In
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, Luke's variational principle is a
Lagrangian variational description of the motion of
surface waves
In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at ...
on a
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
with a
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
, under the action of
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. This principle is named after J.C. Luke, who published it in 1967.
[
] This variational principle is for
incompressible and
inviscid potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
s, and is used to derive approximate wave models like the
mild-slope equation,
[
] or using the
averaged Lagrangian approach for wave propagation in inhomogeneous media.
[
]
Luke's Lagrangian formulation can also be recast into a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
formulation in terms of the surface elevation and velocity potential at the free surface.
[ Originally appeared in '' Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki'' 9(2): 86–94, 1968.][
][
] This is often used when modelling the
spectral density
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
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
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
effects, and by using
Clebsch potentials to include
vorticity.
[
]
Luke's Lagrangian
Luke's Lagrangian formulation is for non-linear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
surface gravity waves on an— incompressible, irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
and inviscid—potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
.
The relevant ingredients, needed in order to describe this flow, are:
* is the velocity potential,
* is the fluid density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
,
* is the acceleration by the Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector qu ...
,
* is the horizontal coordinate vector with components and ,
* and are the horizontal coordinates,
* is the vertical coordinate,
* is time, and
* is the horizontal gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator, so is the horizontal flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
consisting of and ,
* is the time-dependent fluid domain with free surface.
The Lagrangian , as given by Luke, is:
From Bernoulli's principle
Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. For example, for a fluid flowing horizontally Bernoulli's principle states that an increase in the speed occurs simultaneously with a decrease i ...
, this Lagrangian can be seen to be the integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
of the fluid pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
over the whole time-dependent fluid domain . This is in agreement with the variational principles for inviscid flow without a free surface, found by Harry Bateman.
Variation with respect to the velocity potential and free-moving surfaces like results in the Laplace equation for the potential in the fluid interior and all required boundary conditions
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
: kinematic boundary conditions on all fluid boundaries and dynamic boundary conditions on free surfaces.[
] This may also include moving wavemaker walls and ship motion.
For the case of a horizontally unbounded domain with the free fluid surface at and a fixed bed at , Luke's variational principle results in the Lagrangian:
The bed-level term proportional to 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 surface gravity waves on a potential flow.
Derivation of the flow equations resulting from Luke's variational principle
The variation in the Lagrangian with respect to variations in the velocity potential Φ(''x'',''z'',''t''), as well as with respect to the surface elevation , have to be zero. We consider both variations subsequently.
Variation with respect to the velocity potential
Consider a small variation in the velocity potential .[ Then the resulting variation in the Lagrangian is:
Using ]Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integrands ...
, this becomes, in case of constant density :[
The first integral on the right-hand side integrates out to the boundaries, in and , of the integration domain and is zero since the variations are taken to be zero at these boundaries. For variations which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitrary in the fluid interior if there the Laplace equation holds:
with the ]Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
.
If variations 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:
Similarly, variations only non-zero at the bottom result in the kinematic bed condition:
Variation with respect to the surface elevation
Considering the variation of the Lagrangian with respect to small changes gives:
This has to be zero for arbitrary , giving rise to the dynamic boundary condition at the free surface:
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
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
structure of surface gravity waves on a potential flow was discovered by Vladimir E. Zakharov in 1968, and rediscovered independently by Bert Broer and John Miles:[
where the surface elevation and surface potential — which is the potential at the free surface — are the canonical variables. The Hamiltonian is the sum of the kinetic and ]potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
of the fluid:
The additional constraint is that the flow in the fluid domain has to satisfy Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delt ...
with appropriate boundary condition at the bottom and that the potential at the free surface is equal to :
Relation with Lagrangian formulation
The Hamiltonian formulation can be derived from Luke's Lagrangian description by using Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integrands ...
on the integral of :[
with the value of the velocity potential at the free surface, and the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:
The Hamiltonian density is written in terms of the surface potential using Green's third identity on the kinetic energy:][
]
where is equal to the normal derivative of 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 and free surface — the normal derivative is a ''linear'' function of the surface potential , but depends non-linear on the surface elevation . This is expressed by the Dirichlet-to-Neumann operator , acting linearly on .
The Hamiltonian density can also be written as:[
with the vertical velocity at the free surface . Also is a ''linear'' function of the surface potential through the Laplace equation, but depends non-linear on the surface elevation :][
with operating linear on , but being non-linear in . As a result, the Hamiltonian is a quadratic functional of the surface potential . Also the potential energy part of the Hamiltonian is quadratic. The source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape .][
Further is not to be mistaken for the horizontal velocity at the free surface:
Taking the variations of the Lagrangian with respect to the canonical variables and gives:
provided in the fluid interior satisfies the Laplace equation, , as well as the bottom boundary condition at and at the free surface.
]
References and notes
{{physical oceanography
Fluid dynamics