∂
∂
∂
∂
(
)
(
)
(
)
(
)
Ω
=
^^
^^
^^
^^
u 'w' ;
v 'w'
u 'w' ;
v 'w'
(15)
∂z
∂z
proto ∂z
∂z
^
^
^
^
model
where Ω = NX /NZ. This implies that the vertical variation of the turbulent
velocity products u′w′ and v′w′ is greater in the distorted model than in the
prototype.
Conversely, the two nonsimilar turbulent Reynolds stress terms contained in
the vertical momentum equation are smaller in the model than they should be by
a factor equal to the inverse of the geometric distortion, i.e.,
1∂
∂
∂
∂
(
)
(
)
(
)
(
)
=
^^
^^
^^
^^
u 'w' ;
v 'w'
u 'w' ;
v 'w'
^
(16)
Ω ∂x
∂y
proto ∂x
∂y
^
^
^
model
-
In this case the horizontal variation of the turbulent velocity products ′w′ and
v′w′ is less in the distorted model than in the prototype.
Anticipated Scale Effects
Several generalizations can be made regarding anticipated turbulent scale
effects in geometrically distorted models by assuming the turbulence is
homogeneous in the horizontal and vertical directions. As mentioned, turbulence
generated by flow interaction with solid boundaries will be in similitude in a
geometrically undistorted physical model.
As an example, consider a fluid jet exiting a circular orifice into an ambient
fluid. The turbulent jet spreads out uniformly in both horizontal and vertical
directions with distance from the orifice. An undistorted model of this jet would
also have a circular orifice and would also spread out uniformly in the
downstream direction. It should be expected that time-averaged velocity
measurements taken at any location in the model turbulent jet and scaled using
the Froude velocity scale would be the same as averaged velocities measured at
the corresponding location in the prototype.
The same circular orifice would be represented in a distorted model as an
oval with its major axis aligned vertically. As the jet exits the orifice, it will form
an oval-shaped turbulent jet. The jet will expand, and anticipate its expansion to
be the same in the vertical plane as in the horizontal plane as mass is entrained
into the jet. This means the lateral vertical and horizontal entrainment velocity is
the same in the model. However, the distorted model implies that vertical and
horizontal velocity scales are different. Therefore, when the time-averaged
velocities are scaled to prototype and compared to averages at the corresponding
location, expect to see a difference due to dissimilar lateral entrainment velocity.
The magnitude of the error would not be too significant because the principle
flow direction is horizontal. The scaled-up cross-jet velocity profiles would be
similar to the prototype in magnitude, but we should expect some error in the
profile shape.
30
Chapter 4 Turbulence Scale Effect in Distorted Models
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