Simple turbulence models for supersonic flows - Bodies at incidence and compression corners

Abstract

Parabolized Navier-Stokes (PNS) predictions of turbulent flows at supersonic and hypersonic speeds past two sphere-cones and a cone-cylinder-flare are used to evaluate simple turbulence models. Modifications to an algebraic turbulence model are proposed to improve predictions for flow on bodies at incidence. Predictions using a simple modification for the length scale and a model based upon Bradshaw's extra-strain-rate hypothesis are compared with measurements of supersonic and hypersonic flows. The modifications lead to significant improvements in predicted wall shear stress for a Mach 3 flow over a sphere-cone at moderate angle of attack. In addition, predictions of hypersonic flow past a cone-cylinder-flare are used to compare the algebraic CebeciSmith turbulence model with the nonequilibrium Johnson-King half-equation model. Predictions are compared with measurements of surface pressure and heat transfer for isothermal-wall flow at Mach 9.22 for a cone-cylinder-flare. The algebraic model predictions are quite satisfactory and no significant improvement is achieved with the nonequilibrium model for this flow with no streamwise separation.z a ,w Nomenclature = freestream speed of sound = empirical constants in the van Driest damping, Eqs. (4) and (17) = van Driest damping, Eqs. (4) and (17) = dimensionless effective conductivity, K e /K x , Eq. (2) = characteristic or reference length = freestream Mach number, V = dimensionless normal distance from wall, = dimensionless wall coordinate, Eq. (5) = dimensionless pressure, p/p^Vl, = Prandtl number = radial distance from body axis, Eq. (8) = freestream Reynolds number, p^V^L/jl= Stanton number based on freestream conditions, Eq. (25) = temperature = dimensionless velocity components in the Cartesian coordinates (x,y,z), N/L = dimensionless velocity tangent to the surface (excluding crossflow), U/V= dimensionless total velocity, (u 2 + v 2 + w 2 ) 1/2 = dimensionless velocity tangent to the surface (including crossflow), V T /V= freestream total velocity = dimensionless crossflow velocity, W/V= dimensionless physical Cartesian coordinates, x/L,y/L,z/L = angle of attack = Klebanoff intermittency factor, Eq. (9) . Member AIAA. tAssociate Professor, Department of Mechanical Engineering. Senior Member AIAA.d* = dimensionless kinematic, or "incompressible," displacement thickness, Eq. (7) 6 C = cone half-angle K = von Karman constant JM, = dimensionless coefficient of viscosity, \L/\L™ Ht = dimensionless eddy viscosity, /1,/jIoo He = dimensionless effective viscosity, £ e //Ioo, Eq. (1) £,i7,f = dimensionless computational coordinates in the streamwise (axial), radial (normal), and circumferential directions p = dimensionless density, p/pT -dimensionless total Reynolds shear stress (divided by density), Eq. (16) T W = dimensionless wall shear stress, r^/^^V^/L) 4> = meridian angle, 0 = 0 deg windward, 0= 180 deg leeward Subscripts and Superscripts oo = freestream e = edge of boundary layer eq = equilibrium value m = location of maximum in r ti = inner region of turbulent...