II. PRINCIPLES OF SIMILARITY, DIMENSIONAL ANALYSIS, AND SCALE MODELS 



by 

 B.Y. Hudson and G.H. Keulegan 



The requirements for similarity between hydraulic scale models and 

 their prototypes can be established on the bases o£ dynamical consider- 

 ations, dimensional analysis, and differential equations. The use of 

 scale models in the solution of hydraulic problems is based on the appli- 

 cation of several relationships generally known as the laws of hydraulic 

 similitude. These laws, which are based on the principles of fluid me- 

 chanics, define the requirements necessary to ensure correspondence be- 

 tween flow conditions of a scale model and its prototype. In a majority 

 of hydraulic engineering problems, the degree of correspondence is 

 limited because it is not possible to obtain a model fluid that has the 

 required viscosity, surface tension, and elastic modulus to obtain exact 

 similitude unless the linear scale is such that the model is as large, 

 or nearly so, as its prototype. Although complete similitude is not 

 usually feasible, the laws for complete similitude are known and, from 

 experience, it is known that to impose complete similarity in model tests 

 is unnecessary. A very important part of a model engineer's job is to 

 justify his selected departures from complete similarity and, when neces- 

 sary, apply theoretical corrections to compensate for them. 



1. Dynamic Similarity . 



Dynamic similarity between a model and its prototype involves geo- 

 metric and kinematic similarity and Newton's laws of motion. If parts 

 of a model have the same shape as the corresponding parts of the proto- 

 type, the two systems are geometrically similar, and the following rela- 

 tion exists between corresponding linear dimensions: 



Ln, = L,Lp . C2-1) 



The subscripts m and p refer to model and prototype, respectively, 

 and Lj. is the scale of length. (This definition shows that the often- 

 used term "small-scale model" is difficult to define. The term "scale 

 model" is perferred--the size of the model compared with its prototype 

 is indicated by the value of L_..) 



Kinematics deals with space-time relationships; thus, kinematic simi- 

 larity indicates a similarity of motion between model and prototype. Two 

 particles, one in the model and the other in the prototype, that corres- 

 pond to each other are said to be homologous. Kinematic similarity of 

 two systems is obtained if homologous particles are at homologous points 

 at homologous times (American Society of Civil Engineers, 1942), The 

 time intervals in the two systems must have a constant ratio, 



Tm = TJp (2-2) 



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