An Unsteady Cavity Flow 



have 



f, ~— iL_(B,e'"' + B,e-'*^V' as |t|-^oo. (113) 



T 

 47r{Kco) 



From (113) we may derive the following results: (I) From (110) 

 we may conclude that f, °^ cKco. (ii) For r lying on the real T-axls, 

 which corresponds to the cavity wall of the basic flow, f | is purely 

 imaginary; this indicates that near the point at infinity the perturbation 

 velocity W| = 9f|/9z is always perpendicular to the original cavity 

 wall, (iii) If we recall that r = O^t}^^) as |t| -^ oo, we can see that 

 the perturbation velocity is of order |z|" for large values of |z|; 



this, together with the result stated in (li) , implies that the unsteady 

 free surface displacement tends to zero as | z | -* oo. (iv) Along the 

 imaginary T-axis , which corresponds to the line of symmetry of the 

 flow, f| is purely real; this means that there is no velocity component 

 normal to the line of symmetry which, of course, is what we should 

 expect. 



It should be pointed out here that the order of magnitude and 

 the direction of the perturbation velocity on the free surface near 

 the point at infinity agree with the results obtained by Wang and Wu 

 [1963] in the study of small-time behavior of unsteady cavity flows. 



Finally, we shall investioiate the behavior of the solution (87) 

 near the separation point t = i. From (4), (35) and (5), the pertur- 

 bation velocity w, may be written as 



9f, Wo 9f| n t A\ 



Equation (114) indicates that the singular behavior of w, near t = 1 

 can be studied from that of 9f /9t near t = 1. Let us now differ- 

 entiate fi given by (87) with respect to t. The differentiation of 

 f| with respect to T may be viewed as consisting of four parts; the 

 differentiation of the t appearing in the limits of Integration, the 

 differentiation of the factors exp (±iKcoT^) , the differentiation of the 

 Airy functions with respect to t, and the differentiation of the factor 

 in front of the curely brackets in (87). Only the latter two parts 

 produce terms of the form a (t - i)'^'^ near T = 1; all the other 

 parts either give zero or a finite contribution to w,. Therefore, 

 condition (34) and the condition that the pressure Is Integrable over 

 the plate AB are satisfied. 



Since the solution given by (87) behaves properly at Infinity and 

 at the separation point, we conclude that It Is the solution of the 

 problem; no additional solution of the homogeneous equation, as 

 shown in (93) needs to be added. 



213 



