Practical Ship Hydrodynamics Episode 13 docx

The patch method introduces basically three changes to ordinary panel methods: ž ‘patch condition’ Panel methods enforce the no-penetration condition on the hull exactly at one collocati

Boundary element methods 231

usually the panel centre The resistance predicted by these methods is for usual discretizations insufficient for practical requirements, at least if conventional pressure integration on the hull is used S¨oding (1993b) proposed therefore a variation of the traditional approach which differs in some details from the conventional approach Since his approach uses also flat segments on the hull, but not as distributed singularities, he called the approach ‘patch’ method to distinguish it from the usual ‘panel’ methods

For double-body flows the resistance in an ideal fluid should be zero This allows the comparison of the accuracy of various methods and discretizations

as the non-zero numerical resistance is then purely due to discretization errors For double-body flows, the patch method reduces the error in the resistance

by one order of magnitude compared to ordinary first-order panel methods, without increasing the computational time or the effort in grid generation However, higher derivatives of the potential or the pressure directly on the hull cannot be computed as easily as for a regular panel method

The patch method introduces basically three changes to ordinary panel methods:

ž ‘patch condition’

Panel methods enforce the no-penetration condition on the hull exactly at one collocation point per panel The ‘patch condition’ states that the integral

of this condition over one patch of the surface is zero This averaging of the condition corresponds to the techniques used in finite element methods

ž pressure integration

Potentials and velocities are calculated at the patch corners Numerical differentiation of the potential yields an average velocity A quadratic approximation for the velocity using the average velocity and the corner velocities is used in pressure integration The unit normal is still considered constant

ž desingularization

Single point sources are submerged to give a smoother distribution of the potential on the hull As desingularization distance between patch centre and point source, the minimum of (10% of the patch length, 50% of the normal distance from patch centre to a line of symmetry) is recommended

S¨oding (1993) did not investigate the individual influence of each factor, but the higher-order pressure integration and the patch condition contribute approximately the same

The patch condition states that the flow through a surface element (patch) (and not just at its centre) is zero Desingularized Rankine point sources instead

of panels are used as elementary solutions The potential of the total flow is:

i

iϕi

 is the source strength, ϕ is the potential of a Rankine point source r is the distance between source and field point Let Mibe the outflow through a patch (outflow D flow from interior of the body into the fluid) induced by a point source of unit strength

232 Practical Ship Hydrodynamics

1 Two-dimensional case

The potential of a two-dimensional point source is:

ϕ D 1

4ln r

2

The integral zero-flow condition for a patch is:

V Ð nxÐl C

i

iMiD0

nx is the x component of the unit normal (from the body into the fluid),

l the patch area (length) The flow through a patch is invariant of the coordinate system Consider a local coordinate system x, z, (Fig 6.11) The patch extends in this coordinate system from
s to s The flow through the patch is:

M D

 s

s

zdx

z

xq,zq

x

Figure 6.11 Patch in 2d

A Rankine point source of unit strength induces at x, z the vertical velocity:

z D 1

2

z
zq

x
xq 2Cz
zq 2

Since z D 0 on the patch, this yields:

M D

 s

s

1 2

zq

x
xq 2Cz2q dx D

1 2arctan

lzq

xq2Czq2
s2

The local zq transforms from the global coordinates:

zqD
nxÐxq
xc
nzÐzq
zc

xc, zcare the global coordinates of the patch centre, xq, zq of the source

Boundary element methods 233

From the value of the potential  at the corners A and B, the average velocity within the patch is found as:

ED B
A

jExB
ExAjÐ

E

xB
ExA

jExB
ExAj i.e the absolute value of the velocity is:



s D

B
A

jExB
ExAj

The direction is tangential to the body, the unit tangential is ExB
ExA /

jExB
ExAj The pressure force on the patch is:

 Ef D En



pdl D En;

2



V2Ðl



E2dl



Eis not constant! To evaluate this expression, the velocity within the patch

is approximated by:

EDa C bt C ct2

t is the tangential coordinate directed from A to B EvA and EvB are the velocities at the patch corners.The coefficients a, b, and c are determined from the conditions:

ž The velocity at t D 0 is EvA: a D EvA

ž The velocity at t D 1 is EvB: a C b C c D EvB

ž The average velocity (integral over one patch) is Ev: a C 12b C13c D Ev This yields:

a D EvA

b D6Ev
4EvA
2EvB

c D
6EvC3EvAC3EvB

Using the above quadratic approximation for Ev, the integral of Ev2 over the patch area is found after some lengthy algebraic manipulations as:



E2dl D l

 1

0

E2dt D l Ð



a2Cab C1

32ac C b

2 C 1

2bc C

1

5c 2



Dl Ð



E2C 2

15EvA
Ev C EvB
Ev

2
1

3EvA
Ev EvB
Ev



Thus the force on one patch is

 Ef D
En Ð l Ð



Ev2
V2 C 2

15EvA
Ev C EvB
Ev

2

1

3EvA
Ev EvB
Ev



234 Practical Ship Hydrodynamics

2 Three-dimensional case

The potential of a three-dimensional source is:

Figure 6.12 shows a triangular patch ABC and a source S Quadrilateral patches may be created by combining two triangles The zero-flow condition for this patch is

VEa ð Eb 1

 i

iMiD0

S

C

A

B c

b

a

Figure 6.12 Source point S and patch ABC

The first term is the volume flow through ABC due to the uniform flow; the index 1 indicates the x component (of the vector product of two sides of the triangle) The flow M through a patch ABC induced by a point source

of unit strength is
˛/4 ˛ is the solid angle in which ABC is seen from S The rules of spherical geometry give ˛ as the sum of the angles between each pair of planes SAB, SBC, and SCA minus :

˛ D ˇSAB,SBCCˇSBC,SCACˇSCA,SAB


where, e.g.,

ˇSAB,SBCDarctan
[ EA ð EB ð  EB ð EC ] Ð EB

 EA ð EB Ð  EB ð EC jEBj Here EA, EB, EC are the vectors pointing from the source point S to the panel corners A, B, C The solid angle may be approximated by AŁ/d2 if the distance d between patch centre and source point exceeds a given limit

AŁis the patch area projected on a plane normal to the direction from the source to the patch centre:

E

d D 13 EA C EB C EC

AŁD 1

2Ea ð Eb

E d d With known source strengths i, one can determine the potential and its derivatives r at all patch corners From the  values at the corners A, B,

C, the average velocity within the triangle is found as:

ED r D A
C

E

n2AB nABE C

B
A E

n2AC nACE

Boundary element methods 235

with:

E

nABD Eb
c Ð EE b

E

c2 Ec and nEACD Ec

E

b Ð Ec E

b2 E b

With known Ev and corner velocities EvA, EvB, EvC, the pressure force on the triangle can be determined:

 Ef D En



pdA D En;

2



V2ÐA



E2dA



where Ev is not constant! A D 12jEa ð Ebj is the patch area To evaluate this equation, the velocity within the patch is approximated by:

ED EvCEvA
Ev 2r2
r C EvB
Ev 2s2
s C EvC
Ev 2t2
t

r is the ‘triangle coordinate’ directed to patch corner A: r D 1 at A, and

r D0 at the line BC s and t are the corresponding ‘triangle coordinates’ directed to B resp C Using this quadratic Ev formula, the integral of Ev2 over the triangle area is found after some algebraic manipulations as:



E2dA D A Ð



E2C 1

30EvA
Ev

2C 1

30EvB
Ev

2C 1

30EvC
Ev

2

1

90EvA
Ev EvB
Ev

1

90EvB
Ev EvC
Ev

1

90EvC
Ev EvA
Ev

Numerical example for BEM

7.1 Two-dimensional flow around a body in infinite fluid

One of the most simple applications of boundary element methods is the computation of the potential flow around a body in an infinite fluid The inclusion of a rigid surface is straightforward in this case and leads to the double-body flow problem which will be discussed at the end of this chapter

We consider a submerged body of arbitrary (but smooth) shape moving with constant speed V in an infinite fluid domain For inviscid and irrotational flow, this problem is equivalent to a body being fixed in an inflow of constant speed For testing purposes, we may select a simple geometry like a circle (cylinder

of infinite length) as a body

For the assumed ideal fluid, there exists a velocity potential  such that

ED r For the considered ideal fluid, continuity gives Laplace’s equation which holds in the whole fluid domain:

 D xxCzz D0

In addition, we require the boundary condition that water does not penetrate the body’s surface (hull condition) For an inviscid fluid, this condition can be reformulated requiring just vanishing normal velocity on the body:

E

n Ð r D0

E

nis the inward unit normal vector on the body hull This condition is mathe-matically a Neumann condition as it involves only derivatives of the unknown potential

Once a potential and its derivatives have been determined, the forces on the body can be determined by direct pressure integration:

f1 D

S

pn1dS

f2 D

S

pn2dS

236

Numerical example for BEM 237

Sis the wetted surface p is the pressure determined from Bernoulli’s equation:

p D

2V

2r2

The force coefficients are then:

CxD f1

2V

2S

2V

2S

The velocity potential  is approximated by uniform flow superimposed by

a finite number N of elements These elements are in the sample program DOUBL2D desingularized point sources inside the body (Fig 6.10) The choice of elements is rather arbitrary, but the most simple elements are selected here for teaching purposes

We formulate the potential  as the sum of parallel uniform flow (of speed V) and a residual potential which is represented by the elements:

iϕ

i is the strength of the ith element, ϕ the potential of an element of unit strength The index i for ϕ is omitted for convenience but it should be understood in the equations below that ϕ refers to the potential of only the ith element

Then the Neumann condition on the hull becomes:

N



iD1

iEn Ð rϕ D Vn1

This equation is fulfilled on N collocation points on the body forming thus

a linear system of equations in the unknown element strengths i Once the system is solved, the velocities and pressures are determined on the body The pressure integral for the x force is evaluated approximately by:

S

pn1dS ³

N

 iD1 pin1,isi

The pressure, pi, and the inward normal on the hull, ni, are taken constant over each panel si is the area of one segment

For double-body flow, an ‘element’ consists of a source at z D zq and its mirror image at z D zq Otherwise, there is no change in the program

238 Practical Ship Hydrodynamics

7.2 Two-dimensional wave resistance problem

The extension of the theory for a two-dimensional double-body flow problem

to a two-dimensional free surface problem with optional shallow-water effect introduces these main new features:

ž ‘fully non-linear’ free-surface treatment

ž shallow-water treatment

ž treatment of various element types in one program

While the problem is purely academical as free surface steady flows for ships

in reality are always strongly three dimensional, the two-dimensional problem

is an important step in understanding the three-dimensional problem Various techniques have in the history of development always been tested and refined first in the much faster and easier two-dimensional problem, before being implemented in three-dimensional codes The two-dimensional problem is thus

an important stepping stone for researchers and a useful teaching example for students

We consider a submerged body of arbitrary (but smooth) shape moving with constant speed V under the free surface in water of constant depth The depth may be infinite or finite For inviscid and irrotational flow, this problem is equivalent to a body being fixed in an inflow of constant speed

We extend the theory given in section 7.1 simply repeating the previously discussed conditions and focusing on the new conditions Laplace’s equation holds in the whole fluid domain The boundary conditions are:

ž Hull condition: water does not penetrate the body’s surface

ž Kinematic condition: water does not penetrate the water surface

ž Dynamic condition: there is atmospheric pressure at the water surface

ž Radiation condition: waves created by the body do not propagate ahead

ž Decay condition: far ahead and below of the body, the flow is undisturbed

ž Open-boundary condition: waves generated by the body pass unreflected any artificial boundary of the computational domain

ž Bottom condition (shallow-water case): no water flows through the sea bottom

The decay condition replaces the bottom condition if the bottom is at infinity, i.e in the usual infinite fluid domain case

The wave resistance problem features two special problems requiring an iterative solution:

1 A non-linear boundary condition appears on the free surface

2 The boundaries of water (waves) are not a priori known.

The iteration starts with approximating:

ž the unknown wave elevation by a flat surface

ž the unknown potential by the potential of uniform parallel flow

Numerical example for BEM 239

In each iterative step, wave elevation and potential are updated yielding successively better approximations for the solution of the non-linear problem The equations are formulated here in a right-handed Cartesian coordinate system with x pointing forward towards the ‘bow’ and z pointing upward For the assumed ideal fluid, there exists a velocity potential  such that EvD r The velocity potential  fulfils Laplace’s equation in the whole fluid domain:

 D xxCzz D0

The hull condition requires vanishing normal velocity on the body:

E

n Ð r D0

E

nis the inward unit normal vector on the body hull

The kinematic condition (no penetration of water surface) gives at z D : r Ð r D z

For simplification, we write x, y, z with zD∂/∂z D0

The dynamic condition (atmospheric pressure at water surface) gives at z D

:

1

2r2Cgz D 12V2

with g D 9.81 m/s2 Combining the dynamic and kinematic boundary condi-tions eliminates the unknown wave elevation z D :

1

2r Ð rr2CgzD0

This equation must still be fulfilled at z D  If we approximate the potential

and the wave elevation  by arbitrary approximations  and , linearization about the approximated potential gives at z D :

r Ð r12r2C r Ð r   C r   Ð r12r2 C gz D0

and    are developed in a Taylor expansion about  The Taylor expan-sion is truncated after the linear term Products of    with derivatives of

  are neglected This yields at z D :

r Ð r12r2C r Ð r   C r   Ð r12r2 C gz

C[12r Ð rr2Cgz]z   D0

A consistent linearization about  and  substitutes  by an expression depending solely on ,  and  For this purpose, the original expression for  is also developed in a truncated Taylor expansion and written at z D :

 D 1

2gr

2C2r Ð r C 2r Ð rz    V2

   D 

1

22r Ð r  r2V2  g

g C r Ð rz

240 Practical Ship Hydrodynamics

Substituting this expression in our equation for the free-surface condition gives the consistently linearized boundary condition at z D :

rr[r2C r Ð r] C 12rrr2CgzC[12rrr2Cgz]z

g C r Ð rz

ð12[r2C2r Ð r  V2]  g D 0

The denominator in the last term becomes zero when the vertical particle acceleration is equal to gravity In fact, the flow becomes unstable already at 0.6 to 0.7g both in reality and in numerical computations

It is convenient to introduce the following abbreviations:

E

a D 1

2r r

2 D



xxxCzxz

xxzC Czzz



B D[

1

2rrr2Cgz]z

[rEa C gz]z

g C a2

g C a2

2

xxxzC2zzzzCgzz

C2[xzxzzCxzÐa1CzzÐa2]

Then the boundary condition at z D  becomes:

2Ear C xzxz C 2xxxC2zzzCgzBrr

D2Ear  B12 r2CV2  g

The non-dimensional error in the boundary condition at each iteration step is defined by:

ε DmaxjEar C gzj/gV

Where ‘max’ means the maximum value of all points at the free surface For given velocity, Bernoulli’s equation determines the wave elevation:

z D 1

2gV

2r2

The first step of the iterative solution is the classical linearization around uniform flow To obtain the classical solutions for this case, the above equation should also be linearized as:

z D 1

2gV

2Cr22rr

However, it is computationally simpler to use the non-linear equation The bottom, radiation, and open-boundary conditions are fulfilled by the proper arrangement of elements as described below The decay condition – like the Laplace equation – is automatically fulfilled by all elements

238 Practical Ship Hydrodynamics< /small>

7.2 Two-dimensional wave resistance problem... class="text_page_counter">Trang 10

240 Practical Ship Hydrodynamics< /small>

Substituting this expression in our equation for the free-surface... types in one program

While the problem is purely academical as free surface steady flows for ships

in reality are always strongly three dimensional, the two-dimensional problem

is