When the given cavitation number a is smaller than a x for a minimum leading-edge cavity 

 thickness, all the physical quantities corresponding to o^ are taken as an approximation in order to 

 have a consistent application of the cascade theory. This situation occurs near the blade tip where a 

 and o M are both smallest, but the physical quantities for a given foil with an infinite cavity seem to 

 differ very little between the two cases near the blade tip where the solidity is small' 21 '. 



INFLOW ANGLE INFLUENCED BY CAVITIES 



Blade-cavity interference effects may have to be considered not only for cavity drag but also for 

 induced velocity on the blades. The induced velocity determines the angle of inflow velocity or the 

 hydrodynamic pitch angle py The angle 0j determines the regular helical surface where the trailing 

 vorticity is usually located' 10 '. For a subcavitating propeller, the angle j3j is determined completely 

 by trailing vortices on the helical surface. The inflow velocity at the propeller plane is the vector 

 mean velocity of the velocities far ahead and far behind the propeller. The Kutta-Joukovsky theorem 

 for the propeller' 24 ' holds with the vector mean velocity and not with the velocity far upstream as in 

 two-dimensional theory. The finite blade thickness, which can be represented by a source distribution, 

 can influence neither the flow velocity at infinity nor the vector mean velocity because the source 

 effect on the perturbation decays very rapidly with the distance. The finite-cavity thickness can be 

 represented also by a source distribution. Therefore, in a supercavitating propeller with finite cavity, 

 the cavity cannot influence the hydrodynamic pitch angle directly when the vortex and cavity source 

 are superposed for the propeller representation. 



When a long cavity is considered, the infinite-cavity cascade theory shows that the downstream 

 flow deflection, or exit angle, is sizable. Thus, the long cavity may induce a sizable contribution on 

 the inflow angle. This effect can be seen in the considerable increment of shock-free angle of two- 

 term-camber foil in cascade' 21 '. Therefore it may be worth trying in the final design a pitch angle of 

 blade cavity different from a wake angle 0j where the trailing vortices are located' 9 '. 



For supercavitating propellers with long cavities the problem of determining (3j is more compli- 

 cated than for subcavitating propellers because the inflow angle is influenced by flow retardation as 

 well as a cascade effect in addition to the vortex induced velocities. Fortunately, the effect of flow 

 retardation tends to cancel the cascade effect I 4,22 '. In practice, even a long cavity does not extend 

 very far downstream while the trailing vortices extend to infinity. Therefore the present approach is 

 to consider the cavity essentially finite and to employ a cascade theory to approximate the inner 

 flow of the propeller, considering that 0j is influenced only by the vortices. For subcavitating 

 propellers the use of two-dimensional airfoil theory requires determination of an angle of attack with 

 respect to py Two-dimensional theory may be considered then as an inner flow theory which is 

 imbedded in propeller theory. 



Likewise, when a cascade model is applied to a blade section of a supercavitating propeller, the 

 problem is reflected in how to match the inflow velocity of the propeller and the cascade velocity 

 field. That is, the hydrodynamic pitch angle 0j must be matched to the vector mean of the velocities 

 far upstream and far downstream of the cascade. From momentum considerations in the cascade' 4 ', 

 the far downstream velocity angle 5 is 



6 = -(cC L cos7)/(2d) (5) 



which is a function of C L only. Therefore, the vector mean velocity is deflected from the velocity 

 from far upstream at an approximate angle' 4 ' of 6/2. This direction of the cascade velocity has to 

 coincide with the direction of the hydrodynamic pitch angle 0, of the propeller. In this way, the 

 cascade effect is fully manifested in the angle of attack of the blade. That is, the angle of attack 

 becomes much larger than when an isolated supercavitating foil is considered. 



