Three-dimensional numerical simulation of compressible, spatially evolving shear flows

Recent numerical simulations of unforced, subsonic, compressible, spatially evolving two-dimensional shear layers [1,2] have been extended to three dimensions [3] and used to investigate planar shear flows, such as the mixing layer, the plane jet, and the wake behind a rectangular cylinder. In this paper we address the numerical issues of resolution and boundary conditions which are important for the validity of the studies of physical mechanisms in these simulations. The numerical model solves the time-dependent , inviscid conservation equations for mass, momentum, and energy in three dimensions in order to examine the evolution of large-scale coherent structures. A conserved passive scalar is convected with the velocity field in order to examine entrainment. The equations are solved numerically using a fourth-order phase-accurate Flux-Corrected Transport (FCT) algorithm [4] and directionand timestep-splitting techniques on structured grids. This approach has been shown to be adequate for simulating the high-Reynolds-number vorticity dynamics in the transition region of free flows and reproduces the basic large-scaie features of the flow observed in the laboratory experiments, e.g., the asymmetric entrainment [1], the distribution of merging locations [2], and the spreading rate of the mixing layers [6]. No subgrid modeling other than the natural FCT high-frequency filtering has been included. The nonlinear properties of the FCT algorithm effectively act as a subgrid model by maintaining the largescale structures and numerically diffusing the wavelengths smaller than a few computational cells. As a consequence, the small residual numerical viscosity of the algorithm mimics the behavior of physical viscosity in the limit of high Reynolds numbers. This was shown, for example, by Grinstein and Gnirguis [5], who obtained an upper bound on the effective numerical viscosity by comparing the spread of the mixing layer with that predicted by boundary layer theory for an incompressible laminar mixing layer. In the numerical solution, the mixing layer was forced to remain laminar by constraining the cross stream velocity to be zero; this is a reasonable condition in the limit of high Reynolds number Re, where v / u = O ( 1 / R e ) . Figure 1, reproduced from ref. 5, shows the values of the streamwise velocity at several streamwise locations plotted as a function of the similarity variable, ~} = y/[uez/Uo]1⁄2, where Uo is the velocity of the faster stream. The value of the viscosity, ue, was adjusted to obtain the best fit to the laminar mixing layer solution, shown as a solid line. Based on this value of ue, an effective Reynolds number, R~ = ~U/ue, was evaluated, where U" is the mean free-stream velocity and ~ is the initial mixing layer thickness at x = 0. Since numerical diffusion is affected by the grid size, a uniform grid spacing, Ay~ was used in the cross stream direction. The results shown in Fig. 1 correspond to a simulation where the initial condition is a step function velocity profile at x = 0, in which case, $ = Ay, leading to an effective Reynolds number, Re ~ 1200. This is actually a lower bound for the effective Reynolds number, since u~ becomes smaller when the transition from one velocity to the other in the initial velocity profile takes place over scales of more than a few computational cells. Further details are discussed on ref. 5. For 3D simulations of shear flows, inflow and outflow boundary conditions are imposed in the stream-