Hydraulic Structures: Fourth Edition

678 MODELS IN HYDRAULIC ENGINEERING

(a) Geometric, kinematic, dynamic and mechanical similarity

Geometricsimilarity is similarity in form, i.e. the length scale Mlis the

same in all directions.

Kinematicsimilarity denotes similarity of motion, i.e. similarity of

velocity and acceleration components along the x,y,zaxes;Mu,M,Mw

are all constants (not necessarily equal); the same applies for the accelera-

tion scales.

Dynamicsimilarity denotes similarity of forces; thus with Mmas the

scale for mass and Mtfor time we can write

MPxMmMaxMmMlxM^ t^2 MmM^2 uMlx^1 constant (16.1)

(etc., in the yandzdirections).

Mechanicalsimilarity is an all-embracing term including geometric,

kinematic and dynamic similarity, i.e. Ml,MandMPare all constants, the

same in all directions. Mechanical similarity can be defined as follows: two

formations are (mechanically) similar if they are geometrically similar and

if, for proportional masses of homologous points, their paths described in

proportional times are also geometrically similar. This definition, based on

Newton’s law, thus includes geometric similarity of the two formations, the

proportionality of times and the geometric similarity of the paths travelled

(kinematic similarity) as well as the proportionality of masses and thus

also of forces (dynamic similarity).

Thetheory of similarity, leading to dimensionless numbers and

scaling laws, can be elaborated in three ways. The first determines the cri-

teria of similarity from a system of basic homogeneous (differential) equa-

tions which mathematically express the investigated physical phenomena.

The second path leads to the conditions of similarity through dimensional

analysis carried out after a careful appraisal of the physical basis of each

phenomenon and of the parameters which influence it. The combined use

of physical and dimensional analyses is often the best route to a successful

formulation of similarity criteria. The third way could be denoted as the

method of synthesis (Barr, 1983; Sharp, 1981).

An example of the first route – the use of physical laws and govern-

ing equations – is the formulation of the scaling laws (criteria) by writing,

for example, the Navier–Stokes equations, both for the model and the pro-

totype, and inserting the scales of the various parameters onto one set of

the equations. In this way we arrive from

u  w Y 

1

   ...

(16.2)

∂^2 



∂z^2

∂^2 



∂y^2

∂^2 



∂x^2

μ



∂p



∂y

∂



∂z

∂



∂y

∂



∂x

∂



∂t