Modeling of nonparallel effects in temporal direct numerical simulations of compressible boundary-layer transition

One important alternative to spatial direct numerical simulation (SDNS) of a growing boundary-layer transition is a temporal direct numerical simulation (TDNS), where the flow is assumed to be locally parallel and the transition develops in time. To model nonparallel effects of a growing boundary layer, the TDNS allows the boundary layer to grow in time. This approach has been shown to be effective for an incompressible boundary layer. For a compressible boundary layer, however, a simple application of this approach has been found to be insufficient. To investigate this issue, we first split the variation of the flow field in the streamwise direction into a slowly evolving part and a fast and small-scale fluctuation part. By Taylor-expanding the slowly evolving large-scale part, this study shows that the Navier-Stokes operator can be reformulated as a power series of the perturbation parameter (x-x₀), yielding one set of equations for each power. Each set of these equations has a periodic solution in the streamwise direction, and therefore a modified TDNS method can be employed to solve these equations. Only the first set of the equations is considered in the applications presented. During the linear stage of transition, the results from this extended formulation show a significant improvement over those from the previous parallel flow formulation, especially for second modes which have short wavelengths. The results are well comparable with those from parabolized stability equations (PSE) and SDNS. A good agreement between this extended formulation and SDNS results is also demonstrated at the nonlinear stage.