The prediction of transonic flows using a potential method

Transonic flows are simulated within convergent divergent nozzles and within turbomachinery blade rows. The flow is represented by the conservative full potential equation approximated by a nine-node central-difference scheme, which is third order accurate. Artificial viscosity is included into the central-difference approximation of the potential equation, in regions where the flow is locally supersonic. The approximation of the potential equation by central-differences, with an artificial viscosity term included, is equivalent to the approximation by upwind differences and ensures that the upwind nature of the domain of dependence of supersonic flows is correctly modelled. The exact form of this artificial viscosity term is derived and contains third order derivatives of velocity-potential. The inclusion of artificial-viscosity allows the potential equation to be approximated everywhere by central-differences and the flow equation is everywhere elliptic. The Neuman boundary-condition is applied, along solid surfaces, if an inviscid solution is desired. Viscous effects are incorporated by the modification of this condition so as to allow a transpiration flow through the solid surfaces. A standard Successive-Line-Over-Relaxation technique, developed for the solution of simultaneous elliptic equations, is used to solve the discretized potential flow equations. Predictions are presented for both the inviscid and the viscous-corrected potential codes applied to the simulation of transonic flow through nozzles and cascade blade-rows. Comparisons are made with other theoretical models and with experimental data. The problem of non-uniqueness is considered and an estimate of numerical error is made by the application of the inviscid code with two computational grids of different levels of refinement. The stability of this potential code is examined and is found to depend on the level of smearing of the shock discontinuity predicted by the theoretical model.