Velocity and enthalpy distributions in the compressible turbulent boundary layer on a flat plate

Abstract

The object of this paper is to present a unified account of the distributions of velocity, shear stress and enthalpy in the compressible turbulent boundary layer on a flat plate. As a start, the set of velocity profiles measured over a range of heat-transfer conditions at Mach numbers between 5 and 8 by Lobb, Winkler & Persh (1955) is examined. It is found that by plotting in terms of the Howarth variable $\eta = \int ^y_0 (\rho|\rho_ 0)dy$, the outer parts of the profiles for different Mach numbers are brought together on a single curve of the approximate form u/u_∞ = (η/Δ)^1/n, Δ being the transformed boundary-layer thickness. By evaluating the reference density ρ₀ and kinematic viscosity ν₀ at the so-called ‘intermediate’ enthalpy (Eckert 1955) the innerparts of the profiles can also be collapsed, although less completely, to fit a ‘law of the wall’ u/u_τ = A log (ηu_τ/ν₀ - c) + B. Here u_τ = (τ_w/ρ₀)^½, and A, B and c are the same constants as in incompressible flow.

These properties provide a physical starting point from which the remaining features of the mean flow can be calculated. By substitution of appropriate stream functions in the equation of motion the distribution of shear stress r in inner and outer regions is found; this approximates to the form $\tau |\tau_w = 1 - (u|u_\infty)^{(n+2)}$ over the whole layer. A relation between the distributions of enthalpy and shear stress is then found from the energy equation, using a turbulent Prandtl number a which is assumed constant across the layer to relate eddy conductivity to eddy viscosity. The final expression is similar in form to Crocco's integral for the laminar boundary layer with α taking the place of the laminar Prandtl number σ, but contains two extra terms proportional respectively to (α − σ)c_f and (α − σ)c^½_j, which represent the effect of the inner viscous regions.

The enthalpy integral is evaluated using the stated velocity-shear relation, and an expression which agrees well with the available experimental data is found for the heat-transfer coefficient as a function of recovery factor and skin-friction coefficient. It is also found that the usual quadratic enthalpy-velocity relation, exact for α = σ = 1, remains an acceptable approximation for Prandtl numbers considerably different from unity.