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o» U) • - UI CD o BIBLIOTHEEK TU Delft P 1301 1182O S C I L L A T I N G S L E N D E R S H I P S AT F O R W A R D SPEED
OSCILLATING SLENDER SHIPS
AT FORWARD SPEED
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP
AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS IR H. J. DE WIJS
HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP VRIJDAG 17 DECEMBER DES MIDDAGS TE 4 UUR
DOOR
WILLEM JOOSEN
WISKUNDIG INGENIEUR
GEBOREN TE BERGEN OP ZOOM
H. VEENMAN & Z O N E N N . V . - W A G E N I N G E N - 1965
Dit proefschrift is goedgekeurd door de promotor
C O N T E N T S
SUMMARY 8 CHAPTER I. INTRODUCTION 9
1. General introduction 9 2. The thin ship theory 11 3. The slender ship theory 12 4. The effect of forward speed 14
CHAPTER II. FORMULATION OF THE PROBLEM 15
1. Co-ordinate system and definitions 15 2. Linearization of the equations and conditions 17
3. Outline of the approach 19 4. The existence and uniqueness of the solution 22
CHAPTER III. THE STEADY STATE CASE FOR HIGH FROUDE NUMBER 25
1. The linearized velocity potential 25 2. The linearization of the free surface condition 28
3. The wave resistance 30
CHAPTER IV. THE CASE OF HIGH FROUDE NUMBER AND LOW FREQUENCY . . 33
1. The velocity potential 33 2. The asymptotic expansion 34 3. The limiting cases for ^L = 0 and Pg = 0 38
4. The critical value y = 0.25 40
CHAPTER V. THE CASE OF LOW FROUDE NUMBER AND HIGH FREQUENCY . . 46
1. Introduction 46 2. The steady state problem 46
3. The unsteady case 48 4. The case of Froude number zero 55
CHAPTER VI. THE ADDED MASS- AND DAMPING COEFFICIENT FOR THE
HEAV-ING MOTION 57
1. The hydrodynamic forces 57 2. The low frequency approximation 58
3. The high frequency approximation 59
CHAPTER VII. NUMERICAL RESULTS 60
APPENDIX 66 REFERENCES 69
A B S T R A C T
A potential theory is developed for slender ships travelling and oscillating at the free water surface. For the steady state the linearized problem is solved and a non linear correction term is derived. Ship forms with minimum wave re-sistance are obtained. Due to the correction term the curve of sectional areas is not symmetrical, this in contrast with previous results. For the unsteady problem two different approximate theories are derived. One for the case of high Froude number and low frequencies and another for low Froude number and high frequencies. The effect of forward speed on added mass and damping coeffi-cients is considered. Finally the numerical results are compared with ex-perimental data.
CHAPTER I
I N T R O D U C T I O N
1. GENERAL INTRODUCTION
The object of the present dissertation is to derive under certain conditions, a systematic and self-consistent theory for the motion of a partially immfiersed slender body at the surface of a fluid.
The problem of the motion of a solid in an unbounded liquid is in general difficult to solve. The reason is that the Navier Stokes equations of motion, by which the disturbance in the fluid is mathematically characterized, are very intractable. A way to simplify the problem is to assume that no viscous effects are present. The effect of the fluid friction, which is most important close to the body, especially for the calculation of the viscous drag, can then be treated separately with the aid of the boundary layer theory.
A further restriction that can be made is to assume that the medium is incompressible. Of course this assumption is only admissable if the disturbance velocities are small in regard to the velocity of sound in the fluid.
In the following the attention will only be paid to the study of those aspects of the problem for which the viscosity and compressibility effects are negligible. Taking into account the Kelvin theorem on the conservation of circulation, it is obvious that a motion, which is irrotational at one moment of time, will remain irrotational. The theorem is still valid if an external conservative force is present, such as gravity.
Under these circumstances the motion of the fluid is characterized by the existence of a single valued velocity potential function <I>. It can be shown [1] that in an unbounded fluid O is uniquely determinate save for a constant if it fulfils the following conditions:
a. The Laplace equation or what is equivalent, the equation of continuity. b. The normal derivative of <I> at any point of the surface of the body is equal
to the normal velocity of the surface at that point (Neumann problem). c. All derivatives of <I> vanish at an infinite distance in any direction from the
body, due to the condition of finite kinetic energy at infinity.
By application of Green's theorem on the function O an integral equation for O is obtained. The proof for the existence and uniqueness of the solution can be found in the textbooks [1].
Another formulation of the Neumann problem is to express O in surface integrals over a source distribution on the body. The unknown distribution is determined by an integral equation which is derived from the boundary condi-tion on the body. From this Fredholm equacondi-tion a unique solucondi-tion can also be obtained [1].
Recently HESS and SMITH have solved the problem of an arbitrary body by a numerical method using source distributions. The only restriction in their program is that the surface of the body must have a continuous normal vector [2].
In the past several approximate solutions had been obtained by assuming certain geometrical restrictions. One of the best known concepts is applied in the so called lifting surface theories. It is assumed there that the body is thin, only slightly cambered and situated in a uniform flow. If the camber and thick-ness are characterized by a small parameter the velocity potential and boundary conditions can be expanded in terms of this small parameter. It can be proved that for the derivation of the first order term in the series of the potential it is consistent to put source and dipole distributions on the mean plane of the surface. After calculation of the source and dipole strength the pressure distri-butions along the surface can be obtained. However, no resultant force on the body is found. In order to bring the mathematical model in closer agreement with the actual physical situation the condition that the potential is single valued is dropped. In this way the concept of a circulation flow around the surface enters the theory and leads to a lifting force. The strength of the circula-tion is determined by the condicircula-tion of smooth flow at the trailing edge of the surface.
Another shape, on which further simplifying assumptions can be imposed is the slender body. Here the dimensions in the two lateral directions with regard to the main flow are small in regard to the third. The same procedure, as used in the thin body theory, can be applied in this case. However, a difficulty arises. In thin body theory it was possible to satisfy the boundary condition on a flat plate. In the slender body theory the body shrinks down to a line in the limit case. In this case it is difficult to apply boundary conditions on a line for the three dimensional diff'erential equation. This difficulty can be avoided by representing the body either by a line distribution of sources, if the cross sec-tion shape is circular, or by a distribusec-tion of multipoles for an arbitrary shape. The integrals for the potential are written in a co-ordinate system, where the co-ordinates transverse to the body are stretched in the ratio of the reciprocal of the slenderness parameter. In these new co-ordinates the body remains finite if the limit is taken for the slenderness parameter tending to zero. After expansion of the potential function the boundary condition can be applied and taking into account the first order term only, it appears that the source strength is proportional to the axial derivative of the curve of sectional areas. A necessary condition in this approach is the fact that the body is sharply pointed.
The fact which makes the present work much more complicated in comparison with the theories mentioned above, is the presence of a free surface. The subject of water waves engaged the attention of many mathematicians and physicists during the last century. A review of their results is presented in an excellent survey by WEHAUSEN and LAITONE [3]. Only those aspects will be given here which are of interest to the problem discussed.
In addition to the conditions stated in the foregoing the velocity potential must satisfy another boundary condition at the free surface. This non linear condition is originated from two others by eliminating the elevation of the surface. The first condition is a kinematic one and prescribes that there is no
transfer of ffuid particles across the surface. As second condition the pressure is assumed to be constant along the surface. In order to keep the problem linear it is necessary that the slope of the waves is small.
Fundamental potential functions can be constructed, which have the same singular behaviour as travelling or pulsating sources and dipoles in an unbound-ed munbound-edium and which satisfy in addition the linearizunbound-ed free surface condition. In order to determine these functions uniquely the usual boundedness condi-tions at infinity are not sufficient. It is therefore necessary to impose at infinity the so-called radiation or Sommerfeld condition, which excludes the existence of incoming waves generated at infinity. A more general way to ensure the uni-queness of the solution is given by STOKER [4]. He formulated the problem as an initial value problem by assuming the medium originally at rest. When the time tends to infinity the solution tends to the desired steady state solution. Then the only conditions needed are those of boundedness.
If a solid body of finite dimensions is moving at the free surface, the problems of existence and uniqueness of the solution for the potential are in general not yet solved. A discussion on this subject will be postponed till chapter two. In a long series of papers Havelock has treated problems related with the motion of special bodies at the free surface by representing them by source or dipole distributions.
The same restrictions for the body shape as in the case of an unbounded medium can be made for the free surface problem. The next two sections deal with the essential features of the thin body and the slender body theory in relation to the free surface eff'ect.
2. THE THIN SHIP THEORY
The study of the thin ship model was initiated by MICHELL'S paper in 1898 [5]. He derived a formula for the wave resistance of a ship moving at finite speed in smooth water. The linearized free surface condition was used and therefore it was necessary to assume the ship to be thin in order to ensure that the gene-rated waves are small. Michell assumed a source-distribution at the center plane of the ship. At this plane the linearized boundary condition was satisfied in order to obtain the source strength. After derivation of the formula for the velocity potential the wave resistance could be obtained from energy flux con-siderations at infinity.
Especially the last years it was sometimes felt that Michell's assumptions were adhoc approximations and that in addition to the condition of small beam/length ratio other conditions were imposed in relation to the other linear dimensions in the problem i.e. the draft and the ratio between the square of the ship speed and the acceleration of gravity (MARUO and VOSSERS [6], [7]). Recently WEHAUSEN [8], showed, however, that Michell's formula can be ob-tained from the original three-dimensional formulation by a rigorous asymp-totic expansion with respect to the small beam/length ratio only.
for a comparison of experiment and theory. A striking agreement was seldom found, however, the main difficulty being the experimental determination of the wave resistance. In most cases this is obtained by subtracting the estimated viscous resistance from the measured total resistance. Following this procedure it is moreover assumed that no interaction effects occur. In order to check Michell's theory it seems to be more reasonable to compare its results with those obtained from a more exact theory. There are two approximating condi-tions in Michell's theory: the linearization of the boundary condition on the ship's hull and the linearization of the free surface condition. The only way to construct an improved consistent theory is to drop both assumptions, which leads to a very difficult and intractable problem. Another possibihty is to maintain only the linearized free surface condition. Then a two dimensional integral equation for the velocity potential can be set up by Green's theorem. This approach, however, has not yet been made. For a further discussion of these linearization problems is referred to OGILVIE'S paper [9].
The first attempt to a rational approach of the unsteady thin ship problem was undertaken by PETERS and STOKER [10]. They treated a ship in sinusoidal head waves, with an amplitude of the same order of magnitude as the beam. Peters and Stoker started from the exact non linear problem and assumed an expansion of all variables in perturbation series in powers of the small para-meter, the beam/length ratio. These expansions are all substituted in the various conditions and equations and the terms are all arranged according to powers of the beam/length ratio. It appears that for the lowest order of this parameter in the resulting differential equations of motion for heave and pitch, the added mass and damping terms are not present. Therefore the solution predicts an undamped resonance. The spring constant is the hydrostatic restoring force and the disturbing force is the socalled 'Froude Krylov' force. The latter is the force obtained by integrating the pressure in the incident waves over the hull. In order to avoid this difficulty NEWMAN assumed more than one small para-meter in the problem [11]. He introduced two other parapara-meters; one related to the order of magnitude of the incident waves and another related to the order of magnitude of the unsteady motions. It is then possible to express the last parameter in the former two parameters. This relation is different for the two cases, viz. at resonance and at non-resonance. At and near resonance the added mass and the damping forces are dominating. Unfortunately, however, the resulting expressions are rather complex.
3. THE SLENDER SHIP THEORY
The slender body theory is based on the assumption that both beam and draft are small compared with the length of the ship. The procedure to construct a slender ship theory is essentially the same as for a slender body in an un-bounded medium. The derivation of the theory, however, is much more com-plicated due to the free surface effect.
by VOSSERS [7]. Although his general approach has proved to be very useful the elaboration of the theory is not correct at some places. The general concept is the formulation of an integral equation for the velocity potential by means of Green's theorem. This integral equation was simplified by expansion with respect to the slenderness parameter. He concentrated his attention on the problems of the steady advancing slender ship (high Froude number) and the oscillating ship at zero forward speed (high frequency). After his work was published several papers by others appeared.
A different method was applied by URSELL [12] and TUCK [13]. Both authors used the technique of inner and outer expansions for bodies of revolution, a well known method in the theory of viscous flows at low Reynolds numbers. Tuck treated the problem of the steady moving ship and Ursell solved the problem for an oscillating ship at low and high frequencies. The latter derived a second order theory in the slenderness parameter.
JOOSEN [14], [15], formulated both problems for an arbitrary shape of the body with the aid of wave source distributions and obtained the same results as Tuck and Ursell as far as the first order term in the expansions is concerned. Moreover he omitted the condition that the body is sharply pointed. It appears then that the series expansion is not uniformly convergent anymore in the neighbourhood of the endpoints.
Following VOSSERS method, NEWMAN [16], [17], derived the solution for the unsteady motion of an arbitrary pointed body in oblique waves. He restricted himself to low frequencies or in other words to that range of waves, where the wave length is of the same order of magnitude as the ship length. In the lowest order theory for yaw and sway there are three kinds of forces occurring in the equations of motion, the inertia forces, the Froude Krylov force and the motion induced forces. In the latter there is no free surface effect. Moreover no inter-action effects between the sections occur. In fact the result is the already known two-dimensional strip theory. In the first order theory for heave and pitch motions the same difficulties arise as in the thin ship theory. Due to the fact that no added mass and damping terms are present the theory breaks down at resonance. A bounded resonance results, however, from the second order equations. A disadvantage of this result is that in case of high resonant peaks it seems difficult to understand this phenomenon as a second order eff'ect. Nevertheless it is still possible that the theory gives good results, especially if the theory can be extended to the case of forward speed, because only there the reso-nant peak is in the range of frequencies, where the theory is assumed to be valid.
Apart from these usual slender body theories another result is obtained if the motion of a slender body oscillating at high frequencies is considered, more precisely for the case, where the frequency parameter has the order of magnitude of the reciprocal of the slenderness parameter. Joosen proved that with this assumption for the case of heave and pitch Grim's two dimensional strip theory is resulting. GRIM [18] developed his theory some years ago from physical reasoning. The disadvantage of the strip theory is that the three-dimensional interaction effects between the sections are neglected and that it seems impossible
to extend the theory in order to take forward speed effects into account. In order to obtain some insight into the interaction eff"ects Grim proposed to use a strip theory for calculating the singularities representing the ship and then to determine the flow velocities with three-dimensional potential functions. In principle it is of course possible to work out this theory for the forward speed case as well. However, the objection against this approach is that it is not rational and consistent.
After the foregoing discussions and conclusions it is evident that the next logical step in the development of the theory is to include the forward speed effect. But then this theory must be worked out in the same systematic and logical way as for zero forward speed. Before starting the derivation of such a theory a few general remarks will be made on the forward speed effect.
4. THE EFFECT OF FORWARD SPEED
The calculation of ship motions in regular waves has been discussed in a number of papers (KORVIN KROUKOVSKY [19], VASSILOPOULOS [20]). In these papers the two-dimensional strip theory was used to calculate the added mass and damping coefficients (GRIM, TASAI [21]).
The effect of forward speed on the hydrodynamic forces was considered and dynamic cross couphng terms were included in the equations of motion for heave and pitch. Korvin Kroukovsky introduced in his theory terms depending on speed which are, however, mathematically of the second order of magnitude. Vassilopoulos obtained with this theory a good agreement between theoretical and experimental values for the motions. GERRITSMA [22], [23] showed ex-perimentally that the effect of speed on damping, added mass and exciting forces is rather small. The effect on the dynamic cross coupling coefficients, however, appears to be important. Recently he measured the added mass and damping distribution along the length of the ship, during forced oscillation tests with a model moving at various forward speeds. He found a very small effect of speed on added mass distribution and on the total damping force. These values were in striking agreement with the theoretical values, calculated by Grim's strip theory for zero forward speed. The damping distribution varies largely with speed. Gerritsma added to the damping distribution, obtained by the strip theory, the second order correction term of Korvin Kroukovsky and found a good agreement with the experimental data. It seems, however, dangerous to draw conclusions from this fact without knowing the comparison with a theory, which includes all second order terms. Apart from this one may wonder whether there is a first order term to explain the discrepancy.
The object of the following is to look into the details of this possibility by expanding the complete three-dimensional equations and conditions in series of the small parameter in a systematic and self-consistent way. Two different problems will be considered; the case, where the frequency parameter is small and the Froude number is large and the case, where at a high frequency the Froude number is small.
CHAPTER II
F O R M U L A T I O N O F T H E P R O B L E M
1. CO-ORDINATE SYSTEM AND DEFINITIONS
In the co-ordinate system used in the following the (x^, j^j-plane coincides with the mean plane of the free surface and the (y-^^, Zjj-plane is through the midship section. The origin moves with the ship speed V into the same direction as the ship and the Zj-axis is taken positive in upward direction.
The immersed part of the ship at time t, is given by an equation of the type
H(x^, Ji, zi, t) =0
In equilibrium position for / = 0 the hull surface is assumed to have the form
yi = k r^i. ^ J sgn j i
The length of the ship is L, the beam B and the draft T. A cross section contour is indicated by C(x-J. The bow and stern contour are denoted by Fft and Fg respectively. The bow and stern have the shape of sharp wedges. In the next pages BB' and SS' are designated respectively as the bow region and the stern region.
Only heaving and pitching motions of the ship, harmonic in time with angular speed co, will be considered. The displacements and rotations of the ship are supposed to be small with respect to the ship dimensions and are defined by:
r —ICO/ —ï'coi
where CJ is a dimensionless small constant.
Although only the pure harmonical motion is considered here, it is clear that the results of the theory can be used for more arbitrary motions, if the
super-position principle is accepted. The result can be built up by taking the sum of the various frequency components. For further reference see [9].
Since it is assumed that the medium is inviscid, irrotational and incompres-sible, the flow velocities can be characterized by a potential function
^*{xi, yi, Zj, t) in the following way:
with
u (xi, yi, Zj, 0
0*(xi, j i , zi, 0 =
grad O* {xi, y^, z^, t)
- Fxi + (Do(xi, j i , Zi, 0
where ^(^(xi, y^, z^, t) is the disturbance potential and determines the distur-bance velocities due to the presence of the ship in the uniform flow.
The ship is supposed to be slender i.e. the beam/length and draft/length ratio are characterized by a small parameter e.
In order to facilitate the mathematical elaboration of the formulae it appears to be useful to introduce the following dimensionless quantities:
? = -2 ? i yi = ^2'^i % (Xi, J i , Zi, 0 = 2K2 „ « 2 ^ gL ' ' ' - Ig zi = s 2 ^1 A ( ^ i ' ^i) sgn J i = 2tó L ?B = - ^ ^ Ï = 2g g
-^\mx)
- f
PoThe dimensionless co-ordinate normal to the surface of the body is denoted by V. In the new co-ordinate system the transverse and longitudinal dimensions of the ship are of the same order of magnitude.
2. LINEARIZATION OF THE EQUATIONS AND CONDITIONS
In this section the equations and conditions for the potential function <& ('^1, y)i, J^j, d-) will be derived. It is evident that O must satisfy the Laplace equation in the halfspace outside the body
AO = 0
The behaviour of O at infinity will be discussed later. Then the only remaining conditions are the free surface condition and boundary condition on the body. Since no fluid particles can be transported through the surface of the body the condition at / / = 0 is:
or in dimensionless form:
5i H, - y//^. + s^O^, T/^, + cD^, //,, + <b,^ H^^ = 0 (2.1) If the elevation of the free surface is denoted by h (x-,, yi) • _ the kinematic
con-. con-. L 2 dition at Zi = A-— is
' 2
ht - KA,, = 0„,^ - 0„^ /;,. - <D„,, h^^
The dynamic condition at z, = /!•— follows from Bernoulli's law: 2
«f», - y^„., + Sh^+l (0)^,, + 0%, + 02„J + - = 0
L Z p
where P is the pressure, which is assumed to be constant at the free surface. These two conditions take the dimensionless form:
^at!:i
After elimination of h the result is
+ ^L^^ee + ^%%,%. — 2£Y^E.e +<!>:, = 0 (2.2) In order to linearize the problem the expressions must be expanded with
There are two independent small parameters. The first one, the slenderness parameter s, is related to the geometry of the ship and the other one is CT, which is related to the amplitude of the oscillations. It seems difficult to decide apriori in what manner h and ^ are depending on s and a. Therefore it will be asumed here, as usual that Oy and <I>5, are of lower order than h, 0^„^ and O^r^. Con-sequently %^ <D^,5„ 0^^ (D^.e and <D^, O^,^,, 0^, Oi;.^, can be neglected.
Further it is supposed that a Taylor series expansion of O in the neigh-bourhood of (^]^ = 0 exists. Then it appears to be consistent to satisfy the linear-ized free surface condition at (^^ = 0, assuming that h is at most of order s^ In s:
s^i^ee + sPo^E.;, — 2sY<l'eE, + O-^ = 0 at EIi = 0 (2.3) After solving the problem it is of course necessary to check the result, since it is possible that it is in contradiction with the assumptions.
The slenderbody theory in an unbounded medium (WARD [24]) suggests the possibihty of this contradiction. The results show that the expression for O and Oc^ contains terms of order s i n s and terms of order unity, whereas 0^_ is of order unity. Therefore the term C)^^^ must be included in the formula for the pressure on the body. For the linearization of the boundary condition on the body the equation of the hull surface will be expanded with respect to e and a
H(l„ 7)1, ^1, 6) = s{7ii -f{l„ ?;i)} - a((,, - 'ï^.lèke'l 0{^z\ a^)
In order to satisfy condition (2.1) the potential is splitted up into a part depending on time and into a part, independent of time.
0(^1,7]!, ?:„ 6) = YOI(^I, 7]I, ^0 + 02(^1,7)1, !:i, 0) (2.4) With the assumption that 0(1) <^'i,L <^0 (e~^) the relation (2.1) becomes:
4 + ^ 1 . . + k!^K, + Oi^'^ii, '^) = 0 on ^1 = Ml' ^i)
.,0 „ / _ X - , 0
i^Ll [^0 - Uiy,e ' + T^ (^0 - Ui)A.ze
From the last condition it follows that Og must have the form
_ —ie 0)2 (^1, 7]i, ^1, 6) = (Da ($1, 7)1, Q e
For the steady and unsteady part of the problem the following two sets of equations and conditions are now obtained:
For the steady part:
/ so,
+ A . ^ + Oiz') = O,,, + (D,,/,, = - . 4
For the unsteady part: AO2 = 0
— s ^^Og + spo<ï>25,5, + 2/sY<l)2,, + <D2^, = 0 (2.6) ^ 1 + A , ^ + 0(e2) = cl.2.^_ + (D^;/,, = - / ^ 5i (!:o - 4^o^i)yi. +
ov s + ^T {/c.^0 - (^0 - 'Ui)fix] + 3 'ï'iC. (^0 - ^o?i) /c,?.
Moreover the radiation condition must be satisfied by both potentials at infinity (chapter I, 2 and chapter II, 4).
3. OUTLINE OF THE APPROACH
The velocity potential will be formulated in terms of a source singularity distribution on the ship's hull
1
<I>,(5i, ^1, ^i) =fd^fp(^' OG, (^1,7)1, ^1; 5,7), Qd^ i = 1 steady case
—1 c
i = 2 unsteady case (2.7)
where Gi is a Green's function satisfying Laplace's equation, the free surface condition and the radiation condition at infinity. F(^, X^ is the strength of a singularity distribution to be determined from the boundary condition on te body. The existence of a unique solution for F(^, Q will be discussed in the next section.
F is a source distribution multiplied by a factor (1 + / \ + s2/'2^)i/2 since for a surface element dS can be written
dS = (1 + A + sY\)'/2d^d^
It appears to be useful to introduce another definition
<D,. = <p„ + 9,. (2.8) 1 1
+
w i t h , „ ^ / . ^ / F ( U ) { ^
1 dlV(^i - If + s2(7)i -ff + s2(;:i + o^l
1
The form of Gi can be found e.g. in [9]:
0 0
-IH
e{^i + 0 ? + '(El — ~)q COS 0
qe cos{s(7)i —f)q sin 6} %q^ cos^ 6 + 2yq cos 6 + ^^ — q dq
£{!:, + t)q + 1(5, — E)? cos Ü
qe COS{E(7]I—/)g sin 6}
Po^2 cos^ 6 + 2-^q cos 6 + ^^^ — q dq +
Bi M ,
-IH
0 with cos 6is(C, + O ? — '(5i— E)? cos 0
«ye cos{s(7]i —f)q sin 6} Po^"^ cos^ 6 — 2y9 cos 6 + ^z. — ? ^9 1
4y
G](5i,7)i,J;i; ^,-/],0 = [Giili^fiiXi;^,ri,Q] 1^ = 0
(2.10)
If the roots in the denominators are denoted by q^, q^, qs, q^ the contours
M^ and A/g are defined by
-T^
H,
-^3 q4 ^2 • w — " •71 ^2 = 91 = = ^ 2 = ?1 = ? 2 = ?3 =-— 2y cos 8 + /V4y cos 6 -— 1 2po cos2 0
— 2y cos 6 — /V'4y cos 9 — 1 2po cos2 0 — 2y cos 0 — V1 — 4y cos 0 2% cos2 0 — 2y cos 0 + V l — 4y cos 0 2po cos2 0 + 2y cos 0 — V1 + 4y cos 0 2Po cos2 0 + 2y cos 0 + V l + 4y cos 0 cos 01 > 4y f cos 01 < 4y ^* 2% cos2 e
The aim is the derivation of the lowest order term of 91 and 92 in the series expansion with respect to s. In other words the limiting values of 9^ and 92
will be derived when s tends to zero. One may expect, however, that this theory will give a good approximation for the case of small finite s as well.
It is quite easy to obtain the first order terms for the potential 90. Changing the order of integration in the first integral the result after one integration with respect to E, becomes:
Fa, 0 dK
HW
5)2 + ,2(^^_/)2 + ,2(j;^_;;)2 ^=1- 2
J F ( 5 I , 0In s V(7ii-/)2 + «;;=02 dl +j\n 2 \^,-?,\d^ JF^i^^X) d^ -f
c —1 C 1- ƒln 2|5i - mfm> Qd^ + jpil, 0 HI - 5i +
+ V(?i - If + s2(7)i -ff + z^ (JIi - 0^} dl + JF{1, 0 In {?, - I +
+ V(5i - If + S2(7), - / ) 2 + s2(^i _ 0^} ^^ + O (s In s) (2.11) During the derivation it was assumed that the bow and the stern region are of order s. If these regions are of order unity the result can be written in the form:
2 ƒ F(5i, 0 In s V(7ii -ff + (Ci - 0^
dl
+ ƒ sgn (5i - I) In 2|?i - l\dl~JF{l, Qd^
—1
For 9o is obtained
9o = - 2 ƒ F(5i,0 {In V(7ii-/)2 + (!:,-02 _ in V(^ff+l^,+^f} d^
c
+ 2ƒF(^,0 ln{^-^x + V(?i-^)2 + s^ (7)i-/)2 + s2(^i-02} +
- In { ^ - ? i + V ( ? i - ö 2 + zHfi.-ff + s2(^i+0^}] ^E: +
where the last two integrals can be neglected, if the body is sharply pointed. Before starting the derivation of the lowest order terms in 91 and 93 it is necessary to introduce some statements as to the order of magnitude of a, % and ^L with respect to s. As was mentioned earlier three cases will be considered
1 a = o Po = 0(1) IL = O (2.13) 2 CT = sa Po = 0(1) U = 0(1) a « 0(1) (2.14)
3 a = s^a po = sPi li. = IB^^ PI = 0(1), In = 0 ( 0 a < 0 ( 1 ) (2.15) where a is a constant factor.
The first problem is the steady state case for high Froude numbers. The second case is related to the problem of low Froude number and high fre-quencies and one may expect that it corresponds with a ship moving in head waves of small length. The third case deals with the problem of high Froude number and low or moderate frequencies and it seems to be a good approxima-tion for a ship moving in following waves with moderate length. The first problem will be treated in chapter III the second problem in chapter IV and the third problem in chapter V. In the next section the existence and uniqueness of the solution will be discussed as far as the lowest order terms are concerned.
4. O N THE EXISTENCE AND UNIQUENESS OF THE SOLUTION
There are several possibilities to reduce the Neumann problem to an integral equation.
The potential has to satisfy: 1. the Laplace equation.
2. the condition that the normal derivative at the hull is equal to the normal velocity component.
3. the condition that the derivatives in any direction are vanishing at infinity. 4. the free-surface condition.
5. the radiation condition at infinity.
The constant value, which can always be added to the solution of a Neumann problem is taken as zero. This causes no loss of generality since only the induced velocities are physically of interest and these are not affected by the constant part of the velocity potential.
One possibility is to start with the application of Green's theorem 47rO(xi, >»!, Zi) = / / <b(x, y, z) Gn{x^,, J'l, Zi; x, y, z) dS +
— / / ^nix, y, z) G(xi, j i , Zi; x, y, z)dS
s + s, + s, + s,
/
- S :
The Green's function G satisfies Laplace's equation and behaves in the neigh-bourhood of (xi, yi, z-^) as:
1
\/(xi - xf + {y, - yf + (zi - zf
Since O should satisfy the radiation condition the same condition must be imposed on G. At infinity G tends to zero of sufficient order to make the integrals over S^ and ^3 vanishing.
For a steady forward motion the free surface condition reads
If G also satisfies
gO,. + K2(D,.,_ = 0 at zi = 0 g O , + F^G,,, = 0 at zi = 0
the integral over S-^ can be integrated with respect to x^ and the result in dimensionless co-ordinates becomes:
47rO = f f(t>G„dZdl — fj'<^,.Gd^d^ +sPo A o G . — O^G) dr^
Another possibility, which is often used in potential theory [1] leads to an integral equation which is sometimes more convenient for computation and
it will therefore be used in the present work. A solution of a Neumann problem can be represented in terms of a source distribution on the hull surface. In this case the sources will be Kelvin sources, which satisfy the free surface condition and the radiation condition.
<l>(^i. >ii, Ci) = ƒ ƒ n ? , 0 G(?i, 7)1, ^1; 1,7), D dldl
s
where F (^, Q 'S to be determined from condition 2. Assuming that a solution for the function F exists O is expanded close to the body in a series of terms in s. It will be proved in the next chapter that the first order term has the following form:
0(E
1, 7)1, til) = - 2 ƒ F(5i, 0 {In s V(7)i -ff + (^1 - X,f +
c
+ In sV(7ji -ff + (!:i + o n ^t: + r(^i)
where F(^i) is a function of ^i only. The unknown source strength is determined by the integral equation
Sv V\ + f\
-2~JF(1,X) {lnsV(7ii-/)'^ + ( ! : i - 0 ^ + In sV(7)i-/)2 + (^^ + 0^} rfC c
SO
where i ^ is a known function on the hull. This equation is the same as for the ov
two dimensional problem in an unbounded medium. It is well known that this problem has a unique solution. Thus it can be concluded that as far as the first order term is concerned, F exists and is uniquely determined. The bow and stern region however must be excluded from this consideration, if the ship is not sharply pointed.
CHAPTER III
T H E S T E A D Y C A S E F O R H I G H F R O U D E N U M B E R
I. THE LINEARIZED VELOCITY POTENTIAL
It is assumed that the velocity parameter Po is of order unity and that the ship is sharply pointed. According to (2.8) the velocity potential is given by:
^iKi> •^i> Q = 9o(5i. ^1, ?i) + 9i(?i> "^i- ^i) where 90 follows from (2.12). The second term becomes:
I
cpi(?i, v)i, !:i) = jdlJFil, OGi(?i, 7)„ ^1; I, 7), 0^^ (3.1)
with
""^--hP'!
s{Ki + O ? + "£(•01 — / ) ? sin 0 + l ( | , — 'Ol cos '
e dq
Po<? cos* 0 — 1
Re means that the real part of the integral is to be taken and L represents the
contour in the ^-plane ^^ |-h
It appears to be useful to change the form of Gi slightly. Using the expression: 1
V ( 5 i - ?,f + £*(7]i - 0 ' + ^%^1 + Kf
l%f''!
e(C, + 0 ? + '£(^1 —f)q sin 0 + l(5i — %)<l COS 0e dq Gi becomes:
1
'~vW^WT^^^ff+^^+W^
2Po C'^r, fe qcos^'E(Ci + K)l + ffitfl. —Jig sin 0 + i(5i — £)g cos e
^ -J J
Po? cos* 0 — 1 dqPutting this expression in (3.1) the first order term in the series expansion can easily be obtained. The first term of Gi leads to a result already derived, see (2.11). On the second term in Gi an integration with respect to ^ can be applied. Omitting higher order terms in s the result has the form:
Ti(?i> ^1, ^i) = - 4 ƒ F(?i, O In £ V ( 7 i i - / ) * + (^i + 0* ^C c
+ 2 ƒ In 2|5, - l\dl ^ ƒ F(?, 0^? +
—1 c 1- 2 ƒ In 2|^i - l\dl ^ JF{1, Qdl^
+
El C 1jdiJF{i,x:)g^{i^,i)di
with+
giili, l) = 2po
'^Reifd^r.
•^ -nJ J
•IXi — o ? cos 6
e cos 0
Po9 cos* 0 — 1
dq
Introduction of the new variables
q cos^ Q = p cos 0 = 1
coshT leads for g^ (5i, l) to
gi(5i, Q = ^^ Re i
i(^i — ^P cosh T
C dp r •(?.-5)/'
The integration contour L can be changed by closing it to the upper side if 5i — 5 > 0 and to the lower side if ^i — ^ < 0. The separation of the ^-interval is made in order to ensure the convergence of the integrals.
2po c idp r -ffi.-/
(?, — tip cosh Tcoshx 2 I r idp /• (5,-5)^ coshx r p. '•"'"\ or
4p„
rup% - 51)
, , , , 4p„ /-/Co(p|^i - ^1) , , ^ n ^i(5i, 5) = - J ^2^^2 + 1 ^/' ? i - 5 > 0 o4Popo(/>|Si-^|) /|?,_5K
owhere Fo i^ the Bessel function of the second kind and Kf^ the modified Bessel function of the second kind. The integrals can be reduced further by the rela-tion:
oo
0
Here /?o '^ the Struve function. Finally the form of the velocity potential becomes:
<ï>i(5i. >li, !:i) = - 2 ƒ A^i, 0[ln sV(7ii -ff + T ^ i ' ^ ^ * + 'c
+ In £V(7)i - ƒ)* + (!:i + 0^) rfj; +
1
+ 2 ƒ 5^(0 sgn (5i - ^) In 2|5i - l\di +
—1 — 1 1 - 4 7 : ƒ 5 5 ( 5 ) y o ( ^ ^ ) r f 5 (3.2) with
^(5) = ƒ ^(5.0 ^c
The first two terms can be considered as the potential associated with the motion of the double body in an unbounded medium. The last two terms
represent the free surface eff'ect. The source strength F is determined by (2.5) ^ . . . + ^ i c i ^ = - 4 ( ^ 1 ' ^i)
Since the free surface contribution depends on ^ only, this leads to the integral equation
lim 2 fn^i, 0 ;^~^+ilV~^n2 '^^ = ^^.(^- ^i)
(v)i ÏO -> c J {T\X ~ff+ {Ki — ^f
c + c
C is the contour reflected with respect to the 5, 7) plane.
4.rF(?i, ^i) + 2 Tn^i, 0 '!'~i^^'~^^/'; dK = 4(?i, ^i)
J (fli—ff + (Ki — Kf
c + c
where the principal value of the integral is to be taken. An integration is carried out over the area of the double body. After changing the order of integration the result becomes:
27. ƒ F(5i, KM, + ƒ F(5, QdKp^^^^^l±^^^^^ dK, = A,Sl,) (3.3)
c + c c + ~c c + c
where A (^i) is the area of a cross section. The inner integral is the potential of a doublet distribution along the contour with strength unity and equals zero outside the contour. Therefore
jFil„K)dK=^A,^{l,)
c + c
and
5(0 = 1^,(5) (3.4)
2. THE LINEARIZATION OF THE FREE SURFACE CONDITION
The a priori assumption made in section (/, 2) will be discussed here. In deriving the free surface condition it was assumed that the quadratic terms in the Bernoulli pressure formula are negligible or more in detail the term con-taining 0*i^j. From the expression (3.2) it follows that the velocity potential Oi consists of a term of order In s and another term of order unity. The derivative in (^i-direction is at least of order e In e. The derivative in 7)i-direction, however, is of order unity
0„, - - 2 ƒ^(5x. 0 l^^__^+(^^_^. + (7)i-^* + (^x+0^) ^^
This expression is the same as for the unbounded medium problem, where it is known that the linearized form for the pressure on the slender body is not satisfactory.
At a first glance this could lead to the conclusion that the use of the linearized free surface condition is not consistent and that the condition should read:
I»!?, + sPo*5i5iE. - ^Po ( ^ «>^.. + «>^..<l>i.i.i) + ö(s* In s) = 0 If, however, 7)1 or Ki are of order £~i the order of magnitude of O,^^ and 0]„^ is not unity but s. Taking the correct limit for s tending to zero one must there-fore distinguish between the near field behaviour and the far field behaviour of the conditions:
0,j-^ = 0 near field - 'I'lc + Po^iEE = 0 far field
In the theory developed here, the latter expression for the whole field is used. Since in the near field the first term dominates over the second, the correct first order term of Oi is obtained. It can be concluded therefore that the use of the linearized free surface condition leads to a mathematically consistent theory. Although not being consistent with the theory we shall try to obtain some indication of the non-linear free surface effect.
For the velocity potential is taken the function
Oi(?i, 7)1, J^i) + g(Ei) in the near field Oi(^i, 7)1, Q far field The free surface condition becomes:
<l>ic. + sPo<ti5i5. + ^Po ( ^ (fsi - ^'1.1) - '^\.'^irJi = 0 near field <I>ic. + sPo<I>i5.c. = 0 far field By determining g (^) in such a way that
| ( ^ 5 . - ^ ^ i ) - * ï ' ^ . * I ^ . . , o . = 0 (3.5) the free surface condition becomes in the whole field
^K, + sPoOiE.s, = 0
and the function Oj can be obtained as before. The only problem is to satisfy (3.5). The boundary condition on the body gives on the fine of intersection of the body with the free surface
<I>i..=4(?x,«)
In order to obtain a rather rough approximation for g it is assumed that the last expression is valid in the whole region close to the body. Consequently O^^ can be neglected. Then the result becomes
g, = / V ^ , o )
This result leads to the conclusion that a non-linear correction term can be added to O, which is only approximately valid and has the magnitude
ga,)=Jp,a,o)d?,
—1It seems somewhat unrealistic that the correction term is of order unity and does not depend on the Froude number. If the exact free surface condition could be satisfied it can be expected that the resulting term is of higher order and depends on the speed. Due to the approximation used in the procedure above these characteristics of g have disappeared. Nevertheless it is felt that g will give at least an indication of the non-linear effect.
3. THE WAVE RESISTANCE
The wave resistance of a slender ship can be calculated by integrating the pressure over the surface of the ship:
— 1 c
where the pressure P follows from Bernoulli's law which is for the steady case
F = s2pF*J0,.. + l0V,. + l0*,,,}
After integrating once with respect to ^j the dimensionless form for the wave resistance becomes:
_ 1
^ = | p ^ = ~ lY^^i/4(5i. ?i) '^id^v %, ^i) dK^
— I cThe only contribution to the wave resistance arises from that part of Oi which depends on the Froude number:
1 1
R = z^fA,,J?,,)^-jA,,a) sgn (5i - ^ ) { / / „ ( f c ^ ) - Yo(^^^) j dK +
1
fA,,moi^^^)dl] dl,
or 1 1R = - ^ y ^5.5.(U^5i ƒ ^55(5)^o(^^^') dl (3.6)
—1 —1By using the integral representation for the FQ function this expression can be written as
where
R = TTS* ƒ {rV) + J*(T)} di:
/•(T) = ƒ A^^il) cos I — cosh TJ </?
— I 1
J(T) = / .455(5) sin (— cosh T I dl
^0
—1
It is clear that this function is always positive. It can be shown that the non-symmetrical ship form increases the wave resistance without change of the displacement. Therefore the form for minimum resistance should be symmetric. According to the non-linear behaviour of the velocity potential, however, a term can be added to the formula for the wave resistance. This correction term is
1
Ai? = - -JA4^1)J\{1, O) dl (3.7) —1
A/? is zero if the curve of sectional areas is symmetrical with respect to ? = 0 • Ai? can have a negative value for ship forms with a non-symmetrical curve of sectional areas. In that case the wave resistance is decreased by an amount of Ai?. It is therefore to be expected that the optimal ship is not symmetrical any more, if the A/? is taken into account. The optimal form will be determined by a method based on the variational calcules. The conditions under which the wave resistance is minimized are
^(1) = A{- 1) = 0
A^{\) = A,S—\) = 0 1
fA{l)dl = D (3.8)
2/\(5, o) = k{l), a known function
If P /z (l) is a small variation on the function A (I) with A (1) = A (-1) = h^ (1) =
h^ (-1) = 0, then the condition for minimum resistance at a fixed
displace-ment is
1 (i? + Ai? + XD) = 0 (3.9)
where A (l) is replaced by A (l) -f ^h (I). After a partial integration (3.9) can be written as
1 1
-fKlo[2 ^jA,,,l)Y,(^)di -1-^ A.(5..)-x
—1 —1This relation must hold for any arbitrary h Hi) and therefore
1 ^ 2 f / l ï ï | \ 1 2 ^ 2 j ^55(5)>'o(^^jö^5 - 2 ^Ei(?i) - X = O rf?i=0 or 1 4i
j 4 5 ( 5 ) l ' o ( ^ ^ ' ) ^^5 = yA:(5)rf? + al^ + bli + c (3.1 .10)
\ H'O / •'J
—1 —1
with the prescribed conditions (3.8)
1 fA.,{l)dl = 0 —1 1 flA.^(l)dl = 0 (3.11) fl'A^^&dl = 2D
From the relations (3.10) and (3.11) the function A^^ (I) can be calculated. The curve of sectional areas follows then from:
5, 5
Aili) =fdlfA,Xs)ds -1 —1
CHAPTER IV
T H E C A S E O F H I G H F R O U D E N U M B E R A N D LOW F R E Q U E N C Y
1. THE VELOCITY POTENTIAL
In this chapter the derivation of the theory is restricted to the linearized problem. The Froude number and frequency parameter are both supposed to be of order unity. This case seems to be a good approximation for the problem of a ship moving in waves with moderate wave length or more precisely waves which are characterized by the condition that the parameter --— is of order unity. Here w is the frequency of encounter
& = direction of the waves with respect to the positive ^-axis. X = wave length
The steady part of the velocity potential 91 is already derived in the foregoing chapter:
9i(?i. ^1. ^1) = - 4 f Fill, 0 In ^V(rti-ff + (Ki-Kf dX, + c
I
+ 2Jdl^JF{l,l)d-C^gn ill - I) In 2|5i - 5| + ^ //„ ^^' ~ ^
— 1 c
(4.1) The unsteady part of the potential 92 is again of the form
1
'P2(5i, ^1, Q = jdl^Fil, DGlli, 7)1, ^1; 5, 7), 0 dl (4.2)
-1 c
where G, for this case is written as:
Gg = + G3 + G4
V(5i - If + s*(7)i -ff + s* (!:i - If
with
9, 00 5(^, + Qq + ,(F, _ 5 ) , cos 6
- 2 ƒ• CAnjq, 0)e cos{£(7)i —f)q sin 0}
^J J Po9' cos* 0 + 2yq cos Q + II - q ^
2 r^ P
e(Si + O ? + "(Si - E)? cos (^? + (4.3)
Pog-* cos* 0 + 2 y? cos e + 5L — ^
e, M,
ü _ eKi + Z)g — iXE, — E)? cos 6
2 r^^ rg^(g, 0)e cosis(7)i — / ) g sin 0} ^
•^J J Po^ cos* 0 — 2y^ cos Q + II — q n = 3, 4
As = Bs= IL
A^ = Po9* cos* 0 + 2yq cos 0
•Ö4 = Po*?^ cos* 0 — 2y^ cos 0
2. T H E A S Y M P T O T I C E X P A N S I O N
The determination of the first order term of 92 is possible by application of the same procedure as used in chapter III. The term containing G4 is integrated with respect to I. It is assumed that the ship is sharply pointed and therefore
/F(l,Orf^=/F(-l,0^^ = 0
c c
The first order term in 92 can easily be obtained by putting s = 0 in the formulae containing G», since the integrals remain all convergent. The result becomes:
CP2 = - 4 ƒ F ( ? , 0 In s\/(7)i -ff + ill + 0* +
c
1
+ 2 ƒ sgn ill - I) In 2|5i - l\dl ^f HI, 0 dK +
—1 c
+ jdlJFd, X,)G,dl + jdl G, ^JFH, QdK (4.4)
—1 c —1 c where Oi «3 ((Ei — E)<7 c o s 0 ^ 2 /• _,^ r'^»(9. ö)e (/^On
-.ƒ-ƒ;
Po9* COS* 0 + 2yg cos Q + II — q0 Ü
~ 1(^1 — ^)? cos e
2 r'.n r^»(^' Ö)e dq
2 r\ r^
Po?* COS* 0 + 2yq cos 0 + 5i, — q
61 A/, 34
1 —!(Ei — 5)? cos O
_ 2 r ^ g j-Bniq, Q)e ^
T^J J Po9^ COS* 0 — 2y9 cos 0 + 5i — q-o Af,
with
/tg = B3 = IL
Ai = — i{%q cos 0 + 2y) B^ = i{%q cos 0 — 2y)
After changing the integration contour in the manner as was done in chapter III the result for the Green's function is:
l f ^ - ^ i < 0
Ü o» — l ï i — 5 1 ? COS 0
r _2i r'rMiq,Q)e . ,
" - ^J «" j _ p^^2 cos* 0 + liyq cos Q + IL- iq ^ '^
0 0
—l4i — 51? COS 0
Po?* COS* 0 + 2/y? cos 0 + ?L + / ? ' + - j dQJ _ „ „2 „^„2 o r -..: o I g„ I ,J9 +
0 0
••(5i - 5)?i cos 8 dQ
41 j Aniqi + 4/ ƒ Aniai, 0)e ••2 _ - ( E l - 4)?. cos 0 d% V l — 4y cos 0 * e/0 \/4y cos 0 — 1 - 4 ƒ ^n(?i, 0)e \ = (4.5) 0 and if 5 — ^j > 0: Ü " - 1 ( 5 , - 5 1 ? cos 0
2' r'.« C^ni- iq, 0)e
nj J — Po?* cos* 0 — 2/y? cos Q + II + iq
0 0
-|5i — 51? cos 0
-U-0
Po?* cos* 0 — 2jy? cos 0 + ^L — iq0 0 2' r^Q CBréjq^ J ~ %q^ CO 0
ƒ
2^ - .:(5, - 5)?» cos 0 cl% Aniq^, 0)e / + V l — 4 y cos 0 0, '2 _ ; (4, _ 5)„ cos 8 rfe 7-^? +4/ƒ 5„(?3, 0)e
V l + 4y cos 0ƒ
2 —; (I, — ï)q, cos 8 dQ B„iq„ 0)e Vl + 4y cos 0 o -1 (5i — 5)?! cos o d Q V4 y cos 0 — 1 4jAn(q^,Q)e . , . " \ (4.6)The first two integrals of (4.5) and (4.6) can be reduced to a more simple form by the substitution ? cos Q = p oo ^ 4 r -15.-51/. r2 dQ Gn^^^ = ^J Rnip) e dp J ^5^_p^^2+2/y/,sgn(5i-^)}*cos*0+p* 0 0 where Ra = IL Ri = 2/y + %p sgn (l, — I)
After integration with respect to 0:
- 1 5 i - 5 b
V{IL - ^P' + 2iyp sgn ( 5 i - 5)}^ + P' dp
0
With the notations
^1 = (II - %p'f - (4y* - l)p*
^2 = 4y/>(^L — Po/J*) sgn (^i — ^) the final form becomes:
G„") = V 2 r^„(/7)j(A/wi* + m\ + Wi)V2
| 5 i - 5 | / .
dp
i~- — . \e dp
— / sgn W2(V m*i + m*2 — m^ ii^—j-- — (4.7) This function has a logarithmic singularity for I = l,. The integrals in (4.5)
and (4.6) originating from the residues, can be transformed into a form, which is more convenient for computation. Introducing the new variable 2Po?i cos 0
—;—; z- = T and denoting:
n, I, c, d
^ 4 = - / ( ^ - 2 y ) C4 = ' - ( ^ T - 2 y ) ^4 = ' ( o ^ - 2 ï L, = - / ( ^ ^ + 2y) c = 1 + 2y — V l + 4y </ = 1 + 2y + V l + 4y /j = 1 — 2y — V l — 4y"l . i f T < i / = 1 — 2y + V l — 4yJ A = / = 2 y i f T > i the result is for 5 — ?i < 0: 1 jA
ƒ
23 / / » ( T ) . ' ^ • - ^ ' ^ rfT V(/iT + 2y)*—4/!*T* U for 5 - ?i > O 1 _ ie ._ G„(2)= ^rcj'cni-z)e O+ mjiri-ef e
- 2 3 " ,2f.«-23, <;i — i;;T
Vö
- 5 ) T V(/T ( 5 , - 5 ) T rfr ;T—2y)*-rfT+
2y)*-dz - 4C2T* O-4/*T*
+
-^idfüni-eje ,_ ;";^ ^ ^ ; ^ „ (4.8) 1 V(rfT—2y)*—4t/V For the last integral of (4.5) and (4.6) the new variable1 — 2y cos cos 0 is introduced.
8 = 4y - 1 N^ = IL
N, = - 1 V S ( 8 T + 1 ) ( 1 - ^ sgn H, - l) - { (ST + 2y + 1)
the following expression is obtained I f y > i _ ^ ( , _ 2 , ) ( , . _ „ > - ' ^ • ^ ' { ^ / . ( « x + . ) ( . - x ) - . - 8 . s ^ ( 5 i - 5 ) }
^ " ' ' - - ' ^ ° i^«^^)^ V ( s ^ T l ) W ^ ï ) ( r ^ T .
o I f Y < i (4.9) G„(=') = OFrom the formulae it becomes clear that y = | is a critical value. The behav-iour of the velocity potential in the neighbourhood of this value will be dis-cussed in (IV, 4). First the limiting values for II = 0 and Po = 0 will be derived and compared with previous results. Although II and Po are both supposed to be of order unity it will appear that the limiting value for one of the parameters tending to zero leads to a result, which is also obtained if the parameter is taken equal to zero before starting the asymptotic approximation.
3. THE LIMITING CASES FOR ^L = 0 AND po = 0
The limiting value of G« for II = 0 can easily be obtained by putting II = 0 and y = 0 in the expressions for G». The result becomes:
Gjd) = Gg**) = GaP) = G4(3) = 0 ~ —I5i—SI/I ~ _ ± s i n / , -G4(l) = 2 P o S g n ( ^ l - ^ ) ƒ ^ ^ = = = ^ ; , = 2 s g n ( ^ l - 5 ) ƒ e '° ^T = 0 0 . f o r ^ - 5 i < 0 : G4<*) = 0 S (5i - 5) T - ^ (5, - 5)T
5 - ?i > 0:
G4(*)
= 4 f-.
dx
+ 4 f
^. dz
J
VT*— 1 J
VT*— 1
- ' - « ( ^ ) ' 38The final result is therefore: G3 = 0
and this is in agreement with (3.2), the free surface effect of the velocity potential for the steady advancing slender ship, without the non-linear correction term. If Po = 0 for Gn is obtained GiW = G4(*) = G3(3) = G4<3) = 0 oo -\il-Z\P G3W = 21L f % dp = nlL{H,ilL\li - l\) - Y,(IL\II - m 0 1 ; 5z. (El — 5 ) T dT l - l i < ^ G3<^) = 4/ ^ . ƒ ^ ^ ? - ^1 > 0 Po ^ Po • — !5f(5i — 5 ) T 00 • ^ 1 ^ ^ ^ 1 ^ ' ^ Sin — - — T cos —r T G3<*) = 4i5i, K , £/T + 8« lim / ^ V l - T * R _^n •/ Po->Or ^ 1 ^ T
The last integral is zero, except for 5i — 5 = 0. It is clear, however, that this term does not produce a contribution to the velocity potential and therefore it is correct to take
1 , 5 i l 5 , - 5 | T
0
= 2 TO IL{J<,{IL\1, - 1 \ ) + / H^{IL\1, -1\))
and for G» is obtained
G3 = v:lL{2i J^{IL\II - l\) - H,{IL\1, - l\) - Y,{IL\II - l\))
G4 = 0
From (4.4) follows the expression for the potential 92
92(^1, >)i, Q = - 4 ƒ F(?i, 0 In sV(7)i -ff + (^, + 0* dl +
c
1
+ 2jsgnil,-l)ln2\li-l\dl-^^jF(l,^dl + —1 c
+ ^lLj{2iJoilL\li-l\)-H,{lL\li-l\)-Y^{l4li-l\)} dl
—1JF{l,l)dl (4.10)
c
The final form for G3 was earlier obtained see [12], [14], [16]. It can easily be derived from the original form (4.2). Putting Po = 0 in (4.3) G3 can be written as:
s(Ci + V)p P-li e • / o ( p V ( ^ i - ? ) * + e*(7),-/)*) up L
G3 = ^ILJ^^—""^''"!: T ' "^^^^-^ dp
Lwhere L represents the contour | Taking the limit for s tending to zero the result is:
- ^ ^ /
• / o ( p | ^ i - ^ | )
P-IL
G3 = HIL I p— dp L
= 7zlL{2i UIL\II - l\) - HoilLllr - l\) - YOULU^ - l])}
4. THE CRITICAL VALUE y = 0.25
The formulae (4.8) show clearly that if y approaches to 0.25 the velocity potential has a singular behaviour. The term in the Green's function G« which becomes singular in the neighbourhood of y = 0.25 is G^^^':
With X = 2V1 — 4y, (x = 2V4y — 1
H3 = IL, Hi = —l i
and y tending to 0.25 from the lower side:
ik^^'-^' r' d-z
G„(*) <^ 16/ Hn e ƒ ^ ^^ . 5 - 5i < 0 J V(^ + 1)* + 4T {1— (1 — X ) T } *s.'" ^' r dl in '^ 1 r d-z „ „ „ J V | ( T + 1 ) * + 4 T H ( 1 + X ) T - 1 } ' V { ( T + 1 ) * + 4 T } { ( 1 + X ) T - 1 }and if y approaches to 0.25 from the upper side:
43o f d-z
G„<*) ^ 16/ Hn e f ,^ ^ = ^ 5 — ^1 < 0
7 V { ( T + 1 ) * + 4 T > { ( T - 1 ) * + [X*} 0
_ (5, _ E) ""
4f». r d-z
Gn'-^) ^ 16/ Hn e ƒ , = 5 — ^1 > o _7 V { ( T + 1 ) * + 4 T } { ( T - 1 ) * + [X*}
ForX - > 0 or [i. ^ 0 the final result becomes:
Gn ^ G„<*) ^ —2V2i Hne ' In 11 — 4y|
Therefore it can be concluded that the d a m p i n g a n d added mass coefficients tend t o infinity when y tends t o 0.25.
This p h e n o m e n o n has been examined by various authors H A S K I N D [25],
B R A R D [26], H A N A O K A [27], a n d H A V E L O C K [28] for a thin body. Havelock mentions that t h e remarkable behaviour of the wave motion at the critical point could be illustrated in considering the two-dimensional problem of a line source at the surface of the water. In the following the velocity potential of such a line source will be derived.
A source of strength unity is situated in the origin. The source pulsates harmonically with frequency r - a n d the incoming ffow has the velocity V in the
2-K
positive x-direction. By application of the result, derived in the appendix, t h e velocity potential, owing t o the source can be written as
yp + ixp yp — ixp —iai
e dp f e dp\ e
9(x, y) = - [f~ -^ + ƒ
M, —P^ + 2yp-T h e poles of the integrands a r e :
M, —-P^ + 2yp + — — P M, - - p * — 2y/7 o o O 1—2y — A / n ^ g "i- 2 K* 1—2y + V l — 4 y g «2 - 2 K* ~ _ 1—2y-t-/V4y—1 g^ ki—Pi— 2 j/2 - _ 1 — 2 y — ; V 4 y — 1 g^ «2 — P 2 — 2 ' J/*
T > i
l + 2 y — V l + 4 y g l + 2 y + V l + 4 y g «^3 — 2 K* 2 K* After changing the integration contour in a way as done in chapter III for the integrals there the result becomes:f o r x < 0 :
CO iyg — \x\g oo — i y q — \x\q
'•"' f e dq re dq
9(x, y)e =—ij —fi —^ + ij —F72 772 + 0 ?* — 2 / y ? H iq o ?* — 2 / y ? H \-iq
27t/ f-' — ")*• 27r/ (> — '*)*« H , e , e Ri Ri V l + 4y 27t/ (y + ")*» V l + 4y + ^1 V l — 4 j
i f Y < i
2v: (y + '"h' e if y > J V 4 y — 1 for X > 0: icöi r 9(x, j)e = / / o-'f
—iyq — \x\q e dq K* , (O* ?* + 2/y? H \- iq g g e dq F* ^ /72+
+ Ro g ?* + 2/y? -\ iq g R, 27t/ (> + •^)*i e if T < 4 R,= V l — 4 y — 2n; (.» + ")p' ' V 4 Y — 1 i f T > iAt large distances from the source the velocity potential behaves as represented in the scheme below:
T < i
Y > i
X < 0 9 = Wg + W3 + ^4 9 = W3 + W4 X > 0 9 = Wj 9 = 0 where: 271/ *1> + '(*!'' ~ '•''' w, = -. e V l — 4 y2v:i ^ty + i(*!-ï — w')
w, = , e V l — 4y
V l -f 4y 42
271/ *<> — '(*<* + "')
Wi = e
Vl + 4y
For better understanding of the wave phenomena it seems easier to consider the waves in a co-ordinate system, which moves with the velocity V in the direction of the undisturbed ffow:
X + Vt
Then the result becomes:
27:/ k,y + i{k,-x - o>,t) gi k i V \
27t/ k,y + i{k.x-<.Hl) gl k^V^
" ^ = ^ 1 ^ 1 ^ '
« * = M T + —) -.2
gl ^ A : i K * \ r / ^ / C 2 K * \ 27r/ k,y - i{k,x + <.,,t) g/ ^g[/2 •^3 = 7 7 ^ = ^ ^ " 3 = 7 z TV r T 4 y ^ ^ " K
/ A:3K*\ 1-1 +
vr
V i
V i
-+
4y-V
- 4 y - 4 y - 1 271/ * . j v - A i , ? - w , 0 g / ^ ^ 4 ^ * ^ W4 = ^ e C04 = - — y H I c 4 = ,VI + 4y * F\ ' ^ / * 1 + Vl + 4y
Wj and W3 are outgoing waves travelling in the direction of + CXJ and — 00 respectively, w^ and 1^4 are incoming waves from — 00. The condition for progressing waves in a liquid of infinite depth is satisfied:
kjg = CO*,
The phase velocity of the waves follows from
and the group velocity becomes Cgj = \ q. For y < 0.25 the four different waves exist. If y > 0.25 then w, and Wg disappear. The amplitude of these two waves becomes infinite at the critical point y = 0.25 and therefore also the kinetic energy in the waves per unit of length tends to an infinite value. This becomes clear by considering the fact that the group velocity of these two waves equals V and consequently the energy cannot be carried away towards infinity.
Apart from the above derivation the essential features of the wave system can be explained by a rather simple consideration. Let to be the frequency of an oscillating source in a uniform ffow with a velocity V. Then the disturbances originated by the source have the same frequency co in a co-ordinate system fixed to the source. We focus our attention to solutions of the problem, which have the character of progressing waves. Two different kinds of solutions are possible:
waves moving in the positive x-direction with a velocity potential
O = cos (wx — w?) (4.11) and waves moving in the negative x-direction
O = cos (mx + (Of) (4.12) where m is the wave number, which is related to the wave length X as
27t
m = —
X
Let us now consider these waves in a co-ordinate system moving in positive x-direction at speed V, which means that the velocity of the medium is zero in this system. The frequency of the waves is now denoted by [x, the wave number by k and the phase velocity by c. The relation between these quantities is
[X. = ck
For waves of the type (4.11) the relation between co and k becomes w = (c — V)k
or
CO = ( l j [X (4.13) For waves with a phase velocity, which is independent of the frequency, such
as accoustic waves, only one frequency [i is possible:
yx-i
^ = c o ( l - ^ )
Gravity waves however have to satisfy the dispersion relation, which reads in this co-ordinate system
(x* = kg and consequently
c = ? (4.14)
[J-The frequency [x can be obtained by substituting (4.14) in (4.13)
y 2
CO = [X — [X''
g
This equation leads to two solutions
1 — V l — 4coI7g
[X ^ CO, =
and
1 + V l —4<x>V/g
IX = COp =
2V/g
In the co-ordinate system, which is fixed to the source, these two waves have the same frequency co but diff"erent wave length,X. For small values of F the behaviour of coi is:
co*F
COi = CO + +
g
In the limiting case for V equal to zero: coi = CO and CO2 does not exist.
This is evident since the two co-ordinate systems are identical then. For small values of co the behaviour of cog becomes
C 0 2 = - + C 0 +
and thus for co = 0 is obtained: ,
g
0 and CO 2 V
This is in agreement with the well known fact that a non-oscillating source
g
travelling at speed V creates a wave with frequency —
For waves of the type (4.12) the relation between co and [x is - " = ( 1 - 7 ) ^ which leads to and [J- = t^ = coK If — = i it follows that ix g and therefore: c = = 2V C03 = C04 = = COi
vr
vr
= C02 + 4coF/^—1 2Vlg + 4coK/g + 1 2Vlg g 2VConsequently the group velocity of these waves becomes equal to V, which is exactly the speed of the source. It is clear therefore that no waves can exist ahead of the source.
CHAPTER V
T H E C A S E O F L O W F R O U D E N U M B E R A N D H I G H F R E Q U E N C Y
1. INTRODUCTION
In this chapter the free surface contribution in the velocity potential will be derived for another speed and frequency range than in the forgoing chapter. Froude number is defined as po = sPi and the frequency parameter as II =
IBZ~^, where both Pi and IB are of order unity. The supposed order of magnitude of the Froude number seems to be in good agreement with the range of most practical interest. The values of the frequency parameter correspond to small wave lengths in regard to the ship length. It is expected that the main result for this case will be the two-dimensional strip theory, which is already used in practice. Its use has always been justified by physical reasoning but it is clear that it must be possible to derive the theory by a more rigorous mathematical procedure.
In order to take into account additional effects such as speed and interaction between the sections the starting point is kept more general and therefore it is not assumed that the ship is sharply pointed. Firstly the steady state problem will be dealt with and after that the advancing oscillating slender body.
2. THE STEADY STATE PROBLEM
As derived in chapter III the velocity potential consists of two parts. The first part is the unbounded medium contribution and the second part is the free surface effect:
93(?i, ^1, ^i) =^^'fd^fF(l' OdlJdQ
—1 C — 2 (i;, + Qq + i(-o, —ƒ)? sin 0 + - (5, — 5)? cos e
/
e q cos* 0
Pi? cos 0 - 1 ^^ ^^-'^ L
After changing the order of I- and ^-integration, a partial integration with respect to I can be carried out. Omitting higher order terms the result is
