where the subscripts *p *and *m *represent prototype and model, respectively. The

requirement given by Equation 10 can be expressed in terms of scale ratios as:

= V

(12)

where, by definition, a scale ratio is the ratio of a parameter in the prototype to

the value of the same parameter in the model.

Because the scaling ratio for horizontal and vertical velocity should be the

same, the scaling requirement of Equation 4-12 can only be fulfilled when *N*X =

Therefore, turbulent Reynolds stress terms containing this dimensionless

coefficient will not be in similitude in a geometrically distorted model, and this

introduces a scale effect. From Equations 4-6 through 4-9 the following can be

concluded:

turbulence terms in similitude

() ( ) ( ) () ()

∂

∂

∂

∂

∂

(13)

∂*x*

∂*y*

∂*x*

∂*y*

∂*z*

turbulence terms not in similitude

∂

∂

∂

∂

(

)

(

)

(

)

(

)

(14)

∂*x*

∂*y*

∂*z*

∂*z*

The four turbulence terms involving squares or cross products of the horizontal

turbulent velocity fluctuations and the term containing the square of the vertical

turbulent velocity fluctuation are in similitude in distorted models, whereas the

four Reynolds stress terms containing the cross-product of horizontal and vertical

velocities do not fulfill the requirement and represent the potential scale effects.

The two nonsimilar turbulent Reynolds stress terms contained in the

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

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

1

For example, the scale ratio of the characteristic horizontal length *X *is:

Value of X in prototype

Value of X in model

29

Chapter 4 Turbulence Scale Effect in Distorted Models