Download 4 METHODS TO CALCULATE THE 2D POTENTIAL COEFFICIENTS

4 METHODS TO CALCULATE THE 2D POTENTIAL COEFFICIENTS
Objective of the Chapter:
Presentation of methods to numerically solve the radiation two-dimensional
boundary value problem.
Two-dimensional radiation results include:
Radiation potential in the fluid
Velocity of the fluid
Free surface elevation
Hydrodynamic pressures
Added masses and damping coefficients
NUNO FONSECA - IST
4.1 – General Description of 2D Methods
(A) Methods Based on the Conformal Mapping and Ursell Theory
(a1) Circular Cylinder:
Ursell (1949) solved the linear boundary value problem of the circular
cylinder oscillating on the free surface. The velocity potential is represented
as a sum of an infinite set of multi poles satisfying the free surface
boundary condition and each being multiplied by a coefficient in order to
satisfy the body boundary condition.
Reference:
Ursell [1949], On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid,
Quarterly Journal of Mechanics and Applied Mathematics, Vol. II, 1949.
(a2) Lewis transformation
Several researchers combined the Lewis conformal mapping transformation with
the Ursell method to obtain the solution for more realistic ship cross sections.
Lewis transformation:
Transforms the ship cross sections to the circle using a scale factor and two
mapping coefficients
The mapped cross section will maintain the draft, the breadth and the area of
the ship cross section
References:
Tasai [1959], On the Damping Force and Added Mass of Ships Heaving and Pitching,
Technical Report, Research Institute for Applied Mechanics, Kyushu University, Japan,
1959, Vol. VII, No 26.
Tasai [1961], Hydrodynamic Force and Moment Produced by Swaying and Rolling
Oscillation of Cylinders on the Free Surface, Technical Report, Research Institute for
Applied Mechanics, Kyushu University, Japan, 1961, Vol. IX, No 35.
(a3) Multi-parameter Conformal Mapping
More than two mapping coefficients can be used to represent the ship cross
sections more accurately.
Example reference:
Tasai [1960], Formula for Calculating Hydrodynamic Force on a Cylinder Heaving in the
Free Surface, (N-Parameter Family), Technical Report, Research Institute for Applied
Mechanics, Kyushu University, Japan, 1960, Vol. VIII, No 31.
(B) Frank Close Fit Method
• Solution is valid for arbitrary cross sections, partly or fully submerged
• The velocity potential is represented by a distribution of sources over the
mean submerged cross section
• Green functions satisfying the linear free surface boundary condition are
used to represent the velocity potential of unit strength sources
• The density of the sources is an unknown function to be determined from
integral equations obtained by applying the body boundary condition
Reference:
Frank [1967], Oscillation of Cylinders in or below the Free Surface of Deep Fluids,
Technical Report 2375, 1967, Naval Ship Research and Development Centre,
Washington DC, U.S.A.
Numerical Solution for Frank close fit method
Boundary value problem
• ‘Q’ is a point on the
surface of the body –
fundamental source point
or influencing point
• ‘P’ is a point in the fluid
domain – the influenced
point
Applying Green theorem to the fluid volume together with the boundary
conditions one obtains the Fredholm integral equation of the second kind:
Numerical Solution for Frank close fit method
The boundary integral equation is solved numerically:
• The contour of the cross section is divided into a finite number of
segments, N
• On each segment the distribution of sources have an unknown constant
strength, while the normal velocity is known
• The former equation reduces to a set of 2N linear algebraic equations
• The major difficulty is on the evaluation of the Green function
• The numerical solution of the Green function includes some instabilities
at certain frequencies where disparate results arise (irregular
frequencies)
Cross section of a fishing vessel discretized for application of the Frank method
0.0
-1.0
-2.0
-3.0
-4.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
Advantages and Disadvantages of Conformal Mapping Methods
• Smooth solutions over all frequency range (no irregular frequencies)
• Sharp corners are not well represented
• Sections with very low sectional area coefficient may not be well represented
• Do not deal with fully submerged cross sections (like bow bulbous)
Advantages and Disadvantages of Frank Close-Fit Method
• Applicable for arbitrary cross sections, including sections with sharp corners
and fully submerged sections
• Have numerical problems at certain frequencies where the solution diverges
(irregular frequencies). There are ways of reducing this problem
4.0
3.0
Frank
B33
2.0
Conformal
mapping
1.0
0.0
0
5
10
15
ω Lpp / g
Figure 3 Non-dimensional heave
damping coefficient
20