Panel Discussion 



wave resistance. The converse effect of the waves on the viscous flow is more 

 difficult to observe but no less surprising. .■ ,. , ^ r 



The earliest work in this area was Havelock's inclusion of the displacement 

 thickness on the effective beam which led, from Michell's integral, to an estimate 

 of the effects of viscosity on the wave resistance. Wu (1963) included the effects 

 of the wave-field pressure gradient on the boundary layer for the case of a flat 

 plate, and with wake effects ignored. Experimental investigations of this theory 

 are in progress. Webster and his collaborators (1967) used a boundary -layer 

 formulation, in conjunction with Guilloton's method, to compute the separation 

 point on the hull as a function of Froude and Reynolds numbers. 



It seems likely that, for reasonably full ships where the disagreement be- 

 tween theory and experiment is most severe, the effects of separation will domi- 

 nate those of the relatively thin boundary layer upstream of the separation point. 

 With this in mind Milgram (1968) has recently used Michell's integral, with em- 

 pirical values of the wake geometry, to determine the wave resistance of a sim- 

 ple hull form. Generally speaking, the computed values of the wave resistance 

 are decreased by the presence of the wake, at those speeds where the wave re- 

 sistance is a decreasing function of the Froude number, and unaffected by the 

 wake at other Froude numbers. 



The effects of the wake can be idealized, from another viewpoint, by con- 

 sidering the characteristics of wave propagation in a shear flow. This problem 

 has been considered by van Wijngaarden (1968), as well as in the earlier papers 

 by Brooke-Benjamin (1959) and Kolberg (1958). 



If the Reynolds number is sufficiently small and the body streamlined, so 

 that laminar attached flow can be assumed, it is possible to attack the boundary- 

 value problem for the solution of the Navier-Stokes equations, including the free- 

 surface boundary condition. This has been done by Dugan (1968) for the re- 

 stricted case of a submerged two-dimensional horizontal plate, moving at suffi- 

 ciently low velocities that the Oseen linearization of the Navier-Stokes equations 

 can be made. With the additional assumption that the plate is deeply submerged, 

 the following equation is obtained for the drag coefficient: 



^ ' y - 1 + ?n(R/16) 



Here, R and F are the Reynolds and Froude numbers based on the chord length 

 of the plate, d is the ratio of the depth of submergence to the chord length, and y 

 is Euler's constant. This result is valid for large values of F and d, and is 

 asymptotic to the non-free- surface results when F^'d^-co. 



The interaction of waves and viscous wakes has been considered by Lurye 

 (1968), using the Oseen equations and the linear free-surface conditions. A par- 

 ticular "singularity" solution is obtained, and more general flows can be gen- 

 erated by superposition. The Oseen equations have also been used by Nikitin 

 and Gruntfest (1966) to find the wave resistance of a moving pressure distribu- 

 tion in a viscous fluid. (I am indebted to Professor Weinblum for calling atten- 

 tion to this work, and also to the subject of propeller and rudder effects on wave 



1551 



