Abstract:
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.