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LOW FROUDE NUMBER EXPANSIONS FOR THE WAVE PATTERN AND THE WAVE RESISTANCE
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft
op gezag van de Rector Magnificus,
Prof.Dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen
op dinsdag 14 aprIl 1987 te 14.00 uur
door
Franciscus Joseph Brandsma
wiskundig ingenieur geboren te Utrecht
0. INTRODUCTION. 1
THE USUAL LINEARIZATION AND LOW FROUDE NUMBER
NON-UNIFORMITIES. 3
1.1 The nonlinear free surface condition. 3
1.2 The linearized free surface condition. 6
1.3 Low Froude number non-uniformities for the two-dimensional
problem. 7
1.4 The three-dimensional solution and Its low Froude number
expansion. 10
A LINEARIZATION OF THE FREE SURFACE CONDITION IN SLOW SHIP
THEORY. 15
2.1 A formal low Froude number expansion. 15
2.2 The boundary layer approach. 17
2.3 The problem for the perturbation potentIal. 18
2.4 The free surface condition and the wave-height. 20
LOW FROUDE NUMBER SOLUTIONS OF THE TWO-DIMENSIONAL
PROBLEM. 23
3.1 The two-dimensional problem for the perturbation potential. 23
3.2 The problem for a totally submerged body. 25
3.3 Construction of the Green's function. 27
3.4 Some rults for a submerged body. 31
3.5 Surface piercing bodies. 34
3.6 The wave-height far behind the body. 37
A MULTIPLE SCALE APPROACH. 41
4.1 The two-scale problem for the source potential. 41
4.2 The solution for the source potential. 43
4.3 The wave potential and the wave-height function. 45
4.4 The far-field expansion. 47
4.5 The near-field expansion. 49
4.6 Bow- and stern-wav. 52
4.7 Excitation coefficients for a class of hull shapes. 55
5.2 The phase function. 63
5.3 Numerical ray-tracing. 66
5.4 The final numerical procedure for the calculation of raypaths. 69
5.5 Some rults.
725.6 Calculation of the phase function, andsome rults for the
wave-fronts. 75
5.7 The amplitude function. 77
5.8 Excitation coefficients. 81
WAVE PA'TTERNS AND WAVE RESISTANCE. 85
6.1 Calculation of wave patterns. 85
6.2 Some rults for the free surface elevation. 88
6.3 The wave resistance exprsed as a quadratic functional of the disturbance
potential. 88
6.4 The wave resistance calculated from the free wave spectrum. 93
6.5 Calculation of the free wave spectrum from the far field solutionfor the
wave-height function. 95
6.6 Results for the wave spectrum. 97
6.7 Results for the wave resistance. 99
CONCLUSIONS. 103
REFERENCES. 105
SAMENVAI-rING. 107
0. INTRODUCTION.
One of the problems in the field of design is to predict the resistance of a full
ship-form from its designed lines, in order to keep the resistance as low as possible. A
component of the total resistance working on a ship, which may be reduced by an efficient design, is the resistance due to the wave making.
The problem of predicting the wave resistance for full forms, with help of model tests as well as with help of an analytical theory, has been investigated throughout many years. The work done with respect to the development of an analytical method for the calculation of wave resistance was initiated by Michell 1898. The perturbation scheme he introduced in order to obtain an approximate solution for the "exact" problem (which is too difficult to solve in closed from), is known as the "thin" ship theory. This theory led to successful predictions for the wave resistance in various cases. Later several refinements of this theory were worked out, f.i. in order to make the method also suitable for slender ships with "blunt" bows and stems in which case the original thin ship theory did not work fine.
With the introduction of large ships in the past years, the attention has been focussed on another failure of the Michell theory. For "slow" ships, that is with the Froude number based on the length of the ship as a small parameter (which is the case with the modern
very large crude carriers), this theory leads to poor results compared to measured
resistances even when thin ships are considered.
The aim of this thesis is to develop a computational method from which reliable
estimations can be obtained for the wave resistance of full ship forms when small values of the Froude number are considered.
The work presented here can roughly be divided into two parts. The first part,
consisting of chapters 1-3, is used to develop a consistent perturbation scheme in order to formulate a simplified problem of which the solution should give a good approximation
for the nonlinear waves in case of small Froude numbers. Solutions for the two-dimensional problem will be given. In the second part, consisting of chapters 4-6,
solutions for the three-dimensional problem will be constructed, and finally the desired low Froude number approximations for the wave resistance will be obtained. A more detailed enumeration of the contents of each chapter is given below.
The "exact" problem for the wave making of a ship at a calm sea will be formulated in chapt.1, and the condition to be satisfied at the free surface is shown to be a non-linear one. Then the Michell theory will be discussed which leads to a problem in which the "linearized free surface condition" occurs. The main results of this usual thin ship theory will be summarized together with the low Froude number expansions of these results as given by several authors. The non-uniformity introduced in this linearization, in case of low Froude numbers, will be shortly demonstrated.
As indicated by several authors, the usual linearization, which is carried out around the solution of an uniform stream, should, in case of low Froude numbers, be replaced by a linearization around the so called "double-body" solution. It will be shown in chapt.2 that such a linearization may be derived in a natural way from the application of the method of matched asymptotic expansions to the exact problem. The advantage of the method as used in chapt.2 is the fact that the importance of the different terms in the free surface condition becomes more clear. The final problem formulated for the first term in a
formulation used by most other authors.
For the two-dimensional case the solution of the problem as formulated in chapt.2 can be obtained in a way which is more or less similar to the method of solution in the usual linearized theory. Such a solution is given in chapt.3 where a generalized free surface condition is considered which makes it possible to show the influence of the additional terms incorporated in the present theory.
For the three-dimensional case the low Froude number solution can not be derived as straightforward as in the two-dimensional case. For that reason perturbation techniques are used. In chapt.4 the so called method of multiple scales is considered and it is shown that a solution derived in that way only will be valid in the region close to the ship's hull (in contrast to the ideas of other authors who used similar approximations in order to obtain a far field solution). When the "two-scale" solution, which is represented as a source distribution along the whole free surface, is expanded for low Froude numbers, it can be shown that the main contributions arise from excitation at corner points of the waterline, which usually are the stagnation points at bow and stern. Results for this near field solution for the bow- and stern wave systems will be given for some hull shapes.
However, in order to find an approximation for the wave resistance a far field solution for the wave-height is needed. A perturbation technique which turned out to be suitable for the continuation of the wave solution to points far away from the ship, when large
wave numbers are considered, is the "ray-method". This method will be applied in
chapt.5. Because of the fact that the equations for the phase function and the amplitude function of the ray solution only can be solved numerically in the general case, in this chapter a computer program is developed in order to calculate these quantities. For each step taken in the ray method, the realisation of this step in the final numerical method will be discussed, and each time some results will be given.
In order to complete the solution for the wave-height, found with help of the ray
method, at the points where the waves are generated excitation coefficients should be provided. These coefficients will be obtained from the two-scale near field solution. This matching of the ray-solution with the two-scale solution will be carried out in chapt.6.
The computer program developed in this work is completed with routines for the
calculation of the wave pattern and of the far field wave-spectrum. The final goal of this thesis then is reached by the calculation of the wave resistance which may be derived from the wave-spectrum. Results will be shown for some theoretical hull shapes, and comparision is made with other methods.
In chapt.7 some general conclusions with regard to the method as presented In this work will be stated.
1. THE USUAL LINEARIZATION AND LOW FROUDE NUMBER NON-UNIFORIVIITIES.
In this chapter first the nonlinear problem is formulated which has to be solved in order to calculate the wave-height and the wave resistance of a ship sailing at a constant speed in a calm sea.
Then the usual linearization is introduced which is based on the assumption that the disturbances created by the ship are small perturbations of the flow field of an uniform stream, which is expected to be valid for ships with small beam-length ratio. Results found for the first order approximation, from which good estimations were derived for the wave resistance in various cases, will be summarized.
However, when small values of the Froude number are considered, non-uniformities will occur in the solution of the linearized problem. These non-uniformities will be demonstrated for the two-dimensional problem, where an explicit solution of the second order term in the linearization can be found. It can be shown that for fixed values of the slenderness parameter the ratio of the second order term and the first order term becomes infinitely large when the Froude number tends to zero.
For the three-dimensional problem low Froude number expansions of the first order solution of the linearized problem will be discussed. A correction will be considered, in order to take care for the violation of the exact boundary condition at the hull by the
usual linearized theory. This correction seems to give a first explanation for the
discrepancy between the low Froude number expansion of the wave resistance calculated with help of the linearized theory and those obtained from measurements. However it is expected that in the same way as in the two-dimensional case, low Froude number non-uniformities are introduced because the exact free surface condition has been violated. This violation is most severe near the stagnation points at bow and stern. But when the solution of the linearized problem is analysed for small values of the Froude number, these points are just the points where the main contributions to the wave solution are generated. Hence, the validity of the application of the usual linearized free surface condition in slow ship theory becomes questionable.
1.1 The nonlinear free surface condition.
The problem to be considered is that of a ship sailing at a constant speed U at the "free surface" of a sea which is assumed to be infinitely extended. Also it is assumed that no other disturbances of the free surface are present than those generated by the ship. A Cartesian coordinate system is introduced, moving with the ship. Hence within this
system the ship is at rest and seems to be placed in a uniform stream with U as the magnitude of the velocity at infinity. A steady wave pattern behind the ship can be
observed in this system.
The x-axis is chosen along the free surface at rest, in the direction of the unperturbed incoming velocity field. The y-axis is chosen perpendicular to the free surface at rest, positive in upward direction, hence the z-axis is chosen along the free surface at rest. See also fig.l.1.
The position of the free surface S is determined by:
y= h(x,z)
(1.1.1)Fig.11 The choice for the coordinate system.
velocity potential (x,y,z) can be introduced which is related to the velocity vector by:
11= V (1.1.2)
The law of continuity then states that should be a solution of Laplace's equation in the
fluid region:
= 0
for yh(x,z)
(outside the ship) (1.1.3)In order to solve the problem for a proper set of boundary conditions should be given. The condition at infinity will at this stage be given as:
Ux + "wave solution"
for IxIc'o
(1.1.4)It may be clear that Ux occurs in this condition in order to find the correct velocity of the unperturbed field at infinity. A more mathematical specification of the condition for to
be partly a "wave solution" at infinity should be given by a proper boundedness condition together with a radiation condition which states that in a fixed coordinate system only waves may be found travelling away from the points where they are generated. As will be discussed in chapt.3, a "radiation type" condition should also state that no waves are
admitted far in front of the ship. At this moment these conditions will not be given
explicitly.
On the ship's hull Sb the kinematic condition is given by:
fl=O onS (1.1.5)
The same condition has to be satisfied at the free surface, which gives:
+
= 0
atyh(x,z)
(1.1.6)Because the unknown free surface elevation occurs in the problem for & an extra
Bernoulli's equation applied to the points at the free surface, is suitable for this purpose.
When a constant pressure is assumed above the free surface, and no wavea are admitted
far in front of the body, hence hi 0 there, the following condition is derived:
U2= gh+
IVI2
at y=h(x,z) (1.1.7)in which g is the acceleration of gravity.
The kinematic- and the dynamic condition usually are combined, leading to the
following nonlinear boundary condition to be satisfied at the free surface:
+![2
xx y.yy+ + + = 0 at y=h(x,z) (1.1.8)
with h implicitly given by:
h(x,z) = _
[U2(x,h,z)(x,h,z)&(x,h,z)]
(1.1.9)The aim of the work presented here, is to give a consistent approximation for the solution of the nonlinear problem stated above, in case of "slow" ships, that is for small values of the Froude number which is defined by:
U Fn=
(Lg)½ (1.1.10)
in wich for the length scale L the length of the ship has been taken.
In the past years many effort has been put into the development of a numerical method in order to solve the whole nonlinear problem directly (see f.i. Salveaen&v.Kerczek 1976 and Korving&Hermans 1976). These numerical methods work fine when a pressure distribution at the free surface is considered instead of a real ship form. Difficulties arise when real ship forms are involved especially for points of the waterline Cf because there both the boundary conditions (1.1.5) and (1.1.8) have to be satisfied. For low Froude numbers as considered here, an extra problem arisea because, with the wave number of the generated waves of order O(g/U2), the discretisation has to be carried out with a very fine mesh throughout the whole computational domain, in order to incorporate the rapid variations of the solution. For that reason this approach will not be followed here.
In this work perturbation techniques will be looked for which will lead to more or leas simplified problems of which the solution is expected to give a good approximation for the nonlinear wave solution In case of low Froude numbers.
The perturbation technique which was used in the past most frequently
is alinearization of the velocity potential around the potential of an uniform stream, leading to the so called "linearized free surface condition".
The low Froude number
approximations which are found for solutions satisfying this linearized condition will be discussed in this chapter.
1.2 The linearized free surface condition.
In the usual linearized theory, the "slenderness" parameter E Is Introduced as the beam-length ratio:
E B/L (1.2.1)
For small valu of e it then is expected that the disturbance potential induced by the presence of the ship is small compared to the potential of an uniform stream, throughout the greatest part of the fluid region. Hence the following expansion is suggted:
çb = Ux + e2ct2 + (1.2.2)
When this expansion is used in the expression for the free surface elevation it can be seen that:
h(x,z) = E
--
j(x,h,z) + O(E2) (1.2.3)and it may be concluded that h-O( e). Then all quantities which should be evaluated at y=h(x,z) may be expanded into Taylor series around y=0. When the expansion of b is substituted into the problem derived in the previous section and the terms with equal powers of are equated, the following problem can be stated for 1:
for yO
&fl=O
OflS,-+ "wave solution" for I x I .- co
ly + '1xx = 0
aty=0
(1.2.4)and introducing the expansion for h(x,z) by:
h=
eh1+ E2h2+ (1.2.5)the lowest order term may be calculated, once is known, by:
h1(x,z) = - 1(x,0,z) (1.2.6)
Solutions for this linearized problem have been obtained in the past and were successfully
used in order to obtain reliable estimations of the order of magnitude of the wave resistance in various cases, even for ships of which the slenderness assumption did not hold good. These solutions will be discussed in the following sections.
However, in the expansion as used here non-uniformities are introduced for realistic
hull-shapes. One of these non-uniformities is a result of the violation of the small
perturbation assumption for points near the stagnation points at bow and stern. This kind of non-uniformity is the same as in the thin airfoil theory. For wedge shaped bows with
calculated wave resistance can still be seen as a good approximation. For "blunt" bows, this kind of non-uniformity becomes more severe.
Another type of non-uniformity, which is important here, is introduced when small values of the Froude number are considered. This kind of non-uniformity will occur both for sharp bows and for blunt bows, hence even when hull shapes are considered for which the linearized theory may be expected to give good results for the wave resistance, a low Froude number expansion of the solution will give rise to difficulties (see f.i. Dagan 1975 and Ogilvie 1976). In the following sections these low Froude number approximations will be discussed.
In order to investigate the nature of this last type of non-uniformity, attention should be paid to the solution for the second term of the expansion (1.2.2). For convenience sake this will be done for the two-dimensional case, where bodies are considered which are infinitly long in z-direction. The problem to be solved for 2 can be obtained following the same procedure as used for t and the final result, for the two-dimensional case, is written as:
with: D2(x) = - + 21yl1xy + - 1x1xxyJyO (1.2.8)
The second term of the expansion for the free surface elevation then is given by:
h2(x) = -
[2x + (j+y) -
- &x&xyIyO (1.2.9) In the next section solutions will be given for cf and 12 and the first two terms of the solution for the wave-height will be evaluated far behind the body. It will be shownthere that the ratio h2/ h1 will tend to infinity for Fn 0, when a fixed value of e is considered. This kind of non-uniformity is the same as in other perturbation problems where two different small parameters occur.
1.3 Low Froude number non-uniformities for the two-dimensional problem.
The non-uniformity introduced in the usual linearized theory can most easily be
demonstrated when two-dimensional bodies are considered. Several authors, f.i. Tuck
1965 Salvesen 1969 and Dagan 1975, showed this non-uniformity by solving the
problems for the first two terms of expansion (1.2.2). Some results will be summarized
below.
All of them considered submerged bodies of cylindrical shape, infinitely extended in the z-direction. The solution for t may be represented by a source distribution along the body's boundary. For such a solution then the influence function of a source of unit
2xx + 2yy =
for yO
2n°
onSb- "wave solution"
for Ix-.00
strength beneath the free surface is needed. This source solution, which has to satisfy the
linearized free surface condition given in (1.2.4) and has to show a proper wave
behaviour, can be found with help of complex analysis. Introducing: zxiy and denoting the location of the source by s= +i the rerult is given by (see f.i. Kochin,Kibel&Roze
1964):
G(x,y;,) =
ReE_log{(z-)(z-')} - ±.exp[-ikzlf
exp[iktldt (1.3.1)in which the wave number k Is given by: k-gf U2, and * denotes the complex conjungate. The solution for then can be written as:
=
fa (s)G(x,y(s),r (s))ds
(1.3.2)The source strength a has to be determined from the condition at the body's boundary. With the body's contour given by y= - d±f(x), with f-O( 1), in the linearized theory the body is approximated by a straight line between 0 and L at depth d. Then a has to be solved from:
a (s)G(x,- d+0;s,-d)ds
= EUf (1.3.3)For low Froude numbers (hence large k) it then can be easily shown that for a the usual thin airfoil source strength a 2 EUf can be taken approximately. The error introduced at the body's boundary decays as exp[-kdl for increasing values of k.
In order to keep the calculations simple, here a body is considered which contour can be approximated by the closed streamline generated by a source and sink of strength UL located at (0,-d) and (L,-d) respectively, which are placed in a uniform stream. It then is obtained that:
(z-zo)(z-z)
1(x,y)= UL Re[ log ±
(ZZi)(ZZ)
z z 1 1 ± exp[-ikzJf
- ,-}exp[iktldt]
ITt-z
t-z0
(1.3.4)in which z0=-id and z1=-L-id.
From this expression the first order approximation for the wave-height far behind the body can be obtained by differentiation with respect to x and taking x- +oo (see also (1.2.6)). For the integral then the countour can be closed In the upper half of the complex t-plane and only the residues of the poles at t=z and t=z give a contribution resulting
into:
h1(x) - Re[-2L(exp[ikLl- 1)exp[-kd]exp[-ikx] I for x- oo (1.3.5)
h1(x) - A1cos(kxc) with: A1 = 4L[1+cos(kL)Iexp[kd] (1.3.6) In order to get the second order approximation for the problem for 2 as given in (1.2.7) has to be solved. It can be shown that the solution for b2 satisfying Laplace's equation and the free surface condition from (1.2.7), may be writtenas:
+00
2(x,y) = . f D2(s)G(x,y;s,0)ds (1.3.7)
with G the source solution as in (1.3.4) and D2, given by (1.2.8), completely determined by the solution for . The error introduced with respect to the condition at the body's
boundary, again can be shown to decay as exp[-kd} for Increasing wave number k.
Evaluation of 2 is a very tedious task. With use of the expression for c as given In (1.3.4) and the expression for G, it can be shown that for large k the main contribution to the the far field expression for 2 is given by (see Dagan 1975):
2(x,y) - Re[ 2bkL2exp[kdl(exp[ikL]-1)exp[ikxJ1 (1.3.8)
with: b= I(½+2C+1n4+ln[(1+d2)/d2]) in which C Is Euler's constant. For the second order term in the expansion for the free surface elevation (see (1.2.9)) it then is finally obtained that for large k:
h2(x) - A2cos(kxc) for
x
with: A2 - O(kL2exp[kdl) (1.3.9)Using the expressions for h1 and h2 in the expansion for the free surface elevation the result may be summarized as:
li(x)- (B1 + B2 + 0(E3)) cos(kxc)
for xoo
(1.3.10)with: B1 - O(eLexp[kdj) and B2 - O(E2kL2exp[kd]) for Fn-+0
Although both B1 and B2 will vanish for Fn* 0,
theirratio will be of order
B2/ B1-O( EFn2) and will tend to infinity for a fixed value of E.
Hence a non-uniformity arises for Fn+ 0. For the body shapeas considered above, the usual linearized theory will only be valid if c-O(Fn2).
For general two-dimensional submerged bodies of finite thickness the same sort of result can be obtained by considering the source distribution r=2EUf. It then can be shown, as was done by Dagan 1975, that the non-uniformity depends on the shape of the leading edge of the body (by means of the strength of the singularity at the leading edge introduced by the thin airfoil approximation). For the order of magnitude of the ratio B2/B1 it was found there that, B2/B1-O(EFn2) for a body with a blunt bow which may be described by the "source-sink" model as above, B2! B1-O( EFn1) for a body with a leading edge of elliptical shape, and B21B1-O( ElnF) for a body with a wedge shaped bow. Hence the non-uniformity becomes weaker as the shape of the body becomes finer, but will always be present for any bow shape.
It then may be clear that an expansion of the solution of the usual linearized problem for low Froude numbers will give rise to non-uniform results for any hull shape, even when the slenderness approximation may be assumed to be valid. It was suggested by
Dagan 1975 that this kind of non-uniformity can be removed by introduction of a
coordinate straining in the solution of the linearized theory, just as in the usual thin airfoil theory. However an extension of such a method to the three-dimensional case will be extremely difficult, because the second order solution of the linearized problem, needed for such an approach, then only can obtained numericzlly.
In the work presented here the approach of matched asymptotic expansions will be used, as suggested by f.i. Ogilvie 1968, for the derivation of a uniformly valid first order solution. The problem to be solved for such a solution will be derived in chapt.2.
1.4 The three-dimensional solution of the linearized problem and its low Froude number
expansion.
In the three-dimensional case, the wetted part of the ship's hull Sb (see fig.1.1) is represented by z=-f-ef(x,y) for g(x)y0 with g(0)=g(L)=0. With < <1, the "thin"
ship approximation, which led to the linearized problem stated in section 1.2, will be
valid.
In the usual linearized theory of the wave making problem, which was developed by
Michell 1898, a source distribution along the centerplane of the ship
(0(xL ,g(xjy0 and z=0) is considered. The influence function of
a source of unit strength at L= ( , )T wichsatisfies the linearized free surface condition given in (1.2.4)and which shows a proper wave behaviour, is given by (see f.i. Wehausen&Laitone 1960):
4
1{_+-L+Re[2-Efsec2O.
4irr
r1 2 exp[v(y+-)1exp[iv{(x)cosO+(z)sinO}]dvde I } ksec2& i0in which r= ixLI , r1= ix_(,_i)T I, and k again is the wave number k=g/U2. This
source solution is usually referred to as the Kelvin source.
For the source strength then also the thin airfoil approximation a = 2 eUf is taken and it is assumed that i may be written as:
1(x,y,z) = 2Uf f
0 g(e)
(1.4.1)
(1.4.2) For the first term in the expansion of the free surface elevation it then is derived:
h1(x,z) = - -
f f
(1.4.3)0 g()
The first order approximation of the wave resistance can be found when the x-component of the pressure force induced by is integrated over the ship's hull (here approximated
R =
4pUZE2f J' f,(x,y)f f
0 -g(x) 0 -g(E)
Using the properties of G, the expression for R can be written in the following form (see f.i. Wehausen 1973):
R=
4pgk fII(u,k)12v2(v2_l)_½dvwith: I(v,k)
= f f f(x,y)exp[ivkx+v2ky]dydx
0 -g(x)
The integral (1.4.5) is often referred to as the Michell integral. In the past, much work has been done to obtain accurate numerical evaluations of this Michell integral, and good agreement was found between measured wave resitances, and those approximated with
help of this integral. Even for ship hull's for which the validity of the thin ship
approximation is questionable the Michell theory gave good results except for the low
speed region.
The expressions for the first order approximations for the wave-height and the wave resitance can be expanded for large values of k (small values of the Froude number). The following steps then have to be considered. For large k it can be shown that the main contribution to G is given by the residue at the pole in the integral over v in (1.4.1). Using this main contribution of G in the expression for the wave-height, the integrals
over and i in (1.4.4) can be expanded for large k with help of integration by parts. In this way it then is shown that the main contributions to the wave-height result from the excitation at the corner points at bow and stern (0,0,0) and (L,0,0) respectively. The remaining integral over e (see (1.4.1)) then finally can be expanded for large k by means of the method of stationary phase. Details of the procedure sketched above can be found in Keller&Ahluwalia 1976. The low Froude number approximation for h1, is rather complicated and will not be given explicitly here. More details can be found in chapt.4 and chapt.6 where the results of the method presented in this work are compared to those
of the linearized theory. The most interesting features of the low Froude number
expansion may be summarized as follows:
In a point (x,0,z) four different wave contributions may be found. Two generated by the stagnation point at the bow, and the other two generated by the stagnation point at the stern. The region where these contributions are found is determined by:
i Li
8" for bowwaves and
I I 8 for sternwavesx
xL
which means that the wave contributions found in this way are restricted to wakes
generated
by bow and
stern, bounded by straight lines with direction angleYK=arctan(8_½) 19°28, the half angle of the Kelvin wave pattern (which is the pattern
generated by the point source (1.4.1)).
The low Froude number expansion for the wave resistance for hulls which intersect the free surface vertically at bow and stern, is given by (see f.i. Wehausen 1973):
(1.4.4)
(1.4.5)
(1.4.6)
(1.4.7)
When this expression for the wave resistance is compared to experimental data for
realistic ship hulls, it is found that for low Froude numbers, for which this expression is assumed to be valid, the actual wave resistance is highly over-estimated by (1.4.8). Although the discrepancy becomes larger when the beam length ratio increases, also for very thin ships for which the slenderness approximation is valid, a considerable difference between the measured results and those of the low Froude number expansion of the Michell integral can be observed.
One of the reasons for this difference may be the non-uniformity introduced by the assumption that a source distribution along the centerplane had to be considered, because in that way the exact condition at the hull has been violated. As mentioned before, this is
the same type of non-uniformity as introduced in the usual thin airfoil theory. A
correction for this violation can be found when a source distribution is considered over the actual ship hull as done by Brard 1972.
A proper application of Green's theorema to the fluid region with (1.4.1) as the Green's function leads then to the following expression for (see Brard 1972 and also Hermans
1980):
E1(x,y,z)
ffa ()G()dS(.) +
- .. /na
(1.4.9)in which (n,n,n)T denotes the normal vector to Sb. For the source strength
a then a
complicated integral equation has to be solved. This will not be done here. Only the
consequences of this approach to the results of the thin ship approximation will be
discussed. When now the thin airfoil source strength a 2 EUfX is used and the ship's hull
again is approximated by It's centerplane, the expression for the first term in the
expansion for the free surface elevation derived from (1.4.9) is given by:
h1(x,z) = -
f f
+± . (1.4.10)
where use have been made of nd= E2f?(l± E2f/)½d_E2f?de.
When this is compared to the expression found in (1.4.3) it can be concluded that that the line integral in (1.4.10) can be seen as a correction term, correcting for the violation of the exact condition at the hull. When now an expansion is made for large k it can be easily seen that the contribution of the line integral will be of the same order in k as the contribution of the surface integral but with opposite sign (see f.i. Hermans 1980). For both integrals the main contributions again are found resulting from bow and stern. But then the correction term indeed gives rise to a decrease in wave-height and consequently
also a lower wave resistance. However with respect to the small paremeter e the
contribution of the line integral to the fr surface elevation h(x,z) still will be of order O( e3) for ships with a sharp wedge-shaped bow and stern (with f-O( 1) there), which means that also higher order terms in the expansions for b and h should be incorporated. For blunt bows the value of at bow and stern will be proportional to E with p> 0. It
then may be clear that the decrease in wave-height will become more significant. It was shown by Brard 1972, that in case of an elliptical cylinder, for which the exact surface source density is known, the low Froude number contribution of the integral over the hull in (1.4.9), is completeley canceled out by the contribution of the integral along the waterline.
Besides the difficulties for low Froude numbers shown above, it may be expected that the violation of the exact frea surface condition, when the linearized free surface condition is used, lead to the same type of non-uniformities as shown in the two-dimensional case.
It may be clear that this violation will be most severe in the region where the 1argt
deviations from the uniform stream occur, that means for points near the bow and stern.
But it was shown above that in case of low Froude numbers, the most important
contributions were generated just by these points. Hence it is advisable to reconsider the linearization, described in section 1.2, when small values of the Froude number are involved. That will be done in the next chapter.
2. A LINEARIZATION OF THE FREE SURFACE CONDITION IN SLOW SHIP THEORY.
In this (short) chapter a free surface condition is derived, to be used when small values of the Froude number are considered. Such a condition has been derived before by several authors, in most cases as a result of a linearization around theso called "double-body" solution. However, most authors like Ogilvie 1968, Baba 1976 and Maruo 1980, used
estimations of the order of magnitude of the different terms beforehand which are questionable. The most consistent method to derive a correct low Froude number
problem, seems to be the method of matched asymptotic expansions as introduced by Hermans (1974 and 1980) for this kind of problem. That approach will also be used here, however some corrections are made, and the final result will be different. As a direct consequence of the matching procedure it is shown that the concept to be used isquite similar to the process of linearization around the double-body solution. This in contrast to the approach proposed by Keller 1979, who used more terms of theso called "outer" expansion as a start for the boundary layer solution.
The most important difference between the free surface condition derived here and those of Hermans and other authors, can be found in the terms which contain a first order derivative of the perturbation potential. The justification for incorporating these terms (with correct coefficients) in the lowest order equations, willbe postponed untill the next chapter. The coefficients for these terms found here are in agreement with the work of Eggers (1980 and 1981), however he used coefficients in thepart with the second order derivatives which are different from the result presented here.
The final result is a free surface condition, linear in the perturbation potential, in which all terms can be evaluated in the plane which coincides with the location of the free surface in absence of the ship. This free surface condition will be used in the remaining part of this work.
2.1 A formal low Froude number expansion.
Again the nonlinear problem for the velocity potential is considered
= 0
for yh(x,z)
+{2
+ ± 2 ++ 24X). +
±= 0
at y=h(x,z) with: h(x,z) =[U2(x,h,z)b(x,h,z)(x,h,z)]
OIl Sb- Ux + "wave solution"
for II+c
(2.1.1)but now with the Froude number Fn=U/ (gLY as a small parameter. In fact, with L and g fixed the velocity of the ship U is assumed to be small and the problem is often referred to as the low speed problem.
With I V 1-0(U) as a consequence of the that h- 0(U2). But then again an expansion all quantities to be evaluated at yh(x,z).
Doing so, and equating the terms with problems are derived:
o=O
foryO
h0 = __ [U2 - cJ(x,O,z) - b(x,O,z)]
2g
A formal low Froude number expansion can be made Introducing the following
perturbation series:
h=h0+h1+h2+
(2.1.2)with:
- o()
and h1 - o(h1)
for U-O.conditions at infinity, it follows from (2.1.1) into Taylor-seri around y=O can be used for the same order of magnitude, the following
(2.2.3)
0
for IxI-.00
(2.1.4)in which the functions P1 are completely determined by the lower order terms of the expansion for , and it can easily be shown that P-0(U2'1) and I V bj I -O(u2' 1) for U-4 0.
The terms in the expansion for the free surface elevation, h1, can be computed with help of the solutions for at y0. For instance:
(2.1.5) The solution for is the well known "double-body" solution, which is the solution for
the potential flow around the body consisting of the wetted part of the ship and its
mirror image with respect to the plane y=0, which is uniform at infinity. This double-body potential will be denoted by rfrom now on. All terms of the expansion for can
be calculated with help of a source distribution over the free surface and the wetted part of the ship's hull simultaneously.
However, in order to obtain well posed problems for the 's, it has turned out to be
necessary to drop the radiation condition from the equations. This was a consequence of
Oy=°
aty=O
ofl=O OflSb -+ Uxfor IxI-x
and for i 1:=O
foryO
= P1(x,z) at y=O ifl=O OflS,the removal of the second order derivatives from the free surface condition for each to the problem for the higher order terms. It is clear that no wave solutions can be found in
this way. Hence the introduction of the formal perturbation series (2.1.2) leads to
solutions which cannot satisfy all boundary conditions at the same time, and the problem can be seen as a singular perturbation problem. A well known technique for solving this kind of problems is the "boundary layer" approach (see e.g. Van Dyke 1964), which will be used in the next section.
2.2 The boundary layer approach.
The formal (or regular) expansions introduced in the previous section did lead to a
non-uniform rult near the boundary at y=O. Therefor at this boundary a thin "boundary
layer" is introduced, with thickness of order 0(U2) in which a wave solution has to be constructed. Away from y=O (in the outer region), the expansions of the previous section may be used in order to construct an "outer" solution.
As the first term in this "outer" solution, the solution of the nonlinear problem in the limit for the small parameter Fn equal to zero (the zero Froude number solution) can be taken, which was just the double-body solution . However it is questionable whether the solutions of the problems (2.1.4) may be used as higher order terms in the "outer" solution, because generally the boundary value problems for this higher order terms only can be stated as a reault of te matching procedure.
In order to derive the problem to be solved for the "inner" solution the y coordinate has
to be be stretched. From the frea surface condition it may be concluded that an
appropriate stretching is found as:
y' = Fn2y
(2.2.1)For the "inner" solution the following formal expansion is made:
inner
= + 1 + 2 + with
jO(i-1) for UO
(2.2.2)and the double body solution is used as the first term in the "outer" solution:
= + + 2 + (2.2.3)
Now first some attention will be paid to the matching of the two solutions.
A crude form of the matching condition for the first term of each expansion (in which both the potential and its derivatives with reapect to y have to match) is given by (see e.g. Van Dyke 1964):
lim cI
= lim c
andlim Fn20 = lim
(2.2.4)y--O y--O
With ry(X,O,Z) 0, the second condition leads to:
which can be met without problems by any wave like solution.
The first condition of (2.2.4) however, will give rise to problems when a wave solution
,lim Oy' = (2.2.5)
has to be considered for , because r(x,0,Z)-_ Ux for I x I, z I -_ oo. So, in order to satisfy this condition should be unbounded for y"
-
and x I, I z I -+ . It may be clear that such a condition can not be delt with when a wave solution is looked for. For that reason the ordinary boundary layer approach is replaced bysome sort of composite expansion (see O'Malley 1970) in which it is assumed that the first term in the "inner" expansion may be written as a superposition of the first term of the "outer" solution and a perturbation (wave) potential c1 which will vanish outside the boundary layer:= cb + (2.2.6)
The matching condition then leads to:
urn 0 and urn
= 0
(2.2.7)y'._oo y-4-00
In the next section the free surface condition to be satisfiedby this perturbation potential will be derived.
It is now also clear that the incorporation of higher order terms will lead to boundary value problems for the higher order terms of the "outer" solution different from those of (2.1.4). Hence only the first term of the "outer" solution can be used directly. This in contrast with the ideas of Keller 1979, who also used the solution of (2.1.4) for the second term of the regular expansion in his "outer" solution.
The idea of a superposition of the double-body potential and a perturbation potential was used by several authors like Ogilvie 1968, Baba 1976 and many more, but they
derived different formulations for the free surface condition to be satisfied by the
perturbation potential.
The superposition (2.2.6) as a result of matched asymptotic expansions like shown here was introduced by Hermans 1974.
2.3 The problem for the perturbation potential.
The expansion for the "Inner" solution given In (2.2.2) will now be substituted into the equations of (2.1.1). For the first term of the "inner" expansion thesuperposition (2.2.6) will be used.
As mentioned before, the y-coordinate should be stretched for a boundary layer solution. But then it can be seen from the free surface condition that the x- and the z coordinate should be stretched in the same way. However, with the first term of the
"inner" expansion given by jr+, such a stretching easily could lead to misinterpretation of the order of magnitude of the derivatives of this first term, because rwill be a slow varying function but Cl, is expected to be a wave solution which changes rapidly. For that reason the stretching is omitted here.
When the lowest order terms are considered, and theproperties of b are taken into
account it can be seen that the perturbation potential has to be a solution of Laplace's equation in the fluid domain:
= 0
for yh(x,z)
(2.3.2)-. wave solution
for xI-,o
(2.3.3) As a consequence of the matching cZ also has to satisfy the conditions:and cli.O "outside" the boundary layer (2.3.4)
When the 'inner" expansion is substituted in the free surface condition some attention should be paid to the order of magnitude of the different terms. It can be seen that the second order derivatives of 1 only will occur in the lowest order equations when a change
in order of magnitude is admitted proportional to Fn2 when differentiation of cb is
considered. Further the terms in which only derivatives of t occur are of order 0(U3),
which should then also be the order of magnitude of . Hence It assumed that V '1 I o( I V I) for U. 0. That means that quadratic terms in C1 are removed from
the lowest order equation to those for the higher order terms. Consequently the free
surface condition for the first term of the "inner" expansion can be found when the
superposition (2.2.6) is substituded and the terms which are quadratic in 1 are neglected. This can be seen just like a linearization of the free surface condition, but now around the double-body solution
1.
The result of this linearization is given by:+
+(2rxrxx+
ry rXy+ rzcrxz)cx+=
= -
ry rxx+ ryy+ryrzryzl
(2.3.5)at y=h(x,z)
It should be stated here, that in the derivation of (2.3.5) only the assumption is used that the quadratic terms in 1 can be neglected. Most other authors now also neglect the terms
with a first order derivative of , because of the change in order of magnitude as a consequence of differentiation of 4. Although this argument may be valid locally, such an estimation of the orders of magnitude beforehand will not be carried out here. The most important reason for taking these terms into account in the lowest order equations, is the fact that (at least for the two-dimensional case) these terms are essential for the solution to be constructed, as will be shown in the next chapter. It is expected that a removal of these term to the higher order equations will lead to the same type of non-uniformity for the higher order terms as in the asymptotic analysis of ordinary differential equations when "secular" terms are involved (see e.g. Van Dyke 1964).
Finally also the condition on Sb is given here also:
U onSb (2.3.6)
but it should be stated here, that this condition does not really belong to the system of lowest order equations because of the fact that I V 'I I -o( I V rI). Hence a possible error
introduced by omission of condition (2.3.6) as will be done in the next chapters, should be corrected in the problems for the higher order terms. In fact this correction should be
given by the second term of the "outer" solution.
2.4 The free surface condition and the wave-height.
The free surface condition for the perturbation potential (2.3.5) still gives rise to some
difficulties, because all quantities should be evaluated at y=h(x,z), with h implicitly given by the relation in (2.1.1). In this section the procedure of expanding the wave solution will be continued, and finally a simplified free surface condition is derived, to be satisfied by the perturbation potential used in the lowest order term of the "inner" expansion.
The steps to be taken are demonstrated first for the expression for the free surface elevation. When for the lowest order term of the "inner" solution superposition (2.2.6) is
used, and terms which are quadratic in are neglected, the lowest order term of the free surface elevation can be written as:
h0(x,z) =
._
[U2
(x,h,z)(x,h,z)(x,h,z) +
2rx(x,h,Z)x(x,h,z) 2ry(x,h,z)y(x,h,z) +
2rz(x,h,z)4)z(x,h,z)1 (2.4.1)
Because the use of
only was possible for y0, and with h-O(U2), also here an
expansion of r intoTaylor-series around y=O can be used. Only the lowest order terms are taken Into account and use is made of ry(x,0,z)= 0. It Is also convenient to write h0 as a superposition of a part which only depends on r and a function which represents
the wave part. The final result is: h0(x,z) = hr(X,Z) + h(x,z) with:
hr(x,z) = ._ [U2
x,0,z)
2(x,0,z)] (2.4.2)h(x,z) =
-
[rx(X,0,Z)x(X,h1,2)rz()(,0,Z)z(X,h,Z)] (2.4.3) One may now expect that also an expansion into Taylor-series for c around y=0 will be used in order to obtain an explicit expression for h, as done f.i. by Eggers 1981. Howeversuch a procedure will lead to serious problems for the estimation of the order of
magnitude of the different terms. For instance, such an expansion for cI would lead to:
h2
(x,h,z) =
..
Although h-O(U2), it is questionable whether the higher order terms in this expansion may be neglected, because is expected to be a wave solution, with a wave number of order O(U2). Hence differentiation of '1 would lead to a change in order of magnitude
comparable to h'. It then is not clear that a truncation of this expansion after one or
two terms would lead to asymptotically consistent rults.
For that reason a different approach is used here. First of all it is observed that, with
V q 0(U3), hhr will be of order 0(U4). That means that an expansion into Taylor-series around Y='hr may be used which can be truncated after the first term, even when a change in order of magnitude in the differentiation of Z is permitted of order 0(U2)! The
ru1t for h then is given by:
h(x,z) = - --
(2.4.4)Finally, in order to obtain a relation in which all quantities can be evaluated in the same plane, the following coordinate transformation is introduced:
X' = X Y' = - hr(X,Z) Z' = Z (2.4.5)
from which it can be seen that Y=hr corresponds to y'= 0. The final result for the wave-height in the lowest order term of the free surface elevation is then obtained as:
h(x,z)
-
(2.4.6)Now the problem to be solved for c1 wil be reconsidered. With help of the coordinate transformation Laplace's equation tak the form:
+ 3, +
= 0
for y'0
(2.4.7)in which the higher order terms are neglected. It was shown by Brandsma&Hermans 1985, that a correction for this omission of the higher order terms in Laplace's equation, indeed leads only to higher order corrections in the relevant quantities such as
wave-height and wave ristance. Hence, in contrast with the ideas of Eggers 1981, for the
lowest order term of the perturbation potential Laplace's equation, as given in (2.4.7), may be used.
It should be stated here that the coordinate transformation as given above also leads to a slight violation of the condition at Sb, but as pointed out in the previous section this condition is not relevant for the lowest order term of the solution.
Finally the whole procedure of the expansion into Taylor-series and the coordinate transformation, as discussed above, is applied to the free surface condition given in (2.3.5). Again only the lowest order terms are taken into account. For convenience sake the ' is dropped from the new coordinates, and the result is given by:
+ [
rxx
+ 2rxxz +
rzz
+(3rxrxx + 2rzxz +
+(3rzrzz + 213rxrxz + rz1rxx)'IzI= D(x,z) at y=O (2.4.8)
D(x,z) __[rx(x,O,z)hr(x,z)J + _{rz(X,0,Z)hr,Z)1 (2.4.9)
The free surface condition for the perturbation (wave) potential c1 of the lowest order term In the "inner" expansion (2.4.8), derived as a linearization of the nonlinear condition around the double-body solution, is proposed here as a consistent low Froude number condition to be used in order to approximate the nonlinear wave solution in slow ship theory.
The part containing the second order derivatives of c1 is the same as in the low Froude number free surface conditions used by most other authors, like Ogilvie 1968, Baba 1976,
Maruo 1980 and Hermans 1980. With rpect to the terms containing the first order
derivatives, the free surface condition derived here is different from those derived by the
authors mentioned above. Agrment for this terms is found with the rults of Eggers 1981, however he used different coefficients in the part with the second order derivatives, based on the expansion around y=0 also for the perturbation potential, which idea is
rejected here.
it is stated again here that it turned out to be
necessary to incorporate the termscontaining a first order derivative in the lowest order equations with the correct coefficients, in order to obtain reaults which will be uniformely valid. This will be demonstrated in the next chapter where the two-dimensional problem will be treated.
3. Low Froude number solutions of the two-dimensional problem.
In this chapter the low Froude number free surface condition derived In the previous chapter will be used in order to obtain the wave solution in case of body shapes which are assumed to be infinitely long in one direction. The two-dimensional problem is first solved for a totally submerged body of arbitrary cross-section. The problem can be formulated in terms of a source distribution along the free surface. The influence function of a point source (Green's function) is constructed with help of complex analysis. The procedure followed here is an extension of the method used in order to obtain a two-dimensional source solution satisfying the linearized free surface condition, which can be found in Kochin,Kibel&Roze 1964. For the problem to be solved a generalization is made
with respect to the free surface condition, in order to show the influence of the
incorporation of the terms containing the first order derivatives of the perturbation
potential, as proposed in the previous chapter, on the final results.
Results for both the wave-height and the wave resistance are presented, and it is shown that the incorporation of the terms mentioned above is essential for a correct estimation of the order of magnitude of these quantities.
After some adjustments the procedure used for the construction of a Green's function also can be applied to the problem of a surface piercing body. It is shown that then the final result for the wave-height gives no waves far behind the body. This is basicly the same result as found by Hermans 1980 and Maruo 1979 (although the last one gave an incorrect interpretation of his results), who both used different free surface conditions which are shown to be special cases of the generalized problem solved here. The vanishing of the wave contribution at infinity is a direct result of the presence of a stagnation point in the double-body flow, at the intersection of the rear part of the body and the free surface at rest. For the problem of a surface piercing body obviously the double-body
flow is not the correct flow around which the linearization should take place. A
satisfactory model for the flow to be used should predict a shear layer in front of the body as observed in experiments, however, without a reasonable explanation for this phenomena found yet, such a model only can be guessed at.
3.1 The two-dimensional problem for the perturbation potential.
In this chapter cylindrical bodies will be considered of arbitrary cross-section with the axis in the z-direction (see fig 3.1). At this stage, no difference is made between surface piercing and totally submerged bodies. Without dependence on the z-coordinate, the problem for the perturbation potential, derived in the previous chapter, reduces to:
+cIO
fory0
+ 1 2+3rxrxxx] = D(x)
y rx xx with: D(x) _[hr(X)rx(X,0)1 öx hr(x) = 1[U2(x,0)1
at yO
Fig.3.1 The two-dimensional problem.
The free surface condition from (3.1.1) can be written in a convenient form after the introduction of the function X by:
X(x) = !
x,0) (3.1.2)with as result:
+ XI + = D(x) at y=O (3.1.3)
However, in order to show the Influence of the
term containing (D, on the rults,
throughout this chapter a more general free surface condition will be considered given by:
(x,0) ± X(x)(x,0) + X'(x)(x,0)
= D(x) (3.1.4) keeping in mind that, according to the low Froude number free surface condition derived in the previous chapter, the correct value of the coefficient Introduced here, will be:c=3/2.
As mentioned before, most authors remitted these terms to higher order equations, or did use an incorrect coefficient. By using the more general condition (3.1.4) it will be shown that the order of magnitude of the solution as well as of the relevant quantities like wave-height and wave rIstance, will depend strongly on the choice for the value of
oc.
Finally some attention should be paid to the condition at infinity. For the case of the totally submerged body, it will be shown that an extra conditIon is needed at infinity in order to obtain an unique solution. Such a condition of the "radiation" type suitable for
'I=O
onSb- wave solution for lxi - oo
0 and
cI_ 0
"outside" the boundary layer(3.1.1) In which now ris the two-dimensional double-body solution.
this purpose can be given by the requirement that any disturbance created by the body has to die out at infinity in upstream direction, that is, for x-, -. The only justification for such a condition entering the formulation of the problem as given here, is based on the argument that, in realistic physical flows, no waves are observed far In front of the body. A more satisfactory argument can be found in Stoker 1957, where it is shown that when
the steady motion as described here can be seen as the limit for infinite time of an
unsteady motion which has been started at a given time, only solutions will be found which give no waves at infinity in upstream direction.
Hence, a condition will be added which states that c1 vanishes for x- - on:
- 0
for x-. -
and b - wave solution for x-+ +00 (3.1.5) It should be stated here that with c1 vanishing also-O for x- -on.
In the remaining part of this chapter, solutions will be constructed for problem (3.1.1) in which the free surface condition Is replaced by (3.1.4) and the conditions at Infinity are adjusted with help of (3.1.5).
Once the perturbation potential is known, the elevation of the free surface can be calculated with help of:
h(x) = hr(x) + h(x)
in which h1 is given in (3.1.1) and for the two-dimensional problem h may be written
as:
h(x) =
rx(X,0)x(X,0) (3.1.6)3.2 The problem for a totally submerged body.
The body is supposed to be situated totally beneath the free surface. In addition the
restriction is made that the center of the body at (0,-d) with d>0, Is assumed to be
located that far below the free surface, that the whole body lies "outside" the boundary layer region. Hence, for the submergence it is assumed that d> > U21g. With c1 vanishing outside the boundary layer, no attention has to be paid then to the condition on Sb.
In the free surface condition (3.1.4), the function D(x) can be seen as some sort of pressure distribution. With 1 a solution of Laplace's equation, a formulation will now be looked for in which c1 is expressed in terms of a distribution of sources along the free
surface y=O.
In order to do so a "Green's" function, G(,ri;x,z), is introduced as a solution of
Laplace's equation:
G+G, 0
forri0
(3.2.1)except at = x = y in which neighboorhood its behaviour is given by:
G
_L
ln[(_x)z+(_y)2I½
2ir
representing a source of unit strenth at this point.
Fig.3.2 The domain D
Now L2 and L4 will be moved towards +co and
o reapectively and L3
towards the region "outside" the boundary layer. L1 is chosen along the free surface for
0. It is immediately clear that, with -, 0 and 0 outside the boundary layer, no contribution is found from the integral over L3, under the assumption that G (and consequently also G,7) will be bounded there. The same argument holds at L4, where 0, and also no contribution will be found when now G (and consequently also G) is bounded for - oo. In order to get rid of the integral over L2 it is required that, with
and I bounded, G (and consequently also G) should vanish there. Hence, the
conditions for G at infinity will be given by: G bounded for e ,
-GO
for+c
G bounded for I I >> U21 g (3.2.4)
With help of the free surface condition (3.1.4) the remaining integral over L1 can be Written as:
In order to derive a suitable problem for G, Green's theorema is applied to the
rectangular region Dg shown in fig.3.2, with as result:
ccXx,y) =
+ /(G_G)d1) +
+ fGGd
L3 L4
in which: c=O for (x,y) outside Dg.
c=½ for (x,y) on L1, L2, L3 or L4.
c=l for (x,y) inside Dg.
n (3.2.3)
c(x,y) = / G,+(X
+X')G}d - /D G d
(3.2.5) D g L2 L3Integration by parts of the first integral leads to: = a2
c(x,y) =
urn [(l)X'G+X{ GG}
+ a1,a2 00 + fG D d (3.2.6) L1 0Because of X 0 for ±°, and with help of the conditions for 1 and G at infinity, It can easily be shown that the first term of (3.2.6) will vanish. In order to get rid of the second term of (3.2.6) the following condition at the free surface should be satisfied by G:
G +
a-[X'G]
0at i = 0
(3.2.7)The expression for then reduces to:
c(x,y)
=f D(flG(,0;x,y)d
(3.2.8)in which c is given in (3.2.3).
As can be seen, 'zI indeed has been written as a source distribution along the free surface. Before solving the problem for the Green's function, which will be done In the next section, first some remarks should be made about the wave behaviour of G. As can be seen from (3.2.8) 'I' has been written as a superposition of contributions of sources at the free surface. The Green's function G( ,0;x,y) gives the influence of a source of unit strength at the point ( ,0) to the perturbation potential at the point (x,y). When a wave solution for
'1 is looked for, also G should be a wave like function. However, with no waves admitted
far in front of the body, a source at a fixed position at the free surface only may give rise to a wave contribution at points (x,y) "behind" the source, that is with <x. For that
reason for G a wave like behaviour is required for , which seems to be the reverse
of the condition for D' which should be a wave like solution for x +oo. Hence, the
conditions for G at infinity are completed with help of:
G wave solution
for e--°°
and G0 for -*+o
(3.2.9)Once G is known, the wave-height function h can be calculated with help of (see also (3.1.5) and (3.2.8)):
h(x)
2X½(x) f D( )G(e ,0;x,0)d (3.2.10)in which c-'½ had to be taken in (3.2.8) becausecI has been evaluated at y=0.
3.3 Construction of the Green's function.
A Green's function satisfying Laplace's equation and the boundary condition (3.2.7) with the proper behaviour at infinity (see (3.2.4)), can be constructed with help of complex
.d(XF)
Im[i - (l+icr)c)F] = 0
at Im[]=0
d
Now the function f() is introcduced by:
analysis. The method followed here is an extension of the method used to construct a two-dimensional source solution satisfying the linearized free surface condition which can be found f.i. in Kochin,Kibel&Roze 1964. Such an extension for a different problem than stated here for the Green's function was given by Hermans 1974.
When the complex variables z and are introduced by:
z=x+ly
and = + (3.3.1)the Green's function may be written as the real part of a complex function F:
G(,i;x,y) = Re[F(;z)1 (3.3.2)
with F analytic in the half-plane Im[]0 (because G has to be a solution of Laplace's
equation), except at = z in which nelghboorhood it behav like:
F-
1og(z)
(3.3.3)The function X can also be extended into the complex plane with help of:
=
(334)
The condition for G at = 0 given in (3.2.7) then leads to the following condition for F:
Im[id2(XF)
- ic
d(k'F)- .] = 0
at Im[]= 0 (3.3.5)dZ2 d
This condition can be integrated with rpect to , and the constant of integration can be chosen to be zero without loss of generality. Then this condition is written as:
(3.3.6)
.d(XF)
- (l+iyX)F
for Im[]0
(3.3.7)This function will also be analytic In the lower half-plane except at the point z where it behav like:
ix
i(-1)X+1
log(z)
(3.3.8)
2ir(z)
2irBecause of the fact that the imaginary part of f should vanish at the real axis according to (3.3.6), this function can be extended analytically in the upper half-plane with help of Schwarz's symmetry principle:
in which * denotes the complex con jungate.
This extension of f will then be analytic in the upper half-plane except for z where:
i(c-1)X+l
f() -
_________ -log(z )
2ir(z)
2irIn which use has been made of: X()= X). Assuming that f is holomorphic for the
point at infinity, it then is found that:
f() =
2r z
±_
-
_L;) + i(i)X'
log{(-z)I(-z)} +
2r
With f given as above, the function F has to be a solution of the differential equation (3.3.7) in the whole c-plane. The general solution for the homogeneous equation (with f replaced by 0) is given by:
F(;z)= CX
1()exp[_iP ds
X(s)
Variation of the constant rults into the following solution for F:
F( ;z) = iX
)fX(t)f(t)exP[If
]dtAt this point the solution is not unique, because of the fact that for different choices for the path of integration L, different solutions are obtained. Hence at this stage the extra condition mentioned in section 3.1 will be used. In order to get the proper wave behaviour for G (and consequently for ) the real part of F has to vanish for Re[] +oo while for
Re[I* -
the real part of F should be a wave like solution.The proper behaviour as sketched above, imposes the following choice for L. L has to be
situated in the lower half plane, starting at a point with Re[t]-++oo, and when
Re[t] < Re[z] the path has to pass above the singularity at t=z. See also fig.3.3.
Re It]
ig.3.J The path of integration in the complex t-plane.
(3.3.10)
(3.3.12)
(3.3.13)
G(,0;x,0) = Re[
When f is substituted In the expression for F and integration by parts is carried out, the final result for F is written as:
F(;z) =
_1og{Uz)I(z)} +
-
'
t)lO(t_exP[if
-]dt Ti +00 tr
x(t)
int(x+i0) exP[if1dt I
(3.3.14) For calculation of the wave-height the function G has to be evaluated for r= 0 and y 0as shown in the previous section. With Im[] 0 and Im[z] 0 the path of integration
should coincide with the real axis for Re[tI= Re[z]. It is then convenient to take the whole path L along the real axis. With help of (3.3.14) the following expression is derived:
(3.3.15) As can be seen the function G(,0;x,0) depends strongly on the velocity of the double-body flow at the free surface.
This section is concluded with a typical example of G. For the body a submerged circular cylinder with radius r=a was taken with its axis at depth d=3a/2. The value of c was supposed to be = 3/2 as derived In the previous chapter. The function is calculated once for the source point (,0) in front of the location of the body's axis (at (0,-d)), and once with the source point behind it. The results are plotted in fig.3.4.
a
-6 -4 -2 0 2 -2 0 2 4
Fig.3.4 The functioo G s a fonction of c/a for two values of . Ffl0.12.
It can be seen that waves are generated by the source point, and will give rise to a wave solution at points with x> . Remarkable however are also the waves in the solution for G generated in the neighbourhood of x=0, as a result of the disturbance created by the submerged body. These waves are introduced because the function X occurs in the integrand of (3.3.15).
