Theory of Hydraulic Models

The realm of mechanics consists of a variety of concepts such as energy, force, velocity, density, and so on. In the present book, the word ‘mechanics’ refers to classical or non-relativistic mechanics. No limit can be imposed upon the nature and the number of concepts; the progress of science means the birth of new ideas: the introduction of new concepts. On the other hand, each one of this unlimited number of concepts can be defined by means of only three independent entities—length, time and mass—referred to as fundamental entities. It is interesting to note that these three fundamental entities cannot themselves be defined. Nothing in the physical or external world is more obvious to us than what is implied by near and far, earlier and later, and lighter and heavier. We learn these notions, without being taught, simply by living in this world. A concept can be defined only by means of more familiar concepts, yet the forementioned terms remain obvious to us. Any entity that can be measured and expressed in numbers is a quantity. It follows that various mechanical quantities can be regarded merely as compositions of the same three measurable entities: length, time and mass. Let L, T and M be the units for length, time and mass. Since a mechanical quantity a can be considered as a composition of length, time and mass, the unit of a, denoted by [a], must be a function of the fundamental units, i.e.

Since the numerical values of the dimensionless quantities remain the same in all systems of fundamental units, the numerical values of the dimensionless quantities X _j and Π _A , encountered in the preceding chapter, will not change if the system of fundamental units L″, T″ and M″ by which they are expressed is replaced by a different system of units L″, T″ and M″. Two systems of fundamental units can always be related to each other by the ‘transformation formulae’ where the proportionality factors λ _L , λ _T and λ _M are some arbitrary constants. Accordingly, the dimensionless numbers in general (and X _j and Π _A in particular) can be regarded as the quantities which remain invariant with respect to the transformation (2.1).

The stationary flows which belong to this category are essentially those shown schematically in Fig. 1.5a to 1.5h of the first chapter. These flows are either limited by rigid flow boundaries or they are supposed to extend to infinity. We have seen that any dimensionless property Π _A of the flows shown in Fig. 1.5a to 1.5f can be expressed as follows: where ∏_A and Re are and while Γ is the ratio of two lengths

So far as the dimensional nature of characteristic parameters and the meaning of dimensionless variables is concerned, filtration flows are much the same as those forming the subject of the previous chapter. However, the physical picture of filtration flows and the fields of their application are so different from the ‘free’ flows that they are usually studied separately. In accordance with this convention it is intended to analyse filtration flows by introducing this separate chapter. When referring to filtration we refer to the motion of so-called gravitational liquid, continuously occupying the pores of the granular (or porous) medium. Unlike the ‘hygroscopic water’ or ‘pellicular water’, gravitational water (or, in general, fluid) is not subjected to the action of tension forces directed towards the surface of solid particles. Gravitational fluid, which forms the subject of filtration, moves, under natural conditions, because of gravity forces alone. (See the classification and the description of various components of water in soil.^1,2,3,4)

Model tests are usually carried out if the properties involved in the design of a hydraulic project cannot be determined theoretically, or if such a determination cannot be regarded as reliable. Accordingly, with the exception of some investigations mainly for scientific rather than technical purposes, one cannot expect the subject of a model test to be, say, a uniform flow in a regular prismatic channel. Consequently, one must be prepared to face the presence of irregular flow boundaries, and of a complicated, often three-dimensional, structure of the flow. Considering this, scale relations are developed in the following study for the most general case of river and channel flow—non-uniform flow in a non-prismatic channel. The formulation of simpler (special) cases, such as prismatic channels, uniform flows, etc. is then automatically included in the general forms. First, steady state or stationary flows are considered and subsequently the method is generalized to non-steady flows. Since almost all the flows in civil engineering practice are turbulent, the consideration of laminar flows is omitted. In the present chapter, the presence of a rigid bed is assumed, the consideration of a mobile bed (consisting of erodible granular material) being given in Chapter 6. The term rigid bed should not necessarily be understood in the literal sense of the word rigid. If the flow bed consists of loose granular material but the stage of the flow is below the critical stage, corresponding to the initiation of sediment transport, and the granular material is thus not in motion, then for such stages the movable bed is acting as a rigid bed.

Consider the flow in a channel consisting of erodible granular material. The shear stress τ _o acting on the flow boundary varies along the wetted perimeter in a manner shown schematically by the τ _o -diagram in Fig. 6.1.

We now come to the study of similarity as applied to waves, certainly the most difficult subject encountered so far. When the application of the exact method of the theory of similarity becomes impossible, then, as has repeatedly been shown in the preceding chapters, a model is designed using mathematical expressions describing the relationships between the quantities involved. Unfortunately, an adequate mathematical theory exists only for wave motion of an ideal (inviscid) fluid. This theory, as is clear from its definition, cannot provide information on such quantities as shear stress, roughness, viscosity, boundary layer thickness and so on, which are of importance in the design of a model. Reliable information on the oscillatory boundary layer is available solely for viscous (laminar) fluid motion over a smooth plain bed. Considering that in almost all cases of civil engineering practice fluid motion is turbulent, while the bed is neither smooth nor plain, it will be clear how limited is the application of existing reliable methods when practical problems occur. It is small wonder then that, to date, no firm method has been established for the design of a small scale wave and/or tidal model where the influence of roughness, viscosity, shear stress, etc. is important (as in all models with mobile bed). Very often the design depends simply on experience and intuition rather than on any method at all.