A Survey of Unsteady Hypersonic Flow Problems

- 18 -
Then the transformed velocity cconponent u' will be of the same
order of magnitude as the components v and w, and gradients of flow
quantities in terms of x1 will be of the same order of magnitude as those
in the y and z directions. Substituting these transformed quantities into
the equations of continuity, momentum and entropy (with isentropic conditions
along the streamlines) we obtain
and
aP ap ah-4 ah)-+uus+-+-
at a9 ay a2
auf auf au* au* ap
- + us-++-++-+at
axI ay a2 pa+
av av av av 1 ap
- + us -+v-- +w--+-at
a9 ay as P ay
aw an an aw 1 ap
- + us-+v- +w-+-dt
axI ay az P a=
as as as as
- + U6 ---4-v-- +w-at
ad ay az
a(puf)
= -p t . . . (2.8)
ad
au'
= -6%'-, . . . (2.9)
ax*
av
= #,f -
axa 3
. . . (2.10)
an
= -6% -3 . . . (2.11)
a9
as
= #U - . 9.. (2.12)
ai+
where S is the entropy, and is given by S = C, log(p/py) + constant, where
CV is the specific heat at constant volume. If the right-hand sides of these
equations are neglected as they are of second order of smallness, the equations
for v, w, p and p are decoupled from that for u'. The significance of this
is seen more clearly if the equations are now transformed to axes fixed in the
fluid, for if
and
F=t.
2 = x' - mt-3
a a
-=ax'
a2
then equations (2.8) to (2.12) beome:
a a a '
- = -- us -
at a? a2 I
ap abd ah-d
-+-+- =G 0
a% ay az
ih* tit au* 1 ap
-+v-+w-+-- = 0
aZ a;P az p aF
. . . (2.13)
. . . (2.G)
. . . (2.15)
. . . (2.16)