u
v
w
u=
v=
w=
^
^
^
;
;
V
V
W
x
y
z
x= ;
y= ;
z=
^
^
^
(5)
X
X
Z
^ t
p
t= ;
p=
^
T
P
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 accommodate geometrically distorted hydrodynamic
models. For convenience, the same characteristic velocities were used to
nondimensionalize the turbulent Reynolds stresses.
Making these substitutions into Equations 1 through 5, multiplying the
2
continuity equation by X/V, the horizontal momentum equations by X/V , and the
2
vertical momentum equation by Z/W yields the nondimensional equations of
motion.
continuity (nondimensional)
∂u ∂v XW ∂w
^
^
^
+
+
=0
∂x ∂y ZV ∂z
(6)
^
^
^
x-direction momentum (nondimensional)
P ∂p
∂u
∂u
∂u XW
∂u
^
^
^
^
^
X
+u
+v
^ +
= -
^
^
w
2
^
ρV ∂x
∂t
∂x
∂y ZV
∂z
^
^
^
^
VT
v ∂2u
v ∂ 2u vX ∂ 2u
^
^
^
+
^2 +
+ 2 2
^
(7)
XV ∂y 2 Z V ∂z
XV ∂x
^
() ()
∂
∂
∂
(
)
XW
u '2 +
-
u 'v ' +
^
^^
u' ^ '
^
∂z
∂x
∂y
^
^
^
ZV
w
turbulence terms
27
Chapter 4 Turbulence Scale Effect in Distorted Models
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