Free Surface HydrodynamicsNumerical Simulations and More
Finding the Stokes Wave: Low Steepness toHighest Wave
Sergey Dyachenko*, Pavel Lushnikov and AlexanderKorotkevich
Department of MathematicsUniversity of Arizona, Tucson,
Arizona, 85721
TexAMP 2014November 21-23
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Outline
1 Free Surface HydrodynamicsFormulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
2 Numerical Simulations and MoreTravelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Contents
1 Free Surface HydrodynamicsFormulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling WavesTravelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite 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
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Equations of free surface hydrodynamics
Consider a potential flow of ideal fluid in a seminfinite strip(x , y) ∈ [−π, π]x [−∞, η(x , t)] periodic in x-variable.
∇2Φ = 0
with BC and domain defined by:
∂η
∂t+
(−∂η∂x
∂Φ
∂x+∂Φ
∂y
)∣∣∣∣y=η(x ,t)
= 0(∂Φ
∂t+
1
2(∇Φ)2
)∣∣∣∣y=η(x ,t)
+ gη = 0
∂Φ
∂y
∣∣∣∣y→−∞
= 0
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Conformal Map
We introduce the conformal map z(w , t) = w + z(w , t), so that innew variable w = u + iv fluid occupies region [−π, π]x [−∞, 0].
Analyticity of a function z(w) in C− imposes a relation betweenreal and imaginary parts of z on the free surface v = 0:
z(u) = x(u) + iy(u) = x(u) + i H x(u) = 2P x
where H is Hilbert operator, and P = 12
(1 + i H
)is projector.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Hamiltonian
In the absence of viscosity, free surface hydrodynamics constitute aHamiltonian system with:
H =1
2
∫ ∫ η(x)
−∞∇Φ2 dy dx +
g
2
∫η2 dx
H =1
2
∫ΦΦv |v=0 du +
g
2
∫y2xu du
Introduce complex potential Π(w) = Φ(w) + iΘ(w) analytic inC−. Define ψ(u) = Φ(u, v = 0) and we have:
Θ(u) = Hψ(u)
Φv |v=0 = − Θu|v=0 = −Hψu
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
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Equations of motion
Equations of motion are found from extremizing action S:
L =
∫ψηt du −H, S =
∫L dt
And are:
ψt + gy = −ψ2u −
(Hψu
)22|zu|2
+ ψuH
(Hψu
|zu|2
)
zt = zu(H − i)Hψu
|zu|2
The results of simulation of a flavour of these equations isdescribed in further sections.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
What is a Stokes wave?
A Stokes wave is a fully nonlinear wave propagating over onedimensional 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 heightH over a wavelength λ
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
Equation on the Stokes wave
A solution propagating with fixed velocity c:
z(u, t) = u + z(u − ct)
ψ(u, t) = ψ(u − ct)
satisfies: (c2
c20k − 1
)y −
(1
2ky2 + y ky
)= 0
here c0 =√g/k0 - phase velocity of linear gravity waves, and
k = |k| =√−∆
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Contents
Formulation of ProblemHamiltonian FormalismEquations of MotionFinite Amplitude Travelling Waves
2 Numerical Simulations and MoreTravelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Numerical method
We solve equation:(c2
c20k − 1
)y −
(1
2ky2 + y ky
)= 0
with Newton Conjugate-Gradient iterations in Fourier space:
Write exact solution y∗k = y(n)k + δy
(n)k and linearize at y
(n)k :
L0(y(n)k + δy
(n)k
)= L0y
(n)k + L1δy
(n)k = 0
Solve linearized system:
L1δy(n)k = −L0y (n)k
y(n+1)k = y
(n)k + δy
(n)k
These simulations were performed in quadruple precision.Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Stokes Waves
-0.4
-0.2
0
0.2
0.4
-0.5 -0.25 0 0.25 0.5
η(x
)/λ
x/λ
H/λ = 0.140984H/λ = 0.135748H/λ = 0.127549
10-35
10-25
10-15
10-5
-16000-12000-8000-4000 0
|z~k|
k
H/λ = 0.140984
H/λ = 0.135748
H/λ = 0.127549
Figure : Stokes waves computed with Newton-CG method (left) andtheir spectra (right). Simulations with N = 2048 (blue), N = 4096(green) and N = 4194304 (orange) Fourier modes. Black dashed linesare asymptotic decay predicted by theory.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Analytical Continuation
Recall that z(u) is analytic in C−, but does it have an analyticcontinuation into the upper half plane?
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Analytical Continuation
Recall that z(u) is analytic in C−, but does it have an analyticcontinuation into the upper half plane?
Yes! Stokes wave can be written as a Cauchy integral overbranch cut extending from ivc to +i∞
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Asymptotics of Fourier Coefficients
zk =
∫ π
−πz(u) exp(−iku) du
We deform the integration contour into C+ and assume the localbehaviour about ivc to be:
z(w) ∼ (w − ivc)β
It is easy to show that asymptotically
zk → |k |−β−1 exp(−|k|vc)
k → −∞
Result from 1973 by M. Grant shows that β = 12 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
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Simulations in high steepness regimes
0
0.4
0.8
1.2
1.6
2
0 0.03 0.06 0.09 0.12
vc
H/λ Hmax/λ
numerics
10-6
10-4
10-2
1
-10-5
-10-3
-10-1
(H - Hmax)/λ
Figure : Position of the singularity vc as a function of wave steepness.
Our estimate for steepness of highestHmax
λ= 0.1410633± 4 · 10−7
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Limiting Stokes Wave
The limiting Stokes wave a.k.a wave of greatest height has a jumpin derivative ηx and forms a 2π
3 angle on the surface.
The singularity of limiting Stokes appears as a result of coalescenceof more than one singularity, e.g.:
z(w) ∼ f (w) (w − ivc)1/2 + h.o.t.→ w2/3
as vc → 0
where f (w) is some regular function. Finding the limiting Stokeswave from presented equations faces several major difficulties.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Everwidening Spectrum
The Asymptotics of Fourier coefficients as k →∞:
yk ∼1
|k|3/2exp (−|k |vc)
(ky)k ∼1
|k|1/2exp (−|k |vc)
Take a limit vc → 0 and:
(ky)k ∼1
|k |1/2
Divergent!
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Everwidening Spectrum
The Asymptotics of Fourier coefficients as k →∞:
yk ∼1
|k|3/2exp (−|k |vc)
(ky)k ∼1
|k|1/2exp (−|k |vc)
Take a limit vc → 0 and:
(ky)k ∼1
|k |1/2
Divergent!
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Velocity Oscillation
1
1.02
1.04
1.06
1.08
1.1
0.04 0.08 0.12 0.16
H/λ
wav
e sp
eed
L-H 1975
Schwartz 1979 1.0922
1.0924
1.0926
1.0928
1.093
0.138 0.139 0.14 0.141 0.142
H/λ
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
How can we study the analytic properties?
1 Study of Fourier spectrum:
Accurate, but only allows to study the singularities closest tothe real axis
2 Construction of Pade interpolant and analysis of its poles:
Straight-forward construction of Pade interpolant suffers fromcatastrophic loss of precision in finite digit arithmetic.
Solution: Alpert-Greengard-Hagstrom algorithm(AGH) canconstruct Pade approximation using many points on the grid.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Branch Cut
We expand periodic interval u ∈ [−π, π] to infinite intervalζ ∈ (−∞,∞) with auxiliary transform:
ζ = tan(w
2
)Applying Pade approximation we observe that Stokes wave is abranch cut:
z(u) = z(u)− u ≈d∑
k=1
αk
tan(u
2
)− iχk
≈∫ 1
χc
ρ(χ)dχ
tan(u
2
)− iχ
The branch cut spans along the positive imaginary axis from pointiχc to +i∞
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Pade Approximation of Stokes Wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1
ρ
tanh(v/2)
(b)
numerics
analytic
10-25
10-20
10-15
10-10
10-5
0 5 10 15 20 25 30
Ma
x |E
rr|
N, number of poles
(a)
approximation
Figure : (a) Maximum of absolute error between the Stokes wavesolution and its approximation by poles as a function of the number ofpoles; (b) Reconstructed jump on the branch cut using residues andposition of poles from AGH algorithm
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Jump on the Branch Cut
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ρ
tanh(v/2)
H/λ = 0.1409535H/λ = 0.1255102H/λ = 0.1173340
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Numerical simulation of time-dependent problem
The equations on ψ(u, t) and z(u, t) are not optimal for numericalsimulations, instead equations are formulated in terms ofR(u, t) = 1
zuand V (u, t) = iψu
zu:
Rt = i(UR ′ − U ′R
)Vt = i
(UV ′ − B ′R
)+ g (R − 1)
it is convenient to introduce projection operator P = 12(1 + i H),
then U = 2P
(−Hψu
|zu|2
)and B = P
(|Φu|2
|zu|2
)
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Simulation with Simple Pole in V as Initial Data
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Evolution of branch cut corresponding to velocity
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Position of branch cut in the absence of gravity
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.2 0.4 0.6 0.8 1
vc(t
)
t, time
numerics, no hyperviscosity
numerics, with hyperviscosity
prediction
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave
Free Surface HydrodynamicsNumerical Simulations and More
Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem
Conclusions
Analytical properties of Stokes wave are fully determined by asingle branch cut.
High steepness waves have been constructed numerically.
The prediction of oscillatory approach to limiting wave wasconfirmed, with several of the oscillations being well-resolved.
Closed integral equations on ρ(χ) were found.
Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave