Mach-Number-Invariant Mean-Velocity Profile of Compressible Turbulent Boundary Layers

A series of Mach-number- ($M$) invariant scalings is derived for compressible turbulent boundary layers (CTBLs), leading to a viscosity weighted transformation for the mean-velocity profile that is superior to van Driest transformation. The theory is validated by direct numerical simulation of spatially developing CTBLs with $M$ up to 6. A boundary layer edge is introduced to compare different $M$ flows and is shown to better present the $M$-invariant multilayer structure of CTBLs. The new scalings derived from the kinetic energy balance substantiate Morkovin’s hypothesis and promise accurate prediction of the mean profiles of CTBLs.

Figure 1

Wall-normal distributions of $a_{u_i{u}_j}=|\overline{u_i^{\prime }{u}_j^{\prime }}/\overline{u_k^{\prime }{u}_k^{\prime }}|$ versus (a) $y/{\delta }_{99}$ at ${\mathrm{Re}}_{\tau }={\overline{\rho }}_w{\overline{u}}_{\tau }{\delta }_{99}/{\overline{\mu }}_w=575$ and (b) $y/{\delta }_{vw}$ at ${\mathrm{Re}}_{{\delta }_{vw}}={\overline{\rho }}_w{\overline{u}}_{\tau }{\delta }_{vw}/{\overline{\mu }}_w=550$.

Figure 2

$M$ invariance of the density viscosity rescaled Kolmogorov length ${\eta }_{MI}^{+}=(\sqrt{{\overline{\rho }}^{+}/{\overline{\mu }}^{+}}){\eta }^{+}$ at ${\mathrm{Re}}_{{\delta }_{vw}}=550$, where ${\eta }^{+}=({\overline{\mu }}^{+3}/{\varphi }^{+}{)}^{1/4}/\sqrt{{\overline{\rho }}^{+}}$ (inset) is the normal Kolmogorov length and ${\eta }_{MI}^{+}=({\varphi }^{+}/{\overline{\mu }}^{+}{)}^{-1/4}\approx ({\overline{{\omega }_i^{\prime }{\omega }_i^{\prime }}}^{+}{)}^{-1/4}$. The dashed straight line shows ${\eta }_{MI}^{+}=(\kappa y^{+}{)}^{1/4}$ in the log layer.

Figure 3

Global $M$ invariance of density viscosity rescaled Prandtl’s mixing length ${\ell }_{MI}^{+}=\sqrt{{\overline{\rho }}^{+}}{\overline{\mu }}^{+}{\ell }_m^{+}$ at ${\mathrm{Re}}_{{\delta }_{vw}}=550$.

Figure 4

MVPs of the CTBL transformed by the refined viscosity weighted transformation (solid lines) compared to the untransformed MVPs (short-dashed lines) and the empirical MVP of the ITBL [14] (long-dashed dark yellow line) at ${\mathrm{Re}}_{{\delta }_{vw}}=550$. The insets are the enlargement in the wake region. VDT is shown in the lower right inset, and the refined viscosity weighted transformation is shown in the upper left. The arrows indicate that $M$ increases from 2.25 to 4.5 and to 6. See Fig. 3 for the color legend.