Hypersonic expansion wave relations

Consider the flow through an expansion corner as shown in Fig. 4.7.1(b). The expansion fan consists of infinite number of Mach waves originating at the corner and spreading downstream. The notations have their usual meaning and upstream and downstream conditions are denoted by subscripts ‘1' and ‘2', respectively. Let us revisit the exact relations for a Prandtl-Meyer expansion. The relation for deflection angle is expressed through Prandtl-Meyer function .

(4.7.13)

For large Mach numbers, and series expansion can be approximated for the trigonometric functions.

(4.7.14)

Further, simplification of Eq (4.7.14) can be done and the final expression for θ may be written as below;

(4.7.15)

Hypersonic Similarity Parameter

In the study of hypersonic flow over slender bodies, the product of is a controlling parameter which is known as the similarity parameter denoted by K. All the hypersonic shock and expansion relations can be expressed in terms of this parameter. Introducing this parameter, Eq. (4.7.11) is rewritten in the limit of high values of Mach number;

(4.7.16)

Rearranging Eq. (4.7.16), one may obtain a quadratic equation in terms of , which may be easily solved.

(4.7.17)

Within the framework of hypersonic assumption, the hypersonic shock relation for pressure ratio (Eq. (4.7.1), may be reduced in terms of K by using Eq. (4.7.17).

(4.7.18)

Similarly, the pressure coefficient may also be expressed as a function of similarity parameter.

(4.7.19)