The use of tensor notation to develop characteristic equations of supersonic flow

Abstract

The general equations of the steady motion of a non-viscous fluid are given in tensor notation. It is then assumed that one family of co-ordinate surfaces are characteristic surfaces, i.e., surfaces on which the transverse derivatives of the flow-variables are not determined by their values on the surface itself. The condition for this is given by the relation which can be interpreted to give the well-known result that the velocity normal to the surface is sonic. The relation which must then hold between the variables on the surface itself is also determined (characteristic equation). The special cases of axisymmetric and two-dimensional flow are also considered and the results interpreted to give the well-known relationships. As an example, the flow in a simple wave, i.e., a flow in which one fatuity of characteristic lines are straight, is treated in detail. While no new results have been obtained, the authors feel that the extra simplicity resulting from the use of quite general co-ordinates gives a deeper insight into the behaviour of such flows.