the ratio of corresponding quantities in the two systems can be defined. The subscript r is

used to designate the ratio of the model quantity to the prototype quantity. For example, the

length ratio is given by:

Lr = xm / xp = ym / yp = zm / zp

(5.18)

for the coordinate directions, x, y, and z. Equation 5.18 assumes a condition of exact

geometric similarity in all coordinate directions.

Frequently, open channel models are distorted. A model is said to be distorted if there are

variables that have the same dimension but are modeled by different scale ratios. Thus,

geometrically distorted models can have different scales in horizontal (x, y) and vertical (z)

directions, and two equations are necessary to define the length ratios in this case.

Lr = xm / xp = ym / yp

(5.19)

and

zr = zm / zp

(5.20)

If perfect similitude is to be obtained the relationships that must exist between the properties of

the fluids used in the model and in the prototype are given in Table 5.7 for the Froude (gravity),

Reynolds (viscosity), and Weber (surface tension) criteria. The use of this table is presented in

Section 5.9, Problems 5 and 6.

In free surface flow, the length ratio is often selected arbitrarily, but with certain limitations kept

in mind. The Froude number is used as a scaling criterion because gravity has a predominant

effect. However, if a small length ratio is used (very shallow water depths) then surface tension

forces, which are included in the Weber number ( V / σ / ρL ) , may become important and

complicate the interpretations of results of the model. It is desirable that the length scale be

made as large as possible so that the Reynolds number is sufficiently large that friction

becomes a function of the boundary roughness (and essentially independent of the Reynolds

number). A large length scale also ensures that the flow is as turbulent in the model as it is in

the prototype.

The boundary roughness is characterized by Manning's roughness coefficient, n, in free

surface flow. Analysis of Manning's equation and substitution of the appropriate length ratios,

based upon the Froude criterion, results in an expression for the ratio of the roughness which is

given by

1/6

nr = Lr

(5.21)

It is not always possible to achieve boundary roughness in a model and prototype that

corresponds to that required by Equation 5.21 and additional measures such as adjustment

of the slope, may be necessary to offset disproportionately high resistance in the model.

5.41

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