Near-wall variable-Prandtl-number turbulence model for compressible flows

Abstract

A near-wall variable-Prandtl-number turbulence model is developed for the calculations of high-speed compressible turbulent boundary layers. The model is based on the k-e and the 0 2 -e<?0 equations formulated for near-wall flows. These four equations are used to define the turbulent diffusivities for momentum and heat, thus allowing the assumption of dynamic similarity between momentum and heat transport to be relaxed. The Favre-averaged equations of motions are solved in conjunction with the four transport equations for k, €, 0 2 , and e0. Calculations are compared with measurements and with other model predictions where the assumption of a constant turbulent Prandtl number is invoked. Incompressible channel/pipe flows and compressible boundary-layer flows with adiabatic as well as constant temperature wall boundary conditions are considered. Cases where the freestream Mach number as high as 10 and where the wall temperature ratio as low as 0.3 are calculated. The analysis shows that the variable Pr t model yields an asymptotically correct prediction of the temperature variance and the normal heat flux for incompressible flows. In the case of compressible boundarylayer flows, the model calculations are in good agreement with measured mean flow and skin friction for flows with an adiabatic wall and yield substantial improvements in the predictions of mean flow properties compared to the constant Pr t results for cooled wall cases. Nomenclature A += model constant taken to be 45 <*k > bk = coefficients in the expansion for k + in the near-wall region <*uv»b uv = coefficients in the expansion for ~uv + in the near-wall region _ #v0> bve = coefficients in the expansion for vO + in the near-wall region a 2 , b 2 = coefficients in the expansion for 6 + 2 in the near-wall region #e0> b eQ = coefficients in the expansion for e# + in the near-wall region B-constant in law of the wall Qx = model constant taken to be 0.1 C d i = model constant taken to be 1.8 for boundary layers and 2.0 for internal flows Cd2 = model constant taken to be 0 Qo = model constant taken to be 0.72 Q 4 = model constant taken to be 2.2 C ds = model constant taken to be 0.8 Cf = skin-friction coefficient, 2r w /(pU 2 X) ) Ch = heat transfer coefficient, /[(poot/ooC^ -e r )] C d = model constant taken to be 1.5 C e2 = model constant taken to be 1.83 C M = model constant taken to be 0.096 C x = model constant taken to be 0.11 f Wt 2 = near-wall damping function for e equation f Wtf e = near-wall damping function for e e equation fp = near-wall damping function for turbulent momentum diffusivity /x = near-wall damping function for turbulent heat diffusivity H = instantaneous total enthalpy, C P T + l /2U k U k h = half-channel width or pipe radius k = turbulent kinetic energy k + = normalized k, k/u 2 M = Mach number M t = local Mach number, u r /(yRT w ) y2 P$ = production due to mean temperature, defined as -(ue)(d(Q)/dx) Pr = molecular Prandtl number *Pr t = turbulent Prandtl number p -instantaneous pressure q w = heat flux at the wall R = univer...

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