Session OP: Turbulence Theory IV*

Markov–Einstein coherence length—a new meaning for the Taylor length in turbulence

Small scale velocity statistics are measured in different turbulent flows with Reynolds numbers up to $10^6$. The stochastic features of the turbulent signals are investigated by means of Markov processes for the velocity increments evolving along the cascade. The analysis of the validity of the Markovian properties yields to a new small scale coherence length. This coherence length can be seen as analogue to the mean free path length of a Brownian Motion as pointed out in A. Einstein, Ann. Phys.{\bf 17}, 549 (1905). For length scales larger than this coherence length the complexity of turbulence can be treated as a Markov process. We present experimental evidence that this Markov- Einstein coherence length scales with $Re^{1/2}$ and show that it is closely related to the Taylor micro-scale and not to the Kolmogorov dissipation length.   

Geometrical and multi-scale analysis of scalar structures in forced homogeneous turbulence

We present a methodology for the identification and characterization of scalar structures in turbulence consisting of a multi-scale analysis of a turbulence volume data set followed by the eduction of structures of interest and their geometrical characterization. The multi-scale analysis is performed through the curvelet transform (Ying et al, 2005). The eduction of structures is done by isocontouring the volume data sets for different scales. The geometrical characterization is based on the probability density functions of shape index and curvedness (Koenderink, van Door 1992), in terms of area coverage, associated with each structure. This allows a global characterization of the set of structures as well as the study and comparison of relevant groups of structures contained within this set. We present results from the application of this methodology to a turbulence data base obtained from a DNS simulation with $512^3$ points in a periodic cube in which a passive scalar is mixed by a forced, turbulent velocity field in the presence of a mean scalar gradient.   

The effect of large coherent rings on small-scale turbulence

In an experimental partially stirred reactor (PaSR), fluid is injected through a set of 16 pairs of opposed jets and ejected via porous top and bottom walls. The resulting fluid circulation pattern creates pairs of alternately-rotating, large-scale coherent rings. The large-scale properties of the mean flow, i.e., the large-scale vorticity and strain rate, lead to rotating, quasi-2D turbulence at large scales, about 128 Kolmogorov scales in size. In these large structures, turbulence is locally homogeneous and the vorticity field is spatially correlated over a range of scales as large as the rings. This flow configuration is novel, since the vorticity field is usually classified as a small­scale field. The inertial range of second­order structure functions of enstrophy scales as log(r), where, r is the separation between two spatial points, as is the case for 2D turbulence. An analytical approach is developed to statistically describe the enstrophy over the whole range of scales.   

Subgrid scale contributions to Lagrangian time correlations in isotropic turbulence

Lagrangian time correlations or simply LTCs are the correlations of the Lagrangian velocities of one or two particles at two different times. Recently, there have been increasing applications of large-eddy simulation (LES) to turbulent dispersion, mixing processes and particle-laden flows. The applications raise such a fundamental question: can the LES with a subgrid scale (SGS) model predict LTCs correctly? The current existing SGS models are mainly developed in terms of the energy budget equations. As a result, they are able to correctly predict energy spectra. However, they may not ensure the accurate prediction on time correlations. Our previous researches investigate the effects of subgrid scales on the Eulerian time correlations. In present research, we will study the effects of subgrid scales on the LTCs in isotropic turbulence. A direct numerical simulation (DNS) and the LES with a spectral eddy viscosity model are performed for isotropic turbulence. It is observed that the LES overpredicts the LTCs than the DNS. We conclude from the straining hypothesis that an accurate prediction of LES on the enstrophy spectra is most critical to its prediction of the LTCs.   

Time-Dependent Turbulent Scouring

When a water jet plunges into the free surface of a body of water of uniform depth, a turbulent cauldron is established in the body of water under the point of entrance of the jet. If the body of water lies on a granular bed, the turbulent cauldron starts to scour the bed to form a pothole. Under a sustained action of the jet, the pothole deepens until a state of dynamic equilibrium is attained between the granular bed and the turbulent cauldron. We propose a theoretical model to predict the depth of the pothole as a function of time. This model is an extension of a model proposed previously to predict the final, equilibrium depth of the pothole as a function of the power of the jet, the depth of the pool of water, and the size of the grains of the granular bed. Our model yields a first-order, nonlinear ordinary differential equation for the depth of the pothole as a function of the time. At the onset of scouring, where our model can be solved analytically, the depth of the pothole is a power law of the time. As the time tends to infinity, the depth of the pothole tends asymptotically to the same equilibrium depth predicted by the earlier model. For intermediate times, we solve the model computationally and compare the results with a number of experimental data sets.   

Non-robustness of the two-dimensional turbulent inverse cascade

The inverse energy cascade in two-dimensional Navier-Stokes turbulence is examined in the quasi-steady regime, with small-scale, band-limited forcing at scale $k_f^{-1}$. It is found that the forcing Reynolds number, $Re\sim k_{max}^2/k_f^2$, where $k_{max}$ is the maximum resolved wavenumber, plays a crucial role in determining the energy distribution at larger scales. The strength of the inverse energy cascade, or fraction of energy input that is transfered to larger scales, increases monotonically towards unity with increasing $Re$. Moreover, as $Re$ increases beyond a critical value, for which a direct enstrophy cascade to small scales is first realized, the energy spectrum in the energy-cascading range steepens abruptly from a $k^{-5/3}$ to $k^{-2}$ dependence. The steepening is interpreted as the result of a greater tendency for coherent vortex formation in cases when forcing scales are adequately resolved, and is consistent with a small but nonzero net upscale enstrophy flux associated with near inviscid vortex merger. The results suggest the need for a review of the traditional interpretation and analysis of the inverse cascade.