Page 84 - Mechanical Engineers' Handbook (Volume 4)

10 Gas Dynamics 73

Figure 25 Gas flow through converging–diverging nozzle.

A, the exit pressure is also, and sonic flow exists at the throat, but is subsonic elsewhere.
Only at B is there sonic flow in the throat with isentropic expansion in the diverging part of
the nozzle. The flow rate is the same whether the exit pressure is at A or B. Receiver pressures
below B do not affect the flow in the nozzle. Below A (at C, for example) a shock forms as
shown and then the flow is isentropic to the shock, and beyond it, but not through it. When
the throat flow is sonic, the mass flow rate is given by the same equation as for a converging
nozzle with sonic exit flow. The pressures at A and B in terms of the reservoir pressure
p are given in isentropic flow tables as a function of the ratio of exit area to throat area,
0
A /A*.
c
10.3 Normal Shocks
The plane of a normal shock is at right angles to the flow streamlines. These shocks may
occur in the diverging part of a nozzle, the diffuser of a supersonic wind tunnel, in pipes
and forward of blunt-nosed bodies. In all instances the flow is supersonic upstream and
subsonic downstream of the shock. Flow through a shock is not isentropic, although nearly
so for very weak shocks. The abrupt changes in gas density across a shock allow for optical
detection. The interferometer responds to density changes, the Schlieren method to density
gradients, and the spark shadowgraph to the rate of change of density gradient. Density ratios
across normal shocks in air are 2 at M 1.58, 3 at M 2.24, and 4 at M 3.16 to a
maximum value of 6.
Changes in fluid and flow parameters across normal shocks are obtained from the con-
tinuity, energy, and momentum equations for adiabatic flow. They are expressed in terms of
upstream Mach numbers with upstream conditions designated with subscript x and down-
stream with subscript y. Mach numbers M and M are related by
y
x
M 1 kM 2 x M 1/2 M 1 kM 2 y M 1 /2 ƒ(M,k)
k 1
k 1
1
2
2
x
2 x y 1 2 y
which is plotted in Fig. 26. The requirement for an entropy increase through the shock
indicates M to be greater than M . Thus, the higher the upstream Mach number, the lower
y
x
the downstream Mach number, and vice versa. For normal shocks, values of downstream
Mach number M ; temperature ratios T /T ; pressure ratios p /p , p /p , and p /p ; and
0y
x
y
0y
x
y
y
0x
x
density ratios / depend only on the upstream Mach number M and the specific heat
x
x
y
ratio k of the gas. These values are tabulated in books on gas dynamics and in books of gas
tables.
The density ratio across the shock is given by the Rankine–Hugoniot equation