Oblique Expansion Waves

Another class of two dimensional waves occurring in supersonic flow shows the opposite effects of oblique shock. Such types of waves are known as expansion waves . When the supersonic flow is “turned away from itself”, an expansion wave is formed as shown in Fig. 4.5.5(a). Here, the flow is allowed to pass over a surface which is inclined at an angle to the horizontal and all the flow streamlines are deflected downwards. The change in flow direction takes place across an expansion fan centered at point ‘A'. The flow streamlines are smoothly curved till the downstream flow becomes parallel to the wall surface behind the point ‘A'. Here, the flow properties change smoothly through the expansion fan except at point ‘A'. An infinitely strong oblique expansion wave may be called as a Mach wave . An expansion wave emanating from a sharp convex corner is known as a centered expansion which is commonly known as Prandtl-Meyer expansion wave . Few features of PM expansion waves are as follows;

• •  Streamlines through the expansion wave are smooth curved lines.

•  The expansion of the flow takes place though an infinite number of Mach waves emitting from the center ‘A'. It is bounded by forward and rearward Mach lines as shown in Fig. 4.5.5(b). These Mach lines are defined by Mach angles i.e.

(4.5.11)

• •  The expansion takes place through a continuous succession of Mach waves such that there is no change in entropy for each Mach wave. Thus, the expansion process is treated as isentropic.

•  The Mach number increases while the static properties such as pressure, temperature and density decrease during the expansion process.

Fig. 4.5.5: Schematic representation of an expansion fan.

The quantitative analysis of expansion fan involves the determination of for the given upstream conditions of . Consider the infinitesimal changes across a very weak wave (Mach wave) as shown in Fig. 4.5.6.

Fig. 4.5.6: Infinitesimal change across a Mach wave.