II TheLaplace equacon and potencal fluid flow

Transcription

1 THE WAVE EQUATION (3) I Main Topics A The aplace equacon and fluid potencal B AssumpCons and boundary condicons of D small wave theory C SoluCon of the wave equacon D Energy in a wave E Shoaling of waves GG454 1 II Theaplace equacon and potencal fluid flow Consider a square which fluid is flowing across, with no fluid being stored or lost in the square Any increase in the velocity of fluid in the x- direccon (u) across the square must be matched by a decrease in velocity in the y- direccon (v) (1) u/dx = - v/ y () u/dx + v/ y = GG454 1

2 II Theaplace equacon and potencal fluid flow u x + v y = Suppose that the velocices can be given by parcal derivacves of a potencal funccon φ ( 3) u = φ x ( 4) v = φ y SubsCtuCng (3) and (4) into () yields the aplace equacon ( 4) φ x + φ y = ( 5) φ = GG454 3 II Theaplace equacon and potencal fluid flow φ x + φ y = φ x φ φ y φ φ x + φ y φ φ φ 1 φ x φ 5 x φ φ 3 φ 7 Δx Δx Δx Δx φ φ φ φ y φ 6 y φ φ 4 φ 8 Δy Δy Δy Δy φ If Δx = Δy, then φ + φ 1 + φ 3 Δx φ + φ 1 + φ 3 Δx + φ + φ + φ 4 = ( Δx) 4φ + φ 1 + φ + φ 3 + φ 4 = φ = φ 1 + φ + φ 3 + φ 4 4 = φ + φ 1 + φ 3 Δx = φ + φ + φ 4 Δy + φ + φ + φ 4 = ( Δy) So a funccon that sacsfies the aplace equacon has values that average those at nearest neighbors Assuming Δx = Δy GG454 4

3 III AssumpCons and boundary condicons of - D small wave theory A No geometry changes parallel to wave crest (- D assumpcon) B Wave amplitude is small relacve to wave length and water depth C Water is homogeneous, incompressible, and surface tension is nil. D The bo`om is not moving, is impermeable, and is horizontal E Pressure along air- sea interface is constant F The water surface has the form of a cosine wave η = Acos π x t = H cos π x t = wavelength T = wave period GG454 5 IV SoluCon of the wave equacon A General solucons π x (6) φ = H cosh gt ( d + y) π cosh πd sin π x πt H = wave height; = wavelength; d = water depth; C = wave speed; t = Cme; T = wave period (constant); x = horizontal posicon; y = verccal posicon 1 C = gt π tanh πd = CT 3 u = π H π ( d + y) cosh sinh πd 4 v = π H π ( d + y) sinh sinh πd 5 ζ = H π ( d + y) cosh sinh πd 6 ε = H π ( d + y) sinh sinh πd u = horizontal H parccle velocity amplitude* v = verccal H parccle velocity amplitude* ζ = horizontal H parccle displacement amplitude* ε = verccalh parccle displacement amplitude* *FuncCon of wave height, wave period, wavelength, water depth, and distance above bo`om (d+y) GG

4 IV SoluCon of the wave equacon A General solucons 1 C = gt π tanh πd = CT 3 u = π H π ( d + y) cosh sinh πd 4 v = π H π ( d + y) sinh sinh πd 5 ζ = H π ( d + y) cosh sinh πd 6 ε = H π ( d + y) sinh sinh πd B Deep- water solucons (d/ >.5) tanh(πd/) 1 1 C = gt π = CT = gt π 3 u = π H e π y 4 v = π H T e π y 5 ζ = H e π y 6 ε = H e π y * C and depend on T, not water depth d * Amplitudes decrease exponencally with depth (y<) * Wave base: y = - / (e - π =.4) GG454 7 IV SoluCon of the wave equacon A General solucons 1 C = gt π tanh πd = CT 3 u = π H π ( d + y) cosh sinh πd 4 v = π H π ( d + y) sinh sinh πd 5 ζ = H π ( d + y) cosh sinh πd 6 ε = H π ( d + y) sinh sinh πd B Shallow- water solucons (d/<.5) As ω!, cosh(ω)! tanh(ω)! ω 1 C = gd C = gd = CT = gdt 3 u = π H πd = π H πd = C H d = C A d 4 v = π H d + y d 5 ζ = H πd 6 ε = H d + y d * C and decrease as d decreases * v and ε! as y! - d * u and ζ do not change with y GG

5 IV SoluCon of the wave equacon h`ps:// GG454 9 IV SoluCon of the wave equacon GG

6 V Energy in a wave A PotenCal energy in excess of stacc situacon per unit horizontal area 1 Express excess in terms of the potencal energy density ρgy z η E P = ρg ydx d( ) dy z dz ρg ydx d( ) dy z η dz = ρg ydx Integrate the energy density ρgy over the height range!η, and then average that over a wavelength to find average excess potencal energy per unit horizontal area E P = 1 η ( ρgydy ) dx = ρg 1 3 Now express this in terms of wave amplitude A η = Acos( π x ) E P = ρg 1 Acos π x η dx = ρg dx = ρg 1 η 1 π x A cos dy dz dx = ρg η η = mean squared displacement dx = ρg 1 A = ρg 4 A GG V Energy in a wave B KineCc energy in excess of stacc situacon per unit horizontal area 1 Express excess in terms of the kinecc energy density ρg velocity / E K = 1 ( ρ velocity dv ) = ρ z ( u + v ) dx dydz d SubsCtute expressions for u and v [see eqs. (3), (4), and (5)] and proceed as before by integracng verccally and then averaging horiztonally E K = ρ π T sinh( π H ) 1 Acos π x + ρ π 1 Asin π x T sinh( π H ) sinh π ( y + d) dydx d 3 A er considerable algebra (see Kundu (199) for guidance) E K = 1 ρgη = 1 4 ρga cosh π y + d dydx d GG

7 V Energy in a wave C Total energy in excess of stacc situacon per unit horizontal area 1 Horizontally averaged excess potencal energy and kinecc energy are equal E P = 1 ρgη = 1 4 ρga E K = 1 ρgη = 1 4 ρga Total horizontally averaged excess energy is evenly split between kinecc and potencal energy E total = E P + E K = ρgη = 1 ρga GG VI A shoaling wave A B C E total( 1) = E total = B ( B 1 1 ) ρga 1 A = 1 A 1 1 B 1 B ρga 1 As decreases, A increases As B decreases, A increases D Wave steepness = A/ = H/ E Waves get taller and steeper as they shoal because E K decreases and E P increases as water depth decreases (conservacon of energy) F Rule of thumb: waves break where (H/ ) = 1/7 tanh (πd/) 1 Map view of waves nearing shore GG

8 VI A shoaling wave h`ps:// GG Appendices GG

9 Hyperbolic FuncCons sinh( β ) = eβ e β = β + β 3 3! + β 5 5! + cosh( β ) = eβ + e β = 1+ β! + β 4 4! + tanh( β ) = sinh( β ) cosh( β ) = eβ e β e β + e β tanh( β ) = β β β 5 sinh kd cosh k d + y sinh kd sinh k d + y = = 15 + for β < π + e k( d+y) e k d+y = e k( d+y) e k d+y = ( d+y ) ek + e k( d+y) ( d+y ) ek e k( d+y) = ekd e ky + e kd e ky = ekd e ky e kd e ky GG Shallow water and deep water approximacons Shallow water Asβ, sinh β Asβ, cosh β Asβ, tanh β sinh cosh tanh = = 1 = β 1 β Deep water Asβ, sinh β Asβ, cosh β Asβ, tanh β sinh π cosh π tanh π e β e β GG

10 Shallow water and deep water approximacons Shallow water Asβ, tanh β As πd As πd As πd, As πd As πd, β, cosh πd 1, sinh πd πd cosh π d + y sinh πd 1 πd π d + y, sinh π d + y sinh π d + y sinh πd d + y d Deep water Asβ, tanh β As πd π, As πd, tanh( π ) cosh π d + y sinh πd e πd sinh π d + y sinh πd e πd GG References Kundu, P.K., 199, Fluid mechanics: Academic Press, San Diego, California, 638 p. GG454 1