Numerical simulation of compressible homogeneous flows in the turbulent regime

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Abstract
Compressible flows with r.m.s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r.m.s. Mach number Ma, with a transition at Ma = 0.3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256 × 256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum.

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