A Singular Perturbation Method for Non-Linear Water Waves Past an Obstacle

Author: Rispin, Peter Paul Augustine

Year: 1967

Degree: Dissertation (Ph.D.)

Advisor: Wu, Theodore Yao-tsu

Committee Member: Unknown, Unknown

Option: Applied Mechanics

DOI: 10.7907/D0P9-9109

Abstract

The method of matched singular perturbation expansions is used to solve the problem of a steady two-dimensional flow of a perfect fluid with a free surface under the influence of gravity. A flat plate of length ℓ is inclined at an angle [alpha] to the horizontal and its trailing edge is immersed to a depth h below the surface of an otherwise uniform stream of infinite depth, the velocity at upstream infinity being U. A parameter β = gℓ/U² (Froude number F = β-1/2) is assumed small so that the flow separates smoothly at the leading and trailing edges, giving rise to n upward jet and gravity waves in the downstream. An inner solution for the velocity field is obtained which is valid near the plate and an outer solution which holds far away. These are determined through the orders 1, β log β, β, β² log² β, β² log β up to order β², and are matched with one another to these orders. In contrast with linearized planing theory, the depth of submergence can be prescribed as a parameter. The lift coefficient is calculated for several values of α, h/ℓ and β. The results reduce to known ones in certain limits.

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