An analytic characteristic method for steady three-dimensional isentropic flow

Abstract

The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method.

In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system.

The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics.

As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion.