Computation of shock/boundary-layer interactions in bump channels with transport-type turbulence models

In this paper, an explicit time-marching finite-volume scheme has been used together with a number of convergence acceleration techniques such as the multigrid strategy. Two types of turbulence models, a Johnson-King (J-K) model and a two-layer k-ε/k-l model, have been incorporated and modified to model internal compressible flows with multiple walls. Some modifications have been made of the inner layer viscosity formulations of the J-K model in order to improve its predictive capability for flow separation. Partially implicit treatment of the transport-type equations of turbulence in the models is adopted, because the source terms in these equations can cause numerical stiffness when there are flow separation, sharp gradients and high cell-aspect ratio near the solid wall. A two-dimensional arc-bump flow investigated experimentally by Liu and Squire [X. Liu and L.C. Squire, Interaction on curved surface at transonic speed, in: Turbulent Shear/Shock Wave Interactions, IUTAM Symposium Palaiseau 1985 (Springer, Berlin Heidelberg, 1985) 93-104.] was calculated using the J-K model with satisfactory agreement with the corresponding measurement. Although efficient and accurate, it is found that the J-K model lacks the theoretical generality to be extended to model three-dimensional (3D) complex internal flows with multiple walls. Therefore, a two-layer k-ε model is employed for 3D flow computation. Various measures are adopted to ensure stable and convergent numerical solution. A three-dimensional transonic channel flow with multiple shock/boundary layer interactions was studied with the aforementioned two-layer model and numerical methods. The results are compared with experimental measurements [J. Cahen, V. Couaillier, J. Delery and T. Pot, Validation of Navier-Stokes code using a k-ε turbulence model applied to a three-dimensional transonic tunnel, AIAA paper AIAA-93-0293, AIAA, 1993] and numerical results obtained by using a Low-Reynolds-Number (LRN) k-ε model [VUB/FFA, Turbulence Models in EURANUS and the 3D Delery bump, Technical Report SNWP3.3/01, VUB, Pleinlaan 2, 1050 Brussels, Belgium and FFA, P.O. Box 11021, S-161 11 Bromma, Sweden, 1993]. Compared with other (LRN) two-equation models, the two-layer model implemented is promising in modeling very complex 3D internal flows in terms of efficiency, robustness and accuracy. The two-layer model permits uniform distribution of flow properties to be specified as initial condition which makes the simulation easier to be carried out.