24 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



For when equations (12) are satisfied, (f){ -"'--^- ,-^~)=(j)( ,r^), 



SO that <^ cancels out when we divide equation ( ii ) for the full-sized 

 original by the corresponding equation for the geometrically similar 

 model. Speeds which satisfy equations (12) are "corresponding 

 speeds," and when two geometrically similar bodies are run at cor- 

 responding speeds they are " dynamically similar." 



If the speeds are low enough that compressibility may be disre- 



5" . . 

 garded, the value of -^ is ununportant and the condition for corre- 

 sponding speeds, which ensures dynamical similarity, is merely the 

 first of equations (12). If we use only a single medium so that 

 pm=p and vm = v, the condition for corresponding speeds reduces to 



and geometrically similar bodies will be dynamically similar if their 

 speeds are inversely as their linear dimensions. Any great reduction 

 in scale might therefore involve our running the model so fast as to 

 make the effects of compressibility no longer negligible. But if the 

 original is to be run in air while the model can be run in water, this 

 difficulty may be avoided. For under ordinary conditions the kine- 

 matic viscosity of water is from i/io to 1/20 that of air, and for a 

 model of given size the speed required for dynamical similarity with 

 the original is reduced in the same ratio as the kinematic viscosity. 



In practice the foregoing method of experimentation is usually 

 unnecessary. For under ordinary working conditions the resistances 

 of aeroplanes and their separate structural elements are so nearly 

 proportional to the square of the speed, and the effects of compressi- 

 bility are so small, that for practical purposes <f> in equation (11) or 

 in equation (9) may be treated as a constant and equation (10) used 

 for computation, within any ordinary ranges of D and 6". Any speeds 

 may then be regarded as corresponding speeds, and geometrical 

 similarity suffices by itself for dynamical similarity. If the constant 

 K of equation (10) has been determined by experiments on any body 

 of the given shape at any convenient speed, the same value may be 

 used in equation (10) for computing the value of R for a different 

 speed or a different size or both. 



Complete Dynamical Similarity 



The experience with flat plates, showing that even though R is 

 proportional to S- it may not be to D-, warns us to be cautious in 



