*X*

*X*

=

*Z*

(27)

prototype *Z *model

or expressed in terms of scale ratios

(28)

which means a geometrically undistorted model. From Equations 23 through 26

the following can be concluded:

convective acceleration terms in similitude

2

∂*v v *∂*v v v*

∂*v*r vθ ∂*v*r vθ

∂*v*

; ; *v*r θ ; θ θ ; r θ ; *v*z z

;

(29)

∂*r r *∂θ r

∂*r r *∂θ

∂*z*

convective acceleration terms not in similitude

∂*v*

∂*v*r

∂*v v *∂*v*

; *v*z θ ; *v*r z ; θ z

(30)

∂*r r *∂θ

∂*z*

∂*z*

The two nonsimilar convective accelerations contained in the horizontal

direction momentum equations are larger in the model than they should be by a

factor equal to the geometric distortion, i.e.,

∂*v *

∂*v *

∂*v*

∂*v*

^

^

^

^

Ω *v*z r ; *v*z θ

= *v*z r ; *v*z θ

^

^

^

^

(31)

∂*z*

∂*z *proto ∂*z*

∂*z *model

^

^

^

^

vertical gradients of the radial and tangential velocities (*v*r , *v*θ ) are greater in the

distorted model than in the prototype.

Conversely, the two nonsimilar convective accelerations 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 ∂*v*z vθ ∂*v*z

∂*v v *∂*v *

^ ^ ^

^ ^ ^

= *v*r z ; θ z

^

^

;

(32)

Ω ∂*r r *∂θ^ proto ∂*r r *∂θ^ model

^^

^^

35

Chapter 4 Turbulence Scale Effect in Distorted Models

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