Luke's variational principle

In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. [cite journal
author = J. C. Luke,
year = 1967
title = A variational principle for a fluid with a free surface
journal = Journal of Fluid Mechanics
volume = 27
issue = 2
pages = 395–397
doi = 10.1017/S0022112067000412] This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the so-called mild-slope equation,citation
title=Water wave propagation over uneven bottoms
author = M. W. Dingemans
year=1997
series=Advanced Series on Ocean Engineering
volume=13
publisher=World Scientific
location = Singapore
isbn=981 02 0427 2 , 2 Parts, 967 pages. See p. 271.] or using the average-Lagrangian approach for wave propagation in inhomogeneous media.cite book
author=G. B. Whitham
title=Linear and nonlinear waves
publisher=Wiley–Interscience
year=1974
isbn = 0 471 94090 9 p. 555.]

Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.cite journal
author = V. E. Zakharov,
year = 1968
title = Stability of periodic waves of finite amplitude on the surface of a deep fluid
journal = Journal of Applied Mechanics and Technical Physics
volume = 9
issue = 2
pages = 190–194
doi = 10.1007/BF00913182 Originally appeared in "Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki" 9(2), pp. 86–94, 1968.] cite journal
author = L. J. F. Broer,
year = 1974
title = On the Hamiltonian theory of surface waves
journal = Applied Science Research
volume = 29
pages = 430–446
doi = 10.1007/BF00384164] cite journal
author = J. W. Miles,
year = 1977
title = On Hamilton's principle for surface waves
journal = Journal of Fluid Mechanics
volume = 83
issue = 1
pages = 153–158
doi = 10.1017/S0022112077001104] This is often used when modelling the spectral density evolution of the free-surface in a sea state, sometimes called wave turbulence.

Both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects.

Luke's Lagrangian

Luke's Lagrangian formulation is for non-linear surface gravity waves on an — incompressible, irrotational and inviscid — potential flow.

The relevant ingredients, needed in order to describe this flow, are:
*"Φ"("x","z","t") is the velocity potential,
*"ρ" is the fluid density,
*"g" is the acceleration by the Earth's gravity,
*"x" is the horizontal coordinate vector with components "x" and "y",
*"x" and "y" are the horizontal coordinates,
*"z" is the vertical coordinate,
*"t" is time, and
*∇ is the horizontal gradient operator, so ∇"Φ" is the horizontal flow velocity consisting of ∂"Φ"/∂"x" and ∂"Φ"/∂"y",
*"V"("t") is the time-dependent fluid domain with free surface.

The Lagrangian $mathcal{L}$ , as given by Luke, is:

: $mathcal{L} = -int_{t_0}^{t_1} left{ iiint_{V(t)}
ho left [ frac{partialPhi}{partial t} + frac{1}{2} left| oldsymbol{
abla}Phi
ight|^2 + frac{1}{2} left( frac{partialPhi}{partial z}
ight)^2 + g, z
ight] ; ext{d}x; ext{d}y; ext{d}z;
ight}; ext{d}t.$

From Bernoulli's principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain "V"("t"). This is in agreement with the variational principles for inviscid flow without a free surface, found by Harry Bateman.cite journal
author = H. Bateman,
year = 1929
title = Notes on a differential equation which occurs in the two-dimensional motion of a compressible fluid and the associated variational problems
journal = Proc. R. Soc. London A
volume = 125
issue = 799
pages = 598–618
doi = 10.1098/rspa.1929.0189]

Variation with respect to the velocity potential "Φ"("x","z","t") and free-moving surfaces like "z"="η"("x","t") results in the Laplace equation for the potential in the fluid interior and all required boundary conditions — kinematic boundary conditions on all fluid boundaries and dynamic boundary conditions on free surfaces. Cite book
author=Whitham, G.B.
coauthors=
title=Linear and nonlinear waves
publisher=Wiley
year=1974
location=New York
isbn=0-471-94090-9 §13.2, pp. 434–436.] This may also include moving wavemaker walls and ship motion.

For the case of a horizontally unbounded domain free fluid surface "z"="η"("x","t") and a fixed bed "z"=-"h"("x"), Luke's variational principle results in the Lagrangian:

: $mathcal{L} = -, int_{t_0}^{t_1} iint left{ int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)}
ho, left [ frac{partialPhi}{partial t} +, frac{1}{2} left| oldsymbol{
abla}Phi
ight|^2 +, frac{1}{2} left( frac{partialPhi}{partial z}
ight)^2
ight] ; ext{d}z; +, frac{1}{2},
ho, g, eta^2
ight}; ext{d}oldsymbol{x}; ext{d}t.$

The bed-level term proportional to "h"² in the potential energy has been neglected, since it is a constant and does not contribute in the variations. Below, Luke's variational principle is used to arrive at the flow equations for non-linear gravity surface waves on a potential flow.

Derivation of the flow equations resulting from Luke's variational principle

The variation $deltamathcal{L}=0$ in the Lagrangian with respect to variations in the velocity potential "Φ"("x","z","t"), as well as with respect to the surface elevation "η"("x","t"), have to be zero. We consider both variations subsequently.

Variation with respect to the velocity potential

Consider a small variation "δΦ" in the velocity potential "Φ". Then the resulting variation in the Lagrangian is:

: $egin{align} delta_Phimathcal{L}, &=, mathcal{L}(Phi+deltaPhi,eta), -, mathcal{L}(Phi,eta) \ &=, -, int_{t_0}^{t_1} iint left{ int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)}
ho, left( frac{partial(deltaPhi)}{partial t} +, oldsymbol{
abla}Phi cdot oldsymbol{
abla} (deltaPhi) +, frac{partialPhi}{partial z}, frac{partial(delta Phi)}{partial z},
ight); ext{d}z,
ight}; ext{d}oldsymbol{x}; ext{d}t.end{align}$

Using Leibniz integral rule, this becomes, in case of constant density "ρ":

: $egin{align} delta_Phimathcal{L}, =, &-,
ho, int_{t_0}^{t_1} iint left{ frac{partial}{partial t} int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)} deltaPhi; ext{d}z; +, oldsymbol{
abla} cdot int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)} deltaPhi, oldsymbol{
abla}Phi; ext{d}z,
ight}; ext{d}oldsymbol{x}; ext{d}t \ &+,
ho, int_{t_0}^{t_1} iint left{ int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)} deltaPhi; left( oldsymbol{
abla} cdot oldsymbol{
abla}Phi, +, frac{partial^2Phi}{partial z^2}
ight); ext{d}z,
ight}; ext{d}oldsymbol{x}; ext{d}t \ &+,
ho, int_{t_0}^{t_1} iint left [ left( frac{partialeta}{partial t}, +, oldsymbol{
abla}Phi cdot oldsymbol{
abla} eta, -, frac{partialPhi}{partial z}
ight), deltaPhi
ight] _{z=eta(oldsymbol{x},t)}; ext{d}oldsymbol{x}; ext{d}t \ &-,
ho, int_{t_0}^{t_1} iint left [ left( oldsymbol{
abla}Phi cdot oldsymbol{
abla} h, +, frac{partialPhi}{partial z}
ight), deltaPhi
ight] _{z=-h(oldsymbol{x})}; ext{d}oldsymbol{x}; ext{d}t \ =, &0.end{align}$

The first integral on the right-hand side integrates out to the boundaries, in "x" and "t", of the integration domain and is zero since the variations "δΦ" are taken to be zero at these boundaries. For variations "δΦ" which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitrary "δΦ" in the fluid interior if there the Laplace equation holds:

: $Delta Phi, =, 0 qquad ext{ for } -h(oldsymbol{x}), <, z, <, eta(oldsymbol{x},t),$

with Δ=∇·∇ + ∂²/∂"z"² the Laplace operator.

If variations "δΦ" are considered which are only non-zero at the free surface, only the third integral remains, giving rise to the kinematic free-surface boundary condition:

: $frac{partialeta}{partial t}, +, oldsymbol{
abla}Phi cdot oldsymbol{
abla} eta, -, frac{partialPhi}{partial z}, =, 0. qquad ext{ at } z, =, eta(oldsymbol{x},t).$

Similarly, variations "δΦ" only non-zero at the bottom "z" = -"h" result in the kinematic bed condition:

: $oldsymbol{
abla}Phi cdot oldsymbol{
abla} h, +, frac{partialPhi}{partial z}, =, 0 qquad ext{ at } z, =, -h(oldsymbol{x}).$

Variation with respect to the surface elevation

Considering the variation of the Lagrangian with respect to small changes "δη" gives:

: $delta_etamathcal{L}, =, mathcal{L}(Phi,eta+deltaeta), -, mathcal{L}(Phi,eta) =, -, int_{t_0}^{t_1} iint left [
ho, deltaeta, left( frac{partialPhi}{partial t} +, frac12, left| oldsymbol{
abla}Phi
ight|^2, +, frac12, left( frac{partialPhi}{partial z}
ight)^2 +, g, eta
ight),
ight] _{z=eta(oldsymbol{x},t)}; ext{d}oldsymbol{x}; ext{d}t, =, 0.$

This has to be zero for arbitrary "δη", giving rise to the dynamic boundary condition at the free surface:

: $frac{partialPhi}{partial t} +, frac12, left| oldsymbol{
abla}Phi
ight|^2, +, frac12, left( frac{partialPhi}{partial z}
ight)^2 +, g, eta, =, 0 qquad ext{ at } z, =, eta(oldsymbol{x},t).$

This is the Bernoulli equation for unsteady potential flow, applied at the free surface, and with the pressure above the free surface being a constant — which constant pressure is taken equal to zero for simplicity.

Hamiltonian formulation

The Hamiltonian structure of gravity surface waves on a potential flow was discovered by Vladimir Zakharov in 1968, and rediscovered independently by Bert Broer and John Miles:

: $egin{align}
ho, frac{partialeta}{partial t}, &=, +, frac{deltamathcal{H{deltavarphi},\
ho, frac{partialvarphi}{partial t}, &=, -, frac{deltamathcal{H{deltaeta},end{align}$

where the surface elevation "η" and surface potential "φ" — which is the potential "Φ" at the free surface "z"="η"("x","t") — are the canonical variables. The Hamiltonian $mathcal{H}(varphi,eta)$ is the sum of the kinetic and potential energy of the fluid:

: $mathcal{H}, =, iint left{ int_{-h(oldsymbol{x})}^{eta(oldsymbol{x},t)} frac12,
ho, left [ left| oldsymbol{
abla}Phi
ight|^2, +, left( frac{partialPhi}{partial z}
ight)^2
ight] , ext{d}z, +, frac12,
ho, g, eta^2
ight}; ext{d}oldsymbol{x}.$

The additional constraint is that the flow in the fluid domain has to satisfy Laplace's equation with appropriate boundary condition at the bottom "z"=-"h"("x") and that the potential at the free surface "z"="η" is equal to "φ": $deltamathcal{H}/deltaPhi,=,0.$

Relation with Lagrangian formulation

The Hamiltonian formulation can be derived from Luke's Lagrangian description by using Leibniz integral rule on the integral of ∂"Φ"/∂"t":

: $mathcal{L}_H = int_{t_0}^{t_1} iint left{ varphi(oldsymbol{x},t), frac{partialeta(oldsymbol{x},t)}{partial t}, -, H(varphi,eta;oldsymbol{x},t)
ight}; ext{d}oldsymbol{x}; ext{d}t,$

with $varphi(oldsymbol{x},t)=Phi(oldsymbol{x},eta(oldsymbol{x},t),t)$ the value of the velocity potential at the free surface, and $H(varphi,eta;oldsymbol{x},t)$ the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:

: $mathcal{H}(varphi,eta), =, iint H(varphi,eta;oldsymbol{x},t); ext{d}oldsymbol{x}.$

The Hamiltonian density is written in terms of the surface potential using Green's third identity on the kinetic energy:cite journal
author = D. M. Milder,
year = 1977
title = A note on: `On Hamilton's principle for surface waves'
journal = Journal of Fluid Mechanics
volume = 83
issue = 1
pages = 159–161
doi = 10.1017/S0022112077001116]

: $H, =, frac12,
ho, sqrt{ 1, +, left| oldsymbol{
abla} eta
ight|^2};; varphi, igl( D(eta); varphi igr), +, frac12,
ho, g, eta^2,$

where "D"("η") "φ" is equal to the normal derivative of ∂"Φ"/∂"n" at the free surface. Because of the linearity of the Laplace equation — valid in the fluid interior and depending on the boundary condition at the bed "z"=-"h" and free surface "z"="η" — the normal derivative ∂"Φ"/∂"n" is a "linear" function of the surface potential "φ", but depends non-linear on the surface elevation "η". This is expressed by the Dirichlet-to-Neumann operator "D"("η"), acting linearly on "φ".

The Hamiltonian density can also be written as:

: $H, =, frac12,
ho, varphi, Bigl [ w, left( 1, +, left| oldsymbol{
abla} eta
ight|^2
ight) -, oldsymbol{
abla}eta cdot oldsymbol{
abla}, varphi Bigr] , +, frac12,
ho, g, eta^2,$

with "w"("x","t") = ∂"Φ"/∂"z" the vertical velocity at the free surface "z" = "η". Also "w" is a "linear" function of the surface potential "φ" through the Laplace equation, but "w" depends non-linear on the surface elevation "η":

: $w, =, W(eta), varphi,$

with "W" operating linear on "φ", but being non-linear in "η". As a result, the Hamiltonian is a quadratic functional of the surface potential "φ". Also the potential energy part of the Hamiltonian is quadratic. The source of non-linearity in gravity surface waves is through the kinetic energy depending non-linear on the free surface shape "η".

Further ∇"φ" is not to be mistaken for the horizontal velocity ∇"Φ" at the free surface:

: $oldsymbol{
abla}varphi, =, oldsymbol{
abla} Phiigl(oldsymbol{x},eta(oldsymbol{x},t),tigr), =, left [ oldsymbol{
abla}Phi, +, frac{partialPhi}{partial z}, oldsymbol{
abla}eta
ight] _{z=eta(oldsymbol{x},t)}, =, Bigl [ oldsymbol{
abla}Phi Bigr] _{z=eta(oldsymbol{x},t)}, +, w, oldsymbol{
abla}eta.$

Taking the variations of the Lagrangian $mathcal{L}_H$ with respect to the canonical variables $varphi(oldsymbol{x},t)$ and $eta(oldsymbol{x},t)$ gives:

: $egin{align}
ho, frac{partialeta}{partial t}, &=, +, frac{deltamathcal{H{deltavarphi},\
ho, frac{partialvarphi}{partial t}, &=, -, frac{deltamathcal{H{deltaeta},end{align}$

provided in the fluid interior "Φ" satisfies the Laplace equation, Δ"Φ"=0, as well as the bottom boundary condition at "z"=-"h" and "Φ"="φ" at the free surface.

References and notes

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