Dynamical similarity of ideal gas flows

The non-linear nature of Euler’s equation may cause serious mathematical difficulties when solving a problem on the flow of ideal gas. Numerical methods may occasionally be applicable, but sometimes experimental investigation is the only way to determine the distribution of the flow variables. In this case it is very important to know whether the results measured with a particular experimental setup are applicable to another, similar, situation. In general it is assumed that the flow boundaries are geometrically similar for the two processes to be compared. This geometrical similarity is a necessary but not a sufficient condition for the complete similarity of the two flows investigated. It is obvious that an aerofoil of a compressor blade should have the same profile as its counterpart tested in a wind tunnel. The idea of dynamical similarity is originally due to Stokes. Dynamical similarity implies identical solutions of the differential equations for the two cases. The equations must therefore, be identical; the only difference may be a constant multiplier. Dynamical similarity requires that the flow variables of the two cases

should be related by the equations

It also follows that, if R or K is the same, the equation of state is automatically the same in the two cases. For isothermal flow this condition is always satisfied, but for isentropic flow slight differences may occur. If the flow is polytropic the difference can be larger and it is necessary to carry out a more detailed analysis.