Numerical study of a channel flow with variable properties

1. Motivation and objectives In many industrial devices such as heat exchangers, piston engines, or propulsion systems strong temperature gradients arise in the near wall region even if the characteristic Mach number is close to zero. A strong coupling exists between momentum and energy equations caused by variations in the fluid properties, and the classical wall models for incompressible flows are not appropriate. In the 1950’s through the mid 1960’s, many experimental studies focused on the assessment of global quantities at the wall (friction coefficient, Nusselt number) for laminar/turbulent flows with variable properties. Some empirical correlations of engineering interest were derived. More recent studies also deal with velocity and temperature profiles, and it has now reached the point that the supersonic compressible turbulent boundary layer with or without heat transfer is now well documented (see Bradshaw (1977), Fernholz & Finley (1980) and Spina et al. (1994) for reviews). The Strong Reynolds Analogy was introduced by Morkovin (1961) in the context of adiabatic boundary layers and has often been used in turbulence modeling. An extension was proposed by Gaviglio (1987) and subsequently Huang et al. (1995) for use in the presence of heat transfer. Some experimental data support these analogies in the case of a supersonic boundary layer over a cooled or heated wall and low speed flow on a slightly heated wall. Dimensional analysis of the inner layer shows that the law of the wall can be described in terms of two non-dimensional wall parameters, the friction Mach number Mτ = uτ cw and the heat flux parameter Bq = qw ρwCpuτTw , where uτ is the friction velocity √ τw ρw , cw the speed of sound, qw the heat flux, Cp the constant-pressure specific heat, and Tw the temperature at the wall. Two cases, (Mτ ,Bq)=(0.08,−0.05) and (0.12,−0.14), were considered in the DNS study of a supersonic channel flow performed by Coleman et al. (1995). These data were found in Huang & Coleman (1994) to support the validity of the Van Driest (1951) transformation