Direct Simulation of Homogeneous Turbulence and Gravity Waves in Sheared and Unsheared Stratified Flows
Differences in the evolution of stably stratified turbulence with and without mean shear are investigated by means of direct numerical simulations for moderate Reynolds number (Re ℓ = 42.7) and seven values of the Froude or Richardson number. The molecular Prandtl number is unity. In stratified flows without shear, energy decays quickly initially but at reduced rate later when gravity waves dominate the flow pattern. Gravity waves occur when the Ellison length scale reaches about 0.3 to 0.8 times the Ozmidov length scale. The flow becomes anisotropic before the first waves arise. The developed flow is dominated by gravity waves, and turbulent mixing is considerably suppressed, when the Ellison length scale is about six times the Kolmogorov length scale. For sheared turbulence the importance of buoyancy relative to shear forcing depends on the Richardson number Ri. For an initial shear number Sh 0 = 3, we find a critical Richardson number of 0.13 which is smaller than the value 0.25 predicted by linear in viscid theory because of the rather strong dissipation in the present simulations. In subcritical flows (Ri < Ri crit ), turbulence is dominated by shear. If the Richardson number is supercritical (Ri > Ri crit ), the turbulence is controlled by gravity and behaves at large scales as if no shear would be present. But shear causes small-scale turbulence (possibly by wave breaking) and hence the dissipation is larger than without shear. The degree of anisotropy increases with increasing Richardson number but gets limited when counter-gradient fluxes (CGF) of heat and momentum appear in the vertical direction. Temporally oscillating and sign-changing vertical fluxes at large scales have to be distinguished from persistently positive fluxes (p-CGF) at small scales. Both types develop in sheared as well as in unsheared stratified flows. The oscillating flux exchanges energy between kinetic and potential energy reservoirs and can be described by rapid-distortion calculations. The p-CGF is due to an imbalance between kinetic and potential energy sources and sinks at small scales.
KeywordsGravity Wave Direct Numerical Simulation Richardson Number Stable Stratification Stratify Flow
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