2.016 Hydrodynamics Prof. A.H. Techet Fall 2005

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1 .016 Hydrodynamics Reading #7.016 Hydrodynamics Prof. A.H. Techet Fall 005 Free Surface Water Waves I. Problem setu 1. Free surface water wave roblem. In order to determine an exact equation for the roblem of free surface gravity waves we will assume otential theory (ideal flow) and ignore the effects of viscosity. Waves in the ocean are not tyically uni-directional, but often aroach structures from many directions. This comlicates the roblem of free surface wave analysis, but can be overcome through a series of assumtions. To setu the exact solution to the free surface gravity wave roblem we first secify our unknowns: Velocity Field: V (x, y, z,t ) = φ(x, y, z,t ) Free surface elevation: η (x, yt), Pressure field: (x, y, z, t ) version 3.0 udated 8/30/ A. Techet

2 .016 Hydrodynamics Reading #7 Next we need to set u the equations and conditions that govern the roblem: Continuity (Conservation of Mass): φ =0 for z <η (Lalace s Equation) (7.1) Bernoulli s Equation (given some φ ): φ 1 + φ + a t ρ + gz = 0 for z <η (7.) No disturbance far away: φ, φ 0 and = ρ gz (7.3) t a Finally we need to dictate the boundary conditions at the free surface, seafloor and on any body in the water: (1) Pressure is constant across the free surface interface: = atm on z =η. φ = ρ 1 () = atm. (7.4) t V gz + c t Choosing a suitable integration constant, c( t) = z =η becomes atm, the boundary condition on ρ{ φ + 1 V + gη} = 0. (7.5) () Once a article is on the free surface, it remains there always. Similarly, the normal velocity of a article on the surface follows the normal velocity of the surface itself. z =η (x, t) η η z +δ z =η (x +δ x, t + δ t) =η (x, t) + δ x + δ t (7.6) x On the surface, where z =η, we can reduce the above equation to η η δz = uδt + δt (7.7) x t and substitute δ z = w δ t and δ x = uδt the article: to show that the normal velocity follows version 3.0 udated 8/30/ A. Techet

3 .016 Hydrodynamics Reading #7 η η w= u + on z =η. (7.8) x (3) On an imervious body boundary B( x, y, z, t ) = 0. Velocity of the fluid normal to the body must be equal to the body velocity in that direction: φ v nˆ = φ nˆ = = U ( x, t ) n ˆ( x, t ) = U n on B = 0. (7.9) n Alternately a article P on B remains on B always; ie. B is a material surface. For examle: if P is on B at some time t t such that = o B(x, t ) = 0, then B ( x, t ) = 0 for all t, (7.10) o so that if we were to follow P then B = 0 always. Therefore: o DB B = + ( φ )B = 0 on Dt B = 0. (7.11) Take for examle a flat bottom at z = H : φ/ z = 0 on z = H (7.1) II. Linear Waves. Linearized Wave Problem. version 3.0 udated 8/30/ A. Techet

4 .016 Hydrodynamics Reading #7 To simlify the comlex roblem of ocean waves we will consider only small amlitude waves (such that the sloe of the free surface is small). This means that the wave amlitude is much smaller than the wavelength of the waves. As a general rule of thumb for wave height to wavelength ratios ( h / λ, where h is twice the wave amlitude) less than 1/ 7 we can linearize the free surface boundary conditions. Non-dimensional variables can be used to assess which terms can be droed. Non-dimensional variables: η = aη ω t = t u = aω u x = λ x w= aω w φ = aωλ φ Looking back to equation (7.5) we can evaluate the relative magnitude of each term by lugging in the non-dimensional versions of each variable. For examle: dφ = aωλ dφ dt = 1/ω dt dx = λ dx φ Comare φ and V ( ) x to determine which terms can be droed from equation (7.5): Here we see that if h / λ << 17 /, then ( ) a ω ( ) a ( ) = =. (7.13) φ φ λ φ aω λ φ φ φ x x x ( φ φ x ) <<, (7.14) and we can dro the smaller term resulting in linearized boundary conditions which are alicable on z =η. η = φ, z φ + gη = 0, version 3.0 udated 8/30/ A. Techet

5 .016 Hydrodynamics Reading #7 Throughout this discussion we have assumed that the wave height is small comared to the wavelength. Along these lines we can also see why η is also quite small. If we exand φ ( x, zt), about z = 0 using Taylor series: φ φ(x, z =η, t) = φ( x, 0, t) + η +..., (7.15) z it can be readily shown that, the second term, φ η and the subsequent higher order terms z are very small and can be ignored in our linear equations allowing us to rewrite the boundary conditions at z =η as boundary conditions on z = 0. 0.in on z = 0. φ + g φ = 0 (7.16) z 1 φ η = g (7.17) A Solution to the Linear Wave Problem The comlete boundary value wave roblem consists of the differential equation, secifically Lalace s Equation, φ ( xz,, t with the following boundary conditions: φ φ ) = + = 0 (7.18) x z 1) Bottom Boundary Condition: ) Free Surface Dynamic Boundary Condition (FSDBC): φ = 0 on z = H (7.19) z 1 φ η = ( ) on z = 0. (7.0) g version 3.0 udated 8/30/ A. Techet

6 .016 Hydrodynamics Reading #7 3) Free Surface Kinematic Boundary Condition (FSKBC): φ φ + g = 0 (7.1) z Through searation of variables we can solve Lalace s Equation for η and φ. η ( xt, ) = a cos( kx ωt+ ψ) (7.) aω φ ( xz,, t ) = f ( z ) sin( kx ωt+ ψ) (7.3) k ux (, z, t ) = a ω f ( z ) cos( kx ωt+ ψ) (7.4) wx (, z, t ) = a ω f ( z ) sin( kx ωt+ ψ) (7.5) 1 cosh[ kz ( + H)] f( z ) = (7.6) sinh( kh ) sinh[ kz ( + H)] f1 ( z ) = (7.7) sinh( kh ) ω = gk tanh( kh ) disersion relation (7.8) where a, ω, k, ψ are integration constants with hysical interretation: a is the wave amlitude, ω the wave frequency, k is the wavenumber ( k = π/λ ), and ψ simly adds a hase shift. III. Disersion Relation The disersion relationshi uniquely relates the wave frequency and wave number given the deth of the water. The chosen otential function, φ, MUST satisfy the free surface boundary conditions (equation (7.0) and (7.1)) such that lugging φ in to the FSKBC (eq. (7.1)) we get: Resulting in the Disersion relationshi: ω cosh kh + gk sinh kh = 0 (7.9) ω = gk tanh kh ; (7.30) version 3.0 udated 8/30/ A. Techet

7 .016 Hydrodynamics Reading #7 For dee water where H tanh kh 1 so that the disersion relationshi in dee water is: ω = gk (7.31) In general, k as ω or λ as T. Phase and Grou Seed Phase seed, C, of a wave (velocity that a wave crest is traveling at) is found in general using: λ ω g = C = = tanh kh T k k This simlifies for the case of dee water such that. (7.3) C = g (7.33) k Solution to the disersion relationshi in general form can be found grahically. IV. Pressure under a Wave The ressure under a wave can be found using the unsteady form of Bernoulli s equation and the wave otential, (,, φ x zt) : 1 = ρ φ ρv ρ gz (7.34) hydrostatic unsteady nd Order ressure fluctuation term DynamicPressure Since we are only considering the case for LINEAR free surface waves we can neglect all higher order terms. Droing the second order term in the dynamic ressure from equation (7.34), the ressure under a wave is simly φ d ( x, z, t ) = ρ (7.35) version 3.0 udated 8/30/ A. Techet

8 .016 Hydrodynamics Reading #7 aω = ρ f( z) k cos(ωt kx ψ ) (7.36) ω = ρ f ( z )η(x, t ), (7.37) k where ω ( cosh[k z + H )] f( z) = g tanh( kh ) cosh[k z + H )] ( = g. (7.38) k sinh( kh ) cosh( kh ) Therefore the dynamic ressure for all deths becomes ( x, z, d t ) = ρg η(x, t (7.39) cosh[ k( z +H )] ) cosh(kh ) V. Motion of Fluid Particles below a Wave We can define the motion of fluid articles underneath a linear rogressive wave to have a horizontal motion,ζ and a vertical motion, η, such that ζ (x, z, t ) = a f (z) sin(kx ω t) + ψ ) (7.40) η (x, z, t ) = a f 1 (z) cos(ωt k x +ψ ) (7.41) The orbital attern is given by the equation for an ellise: ζ ζ = a (7.4) + f( z ) z f1 ( ) where a is the amlitude of the waves. Plugging in f(z) and f 1 (z) from equations (7.6) and (7.7) we get version 3.0 udated 8/30/ A. Techet

9 .016 Hydrodynamics Reading #7 ζ ζ + (7.43) cosh k ( z +H ) sinh k ( z +H ) a ( sinh kh ) a ( sinh kh ) =1 The horizontal and vertical velocity comonents, u and w, of these articles are simly the time derivatives of the motions: u = ζ w = η (7.44) (7.45) 3. Fluid articles move in ellitical orbits below the surface. In dee water, as H, these ellises become circular. Dee Water Simlification In dee water as H, then f ( z) f 1 ( z) e kz, and the article orbits become circular with an exonentially decreasing radius. All article motion dies out at z λ/. version 3.0 udated 8/30/ A. Techet

10 .016 Hydrodynamics Reading #7 Phase and Grou Velocity Phase Seed The hase seed of a wave is defined as the seed at which the wave is moving. If you were to clock a wave crest you would find that it moves at the hase seed, C = ω/k. Grou Velocity Grou velocity is the seed of roagation of a acket, or grou, of waves. This is always slower than the hase seed of the waves. In a laboratory setting grou seed can be observed by creating a short acket of waves, about 8-10 cycles, and observing this acket as it roagates down a testing tank. The leading edge of the acket will aear to move slower than the waves within the acket. Individual waves will aear at the rear of the acket and roagate to the front, where the ressure forces their aarent disaearance. We can derive Grou Velocity starting with a harmonic surface wave with surface elevation η( x,t ) = a cos(ωt kx ) (7.46) and then looking at that wave as if it was the sum of two cosines with very similar frequencies ( ω ± δω ) and wavenumbers ( k ±δ k ): η( x, t ) = lim { a cos ([ω δω ]t [k δ k x ) + a ] cos ([ω + δω ]t [k + δ k ] x )} (7.47) δ k, δ ω 0 = lim {a cos(ωt kx ) cos(δω t δ k δk, δω 0 x )} (7.48) η (t, x = 0 ) time 4. A wave train (red line) constructed with two waves having similar frequency/wavenumber comonents. This wave aears to be beating. The blue line shows the enveloe or grou outline. version 3.0 udated 8/30/ A. Techet

11 .016 Hydrodynamics Reading #7 In general the enveloe (or Grou") of the wave travels at the grou seed, C g, δω C g =. (7.49) δ k In the limit as δω 0 and δ k 0, the grou velocity becomes the derivative of the frequency with resect to the wavenumber: dω C g = (7.50) dk This can be found using the disersion relationshi from equation (7.30), and then taking the derivative of it with resect to the wavenumber. We can derive the grou seed as follows: d d ω dk { }= dk {kg tanh( kh )} (7.51) dω kgh ω = g tanh( kh ) + dk cosh (kh ) (7.5) dω 1 g kh = tanh( kh ) 1+ (7.53) dk ω sinh( kh )cosh( kh ) = ω =C k 1 kh C g = C 1+ (7.54) sinh kh cosh kh Dee Water: H ω = kg and C g = 1 C (7.55) Shallow Water: H 0 ω = gh k and C g = C (7.56) version 3.0 udated 8/30/ A. Techet

12 .016 Hydrodynamics Reading #7 Wave Energy Potential Energy λ 1 1 g E P = 1 λ 1 ρ η dx = λ 0 ρ ga cos (ωt kx ) (7.57) λ 0 E P = 1 ρ ga 4 (7.58) Kinetic Energy 1 λ 0 1 ρ(u + w ) dz dx (7.59) E k = λ 0 H E k = 1 ρga 4 (7.60) Total Energy er wavelength E = E P + E k = 1 ρga (7.61) Flux of Energy through a Vertical Plane Energy flux: Power = force velocity = ( dz ) u de 0 = H u dz dt (7.6) = 1 ρga ω cos (ωt kx ) k (7.63) Average energy flux over one cycle, assuming dee water, de 1 T de 1 = dt = ρga ω = E C g dt T (7.64) 0 dt 4k version 3.0 udated 8/30/ A. Techet

13 .016 Hydrodynamics Reading #7 Useful References The material covered in this section should be review. If you have not taken or a similar class dealing with basic fluid mechanics and water waves lease contact the instructor. The references below are merely suggestions for further reading and reference. J. N. Newman (1977) Marine Hydrodynamics MIT Press, Cambridge, MA. M. Faltinsen (1990) Sea Loads on Shis and Offshore Structures Cambridge University Press, Cambridge, UK. M. Rahman (1995) Water Waves: Relating modern theory to advanced engineering ractice Clarendon Press, Oxford. version 3.0 udated 8/30/ A. Techet