Conformal Mapping Method for Free Surface Waves
C. Viotti, D. Dutykh
School of Mathematical Sciences, UCD, Dublin, Ireland
(UCD) Toronto, 2013 1 / 27
General Features
PROS
Fully nonlinear (i.e., solves the Euler eqs.)
Fast and accurate, particularly when combined with spectral methods Machine-precision conservation of flow invariants (energy,
momentum, mass) Easy implementation Overturning waves (?) CONS
2D only, periodic only (but periodic does not always mean periodic...) It can be hard to introduce additional physical effects (forcing, damping, dissipation)
RECENT DEVELOPMENTS
Waves over rotational and irrotational currents Arbitrary bathymetry... in progress
History
The general idea about conformal mapping as an approach to water waves dates back to
Byatt-Smith 1970 Schwartz 1974 [1]
Cokelet 1977 [2]
...
The modern formulation of the method stems from the seminal work by Dyachenko et al. 1996 [3]
Then used, consolidated and extended by many others Choi & Camassa 1999 [4]
Li, Hyman & Choi 2004 [5]
Choi 2009 [6]
Milewski, Vanden-Broek & Wang 2010 [7]
(UCD) Toronto, 2013 3 / 27
Rogue waves: an example of what WE use it for
The high efficiency and accuracy of this method make it suitable for computing waves statistics over large ensembles.
If you want to know more about RWs, take a look at Dystheet al 2008[8].
Another example: large-scale wave dynamics
(From Viotti, Dutykh, Dudley & Dias 2013,submitted)
(UCD) Toronto, 2013 5 / 27
LET’S START WITH THE METHOD NOW...
Mathematical formulation
Kinematic and dynamic boundary condition on the free surface:
ζt =−φxζx +φy, φt=−1
2 φx2+φy2
−gζ+γ ζxx (1 +yx2)3/2,
at y =ζ(x,t)
Irrotational incompressible flow in the fluid volume:
∇2φ= 0, inside Ω Bottom boundary condition:
limy→−∞|∇φ|= 0 Deep Water ψ=C aty =−h(x) Finite Bottom
(UCD) Toronto, 2013 7 / 27
Conformal mapping
Ω
y
x
η
ξ ζ(x,t)
h h=h
x =x(ξ, η,t), y =y(ξ, η,t) Conformal mapping:
x+iy =X(ξ+iη) +iY(ξ+iη) =Z(ξ+iη), Z is an analytic function.
Complex potential: φandψ form a conjugate pair in the (x,y) space (harmonic functions + CR), so they do in the conformal space
φ+iψ= Φ(Z(ξ+iη))
Determine Z (ξ, η, t) = X + iY
The harmonic-conjugate pair (X,Y) needs to satisfy Xξξ+Xηη = 0, Yξξ+Yηη = 0 plus Cauchy–Riemann relations
Xξ=Yη, Xη =−Yξ.
The vertical displacement of the free surface and the bathymetry provide known boundary conditions for Y
Y(ξ,0,t) =y(ξ,t), Y(ξ,−h,t) =h(ξ,t), whereas the boundary conditions
X(ξ,0,t) =x(ξ,t), X(ξ,−h,t) =χ(ξ,t), are unknown for X.
(UCD) Toronto, 2013 9 / 27
Determine Z (ξ, η, t) = X + iY
Y is determined by solving the Laplace equation inside a uniform strip with Dirichlet data on both sides ⇒ easily done by Fourier Transform:
Y(ξ,0,t) =y(ξ,t) =
+∞
X
n=0
Yˆneink0ξ, Y(ξ,−h,t) =−h,
It is easy to verify that the most general harmonic function satisfying the above conditions has the form
Y(ξ, η,t) =−ηh h + ˆY0
1 +η
h
+
+∞
X
n=1
Yˆn
sinhnk0(η+h) sinhnk0h eink0ξ.
Determine Z (ξ, η, t) = X + iY
X is determined by enforcingC −R relations:
X(ξ, η,t) = Yˆ0−h
h ξ+x0(t) +
+∞
X
n=1
Yˆnnk0
coshnk0(η+h) sinhnk0h eink0ξ h determines the period in physical space. By choosingh = ˆY0−h the period 2π/k0 is the same in both spaces.
On the free surface we can write in compact notation:
DEEP WATER:x(ξ,t) =ξ−H[y(ξ,t)], H[y(ξ,t)]≡
Z ∞
−∞
y(ξ0,t) ξ−ξ0 dξ0. FINITE DEPTH:x(ξ,t) =ξ−Tc[y(ξ,t)] +x0(t),
Tc[y(ξ,t)]≡ Z ∞
−∞
y(ξ0,t) coth[π(ξ−ξ0)/2h]dξ0.
(UCD) Toronto, 2013 11 / 27
Determine Φ + i Ψ
The harmonic-conjugate pair (Φ,Ψ) is subject to the different set of known boundary conditions
Φξξ+ Φηη = 0, Ψξξ+ Ψηη = 0
Φ(ξ,0,t) =φ(ξ,t), Ψ(ξ,−h,t) =Q(t), The same procedure now yields
φξ =−Ts[ψξ] +U, ψξ =−Ts[φξ− U], where U ≡m(ψ)/h.
Inversion of the mapping:
physical space → conformal space
Typically one needs to assign initial conditions in physical space. Then it is necessary to invert the conformal mapping by numerical iterations.
Want to find:
y(ξ) such that y(x(ξ;y(ξ))) =ζ0(x) Fixed-point iterations
y0 =ζ0(ξ), yn+1 =ζ0(x(ξ;yn)) Linear (i.e., slow) convergence: errn+1 =C errn. (But it typically needs to be done only once.)
(UCD) Toronto, 2013 13 / 27
Evolution equations in conformal variables
An (almost) straigthforward application of chain-rule differentiation leads to the transformed Euler system (for details see, e.g., Choi & Camassa 1999 [4]):
yt = −xξ
ψξ
J
+yξTc
ψξ
J
+yξq(t),
φt = −1 2
φ2ξ−ψ2ξ
J +φξTc
ψξ
J
−gy +xξyξξ−yξyξξ
J3/2 +C(t), xξ = 1−Tc[yξ],
ψξ = −Ts[φξ− U]. (J ≡xξ2+yξ2) Discrete Fourier Transform:
η(x,t) =
N
X
n=−N
ˆ
ηn(t)eink0x, φ(x, η(x),t) =
N
X
n=−N
φˆn(t)eink0x
Nonlinear terms computed pseudospectrally (cost NlogN each) Warning: strong nonlinearities ⇒strong dealiansing
SOME EXAMPLES
(UCD) Toronto, 2013 15 / 27
Random sea simulations (Viotti et al. 2013)
Discrete Fourier expansion:
η(x,t) =
N
X
n=−N
ˆ
ηn(t)eink0x, φ(x, η(x),t) =
N
X
n=−N
φˆn(t)eink0x
Initial condition:
ˆ
ηn= [2P0(nk0)]1/2eink0ϕn, φˆn=ic(nk0)ˆηn, n= 0,1, . . .N
where
ϕn= r.v. in [0,2π], c(k) =p g/k P0(k) = √P0
2πσ0exp
−1
2
k−k0 σ0
2
Benjamin-Feir Index (Janssen 2003) BFI0=√
2sk0
σ0
, s=k0P01/2=k0hη2i1/2
2D random sea - 1
N= 65536, L= 256π, steepness = 0.04
(UCD) Toronto, 2013 17 / 27
Checking conservation laws
0 2000 4000 6000 8000
−1 0 1 2 3 4
5x 10−14 En ergy con servation
t
Error
0 2000 4000 6000 8000
−1
−0.5 0 0.5 1 1.5 2
2.5x 10−11 M omentu m con servati on
t
Error
0 2000 4000 6000 8000
−5 0 5 10 15
20x 10−10M ass con servati on : rel ati ve error
t
Error
2D random sea - 2
N= 65536, L= 256π, steepness = 0.04
(UCD) Toronto, 2013 19 / 27
Checking conservation laws
0 2000 4000 6000 8000
−1
−0.5 0 0.5 1 1.5
2x 10−4 En ergy con servation
t
Error
0 2000 4000 6000 8000
−0.1
−0.05 0 0.05 0.1 0.15
M omentu m con servati on
t
Error
0 2000 4000 6000 8000
−0.1
−0.05 0 0.05 0.1 0.15
M ass con servati on : rel ati ve error
t
Error
Extreme waves
Resolution is not uniform in physical space, this is particularly notable for very steep waves.
Loss of numerical resolution is more critical around wave crests.
58 58.5 59 59.5 60 60.5 61
−0.2
−0.1 0 0.1 0.2 0.3
x grid points surface elevation
(UCD) Toronto, 2013 21 / 27
Extended formulations
Waves over homogeneous shear flow Arbitrary bathymetry... work in progress
Waves over a uniform shear current (Choi 2009) [6]
(From Choi 2009)
Governing eqs.
Mapped governing eqs.
(UCD) Toronto, 2013 23 / 27
Arbitrary bathymetry... in progress!
h y
x
η
ξ
Ω ζ(x,t)
Soliton over submerged step
N= 16384, L= 128π, Fr= 1.15
(UCD) Toronto, 2013 25 / 27
Checking conservation laws
0 50 100 150 200
−1.5
−1
−0.5 0 0.5
1x 10−13 En ergy con servation
t
Error
0 50 100 150 200
−0.02
−0.015
−0.01
−0.005 0 0.005 0.01
M omen tu m con servati on
t
Error
0 50 100 150 200
−4
−2 0 2
4x 10−11M ass con servati on : rel ati ve error
t
Error
THANK YOU!
(UCD) Toronto, 2013 27 / 27
L. W. Schwartz.
Computer extension and analytic continuation of Stokes’ expansion for gravity waves.
J. Fluid Mech., 62:553–578, 1974.
E. D. Cokelet.
Steep gravity waves in water of arbitrary unform depth.
Phil. Trans. Royal Soc. London, 286:183–230, 1977.
A. L. Dyachenko, V. E. Zakharov, and E. A. Kuznetsov.
Nonlinear dynamics on the free surface of an ideal fluid.
Plasma Phys. Rep., 22:916–928, 1996.
W. Choi and R. Camassa.
Fully nonlinear internal waves in a two-fluid system.
Journal of Fluid Mech., 396(-1):1–36, 1999.
Y. A. Li, J. M. Hyman, and W. Choi.
A numerical study of the exact evolution equations for surface waves in water of finite depth.
Stud. Appl. Maths, 113:303–324, 2004.
W. Choi.
Nonlinear surface waves interacting with a linear shear current.
Mathematics and Computers in Simulation, 80:29–36, 2009.
P. A. Milewski, J.-M. Vanden-Broeck, and Z. Wang.
Dynamics of steep two-dimensional gravity-capillary solitary waves.
J. Fluid Mech., 664:466–477, 2010.
K. Dysthe, H. E. Krogstad, and P. M¨uller.
Oceanic rogue waves.
Ann. Rev. Fluid Mech., 40:287–310, 2008.
(UCD) Toronto, 2013 27 / 27
