Solution of Wave-Body Flows, Green's Theorem
Two types of wave body interaction problems are encountered frequently in applications and solved by the methods described in this section: zero-speed linear wave body interactions in the frequency domain in 2D and 3D, forward-speed seakeeping problems in the frequency or time domain in three dimensions (linear and nonlinear).
A consensus has been reached over the past two decades that the most efficient and robust solution methods are based on Green's Theorem using either a wave-source potential or the Rankine source as the Green function. The numerical solution of the resulting integral equations in practice is in almost all cases carried out by panel methods.
Frequency-domain radiation-diffraction. U = 0
Green's Theorem generates a boundary integral equation for the complex potential over the body boundary for the proper choice of the Green function:
For any that solve the Laplace equation in a closed volume .
Define the volume and as follows:
The fluid volume is enclosed by the union of several surfaces
: mean position of body surface
: mean position of the free surface
: Bounding cylindrical surface with radius . Will be allowed to expand after the statement of Green's Theorem
: Seafloor (assumed flat) of a surface which will be allowed to approach
: Spherical surface with radius centered at point in the fluid domain
: Unit normal vector on , at point on
Define two velocity potentials :
Unknown complex radiation or diffraction potential
Green function value at point due to a singularity centered at point .
Two types of Green functions will be used:
Note that the flux of fluid emitted from is equal to .
This Rankine source and its gradient with respect to (dipoles) is the Green function that will be used in the ship seakeeping problem.
Havelock's wave source potential
...Also known as the wave Green function in the frequency domain.
Satisfies the free surface condition and near behaves like a Rankine source:
The following choice for satisfies the Laplace equation and the free-surface condition:
Verify that with respect to the argument , the velocity potential satisfies the free surface condition:
where is the Hankel function of the second kind and order zero.
Therefore the real velocity potential
Represents outgoing ring waves of the form hence satisfying the radiation condition.
A similar far-field radiation condition is satisfied by the velocity potential
It follows that on :
with errors that decay like , hence faster than , which is the rate at which the surface grows as .
It follows that upon application of Green's Theorem on the unknown potential and the wave Green function only the integrals over and survive.
Over , either by virtue of the boundary condition if the water depth is finite or as by virtue of the vanishing of the respective flow velocities at large depths.
There remains to interpret and evaluate the integral over and . Start with :
Note that the integral over is over the variable with being the fixed point where the source is centered.
In the limit as the integrand over the sphere becomes spherically symmetric and with vanishing errors
on known from the boundary condition of the radiation and diffraction problems.
It follows that a relationship is obtained between the value of at some point in the fluid domain and its values and normal derivatives over the body boundary:
Stated differently, knowledge of and over the body boundary allows the determination of and upon differentiation of in the fluid domain.
In order to determine on the body boundary , simply allow in which case the sphere becomes a sphere as :
Note that is a fixed point where the point source is centered and is a dummy integration variable moving over the body boundary .
The reduction of Green's Theorem derived above survives almost identically with a factor of now multiplying the integral since only of the surface lies in the fluid domain in the limit as and for a body surface which is smooth. It follows that:
where now both and lie no the body surface. This becomes an integral equation for over a surface of boundary extent. Its solution is carried out with panel methods described below.
The interpretation of the derivative under the integral sign is as follows:
where derivatives are taken with respect to the first argument for a point source centered at point .
Infinite domain potential flow solutions
In the absence of the free surface, the derivation of the Green integral equation remains almost unchanged using :
The Rankine source as the Green function and using the property that as
For closed boundaries with no shed wakes responsible for lifting effects the resulting integral equation for over the body boundary becomes:
Uniform flow past :
So the RHS of the Green equation becomes: