Scaling of pressure fluctuations in compressible turbulent plane channel flow
G. A. Gerolymos, I. Vallet
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 (Huang et 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.