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lecture20 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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± Lecture 20 - Marine Hydrodynamics Lecture 20 Chapter 6 - Water Waves 6.1 Exact (Nonlinear) Governing Equations for Surface Gravity Waves, Assuming Potential Flow Free surface definition , ( x B 0 ) , , , ( or ) , , ( = = t z y x F t z x y η x y y z Unknown variables Velocity ﬁeld: Position of free surface: Pressure ﬁeld: Governing equations Continuity: Bernoulli for P-Flow: Far way, no disturbance: y , z , t ) = 0 v ( x, y, z, t )= φ ( x, y, z, t ) y = η ( x, z, t )o r F ( x, y, z, t )=0 p ( x, y, z, t ) 2 φ =0 y<η or F< 0 ∂φ 1 2 p p a + |∇ φ | + + gy ; or 0 ∂t 2 ρ / ∂t, φ 0 and p = p a ρgy ±²³´ ±²³´ atmospheric hydrostatic 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

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± ²± ² ³ ´ µ ³´ µ¶ Boundary Conditions 1. On an impervious boundary B ( x, y, z, t ) = 0, we have KBC: ± vn · · ˆ= ∂φ ± ( ± · n x, t )= U n φn = U x, t ) ˆ( ± on B =0 ∂n Alternatively: a particle P on B remains on B , i.e., B is a material surface. For example if P is on B at t = t 0 , P stays on B for all t . B ( ±x P ,t 0 )=0 , then B ( P ( t ) ) = 0 for all t, so that, following P B is always 0. DB ∂B = + ( φ ·∇ ) B on B Dt ∂t example, for a ﬂat bottom at y = h B = y + h = ( y + h 0 = 0 o n B = y + h ∂y =1 2. On the free surface, y = η or F = y η ( x, z, t ) = 0 we have KBC and DBC. KBC: free surface is a material surface, no normal velocity relative to the free surface. A particle on the free surface remains on the free surface for all times. D F D φ η φ∂ η η =0= ( y η ³´µ¶ ∂x ³´µ¶ ∂z ³´µ¶ on y = η ³´µ¶ still vertical slope slope unknown of f.s. of f.s. velocity DBC: p = p a on y = η or F = 0. Apply Bernoulli equation at y = η : 1 2 + |∇ φ | + g η = p a on y = η 2 ³´µ¶ still unknown non-linear term 2
± ² ³ ´ 6.2 Linearized (Airy) Wave Theory Assume small wave amplitude compared to wavelength, i.e., small free surface slope A << 1 λ SWL crest wavelength Water depth h trough Wave height H λ Wave amplitude A Wave period T H = A/2 Consequently φ η , < < 1 λ 2 / T λ We keep only linear terms in φ , η . For example: () | =() + η () | + ... Taylor series y = η y =0 y =0 ±²³´ ∂y keep discard 3

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± ² ³ ´ µ µ µ µ · ³ ´ 6.2.1 BVP In this paragraph we state the Boundary Value Problem for linear (Airy) waves. 2 φ ∂φ + g = 0 t 2 y y = 0 2 φ = 0 y = -h ∂φ = 0 y Finite depth h = const Inﬁnite depth GE: 2 φ =0 , h<y< 0 2 φ ,y< 0 BKBC: ∂φ ∂y ,y = h φ 0 →−∞ FSKBC: FSDBK: = ∂η ∂t + 2 φ 2 + g Introducing the notation {} for inﬁnite depth we can rewrite the BVP: Constant ﬁnite depth h { Inﬁnite depth } 2 φ , 0 2
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lecture20 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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