Turbulent Supersonic Channel Flow: Direct Numerical Simulation and Modeling

a = mean speed of sound, R T 1=2 aw = mean speed of sound at the wall, RTw 1=2 C = turbulent viscosity parameter, T= Lk cp = specific heat at constant pressure, R= 1 ; 1004:5 J= kg K cv = specific heat at constant volume, R= 1 ; 717:5 J= kg K f = uniform body force, w=h h = half channel height hw = wall length scale, h=Re i = turbulence intensity, 2k=3 = U1 k = turbulent kinetic energy, uiui=2 L = characteristic length scale of large-scale eddies, k M = local Mach number, U1=a Mg = gradient Mach number, gS=a Mt = turbulence Mach number, 2k =a M0 = bulk Mach number, u0=aw m = mass fraction of a passive scalar P = production of k, juiukj Ski Pr = Prandtl number, 0.7 Prk = turbulence Prandtl number in the k Eq. (15a) Prkm = molecular Prandtl number in the k Eq. (15a) Prt = turbulence Prandtl number in the mean temperature Eq. (13c) p = pressure, RT R = gas constant, cp cv; 287 J= kg K Re = local Reynolds number, U1h= ReL = turbulence Reynolds number, Lk = Re = Taylor-scale Reynolds number, 20ReL=3 1=2 Re = friction Reynolds number, wu h= w Re0 = bulk Reynolds number, 0u0h= w S = characteristic strain rate, 2 Sik Ski 1=2 Sik = rate-of-strain tensor, 0:5 @Ui=@xk @Uk=@xi Sik = deviatoric part of Sik, Sik Snn ik=3 Sc = Schmidt number, 1.0 Sct = turbulence Schmidt number SL = standardized source rate in the L equation, see (34) S" = standardized source rate in the " equation, see (34) S = standardized source rate in the equation, see (34) S! = standardized source rate in the ! equation, see (25) S!! = standardized source rate in the ! 2 equation, see (34) s = standardized characteristic strain rate, Shw=u T = temperature Tw = wall temperature Ui = velocity vector (i 1, 3) ui = fluctuating velocity vector, Ui Ui (i 1, 3) u0 = bulk velocity, see (5) u = friction velocity, w= w 1=2