Finding the Stokes Wave: Low Steepness to Highest Wave · 2014. 11. 24. · Yes!Stokes wave can be written as a Cauchy integral over branch cut extending from iv c to +i1 Sergey Dyachenko*,

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

[email protected]

TexAMP 2014November 21-23

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave


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



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




Figure : Nearly plane waves in the wake of a boat in Maas-Waal Canal.




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




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.




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



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.




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




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| =√−∆



Travelling Wave solution a.k.a Stokes WaveParameter OscillationEvolution Problem

Contents


2 Numerical Simulations and MoreTravelling 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



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.




Analytical Continuation

Recall that z(u) is analytic in C−, but does it have an analyticcontinuation into the upper half plane?




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∞




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



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




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.




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!




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!




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




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.




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∞




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




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




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

)




Simulation with Simple Pole in V as Initial Data




Evolution of branch cut corresponding to velocity




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




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.


Documents

Finding the Stokes Wave: Low Steepness to Highest Wave · 2014. 11. 24. · Yes!Stokes wave can be written as a Cauchy integral over branch cut extending from iv c to +i1 Sergey Dyachenko*,