Large eddy simulations and subgrid scale motions in stratified turbulence

Abstract

Direct numerical and large eddy simulations (DNS $\&$ LES) of decaying and forced stratified turbulence are studied in this thesis. By defining a test filter scale $k_c$ in the horizontal and vertical directions separately, the energy transfer spectra are investigated. It is shown that stratification affects the horizontal eddy viscosity significantly, by which the non-local energy transfer between large and small horizontal scales are increased. This non-local horizontal energy transfer is around $20\%$ of the local horizontal energy transfer at the cutoff wavenumber $k_c$. In addition, the non-local horizontal energy transfer occurs at large vertical wavenumbers, including the buoyancy wavenumber $k_b = N/u_{rms}$, where $N$ is the buoyancy frequency and $u_{rms}$ is the root-mean-square velocity. The non-local horizontal eddy viscosity decreases and the local eddy viscosity is dominant if the value of the test cutoff $k_c$ varies from large scales to the dissipation scales. Next, the performance of three common subgrid scale (SGS) models, i.e.~the Kraichnan, Smagorinsky and dynamic Smagorinsky models, is investigated in stratified turbulence. It is shown that if the grid spacing $\Delta$ is small enough, the horizontal wavenumber spectra show an approximately $-5/3$ slope along with a bump at the buoyancy wavenumber $k_b$. Our results suggest that there is a maximum threshold on $\Delta$, below which the dynamics of stratified turbulence, including Kelvin-Helmholtz instabilities, are captured. This criterion on $\Delta$ depends on the buoyancy scale $L_b$ and varies with different SGS models: the Kraichnan model requires $\Delta/L_b < 0.47$, the Smagorinsky model requires $\Delta/L_b < 0.17$ and the dynamic Smagorinsky model requires $\Delta/ L_b < 0.24$. In addition, the statistics of the dynamic Smagorinsky coefficient $c_s$ demonstrate that large shear leads to small values of $c_s$ in stratified turbulence (in line with the results for isotropic turbulence). Finally, it is shown that the net down-scale energy transfer in stratified turbulence is a combination of two large values of upscale and downscale energy transfer mechanisms. Overall, our results suggest that stratification changes the dynamics of SGS motions dramatically if the filter scale $\Delta$ is around the Ozmidov scale or smaller; in order to capture the dynamical features of stratified turbulence, LES requires resolution of $L_b$. In addition, when the buoyancy Reynolds number $Re_b \lesssim \mathcal{O}(1)$, the kinetic energy transfer shows some spectral backscatter at intermediate scales that is due to viscous effects and not to the turbulent mechanism.