Quasi-conservative lambda formulation

Abstract

The numerical simulation of inviscid transonic flows by means of a lambda takes in- to account the shock transition from supersonic to subsonic flow conditions, thus allowing the coupling of the supersonic region with the shocked subsonic one and, as a consequence, the upstream movement of the shock. The present methodology is applied to one- and two-dimensional transonic flows. Although the two-dimensional flow calculations involve Cartesian coordinates and are limited to the thin-airfoil approximation, the method can be generalized to arbitrary two- and three-dimensional flow cases. In all of the computed cases, the shock is found to have appropriate strength and position for steady flow conditions and to move upstream properly when a change in the downstream pressure warrants it. A MONO the several numerical methods developed for the JTjLnumerical simulation of transonic inviscid flows, the so- called lambda formulation, recently developed and employed by Moretti 1 and Zannetti and Colasurdo,2 has several very desirable features. The time-dependent, compressible Euler equations are recast in terms of compatibility conditions for characteristic (Riemann) variables along characteristic lines and discretized by means of upwind differences which cor- rectly take into account the direction of wave propagation. In this way a numerical technique is obtained which com- bines the coding simplicity of finite-differ ence methods with the intrinsic accuracy and physical soundness of the method of characteristi cs. Since its first appearance, the lambda formulation has undergone several improvements. In particular, various im- plicit integration schemes3'5 and relaxation methods6'7 have been developed in order to enhance the efficiency of the lambda methodology by removing the Courant-Friedrichs- Lewy (CFL) stability limitation of the explicit schemes previously employed. Also, a significant improvement in ac- curacy is achieved by employing a perturbative formula- tion8'9 such that only the compressibility effects with respect to a suitable incompressible flow solution need to be computed. The lambda formulation has its drawbacks; in particular, numerical experiments have shown that for one-dimensional transonic flows the captured shock cannot move upstream in the supersonic flow region and that this problem persists in some cases for two-dimensional transonic flows, even if the supersonic bubble is embedded in a subsonic region, which would be expected to allow for upstream propagation. This drawback is due to the decoupling of the supersonic region from the subsonic one once the shock wave is established. Recently, the splitting,101 3 which uses governing equations in conservation form and splits the fluxes according to the wave nature of the flow, has been proposed to solve compressible inviscid flows. A comparison of this methodology with the lambda formulation applied to a system of linear hyperbolic equations14 shows that they both lead to identical conclusions, but an analysis of the flux-difference splitting applied to the one-dimensional Euler equations demonstrates that it allows a coupling of the supersonic region with the shocked subsonic one by splitting the waves containing a sonic point, i.e., a vertical characteristic in the (x,t) plane. Such a situation occurs in those mesh intervals where the shock is numerically cap- tured. On the contrary, this basic mechanism is neglected by the classical lambda formulation, but it can be taken into ac- count in order to obtain a modified lambda formulation characterized by a supersonic region coupled with the shocked subsonic one, as done in the present paper. The method will be first described for one-dimensional flows of a test gas with a specific heat ratio equal to unity; the method will then be extended to classical one-dimen- sional nonisentropic flows; and finally to the case of nonhomentropic two-dimensional flows within the frame- work of thin-airfoil theory. The validity and usefulness of the modified lambda methodology will be demonstrated by means of a few example calculations.