Direct Numerical Simulations of Shock Trains

Abstract

Shock wave boundary layer interactions (SBLIs) are physical phenomena which oc- cur in many applications of supersonic flow. One particular category of SBLI in- volves systems of linked shock waves which are arranged within an enclosed duct or channel (such as the internal flow through the inlet of a supersonic engine). Shock trains - as these SBLI structures are known - are the primary focus of this work. In particular, the results of direct numerical simulations (DNS) and implicit large eddy simulations (ILES) of shock trains are presented here in order to provide insights into their behaviour. Simulations are performed with a freestream Mach number of 2.0 and a baseline inflow Reynolds number (based on momentum thickness) of 500. The first part of the research involves a detailed validation of the underlying nu- merical methods. By considering the development of the turbulent boundary layer and formation of the shock train for different grid resolutions and numerical meth- ods we are able to quantify uncertainties in the results. The internal structure of the shock train is found to be less sensitive than its equilibrium location. Following this, a detailed parameter study considers the effect of back pressure, Reynolds number and boundary layer confinement. These parameters are found to have very little effect on the internal structure shock train, including the angle of the leading shock which occurs at the limit of a Mach reflection (between 40◦ and 43◦). By studying the effects of spanwise confinement we are able to show that the sidewalls reduce the strength of the individual shocks, resulting in a lower wall pressure gradient and the shock train length being approximately doubled. This effect is not fully explained by the blockage of the sidewall boundaries. Comparing the wall pressure results of each case with an established empirical model finds generally good agreement. The final part is devoted to investigating the time-dependent behaviour of shock trains. By subjecting shock trains to step changes in back pressure we are able to characterise the response. When correcting for back pressure lag effects we find that the response speed of the leading shock wave is largely independent of the initial conditions and direction of pressure change. The initial response is primarily governed by the back pressure step size, although spanwise confinement also plays an important role. Sinusoidal back pressures are applied to the shock train which allows us to show that, due to a filtering effect, lower forcing frequencies cause larger shock oscillation amplitudes. Back pressure changes are propagated upstream via the subsonic region at approximately Mach 0.3 and the resulting forcing/response lag causes an upstream shift in average shock position, such that the shock train length can be up to 35% higher on average. Lastly, a detailed spectral analysis of the shock train wall pressure reveals a number of flow features including a region of low frequency oscillation below the leading shock.