Mechanics of Fluids

Table 11.2 Changes with distance along the pipe of properties
during adiabatic flow in a pipe of constant cross section
Property Subsonic flow Supersonic flow
Mach number M Increases Decreases
Specific enthalpy h Decreases Increases
Velocity u Increases (!) Decreases
Density ρ Decreases Increases
Temperature T Decreases Increases
Pressure p Decreases Increases (!)
Re = ρud
µ = const.
µ
Stagnation temperature T0 Increases
Constant
Decreases
Constant
Compressible flow in pipes of constant cross-section 533
conditions at the outlet. An initially supersonic flow will also be adjusted to
give sonic conditions at exit: a normal shock will form near the end of the
pipe and the resulting subsonic flow will accelerate to sonic conditions at
the exit. Further increase of length would cause the shock to move towards
the inlet of the pipe and then into the nozzle producing the supersonic flow
so that the flow would become entirely subsonic in the pipe.
These conclusions are valid for all gases. To obtain explicit relations from
the integration of eqn 11.62, however, we assume the gas to be perfect. For
simplicity we also neglect the gravity term δz. Then in the limit as δl → 0
dp
ρ
f dl u
+ udu +
A/P
2
= 0 (11.67)
2
This equation can be integrated only when the number of variables is
reduced, and a solution is most conveniently obtained in terms of Mach
number. As the flow is not isentropic the temptation to use the relation
p/ρ γ = constant must, however, be resisted.
Differentiating the energy equation (11.13) gives cpdT + udu = 0and
division by u 2 = M 2 γ RT then yields
cp
γ RM2 dT du
+ = 0 (11.68)
T u
Also, differentiation of the expression u 2 = M 2 γ RT yields
2 du
u
= 2dM
M
+ dT
T
(11.69)
Eliminating dT/T from eqns 11.68 and 11.69 and noting that cp/R =
γ /(γ − 1) we then obtain
du
u =
dM/M
1
2 (γ − 1)M2 + 1
(11.70)