One-Equation Turbulence Model of Spalart and Allmaras in Supersonic Separated Flows

Introduction S UPERSONIC separated  ows take place in many engineering applications,and in many cases, the state of the separated  ow is unknown. It could be laminar, transitional,or fully turbulent.The one-equation turbulence model of Spalart and Allmaras (SA) has a feature to allow the existence of a stable laminar solution. In other words, the eddy viscosity ot D 0 is a stable solution. This implies that if the  owŽ eld is laminar the SA model can predict a laminar solution. The SA model has a trip term to initiate transition in a smoothmanner. This kind of transition is mainly numerical and has little to cope with the boundary layer’s natural tendency to transition based on local pressure gradients, suction,wall roughness, and so on. The application of the trip term is almost exclusively for a situationwhere the  owupstreamof the trip point is laminar.Downstream, far away from the trip point, this term has no in uence on the solution. It is important to realize that, in the case of relaminarization, the transition process from turbulence to laminar depends mainly on the balance of the production, diffusion, and destruction terms and is independent of the trip point location upstream provided it is far from the location where relaminarizationtakes place. Having said this, because the model does not include any physical argument on the onset of transition, the model is at best able to predict relaminarizationbut cannot be trusted to predict accurately the onset and duration of transition.Nevertheless, this is still better than employing an algebraic model, such as that of Baldwin and Lomax2 (BL), which cannot predict relaminarization except in the casewhere a correct number for the BL parameterCMUTM is known a priori. This parameter is to suppress the incorrectlylarge eddy viscosity generatedby strong local velocity gradients and is a function of local Reynolds number. It is important to make sure that this parameter does not suppress the correct large eddy viscosity. Two examples are given in this Note to demonstrate the preceding discussion. Each example is calculated Ž rst with the BL model, which is then used as an initial solutionwith the SA model. The objective of this procedure is to see whether in a laminar  owŽ eld a fully turbulent  ow created by the BL model can be relaminarizedby the SA model. An initial solution for the SA model can be obtained from a solution with the BL model by converting the eddy viscosity from the BL solution, ot , to the transformed eddy viscosity N o in the SA model, where inside the boundary layer