y-direction momentum (nondimensional)
P ∂p
∂v
∂v
∂v XW ∂v
^
^
^
^
^
X
+u +v +
w = -
^
^
^
^
ρV 2 ∂y
∂t
∂x
∂y ZV ∂z
^
^
^
^
VT
v ∂ 2v v ∂2v vX 2v
^
^
^
+
+
+ 2 2
^2
^2
(8)
XV ∂x XV ∂y Z V ∂z
^
()
∂
∂
∂
()
(
)
XW
v '2 +
-
u 'v ' +
^^
^
^^
v 'w'
∂z
∂x
∂y
^
^
^ ∂
ZV
turbulence terms
z-direction momentum (nondimensional)
P ∂p
∂w
∂w
Z ∂w ZV
∂w
^
^
^
^
^
WT ∂t + XW
+v
+w
= -
^
^
^
u
2
∂x
^
^
ρW ∂z
∂y
∂z
^
^
^
gZ vZ ∂ w vZ ∂ w v ∂ w
2
2
2
^
^
^
- 2 + 2 2 + 2 2 +
^2
(9)
W X V ∂x
X W ∂y
ZW ∂z
^
^
()
ZV
∂
∂
∂
(
)
(
)
w '2
-
u 'w' +
v 'w' +
^^
^^
^
^
∂x
∂y
∂z
^
^
XW
turbulence terms
If two systems are governed by the previous nondimensional equations, then
the solution in terms of the nondimensional parameters will be the same for each
system provided all dimensionless coefficients remain unchanged. This means
complete similitude would be achieved for any free surface hydrodynamic
phenomena governed by the formulation of the Navier-Stokes equations if the
value of each dimensionless coefficient in Equations 6 through 9 remains
constant between prototype and model. Note that all nondimensional terms
without coefficients will be in similitude.
Focusing attention on those groupings labeled "turbulence terms," the only
dimensionless coefficients are (XW/ZV) and its inverse. Therefore, the
requirement for similitude of the differential turbulence Reynolds stress terms
having this coefficient is simply
XW
XW
=
ZV
(10)
prototype ZV model
or rearranged...
X p Wp Z p Vp
=
(11)
X m Wm Zm Vm
28
Chapter 4 Turbulence Scale Effect in Distorted Models
Previous Page
