There is no flow through the free surface kinematic condition, i.e.. Differentiation of the dynamic condition with respect to time and combi-nation with the kinematic condition yields tt
Trang 1Numerical example for BEM 251
Xk
aik
X k+1
n k
i
k +1 z
z
y
y
Figure 7.5 Coordinate system used; sources i are located inside contour
2 There is atmospheric pressure everywhere on the free surface z D (dynamic condition) Then Bernoulli’s equation yields
tC12 r 2g D0
3 There is no flow through the free surface (kinematic condition), i.e the local vertical velocity of a particle coincides with the rate of change of the surface elevation in time:
z Dt
4 Differentiation of the dynamic condition with respect to time and combi-nation with the kinematic condition yields
ttCyytCzztgz D0
This expression can be developed in a Taylor expansion around z D 0 Omitting all non-linear terms yields then
ttgz D0
5 There is no flow through the body contour, i.e the normal velocity of the water on the body contour coincides with the normal velocity of the hull (or, respectively, the relative normal velocity between body and water is zero):
E
n Ð r D En Ð Ev
6 Waves created by the body must radiate away from the body:
lim
jyj!1 DRe Oϕekzei ωe tkjyj
O
Using the harmonic time dependency of the potential, we can reformulate the Laplace equation and all relevant boundary conditions such that only the
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Laplace equation:
O
yyC Ozz D0 for z < 0
Decay condition:
lim
z!1r O D0
Combined free surface condition:
ωe2
g
O
The body boundary condition is here explicitly given for the radiation problem
of the body in heave motion This will serve as an example The other motions (sway, roll) and the diffraction problem are treated in a very similar fashion The body boundary condition for heave is then:
E
nr O D iωen2
radiation condition for with respect to y and z, respectively The resulting two equations allow the elimination of the unknown constant amplitude Oϕ yielding:
i Oz D sign y Ð Oy
O
potentials, see section 6.2.1, Chapter 6 The method described here uses desin-gularized sources located (a small distance) inside the body and above the free surface The grid on the free surface extends to a sufficient distance to both sides depending on the wavelength of the created wave Due to symmetry,
and roll motion, we have anti-symmetrical source strength.) We then exploit symmetry and use source pairs as elements to represent the total potential: O
y, z D
n
iD1
iϕi
2C z zi 2] C 1
2C z zi 2] This formulation automatically fulfils the Laplace equation and the decay condition The body, free surface, and radiation conditions are fulfilled
fulfilled automatically for yi<0
The method described here uses a patch method to numerically enforce the boundary conditions, see section 6.5.2, Chapter 6 The body boundary condi-tion is then integrated over one patch, e.g between the points k and k C 1 on
Trang 3Numerical example for BEM 253
the contour (Fig 7.5):
n
iD1
i
pkC1
p k
rϕinEkdS D iωe
pkC1
p k
n2dS
n
iD1
i
pkC1
p k
rϕinEkdS D iωe ykC1yk
The integral on the l.h.s describes the flow per time (flux) through the patch
mirror image The flux for just the source without its image corresponds to
pkC1
p k
rϕinEkdS D ˛ik
2-Correspondingly we write for the elements formed by a pair of sources:
pkC1
p k
rϕinEkdS D ˛
C ik
˛ik
˛ik D arctan ExkC1ð Exk
E
xkC1Ð Exk
1
The index 1 denotes here the x component of the vector
The other numerical conditions can be formulated in an analogous way The number of patches corresponds to the number of elements The patch conditions form then a system of linear equations for the unknown element
are known, the velocity can be computed everywhere The pressure integra-tion for the patch method described in secintegra-tion 6.5.2, Chapter 6, then yields the forces on the section The forces can then again be decomposed into exciting forces (for diffraction) and radiation forces expressed as added mass and damping coefficients analogous to the decomposition described in section 4.4, Chapter 4 The method has been encoded in the Fortran routines HMASSE and WERREG (see www.bh.com/companions/0750648511)
7.5 Rankine panel method in the frequency domain
7.5.1 Theory
The seakeeping method is limited theoretically to 1 > 0.25 In practice, accu-racy problems may occur for 1 < 0.4 The method does not treat transom sterns The theory given is that behind the FREDDY code (Bertram (1998))
We consider a ship moving with mean speed V in a harmonic wave of
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the steady wave resistance problem described previously and can be solved using similar techniques
The fundamental field equation for the assumed potential flow is again Laplace’s equation In addition, boundary conditions are postulated:
1 No water flows through the ship’s surface
2 At the trailing edge of the ship, the pressures are equal on both sides (Kutta condition.)
3 A transom stern is assumed to remain dry (Transom condition.)
4 No water flows through the free surface (Kinematic free surface condition.)
5 There is atmospheric pressure at the free surface (Dynamic free surface condition.)
6 Far away from the ship, the disturbance caused by the ship vanishes
7 Waves created by the ship move away from the ship For 1 > 0.25, waves created by the ship propagate only downstream (Radiation condition.)
8 Waves created by the ship should leave artificial boundaries of the compu-tational domain without reflection They may not reach the ship again (Open-boundary condition.)
9 Forces on the ship result in motions (Average longitudinal forces are assumed to be counteracted by corresponding propulsive forces, i.e the average speed V remains constant.)
Note that this verbal formulation of the boundary conditions coincides virtually with the formulation for the steady wave resistance problem
All coordinate systems here are right-handed Cartesian systems The inertial
body’s mean velocity V, z points vertically upwards The Oxyz system is fixed
at the body and follows its motions When the body is at rest position, x, y,
The body has 6 degrees of freedom for rigid body motion We denote corre-sponding to the degrees of freedom:
E
u D fu1, u2, u3gT and ˛ D fuE 4, u5, u6gTD f˛1, ˛2, ˛3gT
The relation between the inertial and the hull-bound coordinate system is given by the linearized transformation equations:
E
x D Ex C E˛ ð Ex C Eu
E
x D Ex E˛ ð Ex Eu
Trang 5Numerical example for BEM 255
Then the velocities transform:
ED EvC E˛ ð EvC E˛tð Ex C Eut
ED Ev E˛ ð Ev E˛tð Ex C Eut
Using a three-dimensional truncated Taylor expansion, a scalar function trans-forms from one coordinate system into the other:
f Ex D f Ex C E˛ ð Ex C Eu rxf Ex
f Ex D f Ex E˛ ð Ex C Eu rxf Ex
Correspondingly we write:
rxf Ex D rxf Ex C E˛ ð Ex C Eu rx rxf Ex
rxf Ex D rxf Ex E˛ ð Ex C Eu rx rxf Ex
A perturbation formulation for the potential is used:
totalD 0 C 1 C 2 C Ð Ð Ð
h It is the solution of the steady wave resistance problem described in the
simplicity, the equality sign is used here to denote equality of low-order terms
We describe both the z-component of the free surface and the potential in a
of encounter:
total x, y, z; t D 0 x, y, z C 1 x, y, z; t
D 0 x, y, z CRe O 1 x, y, z eiωe t
total x, y; t D 0 x, y C 1 x, y; t
D 0 x, y CRe O 1 x, y eiωe t
Correspondingly the symbol O is used for the complex amplitudes of all other first-order quantities, such as motions, forces, pressures etc
The superposition principle can be used within a linearized theory Therefore the radiation problems for all 6 degrees of freedom of the rigid-body motions
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and the diffraction problem are solved separately The total solution is a linear combination of the solutions for each independent problem
6
iD1
iui
parts to take advantage of the (usual) geometrical symmetry:
w x, y, z D
w x, y, z C w x, y, z
2
w,s
Cw x, y, z w x, y, z
2
w,a
dDd,sCd,aD7C8
Thus:
1 Dw,sCw,aC
6
iD1
iuiC7C8
from section 7.3 without further comment
ag3
∂
∂z r 0 Eag
1
2 r 0 2Cg 0 D 12V2
Also suitable radiation and decay conditions are observed
The linearized potential of the incident wave on water of infinite depth is expressed in the inertial system:
ik x cos 4y sin 4 kz
eiωe t
DRe Oweiωe t
frequency of encounter k is the wave number The derivation of the
used now in the seakeeping computations, although the average boundary is
at the steady wave elevation, i.e different near the ship This may be an inconsistency, but the diffraction potential should compensate this ‘error’
We write the complex amplitude of the incident wave as:
O
E
Trang 7Numerical example for BEM 257
totalr p p0 D0
condition:
ttotalC1
total
2CgtotalCp
2Cp0
Dtotal
∂
∂t
totalC rtotalr totalDtotalz
tttotalC2rtotalrtotalt C rtotalr 12rtotal 2 C gtotalz D0
instationary terms gives at z D total:
tt 1 C2r 0 r 1 t C r 0 r 12 r 0 2C r 1 r 0
C r 1 r 12 r 0 2 C g 0 z Cg 1 z D0
abbreviations Ea, Eag, and B for steady flow contributions This yields at z D 0 :
tt 1 C2r 0 r 1 t C r 0 aEgC r 0 r 0 r r 1
C r 1 Ea C Eag C Bag3 1 D0
The steady boundary condition can be subtracted, yielding:
tt 1 C2r 0 r 1 t C r 0 r 0 r r 1 C r 1 Ea C Eag C Ba3g 1 D0
1 will now be substituted by an expression depending solely on 0 , 0 0
ttotalC12 rtotal 2CgtotalD 12V2
t 1 C12 rtotal 2Cg 0 12V2C rtotalrtotalz Cg 1 D0
insta-tionary terms and subtracting the steady boundary condition yields:
1 C r 0 r 1 Cag 1 D0
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This can be reformulated as:
1 D 1 t C r 0 r 1
ag3
By inserting this expression in the free-surface condition and performing the time derivatives leaving only complex amplitudes, the free-surface condition
ω2eCBiωe O 1 C 2iωeCB r 0 C Ea 0 C Eag r O 1
C r 0 r 0 r r O 1 D0
The last term in this condition is explicitly written:
r 0 r 0 r r O 1 D 0 x 2xx 1 C 0 y 2 1 yy C z 0 2 1 zz
C2 Ð x 0 0 y 1 xy Cx 0 0 z 1 xz C 0 y 0 z 1 yz Complications in formulating the kinematic boundary condition on the body’s surface arise from the fact that the unit normal vector is conveniently expressed
in the body-fixed coordinate system, while the potential is usually given in the inertial system The body surface is defined in the body-fixed system by the relation S Ex D0
Water does not penetrate the body’s surface, i.e relative to the body-fixed
E
n Ex Ð Ev Ex D0
E
system as:
E Ex D Ev Ex E˛ ð Ev Ex E˛tð Ex C Eut
expressed as the sum of the derivatives of the steady and the first-order potential:
E Ex D r 0 Ex C r 1 E
For simplicity, the subscript x for the r operator is dropped It should be understood that from now on the argument of the r operator determines its type, i.e r Ex D rx Ex and r Ex D rx Ex As 1 is of first order small,
1 Ex D 1 Ex D 1
system:
E Ex D r 0 x C E˛ ð Ex C Eu r r 0 Ex C r 1
Combining the above equations and omitting higher-order terms yields: E
n Ex r 0 x E˛ ð r 0 C E˛ ð Ex C Eu r r 0 C r 1
E˛ ð Ex C Eu D0
Trang 9Numerical example for BEM 259
This boundary condition must be fulfilled at any time The steady terms give the steady body-surface condition as mentioned above Because only terms of
vector identities we derive:
E
nr O 1 C OEu[ Enr r 0 iωen] C OEE ˛[En ð r 0 CEx ð Enr r 0 iωen ] D 0E where all derivatives of potentials can be taken with respect to the inertial system
becomes:
E
nr O 1 C OEu Em iωen C OEE ˛ Ex ð Em iωen C EE n ð r 0 D0
The Kutta condition requires that at the trailing edge the pressures are equal
on both sides This is automatically fulfilled for the symmetric contributions (for monohulls) Then only the antisymmetric pressures have to vanish:
itC r 0 r Oi D0
This yields on points at the trailing edge:
iωeOiC r 0 r OiD0
Diffraction and radiation problems for unit amplitude motions are solved
weight and from integrating the pressure over the instantaneous wetted surface
E
This force could be included in a similar fashion as the weight However, resistance and propulsive force are assumed to be negligibly small compared
to the other forces.)
E
vector):
E
F D
S
p Ex p0 En Ex dS C EG
E
M D
S
p Ex p0 Ex ð En Ex dS C Exgð EG
E
p Ex p0D 1 rtotal Ex 21V2Cgz C totalt Ex
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D 12 r 0 Ex 212V2Cgz
p 0
p 1
system to the body-fixed system This includes a Taylor expansion around the
transformed as usual:
E
x D Ex C E˛ ð Ex C Eu
E
n Ex D En Ex C E˛ ð En Ex
The steady parts of the equations give:
E
S 0 p 0 nEdS C EG D0
E
S 0 p 0 Ex ð En dS C Exgð EG D0
The ship is in equilibrium for steady flow Therefore the steady forces and moments are all zero
E
S 0
[ p 1 C rp 0 E˛ ð Ex C Eu ]EndS E˛ ð EG
E
S 0
[ p 1 C rp 0 E˛ ð Ex C Eu ] Ex ð En dS Exgð E˛ ð EG
where E˛ ð Ex ð En C Ex ð E˛ ð En D E˛ ð Ex ð En and the expressions for EF 0
instan-taneous wetted surface and average wetted surface still has not to be considered
The instationary pressure is divided into parts due to the incident wave, radiation and diffraction:
6
iD1
piui
Again the incident wave and diffraction contributions can be decomposed into symmetrical and antisymmetrical parts:
pw Dpw,sCpw,a
Trang 11Numerical example for BEM 261
of the motions ui, using the vector identity E˛ ð Ex EagD E˛ Ex ð Eag :
E
S 0 pwCpd EndS C
6
iD1
S 0 pinEdS
ui
C
S 0
uEagC E˛ Ex ð Eag EndS E˛ ð EG
E
S 0 pwCpd Ex ð En dS C
6
iD1
S 0 pi Ex ð En dS
ui
Exgð E˛ ð EG C
S 0
uEagC E˛ Ex ð Eag Ex ð En dS The relation between forces, moments and motion acceleration is:
E
F 1 Dm EuttC E˛ttð Exg
E
M 1 Dm Exgð Eutt C IE˛tt
I D
0x 0y 0xz
of inertia and the centrifugal moments with respect to the origin of the body-fixed Oxyz system:
7.5.2 Numerical implementation
Systems of equations for unknown potentials
The two unknown diffraction potentials and the six unknown radiation poten-tials are determined by approximating the unknown potenpoten-tials by a superposi-tion of a finite number of Rankine higher-order panels on the ship and above the free surface For the antisymmetric cases, in addition Thiart elements, section 6.4.2, Chapter 6, are arranged and a Kutta condition is imposed on collocation points at the last column of collocation points on the stern Radia-tion and open-boundary condiRadia-tions are fulfilled by the ‘staggering’ technique (adding one row of collocation points at the upstream end of the free-surface grid and one row of source elements at the downstream end of the free-surface grid) This technique works only well for 1 > 0.4
Trang 12262 Practical Ship Hydrodynamics
For the symmetrical cases, all mirror images have the same strength For the antisymmetrical case, the mirror images on the negative y sector(s) have negative element strength of the same absolute magnitude
Each unknown potential is then written as:
O
O
iϕ
strength ϕ is real for the Rankine elements and complex for the Thiart elements
The same grid on the hull may be used as for the steady problem, but the grid on the free surface should be created new depending on the wave length
of the incident wave The quantities on the new grid can be interpolated within the new grid from the values on the old grid Outside the old grid in the far field, all quantities are set to uniform flow on the new grid
For the boundary condition on the free surface, we introduce the following abbreviations:
fq D ω2eCiωeB
fqx D 2iωeCB 0 x C2a1
fqyD 2iωeCB 0 y C2a2
fqzD 2iωeCB 0 z C2a3
fqxxD 0 x Ðx 0 0 z Ð 0 z
fqxyD2 Ð 0 x Ð 0 y
fqxz D2 Ð 0 x Ð 0 z
fqyy D 0 y Ðy 0 0 z Ð 0 z
fqyzD2 Ð 0 y Ð 0 z
Then we can write the free surface condition for the radiation cases i D
1 6 :
O
i fqϕ C fqxϕxCfqyϕyCfqzϕzCfqxxϕxxCfqxyϕxy
CfqxzϕxzCfqyyϕyyCfqyzϕyz D0
where it has been exploited that all potentials fulfil Laplace’s equation Simi-larly, we get for the symmetrical diffraction problem:
O
i fqϕ C fqxϕxCfqyϕyCfqzϕzCfqxxϕxxCfqxyϕxyCfqxzϕxz
CfqyyϕyyCfqyzϕyz C fqOw,sCfqxOxw,sCfqyOw,sy CfqzOw,sz
CfqxxOw,sCfqxyOw,sCfqxzOw,sCfqyyOw,sCfqyzOw,s D0
...6 Far away from the ship, the disturbance caused by the ship vanishes
7 Waves created by the ship move away from the ship For > 0.25, waves created by the ship propagate only downstream... consider a ship moving with mean speed V in a harmonic wave of
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the... class="text_page_counter">Trang 2
252 Practical Ship Hydrodynamics< /small>
Laplace equation:
O
yyC Ozz