Numerical Simulation of Kelvin-Helmholtz Instability via Direct- and Large-Eddy Simulation
Authors: Thomas S. Lund, Joseph A. Werne
Venue: Proceedings of the 2004 Users Group Conference (2004)

Kelvin-Helmholtz (K-H) instability in a stably stratified background is studied numerically in order to support the AirBorne Laser (ABL) project. The wind shear associated with K-H events produces irregularities in the air density which result in both refraction and scattering of laser beams used for communications and for anti-missile purposes. Optical distortion is measured from the simulations through analysis of the second order structure function of the potential temperature fluctuations. To date, direct numerical simulations (DNS) have been conducted at bulk Richardson numbers of 0.05-0.1 and Reynolds numbers of 2000-2500 using up to 1000×350×2000 Fourier modes, run on as many as 1000 processors of the Cray T3E atERDC. The structure function coefficients, power law exponents, and turbulence inner scale from these simulations are in good agreement with both theoretical predictions and field measurements. An initial phase of a companion large-eddy simulation (LES) study has been completed where the existing DNS has been replicated at lower resolution using dynamic subgrid scale modeling ideas. The results of these simulations show that the first and second order statistics (including the structure function) can be reproduced with acceptableaccurately using meshes that are coarser by factors of up to six in each spatial direction.When the effect of the increased time step is taken into account, the LES can be completed at computational cost that is 1/64 = 1/1296 of that required for the DNS. We also anticipate that it should be possible to produce high-fidelity simulation results at Reynolds numbers that are one to two orders of magnitude higher using LES, which places the simulations within the range of high-altitude wind shear events. The LES studies have revealed several interesting features of the K-H flow and of subgrid-scale modeling for stably-stratified flows. In attempting to replicate the DNS, a strong sensitivity to the precise details of the initial conditions was noted. This sensitivity was quantified through ensembles of runs where 1% random noise was added to the LES initial conditions. The trajectories of the individual members of the ensemble were observed to diverge after a period of 20-40 inertial time units. This high degree of sensitivity suggests that,if possible, ensembles of numerical simulations should be used to infer the characteristics ofthe K-H flow for a particular set of bulk parameters. A second interesting feature found in the combined DNS/LES studies is that the effective turbulent Prandtl number is a function of filtering scale and is not all that well represented by the commonly-used gradient Richardson number based parameterizations. The dynamic subgrid-scale model does not use a turbulent Prandtl number, but rather makes independent estimates for the eddy viscosity and eddy diffusivity. The turbulent Prandtl number inferred from the ratio these two quantities was found to be in close agreement with the values extracted via filtering of the DNS data, independent of filtering scale.