3. OSCILLATORY BOUNDARY-LAYER FLOW 



Theoretical Considerations 



(i) The Laminar Case. The problem of the boundary-layer flow 

 can be treated in the ordinary fashion. According to the fundamental 

 principle the flow field outside the boundary layer can be described by 

 means of the irrotational theory while the complete equation of motion 

 within the layer is simplified with the help of dimensional arguments. 

 The boundary conditions of the simplified equation are set so as to 

 satisfy the nonslip requirement at the solid boundary and the continuity 

 of the velocity components at the outer edge of the boundary layer. Lin 

 (1957)* has presented a solution to the more general problem in which the 

 flow both inside and outside the boundary layer has a mean steady com- 

 ponent as well as an oscillatory one. In the present case the two flows 

 have oscillatory components only. The equations of motion will be 



2H + u 



2H + v 



Su 



= 



at 



ox 



dy 





1 oP .. d 2 u 



(3-1) 



and 



o^t 







B^ 



-^—— 



+ 



u i 



— — 



at 





dx 



p ax ay 



au,- i ap 



r i t 2 = — r- (3 - 2) 



1 ay p ax 



inside and outside the boundary layer respectively. Continuity is 

 assumed to be satisfied individually 



(3-3) 



(3-4) 



Equations (3-2) and (3-4) describe a two-dimensional inviscid un- 

 steady and incompressible flow. The solution of these two equations under 

 a specific set of boundary conditions will define the velocity field in 

 the fluid as a function of time and space, provided of course, that the 

 effect of viscosity is negligible. In the problem at hand the first 

 boundary condition is a sinusoidal progressive wave at the surface (y = 0) 

 and the second, zero vertical velocity component at the bottom (y = -d) . 

 To the first approximation equation (3-2) may be linearized by omitting 

 the quadratic terms; it can be written then as 



so that 



3u 



ax 



+ 



ay 



and 



3"! 



+ 



d v l = 





ax 





ay 



a«i = i ap 

 at p ax 



(3-5) 



* Numbers in parentheses denote date of reference listed on page 29, 



