Generation of anisotropy in turbulent flows subjected to rapid distortion

A computational tool for the anisotropic time-evolution of the spectral velocity correlation tensor is presented. We operate in the linear, rapid distortion limit of the mean-field-coupled equations. Each term of the equations is written in the form of an expansion to arbitrary order in the basis of irreducible representations of the SO(3) symmetry group. The computational algorithm for this calculation solves a system of coupled equations for the scalar weights of each generated anisotropic mode. The analysis demonstrates that rapid distortion rapidly but systematically generates higher-order anisotropic modes. To maintain a tractable computation, the maximum number of rotational modes to be used in a given calculation is specified a priori. The computed Reynolds stress converges to the theoretical result derived by Batchelor and Proudman [Quart. J. Mech. Appl. Math. 7, 83 (1954)] if a sufficiently large maximum number of rotational modes is utilized; more modes are required to recover the solution at later times. The emergence and evolution of the underlying multidimensional space of functions is presented here using a 64-mode calculation. Alternative implications for modeling strategies are discussed.

Figure 1

Ratio of post- to predistortion energy ${\mu }_3$ for various values of ${\mathcal{L}}_{\text{max}}$ as a function of the distortion factor $c$. BP54 is the theoretical result from Ref. [9]. $c=7.34$ corresponds to $t=1$, which is the initial large eddy turnover time.

Figure 2

Evolution of volume integrated $\ell =4$ modes (top) and volume integrated $\ell =6$ modes (bottom). The $\kappa$ (solid) and $\lambda$ (dashed) contributions are essentially indistinguishable.