Vol. 44 • No. 3 July 1997
Numerical Seakeeping Calculations based on the 3-D G r e e n Function Method
by Hidetsugu Iwashita
C F D in Ship Design - Prospects and Limitations by Lars Larsson
A Post-Accident Study of a Tanker's Structural Damage Sergei V . Petinov, Helena A. Polezhaeva, Alexander I . Frumen, and Anna A . Vakulenko
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Vol. 44 • No. 3 • J u l y 1997
Hidetsugu Iwashita
Numerical Seakeeping Calculations based on the 3-D G r e e n Function Method Ship Technology Research 44 (1997), 111-132
The paper surveys recent Japanese approaches to solve the linear seakeeping problem with forward speed using three-dimensional Green function methods. A fast and accurate algorithm to evaluate the numerically sensitive integrals and a higher-order panel approach are briefly discussed. The numerical application of modern 3-D Green functions yields some new insights. For blunt ships, the results are less satisfactory and the influence of the steady flow becomes more important.
Keywords: seakeeping, forward-speed effect, 3-D Green function, high-order panel, pulsating source
Lars Larsson
C F D in Ship Design - Prospects and Limitations Ship Technology Research 44 (1997), 133-154
The state of the art in Computational Fluid Dynamics (CFD) in naval architecture is reviewed. The focus lies on panel methods and Reynolds-averaged Navier-Stokes methods for resistance and propulsion problems. Navier-Stokes methods for propeUer flows are also discussed. Applica-tions cover a wide range of hull types. LimitaApplica-tions, accuracy, applicability, and user-friendliness of present methods are discussed. Present research covers grid generation, turbulence modelling, and free-surface boundary conditions. Future possibilities include Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS).
Sergei V. Petinov, Helena A. Polezhaeva, Alexander I. Frumen, Anna A . Vakulenko A Post-Accident Study of a Tanker's Structural Damage
Ship Technology Research 44 (1997), 155-162
The damages to the cargo block structure of a tanker demonstrate the interaction of fatigue and cyclic buckling of web panels of a main transverse frame. The fatigue crack reduces local stiffness and redistribution of stresses leads ultimately to buckling. A stochastic model of fatigue crack growth accounts for the scatter of the fatigue life evaluation. The combined action of failure mechanisms accelerates the process and must be considered in identifying limit states of structural damage.
Keywords: fatigue, cyclic loads, crack, budding, structural failure, stochastic model, tanker
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Numerical Seakeeping Calculations based on
the 3-D Green Function M e t h o d
Hidetsugu Iwashita, Hiroshima University^
1. Introduction
The 3-D Green function method is the most popular among 3-D seakeeping methods striving to overcome insufficiencies of strip methods. Another increasingly popular, albeit less mature, approach are Rankine panel methods which could i n principle capture all steady-flow effects, but still have too many unsolved problems to be considered as a practical tool. See Bertram
and Yasukawa (1996) for an extensive discussion of Rankine panel methods.
The 3-D Green function method allows to confine the domain of the integral equation to the ship surface only by introducing a source potential that satisfies the (approximated) con-dition on the free surface and the radiation concon-dition as the Green function. However, for a correct solution at all frequencies, the Green function must incorporate the forward speed. The "forward-speed source" is considerably more complicated than the "zero-speed source" making a fast and accurate evaluation an important problem. This paper summarizes recent Japanese research in this field, including some new insights from numerical applications.
2. Bound2iry Conditions
We consider a ship advancing at constant forward speed U in oblique regular waves w i t h angle of en-counter X, Fig- 1- The ship motion 3?(^je''^^*)(j = 1...6) around the equilibrium position and the ampli-tude A of the' incident wave are assumed small. The incident wave has circular frequency CJQ, wave number iC, and Uf.{— OJQ — KUcosx) is the encounter
circu-lar frequency. A linear theory assuming an ideal fluid is employed. The velocity potential $ governed by Laplace's equation is
incident wave
where
Fig. 1: Coordinate system
$ ( x , y, z-1) = U[-x + Mx, y, z)] + y, z)e"^^'], (1)
(f>0 LOQ
ie Kz—iK{x cos x+y sin x)
(2)
(3)
(pa is the steady, (f) the unsteady velocity potential including the incident wave potential (/)o,
scattering potential (^7 and radiation potentials (f)j{j — 1...6). = 1...7) must satisfy the free-surface boundary condition and the body surface condition:
lim d(t) u rrd \ d ^ = n j + i^m, ( i = L . . 6 ) , ^ = -dn dn j -dn 0 on z = 0, on SH, (4) (5)
where (ni,n2,n3) — n, («4,715,716) = r x n, r = {x,y,z), ( m i , m 2 , m 3 ) = — ( n • V ) V , (m4, m5,m6) = -{n • V ) ( r x V ) , V — V{—x + (j)s) and d/dn = n-W. / / i n (4) is Rayleigh's
artificial viscosity introduced to satisfy the radiation condition at infinity, n is a normal vector pointing into the fiuid. 0^ and V ^ j must vanish at infinite depth.
rrij, which shows the effect of the steady non-uniform flow field on the body surface, can
be evaluated by 0^ satisfying the steady free-surface boundary condition. Recent calculations tend to estimate ruj approximately assuming the free surface is not disturbed by the steady advancing ship motion (double-body approximation).
3. Integral E q u a t i o n and H y d r o d y n a m i c Forces
The use of Green's second identity combined w i t h the boundary condition (4) leads to
or w i t h the source distribution (rj{Q) on the body surface
MP) = - J l MQ)G{P, Q)dS + ^^ £aj{Q)G{P, Q) n,, dy'. (7) P = (x, y, z) and Q = {x', y', z') respectively indicate the field point and the source point, and
KQ — g/U"^- The line-integral term in the second right-hand term of (6) and (7) appears for
surface piercing bodies. Generally its contribution can be assumed to be small for the slender bodies. Though Kashiwagi and Ohkusu (1988), Higo and Ha (1991) indicate its significant effect concerning the singularity of cfxj on the free surface and uniqueness of the solution, we have no confirmation for its correct mathematical treatment at this stage. G{P,Q), which satisfies Laplace's equation except at source point and the same boundary conditions as 0 except for the body boundary condition, indicates the Green function; i t will be discussed later.
Using the expressions (6) or (7), the integral equations w i t h respect to the velocity potential or the source distribution on the body surface take the form:
or
Discretizing the body surface into a finite number of elements, (8) or (9) can be arranged as simultaneous equations, where the unknowns are the velocity potential or the source strength on the body surface within each element. By solving these equations numerically we can determine the solutions. When we use (8), the velocity potential on the body surface is obtained directly ('direct method'); on the other hand, i f we use (9) ('indirect method') we have to calculate the velocity potential on the body surface by using (7).
Once the velocity potential on the ship surface is obtained, the hydrodynamic forces acting on the ship can be evaluated. A t first the unsteady hydrodynamic pressure is given by using
the velocity potential as follows, Timman and Newman (1962):
p{x, y, z) = -piiLOe + UV • V)cl> - p ^ j ^ ^ j i p . . V ) ( V • V) (10)
where 0 - 1 ^ ^ U = W )
where - | ^ . ^ ^ ^ (^ = 4,5,6)"
p is the density of the fluid, and ej{j — 1,2,3) are the unit vectors of x, y, z axes. The last term
of (10) indicates the contribution pressure due to the unsteady motion in the steady flow field. Substituting the incident-wave and scattering potential gA{4>o + 07)/'^o into the unsteady potential (f) i n (10) and integrating over SR yields the wave exciting force Ej acting in direction
j-^ = {i + ILv.w){cPo + h)njdS ( j = 1...6). (11)
pgA UJOJJSH^ «'^e ^
V is the steady velocity field defined in (5). The Proude-Krylov force should be defined clearly
when V is taken into account in the calculation. We define the Froude-Krylov force excluding the interaction between (j)s and (j)o:
L ^ = i ^ f r ( A - l L l \ ^ ^ n j d S ^ i ( [ cPon^dS. (12) pgA iOoJJsH^ lujeox/ JJSH
Considering the radiation potential iu)e^j(j)j{j = 1...6) as </> in (10), the added mass and damping coefficients Aij, Bij acting in the i-th direction due to the j - t h motion are computed as
^ + i Ë i i ^ r f (i + lLv.v)<i>jnidS-\(-)' 11 {fij.V){V.V)nidS. (13)
4. O r d i n a r y Calculation Methods of the G r e e n Function
The Green function is defined as velocity potential G{x,y,z;x',y',z')e^^^ of a pulsating source located at a point Q{x',y',z') and satisfying the linearized free-surface condition (4). By enforcing this condition, which includes Rayleigh's artificial viscosity p, G satisfies also the radiation condition. Furthermore G has to be zero at z ^ - o o . The general form of this Green function is given by
^ ^ ^ \ r ~ ^ ) ~ % A ^ ^ J - f L ikcose-iVer-Kok + ip{k cos 6 - U>e)
where
r = ^J{x-x')^ + i y - y ' y + {zTz'y•
The last term 'wave-term' in (14), is often transformed to a form more convenient for the nu-merical calculation. Typical expressions of the wave-term have been classified by Takagi (1992). The most popular one involves the exponential integral function Ei{u). Many results obtained by using this function have been reported, e.g. Chang (1977), Kobayashi (1981), Inglis and
Price (1981), Inoue and Makino (1989), Wu and Eatock Taylor (1989), Hoff (1990). Although
the final expressions used by these authors differ a little from each other, they commonly include
Ei{u). A fundamental form of G including Ei{u) i n wave-term is given in the following form, Iwashita (1989):
where T ( x , y , z ) = - A | _ ^ P _ i ) i - i 1 IjiKjw) 2 J = l \ / r + ~ 4 r a c ö s ö (16) IjiKjw) = KjFo{Kjw) ia cos Ö V/,(i^,tZ7) FoiKjw) 1 KjW ia sgn(y — y') sin Ö
1
Fo{u) = [ ^ i ( « ) + 27ri/3 • iï[/3Ö(n)] • H[-^{u)] Kj = ^ " ^ „ , [ l + 2Tacosg + ( - l ) ^ - V l + 4 r a c o s Ö ] w = Z + ia{Xcose + Ysme), X = KQ{X - x'), Y = Ko\y
( i = 1...4), (17)
y Z = Ko{z + z')
a = U = 3) H{t) =
it>0)
( t < 0 ) i?i(n)
, Ö —> ao{= cos 1/4T), and 1 ( i = i , 2 )
- 1 ( j = 3,4) ' " 1- 1 ( i ^ 3 ) '
Points to be considered i n the numerical calculation are: 1) reducing the CPU time for the calculation of Ei, 2) singularity of the integrand ( j = 3,4) of 0{l/^y\9 — ao
3) integration of the highly oscillating integrand near 9 = ±TX/2.
1) can be improved by use of Hoff's (1990) algorithm. 2) can be overcome by the comple-mentary scheme of Wu and Eatock Taylor (1989), where the numerical integration is performed after excluding the singularity at Ö = ao and the excluded term is added after analytical in-tegration. 3) is an intrinsic problem i n this numerical calculation of the Green function. The integrands Ji and I3 oscillate w i t h high frequency as the amplitude of Ki and Ks become infi-nite at9 = ±TT/2. Especially i f is small (near the free surface), the numerical integration breaks down due to high oscillations and large amplitudes of the integrand. Another approach is necessary to overcome this difficulty.
5. A Fast Computation M e t h o d of the G r e e n Function
Although the expression i n (16) seems to be a single integral, i t includes the function Ei which prevents the fast numerical integration. This problem is solved by using a single integral expression derived by Bessho (1977). The integrand i n this expression includes only elemen-tary functions. However, the numerical integration must be performed i n the complex plane.
Iwashita and Ohkusu (1989) developed a new numerical scheme for this integration, which is
fast and accurate. Furthermore the method allows to evaluate the Green function even i f the field point and the source point are both close to the free surface.
The single integral expression of the Green function derived by Bessho (1977) is:
T{X,Y,Z)= [
V l + 4 r COSÖ
fc2efc2^_sgn(cos0)fcie*=i^ where
ip--2cos2ö (1 -F 2TCOSÖ ± V l + 4 T C O S 9 ) , w = Z + i{Xcos9 + YsinO),
X
cos V X 2 + y 2 ' sinh -1
\Z\ cos -1
V x 2 + r 2 ' The partial derivatives w i t h respect to x, y, z are
1 VT{X,Y,Z)= sgn{y-y') 1 a I/AT (4r > 1) -i cosh ^ l / 4 r (4r < 1) KoT{X,Y,Z) (18) (19)
where f{X,Y,Z)= [ " Ja—TT I COS 9 1 d9
V l + 4 r cos 6 fc|e*=2^ -sgn(cosÖ)fcfe'=i'^l + f o ( X , y , Z ) , (20)
f o ( X , y , Z ) = -l 2 y 2 + ^2 + 7- x\z\ X2 + y 2 (X2 + y 2 ) V J t 2 + y 2 + ^2 x 2 + y 2 ^ (X2 + y2)VJs:2 + y 2 + z2 1 V ^ 2 + y 2 + _^2
Expressions (18) and (20) are genuine single integrals i n the sense that their integrands are expressed w i t h elementary functions. We perform the integrations of (18) and (20) along a path in the complex plane. Difficulties arise w i t h the numerical integration of the terms including
k\. These are due to severe oscillations of the integrands near 9 = ±7r/2 the amplitude of which
increase without limit as Z —> 0.
Iwashita and Ohkusu (1989,1992) developed a 'Numerical steepest descent integration
method' to evaluate (18) and (20) i n the complex plane. The steepest descent line is gen-erally a curved line on which the imaginary part of the argument of the exponential function i n (18) containing ki keeps a constant value, and the real part decreases most rapidly. Along this line, the integrand decreases rapidly without oscillations. The numerical integratioti method is therefore easily applied without losing numerical accuracy and CPU time, and quite robust even i f Z approaches zero. The steepest descent line is approximated by the finite number of segments and searched numerically.
6. Another C a l c u l a t i o n M e t h o d of the G r e e n Function
The most general application of the Green function method is the constant panel approxi-mation, assuming that the velocity potential or the source density is constant over each panel. I n this case, the Rankine term of (15) is evaluated analytically by using the methods of Hess
and Smith (1964), Webster (1975), or Newman (1986). On the other hand, a monopole
approx-imation is generally employed for the wave term: the term is evaluated at the center of each panel and is multiplied by the area of the panel. For a more accurate evaluation of the wave term, it can be considered to integrate (16) or (18) analytically over the panel. For this purpose expressions have been derived by e.g. Iwashita (1989) and Iwashita and Ohkusu (1992),
The panel integral of (16) takes the fohowing form, Iwashita (1989): (P) {r;ri,--- ,rN\9) -d9 (21) 2 j = l where N FQ{KjWi) + \ogwi v i p = 0 1=1 ia cos 0 ia sin 9 1 N Qi){Qi+i-Qi)' Qi){Qw.-Qi)'
Fo{u) = e"[£;i(n) + 2nip • H[pQ{u)] • /f[-3?(«)]J , Kj = (1 + 2Ta cos 9
+ ( - l ) - V l +
4 r a c o s Ö ) ,wi = P + Qi, P = z + ia{xcos6 + ysm9), Qi = zi — ia{xi cos0 + yi s i n 6 ) , r^{x,y,z), ri = {xi,yi,zi), QN+I = Qi (l^l-.N).
N{= 3,4) is tiie number of vertices of the panel, and Si the area of the triangle composed of
vertices I — 1, I and / + 1 (taken modulo N). On the other hand, the panel integral of (18) is,
Iwashita and Ohkusu (1992):
11 Tir- rAd.^ =S"(-1V\^'?.^, I ^^^^^ ^ V 1 /•/• 2 N ] j J j { r , r i ) d S = j y - i y J 2 ^ S i I cost) 1(3 sin 9 1 {Qi+N-i - Qi){Qi+i - Ql) d9 K ^ J o § ui{r,rr,)Ji{r,rr,)dr] (22) where
P — z + i{xcos9 + Py sin 9), Qi = zi - i{xi cos9 + /?y; s i n 9 ) , r^{x,y,z), ri = {xi,yi,zi), Vr, = ri + r]{ri - n ) ,
^ W = /, , . ^ . </5'('^> f l ) = -7r/2 + y3(r, n) - i£{r, n),
(p{r,ri) = cos
V l + 4 r COSÖ
1 X — xi
y/{x - xi)'^ + {y- yiy Si
e{r, ri) = sinh -1 \z + zi\
^/{x - xi)'^ + (y - yi)'^
( Q U - Q m i i - Q i ) '
_ d<j){r,rr,) _ XY' -X'Y .Z'jX^ + Y^) - ZjXX' + YY') Mr,rr,)- - x 2 + y 2 + ^ ( x 2 + y 2 ) V x 2 + y 2 + z 2 '
( X , y , Z) = [x - - ??(3;i - X l ) , P{y - yi) - /97/(yi - y i ) , z + zi+ r]{zi - zi)],
iX', Y', Z') = [-{xi - X l ) , -/?(y( - y i ) , - zi], Ql ^ zi- i{xi cos ( f ) + fSyi sin </>),
y V x 2 T y 2 T z 2 + i X Z . - X \ / X 2 + y 2 + ^2 + ^ y ^
^ ^ ^ ^ = ^ 2 ^ 7 2 . «^^«^^ ^ 2 ^ 7 5 • (-1)^ must be omitted i n (22) when integrating the range [a — TT, —7r/2] for j = 1. /3 — 1 if y — yi > 0 and /? = — 1 i f y — yj < 0. For practical calculations, it is more convenient to modify
expression (22) to: i l / / TdS=j2i-^y C Y.W[c,ci>{T,ri)] 1 ikj cos 9 ikj sin 9 k. 2Si ^e''iiP+Q'] dt 2 / - i ^ K ^ J o § 1 0 0 LO where W[c, Hr, ri)]
= ' ,
xit, c, <t>) = A : , ( Q , + i v - i - Q 0 ( < 3 m - Q 0 i , = ^ ( , , , , ^ ) nir,rr,)Jiir,rrj)dT] (23) {(j){r, ri) — c)t + (f){r, ri) -f cThis expression accelerates the computation i f the summation w i t h respect to / is performed i n one subroutine for evaluating the integrand at a same time.
7. N u m e r i c a l E x a m p l e s of the G r e e n Function
Table I shows CPU time (FACOM M780/20) required for evaluating the Green function and its gradient w i t h our scheme for a specified accuracy, for r = 0.3, Y{= Ko\y — y'\) = 25.0 and three different Z values. The calculation time is average over 400 points i n the interval -125 < X ( = Ko{x - x')) < 125.
Table I : Absolute accuracy and CPU time in IQ-^s for calculating
the Green function
e Z/1.25 e - 1 - 1 0 - 2 _ i o - 3 10-4 1.21 1.36 10-^ 1.41 1.56 lo-'^ 1.59 1.76 10-7 1.77 1.93 10-8 1.88 2.02
Fig. 2: Wave contour due to the wave source [top:
Fn = 0.5, K\z'\ = 0.2, bottom : F„ = 0.5, K\z'\ =
0.3,1
Fig. 2 shows the wave el-evation C = —{l/g){iu>e —
Ud/dx)G generated by a point
source located at {x',y',z') =
(0,0,2'). The conditions are
Fn = U / y / ^ \ = 0.5 and K\z'\ = ujl\z'\/g = 0.2 and 0.3.
The calculation was performed for about 5,000 points: 100 x values i n the interval -10|2'| <
X < 10\z'\, and 50 y values w i t h i n 0<y< I0\z'\.
For the case K\z'\ = 0.2 ( r = Fn\/K\z'\ < 0.25) we see the k2 wave system propagating forward of the source; this wave system disappears for r > 0.25 {K\z'\ — 0.3). The wave contours agree well w i t h Kobayashi's (1981) results computed by using the exponential integral function.
Table I I lists the computed Green function and its derivatives. The source panel is a triangle composed of three vertices n = (-0.3,0.5,-1.01), r2 = (1.0,0.2,-0.98) and
rz = ( 0 . 2 , - 0 . 5 , - 0 . 9 9 ) , and the field point is located at rp = (1.0,2.0,-0.2). The
Rank-ine term is excluded i n this calculation, and the conditions are KQ — 10 and r = 0.2,0.3. The 'Monopole method (Bessho)' is based on (18) and (19), where the monopole is at the center of the triangle. We can use these results as benchmark. The 'Panel method (Bessho)' means the calculation based on (22) obtained by integrating (18) and (19) over the triangle analytically, whereas the 'Panel method {EiY is based on (21) obtained by integrating (16) over the triangle analytically. I n all the calculations based on Bessho's expression the steepest descent integration scheme was applied, and four decimals accuracy was required for numerical integrations.
The results of the 'Panel method (Bessho)' and the 'Panel method (£^1)' coincide w i t h each other within the required accuracy, but they differ from the result of the 'Monopole method (Bessho)' especially the derivatives. This implies a significant effect of the inclination of the panel which can not be considered by the monopole method.
(Bessho)' is indicated. Generally the 'Panel method' needs more time to that of the 'Monopole method (Bessho)'. The difference between the 'Panel method (Bessho)' and the 'Panel method (£•1)' arises from the differences of the expression for G and from the numerical integration scheme. Bessho's expression and the steepest descent integration are fastest. A l l the calculations in this table were performed on EWS Apollo DN3500.
Table I I : Comparison of the Green function per panel area determined by 3 different methods (a) KQ = 10.0, r = 0.2 (b) KQ = 10.0, r = 0.3
Real Imag Time Real Imag Time
Panel method (Bessho) G -0.32482E-1 -0.13008E-0 4.3 Panel method (Bessho) G -0.23814E-1 0.17983E-1 5.4 Panel method (Bessho) Gx -0.64147E-1 0.19134E-1 4.3 Panel method (Bessho) G. 0.41697E-1 -0.92392E-2 5.4 Panel method
(Bessho) Gy -0.46416E-1 0.39403E-1 4.3
Panel method
(Bessho) Gy 0.34072E-1 0.20539E-1 5.4 Panel method (Bessho) -0.52208E-2 -0.83300E-1 4.3 Panel method (Bessho) G. 0.32710E-2 0.43742E-1 5.4 Panel method {E{) G -0.32477E-1 -0.13008E-0 4.3 Panel method G -0.23814E-1 0.17983E-1 18.7 Panel method {E{) -0.64150E-1 0.19122E-1 4.3 Panel
method G. 0.41697E-1 -0.92395E-2 18.7 Panel
method
{E{) Gy -0.46423E-1 0.39404E-1 4.3
Panel method Gy 0.34072E-1 0.20539E-1 18.7 Panel method {E{) -0.52117E-2 -0.83302E-1 4.3 Panel method G. 0.32710E-2 0.43742E-1 18.7 Monopole method (Bessho) G -0.32071E-1 -0.13233E-0 1.0 Monopole method (Bessho) G -0.22206E-1 0.19546E-1 1.0 Monopole method (Bessho) G:. -0.66616E-1 0.18854E-1 1.0 Monopole method (Bessho) Gx 0.42497E-1 -0.13014E-1 1.0 Monopole method
(Bessho) Gy -0.47245E-1 0.40529E-1 1.0
Monopole method
(Bessho) Gy 0.35472E-1 0.18154E-1 1.0 Monopole method (Bessho) G. -0.50087E-2 -0.85499E-1 1.0 Monopole method (Bessho) G. 0.77003E-2 0.45381E-1 1.0
8. Discretization of the B o d y Surface
We discretize the body surface into panels to solve the boundary value problem numerically. The most popular panel is plane, and physical variables are treated as constants within each panel ('constant panel'). The algorithm is simple, but many panels are necessary to get accurate solutions.
Recently more accurate higher-order panels have been applied for the 3-D Green function method. The isoparametric element is one of the typical higher-order panels. Both the geometry of the body surface and the physical variables on each element are approximated by higer-order polynominals. I t is not possible to guarantee the continuity of the derivatives of the potential at the panel boundaries. Many elements are still necessary if high accuracy is required.
A spline element can be effectively applied to overcome this problem. Sclavounos and Nakos
(1988) initially applied the spline element for their Rankine panel method to express the velocity
potential distribution on the free surface, but the application was confined to plane elements because of the difficulty of integrating the Rankine source, which includes a singularity, over curved elements. Iwashita et al. (1994) applied a spline element for the curved panels on the ship surface and showed the efficiency of this element by comparing it w i t h isoparametric or constant elements.
8.1 Discretization based on the Constant P a n e l Approximation 8.1.1 Integral E q u a t i o n
We will use an indirect method following (9). Most of the calculations reported up to now were performed by this method because of its convenience i n estimating the hydrodynamic forces. We discretize the ship surface into N plane elements and assume that the source density is constant i n each panel. We can then represent (9) as
f - f : [ni . VGh - • d^] = ^ for k ^ 1...N. 1=1
(24)
Gil and G^ designate following expressions involving G^^^ and G^'^\ which are the first and
second right-hand term i n (14), respectively:
^Gh= [[ V G ( « ) [ r G f c , n ] d 5 + / / VG^''^[rGk,ri]dS, JJASI JJASi = 11 VG^''\rGk,ri]dS + VG^'^\rGk,rGi]Si, JjASi V G ^ H / VG^^\rGk,ri]dy', JAL, lALi ^^G^^^[rGk,rGi]Yi,
where index k and I indicate panel number, and
(25) (26) (27) (28) ajk 4>jk ASk Sk source density, Yk velocity potential, 71^;'^ surface element, line element, rGk area,
dy' of line element,
X component of unit normal vector,
unit normal vector,
position vector of field point on SH, position vector of source point.
The panel integrations involving G^-^^ in (25) to (28) can be performed exactly, see e.g. Hess and
Smith (1964), Webster (1975), and Newman (1986). On the other hand, as for the integration
of G^'^\ the monopole estimation is generally applied using (26) and (28). The exact integration scheme given i n chapter 6 is also applied sometimes.
8.1.2 H y d r o d y n a m i c Forces
After solving the discretized integral equation (24), the velocity potential and its partial derivatives must be calculated by transforming (7) into a discretized form:
^<t>ik = -j2^ji\^Gli-'^VGVi\ iork = l...N. (29)
1=1 ^ ^ 0 J
The hydrodynamic forces can be obtained by estimating the pressure from these values and integrating i t over the ship surface. E.g. from (11) the wave pressure is computed as
Pk .'^e/, , U
-i—(l + —Vu-V)[ct>ok + hk] forfc = l...Ar. (30) pgA ujQ V iuje
Integrating the wave pressure over the ship surface gives the wave exciting force direction j:
8.2 Discretization based on the H i g h e r - O r d e r PEinel Approximation 8.2.1 Sphne E l e m e n t
We consider the global coordinate system oxyz and the local coordinate system o'^ijC, Fig. 1. ^ is taken along the hull surface from stem to stern, and rj means the direction from the waterline to the bottom; C points into the same direction as n. Then using a spline function, the position vector r — {x, y, z) can be expressed i n the following form as the function of ^ and 77:
H^'V) ^T.T.i'^kuPkUlkóBkimiv)- (32)
fe=l1=1
and A''^ are the grid numbers corresponding to the direction ^ and 77, and Bk{^) and Bi{ri) designate the B-spline function, a^i, Pkl and 7 ^ ; are the spline coefficients which can be uniquely determined from the global coordinate of the panel nodes. The unit vectors e^, e,, and along ^, 77 and C, are calculated from
dr 1 dr
=
ml
dr 1 dr drj=
— X Cj, f 1 dx 1 dy 1 dz\ _ . , > /i2 dr] / i 2 dr]' h2dr]' (33)w i t h hi = | 9 r / ^ | , /12 = \dr/r]\, h^ = \dr/^\. The velocity components can be transformed from local to global directions:
/d(t>/dx\ / e ^ i e^2 e^3 \ ~ V ö < / ) / / i i ö C \ /Cn Cu Ci3\ /d(l>/hidC\ d(f)/dy = e^i e^2 e^s d(f>/h2dr] = C21 C22 C 2 3 d4)/h2dr] . (34) \dct>/dzj Veci ec2 e^3 / \dct>/hzdC) VCsi C32 C 3 3 / \d<j)/h3dC.
Similarly to the geometrical variables, the velocity potential and its normal derivatives on the ship surface can be expressed also by the spline function. By introducing the spline coefficients Ajti and 9ki, it follows that
Nr,
<l>^EE^MOBi{r]), (35)
fc=i 1=1
^ - E E ^ M O B i i v ) . (36) k=i 1=1
Oki is known because d4>ldn is generally known i n the 3-D Green function method. \ki is the
unknown corresponding to 0.
Using the relation d4>/dn = dcjx/hzdC, on the ship surface, the fluid velocity i n x-direction is derived from (34), (35), and (36) i n the form
Yx
=
E E h ( ^ ^ U O S i ( r / )
+
^ B u m i i r i ) ) + euC,,Bu{m{r])\. (37) /c—1 / — I8.2.2 Integral E q u a t i o n
We used (9) i n the previous section as an example of the discretization. I n this section, we will focus our attention on (8) i n which the velocity potential is unknown, and discretize (8)
w i t h spline elements. When we express the body geometry and the velocity potential on it by sphne functions, (8) can be discretized as follows, considering (34), (35), and (36):
Ni Nr, ƒ ƒ S f c ( 0 5 K ^ ) ^ ^ ^ ^ % ^ ^ / i i ( C , ^ ) / ^ 2 ( e , r ? ) d ^ d r ? NL J — i B , m M ( ^ ^ ^ ^ % ^ + 2 7 Ï ^ G ( - ) [ n , r(e, 1)]) } n . ( e , 1) /.r(e, 1) f [ Bk{0Bi{v)G^''\ri,r{^,v)]hi{^,v)h2{tv)d^dr] Ni Nr, fc=i 1=1 + E / ' ^ ' r^'öfc(C)5i(r?)G(^)[r,,r(C,7?)]/ii(e,r/)/i2(C,7?)ded7? J=l
É CisBk{OBi{l)G^'^^[ri, r(e, l ) ] n , ( ^ , 1) /ii(e, 1) d^
for I = l...N^ • Nr,. (38)
G was split into G(^) + G(^) as before, r j is the position vector of the I-th field point on SH, ^fc,
rj'i designate the region of ^, r] forming the A;-th, Z-th spline panel, ^ j , r]j are the local coordinates
of the lower left vertex of panel J, A'^ is the number of elements, and NL the number of element along the free surface.
The line-integral term multiplied by I/ATQ is not to be calculated if the panel does not touch the free surface. (38) includes the integration of the Green function multiplied by the B-spline function. For the wave-term G^-^\ a numerical integration scheme such as Gauss four point formula is applicable. On the other hand, the Rankine term G^^^ should be evaluated carefully because i t is singular. Iwashita et al. (1994) used a numerical integration. The integration of the Rankine part multiplied by the B-spline function can be expressed generally in the form
J= f{xuX2)dxidx2. (39)
1 3
The integrand f{xi,X2) has a singularity of 0 ( l / e 2 ) or 0 ( l / e 2 ) [e — \ri-r\) when r / coincides w i t h a vertex such as r ( a i , 02). When ƒ ( x i , X2) does not have singularities, i.e. r / is out of the integral range, the numerical integration scheme based on the Clenshaw-Curtis method Torii
et al. (1978), is applied. When f{xi,X2) has a singularity, i.e. r j coincides w i t h a vertex, the
following transform is performed:
, ( f e i - Q l ) ( b 2 - a 2 ) / - ° ° r ^ N 1.5coshti 1.5cosh^2
4 J-oo J-oo cosh'^(1.5smhii) cosh'^(1.5smht2)
where
Xl = [(61 - a i ) t a n h ( 1 . 5 s i n h t i ) + (61 + ai)]/2,
X2 = [{b2 - a2)tanh(1.5sinhi2) + (^2 + ^2)1/2,
and the field point is shifted a little ( 0 ( 1 0 " ^ ° ) ) into the normal direction, before the Clenshaw-Curtis method is used. Five decimals accuracy is appropriate for this numerical integration.
8.2.3 H y d r o d y n a m i c Forces
After determining ttie Xku the velocity potential and its derivatives are evaluated by (34) and (35). The wave pressure at each grid point is obtained by substituting these (/)7, V(^7 and the incident wave potential (/«o, V(/)o into (10). Here the spline approximation is also applied for the wave pressure distribution:
^ - ' ^ i ^ h M O B ^ n Y (41)
The wave exciting force is computed by using 6ki obtained from (41) as
A E EE^fc^^'^(05K^)ni(e,r/)/ii(e,r?)/i2(^,r?)d^c?r? ( j = 1...6). (42) Since this integration does not have singularities, the previous numerical integration scheme is easily applied. The determination of the added mass and damping coefficients in (13) is analogous.
9. N u m e r i c a l E x a m p l e s
9.1 Submerged B o d y and Slender Ship
The numerical diffculties of the wave making prob-lem of a ship w i t h forward speed originate in the 'surface piercing' of the body which complicates the numerical calculation of the kernel and the analy-sis of the line-integral. These difficulties disappear for submerged bodies. Fig. 3, Iwashita and Ohkusu
(1989,1992), shows the added resistance in waves (a
second-order force) of the submerged prolate spheroid advancing i n following waves without wave-induced motion. The added wave resistance is large for low frequencies and negative in a narrow frequency range. These features were not realized before by the rational numerical method in which the forward-speed effect and the three-dimensional effect are taken into account accurately.
Numerical results concerning surface piercing bod- ~4O~2D'—to ' 05 A/i-ies refer mostly to slender ships. The assumption of
slenderness allows to ignore the line-integral term and pig. 3: Added resistance of a submerged the steady flow disturbance, and therefore makes the prolate spheroid in following waves application of the 3-D Green function method easy.
Figs. 4 and 5 show typical results obtained by Inoue and Makino (1989). The numerical results for their mathematical model of L/B = 8, which has fine bow and stern, are compared w i t h experimental and w i t h other numerical results based on the strip method and on the interpola-tion theory of Matsunaga and Maruo (1981). The 3-D Green funcinterpola-tion method seems the most accurate for a wide frequency range, including the low frequency range where the discrepancies between the experimental and other numerical results are relatively large, notwithstanding that the 3-D Green function method used the indirect method based on (9) and omitted not only the line-integral term, but also the steady disturbance on the body boundary condition.
4. O < 2, O 0 . 1 5 !> 0. 10 0. 05 Present Kethod N.S.H. — Slender ship th. by HatBunaci 0 E X F. by HatBunaci 3 4
\
Present Methwi N.S.N. — Slender ship th. by Mitaunaca O E X P. by Mitaunaca 3 4 3. O (O 2. O 1. 0 " 0 . 15 j s > a 0. 10 0. 05 Present Kethod N.S.K. Slender by KatAunaca ship th. by KatAunaca O E X P. by KatAunaca 3 4 Present Method K.S.N. Slender ship th. by O C X P. byFig. 4: Added mass and damping coefRcients in heave and pitch of a mathematical model ship {L/B = 8) at F„ = 0.2 0. 4 S 0. 2 Present Method n.s.M. — Slender ship th. by NatsunaKS O E X P. by NatsunaKS 180 •1 80" 0. 10 » 0 . 05 Present Method M.S.M. — Slender ship th. by Matsunaia O EXP. by Matsunaia I F n ^ O . 2~ O I S O * 180' TT O
-e-Fig. 5: Wave exciting heave force and pitch rrioment of a mathematical model ship {L/B = 8) at F„ = 0.2
9.2 B l u n t Ship
A n advantage of the 3-D Green function method is its wide rage of applicability w i t h re-spect to hull form, forward speed, and frequency. The 3-D Green function method will only be ac-cepted in practice i f i t can be applied to cases where the strip method and the slender-body theory fail. Therefore, the 3-D Green function method has been applied to blunt ships and wide frequency ranges, including oblique short waves to which other common methods can not be ap-plied. This research discovered some problems which did not appear for slender ships.
L / B = 5.1 B / d = 3.1
N = 1200
Fig. 6: Grid system of a blunt ship
Iwashita et al. (1992,1998,1994) applied the 3-D Green function method to a blunt ship
(Fig. 6) advancing in oblique short waves. The chain line in Fig. 7 shows the wave exciting forces obtained by the indirect method. The discretization used constant panels, and the steady disturbance (f>s was omitted in the analysis. This method was used also by Kobayashi (1981) and Inoue and Makino (1989), except that the line-integral term was taken into account i n the present calculation. The numerical results for vertical forces agree much less than the results for the slender ship. The discrepancy becomes remarkable as the incident wave length becomes shorter. Iwashita et al. (1992) performed the same calculation also for long waves ( A / L = 1) and confirmed good agreement between numerical and experimental results. Therefore the discrepancy arises from the application of the 3-D Green function method to the blunt hull ship at high frequency.
Slarboad side
W.L,
Sectional hull fonn Pressure gages
W.L.
Iwashita et al. (1993,1994) investigated several items
which might cause this problem: the resolution of the grid, the calculation method of the kernel, the effect of the ship hull form, and the line-integral. The problem was settled by solving (8) instead of (9), i.e. by using the direct method. This suggests irregular frequencies as a cause of the prob-lem. They employed also the spline element of chapter 8 for solving (8). The solid and the bold dotted lines in Fig. 7 show the results obtained by using this direct spline method w i t h and without, respectively, the line integral. The results show that the direct method is preferable and that the line integral should be included.
W.L. Fig. 9 shows the wave pressure distribution on the blunt
ship. The experimental data were measured on the hull sur-face. Fig. 8. For A / L = 0.3, x = 60° where the wave excit-ing forces determined by the indirect method are overesti-mated (Fig. 7), the wave pressure at the stern part is also over-estimated in this method. The direct method improves both the wave pressure results and the wave exciting forces (Fig. 7). The discrepancy observed at the bow part i n head waves is probably caused by the steady wave around the bow which cannot be considered in this calculation. However, the direct method predicts well the wave exciting forces and the wave pressures on the ship side.
Fig. 8: Location of the measured points of the wave pressure
I E , l / p g a ' B L I E I / p g a ' B L 0.15 U L = 0 . 3 / \ i fl
n
\ i r ] HA \ „ I i ..A..-.«.--t xyL=o.5 - - - Fn=0.2 kxSred mothod | Fn=0.2 dirwa method (with tno hlograJ) Fn=0.2 dlrwa irwlhod (wtlhoul KM inlsgral) Fn=0.0 d r « l molhod • •xperimsm (Fn=0.2) O «qjarimBfit (Fn=O.0)(a) Exciting surge force
l E J / p g a ' B L l E ^ I / p g a - B L 0.4 j 1 V U 0 . 3 1 ^ 1 i J i i.„». 1 I f V * 0.4 0.2 XA=0.5 !
r
•
-yis^ / 1 \ 1 ƒ j \ ..M. J. ' X * } z(tiog.) 180 90 X(t<8g.) 180 r 1 !.^uiTT.
V ' *1
V ° V * J i i IE l / p g a « B L(b) Exciting sway force
IE I / p g a ' B L 0.2 yi=o.3 ^
i
V.V\
1M
r 5 ^ • 1 0.6 0.4 ! 1 V L = 0 . 5 CS, II ^ V • --^ 1 > • — i 1 90 X(<isg.) 1 8 0 90 X (deg.) 180 1 o^^.^ • k — ' q . . . /i.X.,....|.,.^:.a.:J * 1 1 1 ^ • • f m = =(c) Exciting heave force
Fig. 7: Wave exciting forces on a blunt ship advancing in oblique short waves at F„ = 0.0 and 0.2
PI pg<
X=Odeg.
Ord.2 P / pgK Ord.5
• dirccl method (wiih line integral) • indirect method sirip method expeiiment U = 6 Odeg. \ 1 •••'/ ——^ f 9 o i o o T n ° 2.0 I, 1.5 X = 12 Odeg. \ ' " ' X=18 Odeg. T ' -90 -60 -30 0 30 60 ?0 (deg.) P / / ' g f Ord.2
X=Odeg. • direct meUiod (with line integral) • indirect nwlhod strip mcliiod experiinent X=60 deg. 1 X=12 3 deg. », \ ? V-c 1 O - n-. X=180 deg. j -90 -60 -30 0 30 60 90 (deg.) , P / P B ^ Ord.9 X=60 deg. O . 1 X=120 deg ,^ 1 ° " "(deg.r (a) A/L = 0.3 P / g ? Ord.5 x=o leg. i { 1 <- i J n X=6(j deg. J5. -90 -60 -30 (b) A/L = 0.5 x=o deg.
i
Si X=6C deg. , 8\ °
>f
Q---rr-"Oj^-IP -90 -«0 -30 0 30 60 90 (deg.) p/ P%K Ord.9 X=0 deg. X=60 deg. 3 0 ? X=12 Odeg. X o ° V¬ ° V -1 o . 0 1—O— X=18 Odeg. ° o A. p 9 / —r\~-)f^ 0 — ° _ t O ^ O -90 -«O -30 O 30 60 90 (deg.)Fig. 9: Wave pressure distributions on a blunt ship surface in oblique waves (JF„ = 0.2)
Fn-0.2
{wtlh lino Inlsgml.wtlh sJeady dlEturbanos) Fn=0.2
(wtlh line Inlegrsl.wllhout deady dltlurbanc^]
B ^ / p V c o
0.0 2.0 4.0 6.0 8.0 KL 10.0
B j j / p V c o L ^
0.0 2.0 4.0 6.0 8.0 KL 10.0 0.0 2.0 4.0 6.0 8.0 KL 10.0
Fig. 10: Added mass and damping coefficients in heave and pitch for a half-immersed prolate spheroid of (L/B = 5) at F„ = 0.0 and 0.2 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3
(wtih In* Inlflgral.wflh eyêsdy ditlurbanc*) Fn-0.2
(wllh line IntogrBl.wtihout Gte&dy dl«tiirbenc«) Fnrf.2
(wllhoul line Integral.wllh steady dteturbancfl) Fn.0.2
(wtthoul line Inteoral.wtthoul ctesdy dIstuflMince) O experiment
8.0 KL 10.0 8.0 KL 10.0
F i g . l l : Coupled added mass and damping coefficients between heave and pitch for a half-immersed prolate spheroid of (L/B = 5) at F„ = 0.0 and 0.2
Figs. 10 and 11 siiow tlie added mass and damping coefRcients of the half-immersed prolate spheroid. The effect of the steady disturbance (/«^ on the ship boundary condition is significant especially at low frequencies ( r = UuJe/g < 0.25). The comparison with the experiments shows the necessity of considering ^s- The importance of the line-integral term is also observed near T = 0.25. 9.3 Unsteady Wave F i e l d Yjl = 0.4 1 E x p e rii l e n t j 1 rii 1 |3 D ( en function jnetliod | 1 1 1 R a n k i n e p a n e l m e t l i o d ( N - K ) | 1 [ R a n k i n e p a n e l m e t h o d ( D - D ) j 1 - 1 . 5 0 XI{I,I2) 2 - 1 - 5 0 A 7 ( L / 2 ) 2
Fig. 12: Heave radiation-wave profile of a half-immersed prolate spheroid
[Fn = 0.2, KL = 10.0,
bold line: cos-component, thin line: sin-component]
The 3-D Green function method can pre-dict not only the forces acting on the ship, but also the unsteady wave field around i t . Fig. 12 shows the radiation wave due to the heave mo-tion of the half-immersed prolate spheroid. The bold solid line and the thin solid line indicate the cosine and sine components of the unsteady wave respectively. The results of the Rankine panel method based on 'Sclavounos and Nakos' method are also given. ' N - K ' in Fig. 12 means the results obtained by using the same free-surface boundary condition and the body boundary con-dition as adopted i n the 3-D Green function method. Therefore the steady disturbance is taken into account only i n the body boundary condition assuming the double-body fiow. 'D¬ B ' is obtained assuming the double-body flow as the steady flow fleld not only i n the free-surface boundary condition but also the body boundary condition.
The k\ wave system (short waves) is observed clearly in the 3-D Green function method, but not i n the Rankine panel method. Probably the grid resolution is insufficient i n the Rankine panel method. The short waves observed i n the results of ' D - B ' should be considered as different from the ki wave system itself, since the reso-lution due to the grid system is the same as i n method ' N - K ' . The wave fleld near the bow part, however, seems to be improved a little bit by in-troducing the double-body flow assumption.
10. Yasukawa-Sakamoto T h e o r y
Yasukawa and Sakamoto (1991) developed a new theory based on the 3-D Green function
method extending Ogilvie's (1968) and Baba's (1976) approach for the low-speed wave making problem to seakeeping. This theory can take into account the effect of the steady non-uniform flow in the free-surface condition, but not i n the ship surface condition (mj-term).
We assume a small forward speed J7 of a ship and the order of several variables as assumed by
Sakamoto and Baba (1986): LÜQ = 0{U-^), A = 0{U^), K = 0{U-^), cp = U[-x + <Ps] = 0{U),
(j) — 0{U^), = 0{lT^). Based on these assumptions the free-surface condition can be derived in the form
-aie</)j 4- 2ioJe{uo (t>jx + Vo (l)jy)+ul (t)jxx + ^UQVO (f)jxy + (pjyy + g (t)jz
\ \ ( ^ r ^ J r ' ^ - - - C o (43)
I
9l{x, v) b = 7)where
J ( x , y ) = - [ L : - ^ K - L : H c o s x + ^ o s m x ) } ' ] • e - ' ' ^ ( " ' ° ' ^ + ^ ' ' " ^ ^ (44)
[uQ, Vo) = {ipx{x,y,0), ipy{x,y,0)) shows the velocity field of the double-body flow at the
undisturbed free sui'face, and I{x, y) appears because the incident wave potential 0o does not satisfy the free surface condition in the presence of the steady disturbance. Co is the free surface elevation corresponding to the pressure distribution of the double-body flow and is generally small except near the stagnation points. Therefore we enforce the free surface condition (43) on z = 0.
According to the theory, the body boundary condition can be derived as
^ = n , f o r ( , = 1...6), ^ = - ^ onSH. (45)
The effect of the steady flow usually represented i n the rrij terms does not appear because i t is of a higher order on the ship surface.
Introducing a function G* (P, Q) which satisfles the homogeneous free surface condition of (43), and applying Green's second identity, (j)j can be expressed as
UP) =-11 MQ) G*{P, Q)dS + - I f MQ) ^*G*{P, Q)dx dy 0 for j = 1...6, I{Q)G*{P,Q)dxdy iovj = 7. lis, (46) iSp
Oj is a source distribution on the ship surface, and C* is a differential operator:
r = -ul - 2iup{uo{Q)^ + voiQ)^} + {MQ)^ + M Q ) ^ y + g^^- (47)
Analogous to the Green function i n chapter 4-6 G*{P, Q) i n (46) is defined as
- 1 - 1 -1 ^ n roo r^ r r'K (• . pk[{z+z')-iTUx] , k[(z+z')-im2\
a-(p.Q) - l i l - ^ u L L L ^ W . . - ( . . A ^ o c o s . ) ^
(48) where
"^'X^ix- x') cos{0 ± Oo) ±{y- y') sm{e ± 0o),
^2 J
{ ; : > : ; : { . - o ^ - ? - » - - s
-G*{P, Q) depends not only on the relative position of P and Q, but also on the steady velocity
field {uo,vo). A new reduced frequency TQ depending on ( ^ 0 , ^ 0 ) is used instead of the usual r .
The function G*{P,Q) i n (48) is more exact than the ordinary Green function G{P,Q) i n (14) because includes the effect of the double-body flow near the source point. The new reduced frequency TQ is a function of the fleld point and reflects the effect of the steady velocity field. Unlike i n the ordinary Green function, unsteady waves propagating forward can exist near the source point even for r > 0.25. Fig. 13 shows the wave field due to the point source potential G*(P, Q) for T = 0.275. I t may be compared w i t h that of the ordinary Green function G(P, Q) in Fig. 2, Yasukawa and Sakamoto (1991). The second term i n the right hand side of (46) was omitted i n the calculation. Although r is smaUer than 0.25, G*{P,Q) includes waves of high
steepness before the source point. This result confirms experimental observations by Ohkusu
(1984) for forward propagating waves near the bow part of a blunt ship even for r > 0.25.
These waves break i n front of the bow.
A33/ P V a.or 0.15 o Exp. —»— Present C a l . R.S.M. 0.1 Strip Method A55/PVL2 o Exp. - — Present C a l . R.S.M. Strip Method 10. 15. 0. 10. 15. <o\/e o Exp. — — Present Cal. \ R.S.M. \ Strip Method 10. 15. 0. 10. 15. to\/g Fig. 13: To field (top) and
wave elevation corresponding to G* (middle) and to G (bottom)
Fig. 14: Comparison of added-mass and damping coefficients in surge, heave and pitch of the half-immersed elHpsoid (L/B = 4, B/2d 1.25 at
F„ = 0.3)
X=180° , A / U 0 . 3
Fig. 14 compares added mass and damping coefRcients for an half-immersed ellipsoid computed by this method w i t h experimental data, Kobayashi (1981), and results of a Rankine panel method, Takagi (1993). The Green function results were obtained without the second and third terms i n the right hand side of (46), and G* was evaluated by Hoff's method. The results seem relatively good compared with other numerical results. However, a more exact calculation including the integration over the free surface i n (46) would be desirable.
Fig. 15 shows the hydrodynamic pressure distribution on a V L C C compared to experimental and strip method results of Tanizawa et al. (1993). The horizontal axis in the figure means the angle from the free surface and X = 0°, 180° correspond to weather-side and lee-side,
respectively. The computation did not include the ship motion as i t is negligible for A / L = 0.3. The results show that the pressure field was improved especially on the bottom compared to the strip theory.
90 120 150 180
0 (deg) Fig. 15: Comparison of the hydrodynamic pressure distribution on
a VLCC in head waves of A/L = 0.3 (F„ = 0.3, S.S.4)
11. Conclusion
Although the history of research on the 3-D Green function method i n seakeeping is fairly long, comprehensive numerical studies started just recently. They showed some previously unnoticed problems. Attempts to settle this problem are still i n progress. The main subjects of future research are:
(1) Numerical examples for many ship forms should be produced. To be practical, the 3-D Green function method should be robust.
(2) For blunt ships the effect of the line-integral should be clarified mathematically and a practical method to include it should be indicated.
(3) Efforts to improve the numerical accuracy not only of first-order, but also of second-order hydrodynamic forces such as the added resistance are necessary.
References
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BERTRAM, V.; YASUKAWA, H. (1996), Rankine source metfiods for seakeeping problems, STG-Jahrbuch
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MATSUNAGA, K.; MARUO, H. (1981), On the Radiation problem of slender ships with forward velocity, J. Soc. Nav. Arch. Japan 150 (in Japanese)
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18th Georg Weinblum Memorial Lecture
CFD i n Ship Design — Prospects and Limitations
Lars Larsson, Chalmers University of Technology^ 1. Introduction
Computational fluid dynamics (CFD) is becoming more and more popular for analysing flow problems in almost all branches of industry. The absolute accuracy is still limited, particularly for high Reynolds numbers, but physical insight into the problem may often be gained and used to improve the design. In this respect CFD is often superior to model tests, where the absolute accuracy is higher, but where the amount of information needed to guide the designer is more limited.
The present paper deals with CFD applications to ship hydrodynamics. Here, inviscid nu-merical methods have been used for a long time in propeller design and seakeeping calculations. Viscous methods have been less accepted in ship hydrodynamics than e.g. in aerodynamics, but during the past years, many shipyards, ship owners, and towing tanks have acquired also viscous methods to aid the design process.
2. Present use of C F D in hydrodynamics 2.1 Classification of methods
CFD should include all computational methods for fluid flows, but is most often confined to Navier-Stokes methods. In my opinion, this definition is too restrictive. Fig. 1 summarizes CFD in ship hydrodynamics including all methods. Simple potential flow methods are e.g. Michefl's wave resistance theory, lifting line theories in propeller design, or strip theories in seakeeping. The last two have been useful for many years and may be considered mature tools. Panel methods are more advanced and are already established tools in resistance, propeller design, and seakeeping. Boundary layer methods have never reached the usefulness of the potential flow ones, but they have been used for some time for resistance calculations, propeller blade flows, and roll damping. Modern Navier-Stokes methods are just making their way into resistance and flow and propeller effects. Euler methods play a much smaller role in hydrodynamics than in aerodynamics. The reason is that lift is much more important in the latter field. When resistance is the major issue, it seems more appropriate to turn directly to the Navier-Stokes methods rather than trying to couple Euler and boundary layer methods. In the following, the emphasis is on panel methods and Stokes methods for resistance and flow and Navier-Stokes methods for propellers and cavitation.
2.2 Codes regularly used in design
Computer codes which are regularly used in ship design are listed below^:
^ Inviscid free-surface codes: DAWSON, RAPID, SHIPFLOW, SHALLO, VSAERO, SWAN, SPLASH, SLAW
- Viscous double-model: SHIPFLOW, PARNASSOS, RANSTERN, NICE - Viscous free-surface: T U M M A C , W I S D A M
DAWSON and RAPID, developed by Raven (1988), (1996), have been used extensively in com-bination with model testing at M A R I N to improve hull designs. DAWSON uses a linearized free-surface boundary condition, while RAPID solves the fully nonlinear problem. SHIPFLOW,
Larsson (1993), Larsson et al. (1989,1990,1992a,1992b), contains both inviscid and viscous
methods and is widely used. The inviscid part can also solve the fully nonlinear problem. 'Dept. of Naval Arch, and Ocean Eng., S-41296 Gothenburg, Sweden
Resist. & Flow Prop. & Cavit. Seakeeping Manoeuvring Simple Pot. Flow Methods
Traditional Traditional Traditional Traditional
Panel Methods Modern, Established Modern, Established Modern, Established Modern, Not Established Boundary Layer
Methods Traditional Traditional Traditional —
Navier-Stokes Methods
Modern, Not Established
Modern,
Not Established Future Future Fig. 1: Use of CFD in ship hydrodynamics. Bold face indicates areas discussed in this paper
SHALLO, Jensen (1988), Jensen and Bertram (1993), has been used by HSVA during many years. VSAERO is a well-known code in aerodynamics, originating from NASA, but since several years marketed by Analytical Methods Inc. A free-surface capability was introduced by Maskew (1989). The code has been updated for unsteady free-surface problems, Maskew
et al. (1993). Due to its capability to include l i f t , VSAERO has been popular in America's
Cup design. The same is true for SPLASH, Rosen et al. (1993). During the 1995 campaign, SPLASH was used extensively in one of the Australian America's Cup syndicates. SWAN and SLAW use only linearized free-surface conditions. SWAN, Sclavounos and Nakos (1988), seems to be mainly used for unsteady calculations. However, during the 1992 and 1995 America's Cup campaigns, i t was used in the PACT syndicate also for wave resistance, Sclavounos (1996). A special technique takes the nonhnear effects of the large overhangs of the yacht into account. SLAW, Letcher et al. (1989), was developed for the 1992 America's Cup, but is now used by some American yards, Stromgren (1995).
SHIPFLOW includes viscous flow methods for the double-model case. Over the forward part of the huh the flow is computed by a boundary-layer theory while the stern flow is obtained from a Navier-Stokes solver^, Larsson (1993). PARNASSOS is a Navier-Stokes code developed at M A R I N , Hoekstra (1989), Eca and Hoekstra (1996). Both codes seem to be used regularly for practical design work, although not at all as frequently as the corresponding potential flow methods. RANSTERN, Ju and Patel (1991,1992),\s also used to some extent at shipyards. In Japan, the codes developed at the Ship Research Institute have been used at the yards for some time. NICE, Kodama (1992) was extended to include a free surface by Liu and Kodama (1993). The best-known viscous free-surface code T U M M A C , Miyata et al. (1985), is used at several shipyards in the Far East. Another popular code from the same department is W I S D A M , Miyata
et al. (1992), Miyata (1996).
There are many other codes in the literature, see e.g. the two CFD workshops held in 1990s,
Larsson et al. (1991), Kodama et al.
(1994)-2.3 T y p i c a l applications
The most common test cases for CFD methods are the HSVA tanker and the Series 60,
CB = 0.60 hull used at the two CFD workshops. Also popular is the Wigley hull, defined by
parabolic sections and waterlines. Together with the fast monohull "Athena" these three hulls were used in the cooperative experimental program coordinated by the Resistance Committee of the I T T C . A t the 20th I T T C conference in 1993, the Resistance Committee suggested in ^Except when specifically stated, Navier-Stokes methods mean here methods based on the Reynolds-averaged Navier-Stokes equations (RANS)
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