Page 294 - Aerodynamics for Engineering Students

276 Aerodynamics for Engineering Students

The equations of conservation and state for quasi-onedimensional, adiabatic flow
in differential form become

o
-
-- d(pA) (for conservation of mass)
dx
where u is the streamwise, and only non-negligible, velocity component.

dP
d(P$A) + A- dx = 0 (for momentum)
dx

(for the equation of state)

Expanding Eqn (6.3) and rearranging,
dp du dA
-+-+-=o
P u A
Similarly, for Eqn (6.6)

dp dp dT -0
P P T

From Eqn (6.4), using eqn (6.3)
puAdu+Adp=O
which, on dividing through by dA and using the identity M2 = u2/a2 = pu2/(yp),*
using Eqn (1.6d) for the speed of sound in isentropic flow becomes

(6.10)

Likewise the energy Eqn (6.5), with cpT = d/(-y - 1) found by combining Eqns (1.15)
and (1.6d), becomes
- -(y - l)M 2dU ; (6.11)
dT
=
T
Then combining Eqns (6.7) and (6.8) to eliminate dp/p and substituting for dp/p and
d T/ T gives
du dA
(M -1)-=- (6.12)
u A
* M is the symbol for the Mach number, that is defined as the ratio of the flow speed to the speed of sound
at a point in a fluid flow and is named after the Austrian physicist Ernst Mach. The Mach number of an
aeroplane in flight is the ratio of the flight speed to the speed of sound in the surrounding atmosphere (see
also Section 1.4.2).