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2.4 Reduced Forms of the Navier-Stokes Equations 61

which provides a good approximation to some real incompressible flows at high
Reynolds numbers where the viscous effects are negligible, as is sometimes the
case when there is no flow separation on the body, as we shall discuss in Chap-
ter 6.
The velocity potential also exists for compressible flows; in this case, however,
in addition to the requirements imposed by Eqs. (2.4.18) and (2.4.19), there is
an additional requirement on density in the continuity equation since density
is not constant but varies. Using the irrotationality condition, which implies
isentropic flow, density can be expressed as a function of temperature alone
1
Q = const.T ^ 7 - 1 ) (2.4.21)
and with the definition of speed of sound a (= y/jRT), it can be written as

2
7 1
Q = const.a ^ - ) (2.4.22)
substituting Eq. (2.4.22) into the continuity equation (2.2.12b), and expanding
the resulting equation, the continuity equation becomes

z
z
_?_ f du dv\ u(a )^- 1 d , 9x v(a )^~ l d , 9x /r> 4 _ ,

for a two-dimensional steady flow. Assume adiabatic flow so that total enthalpy
is constant.
2
a 2 + ^-—- (u 2 + v ) = const (2.4.24)
and substitute Eq. (2.4.24) into Eq. (2.4.23). After simplification,
o f du dv\ f du dv\ f du dv\ ^ . A , .
U
+
U
° U? + dy) ~ [^ +V 8- )~ V [ 0-y ^y) = ° (2A25)
X
With the definition of velocity potential, Eq. (2.4.19), Eq. (2.4.25) can be written
as
2
2
2
2
d (\) d (f) _ 1 d(j)\ 2 d^ fdf\ ^ dcf)d(f) d <t>
(2.4.26a)
2
2
2
2
2
dx dy a dx J dx \dy J dy dx dy dxdy
or as
2 2 2 2 2
u * d (j) I v \ d (j) uv d
2
I " a J dx 2 ' 1 a 2 I dy 2 a 2 dxdy
Equation (2.4.26) is known as the full potential equation for a two-dimensional
steady flow with no body force. Unlike the Laplace equation to which it re-
duces as a —> oc (incompressible flow), it is nonlinear. Assuming that, with
Moo = VWttoo,
^ ( T M <<X 1 , M < ( T M <1> ^ ( ^ ) « 1 (2.4.27a)
<
' ° ° V K J ' ~ U O J ^ ' *"°°Vv«