A domain decomposition technique for small amplitude wave interactions with shock waves

Abstract

In this paper, a domain decomposition technique in the finite volume framework is presented to propagate small amplitude acoustic and entropy waves in a linearized Euler region and simulate the interaction of these waves with an initially steady normal shock in a nonlinear region. An overset method is used to two-way couple the linear and nonlinear regions that overlap each other. Linearized solvers alone cannot capture this interaction due to the discontinuity encountered at shocks. On the other hand, nonlinear solvers based on second order shock-capturing schemes will result in excessive dissipation and dispersion for the small disturbances. The domain decomposition technique provides a good balance between minimizing dissipation and dispersion errors while enabling nonlinear shock-acoustic interactions. To preserve low dispersion and dissipation, a DRP scheme is used to simulate the incoming and outgoing waves in the linear region. To capture the shock wave interaction and motion, a hybrid central-upwind flux scheme is used in the nonlinear region that contains the shock. Grid sensitivity studies for an acoustic wave propagating in stationary flow were performed to compare the linear, nonlinear, and domain decomposition solvers. The nonlinear solver required ten times the mesh resolution to achieve similar accuracy as the linear solver, resulting in a forty-fold increase in computational time. For modest cell size ratios, the domain decomposition solver reduced the computational time by a factor of three compared to the nonlinear solver while achieving similar accuracy. Interaction of standing shocks with acoustic and entropy waves of amplitudes ϵ=±10−2 and ±10−5 was investigated using the domain decomposition technique. The numerical results for ϵ=±10−2 compared well with the linearized interaction analysis (LIA) with less than 3% discrepancy in terms of the amplification factors. The domain decomposition technique acts as a low pass filter that averages the post-shock oscillations generated by the slow-moving shocks in the nonlinear region, resulting in the correct amplification factors in the linear region. For the smaller amplitudes of ϵ=±10−5, the amplification factors deviated from LIA predictions by up to 70%. Numerical results suggest that the large discrepancy for the small amplitude cases is due to insufficient mesh resolution for capturing extremely slow-moving shocks.