Shock Wave--Turbulence Interactions

Abstract

Canonical shock / isotropic turbulence interactions (SITIs) are studied using direct numerical simulation (DNS) and perturbation analysis. Flow parameters include unprecedentedly high turbulence Mach numbers (Mt <= 0.7) and up to 15% dilatational turbulent kinetic energy (TKE).

These extreme conditions necessitate shock-capturing throughout the entire domain and the use of DNS inflow data from auxiliary forced isotropic turbulence simulations.

Three aspects of the DNS results are of particular interest: unprecedentedly high streamwise Reynolds stress amplification; mean flows that differ from classical solutions; and Reynolds stress anisotropy opposite to the predictions of linear theory.

These DNS results are elucidated by perturbation analyses. In high-Mt flows, both solenoidal and dilatational incident modes are important, but no existing work handles these modes in a convenient, unified way. Therefore, existing inviscid linear interaction analyses (LIAs) are reformulated in a general framework allowing any incident mode type, any inclination angle, and any shock obliquity. Integrated results are presented for isotropic incident fields of turbulence, sound, and entropy spots. LIA remains remarkably accurate (within 10% for TKE amplification) even for the strong turbulence considered here.

The new LIA is used as a starting point for second-order “quadratic interaction analysis” (QIA) and viscous LIA. QIA improves the mean predictions of classical theory; viscous LIA offers a possible explanation for a reported failure of single-wave LIA near so-called critical angles.

The anomalous post-shock Reynolds stress anisotropy is explained by the lengthscales of emitted vortical waves as a function of angle. The post-shock waves carrying the majority of the streamwise Reynolds stress are of longer wavelength than those carrying the transverse stress. This implies different timescales, with smaller scales losing the “memory” of their initial anisotropy faster than larger scales lose theirs. A model based on this understanding predicts anisotropies close to those observed.

Finally, the smoothing effects of shock motion (SM) are combined with QIA to give “SM-QIA” theory. The resulting mean pressure profiles closely match DNS data through the shock and downstream until the effects of nonlinear terms become important; SM-QIA thus provides an analog to the classical Rankine–Hugoniot shock-jump conditions applicable even in turbulent flows.