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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 2
WAVY/FREE-SURFACE FLOW: PANEL METHODS 2

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Numerical Analysis of Nonlinear Ship Wavemaking Problem by the Coupled Element Method
X.W.Yu,1 S.M.Li,2 and C.C.Hsiung1
(1Technical University of Nova Scotia, Canada, 2Wuhan University of Water Transportation Engineering, PRC)
Abstract
In this paper, the free surface condition for the ship wavemaking problem is analyzed and simplified with a new order analysis for the ship wavemaking potential based on the slow-ship theory. The total velocity potential is expressed as Φ=x+φr+φ, where φ is the wave disturbance potential, and φr is the double-body disturbance potential with the order and k>1, so that the nonlinear free surface condition can be simplified and calculated on z=0. In the numerical calculation, the coupled element method is applied. The flow domain is divided into inner and outer regions. The finite element method is used with the simplified nonlinear free surface condition for the inner region and the Green function method is employed with the linear free surface condition for the outer region. In the outer region, the Kelvin source function is used as the Green function, so that no numerical treatment of the radiation condition is needed. Numerical calculations were carried out for a cylinder, a sphere, a Wigly model and a Series 60 Block 60 ship model. The computed results agree well with the experimental results.
Nomenclature
a
: radius of a cylinder or a sphere
A,B,C,D,E,F,H,P,Q,R,S,U,V,W
: order groups or coefficient matrices
CB
: block coefficient
CL
: lift coefficient
Cw
: wave resistance coefficient
Cx
: midship section coefficient
Cs
: wetted surface area coefficient
D
: flow domain
D1
: inner region of the flow domain
D2
: outer region of the flow domain
Fn
: Froude number
g
: gravitational acceleration
G
: Green's function
L,B,T
: ship length, beam, and draft, respectively
LB
: intersection of ship hull and free surface
L1,L2,L3
: path of the line integral
: unit outer normal vector
N
: total number of nodes
Ni
: shape function
p
: pressure
Rw,L
: wave resistance and lift on a body, respectively
SB
: wetted surface of ship
S1
: boundary of the inner region
S2
: boundary of the outer region
SF1
: free surface of the inner region
SF2
: free surface of the outer region
Sj
: interface of the inner and outer regions
S∞
: boundary surface at infinity
U
: ship speed
x,y,z
: Cartesian coordinate system
α
: solid angle at a control point
η
: wave elevation
Φ
: total velocity potential
: disturbance potential
φ
: wave disturbance potential
φr
: double-body disturbance potential
ρ
: fluid density

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
1
Introduction
Normally, ship wavemaking is a highly nonlinear problem. The major difficulty in this problem lies in the nonlinear boundary condition at the unknown location of the free surface. A basic approach to deal with this nonlinear problem is to employ the perturbation analysis. Almost exclusively a linearized condition is applied at the free surface, and in most cases the solution is described by a superposition of complicated singularities that satisfy this linearized free surface condition. Based on the assumption of small Froude number the low-speed ship theory, which takes account of the nonlinear effect on the free surface condition, has been developed. The perturbation process was applied to the zero-Froude number flow field instead of the free stream. The series expansion with respect to the wave elevation was carried out. The small parameter of Froude number was introduced to simplify the nonlinear free surface condition. This perturbation analysis with a quasi-analytic method, which was developed by Baba [1] for the low-speed flow past a blunt ship bow, gave a good agreement of computed and experimental results of wave-resistance coefficient over a range of low Froude numbers. Based on the same perturbation method, in 1977, Dawson[2] developed a numerical method by distributing the Rankine sources on the body surface and on the local free surface around the body. A wave field was superimposed on the double-body flow. The source strength distribution on the body surface and on the local free surface was obtained by satisfying the body boundary condition and the free surface condition. The radiation condition, which states that the ship waves occur only behind the ship, was replaced by a one-side finite difference operator for the second derivative of the potential in the direction of the double-body streamlines appearing in Dawson 's free surface boundary condition. Good numerical results were obtained by Dawson's method despite the fact that the numerical treatment of the radiation condition has no theoretical support.
The finite element method is known to be flexible for the nonlinear problem and for the boundary value problem with a complicated boundary. Therefore many researchers have developed numerical methods based on the finite element method to solve the free surface flow problems. Bai[3] developed the localized finite element method for calculating wave resistance of a body moving in a channel. Recently Bai and others [4] successfully used the finite element method to solve the nonlinear wavemaking problem in shallow water, but his method was not applied to the infinite domain. Eatok-Taylor and Wu[5] used the coupled element method to calculate wave resistance and lift on 2-D submerged cylinders, but did not cover the 3-D ship wavemaking problem.
In this work, a new order analysis of the wavemaking potential is developed. As a result, a simplified free surface condition, which takes account of the effect of nonlinearity but is different from Dawson's free surface condition, is obtained. The coupled element method is employed to solve the 3-D ship wavemaking problem. In the numerical computation, the flow domain is divided into two regions as shown in Fig.1: the inner region originates around the ship hull and is bounded by surfaces Sj+SB+SF1; the outer region is outside the interface Sj and is bounded by surfaces Sj+SF2+S∞. In the inner region the effect of nonlinearity of the free surface is very important, so that the finite element method is used together with the simplified nonlinear free surface condition. The effect of nonlinearity is negligible in the outer region far from the ship hull, where the Green function method is employed with the linear free surface condition. The Kelvin source function is adopted as the Green function, consequently, the radiation condition is satisfied exactly. By matching the solutions of the inner and outer regions at the interface of two regions, the solution for the entire flow field can be found. Numerical computations have been carried out for a submerged cylinder, a submerged sphere, a Wigly model and also a Series 60 Block 60 ship model. The computed results agree well with the experimental results.
2
The Free Surface Condition
2.1
Exact Mathematical Expression for the Wavemaking Potential
The coordinate system and flow domain are shown in Fig. 1. It is assumed that the fluid is ideal and incompressible, and the flow is irrotational and steady. The Froude number is defined by , where U is the ship speed, g the gravitational constant and L the ship length. The

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
total velocity potential is written as
(1)
in which must satisfy the Laplace equation
in domain D (2)
subject to the following boundary conditions:
on the free surface η(x,y) (3)
on the free surface η(x,y) (4)
on the body surface SB (5)
(6)
The problem described by (2)—(6) is the exact mathematical representation of the boundary value problem of the ship wavemaking potential.
2.2
Simplification of the Free Surface Condition
The ship wavemaking problem described by (2)–(6) is a highly nonlinear problem. As well-known, the difficulty lies in that the free surface condition is nonlinear and must be satisfied on the unknown surface. Before trying to solve the nonlinear problem, we have to simplify the free surface condition. It is assumed that the Froude number, Fn, is sufficiently small.
By substituting (3) into (4) and keeping the terms of the products of derivatives of
(7)
after applying Taylor's expansion to the wave elevation and again keeping the products of derivatives of , we obtain
(8)
The disturbance potential can be further decomposed into two parts,
(9)
where φr denotes the double body disturbance potential and φ the wave disturbance potential.
If we assume that:
(1)
for φr, i=1, 2, 3
(2)
for φ, i=1, 2, 3
Substituting (9) into (8), we obtain
(10)
If (10) is classified according to the following orders

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
From above, these different groups generally differ from one another with their orders of magnitude. It is clear that:
1. for the terms with the wave potential φ, no matter what value n is, the following is always true:
n−2<n+k−2<n+k
which means that group D is smaller than group A by , and group E is smaller than group D by
2. for the terms only including the double-body disturbance potential, no matter what value k is, the following always exists
k+2<2k+2
which means that group C is smaller than group B by
The order of magnitude of each group with the difference between n and k is shown in Table 1.
Table 1 The order of magnitude with the difference between n and k
GROUP
ORDER
n=k+5
n=k+4
n=k+3
n=k+2
A
n−2
k+3
k+2
k+1
k
B
k+2
k+2
k+2
k+2
k+2
C
2k+2
2k+2
2k+2
2k+2
2k+2
D
n+k−2
2k+3
2k+2
2k+1
2k
E
n+k
2k+5
2k+4
2k+3
2k+2
F
2n−4
2k+6
2k+4
2k+2
2k
From Table 1, it can be seen that:
Group E is the first one to be neglected.
With a decrease of the difference between n and k, group F becomes more influential and group C becomes less influential.
In addition to groups A and B, the first group to be considered is group D.
The value of k has no effect on the choice of the terms in the free surface condition.
The difference between n and k represents the magnitude of the wave potential. The large difference shows the small effect of wave potential, and the small difference means that large wave has been made by the ship at high speed. Since we assume that the ship speed is low, the small difference will lead to invalidation of the assumption, therefore in the numerical calculation we only keep groups A, B, C and D in the free surface condition:
(11)

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
If only the group A is kept in the free surface condition, the well-known linear free surface condition is obtained:
(12)
3
The Coupled Element Method for the Ship Wavemaking Problem
In the boundary value problem of ship wavemaking, the wave disturbance potential should satisfy:
∇2φ=0 z≤0 (13)
z=0 (14)
on SB (15)
(16)
where for the linear ship wavemaking problem, , f2(x,y,z)=−nx; and for the nonlinear problem, it is easy to obtain f1(x,y) from (11), and f2(x,y,z)=0.
As mentioned before, for numerical calculation, the flow domain is divided into two regions. In the inner region around the ship hull, the finite element method is used to solve the nonlinear problem; whereas, the effect of nonlinearity is negligible in the outer region far from the ship hull, thus the Green function method is adopted to solve the linear problem. By matching the solutions of inner and outer regions at their interface, the solution for the entire flow field can be obtained.
The flow domain is shown in Fig. 1. D1, with the boundary S1=SB+SF1+Sj, represents the inner region where the wave potential is defined by φ1. D2, with the boundary S2=Sj+SF2+S∞, represents the outer region where the wave potential is defined by φ2. On Sj, the interface of two regions, φ1 and φ2 should satisfy the matching conditions:
φ1=φ2 on Sj (17)
on Sj (18)
3.1
The Green Function Method for φ2 in the Outer Region
In the outer region D2, the linear free surface condition (12) is used. Applying the Green function method to the potential φ2 in D2 gives
(19)
Using the matching conditions (17) and (18), on the surface Sj, we obtain
(20)
where α is the solid angle at the control point p on Sj and G is the Kelvin source function[8]
(21)
If φ1 and in D1 are now approximated by
(22)

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
(23)
where Ni is the so-called shape function. If the problem we deal with is a Neumann-Kelvin problem of a submerged body, we can choose the inner region under the free surface, then the line integral on the line L1+L2 will disappear. Eq. (20) can be simply rewritten in a discretized form
(24)
or, identically,
(25)
where the element of [A] is
(26)
the element of [B] is
(27)
In a general case, the line integral on L1+L2 exists, then Eq.(20) can not be written as (24). We have to pay special attention to the treatment of the line integral.
3.2 Auxiliary Equations for Computing the Line Integral on the Free Surface
When the boundary of the inner region D1 includes the free surface SF1, the line integral on L1+L2 exists(please see Fig. 1). Since the Kelvin source function is not defined on the free surface when control points are the nodes on the L1+ L2, the Kelvin sources cannot be distributed on these points. After discretization, the number of equations is not equal to the number of nodes on the interface Sj, then the linear equation system is not in a closed form. It can be expressed by the following matrix form
(28)
where [φ1] is the unknown potential vector at the nodes on the surface Sj under the free surface, and [φ10] is the unknown potential vector at the nodes on the path of the line integral. [D] and [F] are the coefficient matrices including the contribution from both surface integrals and line integrals on the interface elements, [E] and [H] are the coefficient matrices only related to line integrals on L1+L2.
Since the system equation expressed by Eq.(28) is not in a closed form, we have to establish auxiliary equations from the relationship between φ1 and on L1+L2. The potential φ1 has to satisfy the following conditions: φ1 is continuous in D1, and the continuity is extended to the boundary of D1; in addition, φ1 has to satisfy the Laplace equation and the free surface condition.
By the finite element approach, we can write
(29)
(30)
then we have
(31)
For example, on L2, we have
(32)
(33)
In the outer region D2, we have the linear free surface condition:
After discretization, we can write the matrix form
(34)
combining (28) and (34), we get
(35)

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
we can also write (35) as
(36)
where
where N is the total number of nodes on the Sj, and M is the total number of nodes on the elements only along the line integral path plus the total number of nodes on the Sj.
3.3 The Finite Element Method for φ1 in D1
According to the Galerkin method, for every Ni, φ1(x,y,z) has to satisfy
(37)
By the Green thoerom, it is clear that
(38)
where S1=Sj+SB+SF1.
(38) becomes
(39)
a. the Neumann-Kelvin problem
For the Neumann-Kelvin problem
(40)
f2(x,y,z)=−nx (41)
(39) becomes
(42)
From (36),
(43)
(44)
Substituting (43) and (44) into (42), we write (42) as a matrix form
[U+V][φ1]=[W] (45)
where the elements of [U] are
(46)
The matrix [V] expresses the matching conditions. Its elements are
(47)
[W] is the known vector,
(48)

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
b. the nonlinear problem
For the nonlinear problem, refer to Eq.(11) and Eq.(14)
(49)
For Eq.(15)
f2(x,y,z)=0 (50)
Similarly as (45), for the nonlinear problem we can also obtain
[U+V][φ1]=[W] (51)
where
(52)
(53)
(54)
By solving (51) we can obtain φ1 for the inner region.
4 Numerical Results
The numerical calculations have been carried out for wave resistance of a 2-D submerged cylinder, a 3-D submerged sphere, a Wigly model and a Series 60 Block 60 model.
From the Bernoulli integral
(55)
for the nonlinear problem, =φr +φ1.
The wave resistance and lift are calculated by integrating pressure over the wetted body surface.
(56)
(57)

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
4.1 The Submerged Cylinder
For a submerged cylinder the wave resistance and lift have been calculated by the coupled element method. The inner region D1 was discretized with 8-node isoparameter quadrilateral elements. The nonlinear free surface condition is
(58)
and the 2-D Green function[8] for the outer region is
(59)
where
r=[(x−x0)2+(z−z0)2]½
r1=[(x−x0)2+(z+z0)2]½
The computed results are compared with results from Havelock's analytical solution and Eatock-Taylor and Wu's numerical results as shown in Fig.3. For the Neumann-Kelvin problem, our numerical results agree very well with the referred results. The solution with the nonlinear free surface condition is not too different from the solution with the linear free surface condition.
4.2 The Submerged Sphere
Using the coupled element method, the Neumann-Kelvin problem has been solved for a submerged sphere. The inner region D1 was under the free surface. 20-node isoparameter hexagon elements were used to discretize the inner region. The comparison between the numerical result and Havelock's analytical solution is shown in Fig. 4. They agree very well.
4.3 Wigly Ship Model
The Wigly ship hull of mathematical form is expressed by
(60)
where
with
CB=0.444 CP=0.667
Cx=0.667 Cs=0.661
The numerical results for the wave resistance coefficient are compared with the experimental results, shown in Fig. 5. Comparing with the solution with the linear free surface condition, the numerical solution with the simplified nonlinear free surface condition is more agreeable with the experimental results.
4.4 Series 60 Block 60 Ship Model
In the numerical calculation, the inner region D1 is chosen as −1.25L≤x≤1.25L, 0≤y≤ 0.5L and −0.06L≤z≤0. We discretized D1 into 120 quadratic isoparameter hexahedron elements with 819 nodes. The computed wave resistance coefficient curves are compared with other known results, shown in Fig. 6.
5 Discussion and Concluding Remarks
In the present study, based on the low-speed ship theory, an investigation has been made into the free surface conditions. Through a new approach of the order analysis, a simplified nonlinear free surface condition, which is different from Baba's and Dawson's, has been obtained.
The coupled element method has been introduced for numerical calculation. Efforts have been made for the treatment of the line integral along L1+L2 (Fig. 1). Auxiliary equations on the path L1+L2 of the line integral were established by using the free surface and continuity conditions to overcome the difficulty arisen in computing the line integral. Also, by using the Kelvin source function as the Green function in the outer

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Fig. 1—Top view of surfaces defined for a transom-stern ship.
(14)
Therefore, the final form of the integral equation is
(15)
Transom Stern
Flow past a ship with a transom stern is now considered. The flow is assumed to separate smoothly at the hull-transom intersection. As before SB denotes the hull surface beneath the mean free-surface level. In this case, however, SF denotes the mean free-surface level only for that portion which is not directly behind the stern. There is an additional surface SE which extends from the separation line at the hull-transom intersection to downstream infinity. Eventually, it would be desirable for SE to vary with respect to distance downstream from the hull so that the surface approaches the mean free-surface level downstream of the stern. In this paper, however, SE does not vary in x and so forms a cylindrical surface downstream of the transom stern. It is assumed that is a smooth surface. The surfaces, viewed from above, are depicted in Fig. 1.
Under the same conditions that were required in the case of a cruiser stern, Green's second identity can be used to obtain the integral equation
(16)
which is valid for . For , the first integral is a principal-value integral; for , the third integral is a principal-value integral; otherwise, the integrals are regular. To continue as before, the normal derivative in the last integral must be expressed in terms of the potential and its tangential derivatives on the boundary. If the surface is given by the equation
z−E(x,y)≡0, (17)
then the unit normal directed into the fluid is given by
(18)
and the last term of the integral equation can be rewritten as
. (19)
Here S′E refers to the projection of SE onto the mean free-surface level.
On the actual free surface z=Z(x, y), which is to be distinguished from the surface z=E(x, y), the nonlinear kinematic and dynamic free-surface boundary conditions are given by
(20)
and
(21)
respectively. It is assumed that , although it might be argued that y is not negligible in the vicinity of deep transoms with steep side walls.

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
After discarding these terms, the dynamic free-surface boundary condition becomes
x+Z/F2=0 on z=Z(x,y). (22)
Expanding the factors involving the potential in eqs. (20) and (22) in Taylor series about z=E(x, y) and dropping all terms of the series expansions except the lowest order terms, we obtain
(23)
and
(24)
on z=E(x, y). Eq. (24) can be differentiated with respect to x and combined with eq. (23) to obtain
(25)
on z=E(x,y) which, if we assume , becomes
(26)
on z=E(x,y). This equation is used to eliminate ζ from the the right hand side of eq. (19):
(27)
If , then eq. (27) becomes
(28)
and eq. (16) becomes
(29)
which is almost the same as eq. (12). The application of Stokes' theorem proceeds as it does for the case of a cruiser stern and we obtain the integral equation
(30)
This equation is the final form of the integral equation for the case of a transom stern hull.
Free-Wave Spectrum and Wave-Pattern Resistance
We wish to compute the magnitude of the free-wave spectrum whose sine and cosine components (u) and (u) are defined in eq. (35) of Eggers et al. (8). The definitions assume that all lengths have been normalized by the inverse of the fundamental wave number k0. The same normalization applies to , , and the magnitude of the spectrum. The transverse wave number u has been normalized by k0.
The free-wave spectrum is calculated from a pair of transverse wave cuts by a method derived by Sharma (9). Sharma, in contrast to Eggers et at., retained dimensions. Taking into account this difference, and are reproduced here in terms of the nondimensional variables of this paper. If the two transverse wave cuts are Z1(y)=Z(x1, y) and Z2(y)≡Z(x2,y), then
(31)
and

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
(32)
where
(33)
(34)
and
(35)
for k=1,2.
Using eq. (2.49) of Sharma, the wave resistance RW can be written in terms of and . The corresponding nondimensional wave-pattern resistance CWP is given by
(36)
where S is the wetted area and L is the length of the hull.
NUMERICAL IMPLEMENTATION
The hull surface and the finite portion of within the finite computational domain are approximated by strips of flat rectangular panels. Strips on SE originate at the hull-transom intersection; strips on SF originate at the computational boundary upstream of the hull. For the time being, the upstream panel in each strip of SF is assumed to have approximately twice the longitudinal dimension of those panels immediately downstream from it.
On each panel, the perturbation potential and its derivative ∂/∂nξ in the surface integrals of integral eq. (30) are assumed to be uniform and equal to their values at the centroid of the panel. Surface integrals over panels are then evaluated analytically.
On the most forward panel of each longitudinal strip of SF, the surface integral and the line integrals along the upstream edge of the strip in the last three terms of the integral equation are combined. On and around this panel ξ is set to zero. Then, when the range of integration for the integrals is restricted to this panel and its edges, the sum of the last three terms in the integral equation on this panel becomes
Here the common multiplicative factor of F2 has been deleted. The numbers 1 through 4 refer to the vertices in clockwise order around the panel when the panel is viewed from above. The side joining vertices 1 and 4 is at the upstream edge of the computational region. The first and third integrals are zero when the edges joining vertices 1 and 2 and vertices 3 and 4 are parallel to the x-z plane.
Similar treatment of the line and surface integrals at the downstream boundaries of the computational domain is employed except that ξ in the second line integral is discarded. Since disturbances propagate downstream, this treatment of ξ at the downstream boundary should not affect the flow near the ship if the downstream boundary is far enough downstream.
Hull-waterline integrals are split into a part along the hull-transom intersection and a part along the sides of the hull. The numerical treatment of the two parts differs. Along the sides of the hull in the first line integral of eq. (30), is assumed to be uniform over the edge of a panel where its value is extrapolated from function values at the centroids of two nearby free-surface panels. The derivative of the potential in the second line integral is approximated by finite differences as is shown in Fig. 2 and is assumed to be uniform over the edge of a panel. In particular, the derivative ξ at the waterline point a is approximated by a linear combination of the potential at a and at three points upstream of a; this differencing scheme is the same one as was used by Dawson (10). The potential at each of the points a through d is in

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Fig. 2—Top view of free-surface panels adjacent to the hull. The x-derivative of the potential at a is a linear combination of the potential at the points a, b, c, and d. The potential at a is in turn a linear combination of the potential at the centroids of panels 1 and 5. Similarly, the potential at each of the points b, c, and d is a linear combination of the potential at the centroids of two numbered panels. Thus the derivative of the potential at a is a linear combination of the potential at the centroids of the panels numbered 1 through 8.
turn expressed as a linear combination of the potential at the centroids of two free-surface panels. Thus the derivative at each waterline point is approximated by a linear combination of the potential at the centroids of eight free-surface panels.
At the hull-transom intersection along the assumed separation line which corresponds to the upstream edge of SE, an approximation to ξ based on hull geometry and a linearized form of Bernoulli's equation is available since it is known that the pressure there is atmospheric pressure. With this approximation, the last line integral of eq. (30) is evaluated analytically along the intersection. The potential along the separation line in the other line integral is set to the potential at the centroid of the free-surface panel directly behind the stern.
On the hull surface SB, collocation points are placed at the centroids of panels. For each panel on , there is associated a collocation point located at the centroid of the panel just upstream of it. This collocation point shifting is a convenient way of enforcing the radiation condition without the use of upstream finite differences.
At the upstream end of SF, where there are no upstream panels, the collocation point is shifted forward a distance approximately equal to the longitudinal dimension nearby free-surface panels. The upstream panel has two collocation points on it because it has twice the longitudinal dimension of the free-surface panels immediately downstream from it. All collocation points associated with SF thus lie on SF.
At the upstream end of strips of panels originating at the hull-transom intersection, collocation points are not associated with panels closest to the
Fig. 3—Top view of some of the panels in the neighborhood of a transom stern. Panels to the left of the heavy line are on the hull surface; panels to the right are on the surface about which the free-surface boundary conditions are linearized.
transom. Instead, equations that would be associated with these collocation points are replaced with equations specifying free-surface depth and slope in terms of the hull geometry at the transom stern. This treatment is based on the work of Sclavounos and Nakos (11) which showed that wave height and slope must be specified at the forward edge of a truncated computational free-surface domain. The specified free-surface elevation and slope ensure that the free-surface elevation and slope at the transom match the hull depth and hull shape. For a linearized problem, these are conditions on x and xx at the transom. After numerical experimentation, it was decided to enforce conditions on x at two successive panels immediately behind the stern rather than to use the direct approach of setting x at the centroid of one panel and xx at the centroid of a second panel. At the centroids of the two successive panels, x is set by means of finite difference equations to values obtained from truncated Taylor series expansions involving x and xx at the transom. An example of such a difference equation for strip j in Fig. 3 is
ij−i−1,j=(xij−xi−1j).
where is the midpoint of the forward edge of the j-th strip on SE and the other, subscripted x-values refer to the centroids of panels immediately downstream from the hull-transom intersection.
For free-surface paneling behind a transom stern, where smaller paneling is usually used, there is a lot of noise in the solution. An examination of columns of the final matrix showed that there was much more upstream-downstream symmetry in the matrix near diagonal elements corresponding to panels be-

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
hind the stern than elsewhere on the free surface. To produce more upstream-downstream asymmetry near these diagonal elements, a small additional shift downstream was used for the collocation points in the strips of panels behind the transom stern. The additional downstream shift succeeded in damping the numerical noise. For all the results presented in this paper, the additional shift downstream was applied only in strips directly behind the transom stern and it amounted to 0.0002L where L is the length of the hull. This value was determined by numerical experimentation so as to minimize the additional downstream shift of the collocation points and at the same time provide enough numerical damping to produce the desired effects.
A full, nonsymmetric system of linear equations is obtained. To solve it, the system is first row scaled; then an accelerated block Gauss-Seidel solver is used to obtain a solution. Smaller Froude numbers and smaller panel sizes tend to increase the number of iterations required in the solver, but convergence is almost always obtained. If not, a change in paneling seems to help convergence.
PREDICTIONS
Athena Hull
Plots of the computed Kelvin wave pattern and the amplitude of the free-wave spectrum obtained from the computed wave pattern are presented for speeds corresponding to the Froude numbers 0.48 and 0.4. The first speed is especially interesting because the wave elevation immediately aft of the transom stern can be compared with published measurements.
The body plan of the Athena hull and the paneling on and near the hull are depicted in Fig. 4 and Fig. 5. Since the flow configuration is symmetric about the center plane, only the starboard half is paneled. In this case, the starboard half of the hull is paneled with 8 longitudinal strips of 50 panels whose longitudinal dimensions are nearly uniform (Δx≈0.02). The widths of these strips at each longitudinal station along the hull are nearly uniform. Free-surface paneling covers a region extending from one ship length upstream to nearly one and a half ship lengths downstream from the hull, extends laterally from the center plane to one ship length away, and is swept back at a 45º angle from the center plane. Except upstream of the hull where the longitudinal dimension of free-surface panels is gradually increased, free-surface panels have approximately the same longitudinal dimension (Δx=0.02) as that of the hull panels. Adjacent to the hull, free-surface panels have aspect ratios nearly equal to unity and match the hull panels at the waterline; the widths of the strips gradually increase with increasing distance from the center plane. Be hind the transom, eight strips of panels lying on a cylindrical surface originating at the stern and extending downstream are used. There is no variation in the depth of these strips with respect to distance downstream from the transom. At the stern the forward edges of these strips match the downstream edges of the strips of panels on the hull. Due to the narrowness of the strips behind the stern, the longitudinal dimension of the panels is halved to avoid numerical problems that occur with panels of aspect ratio greater than two. In this paneling arrangement, the total number of panels used on the starboard half of the configuration is 3880.
Fig. 6 presents a contour plot of the computed Kelvin wave pattern in a neighborhood of the high-speed Athena hull at Froude number 0.48. Figs. 7a and 7b present the same data in close-up contour plots of the computed and measured wave elevation near the transom stern. The computed elevation behind of the stern is slightly deeper and rises more quickly toward the mean free-surface level than is indicated by the measured results of Fig. 7b. However, the fact that the two contour plots agree as well as they do is encouraging. Fig. 8 shows the amplitude of the free-wave spectrum for this flow configuration. The wave resistance computed from this spectrum is 0.0015, higher than the experimental result of 0.0013 reported in (12) for the slightly higher Froude number 0.484.
A contour plot of the computed wave elevation and a plot of the computed amplitude of the free-wave spectrum for Froude number 0.4 appear in Fig. 9 and Fig. 10. The corresponding computed wave resistance is 0.0012.
Model 5415
The body plan for Model 5415 is shown in Fig. 11. Comparison with Fig. 4 shows that with respect to the midship section, the transom stern on the Athena hull is larger and more rectangular than on Model 5415. The two hulls differ mainly in that Model 5415 has a bow dome. This difference, however, is irrelevant to the content of this paper. Paneling for the hull and the surrounding free surface is depicted in Fig. 12. The paneling is similar to the paneling used for the Athena hull. Experimental data measurements for this model at several speeds are available in a report by Ratcliffe and Lindenmuth (13). The measurements are presented in the form of plots of the wave spectrum and contour plots of the measured wave elevation. Two speeds corresponding to Froude numbers 0.414 and 0.25 are considered here. For the slower speed, measured free-surface height is available in the region immediately behind the transom stern.
Contours of the computed wave elevation near the stern of this hull for a speed corresponding to

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Fig. 4 — Body plan of the Athena hull.
Fig. 5 — Top view of the panelling of the athena hull and the free surface near the hull.
Fig. 6 -Contours of the computed free-surface elevation for the Athena hull at F = 0.48. Solid lines indicate positive nondimensional wave elevation Z/F2 = 0.005j for j = 1,2, etc. Dashed lines indicate negative wave elevation Z/F2 = -0.0005j for j = 1,2, etc. The zero contour level is not drawn. (b) Measured
Fig. 7 -Contours of the Kelvin wave pattern near the transom stern of the Athena hull at F = 0.48. (a) Contours of the computed wave elevation. (b) Con-tours of the measured wave elevation. [Measured data from (12).] T he contour levels are the same as in Fig. 6.
Fig. 8 - Amplitude of the free-wave spectrum from a pair of transverse wave cuts at 1.15 and 1.20 ship lengths aft of the midship section of the Athena hull for F = 0.48.

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Fig. 9—Contours of the computed free-surface elevation for the Athena hull at F=0.4. Solid lines indicate positive nondimensional wave elevation Z/F2=0.005j for j=1, 2, etc. Dashed lines indicate negative wave elevation Z/F2=−0.005j for j=1, 2, etc. The zero contour is not drawn.
Froude number 0.414 are shown in Fig. 13 and Fig. 14a. The computed contours near the transom are compared with the corresponding contours obtained from measurements in Fig. 14. (In Fig. 14b, the contours immediately behind the transom are computed, not measured, since no measurements were obtained in this region.) From Fig. 14b it appears that the peak behind the stern is nearly in the correct position. The measured contours of the wave elevation in Fig 14b show a sharp gradient of the wave elevation that starts approximately 0.1 ship length aft of the stern and lies along a ray at about the Kelvin angle away from the center plane. In the computed contours of Fig. 14a there is a corresponding surface height gradient, but is not as sharp. The peak of the computed free-surface elevation here corresponds to level 0.08F2 whereas the measured wave height corresponds to level 0.085F2. A comparison of the amplitude of the free-wave spectrum obtained from computed wave elevations along transverse wave cuts and from measured wave elevations is presented in Fig. 15. The wave resistance computed from the amplitude of the wave spectrum shown in this figure is 0.0018, which is lower than the value of 0.0024 obtained from experimental data.
For Froude number 0.25 contours of the computed free-surface height are shown in Figs. 16 and 17a. The corresponding amplitude of the free-wave spectrum is plotted in Fig. 18. A close-up comparison of the contours of computed and measured wave elevation near the stern of Model 5415 at Froude number 0.25 is provided in Fig. 17. For this speed, measurements are available in the region immediately behind the stern. The positions of the computed and measured peaks behind the stern agree fairly well. Outside the area immediately behind the stern, as is the case for the higher speed, there is a sharp gradient in the measured wave elevation along a line starting
Fig. 10—Amplitude of the free-wave spectrum obtained from a pair of transverse wave cuts at 1.15 and 1.20 ship lengths aft of the midship section of the Athena hull for F=0.4.
at the stern and radiating downstream from the hull at approximately the Kelvin angle. The large gradient is also present in the computed results, but it is not as sharp. At the peak elevation near the sharp gradient, the computed nondimensional free-surface height is 0.08 F2. Except for a very small area where the measured elevation is higher, this is the nondimensional wave height of the highest measured contour level plotted. The wave resistance computed from the magnitude of the wave spectrum is 0.00042, which is higher than the value of 0.00037 obtained from experimental data.
If paneling behind the stern is placed on the mean free-surface level rather than on the cylindrical surface surface extending downstream from the hulltransom intersection and if no strips of wake panels originating at the transom are used, then a different solution is obtained. In this case finite differencing to set the free-surface depth and slope immediately behind the stern is based on the potential solely at the centroids of free-surface panels and thus excludes the potential on the hull. Contours of the wave elevation near the stern for Froude numbers 0.25 and 0.414 are shown in Figs. 19 and 20. These are to be compared with the previous results depicted in Figs 17a and 14a, respectively. There is a visible difference in the contours. There is less of the sharp gradient that is seen in the contours of the experimental measurements. This difference is more obvious for the lower speed. Corresponding plots of the amplitude of the free-wave spectrum are given in Figs. 21 and 22. There is not much difference at Froude number 0.414.
CONCLUSION
It has been shown that it is possible to linearize the wave resistance problem about a surface that originates at the intersection of the hull and a transom

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Fig. 11 - Body plan of Model 5415.
Fig. 12 -Top view of the paneling on Model 5415 and on the surrounding free surface.
Fig. 13 - Contours of the computed Kelvin wave pattern for Model 5415 at F = 0.414. Solid lines indicate positive nondimensional wave elevation Z/F2 = 0.005j for j = 1, 2, etc. Dashed lines indicate negative wave elevation Z/F2 = -0.005j for j = 1,2, etc. The zero contour level is not drawn.
Fig. 14 - Contours of the free-surface elevation near the transom stern of Model 5415 at F = 0.414. (a) Contours of computed wave elevation. (b) Contours of measured wave elevation except aft of the transom where contours of computed wave eleva-tion are presented (and are identical to the corre-sponding contours in (a)). Contour levels are the same as in Fig. 13
Fig. 15—Comparison of the amplitude of the free-wave spectrum for Model 5415 at F=0.414 obtained from a pair of computed transverse wave cuts (—) and from experiments (▲).

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Fig. 16—Contours of the computed free-surface elevation for model 5415 at F=0.25. Solid lines indicate positive wave elevation Z/F2=0.005j for j= 1,2, etc., and the dashed lines, negative wave elevations Z/F2=–0.005j for j=1,2, etc. The zero contour is not drawn.
Fig. 17—Contours of the free-surface elevation near the stern of model 5415 at F=0.25 (a) Contours of computed wave elevation. (b) Contours of measured wave eleveation. Contour levels are the same as in Fig. 16.
Fig. 18—Amplitude of the free-wave spectrum for Model 5415 at F=0.25 obtained from a pair of computed transverse wave cuts (—) and from experiments (▲).
stern and which extends downstream from this intersection. The computed wave height is different from the wave height obtained using a model in which panels behind a transom stern are placed on the mean free-surface level and the transom is left open. For the few results presented here, some features present in experimental results show up more clearly when panels behind the stern are placed on the surface originating at the hull-transom intersection.
To compute the solution of the wave resistance problem, we have modified an existing Rankine singularity code that had been based on the use of distributions of sources and dipoles on the mean free-surface level. The finite-differencing scheme in the code was replaced by one in which analytic differentiation and collocation point shifting are used to enforce the radiation boundary condition. In so doing free-surface integrals have been replaced by a combination of free-surface and hull-waterline integrals. The hull-waterline integrals are analogous to those customarily seen in Havelock singularity methods only in that similar mechanisms are used to arrive at them; they are otherwise completely different. The wave height appears to be calculated more accurately by using this scheme than by using the scheme in the original code.
Difficulties can be expected in this method when the wave slope is enforced at a transom stern. Several causes can be singled out. First, obtaining the hull slope is problematic since there is typically a rapid variation in the hull shape at the stern. Second, developing an accurate numerical scheme that enforces the slope on a curved surface is complicated. Here part of the latter problem has been eliminated by using a cylindrical surface beneath the mean free-surface level without longitudinal variation in depth downstream of the hull-transom intersection.
Further developments might be made. The first involves putting panels behind a hull-transom intersection on a curved surface that rises toward the mean free-surface level. This was tried, but anomalies in the contours of the computed wave elevation directly behind the transom led to the simpler approach of putting panels on a cylindrical surface originating at the intersection. Perhaps the linearization scheme should be reexamined, or perhaps the accuracy to which higher derivatives of the potential due to source and dipole distributions on curved surfaces can be computed should be reevaluated. As a second refinement, the nonlinear zero-pressure Kutta condition, Bernoulli 's equation with the pressure set to atmospheric pressure, could be satisfied at the transom stern. This nonlinear Kutta condition should be used in conjunction with a varying depth in the surface originating at the hull-transom intersection about which the free-surface boundary conditions are linea-

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Fig . 19 - Contours of computed free-surface elevation for Model 5415 hull at F = 0.25. Paneling behind the transom stern has been placed on the mean free-surface level. Finite differencing at the upstream end of strips of panels originating at the stern involves the disturbance potential only at the centroids of free-surface panels instead of at the centroids of neighboring free surface and hull pan-els. Solid lines indicate positive wave elevation Z/F2 = 0.005j forj = 1,2, etc. and dashed lines, neg-ative elevation Z/F2 = -0.005j forj = 1,2, etc. The zero contour level has not been drawn.
Fig. 20 - Contours of computed free-surface elevation for Model 5415 hull at F = 0.414. Paneling aft of the transom stern has been placed OII the mean free-surface level. Finite differencing at the upstream end of strips of panels originating at the stern involves the disturbance potential only at the centroids of free-surface panels instead of at the centroids of neighboring free-surface and hull panels. The contour levels are the same as in Fig. 19.
Fig. 21 - Amplitude of the free-wave spectrum obtained from a pair of computed transverse wave cuts at 1.15 and 1.20 ship lengths aft of the midship of Model 5415 Froude number 0.25 (—) and the corresponding spectrum obtained from experimen-tal data (▲). Paneling aft of the transom stern has been placed on the mean free-surface level. Finite differencing at the upstream end of strips of panels originating at the stern involves the disturbance po-tential only at the centroids of free-surface panels instead of at the centroids of neighboring free-surface and hull panels.
Fig. 22—Amplitude of the free-wave spectrum obtained from a pair of computed transverse wave cuts at 1.15 and 1.20 ship lengths aft of the midship section of Model 5415 at F=0.414 (—) and the corresponding spectrum obtained from experimental data (▲). Paneling aft of the transom stern has been placed on the mean free-surface level. Finite differencing at the upstream end of strips of panels originating at the stern involves the disturbance potential only at the centroids of free-surface panels instead of at the centroids of neighboring free surface and hull panels.

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rized. The combination may be especially important for deep transoms.
Although, according to Lechter (6), the method of collocation point shifting and analytic differentiation may break down at low Froude numbers, several enhancements might be made to the Rankine singularity method that was used to obtain numerical approximations to free-surface potential flows. They are related to the more general problem of calculating the linearized free-surface potential flow near arbitrarily shaped ships advancing into calm water. The first would be to eliminate the finite differencing at the hull waterline. This would eliminate all finite differences except those at the hull-transom intersection. Second, the relationship between free-surface paneling and collocation point shifting should be clarified. When a second small longitudinal shift downstream is used, damping is introduced into the wave pattern. Further, a more or less rectangular grid of panels on the free surface damps the waves in the Kelvin wave pattern more than the arrangement used in this paper, in which the computational free-surface domain is swept back at a 45-degree angle. The effect is most noticeable in the diverging waves seen in contour plots. Finally, it might be useful to consider whether something similar to the DtN exact boundary condition of Keller and Givoli (14) could be used for the outer boundaries of the computational domain to handle the boundary conditions there more accurately and to reduce the size of the computational domain.
ACKNOWLEDGMENT
This work was supported in part by the Applied Hydromechanics Research program of the Applied Research Division of the Office of Naval Research.
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