Two-Dimensional Steady Bow Waves in Water of Finite Depth

Citation

Kao, John (1998) Two-Dimensional Steady Bow Waves in Water of Finite Depth. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tp0w-sd61. https://resolver.caltech.edu/CaltechETD:etd-01222008-091958

Abstract

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In this study, the two-dimensional steady bow flow in water of arbitrary finite depth has been investigated. The two-dimensional bow is assumed to consist of an inclined flat plate connected downstream to a horizontal semi-infinite 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 α (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, β, must always be equal to 120°, otherwise no solution can exist. Moreover, we further find that the local flow solution does not exist if α > 60°, and that on the inclined plate there exists a negative pressure region adjacent to the stagnation point for α < 30°. Singularities at the stagnation point and the upstream infinity are found to have multiple branch-point singularities of irrational orders.

A fully nonlinear theoretical model has been developed in this study for evaluating the incompressible irrotational flow satisfying the free-surface 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 semi-circle 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° < α < 60°, a draft Froude number Fr_d less then 0.5, and a water-depth Froude number Fr_h less than 0.4.

The dependence of the solution on these key parameters is examined. As α decreases for fixed Fr_d and Fr_h, the free surface falls off more steeply from the stagnation point. Similarly, as Fr_d increases, the free surface falls off quickly from the stagnation point, but for decreasing Fr_h it descends rather slowly towards the upstream level. As Fr_h 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.

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