Session D9: Focus Session: Turbulence

Turbulent Viscosity Coefficient in 3-Dimesional Turbulence

A new model for the large-eddy simulation (LES) is proposed. The LES has been accepted as the standard formalism for calculating observables concerning turbulence. In the application of the LES, several models are chosen. The purpose of the present paper is for us to propose one of the most physical models. The LES is usually written down as \begin{equation} \label{eq1} \frac{\partial U_i }{\partial t}+(\vec {U}\cdot \vec {\nabla })U_i =-\frac{1}{\rho }\frac{\partial P}{\partial x_i }+\nu _0 \Delta U_i -\frac{\partial Q_{ij} }{\partial x_j }. \end{equation} The above equation is rewritten as \begin{equation} \label{eq2} \frac{\partial U_i }{\partial t}+(\vec {U}\cdot \vec {\nabla })U_i =-\frac{1}{\rho }\frac{\partial P}{\partial x_i }+\nu \Delta U_i \end{equation} and $\nu $ is referred to as turbulent viscosity coefficient. The statistical mechanical method by Helfand[1] is extended replacing the relationship between the thermal velocity and the kinetic viscosity coefficient by the one between the turbulent velocity and the turbulent viscosity coefficient[2]. The major assumption here is the Gaussian statistics for the turbulent velocity. The concrete calculation using the lattice Boltzmann method is shown for 3-dimesional turbulence. [1] E. Helfand, Phys. Rev. 119,1(1960). [2] H. Shibata, J. Phys. Soc. Jpn. 76,024002(2007).   

Exploring the dynamics of the velocity gradient tensor

The dynamics of the velocity gradient tensor is investigated by means of analytical and numerical computations. Our starting point is the Lagrangian evolution equation of this tensor and a model for the pressure Hessian and viscous term proposed in Chevillard and Meneveau ({\it Phys.~Rev.~Lett.}~{\bf 97}, 174501, 2006). The model is based on the Recent Fluid Deformation (RFD) closure, which was introduced in order to overcome the unphysical finite-time blowup of the Restricted Euler model that neglects anisotropic pressure Hessian effects. Using matrix exponentials, the RFD closure takes into account both the geometry and the dynamics of the recent history of the deformation of a fluid particle, and requires the specification of a decorrelation time scale $\tau$. When this time scale is chosen too short (or, equivalently, the Reynolds number is too high), unphysical statistics are observed in the model. In order to understand this model in greater detail, the original, full matrix-exponential-based model is compared with its power- series expansion for small $\tau$. In particular, the time evolution in the so-called $R$-$Q$ plane is studied for the two approaches, and also, the effects of adding a Gaussian white noise are examined.   

The Generalized Fractal Dimensions of a 2-D Compressible Turbulence

Steady-state turbulence is generated in a tank of water 1m x 1 m x 0.3 m and the trajectories of particles floating on the surface are tracked in time. Initially the floaters are uniformly distributed. As time goes on they coagulate and form a fractal structure. The surface pattern reaches a steady state in approximately $t^*$ = 1 s. In the time interval $0 \alt t \alt 2t^*$, measurements are made of the generalized fractal dimensions $D_q(t)$ of the floating particles starting with the uniform distribution $D_q(0)$ =2. In the steady state, the pattern formed by the floaters continues to fluctuate at a time scale dictated by the underlying turbulent flow. This time scale is also of the order of 1 s. To understand the origin of the coagulation phenomenon, one must remember that the floaters form a compressible system, unlike the water molecules that drive them. The time evolution of the $D_q(t)$ are measured for a range of $q$ less than 10. The coagulated particles form into string-like structures having values of $D_q$ ranging down to approximately 1.5.   

Particle Dynamics in Turbulence

The interaction between particles and turbulence features in many environmental and engineering problems, e.g., the formation of rain, the dispersion of particulate pollutants, and sedimentation in rivers and oceans. In addition, tracer particles are routinely used in scientific research to study the flow itself. Understanding the behavior of particles in turbulent flows is not only an important practical problem, but also an intriguing scientific challenge. Our group has developed a three-dimensional Lagrangian Particle Tracking (LPT) system. Using high speed CMOS cameras, the system is capable of following simultaneously hundreds of particles in a turbulent flow with Taylor microscale Reynolds number $R_{\lambda}$ up to $10^3$. The LPT measurements provide both single- and multi-particle statistics following Lagrangian trajectories, at temporal resolutions better than the Kolmogorov time scales of the turbulence. Using the LPT system, we investigated the Lagrangian properties of turbulence by tracking tracer particles seeded in the flow. In the study of turbulent relative dispersion, our measurement of the separation of pairs of fluid elements in turbulence demonstrated that only when the separation between a time scale related to the initial separation between the pair and the turbulence integral time scale is large enough, or equivalently, at very large Reynolds numbers, the long-believed Richardson's $t^3$ law may be observed. Furthermore, measurements of multiple particles in the flow showed the evolution of geometric structures in turbulence. Due to its ability to follow individual particles, the LPT system is an ideal tool to study the behavior of non-tracer particles in turbulence. The inertial particles have density different from the fluid, but size smaller than the Kolmogorov length scale of turbulence. On the other hand, neutrally buoyant particles with size larger than the Kolmogorov scale behave very differently from inertial particles. We will present results from both cases.   

Craig's XY-distribution and the statistics of Lagrangian power in two-dimensional turbulence.

We study the probabilility distribution function (PDF) of injected power in numerical simulations of stationary 2D turbulence in the Lagrangian frame. The simulation mimics an electromagnetically driven fluid layer, a well-documented system for generating 2D turbulence in the laboratory. The forcing and velocity fields in the numerics are close to Gaussian, but the injected power PDF is sharply peaked at zero (suggesting a singularity) with asymmetric exponential tails. Large positive fluctuations are more probable than large negative ones leading to a net positive mean energy input. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behaviour and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity and compare the measured PDFs with theoretical calculations.   

Multiscale Sample Entropy of 2D Decaying Turbulence

Kolmogorov-Sinai entropy has been used to quantify degrees of complexity of spatiotemporally chaotic systems. However, it is not always convenient to implement in real experiments. Recently a Multiscale Sample Entropy (MSE) measure has been proposed, which allows easier analyses of time series. In this study, we have generated decaying turbulence in a two-dimensional soap film and have measured velocity fluctuations as functions of time and downstream distance using a laser Doppler velocimeter. We performed MSE analysis and found there is a time scale $\tau _0 $ at which the MSE is maximized. The value of $\tau _0 $, which correlates well with the large-eddy turn-over time, gets larger as turbulence decays. Other aspects of 2D turbulence are also analyzed using the velocity time series.   

Conformal invariance in two-dimensional turbulence

We show that some features of two-dimensional turbulence display conformal invariance. In particular, the statistics of vorticity clusters in the inverse cascade is equivalent to that of critical percolation, one of the simplest universality classes of critical phenomena. Vorticity isolines are therefore described by Stochastic Loewner Equation curves $SLE_{6}$. This result is generalized to a class of 2d turbulent systems, including Surface Quasi-Geostrophic turbulence (which corresponds to $SLE_{4}$) and Charney-Hasegawa-Mima turbulence. The picture emerging from our results is that conformal invariance may be expected for inverse cascades in two-dimensions therefore opening new perspectives in our understanding of 2d turbulent flows. References:\newline D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, Nature Physics {\bf 2} 124 (2006) \newline D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, Phys. Rev. Lett. {\bf 98} 024501 (2007)   

Mixing and entrainment of oceanic overflows: Implications for global climate evolution

Oceanic overflows are important elements of the Earth's global thermohaline circulation but the mixing and entrainment that occur for such overflows is poorly understood. In particular, as overflow water moves down an inclined slope its stability is governed by the competition between stratification, which stabilizes the flow, and vertical shear, which tends to destabilize the flow. The properties of our laboratory experiment are designed to mimic oceanic overflows to the extent achievable on laboratory-accessible length scales. The flow exits a nozzle and flows along an inclined plane such that there is gravitational forcing of the flowing gravity current. Velocity and density fields are measured simultaneous using particle image velocimetry and planar laser induced fluorescence. The flow structure and dynamics of mixing at different downstream locations are investigated for a different levels of stratification and shear. The role of turbulence is examined by comparing cases of turbulent and laminar gravity currents. The implication of these results for ocean simulations and for understanding global climate are discussed.   

Experimental Investigation of Homogeneity, Isotropy, and Circulation of the Velocity Field in Buoyancy-Driven Turbulence

We present a direct multipoint velocity measurements of the 2D velocity field in the central region of turbulent Rayleigh- B\'{e}nard convection. The local homogeneity and isotropy of the velocity field are tested using a number of criteria and are found to hold to an excellent degree. The distribution of $\Gamma_r$ is found to depend on the scale $r$, reflecting strong intermittency. Besides, the slight asymmetry of the distribution tails reflects the fact that the velocity circulation structure functions (CSFs) are able to capture anisotropic coherent structures, such as thermal plumes, more effectively than longitudinal structure functions (LSFs) and transversal structure functions (TSFs). It is further found that velocity circulation has the same anomalous scaling exponents as LSFs and TSFs for low-order moments ($p<=5$). Whereas, for high-order moments ($p>5$), the anomalous scaling exponents for circulation are found to be systematically smaller than those of LSFs and TSFs.   

Flow mode transitions in turbulent thermal convection

We report an experimental study of structures and dynamics of the large-scale mean flow in Rayleigh-B\'{e}nard convection cells with aspect ratio ($\Gamma$) 1, 1/2 and 1/3. It is found that both a single circulating roll flow structure and two vertically stacked counter-rotating rolls exist in the three aspect ratio cells. The average percentage of time that the large-scale mean flow spends in the single-roll mode (SRM) and the double-roll mode (DRM) are 87.1\% and $0.8\%$ for $\Gamma = 1$, 69.5\% and 7.9\% for $\Gamma = 1/2$, and 26.7\% and 34.1\% for $\Gamma = 1/3$. Several routes of transitions among the different flow modes are identified. In addition, different structures for the DRM are found and their relative weights are determined. We also show direct evidence that the SRM is more efficient for heat transfer than the DRM. Although the difference is very small, it shows how changes in internal flow state can manifest in the global transport properties of the system. It is also found that the time interval between successive flow mode transitions has an exponential distribution, suggesting a Poisson process for the underlying dynamics. The duration of the flow mode transition is found to be log-normally distributed.   

The properties of elastic turbulence in semi-dilute polymer solutions

We studied elastic turbulence in Karman swirling flow of semi-dilute polymer solution. The concentrations of polymer solution used in the experiment were 100, 300, 500, 700, and 900 ppm, and the velocity fields to calculate the rms of the gradients of the tangential velocity, $\omega_{rms}$, were obtained using PIV. First we checked the saturation of $\omega_{rms}$ in the bulk, which represents the saturation of elastic stress. We found that $Wi_{bulk}=\omega_{rms}\tau$ saturates and approaches to unitary value as the polymer concentration increases. Here $\tau$ is the longest polymer relaxation time. Also we studied existence of the velocity boundary layer which is related to boundary layer of elastic stresses of elastic turbulence. The thickness of the boundary layer is the decreasing function of polymer concentration near the rotating upper plate but independent of concentrations near the wall.   

Geometry of plane Couette flow transitional turbulence

We propose to use a hierarchy of exact unstable invariant solutions of the Navier-Stokes equations -- corresponding to the recurrent coherent structures observed in experiments -- to construct a description of the spatio-temporally chaotic dynamics of turbulent fluid flows as a walk through the space of such structures. This description should allow us to obtain quantitative predictions of transport properties of fluid flows such as bulk flow rate and mean wall drag.   

The heat transfer of water-based Al$_2$O$_3$ nanofluid in turbulent Rayleigh-B\'{e}nard convection

We report experimental measurements of the convective heat transfer in water-based Al$_2$O$_3$ nanofluid in a cylindrical convection cell, which has 19 cm in both height and diameter. The nanofluid has been supplied by Nanophase Technologyies Inc. with an initial volume fraction ($\phi$) 22{\%}. It has been diluted into deionized water to obtain nanofluid of low volume fraction. The nominal diameter of Al$_2$O$_3$ particle is 45 nm. At the fixed heating power, $Q =500 W$, it has been found that the convective heat transfer coefficient ($h=Q/{\Delta T} $, $\Delta T$ is the temperature difference across the cell.) decreases to 2{\%} when $\phi$ varies from 0.03{\%} to 1.1{\%}. At $\phi =1.1{\%}$, we have measured the Nusselt number ($Nu$) as a function of Rayleigh number ($Ra$). It has been found that $Nu$ of nanofluid collapses on the $Nu \sim Ra$ scaling curve of pure water at higher $Ra$ ($4\times10^{9}$ to $1\times10^{10} $). While the deterioration of convective heat transfer has been observed at lower $Ra$ ($8\times10^{8}$ to $4\times10^{9} $), and the deterioration becomes more pronounced with decreasing $Ra$. Additional measurement on the thermal and flow structures is in progress to understand the convective heat transport in nanofluid.