Practical Ship Hydrodynamics Episode 12 pptx

Two-dimensional case For a panel of constant source strength we formulate the potential in a local coordinate system.. the assumption of constant strength, constant pressure, constant no

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Boundary element methods 211

xxD3xx xq /r2

xy D3xy yq /r2

xz D3xz zq /r2

yy D3yy yq /r2

yzD3yz zq /r2

xxzD 2/r2 C5xx dz/r2

xyz D 5xydz/r2

xzz D 5xzdz/r23x/r2

yyz D 2/r2 C5yy dz/r2

yzzD 5yzdz/r23y/r2

6.2.2 Regular first-order panel

1 Two-dimensional case

For a panel of constant source strength we formulate the potential in a local coordinate system The origin of the local system lies at the centre of the panel The panel lies on the local x-axis, the local z-axis is perpendicular

to the panel pointing outward The panel extends from x D d to x D d The potential is then

D

d

2 Ðln

x xq 2Cz2dxq

With the substitution t D x xq this becomes:

D 1

2

xCd

xd

2Ðlnt

2Cz2 dt

4

tlnt2Cz2 C2z arctant

z2t

xCd xd Additive constants can be neglected, giving:

4

xlnr1

r2

Cdlnr1r2 C z2 arctan 2 dz

x2Cz2d2 C4d

with r1Dx C d 2Cz2 and r2Dx d 2Cz2 The derivatives of the potential (still in local coordinates) are:

x D

2Ð

1

2ln

r1

r2

z D

2Ðarctan

2 dz

x2Cz2d2

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212 Practical Ship Hydrodynamics

xxD

2Ð

x C d

r1

x d

r2

xz D

2Ðz Ð

1

r1

r2

xxz D

2Ð2z Ð

x C d

r12

x d

r22

xzz D

2Ð

x C d 2z2

x d 2z2

r22

x cannot be evaluated (is singular) at the corners of the panel For the centre point of the panel itself z is:

z0, 0 D lim

z!0z0, z D

2

If the ATAN2 function in Fortran is used for the general expression of z, this is automatically fulfilled

2 Three-dimensional case

In three dimensions the corresponding expressions for an arbitrary panel are rather complicated Let us therefore consider first a simplified case, namely

a plane rectangular panel of constant source strength, (Fig 6.2) We denote the distances of the field point to the four corner points by:

r1D

x2Cy2Cz2

r2D

x 2Cy2Cz2

r3D

x2Cy h 2Cz2

r4D

x 2Cy h 2Cz2

1

4

3 s

2

x y

Figure 6.2 Simple rectangular flat panel of constant strength; orgin at centre of panel

The potential is:

D

4

h 0

0

1

x 2Cy 2Cz2

d d

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The velocity in the x direction is:

∂

∂x D

4

h

0

x

x 2Cy 2Cz2

3d d

4

h

0

x 2Cy 2Cz2

x2Cy 2Cz2

d

4ln

r3y h r1y

r2y r4y h

The velocity in the y direction is in similar fashion:

∂

∂y D

4ln

r2x r1x

r3x r4x

The velocity in the z direction is:

∂

∂z D

4

h

0

z

x 2Cy 2Cz2

3d d

4

h

0

y 2Cz2

x 2Cy 2Cz2

y 2Cz2

x2Cy 2Cz2

d

Substituting:

x2C y 2Cz2

yields:

∂

∂z D

4

hy /r4

y/r 2

zx

z2Cx 2t2dt C

hy /r3

y/r 1

zx

z2Cx2t2dt

4

arctanx

z

h y

r4

Carctanx

z

y

r2

arctanx

z

y

r1

Carctanx

z

h y

r3

The derivation used:

1

x2Ca2 dx D lnx C

x2Ca2 C C

x

x2Ca2

dx D 1

x2Ca2 CC

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1

x2Ca2

a2

x2Ca2 CC

1

a C bx2 dx D

1 p

abarctan

bx p

ab for b > 0 The numerical evaluation of the induced velocities has to consider some special cases As an example: the finite accuracy of computers can lead to problems for the above given expression of the x component of the velocity, when for small values of x and z the argument of the logarithm is rounded off to zero Therefore, for (px2Cz2 −y) the term r1ymust be substi-tuted by the approximation x2Cz2 /2x The other velocity components require similar special treatment

Hess and Smith (1964) pioneered the development of boundary element methods in aeronautics, thus also laying the foundation for most subsequent work for BEM applications to ship flows Their original panel used constant source strength over a plane polygon, usually a quadrilateral This panel is still the most popular choice in practice

The velocity is again given in a local coordinate system (Fig 6.3) For quadrilaterals of unit source strength, the induced velocities are:

∂

∂x D

y2y1

d12

ln

r1Cr2d12

r1Cr2Cd12

Cy3y2

d23 ln

r2Cr3d23

r2Cr3Cd23

Cy4y3

d34 ln

r3Cr4d34

r3Cr4Cd34

C y1y4

d41 ln

r4Cr1d41

r4Cr1Cd41

∂

∂y D

x2x1

d12

ln

r1Cr2d12

r1Cr2Cd12

Cx3x2

d23 ln

r2Cr3d23

r2Cr3Cd23

Cx4x2

d34 ln

r3Cr4d34

r3Cr4Cd34

Cx1x4

d41 ln

r4Cr1d41

r4Cr1Cd41

1

4

3

3 2

1 1

s

2

x

y

−s /2

s / 2 s / 2

s / 2

−s /2 −s /2

Figure 6.3 A quadrilateral flat panel of constant strength is represented by Hess and Smith as superposition of four semi-infinite strips

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∂

∂z Darctan

m12e1h1

zr1

arctan

m12e2h2

zr2

Carctan

m23e2h2

zr2

arctan

m23e3h3

zr3

Carctan

m34e3h3

zr3

arctan

m34e4h4

zr4

Carctan

m41e4h4

zr4

arctan

m41e1h1

zr1

xi, yi are the local coordinates of the corner points i, ri the distance of the field point x, y, z from the corner point i, dij the distance of the corner point i from the corner point j, mijDyjyi /xjxi , eiDz2Cx

xi 2 and hi Dy yi x xi For larger distances between field point and panel, the velocities are approximated by a multipole expansion consisting

of a point source and a point quadrupole For large distances the point source alone approximates the effect of the panel

For real ship geometries, four corners on the hull often do not lie in one plane The panel corners are then constructed to lie within one plane approximating the four points on the actual hull: the normal on the panel

is determined from the cross-product of the two ‘diagonal’ vectors The centre of the panel is determined by simple averaging of the coordinates of the four corners This point and the normal define the plane of the panel The four points on the hull are projected normally on this plane The panels thus created do not form a closed body As long as the gaps are small, the resulting errors are negligible compared to other sources of errors, e.g the assumption of constant strength, constant pressure, constant normal over each panel, or enforcing the boundary condition only in one point of the panel Hess and Smith (1964) comment on this issue:

‘Nevertheless, the fact that these openings exist is sometimes disturb-ing to people heardisturb-ing about the method for the first time It should

be kept in mind that the elements are simply devices for obtaining the surface source distribution and that the polyhedral body has

no direct physical significance, in the sense that the flow eventually calculated is not the flow about the polyhedral-type body Even if the edges of the adjacent elements are coincident, the normal velocity

is zero at only one point of each element Over the remainder of the element there is flow through it Also, the computed velocity is infinite

on the edges of the elements, whether these are coincident or not.’

6.2.3 Jensen panel

Jensen (1988) developed a panel of the same order of accuracy, but much simpler to program, which avoids the evaluation of complicated transcendental functions and in it implementation relies largely on just a repeated evaluation

of point source routines As the original publication is little known and difficult

to obtain internationally, the theory is repeated here The approach requires, however, closed bodies Then the velocities (and higher derivatives) can be

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computed by simple numerical integration if the integrands are transformed analytically to remove singularities In the formulae for this element, Enis the

unit normal pointing outward from the body into the fluid, c

the integral over

S excluding the immediate neighbourhood of Exq, and r the Nabla operator with respect to Ex

A Rankine source distribution on a closed body induces the following poten-tial at a field point Ex:

Ex D

S

Exq GEx, Exq dS

S is the surface contour of the body, the source strength, GEx, Exq D

1/2 ln jEx Exqj is the Green function (potential) of a unit point source Then the induced normal velocity component is:

vnEx D EnEx rEx D C

S

Exq EnEx rGEx, Exq dS C1

2Exq Usually the normal velocity is given as boundary condition Then the impor-tant part of the solution is the tangential velocity on the body:

vtEx D EtEx rEx D C

S

Exq EtEx rGEx, Exq dS

Without further proof, the tangential velocity (circulation) induced by a distribution of point sources of the same strength at point Exq vanishes:

C

SE x

rGEx, Exq EtEx dS D 0

Exchanging the designations Exand Exq and using rGEx, Exq D rGExq, Ex ,

we obtain:

C

S

rGEx, Exq EtExq dS D 0

We can multiply the integrand by Ex – which is a constant as the inte-gration variable is Exq – and subtract this zero expression from our initial integral expression for the tangential velocity:

vtEx D C

S

Exq EtEx rGEx, Exq dS C

S

Ex rGEx, Exq EtExq dS

D0

DC

S

[Exq EtEx Ex EtExq ]rGEx, Exq dS

For panels of constant source strength, the integrand in this formula tends to zero as Ex ! Exq, i.e at the previously singular point of the integral Therefore this expression forvt can be evaluated numerically Only the length S of the contour panels and the first derivatives of the source potential for each E

x, Ex combination are required

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2 Three-dimensional case

The potential at a field point Exdue to a source distribution on a closed body surface S is:

Ex D

S

Exq GEx, Exq dS

the source strength, GEx, Exq D 4jEx Exqj 1 is the Green function (potential) of a unit point source Then the induced normal velocity compo-nent on the body is:

vnEx D EnEx rEx D C

S

Exq EnEx rGEx, Exq dS C1

2Exq Usually the normal velocity is prescribed by the boundary condition Then the important part of the solution is the velocity in the tangential directions Et and Es Et can be chosen arbitrarily, Es forms a right-handed coordinate system with Enand Et We will treat here only the velocity in the t direction, as the velocity in the s direction has the same form The original, straightforward form is:

vtEx D EtEx rEx D C

S

Exq EtEx rGEx, Exq dS

A source distribution of constant strength on the surfaceSof a sphere does not induce a tangential velocity onS:

C

SEtEx rGEx, Ek dSD0

for Ex and Ek on S The sphere is placed touching the body tangentially at the point Ex The centre of the sphere must lie within the body (The radius

of the sphere has little influence on the results within wide limits A rather large radius is recommended.) Then every point Exq on the body surface can be projected to a point Ek on the sphere surface by passing a straight line through Ek, Exq, and the sphere’s centre This projection is denoted by E

k D PExq dS on the body is projected on dSon the sphere.Rdenotes the relative size of these areas: dSDRdS Let R be the radius of the sphere and Ec be its centre Then the projection of Exq is:

PExq D Exq Ec

jExq EcjRC Ec

The area ratioR is given by:

RD n Ð EE xq Ec

jExq Ecj

R

jExq Ecj

2

With these definitions, the contribution of the sphere (‘fancy zero’) can be transformed into an integral over the body surface:

C

EtEx rGEx, PExq RdS D 0

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We can multiply the integrand by Ex – which is a constant as the inte-gration variable is Exq – and subtract this zero expression from our original expression for the tangential velocity:

vtEx D C

S

Exq EtEx rGEx, Exq dS C

S

Ex EtEx rGEx, PExq RdS

D0

DC

S

[Exq EtEx rGEx, Exq Ex EtEx rGEx, PExq R] dS

For panels of constant source strength, the integrand in this expression tends to zero as Ex ! Exq, i.e at the previously singular point of the integral Therefore this expression for vt can be evaluated numerically

6.2.4 Higher-order panel

The panels considered so far are ‘first-order’ panels, i.e halving the grid spacing will halve the error in approximating a flow (for sufficiently fine grids) Higher-order panels (these are invariably second-order panels) will quadrati-cally decrease the error for grid refinement Second-order panels need to be

at least quadratic in shape and linear in source distribution They give much better results for simple geometries which can be described easily by analyt-ical terms, e.g spheres or Wigley parabolic hulls For real ship geometries, first-order panels are usually sufficient and may even be more accurate for the same effort, as higher-order panels require more care in grid generation and are prone to ‘overshoot’ in regions of high curvature as in the aftbody For some applications, however, second derivatives of the potential are needed on the hull and these are evaluated simply by second-order panels, but not by first-order panels

We want to compute derivatives of the potential at a point x, y induced

by a given curved portion of the boundary It is convenient to describe the problem in a local coordinate system (Fig 6.4) The x- or -axis is tangent

to the curve and the perpendicular projections on the x-axis of the ends of the curve lie equal distances d to the right and the left of the origin The

y-or -axis is ny-ormal to the curve The arc length along the curve is denoted

by s, and a general point on the curve is , The distance between x, y

x ,h

y, h x , y

r

r0

Figure 6.4 Coordinate system for higher-order panel (two dimensional)

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and , is:

r D

x 2Cy 2

The velocity induced at x, y by a source density distribution s along the boundary curve is:

2

d

x

y

s

r2

ds dd

The boundary curve is defined by D In the neighbourhood of the origin, the curve has a power series:

D c2Cd3C Ð Ð Ð

There is no term proportional to , because the coordinate system lies tangentially to the panel Similarly the source density has a power series:

s D 0 C1 s C 2 s2C Ð Ð Ð

Then the integrand in the above expression for r can be expressed as a function of and then expanded in powers of The resulting integrals can be integrated to give an expansion for r in powers of d However, the resulting expansion will not converge if the distance of the point x, y from the origin is less than d Therefore, a modified expansion is used for the distance r:

r2 D[x 2Cy2] 2y C 2 Drf2 2y C 2

rfD

x 2Cy2 is the distance x, y from a point on the flat element Only the latter terms in this expression for r2 are expanded:

r2 Drf2 2yc2CO3

Powers O3 and higher will be neglected from now on

1

r2 D

1

rf2 2yc2 Ð

rf2 C2yc2

rf2 C2yc2 D

1

rf2 C

2yc2

rf4 1

r4 D

1

rf4 C

4yc2

r6f The remaining parts of the expansion are straightforward:

s D

0

1 C

d

2

d D

0

1 C 2c 2d

³

0

1 C 2c22d D C2

3c

2

3

Combine this expression for s with the power series for s :

s D 0 C1 C 2 2

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Combine the expression of s with the above expression for 1/r2:

1

r2

ds

d D

1

r2fC

2cy2

rf4

1 C 2c22 D 1

r2fC

2cy2

rf4 C

2c22

rf2

Now the integrands in the expression for r can be evaluated

xD 1

2

d

x

r2

ds d d

2

d

0 C1 C 2 2 x 1

rf2 C

2cy2

rf4 C

2c22

rf2

d

4[

0

x 0 C1 x 1 Ccc x 0 C2 x 2 C2c20 ]

x0 D

d

2x

rf2 d D

xCd xd

2t

t2Cy2dt D [lnt

2Cy2 ]xCdxdDlnr12/r22

with r1 D

x C d 2Cy2 and r2D

x d 2Cy2

x1 D

d

2x

rf2 d D 2

xCd xd

tx t

t2Cy2 dt D x

0

x Cy0 y 4d

c x D

d

4x y2

rf4 d D 4y

xCd xd

tt x 2

t2Cy2 2dt2

1

y C2d 3xy

r12r22

x2 D

d

2x 2

r2f d D 2

xCd xd

tt x 2

t2Cy2 dt D x

1

x Cy1 y

Here the integrals were transformed with the substitution t D x

y D 1

2

d

y

r2

ds dd

2

d

0 C1 C 2 2 y c2

rf2 C

2cy2

rf4 C

2c22

rf2

d

4[

0

y 0 C1 y 1 Ccc y 0 C2 y 2 C2c20 ]

y0 D

d

2y

rf2 d D 2

xCd xd

y

t2Cy2 dt

D2

arctant

y

xCd

x2Cy2d2

218 Practical Ship Hydrodynamics< /small>

We can multiply the integrand by Ex – which is a constant... class="text_page_counter">Trang 10

220 Practical Ship Hydrodynamics< /small>

Combine the expression of s with the above expression for 1/r2:... which can be described easily by analyt-ical terms, e.g spheres or Wigley parabolic hulls For real ship geometries, first-order panels are usually sufficient and may even be more accurate for the

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