Numerical Simulation of Developing Compressible Turbulent Flow with Heat Transfer

Abstract

This study investigates the effects of wall heating and skin friction on the characteristics of a compressible turbulent flow in developing and developed regions of a pipe. The numerical solution is performed by finite-elementbased finite volume method applied on unstructured grids. A modified -" model with a two-layer equation for the near-wall region and a compressibility correction are used to predict turbulent viscosity. The results show that shear stress in fully developed flow is nearly constant from the centerline up to 75% of the pipe radius, then increases sharply next to the wall, and the ratio of the turbulent viscosity to the molecular one is less than 0.2. Under a uniform wall heat flux condition, the friction factor decreases in the entrance region and will be fully developed after Z=D > 50, but the Nusselt number increases first and then will be fully developed after Z=D > 10. In addition, the heat flux accelerates the developing compressible flow and causes the entrance length to decrease, unlike the incompressible flow. Nomenclatureenergy flux by force and heat, W=m 2 C = triangle area, m 2 c = speed of sound, m=s C = constant D = pipe diameter, m D = destruction, 1=s e = internal energy, J=kg F = inviscid vector flux f = friction factor H = total enthalpy, J=kg H = Heaviside step function k = thermal conductivity, W=m K L = pipe length, m l = length scale, m M = Mach number (u= RT p ) N = viscous flux vector, number of grid nodes Nu = Nusselt number (hD=k) n = unit normal vector P = production p = pressure, Pa Pr = Prandtl number (C p =k) Q = conservative variables vector q 00 = heat flux rate, W=m 2 R = pipe radius, m R = eigenvector matrix Re = Reynolds number ( uD=) r = radial direction S = source vector, W=m 3 T = temperature, K t = time, s u = velocity in the z direction, m=s V = velocity vector, m=s y = distance measured from wall inward, m z = axial direction = specific heat ratio = difference = boundary-layer thickness, m " = turbulent dissipation energy, W=kg = turbulent kinetic energy, m 2 =s 2 = eigenvalues matrix, = dynamic viscosity, kg=m s = density, kg=m 3 = shear stress, Pa = shape function = general function = computational domain Subscripts c = critical, centerline, cutoff fd = fully developed h = discretized computational domain i, j, k = direction, counter in = inlet L = lower cell index n = normal to boundary out = outlet R = upper cell index r, = cylindrical coordinates S = Sutherland constant, compressibility t = turbulent, tangential w = wall 0 = reference 1,2,3 = indices for triangle vertices Superscripts n = time-step iteration = Roe-average quantity