Bvennen and Whitney 



E, Singularities 



Successful numerical treatments of singularities depend upon 

 the availability of analytic solutions to the flow in the neighborhood 

 of that point. For example, at a corner between solid walls the 

 velocity varies as the (ir - P)/^ power of distance from that junction 

 where P is the included angle. If this is ir/2 (as at points C or 

 D, Fig, 4(a)) the variation is linear and thus the numerical estimate 

 of the circulation around the cell (see Section 4) in such a corner is 

 a good one. Where the angle is not ir/2 (D, Fig. 4(c)) errors will 

 occur due to the non-linear variation of velocity, but corrective pro- 

 cedures are easily devised, 



A great deal less is known about the singularities at a junction 

 of a free surface and a solid boundary. If the wall is static and verti- 

 cal (A, Fig. 4(a)) so that X^i = X^ = X^t = 0, etc. , it follows from the 

 equation of motion that if Yq = at t = then it is always zero for 

 ir rotational flow; the tangent to the free surface at the wall is always 

 horizontal. Thus the free surface condition without surface tension 

 is automatically satisfied at such a junction and only weak singular 

 behavior is expected. But a similar analysis of the case when the 

 wall begins to move at t = (remaining vertical) indicates that Ytt 

 must be infinite at the junction (B, Fig, 4(a)) at t = 0, the singularity 

 being logarithmic in space. An extension to t^ has not so far been 

 obtained. One approach might be a Fourier analysis of the step in 

 X^i so that the steadily oscillating solutions of Fontanet [ 1961] could 

 be used. These suggest that Y^^ becomes finite for t > 0, 



FIG. 4(a) 



FIG. 4(c) 



Y=I-HR 



132 



