z-direction momentum
∂vz
∂v
v ∂v
∂v
1 ∂p
+ vr z + θ z + vz z = -
+
(21)
ρ ∂z gravity
r ∂θ
∂t
∂r
∂z
g
acceleration terms
pressure term
where
t = time
r = horizontal cross-channel (radial) coordinate
θ = horizontal along-channel (angular) coordinate
z = vertical coordinate
vr = nonturbulent horizontal cross-channel component of velocity in the r
direction
vθ = nonturbulent horizontal along-channel component of velocity in the θ
direction
vz = nonturbulent vertical component of velocity in the z direction
p = fluid pressure
ρ = fluid density
Appropriate scaling criteria are derived from the equations of motion by
expressing the equations in nondimensional form using the following definitions
and substituting for the independent and dependent variables:
vθ
vr
vz
^ t
vr =
vθ =
vz =
t=
^
^
^
;
;
;
V
V
V
T
(4-22)
r
z
p
r= ;
z=
p=
^
^
^
X
Z
P
where
V = characteristic nonturbulent horizontal velocity
X = characteristic horizontal length
Z = characteristic vertical length
T = characteristic time
P = characteristic pressure
33
Chapter 4 Turbulence Scale Effect in Distorted Models
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