Mean Velocity Scaling of High-Speed Turbulent Flows Under Nonadiabatic Wall Conditions

Abstract

The problem of scaling the near-wall mean velocity profiles of turbulent flows and collapsing them with the law of the wall has been traditionally studied using the conservation of momentum, by employing various levels of assumptions. In the van Driest transformation (“Turbulent Boundary Layer in Compressible Fluids,” Journal of the Aeronautical Sciences, Vol. 18, No. 3, 1951, pp. 145–216), the viscous stress was neglected, whereas in the Trettel and Larsson transformation (“Mean Velocity Scaling for Compressible Wall Turbulence with Heat Transfer,” Physics of Fluids, Vol. 28, No. 2, 2016, Paper 026102), the Reynolds stress was assumed to cancel out. Recent work by Griffin, Fu, and Moin (“Velocity Transformation for Compressible Wall-Bounded Turbulent Flows with and Without Heat Transfer,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 118, No. 34, 2021, Paper e2111144118) used a quasi-equilibrium assumption between turbulence production and dissipation in the log layer and demonstrated success for a wide variety of canonical flows. However, the extent of quasi-equilibrium is not verified, particularly for noncanonical flows. In this work, an alternate transformation is developed using the semilocal gradient of the van Driest transformed velocity, which is principally tied to the dynamics of vorticity transport in the boundary layer. In addition, a modified stress balance is utilized to directly account for and scale the buffer region. The transformation was benchmarked using a similar dataset as in Griffin et al., and comparable performance was obtained. Moreover, at supercritical pressures, the present transformation is found to produce significantly better collapse than existing works.