Kao, John (1998) Twodimensional steady bow waves in water of finite depth. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd01222008091958
Abstract
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In this study, the twodimensional steady bow flow in water of arbitrary finite depth has been investigated. The twodimensional bow is assumed to consist of an inclined flat plate connected downstream to a horizontal semiinfinite draft plate. The bottom of the channel is assumed to be a horizontal plate; the fluid is assumed to be invicid , incompressible; and the flow irrotational. For the angle of incidence [alpha] (held by the bow plate) lying between 0° and 60°, the local flow analysis near the stagnation point shows that the angle lying between the free surface and the inclined plate, [beta], must always be equal to 120°, otherwise no solution can exist. Moreover, we further find that the local flow solution does not exist if [alpha] > 60°, and that on the inclined plate there exists a negative pressure region adjacent to the stagnation point for [alpha] < 30°. Singularities at the stagnation point and the upstream infinity are found to have multiple branchpoint singularities of irrational orders.
A fully nonlinear theoretical model has been developed in this study for evaluating the incompressible irrotational flow satisfying the freesurface conditions and two constraint equations. To solve the bow flow problem, successive conformal mappings are first used to transform the flow domain into the interior of a unit semicircle in which the unknowns can be represented as the coefficients of an infinite series. A total error function equivalent to satisfying the Bernoulli equation is defined and solved by minimizing the error function and applying the method of Lagrange's multiplier. Smooth solutions with monotonic free surface profiles have been found and presented here for the range of 35° < [alpha] < 60°, a draft Froude number [...] less then 0.5, and a waterdepth Froude number [...] less than 0.4.
The dependence of the solution on these key parameters is examined. As [alpha] decreases for fixed [...] and [...], the free surface falls off more steeply from the stagnation point. Similarly, as [...] increases, the free surface falls off quickly from the stagnation point, but for decreasing [...] it descends rather slowly towards the upstream level. As [...] decreases further, difficulties cannot be surmounted in finding an exact asymptotic water level at upstream infinity, which may imply difficulties in finding solutions for water of infinite depth. Our results may be useful in designing the optimum bow shape.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Major Option:  Engineering and Applied Science 
Thesis Committee: 

Defense Date:  4 May 1998 
Record Number:  etd01222008091958 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd01222008091958 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  277 
Collection:  CaltechTHESES 
Deposited By:  Imported from ETDdb 
Deposited On:  15 Feb 2008 
Last Modified:  25 Dec 2012 14:58 
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