High-Order Semi-Implicit Schemes for Unsteady Compressible Flow Simulations

Direct numerical simulation of stability and transition of compressible boundary layers requires high-orderaccurate and computationally efŽ cient numerical methods to resolve a wide range of timeand length scales associated with wave Ž elds in the boundary layers. Explicit methods have been used mainly in such simulations to advance the compressible Navier–Stokes equations in time. However, the small wall-normal grid sizes for viscous  ow simulations impose a severe stability restriction on the allowable time steps in simulations using the explicit method. This requires implicit treatment to our numerical method. Although fully implicit methods are often used in steady- ow calculations to remove the stability restriction on time steps, they are seldom used in transient  ow simulations because the time steps used in time-accurate calculations are often not large enough to offset high computational cost of using fully implicit methods. A high-order-accurate semi-implicit scheme is presented for the direct numerical simulation of the stability and transition of compressible wall-bounded  ows. The efŽ ciency and accuracy of the semi-implicit scheme are evaluated by applying the method to transient  ow simulations of several supersonic and hypersonic wall-bounded  ow stability problems.