Massachusetts Institute of TechnologyProject for Subject 2.29

Formulation and numerical investigation of a new fully nonlinear model for irrotational

wave-current interactions

byAlexis Charalampopoulos

alexchar@mit.edu

2.29 Project

OBJECTIVES OF THIS WORK

• Formulate and numerically study a model for wave-current interactions (shoaling, reflection, focusing) by using

• A fully nonlinear (potential) water-wave solver based on a new efficient Hamiltonian Formulation

Water Waves and Wave-Current Interactions using HCMS 2

2.29 Project

1x

hN

N

Seabed topography

2 2 0z x , in ( , ) ,hD X t 0 1[ , ]t I t t

0h N

on ( )h X

The kinematic 0t N ,

and the dynamic 21 ( ) 02t g ,

boundary conditions on the free surface

0 1( , [ ]) , ,t I tt tX

Formulation of the problem: Differential formulation

Lateral BCLateral BC

Water Waves and Wave-Current Interactions using HCMS 3

Formulation of the problem: Hamilton’s equations

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Hamilton’s equations (Zakharov 1968, Craig & Sulem 1993)

,t G h ,

2

22

12 2

,

1t

Gg

h

x xx

x

where 21[ ,, ]2 X

g dG h x is the Hamiltonian

and , zz zzG h

x xN is the Dirichlet to Neumann (DtN) operator,

defined from the kinematic subproblem 0 10 in [ , ]hD t t

0 on ( )h h X N on ( , )X t

2.29 Project

Formulation of the problem: Hamiltonian coupled-mode formulation

The two nonlinear Hamiltonian evolution equations take the form

22

0 0( ) ( [ ] ) [ ] (| | [1) /, ]b bt hh x x x xz ,

2 2

220 02

1 1[ ] ( [ ] ) [ ]2 2

1 ( | | 1) /2

[ , ]

b b bt g

hh

x x t z

x ,

where 2 [ , ]h is a nonlocal operator, corresponding to 2 ( ; )t x .

Water Waves and Wave-Current Interactions using HCMS 5

Using the representation

2( ; ,( , , ) (( , ) , )) ,nn bn

Z z hz t t z t

xx x

Similar to the one studied in detail in [Athanassoulis & Papoutsellis PRSLA 2017], in conjunction with

Luke’s Variational Formulation (Luke 1967)

1

0

2 2 01 1| | ( )2 2

t

t zt X h

g z dz d dt

x x

2.29 Project

Formulation of the problem: Hamiltonian coupled-mode formulation

Water Waves and Wave-Current Interactions using HCMS 6

The coefficients of the substrate problem are given by

2 1 2L [ , ] A (B , B ) Cmn mn mn mn mnh x x

Amn n mhZ Z d z

, B 2

i iimn n m m n z hh

x xZ Z d z h Z Z

1, 2i

Cmn n m h n m z hhZ Z d z Z Z

N

Nonlocal operator 2 [ , ]h is obtained by solving the substrate kinematic problem (in modal form)

2L ( [ ] )[Z ][ ]mn n b h z b h m hn

h

x x , 2m , Xx

2

( , )nnt

x , Xx

1 22

1A (B , B ) [ ]2n mn n mn mn x m xn hx

N N V Z d z

xx x , 2m

2.29 Project

Formulation of the problem: Hamiltonian coupled-mode formulation

Water Waves and Wave-Current Interactions using HCMS 7

Remark: The evolution equations presented before reduce to Zakharov-Craig-Sulem formulation after the identification (modal form of the DtN operator)

20 02(| | 1) () , )( , [ ] /G h hh x x x

022

0( ) ( ) (| | 1) /[ , ]t h h x x x ,

22 20 02

1 1( ) ( | | 1) /2

[ ,2

]t hg h x x ,

Numerical solution of the substrate problem.The two-dimensional (2D) case

2.29 Project

1. The truncated substrate problem is solved at each time t

2L [ , ] ( [ ] )[Z ][ ]Mmn n b h z b h m hn

h h x x , 2 (1) ( 1)m M ,

2

Mnn

, x X , [ 3totN M equations for 3M n ’s]

1 22

1A (B , B ) [ ]2

Mn mn n mn mn x m xn hx

N N V Z dz

xx x , 2 (1) ( 1)m M

20

( )2 0( ) ( [ ] ) [ ] (| | 1) /[ , ]totN

b bt hh x x x xz ,

( )

2

2

2

2

20 0

1 1[ ] ( [ ] ) [ ]2 2

1 (| | 1) ] /[ ,2

tot

b b bt

N

g

hh

x x t z

x ,

Water Waves and Wave-Current Interactions using HCMS 8

2. Then, the evolution equations are marched in time using (22

) [ , ]totN h

Starting from initial conditions 0 0 0 0( , ) , ( , ) ( , )x t x t , the numerical scheme is implemented in two steps

2.29 Project

Numerical solution of the two evolution equations

Set T( , )U , ( , )U U x t and write the evolution equations as

[ , ]tU t U , 0 0 0( ,0) ( , )U x U , [ , ]x X a b .

Runge-Kutta 4th order: 4 inversions of the substrate problem per time step

1 1 2 3 4( / 6)( 2 2 )n nU U t .

1 [ , ]n nt U , 2 1,2 2n nt tt U

,

3 2,2 2n nt tt U

, 4 3,n nt t U t .

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Remark: For the case of solitary waves, initial conditions 0 0( , ) are derived from [Clamond & Dutykh 2013] as highly accurate solitary wave solution

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Solitary wave – propagating over an undulating bottom

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Solitary wave – propagating over an undulating bottom, reverse simulation

Regular waves interacting with a vortex ring

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Regular waves interacting with a vortex ring

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Regular waves interacting with a vortex ring

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Thank you for your

attention !!

Study of wave-current interactions using a new Hamiltonian formulation

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