Introduction to Water Waves and Simulation of Multi-Scale Nonlinear Wave Phenomena

Lian Shen

November, 2008

Contents• Ocean wave overview• Linear wave theory

– Governing equation; free-surface boundary conditions– Dispersion relationship – Wave kinematics

• Simulation of nonlinear waves– High-order spectral method for multi-scale, phase-resolving, nonlinear

wave simulation

References: • Water Wave Mechanics for Engineering and Scientists by R.G. Dean & R.A. Dalrymple,

World Scientific • Theory and Applications of Ocean Surface Waves by C.C. Mei, M. Stiassnie & D.K.P. Yue,

World Scientific• Marine Hydrodynamics by J.N. Newman, MIT Press

Website: http://www.coastal.udel.edu/faculty/rad

Ocean Wave Overview

Wave Parameter Definition

( )2 2( , ) cos cos2H x tx t a kx t

L Tπ πη ω⎡ ⎤= − = −⎢ ⎥⎣ ⎦

0=t kxa cos=η Lk /2π= kL /2π=

ta ωη cos= t T/2πω = ωπ /2=T

The water surface elevation is given by

, → periodic in , wavenumber , wavelength

→ periodic in , frequency , period

At

0=xAt ,

x .

.

.

Phase velocity or wave celerity: . kTLC // ω==

Linear Wave Theory (2D)

Potential flow:For the majority of ocean water, flow is irrotational . There exists a scalar potential function, called the velocity potential, the gradients of which equals the velocity vector:

Mathematically,

and the integral

does not depend on the particular path from to .

Physical meaning of velocity potential: Flow particles move from low region to high region with a velocity .Using the velocity potential, we replace the three-component vector with a scalar .

φ∇=u

,u wx zφ φ∂ ∂

= =∂ ∂

0)( =∇×∇=×∇= φω u

∫ •=−x

xxdutxtx

0

),(),( 0φφ

00( , ) ( , )

x

xx t u dx x tφ φ= • +∫

0x x

φ

φ

φ∇=uu

φ

Governing Laplace EquationFrom potential flow definition:

ux

∂Φ=∂

zw

∂Φ∂

=

),,( tyxΦ is solved, we can obtain description of the wave field in terms of velocity ( )vu,Once .

, we use continuity equation based on mass conservation To derive a governing equation for ),,( tyxΦ

( )vu,

0u wux z

∂ ∂∇ = + =

∂ ∂i

Substitute into it, we have

022

2

2

2

=∆Φ=Φ∇=∂Φ∂

+∂Φ∂

=⎟⎠⎞

⎜⎝⎛∂Φ∂

∂∂

+⎟⎠⎞

⎜⎝⎛∂Φ∂

∂∂

yxzzxx

The above is called the Laplace equation for ),,( tyxΦ .

Kinematic Free-surface Boundary Condition ),( txz η=

t ( )),(, txzx ppp η=

tt δ+ ( )twztux pp δδ ++ ,

Elevation of the water surface is

Surface fluid particle at time has the location

at time , the location is

.

The particle is still at the water surface, so ),( tttuxtwz pp δδηδ ++=+

Taylor expansion of right-hand-side of the above equation gives

...),(),( TOHtt

tux

txtttux pp +∂∂

+∂∂

+=++ δηδηηδδη

...),( TOHtt

tux

txtwz pp +∂∂

+∂∂

+=+ δηδηηδ

),( txz pp η= 0→tδ

tt

tux

tw δηδηδ∂∂

+∂∂

=

ux

wt ∂

∂−=

∂∂ ηη

ux∂

∂η

0=∂Φ∂

==∂∂ zat

zw

tη

Therefore, we have

Use , let , and neglect H.O.T., we have the Kinematic Free-surface Boundary Condition

Therefore, the Kinematic free-surface boundary condition obtains as

For small amplitude wave, we can neglect the quadratic term

And the Linearized Kinematic Boundary Condition becomes

.

Bernoulli’s Equation

( ) ugzpuutu 2)(1

∇++∇−=∇•+∂∂ νρ

ρNavier-Stokes equations:

( ) ϖ×−∇=×∇×−•∇=∇• uuuuuuuu )(21)()(

21 2Using

N-S equation becomeugzpuu

tu 22 )(1)(

21

∇++∇−=×−∇+∂∂ νρ

ρϖ

( ) )(21 2 gzp

t+−∇=∇∇+

∂∂∇

ρφφFor inviscid, irrotational flows,

021 2 =⎟⎟

⎠

⎞⎜⎜⎝

⎛++∇+

∂∂

∇ gzpt ρ

φφ

)(21 2 tfgzp

t=++∇+

∂∂

ρφφTherefore,

)(21 2 tFgz

tp +⎟

⎠⎞

⎜⎝⎛ +∇+∂∂

−= φφρ

Dynamic Free Surface Boundary Condition

The pressure in the water is given by the Bernoulli’s equation

)(21 2 tfgzp

t=++Φ∇+

∂Φ∂

ρ

),( txz η= )(21 2 tfg

tp ρηρρρ +−Φ∇−

∂Φ∂

−=At the water surface, , the pressure is then

app =

The pressure at the water surface on the water side is balanced by the atmospheric pressure. That is,

, which is a constant. We now have

)(21 2 tfg

tpa ρηρρρ +−Φ∇−

∂Φ∂

−=

ρ/)( aptf =In the above equation, we can set to make things simple.

021 2 =+Φ∇+

∂Φ∂ ηgt

With this we obtain the dynamic free-surface boundary condition:

For small amplitude wave, the quadratic terms can be neglected. And we have the linearized DBC:

00 ==+∂Φ∂ zatgt

η

Bottom Boundary Conditions

If the bottom is not deep, at the water bottom ( dz −= ), we have 0=wpenetrate the ocean bottom).

(water cannot

Therefore, we obtain the bottom boundary condition as

dzatz

−==∂Φ∂ 0

If the bottom is deep, the bottom boundary condition is

0 as z∇Φ→ → −∞

Summary for Linear Wave Boundary Value Problem

Governing Laplace equation: 02

2

2

2

=∂Φ∂

+∂Φ∂

yx

Free-surface boundary conditions:

0=∂Φ∂

=∂∂ zat

ztη

00 ==+∂Φ∂ zatgt

η

The above two equations can be combined to give (by eliminating η ):

002

2

==∂Φ∂

+∂Φ∂ zat

zg

t

The bottom boundary condition:

0 deep water

0 not deep

as z

at z dz

∇Φ→ →−∞∂Φ

= = −∂

For a plane progressive wave, ( ) 2 2( , ) cos cos2H x tx t a kx t

L Tπ πη ω ⎡ ⎤= − = −⎢ ⎥⎣ ⎦

, the solution for Φ is

( )[ ]

cosh 2 / 2 2( , , ) sin4 cosh 2 /

z d LgHT x tx z td L L T

π π ππ π

⎡ ⎤+ ⎡ ⎤⎣ ⎦Φ = −⎢ ⎥⎣ ⎦

Note that the hyperbolic functions are defined as ( )2

coshxx eex

−+= ( )

2sinh

xx eex−−

= ( ) xx

xx

eeeex −

−

+−

=tanh, , .

Solution of Linear Wave Boundary Value Problem

To solve the Laplace equation, we can use the separation of variable method. The solution is expressed as a product of functions of each independent variables

From the lateral periodic boundary condition, potential should be periodic in time, so we can specify

Substitute into the Laplace equation, we have

Dividing through by give us

The two terms depend on two different independent variables, so we can separate them

Solution of Linear Wave Boundary Value Problem (cont.)

Apply the lateral periodicity condition, X(x) should be periodic, so

Using the superposition principle, the potential can be written as

Apply the horizontal bottom boundary condition

This condition doesn’t change with x and t, so

Then we have

Apply the linearized dynamic free surface boundary condition

So the surface elevation is

And

The velocity potential becomes

Dispersion Relationship

ΦWe first discuss the dispersion relation which provides us with important information regarding the dependence of wave length and celerity on wave frequency (or the other ways). Substitute the solution

( )[ ]

cosh 2 / 2 2( , , ) sin4 cosh 2 /

z d LgHT x tx z td L L T

π π ππ π

⎡ ⎤+ ⎡ ⎤⎣ ⎦Φ = −⎢ ⎥⎣ ⎦

002

2

==∂Φ∂

+∂Φ∂ zat

zg

tinto the free-surface boundary condition

2

2 2tanhL hgT Lπ π

=

2

2 1LgTπ

=

L2

2tanh

LTg h

L

ππ=

TLC /=

2tanh

2

gL hL LCT

π

π= =

We obtain the dispersion relation

Say, the wave length is known. We can use the above formula to obtain the wave period

We can further get the wave celerity (phase velocity), :

The above equation shows that waves of different wavelengths travel at different speed. That is how the dispersion relation gets its name.

For deep water , the relation becomes:1 2( , tanh 1)2

h hL L

π≥ →

2 2L gL gTCT π π

= = =

2

2 2L hgT Lπ π

= LC ghT

= =For shallow water , the relation becomes:1 2 2( , tanh )20

h h hL L L

π π≤ →

Fluid Velocity and Acceleration under Wave

( ) 2 2( , ) cos cos2H x tx t a kx t

L Tπ πη ω ⎡ ⎤= − = −⎢ ⎥⎣ ⎦

For a wave , from the velocity potential solution,

( )[ ]

cosh 2 / 2 2( , , ) sin4 cosh 2 /

z d LgHT x tx z td L L T

θ

π π ππ π

⎡ ⎤+ ⎡ ⎤⎣ ⎦Φ = −⎢ ⎥⎣ ⎦

We can obtain the velocity of a fluid particle in the wave, based on definition,

( )[ ]

( )[ ]

cosh 2 / cosh 2 /2 2 2 2cos cos2 cosh 2 / cosh 2 /

z d L z d LH gT x t H x tux L d L L T T d L L T

π ππ π π π ππ π

⎡ ⎤ ⎡ ⎤+ +∂Φ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= = − = −⎢ ⎥ ⎢ ⎥∂ ⎣ ⎦ ⎣ ⎦

( )[ ]

( )[ ]

sinh 2 / sinh 2 /2 2 2 2sin sin2 cosh 2 / cosh 2 /

z d L z d LH gT x t H x twz L d L L T T d L L T

π ππ π π π ππ π

⎡ ⎤ ⎡ ⎤+ +∂Φ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= = − = −⎢ ⎥ ⎢ ⎥∂ ⎣ ⎦ ⎣ ⎦

( )[ ]

2 cosh 2 / 2 22 sincosh 2 /x

z d Lu x ta Ht T d L L T

ππ π ππ

⎡ ⎤+∂ ⎛ ⎞ ⎡ ⎤⎣ ⎦= = −⎜ ⎟ ⎢ ⎥∂ ⎝ ⎠ ⎣ ⎦

( )[ ]

2 sinh 2 / 2 22 coscosh 2 /z

z d Lw x ta Ht T d L L T

ππ π ππ

⎡ ⎤+∂ ⎛ ⎞ ⎡ ⎤⎣ ⎦= = − −⎜ ⎟ ⎢ ⎥∂ ⎝ ⎠ ⎣ ⎦

Fluid Velocity and Acceleration under Wave

Also see the website

http://www.coastal.udel.edu/faculty/rad

Ocean Wave Statistics• Ocean waves consist of progressive waves of

different frequencies, directions, phases, and amplitudes.

• Wave spectrum: ∫∫∞

=π

ωθθωη2

00

2 ),( ddS

Example of Applications: Indian Ocean Tsunami

Wave propagation for tsunami: C=√ (g depth)

Wave length ~ 330 milesWave period ~ 40 minutes

Wave phase speed 500+ mph--equivalent to jet airliner

Wave speed ≠ water speed (2 fps)

High-Order Spectral (HOS) Method for Nonlinear Ocean Wave Simulation

• Governing Laplace equation:

( )2 , , 0z t∇ Φ =x

• Fully nonlinear KBC:

( ) at ,t z x x y y z tη η η η= Φ − Φ − Φ = x

• Fully nonlinear DBC:

( )1 at ,2t ag p z tη ηΦ + + ∇Φ ⋅∇Φ = − = x

• Question:

How to solve the above equations efficiently?

Free-Surface KBC in Surface Potential Form• Surface potential

( ) ( )( ) ( ), , , , at ,η ηΦ = Φ =S t t t z tx x x x (H1)

• Chain rule

( ) at ,η η η∇ Φ =∇ Φ+Φ ∇ ⇒∇ Φ =∇ Φ −Φ ∇ =S Sz z z tx x x x x x x (H2)

( ) at ,η η ηΦ = Φ +Φ ⇒Φ = Φ −Φ =S St t z t t t z t z tx (H3)

( )/ /x ye x e y∇ ≡ ∂ ∂ + ∂ ∂xwhere the horizontal gradient operator:

• Kinematic boundary conditions (KBC) on free surface

( )0 at ,η η η+∇ Φ⋅∇ −Φ = =t z z tx x x (H4)

Substitute equations (H2) and (H3) into (H4)

( ) ( )

( ) ( )

0 at ,

1 0 at ,

η η

η η η η

η η

η+∇ Φ ⋅∇ − +∇ ⋅∇ Φ =

⇒ + ∇ Φ −Φ ∇ ⋅∇ −Φ = =

⇒ =St z

St z z

z

z t

t

x x x

x x x x x

x

(H5)

KBC in terms of surface potential

Free-Surface DBC in Surface Potential Form• Dynamic boundary conditions (DBC) on free surface

( )21 1 at ,2 2t z ag p z tη ηΦ + + ∇ Φ ⋅∇ Φ + Φ = − =x x x (H6)

( ) ( )1 at ,η η η η η⇒ = −∇ Φ ⋅∇ + +∇ ⋅∇ Φ =St z z tx x x x x (H7)

air pressure on free surfaceRewrite equation (H5)

Substitute equation (H7) into (H3)

( )( )( ) ( ) ( )2

1

+ 1 at ,

η η η

η η η η

⇒Φ = Φ −Φ −∇ Φ ⋅∇ + +∇ ⋅∇ Φ

= Φ ∇ Φ ⋅∇ Φ − +∇ ⋅∇ Φ =

S St t z z

S St z z z t

x x x x

x x x x x (H8)

Substitute equations (H2) and (H8) into (H6)

( ) ( )

( ) ( )

( ) ( ) ( )2

2

2

1 1 1 , , at

+ 1

1 1 2 2

,2 2

S St z z

S Sz z z

S St z

a

Sag t

g

p

p z t

η η η η

η η η η η

η η

Φ + + ∇ Φ ⋅

⇒Φ ∇ Φ ⋅∇ Φ − +∇ ⋅∇ Φ +

+ ∇ Φ −Φ ∇ ⋅ ∇ Φ

∇ Φ −

−Φ

+∇ ⋅∇ Φ = − =

∇ + Φ = −

⇒

x x x x

x x

x

x

x x

x

x x x

DBC in terms of surface potential

(H9)

High-Order Expansion• M-order perturbation

( ) ( ) ( )1

, , , ,=

Φ = Φ∑M

m

mz t z tx x

( ) ( ) ( ) ( )1 0

, , , , 0,!

ηη−

= =

∂Φ = Φ = Φ =

∂∑ ∑k kM M m

mSk

m kt t z t

k zx x x

( ) ( ) ( ) ( ), , ε η ε ε= Φ = = Φ =m mak O O OAssume

• Taylor expansion of surface potential about the mean free surface

(H10)

(H11)

By collect terms at each order, we obtain

( ) ( ) ( ), 0, 1, 2,...,Φ = = =m mz t f m Mx (H12)

where

( )

( ) ( ) ( )

1

1

1, 0, 2,3,...,

!

S

k kmm m

kk

f

f z t m Mk zη−

=

⎧ = Φ⎪⎨ ∂

= − Φ = =⎪ ∂⎩∑ x

(H13)

Boundary Value Problem• Laplace equations

( ) ( )2 , , 0 0 1, 2,...,∇ Φ = ≤ =m z t z m Mx (H14)

• Dirichlet boundary condition at mean free surface( ) ( ) ( ), 0, 1, 2,...,Φ = = =m mz t f m Mx (H12)

where ( )

( ) ( ) ( )

1

1

1, 0, 2,3,...,

!

S

k kmm m

kk

f

f z t m Mk zη−

=

⎧ = Φ⎪⎨ ∂

= − Φ = =⎪ ∂⎩∑ x

(H13)

• Neumann boundary condition for deep water wave( ) 0 as 1, 2,...,∇Φ → →−∞ =m z m M (H15)

• Periodic boundary conditions in horizontal directions( ) ( ) ( ) ( )( ) ( ) ( ) ( )

L , , , , 1, 2,...,

L , , , ,

⎫Φ + = Φ ⎪ =⎬Φ + = Φ ⎪⎭

m mx x

m my y

e z t z tm M

e z t z t

x x

x x(H16)

where Lx and Ly are the domain sizes in x- and y-directions, respectively.

Spectral Approach• Eigenfunction expansion

( ) ( ) ( ) ( ) ( )0

, , , 0=

Φ = Φ Ψ ≤∑N

m mn n

nz t t z zx x

For a rectangular domain, the eigenfunction can be represented as

(H17)

( ) ( ), exp | | i 0Ψ = + ⋅ ≤n n nz z zx k k x (H18)

By substituting (H17) into (H12) , the modal amplitudes can be solved successively by spectral method for any prescribed and .

ΦS

η

( ) ( )Φ mn t

Equation (H17) satisfies all but the free surface Dirichlet boundary condition (H12).

• Solving the boundary value problem

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )1 0

L L ( )

0 0

, 0 exp i

1 exp i 0,1,...,L L

= =

= Φ Ψ = = Φ ⋅

⇒Φ = − ⋅ =

∑ ∑

∫ ∫x y

N Nm m m

n n n nn n

m mn n

x y

f t z t

t f dxdy n N

x k x

k x

where is the wavenumber vector for the nth mode.( ),=n xn ynk kk

(H20)

(H19)

Time Advancement of Free-Surface Boundary Conditions• Surface vertical velocity

(H21)( ) ( ) ( ) ( )1

11 0 0

, , , 0!

ηη+−

+= = =

∂Φ = Φ Ψ =

∂∑ ∑ ∑k kM M m N

mz n nk

m k n

t t zk z

x x

• Update of the free surface boundary conditions

By substituting equation (H21) into (H5) and (H9), the free-surface boundary conditions as functions of the modal amplitude are

( ) ( ) ( ) ( )1

11 0 0

1 ,0 0!

ηη η η η+−

+= = =

⎡ ⎤∂+∇ Φ ⋅∇ − +∇ ⋅∇ Φ Ψ =⎢ ⎥∂⎣ ⎦

∑ ∑ ∑k kM M m N

mSt n nk

m k nt

k zx x x x x (H22)

( )

( ) ( ) ( )21

11 0 0

1 1 12 2

, 0!

η η η

η +−

+= = =

Φ + + ∇ Φ ⋅∇ Φ − +∇ ⋅∇

⎡ ⎤∂× Φ Ψ = −⎢ ⎥∂⎣ ⎦∑ ∑ ∑

S S St

k kM M m Nm

n n akm k n

g

t pk z

x x x x

x

(H23)

By integrating equations (H22) and (H23), the new values of and at next timestep are obtained.

ΦS η

Summary of the High-Order Spectral Method

• Procedure of HOS method– Give and as initial conditions

– Solve the modal amplitude successively from equation (H19) by the pseudo-spectral method with fast Fourier Transform

– Integrate equations (H22) and (H23) in time by the 4th-order Runge-Kutta scheme to obtain new values of and

– Repeat the above steps

0ΦS0η

( ) ( )Φ mn t

ΦS η

• Features of HOS method– Almost linear computational effort with mode number N and

perturbation order M: O(M*N*lnN)– Exponential convergence with both M and N– Low global truncation error and large stability region with 4th-order

Runge-Kutta scheme

Simulation Example: L2 Crescent Wave

t=0T

t=6T

t=3T

HOS simulation of the nonlinear evolution of a large-amplitude plane Stokes wave subject to small initial 3D perturbations.

L2 crescent waves have alternating spanwise peak and trough on two successive streamwise peaks.

Su (1982)

Simulation Example: L3 Crescent Wavet=0T

HOS simulation of the nonlinear evolution of a large-amplitude plane Stokes wave subject to small initial 3D perturbations.

L3 crescent waves have a ‘high-high-low’ (HHL) pattern of wave crest amplitudes.

t=3T

t=6T

H H LSu (1982)

Simulation Example: Wind-Wave Interaction

Simulation Example: Kelvin’s Ship Waves

http://resources.edb.gov.hk/physics/articlePic/Optics_Wave/SurfaceWaves_05s.jpg

http://www.wikiwaves.org/index.php/Image:Wake.avon.gorge.arp.750pix.jpg

Simulation Example: Large-Scale Ocean WavefieldDomain: 30km × 30kmEvolution time: 0.5hour

Irregular short-crested wavefield, sea-state ~7 (JONSWAP: Tp = 12s, Hs = 10m, cos2θ spreading)

Wave modes, N = 1.6×107

Nonlinearity order, M = 3

# time steps ~ O(104)

Computing platform: Cray XT3 with 256 processors Simulation time: 22 hours