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Finding the Stokes Wave: Low Steepness to Highest Wave Free Surface Hydrodynamics Numerical Simulations and More Finding the Stokes Wave: Low Steepness to Highest Wave Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Department of Mathematics University of Arizona, Tucson, Arizona, 85721 [email protected] TexAMP 2014 November 21-23 Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Free Surface Hydrodynamics Numerical Simulations and More Outline 1 Free Surface Hydrodynamics Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves 2 Numerical Simulations and More Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Contents 1 Free Surface Hydrodynamics Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Figure : Nearly plane waves in the wake of a boat in Maas-Waal Canal. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Equations of free surface hydrodynamics Consider a potential flow of ideal fluid in a seminfinite strip (x; y) 2 [−π; π]x[−∞; η(x; t)] periodic in x-variable. r2Φ = 0 with BC and domain defined by: @η @η @Φ @Φ + − + = 0 @t @x @x @y y=η(x;t) @Φ 1 2 + (rΦ) + gη = 0 @t 2 y=η(x;t) @Φ = 0 @y y→−∞ Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Conformal Map We introduce the conformal map z(w; t) = w +z ~(w; t), so that in new variable w = u + iv fluid occupies region [−π; π]x[−∞; 0]. − Analyticity of a functionz ~(w) in C imposes a relation between real and imaginary parts ofz ~ on the free surface v = 0: z~(u) =x ~(u) + iy(u) =x ~(u) + iH^ x~(u) = 2P^x~ ^ ^ 1 ^ where H is Hilbert operator, and P = 2 1 + iH is projector. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Hamiltonian In the absence of viscosity, free surface hydrodynamics constitute a Hamiltonian system with: 1 ZZ η(x) g Z H = rΦ2 dy dx + η2 dx 2 −∞ 2 1 Z g Z H = ΦΦ j du + y 2x du 2 v v=0 2 u Introduce complex potential Π(w) = Φ(w) + iΘ(w) analytic in − C . Define (u) = Φ(u; v = 0) and we have: Θ(u) = H^ (u) ^ Φv jv=0 = − Θujv=0 = −H u at v = 0, where −H^ @u is Dirichlet-Neumann operator. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Equations of motion Equations of motion are found from extremizing action S: Z Z L = ηt du − H; S = L dt And are: 2 2 ! − H^ u ^ u ^ H u t + gy = − 2 + uH 2 2jzuj jzuj ^ ^ H u zt = zu(H − i) 2 jzuj The results of simulation of a flavour of these equations is described in further sections. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves What is a Stokes wave? A Stokes wave is a fully nonlinear wave propagating over one dimensional free surface of an ideal fluid. The defining parameters of a Stokes wave is steepness. The steepness is measured as a ratio of crest to trough height H over a wavelength λ Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Equation on the Stokes wave A solution propagating with fixed velocity c: z(u; t) = u + z(u − ct) (u; t) = (u − ct) satisfies: 2 c ^ 1 ^ 2 ^ 2 k − 1 y − ky + yky = 0 c0 2 here c = pg=k - phase velocity of linear gravity waves, and 0 p 0 k^ = jkj = −∆ Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Contents Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves 2 Numerical Simulations and More Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Numerical method We solve equation: 2 c ^ 1 ^ 2 ^ 2 k − 1 y − ky + yky = 0 c0 2 with Newton Conjugate-Gradient iterations in Fourier space: ∗ (n) (n) (n) Write exact solution yk = yk + δyk and linearize at yk : ^ (n) (n) ^ (n) ^ (n) L0 yk + δyk = L0yk + L1δyk = 0 Solve linearized system: ^ (n) ^ (n) L1δyk = −L0yk (n+1) (n) (n) yk = yk + δyk These simulations were performed in quadruple precision. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Stokes Waves H/λ = 0.140984 -5 0.4 H/λ = 0.135748 10 H/λ = 0.127549 0.2 10-15 λ | 0 k ~ (x)/ |z η 10-25 -0.2 H/λ = 0.140984 -35 H/λ = 0.135748 -0.4 10 H/λ = 0.127549 -0.5 -0.25 0 0.25 0.5 0 -4000 -8000 -12000 -16000 x/λ k Figure : Stokes waves computed with Newton-CG method (left) and their spectra (right). Simulations with N = 2048 (blue), N = 4096 (green) and N = 4194304 (orange) Fourier modes. Black dashed lines are asymptotic decay predicted by theory. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Analytical Continuation − Recall thatz ~(u) is analytic in C , but does it have an analytic continuation into the upper half plane? Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Analytical Continuation − Recall thatz ~(u) is analytic in C , but does it have an analytic continuation into the upper half plane? Yes! Stokes wave can be written as a Cauchy integral over branch cut extending from ivc to +i1 Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Asymptotics of Fourier Coefficients Z π zk = z~(u) exp(−iku) du −π + We deform the integration contour into C and assume the local behaviour about ivc to be: β z(w) ∼ (w − ivc ) It is easy to show that asymptotically −β−1 zk ! jkj exp(−|kjvc ) k ! −∞ 1 Result from 1973 by M. Grant shows that β = 2 and it is supported by analysis of the spectra we have found. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Simulations in high steepness regimes 2 1 10-2 1.6 10-4 (H - H )/λ -6 max 1.2 c 10 v -10-1 -10-3 -10-5 0.8 0.4 numerics 0 0 0.03 0.06 0.09 0.12 λ H/λ Hmax/ Figure : Position of the singularity vc as a function of wave steepness. H Our estimate for steepness of highest max = 0:1410633 ± 4 · 10−7 λ Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Limiting Stokes Wave The limiting Stokes wave a.k.a wave of greatest height has a jump 2π in derivative ηx and forms a 3 angle on the surface. The singularity of limiting Stokes appears as a result of coalescence of more than one singularity, e.g.: 1=2 2=3 z(w) ∼ f (w)(w − ivc ) + h:o:t: ! w as vc ! 0 where f (w) is some regular function. Finding the limiting
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