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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