OMAE2012-83170

Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic EngineeringOMAE 2012

July 1-6, 2012, Rio de Janeiro, Brazil

OMAE2012-83170

COMPARISON OF TWO BASE-POTENTIALS FOR A SLENDER-BODY METHOD

Mahmoud AlidadiDepartment of Mechanical Engineering

University of British ColumbiaVancouver, Canada

Sander CalisalFaculty of Engineering

Piri Reis UniversityIstanbul, Turkey

ABSTRACTThe effects of two base-potentials on the accuracy of a

slender-body method are studied in this paper. In the formula-tion for this method which is developed for the slender ships, thevelocity potential is decomposed into a base-potential and a per-turbation potential. Then using an order of magnitude analysis,the three-dimensional flow problem is simplified into a series oftwo-dimensional problems for the perturbation potential. Thesetwo-dimensional problems are solved with the linearized freesurface boundary conditions, using a mixed Eulerian-Lagrangianmethod. Finally for the two base-potentials, the numerical waveelevation along a Wigleyull are compared with the experimentalresults.

NOMENCLATUREU Ship velocityL Ship lengthB Ship beamD Ship draftFR Froude numberϕ Total velocity potentialφ Perturbation velocity potentialη Wave height

INTRODUCTIONThe slender-body methods (e.g. 2D+t) have been adopted

extensively for flow calculation around slender ships at lowFroude numbers, where the flow nonlinearity is minimum.

Numerous researchers at the University of British Columbiaadopted these methods for calculation of wave pattern and waveresistance of slender ships (see [1, 2], [3], [4], [5]). Song andMaruo [6] used a 2D+t method to calculate the bow impact anddeck wetness on a Wigleyhull. Tulin and Wu [7] numericallycalculated the divergent waves generated by a Wigleyhull usinga 2D+t method.

Development of the slender-body methods involves decom-position of the potential into a base-potential and a perturba-tion potential, and then using an order of magnitude analysisto turn the three dimensional flow problem into a series of two-dimensional problems which are faster to solve than the originalproblem.

The terms neglected in the order of magnitude analysis haveto be as small as possible to have an accurate solution for theflow field around a slender hull. However, their magnitude aregreatly related to the base-potential. This paper investigates theeffects of two base-potentials free stream and double body on theaccuracy of a slender-body method developed in the followingsections.

FORMULATION FOR THE SLENDER-BODY METHODA Cartesian coordinate system (x,y,z) is employed with the

origin at the amidship of centerline and on the undisturbed freesurface. The z direction is vertically upward, and the x-axis co-incident with the longitudinal axis of the ship, the bow of whichis oriented in the x-direction (see Fig. 1). This coordinate systemmoves with the ship, with steady velocity U in the x-direction.As the fluid is assumed inviscid and incompressible, the fluid

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motion is irrotational and can be specified by the velocity poten-tial ϕ which satisfies the Laplace equation

∇2ϕ = 0 (1)

where

∇2 =

∂ 2

∂x2 +∂ 2

∂y2 +∂ 2

∂ z2 (2)

The kinematic and dynamic boundary conditions for a steadyfree surface represented by z = η(x,y) are

ϕxηx +ϕyηy−ϕz = 0 (3)12(ϕ2

x +ϕ2y +ϕ

2z −U2)+gη = 0 (4)

The boundary condition on the hull is

∇ϕ ·n = 0 (5)

where

∇ =∂

∂x+

∂

∂y+

∂

∂ z(6)

and n is the normal vector on the hull pointing outwards.The velocity potential ϕ is assumed to be composed of a

perturbation potential φ and a basis potential φ , i.e.

ϕ = φ + φ (7)

where the basis potential φ satisfies the Laplace equation (1), and

φz = 0 (8)

Then the perturbation potential φ must satisfy the Laplaceequation

∇2φ = 0 (9)

Replacing potential ϕ from the Eqn. (7) in the kinematic anddynamic boundary conditions results in

(φx +φx)ηx +(φy +φy)ηy− (φz +φz) = 0 (10)

xy

z

U

FP

AP

FIGURE 1. HULL WITH BODY-FIXED COORDINATE SYSTEM

12((φx +φx)

2 +(φy +φy)2 +(φz +φz)

2−U2)+gη = 0 (11)

Similarly the body boundary condition (5) changes to

φxnx +φyny +φznz =−(φxnx + φyny + φznz) (12)

Slender-Body SimplificationIf we define non-dimensionlized variables

x = x/L

y = y/B

z = z/D

then one can show that the order of the terms in governing equa-tion and boundary conditions are:

φxx = O(

φ

L2x2

)φyy = O

(φ

B2y2

)φzz = O

(φ

D2z2

)φx = O

(φ

Lx

)φy = O

(φ

By

)φz = O

(φ

Dz

)nx = O

( nLx

)ny = O

(n

By

)nz = O

(n

Dz

)ηx = O

(η

Lx

)ηy = O

(η

By

)(13)


Comparing the order of magnitudes in Eqn. (13), we can con-clude that

φxx� φyy,φzz

φxηx� φyηy

φ2x � φ

2y ,φ

2z

φxnx� φyny,φznz

for when O(D)≈ O(B) and B/L� 1.These relations simplify the governing Eqn. (9) to

∂ 2φ

∂y2 +∂ 2φ

∂ z2 = 0 (14)

The governing equation (14) shows that the flow around a slendership can be described by a series of two-dimensional problems inthe cross flow planes along x axis.

The kinematic and and dynamic boundary conditions at z =η(x,y) also convert to

φxηx +(φy +φy)ηy− (φz +φz) = 0 (15)

12(φ 2

x + φ2y + φ

2z −U2)+

12(φ 2

y +φ2z )+

(φxφx + φyφy + φzφz)+gη = 0 (16)

and the body boundary condition for a slender ship changes to

φyny +φznz =−(φxnx + φyny + φznz) (17)

Linearization of the Free Surface Boundary ConditionsThe kinematic and dynamic boundary conditions (15), (16)

can be linearized around z = 0. The Taylor expansion of the Eqn.(15) up to the first order results in

φxηx +(φy +φy)ηy− φz−φz +η(φxzηx + φxηxz)+

η((φyz +φyz)ηy +(φy +φy)ηyz− φzz−φzz) = 0 (18)

Since φz = 0 at z = 0, and z = η(x,y) then

φzx = φxz = φyz = φzy = 0 , ηzx = ηxz = ηzy = ηyz = 0

The above relations simplify the kinematic free surface boundarycondition to

φxηx +(φy +φy)ηy−φz +η(φyzηy− φzz−φzz) = 0 (19)

An order of magnitude analysis shows that

φzz < φzz

After neglecting the nonlinear term

φyzηy < φzz

the linearized kinematic boundary condition becomes

φxηx +(φy +φy)ηy−φz−ηφzz = 0 at z = 0 (20)

Using the same approach the linearized dynamic boundary con-dition becomes

12(φ 2

x + φ2y + φ

2z −U2)+

12(φ 2

y +φ2z )+

φyφy +gη + φzzφzη = 0 at z = 0 (21)

Free Surface Boundary Conditions in Lagrangian FormFor writing the kinematic and dynamic conditions in La-

grangian form the operator ddt is defined as

ddt

= φx∂

∂x+(φy +φy)

∂

∂y+(φz +φz)

∂

∂ z(22)

This choice for the operator leads to the equation

dydt

= φy +φy (23)

for the velocity in y direction.Using Eqn. (22) the Lagrangian form for the linearized kine-

matic and dynamic boundary conditions (20), (21) at z = 0 re-spectively are

dη

dt= φz + φzzη (24)

dφ

dt=

12(U2− (φ 2

x + φ2y ))+

12(φ 2

y +φ2z )−

gη− (φyφzy + φzzφz)η (25)


SOLUTION APPROACH FOR THE SLENDER-BODYMETHOD

The solution approach developed by Longuet-Higgins andCokelet [8] is used for the implementation of the slender-bodymethod. This approach involves a two-step procedure dividedinto an Eulerian boundary element method and a Lagrangianscheme.

Boundary Element MethodThe first step attacks the solution of Eqn. (14) in a cross flow

plane using the velocity potential φ0 known on the free surfacefrom the previous time step, and the body boundary condition(17). From the solution of Eqn. (14) the potential at a point q inthe cross flow plane is (see [9])

φ(q) =1

2π

∫S(

∂φ

∂nlnr−φ

∂

∂nlnr)ds (26)

where S is the boundary for the cross flow plane and r is thedistance from point q.

The boundary S is discritized into N elements with constantdistribution of ∂φ

∂n and φ over each element. For collocationpoints on the elements, placed at the center of elements, the inte-gral Eqn. (26) gives the linear algebraic equation

N

∑j=1

Ai j∂φ

∂n| j +Bi jφ j = 0 i = 1, ...,N (27)

where the coefficients Ai j and Bi j are

Ai j =1

2π

∫jlnrds , Bi j =

12π

∫j

∂

∂nlnrds

Lagrangian SchemeThe second step involves calculation of the velocity potential

and the free surface for the next cross-flow plane. This marchingfrom one cross-flow plane to the next one can be viewed as atime-domain problem where the time step is the amount of timetaken to traverse dx, the spatial step along the longitudinal axis.

Using the Euler method, the values for velocity potential andfree surface location at new cross flow plane are

y(t +dt) = y(t)+ yt(t)dt

η(t +dt) = η(t)+ηt(t)dt

φ(t +dt) = φ(t)+φt(t)dt (28)

where the values for yt , ηt , and φt are determined using Equations(23), (24), and (25) and the time step dt is

dt =dxφx

NUMERICAL RESULTS FOR A WIGLEYHULLThe slender-body method explained in the previous sections

was used to calculate the wave elevation along a Wigleyhull atFroude numbers FR = 0.267 and FR = 0.316. These results wereobtained for the free stream and double body base-potentials, us-ing a computer with Pentium D processor and 4GB of RAM. Thecalculation for each of these Froude numbers took about 4 min-utes in the case of the free stream potential and 10 minutes in thecase of double body potential.

The Wigleyhull is described by the equation

y = Y (x,z) =±B2(1− 4x2

L2 )(1− z2

D2 ) (29)

where L = 2 m, B = 0.2 m and D = 0.125.As the hull is symmetric, only half of the hull was consid-

ered in the calculation. An impermeable wall was placed 35Baway from the body to make sure waves does not reflect into thecomputational domain. The step size was set to dx = 1 mm, and10 equal length elements were used on the hull surface.

For the starting cross section, the values for the wave eleva-tion and perturbation potential were set to zero. On the free sur-face of this cross section 75 panels were put which their lengthsincreased toward the impermeable wall in a geometric progres-sion, with the first panel size D/10 long.

Results for the Free Stream Base-PotentialFor a ship moving with the free stream velocity U , the free

stream base-potential can be described as

φx =Ux (30)

This base-potential simplifies the Lagrangian equations (23),(24), and (25) to

dydt

=φy (31)

dη

dt=φz (32)

dφ

dt=

12(φ

2y +φ

2z)−gη (33)


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Experiment Results

Resutls for Free-Stream Potential

Results for Double-Body Potential

2XL

2ηL

FIGURE 2. WAVE ELEVATION ON THE WIGLEYHULL SIDE ATFR = 0.267

Numerical results for the free stream base-potential are comparedwith the experimental results at Figures 2 and 3. The startingcross section in these simulations was at the bow. The experi-mental results for FR = 0.267 was obtained from a research con-ducted at the department of Naval Architecture and Ocean En-gineering at the Yokohama National University [10]. Measuredvalues for FR = 0.316 are obtained by Maruo and Song for a 6m Wigleyhull model [11]. As the Figures show the numericalresults for free stream base-potential does not match well withthe experimental results. They especially deviate from the exper-imental results in the bow and stern area.

Results for the Double Body Base-PotentialAs Equations (23), (24), and (25) show, the slender-body

method with this base-potential requires the calculation of thedouble body potential. We adopted the boundary element methoddescribed by Katz and Plotkin [12] for this purpose, using 50panels in x direction and 20 panels in z direction.

Adopting the double body potential as the base-potential hadadvantage over the free stream base-potential as we were able tocalculate the potential and free surface at the bow position. Thestarting cross section for this base-potential was at L/4 upstreamof the bow(i.e. x = 3L/4). Since calculation of the double bodypotential for all cross sections was very time consuming, we cal-culated its values for a grid on the free surface (see Figure 4), andthen interpolated the values to obtain the double body potentialfor a cross section. The panel size for this grid was L/50 in thex direction except in bow and stern which was L/100. In the ydirection, 20 panels were used on this grid, where the panel sizeincreased in a geometric progression with the initial size D/20.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Experiment Results

Resutls for Free-Stream Potential

Results for Double-Body Potential

2XL

2ηL

FIGURE 3. WAVE ELEVATION ON THE WIGLEYHULL SIDE ATFR = 0.316

xy

z

stern

bow

x = 3L/4

FIGURE 4. FREE SURFACE GRID FOR DOUBLE BODY POTEN-TIAL CALCULATION

The wave elevation results along the hull for this base-potential are also presented in Figures 2 and 3. These resultsmatch the experimental results better than the results for the freestream base-potential. The values around the bow and stern re-gions are far better than those predicted for the free stream base-potential. This is because of two reasons:

a) The potential and free stream elevation at the bow were set tozero for the free stream base-potential, while we were ableto predict those values using the double body base-potential.

b) Since the double body potential is the same as the potentialfor a flow with FR = 0, it can be argued that the double bodypotential is a better representation of the actual flow than thefree stream flow at low Froude numbers. Therefore one canconclude that the terms ignored in development of the theslender-body method (refer to pages 2 and 3) were smallerfor the double body base-potential.


CONCLUSIONSTwo base-potentials free stream and double body were stud-

ied for a slender-body method developed for slender ships. Thevelocity potential was decomposed into a base-potential and aperturbation potential in this method. Then using an order ofmagnitude analysis the three-dimensional steady flow problemwas transformed into a series of two-dimensional problems inthe cross-flow planes along x axis. The free surface boundaryconditions were then linearized around z = 0. A mixed Eulerian-Lagrangian approach was used to implement the slender-bodymethod.

The slender-body method was used to calculate the waveelevation along a Wigleyhull. The results for the double bodypotential were better match with the experiments in comparisonwith the results for the free stream potential. This is because a)we were able to predict the potential and free surface elevation atthe bow for the case of double body potential, b) the values forthe terms ignored in development of the slender-body methodwere smaller for the case of double body base-potential as it is abetter representation of the actual flow.

REFERENCES[1] Calisal, S., and Chan, J., 1989. “A numerical calculation for

ship bow wave”. Ship Research, 33(1), March, pp. 21–28.[2] Calisal, S., and Chan, J., 1993. “A numerical procedure for

time domain nonlinear surface wave calculations”. OceanEngineering, January, pp. 19–32.

[3] Allievi, A., and Calisal, S., 1993. “A semi-implicit, semi-lagrangian finite element model for nonlinear free surfaceflow”. In 6th International Conference on Numerical ShipHydrodynamics, pp. 195–215.

[4] Wong, L., 1994. “A numerical solution for potential flowsincluding the effects of vortex shedding”. PhD thesis, Uni-versity of British Columbia.

[5] Wong, L., and Calisal, S., 1996. “Numerical algorithms forslender bodies with vortex shedding and density stratifica-tion”. Ship Research, 40(1), March.

[6] Song, W., and Maruo, H., 1993. “ow impact and deck wet-ness: simulation based on nonlinear slender body theory”.In Proceedings of the 3rd International Offshore and PolarEngineering Conference, Vol. III, pp. 34–38.

[7] Tulin, M., and Wu, M., 1996. “Diverging bow waves”. In21th Symposium on Naval Hydrodynamics.

[8] Longuet-Higgins, M., and Cokelet, E., 1976. “The defor-mation of steep surface wavs, a numerical method for com-putation”. In Proceedings of the Royal Society, Vol. A350,pp. 1–26.

[9] Brebbia, C., and Dominguez, J., 1989. Boundary Elements:An Introductory Course. Computational Mechanics Publi-cation, Southamption, UK.

[10] Hayashi, K., and Kunishige, Y., 1988. Measurement of

wave pattern around a yawed ship model. Graduation the-sis, Yokohama National University, Japan.

[11] Maruo, H., and Song, W., 1990. “Numerical appraisal ofthe new slender ship formulation in steady motion”. In Pro-ceedings of the 18th Symposium on Naval Hydrodynamics,pp. 239–257.

[12] Katz, J., and Plotkin, A., 2001. Low Speed Aerodynamics,second ed. Cambridge University Press.


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