Figure1 (a) Iso-surfaces of velocity-gradient invariant (QD) measured in square-duct turbulent flow. QD = - 0.70 (light surfaces), and QD = +0.70 (dark surfaces). The mean flow is along the x-axis. (b) Measured joint-PDF of gradient tensor invariants and . Also indicated are sketches of the local flow structures implied by pairs (RD,QD ). (For more information, see van der Bos, Tao, Meneveau & Katz 2002).

Dynamics of velocity gradient tensor in turbulent flows

F. Van der Bos, B. Tao, C. Meneveau & J. Katz

A recently completed project focused on trying to understand how the small scale eddies of turbulence affect the evolution of the large-scale turbulence. Specifically, we were interested in quantifying how the large-scale velocity gradients (differences in velocity along differerent spatial directions) are affected. We asked the following question: What is the effect of subgrid-scales (SGS) on the evolution of the resolved (filtered) velocity gradient tensor Aij? This tensor is defined as duj/dxi, where uj is the filtered velocity vector. Aij includes both the strain-rate and the rotation (vorticity) and provides a characterization of the flow structure at the smallest of the resolved scales in Large Eddy Simulations of turbulence (LES). One desirable goal for LES and turbulence models is that the topology of (e.g. eigenvalue structure, alignment between vorticity and strain-rates, etc..) be correctly predicted in simulations.

Both DNS and our earlier experimental data (Tao et al. 2002a,b) have shown that Aij follows similar trends as those exhibited by the unfiltered velocity gradient tensor. For instance, a clear preference for axisymmetric extension (i.e. the preferred deformation is to generate pancakes as opposed to tubes), and for the vorticity to be aligned with the intermediate strain-rate eigenvector. These basic trends can be explained by an elegant analysis of the governing equations, as shown by the work of Vieillefosse (1982) and Cantwell (1992). Assuming an isotropic pressure Hessian, the analysis leads to the so-called restricted Euler (RE) system, which is amenable to analytical solution. It tends to a state in which the vorticity is indeed aligned with the intermediate strain-rate eigenvector and the deformation generates pancakes as opposed to tubes, hence explaining the basic observations. However, the solution then goes on to become singular (infinite velocity gradients) in finite time. This unphysical feature is prevented in reality by the deviatoric part of the pressure Hessian.

With former visiting student Fedderik van der Bos (now MS student in The Netherlands) and former PhD student Bo Tao (now Assistant Professor at Purdue University), we show that beside the pressure Hessian, the SGS stress tensor also affects the evolution of the tensor Aij. In Tao et al. 2001a,b, the alignment trends of the SGS stress tensor with respect to the orthogonal basis formed by the eigenvectors of the strain-rate tensor were studied in detail. Here we show that the tensor (the gradient of the SGS force vector) appears in the transport equation of Aij. The force-gradient generates changes in the local velocity gradient tensor in time. Following Cantwell (1992), we find that a substantial simplification occurs when considering the two invariants of Aij, called QD and RD, respectively.

The effects of the SGS motions can be evaluated from data by measuring the conditional averages of various contractions of the SGS force-gradient, conditioned upon these invariants. We show that these conditional averages uniquely characterize the evolution of the joint-PDF of the invariants, and that they can be measured from 3D-HPIV data in turbulent duct flow. Figure 1a shows two iso-surfaces of QD measured from one of the HPIV data sets. The blue (dark) levels are high positive values of QD (where vorticity dominates over strain), indicating tubular topology. The yellow (lighter) regions are where QD is negative (where strain dominates over vorticity), showing a more blob-like structure. Figure 1b shows the measured joint PDF of the two invariants. The long tail at positive and negative indicates the preferred state of axisymmetric extension and the vorticity aligned with the intermediate strain-rate eigenvector (Vieillefosse tail).


Figure 2a shows the vector field in (RD,QD )-space as implied by the Restricted Euler dynamics: the vectors indicate the speed at which restricted Euler dynamics would tend to modify the equilibrium PDF of Fig. 1b. In this plane, the effect of SGS stresses is shown in Fig. 2b. The results show that the SGS stresses have significant effect on the evolution of filtered velocity gradients. In particular, along the Vieillefosse tail the SGS motions oppose the formation of a finite-time singularity that occurs in Restricted Euler dynamics. In other regions, the SGS stress affects the dynamics in non-trivial ways (e.g. in the positive half-plane the SGS effects are almost perpendicular to the restricted Euler dynamics). A-priori tests of the Smagorinsky, nonlinear, and mixed models show that all reproduce the real SGS stress effect along the Vieillefosse tail (i.e. they help in preventing the finite-time singularity formation). However, in other quadrants they show some important differences with the real dynamics. This work sheds new light on the local structure of turbulence, the topology of large-scale structures, and how the small scales (which remain unresolved in LES and must be modeled) affect the dynamical evolution of this topology. For upcoming research, we anticipate using this new analysis technique in studying the effects of large-scale straining and unstraining on relationships between small and large-scales in turbulence.

This work is funded by the Office of Naval Research (Dr. P Purtell, program manager).

References:

Figure 2(a) Vector field arising from the deterministic dynamics in (RD,QD) plane of invariants of filtered velocity gradients (restricted Euler flow). (b): Vector field arising from the gradients of the SGS force, resisting the RE flow in the right lower quadrant.(For more information, see van der Bos, Tao, Meneveau & Katz 2002).

Charles Meneveau, Department of Mechanical Engineering, Johns Hopkins University, 3400 N. Charles Street, Baltimore MD 21218, USA, Phone: 1-410-516-7802, Fax: 1-(410) 516-7254, email: meneveau@jhu.edu