bottom in a rotating fluid

A density driven current was generated in the laboratory by releasing dense fluid over a sloping bottom in a rotating freshwater system. Over a wide range of parameter values, the following four flow types were found: laminar, wave, turbulent and eddy regime. The amount of mixing between the dense and the ambient fluids was measured and its dependence on the Froude number and on the distance downslope was determined for increasing values of the Reynolds number. Mixing increased significantly when passing from the laminar to the wave regime; i.e. with increasing Froude number. We believe that mixing between the dense salty water and the lighter fresh water was caused by breaking waves. We quantified the amount of mixing observed and estimated the value of the entrainment velocity at the interface between the dense fluid and the fresh overlying fluid. The results have been compared with previous laboratory experiments which presented the classic turbulent entrainment behavior and observational estimates of the Mediterranean and Denmark Strait overflow.   

Session KP: Turbulence Simulations: DNS III

Anisotropy of scalar transport in conductive flows at low magnetic Reynolds numbers.

Conductive flows at low magnetic Reynolds numbers are encountered in many industrial applications. For example, in the steel industry, applied magnetic fields can be used to damp turbulence in the casting process. In nuclear fusion devices (Tokamaks), liquid lithium flows are used as coolant blankets and interact with the surrounding magnetic field that drives and confines the fusion plasma. An important characteristic of such flows is the development of a strong anisotropy. This anisotropy can be twofold. Firstly, flows structures tend to elongate in the direction of the magnetic field (morphological anisotropy). Secondly, the magnetic field can severely alter the partition of energy between the components of the velocity field parallel or orthogonal to its direction (even in the case of homogeneous turbulence). In this work we study numerically how the transport of a passive scalar is affected by those two kinds of flow anisotropies. For the regimes considered, it is shown that the morphological anisotropy is the factor that is most influential on the anisotropic scalar transport.   

Energy transfers in MHD turbulence

The knowledge of the interactions between resolved and unresolved scales is required to design accurate subgrid-scale models for large-eddy simulations (LES). Although these interactions have been extensively studied for Navier-Stokes turbulence, much less has been done for turbulent conducting fluids. The objective of this study is to analyze the triad interactions in magneto-hydrodynamic (MHD) turbulence and more specifically the shell-to-shell interactions which are more complex in MHD. Indeed, the total energy can be split into the kinetic and the magnetic energy. The characterization of energy fluxes thus requires a detailed analysis of the energy transfers, not only between different length scales but also between the velocity and magnetic fields. Simulations of isotropic decaying and forced MHD turbulence using $512^3$ Fourier modes have been analyzed. The energy transfers appear to be very much local, i.e. the transfers are dominated by exchanges of energy between structures that have similar sizes. Shell-to-shell interactions have also been decomposed into forward and backward energy transfers. Such a decomposition is not unique and two possible strategies to identify the energy backscatter are suggested. They lead to fairly different diagnostics.   

Anisotropy of MHD turbulence at low magnetic Reynolds number.

Turbulent fluctuations in MHD flows are known to become anisotropic under the action of a sufficiently strong magnetic field. We investigate this phenomenon in the case of low magnetic Reynolds number using DNS and LES of a forced flow in a periodic box. A series of simulations is performed with different strengths of the magnetic field, varying Reynolds number, and two types of forcing, one of which is isotropic and the other limited to two-dimensional flow modes. We find that both the velocity anisotropy (difference in the relative amplitude of the velocity components) and the anisotropy of the velocity gradients are predominantly determined by the value of the magnetic interaction parameter. The effects of the Reynolds number and the type of forcing are much weaker. We also find that the anisotropy varies only slightly with the length scale. The work was supported by the DOE Office of Basic Energy Science and NSF-MRI program.   

MHD Shallow-Water Turbulence on the Sphere

Motivated by astrophysical-geophysical applications, we have performed a series of high Reynolds number simulations of freely- evolving, magnetohydrodynamic shallow-water turbulence (MHDSWT) on a rotating sphere. MHDSWT is the simplest turbulence model that includes the effects of stratification, differential rotation, and magnetic field that can be studied over long durations. A systematic exploration of the full physical and numerical parameter-space shows novel as well as consistent behavior, compared to those of pure hydrodynamic (HD) and 2-D MHD counterparts. In the case without rotation, our simulations show that the turbulent evolution is sensitive to initial conditions, most strongly to the peak of the energy spectrum. With increasing magnetic field strength, the flow field is more susceptible to loss of balance; the field blows up in finite time. In addition, the pronounced anisotropic structures (jets and vorticity bands) observed in differentially-rotating HD systems do not form. An application of the model to the solar tachocline is also presented.   

Acceleration statistics of heavy particles in turbulence

We study, by means of direct numerical simulations, the dynamics of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up to 120 million particles with Stokes numbers, $St$, in the range from $0.16$ to $3.5$ we are able to characterize in full detail the statistics of particle acceleration. We will show that the root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value already at quite small Stokes numbers, that at a given $St$ the normalised acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers and that the tails of the probability density function of the normalised acceleration $a/a_{\rm rms}$ decrease with $St$. Two concurrent mechanisms lead to the above results: particle clustering, very effective at small $St$, and filtering induced by the particle response time, that takes over at larger $St$.   

Dispersion of heavy particles in isotropic turbulence

Particle-laden turbulence is frequently observed in nature such as in the atmosphere or the ocean as well as in many engineering flows. Recently, airbourne micro or nano-scale particles in the atmosphere draw much attention from environmental societies since such small particles cause a lot of environmental problems in the industrialized areas. In order to predict disperion of small particles, we have to understand the mechanism by which laden particles disperses in turbulent environment. In this study, we carried out direct numerical simulation of isotropic turbulence with particles under the Stokes drag assumption for a spherical particle. We consider only oneway interaction so that modification of turbulence by the particles is not taken into account. Particularly, we investigate the evolution of flow varialbe along the trajectory of a heavy particle with and without the influence of gravity. Comparisonal study of statistics between Lagrangian fluid dispersion and heavy particle dispersion is made. These results can be used in the development of a stochastic model for particle dispersion. Detailed results including correlation or time scales associated with dispersion will be presented in the meeting.   

Direct numerical simulation of confined turbulent thermal convection at high Rayleigh numbers

Results from direct numerical simulations of turbulent Boussinesq convection are presented. The flow is computed for a cylindrical cell of aspect ratio 1/2 in order to compare with the results from recent experiments. The results span eight decades of the Rayleigh number, $Ra$, from $2 \times 10^6$ to $2 \times 10^{14}$, and are unique because the Prandtl number is held constant at 0.7 and Boussinesq conditions are strictly enforced. One of the conclusions is that the Nusselt number varies nearly as the 1/3 power of $Ra$ for about 4 decades towards the upper end of the $Ra$-range covered. Another important observation is that, in same range of $Ra$, the large-scale recirculation, often called the mean wind, first weakens and eventually disappears. In the absence of the mean wind, the thermal boundary layers on the lower and upper plates become disconnected from each other, and the heat transfer scaling becomes essentially independent of the plate distance $h$. This is one of the basic assumptions of Malkus (1954) leading to the 1/3 power law of the relation between the Nusselt and Rayleigh numbers.   

Viscoelastic nonlinear traveling waves and drag reduction in plane Poiseuille flow

Nonlinear traveling wave solutions to the Navier-Stokes equations in the plane Poiseuille geometry come into existence at a Reynolds number $Re$ very close to the experimentally observed value for transition to turbulence. These ``exact coherent states'' (ECS) capture the dominant streamwise-aligned counter-rotating pairs of vortices of the near-wall buffer layer. The present work considers the effect of viscoelasticity on these states, using the FENE-P constitutive model of polymer solutions. These effects mirror many experimental observations in fully turbulent flows near the onset of turbulent drag reduction. The mechanism underlying these changes is suppression of streamwise vortices by the polymer forces. Moreover, all mean velocity profiles on the ECS existence curves collapse onto a universal profile, which is insensitive to polymer extensibility, concentration, Weissenberg number or Reynolds number. In experiments with a given fluid, Reynolds and Weissenberg numbers ($Wi$) are linearly related. In this situation, we predict that there exist experimental conditions where these states cannot exist at all. Since polymer additives do not relaminarize turbulent flow, these results imply that there must exist other nontrivial states in turbulent viscoelastic flow that exist at high $Wi$.   

Energy Analysis of Turbulent Flow Using Bi-Orthogonal Wavelets

Turbulent flow exhibits many different length and time scales. Hence in order to have a deeper insight into turbulence it is important to study energy transfer at discrete scales and between scales. Wavelets offer some potential for the analysis of energy transfer in turbulent flow. This is mainly due to their locality and scalability property. Wavelet representations have a natural built-in adaptivity through their ability to express and separate structures in a flow at different scales. Combined with the concept of the energy cascade, this feature makes wavelets a powerful tool for analysis compared to conventional discretizations. Wavelets are traditionally associated with orthonormal bases. A closer look reveals that orthogonality is often convenient but not essential. So in order to have more flexibility we have used the concept of bi-orthogonality for resolving different energy terms in the turbulent kinetic energy equation. This approach appears sometimes better suited, and offers interesting new combinations of concepts. The present work involves a wavelet decomposition of the terms in the turbulent kinetic energy transport equation of a fully developed channel flow, and a study of the behavior of the important terms. A detailed study of the energy transfer term is performed. An attempt is made to identify some well-known structures in turbulent channel flow at different scales of decomposition. The dynamics of these coherent structures is studied based on their contribution to energy transfer in the flow at discrete scales and over a period of time.   

Parametric Dependence of Homogeneous Turbulent Shear Flow on Reynolds Number and Shear Parameter

The combined role of the Shear Parameter, $S^{*} = S k / \epsilon$, and the Reynolds number in homogeneous turbulent shear flow is studied using direct numerical simulations (DNS). The parametric investigation involves DNS of $256^{3}$, $512^{3}$ and $1024^{3}$ with a constant shear parameter, $S^{*}$, between 1 and 100. Particular attention is given to velocity derivatives (strain and rotation) and higher-order structure functions and their scaling with the two parameters. This study is in line with some recent results reported by Schumacher, [Phys. Fluids 16, 2004]. Due to the high level of shear investigated, a new algorithm that avoids the remeshing step is used. The 20\% - 40\% loss in kinetic energy and dissipation rate reported by Lee \emph{et al.} [JFM, 216, 1990] using the Rogallo code is consequently avoided.   

Numerical Methods for Scalar Transport Equation

Scalar transport phenomenon is known to generate small scale fluctuations. Consequently, in high Schmidt number regime it creates numerical difficulties in computer simulations. Besides that, the dissipative and dispersive nature of most numerical schemes aggravates the inaccuracy of such simulations. In the present work, two second order non-dissipative numerical schemes are implemented. These methods are optimized for the scalar transport equation and consider dispersion characteristics of numerical schemes, stability and computational efficiency. In addition, the advection of passive scalar is studied in Lagrangian framework. It is demonstrated that Lagrangian approach is very suitable for this problem when the Schmidt number is infinity. To ensure adequate resolution a particle destruction and creation algorithm is implemented. The effect this algorithm is studied in terms of overall accuracy and numerical dissipation. Passive scalar transport in a 2D Green-Taylor vortex is chosen as a benchmark problem where these new numerical techniques are applied. Results are compared in terms of dispersion, energy cascade and computational efficiency.   

A numerical scheme to simulate arbitrary shaped resolved particles in complex flows

We present a numerical scheme for the simulation of freely moving rigid particles in complex flows. The approach is based on the work by Patankar (2001) and Sharma \& Patankar (2004). The entire fluid-particle domain is assumed to be a fluid and the flow inside the particle domain is constrained to be a rigid body motion using an additional rigidity constraint in the context of a fractional step scheme. The particle is assumed to be made of material points moving on a fixed background mesh where the fluid flow equations are solved. The original scheme is further modified by introducing mollification kernels typically used in particle methods to interpolate between the particle material points and the fixed mesh. We evaluate the accuracy of the scheme together with the interpolation operator. This scheme is used to simulate a range of single and multiple particle problems in laminar flows. Some preliminary computations of particles in turbulent flows will also be presented. Application and extension of the scheme for the simulation of large number of resolved/deformable particles in complex turbulent flows will be discussed.   

Session KQ: Wake Stability

Wake development and control for an airfoil with blunt and divergent trailing edge

The wake development downstream of an airfoil with a blunt and divergent trailing edge is experimentally investigated with conventional hot-wire anemometry. Two distinct wake development regions are identified. (i) a near-wake region where the vortex shedding is robust, the wake is highly asymmetric and the wake mean flow direction is curved; (ii) a far-wake region where momentum thickness reaches an asymptotic value, distributions of mean flow and turbulence quantities are almost symmetric, curvature of the mean flow becomes negligible and self-preserving state is reached. The effect of attaching rectangular vortex generators to the pressure and suction sides of the blunt trailing edge on the vortex shedding phenomena is quantified. The results clearly indicate vortex shedding suppression when the vortex generators are placed at a distance that equals twice the integral length scale in the spanwise direction. Based on these results, it is concluded that the streamwise components of the horseshoe vorticies generated by the vortex generators are responsible for the early suppression of the von Karman rolls; hence weakening the vortex shedding and accelerating the flow transition toward the far wake state. The effectiveness of this mechanism depends on the vortex generators placement in the spanwise direction.   

Experimental Measurements of a Model Submarine Wake

High resolution stereo-PIV measurements were made over ten body lengths downstream of a 1/18$^{th}$ scale submarine model in the Deep Water Tow Basin at NSWCCD. The submarine model is an unclassified generic submarine shape (ONR Body-1) composed of an axisymmetric body, four stern appendages (control surfaces) and a propeller. This body is 5.8 m long, 0.49 m in diameter. Block gages on the struts measured streamwise force on the body and provided loading details for setting propeller speed. The model was towed through a stationary laser sheet oriented perpendicular to the tow direction to obtain three-dimensional velocity fields. The objective of the study was to quantify the submarine wake and rate of decay of the coherent vortices. These data will be used in conjunction with measurements obtained on a model towed array to validate computational models for array shape and dynamics. Results with and without the propeller will be presented. Approximately 40 instantaneous vector fields were obtained for each location. Mean and fluctuating streamwise and cross-stream velocities and vorticity were computed.   

Critical properties of forced wakes

We present direct numerical simulations of a flow behind an oscillating cylinder around its axis, at moderate Reynolds number. This flow geometry represents the very typical situations observed in flow control studies. We worked on lock-in and non lock-in regimes and, in this latter case, we analyzed the critical behavior of the global mode as a function of the forcing amplitude, as well as the forcing frequency. We confirmed the results of previous experimental works dealing with the scaling properties of theses global modes, scaling with the growth rate of perturbations, themselves depending on forcing parameters. Owing to the scaling, we have been able to renormalize the global modes.   

Late-wake Vortices of Maneuvering Bodies in Stratified Fluids

Laboratory experiments were conducted in a large flow facility to investigate the formation and evolution of large vortices that form in stratified late wakes of maneuvering and self-propelled bodies. The maneuvers included acceleration, deceleration and turning, whence a significant momentum is imparted to the fluid. Previous small-scale experiments conducted at Re = 1000 [Phys. Fluids, 1999, 11(6), 1682] showed that the late-wake vortices of maneuvering bodies are much bigger and different in dynamics and morphology from that of the steadily moving bodies. The present experiments delved into such differences at much higher Reynolds numbers (Re = 50,000). New findings include: the effect of internal wave radiation on momentum balance in the wake, surface signatures of stratified wakes, and transverse-propagating dipoles that form during the motion of bodies in curved paths. State of the art flow diagnostic techniques were employed for flow measurements, and the flow phenomena and measurements were explained using simple dynamic arguments and phenomenological models.   

Vorticity generation and K\'arm\'an street in the flow past a magnetic obstacle

A numerical investigation of a 2D flow of an incompressible electrically conducting viscous fluid past a localized zone of applied magnetic field, denominated a magnetic obstacle, is carried out. The applied field is produced by the superposition of two small parallel magnetized square surfaces uniformly polarized in the normal direction. Using the low magnetic Reynolds number approximation, it is shown that the flow past a magnetic obstacle may develop vortical structures and eventually instabilities similar to those observed in flows interacting with bluff bodies. In the small zone where the oncoming uniform flow encounters the non- negligible magnetic field, the induced electric currents interact with the field, creating a non-uniform Lorentz force that opposes the flow and creates vorticity. Numerical computations have been conducted for Reynolds numbers $Re=100$ and 200, and Hartmann numbers in the range $1 \le Ha \le 100$. Under these conditions, a wake is formed behind the obstacle. It may display two elongated streamwise vortices that remain steady as long as the Hartmann number does not exceed a critical value. Once this value is reached, the wake becomes unstable and a vortex shedding process similar to the one observed in the flow past bluff bodies is established.   

The effect of buoyancy on vortex shedding in the wake of a heated circular cylinder

Particle Image Velocimetry (PIV) and Planar Laser Induced Fluorescence (PLIF) techniques were used to conduct velocity and temperature measurements in the wake of a heated cylinder to investigate the effect of buoyancy on wake behavior behind the heated cylinder. The experiment was conducted with the heated cylinder installed horizontally in the middle of a vertical water channel and approaching forced flow being downward, which results in the direction of buoyancy force being opposite to that of the approaching forced flow. The temperature and Reynolds number of the approaching forced flow were held constant during the experiment. The temperature of the heated cylinder was adjusted to let the Richardson number, which represents a ratio of buoyancy to inertial forces, varying from 0 (unheated cylinder) to about 3.8. The PIV and PLIF measurement results show that the wake behavior behind the heated cylinder changes dramatically as the Richardson number increasing. The effect of buoyancy on the wake behavior is quantified in terms of mean and fluctuation velocity and temperature distributions, vortex shedding pattern and frequency, wake closure length, drag coefficient and averaged Nusselt number of the heated cylinder.   

Direct numerical simulation of lock-on phenomenon in the wake of a circular cylinder

Lock-on phenomenon in the wake of a circular cylinder is investigated at the Reynolds number of 360 using direct numerical simulation. To induce lock-on, a streamwise velocity perturbation with a frequency of twice the natural shedding frequency, is superimposed on the mean velocity. The Reynolds stresses are investigated to analyze the streamwise force balance acting on the recirculation region. In the perturbed flow, the base pressure is shown to decrease mainly due to the reversal of the Reynolds shear stress. It is also shown that, with the perturbation, the strength of the primary vortices increases whereas that of the secondary vortices decreases significantly. Further, the wavelength of the secondary vortices increases by 2.5 times under the lock-on condition, which causes the three-dimensional vortical structure in the non-perturbed cylinder wake to become a two-dimensional one.   

Stability of inviscid vortices behind a circular cylinder

In a previous work (JFM 409(2000), 13-27 famillies of vortex patches in equilibrium with flow past a circular cylinder which is uniform at infinity were found using iterations for a nonlinear Poisson equation. These included desingularizations of the Foppl pairs. In this work we study the stability of these vortices with respect to two dimensional perturbations. In order to do this we have formulated a curve perturbation algorithm, based on the ideas of contour dynamics, which sets the normal component of velocity at a point on the boundary of the vortex patch equal to zero. The discretization is solved by a version of Newton's method; the Jacobean is factored using the singular value decomposition and a generalized inverse with the smallest singular value removed is used in the Newton iteration. This is necessary because there is always a small singular value due to the fact that there is always a nearby solution vortex in the familly. The Foppl familly is always neutrally stable with respect to symmetric perturbations in nthe sense that all of the eigenvalues are on the imaginary axis. When non symmetric perturbations are allowed there is exactly one unstable mode. A perturbation in the direction of this eigenvector implies a roll suggestive of Karman vortex shedding.   

Session KR: Turbulent Boundary Layers: General

Characterizing Coherent Structures in Supersonic, Turbulent Boundary Layers

Using a direct numerical simulation database, we present a preliminary characterization of the properties of coherent structures in turbulent boundary layers at Mach numbers from 0.3 to 7. The attributes of organized turbulent motions, such as angle, length scale, convection velocity, and internal structure, as well as their variation with Mach number, are the subject of ongoing research. Moreover, there are little quantitative data on how the length scales of the streaky structures at the wall vary with Mach number. We describe a strategy for characterizing the instantaneous properties of coherent structures that are captured using various identification criteria. Structures that have been implicated in the production of turbulent stresses, such as the ``legs'' and ``heads'' associated with horseshoe vortices, as well as the wall streaks, are investigated.   

A Compressible ``Poor Man's Navier--Stokes'' Equation Discrete Dynamical System

Starting from the 3-D compressible Navier--Stokes equations we outline the derivation of a discrete dynamical system (DDS) known as the ``poor man's Navier--Stokes (PMNS) equation'' which we propose to use as the fluctuating component in synthetic-velocity forms of sub-grid scale models for large-eddy simulation. The DDS is obtained directly from the governing equations via a Galerkin procedure followed by decimation of all wave vectors but a single, arbitrary one that is incorporated into the bifurcation parameters; these are related to Reynolds, P\'{e}clet and Mach numbers, or the velocity gradients, and thus to flow physics. We provide computational results in the form of regime maps (bifurcation diagrams) to show that the DDS can produce essentially any temporal behavior observed either experimentally or computationally in compressible Navier--Stokes flows as the bifurcation parameters are varied over their ranges of effective behaviors analogous to results reported by McDonough \& Huang ({\it Int.\ J.\ Numer.\ Meth.\ Fluids} {\bf 44}, 545, 2004) for the 2-D incompressible PMNS equations.   

Large-eddy simulation of accelerating boundary layers

Large-eddy simulation of flat-plate boundary layers in favorable pressure gradients (FPG) are performed for two different acceleration parameters. The high-acceleration case is in good agreement with the experimental data by Fernholz and Warnack [JFM, vol. 359, 329 (1998)]. Substantial reduction in turbulent kinetic energy and shear stress production, strong decorrelation of u and v fluctuation, and a reduction of the bursting frequency indicate that the accelerated boundary layer is in a laminar-like state when the pressure-gradient parameter K exceeds a threshold value. Near the wall, the turbulent shear stress becomes negligible compared with the viscous stresses. In the region of peak acceleration the pressure gradient is larger than the turbulent momentum transport and balances the viscous stress; in the low-K case, on the other hand, turbulent transport remains dominant over the pressure gradient. Downstream of this region, the boundary layer has a fast re-transition to turbulence. In the low K case, the boundary layer does not depart significantly from equilibrium. Research supported by the AFOSR.   

Validation of Dual Plane PIV Measurements in Wall Turbulence Using DNS Data

Experimental dual plane particle image velocimetry (PIV) data from a zero pressure gradient flow over a flat plate at friction Reynolds number $Re_\tau = 1160\ $ is compared with direct numerical simulation (DNS) data from a fully developed channel flow at $Re_\tau = 934$. An averaging scheme is implemented to reduce the resolution of the DNS data to that of the PIV data and thus study the effects of averaging inherent to PIV. A vortex core identification algorithm is implemented on all the datasets using the three dimensional swirl $\lambda^{+}_{3D}$, and statistical distributions are computed of the projection angles of vortical structures in the boundary layer. The close match between the PIV, the raw and the averaged DNS data suggest that PIV can be a reliable and accurate technique for statistical analysis and identification of vortex structures in the turbulent boundary layer. In the talk, details of the statistical distributions and the averaging effects will be given.   

Population Trends of Small-Scale Spanwise Vortices in Wall Turbulence

The population trends of prograde and retrograde (counter to the sense of the mean shear) spanwise vortex cores are studied via detailed PIV measurements in the streamwise--wall-normal plane of turbulent channel flow at $\mathrm{Re}_\tau=566, 1184$ and 1759 and in a zero-pressure-gradient turbulent boundary layer at $\delta^+=1401$ and 2347. A vortex extraction algorithm is used to isolate individual small-scale spanwise vortex cores from the background turbulence and the population trends of these vortices are studied as a function of Reynolds number and wall-normal position in both flows. Substantial numbers of prograde spanwise vortices with structural signatures consistent with the heads of hairpin-like vortices are found to populate the inner boundary of the log layer. In addition, a significant number of retrograde vortices also exist, sometimes appearing as isolated structures but often forming counter-rotating vortex pairs with the remaining prograde vortices. Retrograde vortices are found to be most prominent near the outer edge of the log layer of \underline {both} turbulent channel flow and the turbulent boundary layer, indicating that they may be generated locally within the log layer, advected into this region from more-distant wall-normal locations, and/or may be the byproduct of vortex merging. Of particular significance is the observation that the fractions of prograde and retrograde spanwise vortices collapse irrespective of Reynolds-number and flow in the log layer of wall turbulence.   

Measurements of Instantaneous Wall Shear Stresses and Near-wall Structures Using Digital Holographic Microscopy

Flow measurements are conducted near the wall of a square channel at Re$_{h}$=60,000 using Digital Holographic Microscopy. Instantaneous 3D velocity distributions are obtained over a volume of 1.5 x 2.5 x 1.5 mm$^{3}$, corresponding to x$^{+}$=50, y$^{+}$=83, z$^{+}$=50, y being the wall normal direction. The (pixel) displacement resolution is 0.7$\mu $m in the streamwise and spanwise directions and 10$\mu $m in the wall-normal directions. Using PIV guided particle tracking, each reconstructed hologram provides 2000 -- 6000 vectors. The distributions of 2 $\mu $m particles are not uniform, and they tend to cluster in layers at 2$<$y$^{+}<$5, and at 20$<$y$^{+}<$50. Local distributions of wall shear stresses are computed directly from the instantaneous velocity gradients in the viscous sub-layer (0$<$y$^{+}<$5). Preliminary analyses reveal clear correlations between the distribution of local wall-shear stresses and the presence of streamwise flow structures in the buffer layer (5$<$y$^{+}<$50). Current on-going analysis examines the effects of these buffer-layer structures, the local 3-D vorticity distribution and alignment of the strain tensor eigenvectors on the distribution of wall-shear stresses.   

On the measurement of wall shear stress in turbulent boundary layers

Wall shear stress is an important parameter for turbulent boundary layers, both theoretically and practically. Yet highly accurate measurements have proven to be difficult. In this study, we carry out a comparative study of three methods for its measurement, which are all characterized by being exact in principle: the measurement of the mean velocity gradient at the wall, measurement of all terms in the integrated momentum equation, and the oil film interferometry method. All three methods are applied under identical conditions in a number of stream-wise positions in a wall jet facility. Each of the methods exhibits their own unique difficulties. These difficulties, the resulting inaccuracies and some means to minimize them are discussed.   

High accuracy skin friction measurements demonstrated in a wall jet

The demand for high accuracy mean skin friction $C_f$ measurements has increased recently due, in part, to the need to have accurate friction velocities for scaling wall bounded flows. Other areas that will benefit from accurate $C_f$ values are validation cases for computational fluid dynamics and wall-bounded flow control studies. Here we consider a $C_f$ measurement technique capable of high accuracy, oil film interferometry (OFI). Specifically, the steps required for quality measurements in general flows are discussed including image registration and image analysis with a focus on one, two, and multiple image analysis approaches. The methods are applied to interferograms taken in a wall jet flow using OFI, and the results are applied to scaling wall jet velocity profiles obtained using laser Doppler anemometry.   

Mean-Velocity Profile of Turbulent Boundary Layers Approaching Separation

Turbulent boundary layers approaching separation are a common flow situation in many technical applications. Numerous theoretical, experimental and numerical attempts have been made to find the proper scaling for the mean-velocity profile of this type of wall-bounded flow. However, none of these approaches seems to be completely satisfactory, and controversy still persists regarding the behavior of the mean velocity profile of turbulent boundary layers approaching separation. In this talk, we present new water-tunnel experiments of adverse-pressure-gradient turbulent boundary layers that clearly show the breakdown of the logarithmic law. Using these data and experimental results from several independent research groups, we analyze the classical scaling for ZPG TBL and the scaling by George \& Castillo and Zagarola \& Smits for APG TBL. Only the latter can be applied successfully for the outer region of the mean-velocity profile close to separation. It is shown that Zagarola \& Smits' scaling is consistent with the classical two-layer approach, and can be applied to collapse the different data. Analyzing the Reynolds shear stress, the George \& Castillo's scaling shows a reasonably good collapse of the data in the outer region.   

Flow Separation Control for Low-Pressure Turbine Blade using Vortex Generator Jets

Numerical study of flow separation control is conducted employing Vortex-Generator Jets. This strategy is first tested for the flow past a cylinder at Reynolds number (Re) of 13,400, and then applied to flow in a low-pressure turbine (LPT) cascade for the PAK-B blade geometry at Re = 25,000. A fourth-order accurate compact-difference scheme is used along with sixth-order filtering (C4F6). FDL3DI, a research code developed at WPAFB, is used as the flow solver. A blowing ratio of 2.0 with a skew angle of 90$^{\circ}$ and a pitch angle of 30$^{\circ}$ is employed in the simulations for the aforementioned configurations. The control jets are pulsed with F$^{+}$ = 1.0 for the case of the cylinder, and with F$^{+}$ = 2.33 for the LPT case. The results show a significant decrease in drag on the cylinder after the jets are turned on. The total-pressure loss is calculated in the wake region, at x/D = 3.0, and a reduction of 10{\%} is observed. For the LPT case, the implemented flow separation control strategy totally eliminates the separation and leads to 27.5{\%} reduction in wake total-pressure loss.   

Numerical study on the largest scales of fully developed turbulent pipe flow by LES

Large Eddy Simulations of fully developed turbulent pipe flows up to $Re_\tau=2360$ were conducted using a very long streamwise analysis region, which amounts to a hundred times longer than a pipe radius, to investigate the longest streamwise motions typically observed in the logarithmic layer of wall turbulence. In the previous study we conducted an intensive grid resolution study to properly reproduce outer large-scale structures in plane channels and found that sufficiently fine grid resolutions of around $h_x^+\sim 30$ and $h_z^+\sim 20$ for streamwise and spanwise directions in wall coordinate are required in the near wall region. Following this grid resolutions, our new pipe LES were carried out based on the fully conservative finite difference scheme in cylindrical coordinates along with the novel pole treatment developed recently by Morinishi et al. (2003). The structural difference of the obtained large scales between the pipe and the plane channel flows is discussed regarding the universality of the outer large scales, and the $k^{-1}$ spectrum of the streamwise velocity in the context of self-similar structure having been reported in experimental measurements of pipe flows will also be mentioned.   

Relaminarization under stationary vortices

Flow visualization reveals that a turbulent boundary layer is relaminarized when stationary streamwise vortices are introduced. Following a suggestion of Balle, the vortices are stabilized by large streamwise ``Karman'' grooves in a wavy wall. In a water tunnel, upstream vortex generators place a large streamwise vortex in the middle of each groove, at the stationary point where Prandtl's vortex force vanishes. According to a theory by Cotel, the wall fluxes of a turbulent boundary layer should decline to laminar values under such ``persistent'' vortices. The observed relaminarization is consistent with this theory and with previous measurements of heat transfer by Touel and Balle. However, the structure of the transverse flow resembles the cats-eye pattern of a temporal shear layer rather than the anticipated von Karman wake. The cats-eye pattern corresponds to the forced shear layers of Oster-Wygnanski and Roberts, who found that the Reynolds stresses and mixing rate also decline to laminar values.   

Session KS: Compressible Turbulence

Examination of the dominant azimuthal structures in the near-field pressure region of a high-speed turbulent jet

Numerous experiments have been conducted on characterizing the near-field region of the turbulent axisymmetric jet. The modal distribution of the velocity field has been shown to be characterized by a substantial amount of energy, both in the lower Fourier-azimuthal modes (0,1,{\&} 2), as well as the higher modes (4,5,{\&} 6). The near-field pressure region has demonstrated energy in the lower azimuthal modes (0,1,{\&} 2) only. Capturing the signature of higher modes known to be present in the velocity field, in the pressure field, would be valuable in a controls application. In particular if sensed at the jet lip. Experiments are conducted using a jet nozzle 50.8mm in diameter at exit, with a flow temperature of 25$^{\circ}$C, balanced with ambient conditions. Fluctuating pressure measurements are captured by an azimuthal array of 15 \textit{Kulite }transducers, at an exit velocity of \textit{Mach }0.85 (\textit{Re}=9.8E5). The array is repositioned downstream at several streamwise locations in the fully turbulent, high Reynolds number compressible flow field in an attempt to capture the higher modes found in the velocity field. If no signature of higher modes is found in the fluctuating pressure field at locations where the higher modes are present in the velocity field, these experiments will confirm that pressure cannot resolve these higher modal events. A transfer function between the two distributions can also be evaluated.   

Predictions of Transverse Injection of Air or Helium into Supersonic Crossflow.

Predictions of a sonic jet of air or helium injected into a Mach 2 crossflow are presented. The injection is transverse, characterized as a jet in crossflow (JICF). The Wilcox two-equation model was used to model turbulence. The predictions are steady and symmetric about the center of the circular injection hole. Results from the two cases are compared with ensemble-averaged experimental results by Gruber et al. (1996). Predictions of shock structures, including the barrel shock and Mach disk, and the counter-rotating vortex pair that dominates mixing are in good agreement with the experiment   

Aerodynamic Control and Mixing with Ramp Injection

Experiments were conducted to investigate the behavior of a flow and geometry with many features that are potentially useful for a Supersonic Combustion Ramjet (SCRAMJET) engine: a recirculation zone for flameholding, enhanced mixing between fuel and air, and low total-pressure losses. In subsonic flow with no mass injection, the exit velocity and guidewall static- pressure profiles are found to collapse over a large range of inlet Reynolds numbers. Significant control of exit velocity and guidewall pressure profiles is possible via injection through a perforated ramp into the freestream. The control authority on the overall pressure coefficient increases with increasing inlet Reynolds number. Simple control volume analysis bounds the expected overall pressure coefficient for the device. In transonic/low-supersonic flow, the area ratio calculated from measured pressures agrees well with the visual shear-layer thickness, confirming low total-pressure losses. Further flow control is possible through variable heat release from a fast- chemical reaction between reactants carried in the two streams. At the highest heat release studied, mass injection requirements are lowered by, roughly, a factor of two. Measurements of mixing inferred from the temperature rise in reacting flow indicate higher mixing levels vs. classical free shear layers. As in free shear layers, however, mixing levels decrease with increasing heat release.   

Turbulent subsonic particle-laden flow over an open backward-facing step with and without countercurrent shear at {R}e=3000

We present a numerical study of the subsonic particle-laden flow over an open backward-facing step at a Reynolds number of $Re$=3000. The compressible Navier-Stokes equations are simulated without turbulence models using a multidomain spectral method. Point particles are tracked assuming a Stokes flow model. We focus on the compressibility effects and the effect of countercurrent shear on the zero and first order turbulence statistics and flow topology for low to moderate Mach numbers. At $Mach$=0.1 our results compare well with published incompressible numerical simulations and experiments. At $Mach$=0.4 we show that the shear layer reattaches 10\% closer the step than at the nearly incompressible $Mach$=0.1 as a result of transient compressibility effects related to pressure-dilatational dissipation. The application of a moderate level of countercurrent shear destabilizes the shear layer immediately increasing turbulence intensity and particle deposition behind the step, and decreasing the reattachment length with 12\%. Countercurrent shear also creates large structures that shed with a distinct period frequency.   

Large-scale coherence in a supersonic turbulent boundary layer

Wide-field particle image velocimetry measurements were performed in a Mach 2 turbulent boundary layer to study the characteristics of large scale coherence at two wall-normal locations ($y/\delta = 0.16$ and $0.45$). Instantaneous velocity fields at both locations indicate the presence of elongated streamwise strips of uniform low- and high-speed fluid (length $>8\delta$). These long coherent structures exhibit strong similarities to those that have been found in subsonic boundary layers, which suggests an underlying similarity between the subsonic and supersonic regimes. Two-point correlations of streamwise velocity fluctuations show coherence over a longer streamwise distance at $y/\delta = 0.45$ than at $y/\delta = 0.16$, which indicates an increasing trend in the streamwise length scale with wall-normal location. The spanwise scale of these uniform velocity strips increases with increasing wall-normal distance as found in subsonic boundary layers. The large scale coherence is consistent with the presence of ``hairpin packets'' (a model previously proposed for subsonic boundary layers).   

Unsteady Effects in Shock/Turbulent Boundary Layer Interaction at M=2.25

The interaction of a supersonic flat plate boundary layer flow at $Re_{\theta} \approx 4000$ with an impinging oblique shock wave ($M_{\infty}=2.25$, $\beta=32.7^{\circ}$) is analyzed by means of direct numerical simulation of the Navier-Stokes equations. Under the selected conditions the incoming boundary layer undergoes a mild unsteady separation and the incident shock undergoes a severe flapping motion due to the interaction with the large scale structures embedded in the boundary layer. The analysis of the unsteady flow properties indicates that such quantities as pressure are characterized by a broadband spectrum extending to high frequencies, superposed with low frequency oscillations associated with the large scale motion of the separation shock. The main contribution of the present work is to provide a link between the unsteady motion of the shock wave and the unsteady shedding of the large vortical structures generated next to the separation point; such structures are also shown to be responsible for the amplification of turbulence kinetic energy and shear stress across the interaction zone and for the slow relaxation of the boundary layer to an equilibrium state.   

Large Eddy simulation of compressible flows with a low-numerical dissipation patch-based adaptive mesh refinement method

We describe a hybrid finite difference method for large-eddy simulation (LES) of compressible flows with a low-numerical dissipation scheme and structured adaptive mesh refinement (SAMR). Numerical experiments and validation calculations are presented including a turbulent jet and the strongly shock-driven mixing of a Richtmyer-Meshkov instability. The approach is a conservative flux-based SAMR formulation and as such, it utilizes refinement to computational advantage. The numerical method for the resolved scale terms encompasses the cases of scheme alternation and internal mesh interfaces resulting from SAMR. An explicit centered scheme that is consistent with a skew-symmetric finite difference formulation is used in turbulent flow regions while a weighted essentially non-oscillatory (WENO) scheme is employed to capture shocks. The subgrid stresses and transports are calculated by means of the streched-vortex model, Misra \& Pullin (1997)   

Comparison of Modern Methods for Shock Hydrodynamics

The accuracy and efficiency of several methods are compared for simulating multi-fluid compressible flows. The methods include a Godunov scheme, a Weighted Essentially Non-Oscillatory (WENO) method, an Arbitrary Lagrangian Eulerian (ALE) algorithm and a Spectral/Compact (S/C) scheme. Test problems include a compressible breaking wave, Shu's problem, Noh's problem, the Taylor-Green vortex, decaying turbulence, Rayleigh-Taylor instability and Richtmyer-Meshkov instability. The S/C method employs an artificial bulk viscosity for treating shocks and an artificial shear viscosity for modeling turbulence. A polyharmonic operator, applied to the strain rate, imparts spectral-like behavior to the viscosities, thus eliminating the need for ad hoc limiters and/or switches to turn them off in smooth regions, e.g., expansion, uniform compression, solid-body rotation etc. A low-pass filter is applied to the flow variables to reduce aliasing errors. The S/C method is demonstrated to capture shocks as well as the other schemes, while providing superior resolution of small features.   

Local Stencil Adaption Properties of a WENO Scheme in Direct Numerical Simulations of Compressible Turbulence

Weighted essentially non-oscillatory (WENO) methods can simultaneously provide the high order of accuracy, high bandwidth-resolving efficiency, and shock-capturing capability required for the detailed simulation of compressible turbulence. However, rigorous analysis of the local error properties of these nonlinear numerical methods is difficult. We use a bandwidth-optimized WENO scheme to conduct direct numerical simulations (DNS) of two- and three-dimensional decaying isotropic turbulence, and we evaluate the performance of quantitative indicators of local WENO adaption behavior within the resulting flow fields. One aspect of this assessment is the demarcation of shock-containing and smooth regions where the WENO method should respectively engage its adaption mechanism and revert to its linear optimal stencil. Our results show that these indicators, when synthesized properly, can provide reliable and valuable quantitative information suitable for statistical characterization.   

Session KT: Flow Control

Control of Incompressible Flows

The prior art in flow control has not explicitly addressed the nonlinearity in the incompressible Navier-Stokes equations and has hitherto not produced a control strategy with a proof of closed loop stability for the nonlinear plant. We will show that because of the particular structure of the nonlinearities in the Navier-Stokes equations, relaminarisation is achievable in principle with a linear controller. The required controller synthesis problem has been solved in the control literature, subject to standard assumptions of detectability and stabilisability. We present a procedure to generate a controller that guarantees return to laminar flow in a closed or periodic domain, from any flow state and the region of guaranteed stability is not confined to a neghbourhood of small perturbations around laminar flow. The consequence of this is that fully turbulent flow or any mechanism of transition is controlled. The control of a simple 2x2 example that demonstrates the pertinent features is presented. The ongoing work of application to canonical turbulent flows and transition will be addressed with the aim of producing the associated benefits of greatly reduced skin friction.   

Nonlinear Gradient Preconditioning in Problems of Optimal Control and Estimation

Optimal control and estimation of PDE systems usually results in ill--posed inverse problems which are typically solved using iterative methods of optimization. Their rate of convergence is determined by the conditioning of the problem which can be improved using suitable preconditioning techniques. In the case of optimization of PDE systems such preconditioners can be constructed by endowing the optimization space with an appropriate inner--product structure, and the preconditioned gradient is obtained by solving a linear boundary--value problem. In this study we investigate how this procedure can be extended to optimization in general Banach spaces without an inner--product structure. It is shown that such nonlinearly--preconditioned gradients can be obtained via solution of a nonlinear elliptic problem generalizing the familiar Laplace equation. To highlight the utility of such preconditioning techniques for solution of control and estimation problems for nonlinear PDEs we present computational examples obtained for the Kuramoto--Sivashinsky and Navier--Stokes equations.   

On POD Galerkin modeling of turbulent shear flows using 'subgrid' turbulence representations

Low-dimensional POD Galerkin models are developed from LES data of turbulent shear flows, including mixing layers and jets. The key enablers are 'subgrid' turbulence representations to account for unresolved fine-scale fluctuations in the POD. These auxiliary models improve the prediction horizon and long-term statistics. Different calibration techniques are employed to determine the 'subgrid' parameters, e.g.\ modal eddy viscosities (Rempfer 1991). These methods range from physical modal balance equation to mathematical optimal control strategies.   

Active Control of High Speed and High Reynolds Number Jets via Plasma Actuators$^{\ast }$

Localized arc filament plasma actuators developed at OSU are uniquely suited to force high speed and high Reynolds number jets and shear flows. The actuators have high bandwidth ranging from 0 to 200 kHz and high amplitude with prescribed duty cycle and phase. Eight of these actuators were distributed around the perimeter of an axisymmetric nozzle of 2.54 cm diameter and were used to force ideally expanded Mach 1.3 jet with a Reynolds number of about 1x10$^{6}$. Axisymmetirc, helical (with m=1,2, and 4), flapping, and m = $\pm $2 modes were used. The streamwise flow images showed that the jet column mode was forced most effectively around St$_{D}$ = 0.33, which is in line with what other researchers have found. At this Strouhal number, robust and periodic structures were generated. The effects of forcing amplitude were very limited. However, the effectiveness of forcing was strongly affected by forcing frequency and duty cycle. For all the modes of actuation, the optimum duty cycle was 5-15{\%}. Pitot pressure measurements along the jet centerline showed significantly reduced potential core for some forced cases, especially for the forcing frequency around St$_{D}$ = 0.33. From streamwise images and the centerline pitot data, it appeared that helical and flapping modes are best for mixing enhancement. $^{\ast }$Supported by NASA Glenn Research Center and OCAPP.   

Bluff Body Flow Control Using Plasma Actuators

In this study, the use of single dielectric barrier discharge plasma actuators for the control of bluff body flow separation is investigated. In particular, surface mounted plasma actuators are used to reduce both drag and unsteady vortex shedding from circular cylinders in cross-flow. It is demonstrated that the plasma-induced surface blowing gives rise to a local Coanda effect that promotes the maintenance of flow attachment. Large reductions in vortex shedding and drag are demonstrated for Reynolds numbers $\sim $ 10$^{4}${\ldots}10$^{5}$. Both steady and unsteady plasma-induced surface blowing is explored. Results are presented from experiments involving both two and four surface mounted actuators.   

Electric Circuit Model for a Single-dielectric Barrier Discharge Plasma Actuator

It has been shown previously that the lumped-element circuit model correctly describes the temporal behavior of the aerodynamic plasma actuator. To incorporate this model into the Navier-Stokes (N-S) solver, it was modified to include the spatial behavior of the discharge within the plasma. To model this, the single dielectric barrier discharge plasma actuator is represented as a network of electric circuit elements. The electric circuit consists of $N$ elementary subcircuits, each representing a small physical domain with finite width and length. Each subcircuit consists of an air capacitor, dielectric capacitor, plasma resistive element, and diodes with time-dependent properties that govern the presence of the plasma. The results of the simulation are compared to the experimental data of the plasma spatial distribution obtained with a photomultiplier tube. The obtained results are used to provide accurate time-dependent models of the actuator in N-S simulations as well as to optimize the actuator designs to enhance their flow control effectiveness.   

Control of turbulent shear flow structure using Lorentz force actuators

We present experimental results concerning the use of electro- hydrodynamic ``Lorentz Force'' actuators to affect the near- wall flow of a low Reynolds number fully turbulent channel flow. The actuators are used to induce an oscillatory motion near the surface, and their effect on the structure of the turbulent flow is measured using Particle Image Velocimetry. Previous results have shown that certain amplitude and frequency combinations are effective in suppressing the turbulent fluctuations, wall shear stress and Reynolds stresses. We extend these measurements with conditional sampling of velocity data and computation of two-point velocity correlations which indicate that the effect of forcing is to reduce the streamwise scale of the near-wall coherent structures and to sharply reduce the frequency of high- amplitude turbulence-producing ``bursts''. Measurements phase- locked to the forcing are also presented. Lastly, the relaxation of the controlled flow to its uncontrolled state following the removal of the actuation is discussed.   

Suction and Blowing Boundary Layer Measurements on a Novel Micro-pump Equipped Wing

A NACA-4412 airfoil wing section is outfitted with a micro-fluidic pump capable of providing span-wise, leading edge suction and blowing. The experiment is performed in a water tunnel for several free- stream test velocities. The detailed unsteady flow physics of the micro-pump/ boundary layer interaction are explored via velocity profile measurements taken using a laser Doppler anemometer (LDA). Finally, a basis for comparison of the efficiency between this innovative micro-pump and other micro-fluidic devices like synthetic jets is formulated. We attempt to generate micro-pump equivalent synthetic jet performance parameters such as jet momentum coefficient to determine the feasibility of such a micro-pump in applications like flow control and boundary layer separation control.   

Control of a Supercavity-Piercing Fin

Supercavitation provides a means of significantly reducing the drag of an underwater vehicle, thus enabling a dramatic increase in maximum speed. The control of supercavitating vehicles poses unique challenges. Only small regions at the nose (cavitator) and on the afterbody (fins) are in contact with water. Unlike for a fully wetted vehicle, there is an absence of lift on the body. Viable vehicle control options are limited to actuation of the cavitator and fins, and possibly thrust vectoring. Fin control is highly nonlinear due to the interaction of the fin with the cavity wall. Also, the cavity-fin interaction exhibits strong hysteresis effects. Tests were conducted in the high-speed water tunnel at St. Anthony Falls Laboratory with a semi-axisymmetric, ventilated cavity and a single wedged-shaped, 45 degree swept, cavity-piercing fin. Using a variety of fin control experiments, cavity stability and hysteresis effects were studied and compared to theoretical results. Fin torque was measured for different angles of attack with varying cavitation numbers. A closed-loop control experiment with fin responses to upstream/cavity disturbances is being carried out. Simulink models are being used to control the experimental setup and the measured parameters (fin position and torque) are compared to theoretical results.   

Recent progress in the estimation of laminar and turbulent wall-bounded flows

We present recent progress on a range of techniques to estimate the flow state based on a history of noisy measurements from an array of flush-mounted skin-friction and pressure sensors on a wall:\\ (A) {\bf Model predictive estimation} (a.k.a. 4D-var) in a ``{\bf multiscale retrograde}'' framework {\small (Bewley \& Protas, {\it Physica D} 2004)};\\ (B) {\bf Kalman filtering} based on an artificial (but sufficiently smooth) stochastic model for the covariance matrix characterizing the statistics of the state disturbance forcing {\small (Hoepffner {\it et al.}, {\it JFM} 2005)};\\ (C) {\bf Extended Kalman filtering} based on a covariance matrix derived from a DNS of a turbulent flow {\small (Chevalier {\it et al.}, {\it JFM} submitted)}; and\\ (D) {\bf Weiner filtering} derived from DNS-based impulse response functions.   

Session KU: Convection and Buoyancy Driven Flows IV

Test of the steady-state fluctuation theorem in turbulent Rayleigh-B{\'e}nard convection

Local entropy production rate $\sigma({\bf r},t)$ in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in an aspect-ratio-one cell filled with water. The statistical properties of the time-averaged $\sigma({\bf r},t)$ are analyzed and the results are compared with the predictions of the steady state fluctuation theorem (SSFT) of Gallavotti and Cohen. The experiment reveals that the SSFT can indeed be extended to the local variables, but further development is needed in order to incorporate the common dynamic complexities of far-from-equilibrium systems into the theory. *Work supported by the Research Grants Council of Hong Kong SAR under Grant Nos. HKUST603504 (P.T.) and CUHK403003 (K.Q.X.).   

Structure of thermal and velocity boundary layers in turbulent thermal convection

We report a series of experimental investigations of the structure of thermal and velocity boundary layers in turbulent Rayleigh-Benard convection. Our measurements are conducted for Rayleigh numbers $10^8 [Preview Abstract]

The Reynolds number of the large-scale circulation in turbulent Rayleigh-Benard convection

We measured Reynolds numbers $R_e$ of the large-scale circulation of turbulent Rayleigh-B\'enard convection over the Rayleigh-number range $2\times 10^8 \alt R \alt 10^{11}$ and Prandtl-number range $3.3 \alt \sigma \alt 29$ for cylindrical samples of aspect ratio $\Gamma = 1$. For $R \alt R_c \simeq 3times 10^9$ we found $R_e \sim R^{\beta_{eff}}$ with $beta_{eff} \simeq 0.46 < 1/2$. Here both the $\sigma$- and $R$- dependences are quantitatively consistent with the Grossmann- Lohse (GL) prediction. For $R > R_c$ we found $R_e = 0.106~ sigma^{-3/4} R^{1/2}$, which differs from the GL prediction. The relatively sharp transition at $R_c$ to the large-$R$ regime suggests a qualitative and sudden change that renders the GL prediction inapplicable.   

Complex patterns in rotating Rayleigh-B\'{e}nard convection.

Flows induced by thermal buoyancy in rotating systems play an important role in many industrial processes as well as in numerous problems in geophysical and astrophysical fluid dynamics. Thermal convection in a horizontal fluid layer heated from below and rotating about a vertical axis has also become a prime example in theories of pattern formation and of the transition to spatio-temporal chaos. The K\"{u}ppers-Lortz instability occurs in a rotating Rayleigh-B\'{e}nard convection and is a paradigmatic example of spatiotemporal chaos [G. K\"{u}ppers and D. Lortz, J. Fluid Mech. 35, 609 (1969)]. Surprisingly and contrary to this established scenario, Bajaj et al. 1998 [K. Bajaj, et al., Phys. Rev. Lett. 81 (1998)] observed experimentally in a cylinder square patterns in the range of parameters where K\"{u}ppers-Lortz instability was expected. In this work we study numerically square patterns properties by taking into account realistic boundary conditions. The Navier-Stokes and heat transport equations have been solved in the Oberbeck-Boussinesq approximation using an efficient pseudo-spectral technique. All the characteristics of the pattern show that it appears when the flow is laterally confined.   

Re-orientations of the large-scale circulation in turbulent Rayleigh-B{\'e}nard convection

We present measurements of the orientation $\theta_0(t)$ of the large-scale circulation (LSC) of turbulent Rayleigh-B{\'e}nard convection in cylindrical cells of aspect ratio 1. The orientation undergoes irregular reorientations. It contains two types of reorientation events to be called rotation and cessation. Rotation through angles $\Delta \theta$ has a monotonically decreasing probability distribution $p(\left| \Delta \theta \right|) \propto \left| \Delta \theta \right|^{- \gamma}$ with $\gamma \simeq 4$ reminiscent of heavy-tail distributions in many other systems. Cessations involve a brief vanishing of the LSC, followed by a new spontaneous re- organization of the LSC with a randomly chosen new orientation. Thus the probability distribution for cessation is uniform: $p( \left| \Delta \theta \right|) = 1/\pi$. Both rotations and cessations have Poissonian statistics in time.   

Convectons

Simulations of $^3$He-$^4$He mixtures with a negative separation ratio in two-dimensional containers, heated from below, with realistic boundary conditions and moderately large aspect ratio reveal, at supercritical Rayleigh numbers, the existence of 'convectons', i.e., localized states of stationary convection, separated by regions of no convection (O. Batiste and E. Knobloch, Phys. Fluids 17, 064102, 2005). These states exist over a well-defined range of Rayleigh numbers, and different numerically stable convectons may exist at fixed parameter values. When the Rayleigh number is reduced the convectons shrink by eliminating rolls at the edges; if the Rayleigh number is reduced too far no stable convectons are present and the convecton decays to the conduction state before a new convecton regrows in its place. Similar behavior occurs with periodic boundary conditions in the horizontal. The origin and properties of these states will be described.   

Organized flow structures is turbulent thermal convection

The technique of particle image velocimetry is used to study the velocity field in turbulent Rayleigh-B\'{e}nard convection in an aspect-ratio-1 cylindrical cell filled with water. By measuring the 2D velocity vector map in different cross-sections of the cell, we investigate the 3D flow structures and dynamics of the synchronized plume motions. The experiment reveals how thermal plumes synchronize their emissions and organize their motions spatially between the top and bottom plates, which generate highly coherent velocity oscillations in the entire convection box.   

Local temperature fluctuations in turbulent Rayleigh-B{\'e}nard convection with wide-ranging aspect ratios

We report measurements of the local temperature fluctuations in 1-meter diameter cylindrical convection cells with aspect ratio $\Gamma$ ranging from 0.67 to 20 and the Rayleigh number Ra varying from 10$^7$ to 4$\times$10$^{12}$, at the Prandtl number Pr $\approx$ 4.3. Measurements are made at both cell center and the sidewall positions. The results show that the normalized temperature rms has a power-law dependence on Ra for all positions and aspect ratios, i.e. $\sigma/\Delta T\sim$ Ra$^{\alpha}$, where $\Delta T$ is the temperature difference across the convection cell. It is found that for sidewall positions $\alpha$ is approximately the same for most values of $\Gamma$, while it generally increases with $\Gamma$ for the center positions. We also found that the magnitude of the normalized temperature rms at both the center and sidewall is approximately the same for large $\Gamma$ ($\agt$ 10), while for small values of $\Gamma$ the sidewall fluctuations are roughly a factor of 2 larger than the center ones.   

Azimuthal Motion of the Mean Wind in Turbulent Thermal Convection

We report an experimental study of the azimuthal motion of the circulation plane of the mean wind in turbulent Rayleigh-B{\'e} nard convection in water. Measurements were made in both aspect ratio $\Gamma = 1$ and $0.5$ cylindrical cells. The results show that for $\Gamma = 1$ the orientation of the wind fluctuates over an azimuthal angular range of $\sim \pm 100$ degrees about a preferred direction for over 90$\%$ of the time. In contrast, for $\Gamma =0.5$ the orientation of the wind shows no preferred direction. For $\Gamma = 1$ the observed azimuthal motion of the wind is a superposition of a periodic oscillation in short timescale and chaotic fluctuation in longtime scale. For both $\Gamma =$ 1 and 0.5 the apparently stochastic azimuthal motion of the wind generates a net-rotation on average, with the $\Gamma =$ 0.5 cell having a much larger net-rotation rate. Measurements with varying values of the Rayleigh number Ra is made for the $\Gamma =0.5$ case, and it is found that the net rotation rate diminishes with increasing Ra, reaching a vanishing value around $Ra = 1\times 10^{11}$.   

Turbulent Thermal Convection in Liquid Sodium

We investigate heat transport in a low Prandtl number (Pr=0.0096) fluid in a cylindrical cell of aspect ratio $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $ heated from below. We measure temperature fluctuations and velocity (using ultrasonic velocimetry). In this talk we will discuss the velocity and heat flux dependence on temperature drop. We will also present measurements of plate temperature fluctuations stemming from the large fluid thermal conductivity relative to that of the plate.   

Entropy cascade and Bolgiano-Obukhov scaling in turbulent thermal convection

It is interesting to understand the scaling behavior of velocity and temperature fields in turbulent thermal convection. Theoretical ideas suggest Bolgiano-Obukhov scaling when the turbulent dynamics are governed by a cascade of entropy. On the other hand, there were experimental and numerical studies of confined convection which showed results that are inconsistent of Bolgiano-Obukhov scaling. To help shedding light on this issue, we have studied a shell model of turbulent convection whose stationary dynamics are, by construction, governed by a cascade of entropy when buoyancy is significant. We have indeed observed Bolgiano-Obukhov scaling plus corrections. We have further found that the corrections are due to intermittent variations of the entropy transfer rate. By assuming that the moments of the entropy transfer rate have a hierarchical structure, we are able to understand the observed scaling behavior and predict the velocity and temperature scaling exponents.