Hydraulic Structures: Fourth Edition

HYDRAULIC MODELS 679

withYgat the condition

Mg

or

  1 (16.3a–d)

i.e. the condition that the scales of the Strouhal (Sh), Froude (Fr), Euler
(Eu) and Reynolds (Re) numbers must be 1. The dimensionless numbers
thus derived can, however, be criteria of similarity only if the initial equa-
tions have an unambiguous solution. This can only be attained if the equa-
tions are limited by certain boundary conditions which assume the
character of conditions of unambiguity of the solution.
An inspection of equations (16.3a–d) shows that with Mg1 and
using the same liquid for the model as in prototype (M Mμ1) they can
be satisfied only if Ml1, i.e. in a model the same size as the prototype.
Therefore we have to design and operate our models almost always with
approximate mechanical similarity, choosing a dominant force component
(e.g. gravity) and neglecting the effects of the others. To minimize the
resulting scale effects we have to impose limiting (boundary) conditions on
our scales, e.g. choose a scale where in the model flow the effect of viscos-
ity will be negligible.
Taking gravity as our dominant force (correct in most models of
hydraulic structures) results from equations (16.3) with Mg1in

MFr1. (16.4)

The same result can be obtained form equation (16.1) by writing

MaMmMgM M^3 lM M^2 lM^2 .

Equation (16.4) represents the Froude law of similarity. From it we can
obtain all the other required scale factors expressed in terms of the length
scaleMl(Table 16.1).

(b) Scale models of hydraulic structures

In the vast majority of cases, design problems associated with hydraulic
structures as described in the previous chapters are investigated on geo-
metrically similar scale models, operated according to the Froude law of

M



M1/2l

M



(MgMl)1/2

Mμ



M MMl

Mp



M M^2 

MgMl



M^2 

Ml



MMt

MμM



M M^2 l

Mp



M Ml

M^2 



Ml

M



Mt