Mechanics of Fluids

the external pressure. This compression involves oblique shock waves and
the subsequent events cannot be described in one-dimensional terms.
At the design conditions p2 = pG and the pressure in the exit plane of
the nozzle is the same as the external pressure. When the external pressure
is below pG the nozzle is said to be under-expanding and the expansion
to pressure p2 must be completed outside the nozzle. The curve CG is
followed and the exit velocity is that corresponding to pG; the additional
expansion then takes place through oblique expansion (Mach) waves. Here,
too, a one-dimensional description of events is not possible. An extreme
case of an under-expanding nozzle is one having no diverging portion
at all.
The foregoing is illustrated by the photographs of Fig. 11.25 which were
obtained by the use of the schlieren technique (see Section 11.13). The theory
agrees reasonably well with measurements of pressure within the nozzle if
the external pressure is only a little lower than pF. However, separation
of the boundary layer from the walls is greatly encouraged by the abrupt
pressure increase across a shock wave and, with lower values of the external
pressure, the normal shock wave may not fill the entire cross-section. Also,
oblique shock waves may be formed close to the wall.
Example 11.5 A convergent-divergent nozzle designed to give an exit
Mach number of 1.8 when used with helium (γ = 5/3) is used with
air (γ = 1.4) under conditions that produce a normal shock just inside
the nozzle. Determine the Mach number just before the shock and thus
the stagnation pressure at inlet if the absolute pressure beyond the exit
is 30 kPa.
Solution
What remains constant when the gas is changed is the geometry of the
nozzle. From eqn 11.59
Mass flow rate = ρAu = MA(pγρ) 1/2
� � � �
(γ +1)/(γ −1) 1/2
2
= At p0ρ0γ
γ + 1
� �2 A
∴ = p0ρ0γ
� � (γ +1)/(γ −1)
2
At
= 1
=
M 2 pγρ
M 2
� p0
p
γ + 1
� (γ +1)/γ �
2
γ + 1
�
γ −1
1 + 2 M2� (γ +1)/(γ −1)
M 2
� (γ +1)/(γ −1)
�
2
γ + 1
� (γ +1)/(γ −1)
One-dimensional flow through nozzles 529