# 12.3.1: Introduction

MULTILINE START acquiringCaption=0 differenceBraces=1 tmpCaptionString=The definition of the angle for the Prandtl–Meyer function.

*Fig. 12.21 The definition of the angle for the Prandtl–Meyer function.*

As discussed in Section 12.2 when the deflection turns to the opposite direction of the flow, the flow accelerates to match the boundary condition. The transition, as opposed to the oblique shock, is smooth, without any jump in properties. Here because of the tradition, the deflection angle is denoted as a positive when it is away from the flow (see Figure 12.21). In a somewhat a similar concept to oblique shock there exists a "detachment'' point above which this model breaks and another model has to be implemented. Yet, when this model breaks down, the flow becomes complicated, flow separation occurs, and no known simple model can describe the situation. As opposed to the oblique shock, there is no limitation for the Prandtl–Meyer function to approach zero. Yet, for very small angles, because of imperfections of the wall and the boundary layer, it has to be assumed to be insignificant.

*Fig. 12.22 The angles of the Mach line triangle.*

Supersonic expansion and isentropic compression (Prandtl-Meyer function), are an extension of the Mach line concept. The Mach line shows that a disturbance in a field of supersonic flow moves in an angle of \(\mu\), which is defined as (as shown in Figure 12.22)

\[

\mu = \sin^{-1} \left( \dfrac{ 1 }{ M} \right)

\label{pm:eq:mnu1} \tag{1}

\]

or

\[

\mu = \tan ^{-1} \dfrac{1 }{ \sqrt{M^1 -1}}

\label{pm:eq:mu2} \tag{2}

\]

A Mach line results because of a small disturbance in the wall contour. This Mach line is assumed to be a result of the positive angle. The reason that a ``negative'' angle is not applicable is that the coalescing of the small Mach wave which results in a shock wave. However, no shock is created from many small positive angles. The Mach line is the chief line in the analysis because of the wall contour shape information propagates along this line. Once the contour is changed, the flow direction will change to fit the wall. This direction change results in a change of the flow properties, and it is assumed here to be isotropic for a positive angle. This assumption, as it turns out, is close to reality. In this chapter, a discussion on the relationship between the flow properties and the flow direction is presented.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.