682 MODELS IN HYDRAULIC ENGINEERING
(c) River and coastal engineering models
From Bernoulli’s equation using the same procedure as used for the
derivation of equation (16.3), we can obtain
MzMhM^2 MM^2 MlM^ R^1 MM^2 (16.6a–d)
whereMz,Mh,Mare scale ratios for height above datum, depth and local
loss coefficient. Equation (16.6a) indicates that for open-channel models with
non-uniform flow the height and depth scales must be identical, and a tilting
of the model about one of its ends is permissible for uniform flow conditions
only. As we frequently require a different vertical and horizontal (length and
width) scale to achieve a sufficiently high Reynolds number on the model,
and to ensure a fully turbulent flow régime, i.e. MhMl, the discharge scale
from equation (16.6b) (which again represents the Froude law) will be
MQMAMMh3/2Ml. (16.7)
Equation (16.6c) results in
MRMlM (16.8)
and equation (16.6d) in
M1, (16.9)
i.e. local loss coefficients should be the same on the model as in prototype.
This last condition is practically impossible to achieve in distorted river
models for every local loss but can be achieved overall, taking all local
losses (at changes of section and direction) together.
The above equations contain seven variables (Mz,Mh,Ml,M,M,
MR,M); furthermore, MRmust be a function of MhandMl, and MofMR
andMk, where Mkis the roughness size scale:
MRf 1 (Mh,Ml), (16.10)
Mf 2 (MR,Mk). (16.11)
Thus we have the six equations (16.6a), (16.6b), (16.8), (16.9), (16.10) and
(16.11) for eight variables, giving two degrees of freedom. In the design of
the model we can therefore choose only two variables, usually MlandMh
(orMQ).
In the case of a movable bed model, in the first approach we can sub-
stitute for Mkthe variable Md, i.e. the sediment size scale, and if we want
to achieve similarity of incipient sediment motion from the Shields crite-
