Similarity Rule

From Section 9.8, it is seen that the governing equation for compressible flow is elliptic for subsonic flows (that is, for Mx < 1) and becomes hyperbolic for supersonic flows (that is, for > 1). This change in the nature of the partial differential equation, upon going from subsonic to supersonic flow, indicates the possibility of deriving similarity relationships between subsonic compressible flow and the corresponding incompressible flow, and the importance of Mach wave in a supersonic solution. In this chapter we shall derive an expression which relates the subsonic compressible flow past a certain profile to the incompressible flow past a second profile derived from the first principles through an affine transformation. Such an expression is called a similarity law.

If the governing equations of motion could be solved easily, the solution themselves would indicate quite clearly the nature of any similarities which might exist among members of a family of flow patterns. Then there is no need for a separate derivation of similarity laws.

But in the majority of situations, we are unable to solve the equations of motion. However, even though solutions are lacking, we may use our knowledge of the forms of the differential equations and the related boundary conditions to derive the similarity laws.