TRANSCRITICAL FLOW PAST SLENDER SHIPS 
G. -K. .bea 
Nattonal Setence Foundatton 
Washtngton, D. C. U.S.A. 
J. P. Feldman 
Naval Shtp Research and Development Center 
Washtngton, D. C. U Gan 
ABSTRACT 
The transcritical shallow water flow past slender 
ships is analyzed using the method of matched 
asymptotic expansions. A consistent first order 
approximation was derived which is analogous to 
the non-linear transonic equation with the Froude 
and Mach numbers playing similar roles. Solutions 
are obtained for sinkage and trim in the transcri- 
tical region and are compared with experimental 
results, An important result is that both sinkage 
and trim are functions of Froude number as well 
as beam to length ratio in the region where Froude 
number based on undisturbed depth is close to unity. 
INTRODUCTION 
In a series of papers Tuck @M @ developed a systematic 
expansion procedure for the approximate solution to the shallow water 
flow past slender ships. It was pointed out that a close analogy exists 
between this problem and the inviscid slender body aerodynamics pro- 
blem. In fact, Tuck's solution contains the same type of singularity 
that is encountered in aerodynamic theory and we present here an at- 
tempt to remove the singularity which occurs in the transcritical re- 
gion. Thus the shallow water problem that will be examined is concer- 
ned only with steady translational motion of a slender ship and the 
associated surface waves so that viscous and compressibility effects 
are neglected. 
The Froude number a = U, /V gh, where U,, is the free 
stream speed, g is the acceleration of gravity, h is the undisturbed 
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