Marine Hydrodynamics Lecture 19. Exact (nonlinear) governing equations for surface gravity waves assuming potential theory

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1 Marine Hydrodynamics, Fall 2004 Lecture 19 Copyright c 2004 MIT - Department of Ocean Engineering, All rights reserved. Water Waves Marine Hydrodynamics Lecture 19 Exact (nonlinear) governing equations for surface gravity waves assuming potential theory y Free surface definition: y = η(x,z,t) or F(x,y,z,t) = 0 x z B(x,y,z,t) = 0 Unknown variables: Velocity field: v (x, y, z, t) = φ (x, y, z, t) Position of free surface: η (x, z, t) or F (x, y, z, t) Pressure field: p (x, y, z, t) Field equation: Continuity: 2 φ = 0 y < η or F < 0 Given φ can calculate p: t φ 2 + p p a ρ + gy = 0; y < η or F < 0 Far way, no disturbance: / t, φ 0 and p = p a atmospheric ρgy hydrostatic

2 Boundary Conditions: 1. On an impervious boundary B (x, y, z, t) = 0, we have KBC: v ˆn = φ ˆn = n = U ( x, ) ( x, ) t ˆn t = U n on B = 0 Alternatively: a particle P on B remains on B, i.e. B is a material surface; e.g. if P is on B at t = t 0, i.e. so that, following P, B = 0. For example, flat bottom at y = h: B( x P, t 0 ) = 0, then B( x P (t), t 0 ) = 0 for all t, DB Dt = B + ( φ ) B = 0 on B = 0 t / y = 0 on y = h or B : y + h = 0 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 perpendicular relative velocity to free surface, particle on free surface remains on free surface: DF Dt = 0 = D (y η) = Dt y vertical velocity η t x η x slope of f.s. z η z slope of f.s. on y = η still unknown DBC: p = p a on y = η or F = 0. Apply Bernoulli equation at y = η: t φ 2 non-linear term + g η still unknown = 0 on y = η (assuming p a = 0)

3 Linearized (Airy) Wave Theory Consider small amplitude waves: (small free surface slope) SWL Wave height H crest Wave amplitude A = H/2 Water depth h trough wavelength Wave period T λ Assume amplitude small compared to wavelength, i.e., A << 1. λ φ Consequently: λ 2 /T, η << 1, and we keep only linear terms in φ, η. λ For example: () y=η = () }{{ y=0 + η } y () y= Taylor series keep discard

4 Finally the boundary value problem is described by: y = 0 2 φ + g y t 2 = 0 y = -h 2 φ = 0 y = 0 Linear (Airy) Waves Constant finite depth h Infinite depth (1) GE: 2 φ = 0, h < y < 0 2 φ = 0, y < 0 (2) BKBC: y (3a) FSKBC: (3b) FSDBC: = 0, y = h φ 0, y = η, y = 0 y t t + gη = 0, y = 0 (3) 2 φ + g t = 0, y = 0 2 y

5 Constant finite depth h {Infinite depth} { 2 φ = 0, h < y < 0 2 φ = 0, y < 0 } (1) = 0, y = h { φ 0, y } (2) y 2 φ t + g { 2 2 y = 0, y = 0 φ t + g } 2 y = 0, y = 0 (3) Given φ : η (x, t) = 1 g p p a = ρ }{{ t} dynamic t y=0 ρgy hydrostatic { η (x, t) = 1 g t p p a = ρ t dynamic y=0 } ρgy hydrostatic (4) (5) Solution of 2D Periodic Plane Progressive Waves(using separation of variables) Math... (solve (1), (2), (3)). Answer: φ = ga ω sin (kx ωt) cosh k (y + h) cosh kh η = using (4) A cos (kx ωt) where A is the wave amplitude = H/2. {φ = gaω sin (kx ωt) eky } η = A cos (kx ωt) using(4)

6 1. At t = 0 (say), η = A cos kx periodic in x with wavelength: λ = 2π/k units of λ : [L] λ x k K = wavenumber = 2 /λ [L -1 ] 2. At x = 0 (say), η = A cos ωt periodic in t with period: T = 2π/ω units of T : [T ] T t ω = frequency = 2 /T [T -1 ], e.g. rad/sec 3. η = A cos [ k ( x ωt)] units of ω : [ L k k T Following a point with velocity ω ( ω ) k, i.e. x p = t + const. the phase of η does not change: k ω k = λ T V p phase velocity. ]

7 Dispersion Relationship So far, any ω, k combination is allowed. Must satisfy FSBC, substitute φ into equation (3): which gives: ω 2 cosh kh + gk = 0, { ω 2 = gk tanh kh ω 2 = gk } Dispersion Relationship (6) Given k (and h) ω. Given ω (and h),... Dispersion relationship (6) uniquely relates ω and k, given h: ω = ω(k; h) or k = k(ω; h) 1 C kh kh =f(c) tanh kh kh C ω2 h g C kh = from (6) = tanh kh (kh) tanh (kh) obtain unique solution for k In general: (k as ω ) or (λ as T ). λ T = V p = ω g k = tanh kh k { V p = } g k Also: V p as T or λ. For fixed k (or λ), V p as h ( from shallow to deeper water) V p = V p (k) or V p (ω) frequency dispersion

8 Solution of the Dispersion Relationship : ω 2 = gk tanh kh Property of tanh kh: tanh kh = cosh kh = 1 e 2kh 1 + e 2kh = { kh for kh << 1, i.e. h << λ ( long waves or shallow water) 1 for kh > 3, i.e. kh > π h > λ 2 ( short waves or deep water)(e.g. tanh 3 = 0.995) Deep water waves Intermediate depth Shallow water waves or short waves or wavelength or long waves kh >> 1 Need to solve ω 2 = gk tanh kh kh << 1 (h > λ/2) given ω, h for k (h/λ < 1/20 in practice) (given k, h for ω - easy!) ω 2 = gk (a) Use tables or graphs (e.g.jnn fig.6.3) ω 2 = gk kh ω = gh k λ = g 2π T 2 ω 2 = gk tanh kh = gk λ = gh T ( λ(in ft.) 5.12T 2 (in sec.) ) k k = λ = V p = tanh kh λ V p (b) Use numerical approximation V p = ω k = g ω = g 2π λ (hand calculator, about 4 decimals ) V p = ω k = λ T = gh i. Calculate C = ω 2 h/g Frequency dispersion ii. If C > 2: deeper No frequency dispersion λ = 2π g V p 2 kh C(1 + 2e 2C 12e 4C ω +...) k = gh If C < 2: shallower kh C( C C )

9 Characteristics of a Linear Plane Progressive Wave y k = 2π λ ω = 2π T H = 2A MWL A Vp λ h η(x,t) = y x Define U ωa Linear Solution: with: Velocity field: φ = Ag ω cosh k (y + h) cosh kh sin (kx ωt) ; ω 2 = gk tanh kh η = A cos (kx ωt) u = x = Agk ω = Aω U cosh k (y + h) cos (kx ωt) cosh kh cosh k (y + h) cos (kx ωt) v = y = Agk sinh k (y + h) sin (kx ωt) ω cosh kh sinh k (y + h) = Aω sin (kx ωt) U On y = 0: 1 u = U o = Aω cos (kx ωt) tanh kh { u cosh k (y + h) e ky deep water = U o cosh kh 1 shallow water v = V o = Aω sin (kx ωt) = η { t v sinh k (y + h) e ky deep water = V o 1 + y shallow water h u is in phase with η v is out of phase with η

10 Shallow water Intermediate water Deep water Pressure Field: dynamic pressure p d = ρ t ; total pressure p = p d ρgy p d = ρgη (no decay) cosh k (y + h) p d = ρga cos (kx ωt) cosh kh cosh k (y + h) = ρg η cosh kh p d p = ρg(η y) hydrostatic approximation p do same picture as u U o p d = ρge ky η p d ( h) p do = 1 cosh kh p d ( h) = e ky p do p = ρg [ ηe ky y ]

11 V p = gh V p (kh >> 1) p (kh << 1) p p d p s p d Particle Orbit/ Velocity (Lagrangian). Let x p (t), y p (t) be the position of the particle P, then x p (t) = x + x (t); y p (t) = ȳ + y (t) where ( x; ȳ) is the mean position of P. ( x, y) P(x p,y p ) (x,y ) v p v ( x, ȳ, t) ( x, ȳ) u p = dx p dt u = u ( x, ȳ, t) + x ( x, ȳ, t) x + u y ( x, ȳ, t) y +... ignore - linear theory x p = x + dt u ( x, ȳ, t) = x + x cosh k (ȳ + h) = A y = dt v ( x, ȳ, t) = sin (k x ωt) cosh k (ȳ + h) dt ωa cos (k x ωt) sinh k (ȳ + h) dt ωa sin (k x ωt) sinh k (ȳ + h) = A cos (k x ωt) on ȳ = 0, y = A cos (k x ωt) = η x 2 a 2 + y 2 b 2 = 1 or (x p x) 2 a 2 + (y p ȳ) 2 b 2 = 1 cosh k(ȳ+h) sinh k(ȳ+h) where a = A ; b = A, i.e. the particle orbits form closed ellipses with horizontal and vertical axes a and b.

12 crest V p (a) deep water kh >> 1: a = b = Ae ky circular orbits with radii Ae decreasing exponentially with depth ky A trough A ky Ae (b) shallow water kh << 1: A Vp = gh A y a = = const. ; b = A(1+ ) kh h decreases linearly with depth A/kh V p (c) Intermediate depth P V p A Q S Q S R R λ R

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14 Summary of Plane Progressive Wave Characteristics f(y) Deep water/ short waves kh > π (say) Shallow water/ long waves kh << 1 cosh k(y+h) = f cosh kh 1 (y) eky 1 e.g.p d cosh k(y+h) e.g.u, a = f 2 (y) e ky 1 kh sinh k(y+h) e.g. v, b = f 3 (y) e ky 1 + y h

15 C (x) = cos (kx ωt) S (x) = sin (kx ωt) (in phase with η) (out of phase with η) η A = C (x) u = C (x) f Aω 2 (y) v = S (x) f Aω 3 (y) p d ρga = C (x) f 1 (y) y A = C (x) f 3 (y) x A = S (x) f 2 (y) a = f A 2 (y) b = f A 3 (y) b a