Pa 

 Pw 



characteristic linear dimension of the surface 

 roughness of armor units 



mass density of armor imits 



mass density of water in which the structure is 

 situated 



Thus, by the it theorem (see Sec. II), 



i^w' g' Pa' Pw Z^' H' ^' d, (0 . (?c) ' «' ^' ^' 0, d] = 



and 



Vw ^w(5cl P^ {'c\ H d (^c)^ 



„l/2/g\l/2' (Ji/p^ 



d 'X'X'/t 



,«, i3, A,e, D 



(6-la) 



(6- lb) 



The forces imposed on individual armor units by the flow of water 

 caused by wave action are inertia, form drag, and surface drag (viscous 

 shear). The inertia forces result from the pressure gradient in the 

 water (Allen and Russel, 1958). Since gravity forces are also involved, 

 and are predominant for this type of phenomenon, stability models of 

 rubble-mound structures are designed based on Froude's law. For such 

 models the inertia forces, relative to gravity forces, scale down cor- 

 rectly; the form drag forces, relative to gravity forces, scale down 

 nearly correctly, depending on the form of armor unit, its weight, and 

 the size of wave; and viscous forces scale down incorrectly. For the 

 armor units, the viscous forces can be made negligible if the linear 

 scale is selected so that the Reynolds number, R^, is not too small. 

 However, for the smaller imderlayer rock, it is difficult at times 

 (depending on the height of the design wave) to select a scale so that 

 the viscous forces in the underlayers are negligible. 



In equation (6-lb) the Froude number and the density ratio can be 

 combined, since the stability of the armor units is determined to a 

 considerable extent by the relative magnitudes of the form drag and 

 submerged weight of the armor units. Since it can be shown that equa- 

 tion (6-lb) has been solved for the combined terms. 



V„ 



(Pw) 



1/2 



"^"\%)f(p.-P.) 



1/2 



= f 



V„(«c)3 i'c), H d ^^4 ,^„^ 

 ' Va 



m/p. 



(6-lc) 



Equation (6-lc) is the basis for design and operation of stability models 

 for use in general testing of rubble-mound breakwaters, wave absorbers, 

 and jetties exposed to wave action. 



The desired relationships, model-to-prototype, for similarity of 

 stability of the armor units, D^ = Dp, will be obtained if each t\ term 



317 



