On the Linearized Theory of 

 Hub Cavity with Swirl 



G. H. Schmidt 



Technical University of Delft 



and 

 J. A. Sparenberg 

 University of Groningen 

 The Netherlands 



ABSTRACT 



In general, there is a cavity astern of the hub of 

 a ship screw. This cavity is rather stable and is 

 roughly in the shape of a long circular cylinder. 

 There is circulation about it, which occurs in the 

 case of a real screw propeller, when the circulation 

 around the blades at their roots is nonzero. 

 Because the divergence of the vorticity field is 

 zero, this circulation at the roots "flows" down- 

 stream in the form of circulation about the hub. 

 At the end of the hub the flow contracts and the 

 swirl velocity increases. The pressure becomes 

 lower and a cavity forms where the pressure decreases 

 to the vapor pressure. 



We introduce the following simplifications: 

 First, we neglect the influence of the finite number 

 of blades and consider a half infinite axially 

 symmetric hub immersed in an inviscid and incom- 

 pressible fluid. The incoming flow consists of a 

 homogeneous part, parallel to the axis of the hub 

 in the direction of the endpoint, and of a swirl 

 which represents the circulation around the hub. 

 In the upstream direction the hub tends to a 

 circular cylinder while its radius tends to zero 

 towards the end point. Second, our theory will be 

 linear: The difference between the radius of the 

 hub and the radius of the cavity is assumed to be 

 small and quantities which are quadratic in this 

 difference will, in general, be neglected. 



Using these simplifications we determine the 

 shape of the cavity for given values of, for 

 instance, the swirl, the incoming velocity, the 

 ambient pressure, and the vapor pressure. The 

 surface tension is also included in the general 

 formulation of the problem. The more detailed 

 considerations, as well as the numerical calculations, 

 will be confined to zero surface tension. 



One of the unknowns of the problem is the 

 position of the point of separation. This position 

 can be determined by demanding that the pressure 

 exceeds the vapor pressure everywhere on the wetted 



348 



surface of the hub and by demanding that the flow 

 cannot penetrate the surface of the hub. 



The shape of the cavity is roughly a circular 

 cylinder. There are waves on the surface of this 

 cylinder which are, within the limitations of our 

 theory, steady with respect to the hub, and their 

 crests and throughs are perpendicular to the axis 

 of the hub. We will give numerical results for 

 the wavelengths and amplitudes of the waves as 

 functions of, for instance, the incoming velocity 

 and of the shape of the hub . 



1 . INTRODUCTION 



A long cavity generally begins somewhere at the 

 end of the hub of a ship screw. This cavity, which 

 has circulation around it, does not close or widen, 

 it has a rather stable mean value to its radius. 

 The circulation or swirl occurs in the case of a 

 real screw propeller when the circulation around 

 the blades at their roots is not zero. Because 

 the divergence of the vorticity field is zero, this 

 circulation at the roots "flows" downstream in the 

 form of circulation about the hub and then about 

 the cavity. 



In order to gain some insight in this phenomenon 

 we introduce some simplifications. We neglect the 

 influence of the finite number of blades and con- 

 sider a half infinite axially symmetric hub immersed 

 in an inviscid and incompressible fluid. The 

 incoming flow consists of a homogeneous part paral- 

 lel to the axis of the hub in the direction of the 

 endpoint and of a swirl which represents the 

 circulation around the hub. In the upstream 

 direction the hub tends to a circular cylinder 

 while its radius tends to zero towards the and 

 point. Hence, near the endpoint the flow contracts 

 and the swirl velocity increases proportional to 

 the inverse of the radius. This means that the 

 pressure becomes lower and a cavity starts where 

 the pressure decreases to the vapor pressure of 



