Page 290 - 48Fundamentals of Compressible Fluid Mechanics

252 CHAPTER 14. PRANDTL-MEYER FUNCTION

Maximum
turning
slip line

Fig. 14.5: Expansion of Prandtl-Meyer function when it exceeds the maximum angle

14.5 d’Alembert’s Paradox

In ideal inviscid incom-
pressible flow, movement
3
of body doesn’t encoder
1 2
any resistance. This results
4
is known as d’Alembert’s
Paradox and this paradox is
w θ 2
examined here. θ 1
Supposed that a two θ 1 θ 2
dimensional diamond shape 4
body is stationed in a su- 1 2
personic flow as shown in
3
Figure (14.7). Again it is
assumed that the fluid is in-
Fig. 14.7: A simplified Diamond Shape to illustrate the Su-
viscid. The net force in flow
personic d’Alembert’s Paradox
direction, the drag, is
(14.38)

It can be noticed that only the area “seems” by the flow was used in express-
ing equation (14.38). The relation between and is such that it depends on
the upstream Mach number, and the specific heat, . Regardless, to equation

of state of the gas, the pressure at zone 2 is larger than the pressure at zone 4,
. Thus, there is always drag when the flow is supersonic which depends on the

upstream Mach number, , specific heat, and the “visible” area of the object.

This drag known in the literature as (shock) wave drag.