88
CHAPTER 4. HYDRODYNAMIC MODELS
where
V - characteristic nonturbulent horizontal velocity
W - characteristic nonturbulent vertical velocity
X - characteristic horizontal length
Z - characteristic vertical length
T - characteristic time
P - characteristic pressure
Different characteristic lengths and velocities were chosen for the horizontal and vertical directions to leave open the possibility of a geometrically
distorted hydrodynamic model. For convenience the same characteristic
velocities were used to nondimensionalize the turbulent Reynolds stresses.
Different characteristic velocities of the nonturbulent flow could be chosen
for the turbulence terms, but this would only result in the trival requirement
that the different characteristic velocities have the same scale.
Making these substitutions into Eqn. 4.2 and multiplying by X/V yields
the nondimensional continuity equation:
dû dû
tct +
+
dx dy
XVKX dw_ _
ZV J dè ~
(4.7)
Similarly, substitution of the nondimensional parameters into the equations
of motion and multiplying by X/V2 (horizontal direction equations) or
Z/W2 (vertical direction equation) gives
x-Direction
X X
vt)
dû
~dû „dû
~zv J
dû _
f R \
W dz
dx
d2û
+
v \ d2û
XV J d^2 +
vX X
z^v)
d2û
dz2
(4-8)
^(û'2) +
+
ox
dy
ZV J
d
dz
+
y-Direction
X \
~dû_ ^dû_ /~dv
( P \ dp
VT ) di U dx V dy+\ ZV ) W~dê ~ ~ y^V2 ) d^
\ d^û
/
d^û
/ vX X d2v
xv) dî2 + \xv) dy2 + yz^v) 'dè2^
(4-9)
d
dz