Session J41: Vortex Dynamics, Turbulence and Geophysical Flows

Two point correlations between velocity sums and differences in turbulence

In turbulent flows, the universality of small scales has been a subject of ongoing investigation. Recent work has explored the degree to which small scales are independent of large scales by measuring correlations between velocity differences over a distance r (whose variance is dominated by scales near r) and velocity sums over the same distance (whose variance is dominated by large scales). Some correlations between velocity differences and sums are required by the Navier-Stokes equations (Hosokawa, Prog. Theor. Phys. Lett., 118:169, 2007.) This talk will focus on experimental measurements of correlations between velocity sums and velocity differences in a turbulent flow between oscillating grids. We find that these correlations provide an accurate way to measure the energy dissipation rate that complements existing methods based on the third order structure functions. The correlations which are required by Navier-Stokes dynamics do not appear to violate the assumption of independence between the large and small scales, however there are other correlations in our measurements that show clear dependence of the small scales on the large scales.   

Variance of scalar fluctuations using backwards relative dispersion in turbulent channel flows

Temperature fluctuations at a location in a turbulent flow field are brought about by the arrival of particle pairs with different scalar concentrations. Studying backwards relative dispersion can be an alternative way to describe the local variance in scalar fluctuation. This work uses a numerical approach that couples a direct numerical simulation with the tracking of scalar markers to obtain scalar statistics in an infinitely long turbulent channel flow. Focusing on the anisotropic direction perpendicular to the channel walls, the two-particle correlation coefficients are used to determine a Lagrangian material time scale as a function of distance from the wall. Introducing a model that follows Durbin's theory [1], the variance of the temperature fluctuation is calculated by assuming that particle pairs that arrive at a particular location carry with them the mean temperature acquired at the location they were at a previous time. This earlier location is determined by utilizing the Lagrangian backwards timescale. Results obtained from this model are tested at two different Reynolds numbers (at Re$_{\tau }$ = 150 and 300) and for each \textit{Re} case at several different Prandtl numbers (from 0.1 to 1,000). \textbf{References} [1] Durbin, P.A., J. Fluid Mech., 100, 279-302, 1980   

Measuring anisotropy as a function of scale in turbulence using 3D particle tracking

We report the first full 3D experimental measurements of anisotropy as a function of scale in turbulence. From 3D particle tracks obtained with stereoscpic high speed video, we measure the Eulerian structure functions and decompose them into irreducible representation of SO(3) rotation group. This method allows us to quantify the anisotropy in different sectors, specified by $j$ and $m$ of the spherical harmonics $Y_{jm}(\theta,\phi)$, at all scales in the flow. We study a turbulent flow between two oscillating grids in an octagonal tank filled with $1100~l$ of water with $R_{\lambda}=265$. An image compression system processes high-speed video from four cameras in real-time allowing us to acquire huge data sets required for full 3D measurement of anisotropy as a function of scale. Careful selection of a sample of measurements with isotropic orientations is necessary to ensure that anisotropy of the measurement system does not affect the measured anisotropy of the flow. Increasing $j$ sectors show faster decay of anisotropy as scale decreases, consistent with the idea that the small scales should become isotropic at very high Reynolds number. However, conditioning the measured anisotropy on the instantaneous velocity reveals that characterization of anisotropy in an inhom   

Rotation rate of rods in turbulent flow

We present the first time resolved experimental measurements of the motion of small rod-like particles in turbulent flow. The orientation and position of rods are measured using Lagrangian particle tracking with images from multiple high speed cameras in a flow between two oscillating grids. We work at low particle density so rod-rod interaction can be ignored. The probability distribution of the rotation rate of the rods has extended tails indicating the presence of rare events with large rotation rate. Rods rotation rate is determined by the velocity gradients of the flow, so measurements of the rotation rate provide indirect access to statistics of the velocity gradient of the flow as well as the energy dissipation rate. However, tracer rods preferentially sample the flow since their orientation becomes correlated with the local axes of the velocity gradient tensor. The result is that the typical rotation rate of rods is much smaller than it would be if they were randomly oriented.   

Statistics of Macroturbulence from Flow Equations

Probability distribution functions of stochastically-driven and frictionally-damped fluids are governed by a linear framework that resembles quantum many-body theory. Besides the Fokker-Planck approach, there is a closely related Hopf functional method\footnote{Ookie Ma and J. B. Marston, J. Stat. Phys. Th. Exp. P10007 (2005).}; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we generalize the flow equation approach\footnote{F. Wegner, Ann. Phys. {\bf 3}, 77 (1994).} (also known as the method of continuous unitary transformations\footnote{S. D. Glazek and K. G. Wilson, Phys. Rev. D {\bf 48}, 5863 (1993); Phys. Rev. D {\bf 49}, 4214 (1994).}) to find the zero mode. We test the approach using a prototypical model of geophysical and astrophysical flows on a rotating sphere that spontaneously organizes into a coherent jet. Good agreement is found with low-order equal-time statistics accumulated by direct numerical simulation, the traditional method. Different choices for the generators of the continuous transformations, and for closure approximations of the operator algebra, are discussed.   

Motion of a Thread in Compressible Turbulence

Particles that float on a turbulent tank of water form a system that is compressible in the two dimensions on which they move. Here we study, with an overhead camera, the snake-like motion of a 10 $\mu$m thread that floats on the surface. The thread, of length much greater than the integral scale $L_I$ of the underlying turbulence, cannot respond to the small-scale turbulent motions at the surface; its Young's modulus is too large. As a result, the mean curvature of the thread is of the order $1/L_I$. Measured properties include velocity structure functions of the thread $S_{n}(r)$ (including the third moment), the local curvature along the thread (a random variable), and ``Richardson diffusion'' of pairs of points along the thread separated by distances $r$. Supported by NSF Grant DMR 0604477 and the Okinawa Institute of Science Technology.   

Statistical Equilibria of Turbulence on Surfaces of Different Symmetry

We test the validity of statistical descriptions of freely decaying 2D turbulence by performing direct numerical simulations (DNS) of the Euler equation with hyperviscosity on a square torus and on a sphere. DNS shows, at long times, a dipolar coherent structure in the vorticity field on the torus but a quadrapole on the sphere\footnote{J. Y-K. Cho and L. Polvani, Phys. Fluids {\bf 8}, 1531 (1996).}. A truncated Miller-Robert-Sommeria theory\footnote{A. J. Majda and X. Wang, \emph{Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows} (Cambridge University Press, 2006).} can explain the difference. The theory conserves up to the second-order Casimir, while also respecting conservation laws that reflect the symmetry of the domain. We further show that it is equivalent to the phenomenological minimum-enstrophy principle by generalizing the work by Naso et al.\footnote{A. Naso, P. H. Chavanis, and B. Dubrulle, Eur. Phys. J. B {\bf 77}, 284 (2010).} to the sphere. To explain finer structures of the coherent states seen in DNS, especially the phenomenon of confinement, we investigate the perturbative inclusion of the higher Casimir constraints.   

Design and Evolution of Shaped Vortices

We present a novel method for generating vortex lines of arbitrary shapes. We then image their dynamics using a high speed scanning technique which provides three-dimensional information at up to 500 volumes per second. We create a variety of configurations and study the effect of geometry on their evolution.   

Insights into vortex merger using the core growth model

We revisit the two vortex merger problem (both symmetric and asymmetric) for the Navier-Stokes equations using the core growth model for vorticity evolution coupled with the passive particle field and an appropriately chosen time-dependent rotating reference frame. Using the combined tools of analyzing the topology of the streamline patterns along with careful tracking of passive fields, we highlight the key features of the stages of evolution of vortex merger, pinpointing deficiencies in the low-dimensional model with respect to similar experimental/numerical studies. The model, however, reveals a far richer and delicate sequence of topological bifurcations than has previously been discussed in the literature for this problem, and at the same time points the way towards a method of improving the model.   

Entrainment of solid particles by energetic flow events

The focus of this research study is to investigate the utility and applicability of a recently proposed criterion for the prediction of incipient entrainment of sediment particles. Recently introduced theoretical frameworks and stochastic approaches are presented. At near incipient flow conditions the magnitude of energetic flow events follows a power law distribution, over a wide range of frequencies, similar to many other geophysical phenomena. This implies that highly energetic flow structures, which have a good potential of impinging on an exposed particle and displacing it downstream, occur less frequently. This is in agreement with the intermittent and episodic character of particle entrainment observed from mobile particle flume experiments at low flow stages. Further, analysis of synchronous time series of particle entrainment and local instantaneous flow upstream of it, allows for extraction and characterization of the scales and magnitudes that are relevant to the displacement of individual particles. In addition to having a sound theoretical basis, the modeling approach is shown to perform well in accurately defining the condition of incipient motion and various levels of probability of particle entrainment.   

Modeling and dynamics of sand bed vortex ripples

Vortex ripples arise through the instability of a flat sand bed under oscillatory water flow, for example due to wave action at a beach or continental shelf. Fully developed vortex ripples display complex interactions through the mutual influence of fluid flow and bed topography on each other, via sediment transport. We discuss a mechanistic model of ripple dynamics in which the hydrodynamic influence is linearized, and show that this reduced model nonetheless captures many of the ripple dynamics observed in experiments. Cross-sectional profiles of experimentally generated ripples constrain the modeled sediment flux and provide support for our approximations.   

The statistics of wind driven ocean currents

Ocean currents play an important role is the climate system, yet the properties and origin of their statistics is not fully understood. Using the Ekman layer model we show that the statistical properties of the depth integrated surface currents are associated with the temporal correlations of the wind driving the surface currents---when the temporal correlations of the wind are long the probability distribution of the current magnitude is proportional to that of the wind stress. When the temporal correlations of the wind is short the current approaches zero where each component of the current follows a Gaussian distribution such the current magnitude follows the Rayleigh distribution. Using two idealized cases we show that in between these two limits the second (and higher) moment of the current magnitude reaches a maximal value. The results are validated using an oceanic general circulation model.   

Modulation of Atlantic tropical cyclones by El Nino - Southern Oscillation

North Atlantic tropical cyclones (TC) usually form in the northern hemisphere summer and fall with a maximum of activity in September. El Nino--Southern Oscillation (ENSO) has been shown to impact seasonal levels of Atlantic basin TC activity. ENSO is the strongest year to year climate fluctuation on Earth. It originates in the tropical Pacific through coupled ocean-atmosphere interactions mediated by surface wind stress and sea surface temperature (SST) variations. Understanding the effects of ENSO on the seasonal variations of TC activity has important practical consequences for seasonal forecast of hurricanes in North Atlantic. In this study we use the NOAA Extended Reconstructed Sea Surface Temperature, the TC counts in North Atlantic, and the NCEP/NCAR reanalysis data to investigate the relationship between significant ENSO events and TC activity during the hurricane season. Model calculations show that forecasted ENSO SSTA can be used as predictors of TC in North Atlantic region. Such results are illustrated in the context of current efforts to understand climate predictability relevant to North Atlantic tropical storms.   

Fingers and Toes in Miscible Viscous Flows

The displacement of a more viscous fluid by a less viscous one in a porous medium produces complex fingering patterns. To study this phenomenon a Hele-Shaw geometry is used in which the gap-averaged equations for flow between two parallel plates has the same form as Darcy's law for flow in porous media. Our experiments use a radial Hele-Shaw cell as well as a two dimensional porous medium of densely packed granular beads between two circular glass plates, to study viscous fingering in miscible fluids. For immiscible fluids it is known that the most-unstable wavelength for interface growth depends on surface tension, viscosity difference, velocity and plate spacing. In contrast, we find that for \textit {miscible fluids} the large-scale structure (i.e., the ratio of finger length to overall size of the pattern) is set entirely by the viscosity ratio rather than the viscosity difference of the two fluids. We further investigate the role played by other dimensionless parameters in determining the fine structure and evolution of these fingering patterns in the two geometries.   

Thermal plumes in locally heated vertical soap films

A vertical soap film is maintained by injection of a soap solution from the top. The film is then locally heated. Thermal plumes may be observed to rise in the film, depending on the magnitude of the heating and injected flows. The nearly-2D nature of the system allows to visualize the motion of the plumes using an infrared camera. A model is proposed to describe the growth, emergence, and stationarity of the plumes in the film by taking into account both magnitudes of the heating $\Delta T$ and injected flow $Q$. Oscillatory behaviors of both the full-grown plumes size and direction with respect to the vertical direction may also be observed. Particular soap film thickness dynamics shows to be the origin of those phenomena.