Different characteristic lengths were chosen for the horizontal and vertical
directions to accommodate geometrically distorted hydrodynamic models. For
convenience the same characteristic velocity was used to nondimensionalize the
radial, tangential, and vertical velocity components.
Making these substitutions into Equations 18 through 21, multiplying the
2
continuity equation by X/V, the horizontal momemtum equations by X/V , and
2
the vertical momentum equation by Z/V yields the nondimensional equations of
motion in cylindrical coordinates.
continuity (nondimensional)
∂vr vr 1 ∂vθ X ∂vz
^
^
^
^
++
+
=0
(23)
r r ∂θ Z ∂z
∂r
^
^^
^
x-direction momentum (nondimensional)
P ∂p
∂vr vθ ∂vr vθ 2 X ∂vr
∂vr
^ ^
^
^
^
^
^
X
+ vr
+
-
+ vz
= -
^
^
2
^
(24)
r ∂θ
ρV ∂r
∂t
∂r
r Z ∂z
^
^
^
^
^
VT
convective accelerations
θ-direction momentum (nondimensional)
P 1 ∂p
X ∂vθ
∂v
v ∂v
∂vθ
^
^
^^
^^
^
^
vv X
+ vr θ + θ θ + r θ +
= -
^
^
vz
(25)
r ∂θ
ρV r ∂θ
VT ∂t^
∂r
∂z
2
^
^
^
^
^
r
Z
convective accelerations
z-direction momentum (nondimensional)
P ∂p gZ
Z ∂vz Z ∂vz Z vθ ∂vz
∂v
^ ^
^
^
^
^
+ vr
+
+ vz z = -
+
^
^
2
^
(26)
VT ∂t^ X ∂r X r ∂θ
ρV
∂z
∂z V 2
^
^
^
Just as before, if two systems are governed by the 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 cylindrical form of the Navier-Stokes equations if
the value of each dimensionless coefficient in Equations 23 through 26 remains
constant between prototype and model. Note that all nondimensional terms
without coefficients will be in similitude.
For steady flow, the only potential scale effects arise from the convective
accelerations, and the only dimensionless coefficients are (X/Z) and its inverse.
Therefore, the requirement for similitude of the convective accelerations is
simply:
34
Chapter 4 Turbulence Scale Effect in Distorted Models
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