Scaling of pressure fluctuations in compressible turbulent plane channel flow

Abstract

The purpose of the paper is to identify Mach-number effects on pressure fluctuations$p'$in compressible turbulent plane channel flow. We use data from a specifically constructed$(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-matrix direct numerical simulation (DNS) database, with systematic variation of the centreline streamwise Mach number$0.32\leqslant \bar {M}_{{CL}_x}\leqslant 2.49$and of the HCB (Huanget al.,J. Fluid Mech., vol. 305, 1995, pp. 185–218) friction Reynolds number$66\leqslant Re_{\tau ^\star }\lessapprox 1000$. Strong$\bar {M}_{{CL}_x}$effects (enhanced by the increasingly cold-wall condition) appear for$\bar {M}_{{CL}_x}\gtrapprox 2$, for all$Re_{\tau ^\star }$, very close to the wall ($y^\star \lessapprox 15$). Compared with incompressible flow at the same$Re_{\tau ^\star }$, the wall root-mean-square$[p'_{rms}]^+_w$(in wall-units, i.e. scaled by the average wall shear stress$\bar {\tau }_w$) strongly increases with$\bar {M}_{{CL}_x}$. In contrast, the peak level across the channel,$[p'_{rms}]^+_{PEAK}$, slightly decreases with increasing$\bar {M}_{{CL}_x}$. In order to study the near-wall coherent structures we introduce a new wall-distance-independent non-local system of units, based for all$y$on wall friction and the extreme values of density and dynamic viscosity, namely, for cold walls$\{\bar {\tau }_w,\min _y\bar {\rho },\max _y\bar {\mu }\}$. The average spanwise distance between streaks, scaled by this length-unit, is nearly independent of$\bar {M}_{{CL}_x}$at constant$Re_{\tau ^\star }$. Using the in-plane (parallel to the wall) Laplacian$\nabla ^2_{xz}p'$we find that the$(+/-)\text {-}p'$wave-packet-like structures appearing inside the low-speed streaks ($y^\star \lessapprox 15$) with increasing$\bar {M}_{{CL}_x}\gtrapprox 2$are part of a more complex wave system with spanwise extent over several streaks, whose spatial density decreases rapidly with decreasing$\bar {M}_{{CL}_x}$or increasing$y^\star$. These$p'$wave packets appear to be collocated with strong$(+/-)$-$v'$events and could be responsible for compensating towards 0 the negative incompressible-flow correlation coefficient$c_{p'v'}$, with increasing$\bar {M}_{{CL}_x}$very near the wall.