which obviously contradicts the continuity of the flow (and can also be shown mathe- 

 matically untenable if they are considered to overlap). Hi. The streamlines turn back, 

 forming a re-entrant jet which, if continued, must pass through the rear of the body 

 or through the cavity wall (Fig. 1). The last alternative is the only one mathematically 

 consistent with the theory but is of course non-physical. Thus, under the present 

 assumptions, it is impossible in principle to satisfy the physical requirements of steady 

 state cavitation. 



Water tunnel and water entry experiments reveal clearly the existence of the 

 re-entrant jet. However, the jet is greatly weakened by turbulent mixing in the stagna- 

 tion region, and in larger cavities it can be seen striking the cavity walls and then being 

 swept away by the flow. To all intents and purposes these flows may be considered 

 stationary. In smaller cavities, or in flows past obstacles with large afterbodies, the 

 jet can be observed (in high speed photographs) distorting the cavity and giving rise 

 to an unsteady pulsating motion. 



Another approach to the problem of the finite cavity is through the artificial 

 but conceptually simpler Riabouchinsky model. This avoids the non-physical doubly- 

 covered flow plane by introducing the mirror image of the obstacle as a streamline of 

 the flow (Fig. 2). As far as quantitative results are concerned it is a matter of indiffer- 

 ence which cavity model is adopted, for the Riabouchinsky and re-entrant jet flows 

 yield almost identical values of drag and cavity dimensions. From this it appears that 

 conditions at the rear of the cavity have but little effect on gross flow quantities. 



In the two-dimensional case the calculation of these flows is reduced by the 

 hodograph method to a simple exercise in conformal mapping. Consider first the 

 symmetric re-entrant jet flow past a flat plate, shown in Fig. 1 with its image planes. 

 Let / = <f + iip be the complex potential and let the cavity speed be unity. Then one 

 finds: 



s(s 2 - b 2 ) 



f(s) = M , M real, (2) 



.s 2 - h 2 



i / h - .<? \ 



(3) 



where b — — 2/r — h + 2, and the cavitation parameter is given by 



- 4h 3 + h 2 + 4/i - 2 



a = . (-4) 



(h 2 + h+ l) 2 



The expression for the drag coefficient obtained from (2) and (3) by integrating the 

 pressure difference p — p c over the front face is 



"° /l + sV/s-5 



CM = (1 + v) 



1 - s / \ s 2 - h 2 



1 - : : 



1 - s \* / s + b 



1 + s / \ s 2 -h 2 



sds 



sds 



(5) 



This gives the asymptotic formula, valid for small a, 



2tt 

 C D (v)~ (1 + <x). (6) 



4 + 7T 



282 



