Ship Technology Research/Schiffstechnik

Vol. 45 • No. 2 May 1998

Calculation of the Motion of L i q u i d in a Cargo T a n k by Carsten Schumann

A R a n k i n e Source K o c h i n - F u n c t i o n Method to Remove the Negative Ship Wave Resistance

by Hironori Yasukawa

s •H

A K u t t a Condition for Ship Seakeeping Computations with a R a n k i n e P a n e l Method

by Volker Bertram and Gerhard D. Thiart

Viscous F l o w Simulations for Conventional and High-Skew M a r i n e Propellers

by Moustafa Abdel-Maksoud, Florian Menter and Hans Wuttke

n

E s t i m a t i n g Resistance^ and Propulsion for Single-Screw and T w i n - S c r e w Ships

by Klaus Uwe Hollenbach

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SHIP T E C H N O L O G Y RESEARCH/SCHIFFSTECHNIK was founded by K . Wendel i n 1952. I t is edited by H. Coding and V. Bertram i n collaboration w i t h experts from universities and model basins in Berlin, Duisburg, Hamburg and Potsdam, from Germanischer Lloyd and other research organizations

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Vol. 45 • No. 1 • F e b r u a r y 1998

Carsten Schumann

Calculation of the Motion of Liquid in a Cargo T a n k Ship Technology Research 45 (1998), 39-46

A method for computating the free surface flow in a tank is applied to study the response of a fluid at resonance and off resonance cases. The Euler equations for incompressible fluid and transient flow are solved in a finite-volume approach on a non-orthogonal, boundary-fitted, structured grid. The motion of the free surface is calculated with the volume-of-fluid method. Results are presented for a practical example of a bi-loop tank which is ideafised in two dimensions and forced i n harmonic pitch motion. Keywords: sloshing, tank, flnite volume method, VOF, breaking wave

Hironori Yasukawa

A R a n k i n e Source K o c h i n - F u n c t i o n Method to Remove the Negative Ship Wave Resistance

Ship Technology Research 45 (1998), 47-53

A Rankine source method for steady ship wave resistance in conjunction w i t h a Kochin function is proposed to remove negative resistance found by pressure integration for low speed. A wave resistance formula w i t h a new Kochin function is derived in which the velocity potentials on the ship hull and the still water surface are unknown variables. Calculations of the wave resistance are carried out for f u l l hull forms and corresponding low Froude numbers. The method considerably improves the prediction accuracy of the wave resistance compared w i t h model experiments.

Volker Bertram, Gerhard D. Thiart

A Kutta Condition for Ship Seakeeping Computations with a Rankine Panel Method Ship Technology Research 45 (1998), 54-63

A 3-d seakeeping code uses first-order Rankine panels with special numerical integration on the ship hull and Rankine point source clusters above the free surface. The code computes ship motions in a regular wave of small wave height (linearized). The steady flow is captured without simplification by solving the fully nonlinear wave-resistance problem first. For antisymmetric motions, a Kutta condition is enforced using an additional dipole distribution. A new element accounting for the periodically oscillating dipole strength in the free vortex behind the ship is derived. Results for the S-175 containership show that the Kutta condition reduces considerably the overprediction especially of roll motions for resonance in long waves, but has little effect for shorter waves. Keywords: seakeeping, panel method, containership, Kutta condition, dipole element, motion

Moustafa Abdel-Maksoud, Florian Menter and Hans Wuttke

Viscous Flow Simulations for Conventional and High-Skew Marine Propellers Ship Technology Research 45 (1998), 64-71

Different aspects of semi-automatic grid generation and numerical flow calculations for marine propellers are discussed. Block-structured hexahedral grids are designed for use in Reynolds-averaged Navier-Stokes compu-tations. Special emphasis is put on grid topology. Numerical results are presented for a conventional, a skew and a high skew propeller for a range of flow conditions.

Keywords: viscous flow, RANSE, propeller, grid, CFD

Klaus Uwe Hollenbach

Estimating Resistance and Propulsion for Single-Screw and Twin-Screw Ships Ship Technology Research 45 (1998), 72-76

The traditional methods to predict ship resistance and propulsion data based on statistical information are shown to be inadequate for modern ship forms. A new estimation for resistance and propulsive data is developed based on measurements of the Vienna Ship Model Basin for the years 1980 to 1995. The method allows not only to estimate a 'mean resistance', but also a 'minimum resistance' for excellent lines and a 'maximum resistance' for unsuitable lines. The method shows moderate improvement in prediction accuracy for single-screw ships and substantial improvement for twin-screw ships. Formulas for estimating the propulsion factors both for single-and twin-screw ships are also presented.

Keywords: resistance, prediction, preliminary design, propulsion, performance

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Calculation of the M o t i o n of Liquid i n a Cargo Tank

C a r s t e n Schumann, Hamburg Ship Model Basin^

1. Introduction

I n large tanlcs of ships operating in rough seas, the liquid may slosh heavily and damage the tank structure. Therefore several classification societies demand to calculate of the largest natural period of the motion of the liquid in longitudinal and transverse direction. Slosh bulkheads are prescribed i f these periods exceed empirical values for the ship motion periods. Otherwise i t is assumed that sloshing will not occur, and the tank is designed disregarding sloshing loads. The regulations should not be too conservative because bulkheads increase the tank fabrication costs.

In this paper a semi-pressurised gas and chemical tanker (designed by L G A Marine Consult-ing, Hamburg) w i t h tanks of alloyed steel is considered. I n this case bulkheads are extremely expensive due to high material and fabrication costs. I t will be shown that two baffles in the tank distinctly reduce the fluid motion, avoiding a standard bulkhead for the considered case. 2. Description of the ship

The ship particulars are: Loa = 163.4m, B = 24.2m, T = 11.0m, A = 32200t and U = 16.4knots. The liquid cargo is contained in four bi-loop tanks consisting of a cylindrical middle part w i t h horizontal axis and hemi-spherical ends. The tank length is 26.35m and the cyUndrical middle part is 12.35m long. The middle part of each tank has a cross section consisting of two intersecting circles of 14m diameter; the tank has a total breadth of 22m.

The tanks are smooth inside except for two rings which are used to embed the tanks i n the ship. The second tank from the bow is chosen for the present calculations. The total tank volume is 5500m^. The tank is filled up to 8.4m from its bottom with ammonia (680kg/m'^). 3. G e n e r a l solution technique

I n solving vibration problems, the simplest procedure is to calculate the natural periods of the system and compare it to the excitation periods. Practical experience shows that in most cases only the two largest natural periods must be considered, whereas smaller periods can be neglected. For small damping, the system responds with large amplitudes (resonance) i f a natural period and the period of excitation are close to each other. I f these periods differ by a safety margin, the system is assumed safe regarding resonance; otherwise the system has to be modified.

For example the rules of Germanischer Lloyd (GL) yield an upper limit to the natural period of the tank of 6.7s, while the period of the ship's pitch motion is estimated as 7.2s. After a discussion with GL the fitting of a slosh bulkhead was deemed necessary. Therefore, the time dependent fiuid flow in the tank when forcing the ship in harmonic pitch motion was calculated. I n several runs w i t h different pitch periods, the simulation was continued until the fluid motion became nearly periodical. The recorded wave elevations and the kinetic energy of the fluid in each r u n gave information about the fluid motion depending on the period of excitation.

Compared to the above resonance consideration, this approach has the advantage of taking into account the non-linear effects of the fluid motion, the damping caused by flow separation at the rings inside the tank, and the reaction of the fluid in the resonance case. The disadvantage is that the interpretation of the results is problematic. Effects of non-harmonic excitation and the interaction w i t h other degrees of freedom (rolling, surging, etc.) were neglected; so this approach is still a simpliflcation, but nevertheless a progress compared to just using the simple class rules.

To get an idea of the (largest) ship motion, a strip method calculation was made of which only the pitch motion i n long-period head seas was taken into account. Fig. 1 shows the non-dimensional transfer function of the pitch motion at different speed versus wave encounter period. The ship does not move substantially i f the wave length is lower than about half the ship length. Fig. 2 shows Pierson-Moskowitz wave spectra of fully developed seaways for wind speed 10, 20 and 30m/s versus wave period. Comparing both figures (neglecting the difference between wave period and wave encounter period), pitch motion of the ship is expected i f the period of encounter is larger than 4s. At lower wave periods the sea contains nearly no energy and the ships transfer function is also nearly zero.

The tank was assumed to rotate harmonically with 3° amplitude about a transverse axis through the tank centre; smaller amplitudes seem less dangerous, while occurence of larger amplitudes is expected to be seldom in heavy seas. To avoid strong transient effects which would vanish only after a long time, the amplitude was increased slowly from 0 to 3°.

4. Modelling the liquid motion 4.1. M a t h e m a t i c a l model

A n inviscid and incompressible single phase fluid was considered to move only i n the vertical longitudinal plane. The 2D equations of mass and momentum conservation (Euler equations) are posed in a Cartesian tank-fixed co-ordinate system, Bridges (1981). At the free surface the pressure is constant (dynamic boundary condition), and nothing fiows through the free surface (kinematic boundary condition). For a simple numerical implementation the kinematic bound-ary condition is formulated using the 'volume-of-fluid' (VOF) function F, Hirt and Nichols

(1981). A t the wall of the tank and the rings again a no-penetration condition is enforced. This model can predict the flow separation at the sharp edges of the rings inside the tank.

The formulation of the free surface condition allows overturning surfaces, but the model cannot describe the flow accurately i f waves break or if spray hits the wall or the free surface, because the gas in the tank, the viscosity and the surface tension are neglected. During the simulation such cases appeared often, but the mathematical model only allowed an approximate description of this flow. These phenomena cause a damping of the real tank flow and hmits the amplitudes of the fluid motion. The importance of an appropriate computation of the damping effect needs further investigation.

4.2. N u m e r i c a l model

A 'standard' finite volume method (e.g. Cura Hochbaum (1994)) was used to discretise the governing equations by subdividing the domain into small cells using a structured non-orthogonal and boundary fltted grid. Fig. 3. Constant time steps At were used. The unknown variables i n the numerical grid (velocities, pressure and volume fraction F) were defined i n every cell centre. The required values at the cell faces were determined by linear interpolation except for (i) the velocities in the convective momentum fluxes which were approximated by the Hnear upwind scheme, (ii) the volume fluxes, which were calculated by a scheme similar to Rhie

and Chow (1983), and (iii) the function F , which was determined w i t h the upwind scheme, plus a correction term which was derived from the geometry of the free surface, Schumann

(1997).

Fig. 3 shows a discretisation of the partially filled tank. The 2D discretisation was found by intersecting the tank in a plane parafiel to ships longitudinal plane. The rings inside the tank were modelled by blocking some cefis. The free surface moved during the simulation over the stationary grid, which consists of empty cells, f u l l cells and partially filled cells. The VOF-method was used to determine the volume of fiuid in each cell. I n partially filled cells the free surface is constructed as a straight fine. The position of the line is determined such that the filled part of the cefi containes the actual volume of fiuid in the cefi. I n f u l l cells, the govering equations are discretised as in methods without a free surface. Partially filled cells are

treated specially: the continuity equation is not used there, the velocity is extrapolated from the adjacent f u l l cells, and the pressure is set according to the dynamic boundary condition. I n every time step the F function is solved first explicitly by a forward Euler integration in time. Then the momentum equation is integrated over time by an implicit three-level scheme, Ferziger and Peric (1996). The SIMPLE algorithm is used to f u l f i l the continuity equation: starting w i t h an approximate solution of the pressure, the momentum equations are solved. As the resulting velocities do not satisfy the continuity equation, the latter is used to derive an equation for correcting simultaneously the pressure and the velocities to satisfy also the continuity equation.

4.3. Test cases

As a demonstration, two calculations are presented for a non-moving tank without rings, containing a freely oscillating fluid. As a non-equilibrium starting position of the fluid, the free surface was assumed straight but not horizontal. The fluid volume in the tank (195.1m^/m breadth) wets equivalent to 8.4m filling level. The first calculation started w i t h a free surface inclination of 11.3° and was continued for about 18s. Fig. 5 shows the potential energy (zero level corresponds to a horizontal free surface), kinetic energy and total energy in the tank versus time. These values were calculated by integrating over all cells. During the simulation the free surface remained smooth. As expected, the total energy in the tank was nearly conserved. The maximum error in mass conservation became 2 • 10"^ of the fluid mass at the start of the calculation.

The second calculation started w i t h 16.7° inchnation (Fig. 6). A large loss of energy occurred at time t =5s when the breaking wave hit the wall, and this was followed by a spray along the wall which then fell down in little 'numerical' droplets. After the free surface became smoother at t =10s, the rate of energy loss decreased. Numerical errors seem to cause the decrease of energy, since, due to the mathematical model, no losses were expected. The error in mass conservation became 2 • 10~^ times the initial fluid mass.

5. Case studies

For all cases investigated, the same numerical grid w i t h 100 • 50 cells was used (Fig. 3). Starting from the fluid at rest, the tank was forced with the pitching angle

^p = At sm(— t). The amplitude was calculated as

At = r ( l - e - ( * / ^ ) ' ) ,

where is the period of excitation and r = 2Te . . . "iT^ is a constant to control the increase of the amplitude. The duration of every run was about 15 • Te, where Tg ranged from4 to 9s. I n all calculations a time step of At — 0.008s was used. The required CPU time varied from 4 to 9 hours for one run on a standard workstation (HP C160). As values characterizing the fluid motion, the wave elevation at the left and the right end of the tank and the kinetic energy of the fiuid were used. The energy seems most suitable for characterizing the motion because it can be interpreted as a measure of the tank force, and because it is not as sensitive to local disturbances as the wave elevation. The characteristic values were further condensed since for every r u n only the mean maximum double amplitude of the wave elevations hm and the mean maximum energy in the tank were used. Both values are averaged over the time by using the results over the last 30s. For the energy the maximum values and for the wave elevation the difference of the minimum and maximum values in every cycle are used. Resonance cases are characterized by high amplitudes of h^ and e^. Fig. 7 shows a set of velocity vectors and isobars at four time steps in the resonance case w i t h the lowest natural period at Tg = 6.875s. Time records of the energy and the wave elevations are shown in Figs. 8 and 9. The results

converge to nearly periodic motions. The mean kinetic energy is about = 420kNm/m, and the double amplitude is about hm = 7m. This period of excitation can be classified as dangerous because of the large amplitude and velocities: the loads on the tank differ greatly from the static values. Contrary to this, Figs. 10 to 12 show the off resonance fiuid motion at Te=5.0s, records of wave elevation and energy {em — 5.5kNm/m and hm. — 0.5m).

Figs. 13 and 14 show and hm versus the period of excitation. The smallest resonance period is at Te = 6.875s in both figures. To decrease the sloshing motion in the tank, two baffles (height 2.5m) were installed (Fig. 15). These can be build more easyly than a usual bulkhead w i t h holes and girders. They reduced the energy e „ and the amplitudes hm very effectively (Figs. 13 and 14) and increased the resonance period. Fig. 15 shows velocity vectors and isobars which can be compared to the former results with smafi rings in Fig. 7. I t is used the same scaling factor of the velocity vectors in the figures.

Which fiuid motions should be used for dimensioning the tank structure? This question is left to the classification societies because it depends on the saftey margin and is related to the dynamic load factor (i.e. sloshing load / static load) used to design the tank.

6. Conclusion

CFD methods may help to design tank structures to sustain critical conditions. Further improvements could soon make such calculations useful also in praxi. Experiments should be conducted to validate the numerical results and enhance the scope of application of the method. Especially, experiments w i t h a practical tank design in realistic conditions should help to find out whether the computed damping of fluid motions is correct also in case of breaking waves and spray formation. When such questions are clarified, one could use the calculated pressure directly to design the tank structure; however, it is not sensible to base the tank structure on impact pressures determined by this method and lasting only for a few milliseconds, because for such impacts the mathematical model has to include effects like the compressibility of the fluid, the influence of the gas phase and the flexibihty and mass of the tank wafi, which were neglected here.

References

ARMENIO, v.; LA ROCCA, M. (1996), On the analysis of sloshing afwater in rectangular containers: Numerical study and experimental validation, Ocean Engng. 23/8

BRIDGES, T.J. (1981), A numerical simulation of large amplitude sloshing, 3rd Int. Conf Num. Ship Hydrodyn., Paris

CURA HOCHBAUM, A. (1994), A finite-volume method for turbulent ship flows. Ship Technology Research 41/3

FALTINSEN, O.M.; OLSEN, H.A.; ABRAMSON, H.N.; BASS, R.L., Liquid slosh in LNG carriers, Det Norske Veritas Publ. No. 85, Sept. 74

FERZIGER, J.H.; PERIC, M. (1996), Computational Methods for Fluid Dynamics, Springer

HIRT, C. W.; NICHOLS, B. D. (1981), Volume of fluid (VOF) method for the dynamics of free bound-aries, J. Comp. Physics 39

RHIE, C. M.; CHOW, W. L. (1983), Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21/11

SCHUMANN, C. (1997), Computation of the inviscid flow around a ship advancing in calm water with constant speed using a vof-method, 5th Symp. on Nonlinear and Free-Surface Flows, Hiroshima University

SOLAAS, F. (1995), Analytic and Numerical Studies of Slosliing in Tanks, MTA-rapport 1995:107, Norges Tekniske H0gskole Trondheim

1 0.8 0.6 0.4 0.2 O ! ! r 1 ~ i .p' Cl / ' J Sknols / i lOknols -H— i / i 15knots - a - - \ é t f i i i 6 8 10 period of encounter [s]

Fig. 1: Non-dimensional transfer function of pitch motion versus period of encounter, parameter ship speed 1 ! ! 1 ! 30nt/s 20m/s lOrrj/s i ^^WrTTTrrrrTj •"' 6 a 10 period of wave [s]

Fig. 2: Pierson-Moskowitz wave spectra versus wave period, parameter wind speed

Fig. 3: View of the numerical grid Fig. 4: Partial view near the tank wall showing full, empty and partially filled cells

400000 350000 _ 300000 "I 250000 f ; 200000 pi J 150000 100000 50000 0 _{10 15} lime [s] kinetic potcntiaf total " \ U 1'' ' / \ * Ï \ '' f \ ' / \ /'; ;'\ \\ \ \\ y J '' \ / '> 'W \ / \ : \ j\ ,''\ /\ / ';' V \j \I\ \j \j

Fig. 5: Energy in the tank, free oscillating fluid, Fig. 6: Energy in the tank, free oscillating fluid, first example second example

Fig.7: Velocity vectors and free surface (left) and isobars (right) at t =71.Os, t =72.Os, t =73.Os and

t =74.0s for Te = 6.875s; the increment of isobars is 5000Pa/m

0 10 20 30 40 50 60 time [s]

70 80 90 100

Fig. 8: Kinetic energy in the tank, T^ — 6.875s Fig. 9: Wave elevation at left and right tank wall, T = 6.875s

Fig. 10: Velocity vectors, free surface and isobars at t = 72.0s for Tg = 5.0s

Fig. 11: Kinetic energy in the tank, Te = 5.0s Fig. 12: Wave elevation left and right tank wall, Te = 5.0s 500000 Ë E 400000 300000 c QJ "Ë 200000 JX. _c ca 100000 0) E 0 1 origiHal -ö— Ï with bulkheads • f\ —i / , / '''' i , original vith bulkheads • 6 7

period of exiting [s] period of exiting [s]

Fig. 13: Mean kinetic energy in the tank versus Fig. 14: Double amplitude of wave elevation ver-period of excitation T^ sus ver-period of excitation Te

Fig. 15: Velocity vectors, free surface and isobars at t =70.Os and t =72.Os for T^ = 6.875s, modi-fied tank

Discussion by P . C . Sames and T . E . Schellin:

For partially filled tanks, rules of Germanischer Lloyd for classification and construction of seagoing ships (Sections 4 D, Load on Tank Structures, and Section 12 C, Tanks with Large Lengths or Breadths) require the investigation of resonance conditions between ship motions and liquid motions in the tank. This is because critical tank filling ratios may lead to excessive design pressures on the tank structure. The author presents a numerical method to predict fiuid motions in cargo tanks and demonstrates that the installation of partial bulkheads inside the tank can be an effective means to reduce these fluid motions. We compliment the author for developing this method and encourage further improvements. To acquire confldence of numerical predictions for design purposes, however, an experimental validation of numerical results is necessary.

Some questions arise concerning the author's use of his method to predict sloshing motions inside a partially filled tank. How valid is the selected three degree excitation period of the tank? Based on the pitch transfer function shown in Fig. 1 (for ship speed of 15 knots, wave encounter period of 6.8 s, wave length of 160 m), the considered ship may well experience larger amplitude pitch motions of up to six degrees. Numerical and experimental studies of, for example, Armenio and La Rocca (1996) indicate that the exciting amplitude as well as the exciting period affect wave heights inside the tank. Our computations with another tank (50 m long, 50 percent full) showed similar effects as seen in the figure below. This figure depicts maximum wave heights, normalized by the tank height, as a function of the ratio of exciting period to natural period. Maximum wave heights were recorded during the last 30 s of 150 s simulations. Unfortunately, we were not able to predict fluid motions due to large amplitude (4.5 degrees) excitations close to the fluid natural period. In addition, the accuracy of numerical predictions and thus the detection of sloshing strongly depends on the discretization of the fluid domain. We are not certain whether the motion of a viscous fiuid at resonance condition depends on the amplitude of excitation.

To account for relevant design cases when evaluating a tank's structure requires a careful choice of parameters. The analysis needs to consider that resonance periods shift as a result of installing internal structural elements. Also, different filling levels affect fluid motions and must be investigated. Finally, if the analysis shows that internal structural elements dampen fluid motions inside the tank, are the resulting hydrodynamic pressure predictions sufficiently accurate to enable design loads to be specifled to determine whether the structure fails?

0.66 0.75 0.83 0.92 1.00 1.08 EXCITING PERIOD / NATURAL PERIOD

Author's answer

The exitation with 3° is typical for this ship in rough seas and that is the reason why I think it can be used to study the flow in different tank geometries. If the results should be used to design the tank, the exitation has to be more realistic, and I am not sure whether a harmonic exitation of 6° is suitable. But as mentioned above, impact pressure cannot be predicted using this fiuid model.

The dependence of the amplitude of fiuid motion (energy e^) on the amplitude of exitation can be described as follows: For small exitation periods away from the natural period ./ë^ is proportional to the exitation. For bigger amplitudes, ^fë^ rises lower than proportional. This is known from linear and nonlinear theories, e.g. Solaas (1995). Near the natural period the increase of .Je^ is much lower than proportional to the exitation because of damping. If we assume hm ~ ^^'^^ effect is shown also also in your figure.

A Rankine Source Kochin-Function M e t h o d

to Remove the Negative Ship Wave Resistance

Hironori Yasukawa, Mitsubishi Heavy Industries^

1. Introduction

Linearized Rankine Source Method (RSM) codes may calculate negative ship wave resis-tance for low speeds, Raven (1991). A Kochin function, obtained from a far field expression of the velocity potential, would certainly remove the negative wave resistance. However, such a Kochin function has not been derived for double-body linearized fiows. I n this paper, a wave resistance formula w i t h a new Kochin function is derived under the free-surface condition pro-posed by Baha (1976), and a Rankine source method in conjunction w i t h the Kochin function is proposed which is called Rankine Source Kochin-Function Method (RSKFM). I n RSKFM, wave resistance is always positive. Calculations of the wave resistance for f u l l hull forms com-pare RSKFM and ordinary RSM w i t h the pressure integral scheme. R S K F M considerably improves the prediction accuracy of the wave resistance.

2. T h e o r y

I n our coordinate system, x-axis is defined as direction from the ship stern to the bow, y-axis to port and ^;-axis vertically upward. The x — y plane is the still water surface.

The total velocity potential is assumed to be represented as the sum of the double-body fiow potential U{x + (p) and the steady wavy fiow potential (j). The hull and free-surface conditions w i t h respect to (/) are represented as:

[H] [F] on on SH on z (1) (2) where

D{x,y) — ( C o ( a ; , y ) ? / o )- 1 -— ( C o ( a ; , y ) ï ^ o ) , Co(a;,2/) = i^{U^ - u l - v l )

U is the ship speed, Co the wave elevation based on double-body fiow, {UQ,VQ) the velocity component of the double-body fiow on = 0, and g the acceleration gravity. SH denotes the hull surface and n the outward normal unit vector on the hull.

A n analytical expression of velocity potential (f) is derived in the framework of this boundary value problem. From the expression, a Kochin function is obtained. Applying Green's theorem, the velocity potential cf) is expressed as:

Q

G{P;Q)dS ₍₃₎

SF denotes the stih water surface {z = <)). P = {x, y, z) is a field point and Q = ( a ; i , y i , ^ i ) the source point, a is a constant. Rewriting the integration on Sp using the free-surface condition (2) yields:

a m =

- ff

duQ HQ) _{dn Q} G{P;Q)dS- D{Q,)G{P;Qi)dxrdy,

+

Eq.(4) can be regarded as a basic equation for an RSM using Rankine sources as G. Sclavounos and Nakos (1988) apply a higher-order panel method to solve an equation similar to Eq.(4) numerically. The radiation condition of waves is fulfilled numerically.

Eq.(4) includes terms w i t h respect to the derivative of 4) on Sp as unknown variables. The terms get into difiiculties to solve the Eq.(4) and to derive the Kochin function. To eliminate the terms, we consider the transformation w i t h respect to the first operator ( w o g f ^ + ""ü-^)"^

which acts on 4> to that on G. The last term except the term w i t h respect to dG{P\ Qi)/dzi in Eq.(4) denoted by IF, can be rewritten:

IF = ^ ^ ^ ' ^ i ) HQi)dxidyi = II « / - ( Q l ) f ^ O T ^ + ^ ^ O T ^ ' l G[P;Qi)dxidyi JJSF V oxi dyiJ - L - I dy] d

HQi) [uo^ + v o ^ ) G{P;Qi) -G{P;Q,) ( m ^ + vo d

HQi) ( + ) G{P; Q l ) - G{P; Qi) ( n o ^ + vo dxi d dxi _d_ _d_ 'dy-i 4>{Qi) dyi dxi (5) c is the intersection between waterline of the hull and stifi water surface. The following relation is satisfied because of the hufi boundary condition, at least in case of a vertical side at stiU water line of the ship:

uo{Qi)n^i + vo{Qi)ny, = 0 (6) W i t h Eq.(6), the integration terms w i t h respect to c in Eq.(5) vanish:

IF = ' ^ ^ ^ ^ ^ i ) ( ' ' 0 0 ^ + ' ' ° ^ ) HQi)dxidyi

= II <f>{Qi)(uo^+vo

^y

G{P;Qi)dx,dyi (7)

JJsp V 0331 dyiJ The above derivation follows closely ideas of Kashiwagi (1994).

Taking the hull boundary condition into account, an expression of the velocity potential is obtained as:

acf>{P) = 11^ 0 ( g ) ^ ^ ^ ^ ^ ^ ^ r f g - / / ^ D{Qi)G{P;Qi)dx,dy,

+

- If

g JJSF

d d \^ d

dxi dyi) dzi G{P;Qi)dxidyi (8)

Throughout the present formulation, G was not yet defined. Here, the following function is employed:

G{P;Q) = 1

₊

r{P;Q)^ n{P;Q) (9)

'}^'p^l^ I = \l{x-xi? + {v-Vi? + {z^z^Y (10)

Then, 0 is expressed as follows:

dG{P-Q)

2

E q . ( l l ) is another expression of velocity potential using Rankine sources. This equation can be derived only under the Baba's free-surface condition. I f Dawson's (1977) free-surface condition is employed, this expression cannot be derived. Terms w i t h respect to the derivatives of the potential do not appear i n E q . ( l l ) , so the numerical treatment is much easier than that i n Eq.(4).

When defining the Green function G such that it satisfies the ordinary hnearized free-surface condition, 0 is expressed as:

a m = I I m ^ ~ ^ d S - I I DiQ,)GiP;Q,)dxidyi

JJs„ onq JJsp

+ ^ I I HQi) {^X, + 2)G,,,, (P; Q l ) + 2{^x, + G,,y, (P; Q i )

Wy,Gy,y,{P;Qi)\dxidyi (12) if is perturbation potential of the double-body fiow. I n this expression, a hne integral term

does not appear.

Far away from the ship, the double-body fiow becomes uniform. The free wave component of (j) remains. From Eq.(12), a new Kochin function is defined:

H{e) = - jj^ (j){Q)[nx,icose + ny^isme + n,,]kE{e;Q)dS

- JJ^ 0 ( Q i ) [</fa:i(</'xi+ 2)cos^6'-f-2((^^i-M)(^yicos6'sinÖ

+ipl^sm^e]k^E{9-Q,)dxidyi - jj^ D{Q,)E{e;Qi)dxidyi (13)

E{d;x,y,z) = exp{ik{x cos d + ysinO) + zk}, k = KQ/cos^d and KQ = g/U'^.

{nx,ny,nz) is the unit normal on the ship hull surface. Then, the wave resistance Ru, is

represented as:

R. = '-l^r'\H(etJi- (14,

27r y_,r/2 cos-^ 9 '

The wave resistance, depending on the square of the Kochin function i f , is always positive. Obtaining the velocity potential 0 on the huh and on 2; = 0, the Kochin function and the wave resistance can be calculated from Eqs. (13) and (14) respectively.

Theoretically we have to treat an infinite region of the still water surface outside the ship i n the integral over the free-surface i n the Kochin function Eq.(13). I n the numerical calculation, however, it is sufiicient to deal only w i t h the near field because the perturbation velocity of the double-body fiow {(px,Vy) and the D function rapidly approach zero w i t h increasing distance

from the ship.

3. N u m e r i c a l Procedures in R S K F M

Direct panel methods solve dhectly E q . ( l l ) for the unknown potential on the huU and the stiU water surface. The wave resistance is evaluated by Eqs. (13) and (14). For indhect panel methods, the calculation procedure is as follows:

1. Use Baba's free-surface condition.

2. Solve the basic equations w i t h respect to the source strengths on SH and Sp denoted by an and ap respectively.

3. Calculate the velocity potential on SH and Sp from the source strengths using the fol-lowing equation:

4. Calculate the wave resistance using (13) and (14).

I n this paper, the indirect method is employed. To f u l f i l l the radiation condition numerically, the method of Xia and Larsson (1986) is employed. Sinkage and t r i m are taken into account as in Yasukawa (1993).

The wave resistance formula Eq.(13) and the Kochin function Eq.(14) include singularities w i t h respect to cos 6» = 0 i n Ö = ±7r/2. To remove the singularities, we discretize Eq.(14) after integrating E{e; Q) on the panel of SH and Sp in the same way as the surface integration of 1/r in Hess and Smith's method. This weakens the intensity of the singularities.

4. Applications

The wave resistance was calculated for an ore carrier (SR107 hufi form). Table I . Fig. 1 shows two hull surface grids, Fig. 2 two free-surface grids. These grids are employed for all speeds. Fig. 3 compares wave resistance coefficient curves for 3 different grid combinations. Cw is based on V^/^ where V is the ship's displacement. "Kochin" denotes the R S K F M and "Pressure" the pressure integration over the hull. Both calculations use the same boundary conditions. The wave resistance calculated by pressure integration are all negative, and no improvement of the accuracy can be seen even for finer grids. R S K F M has always positive resistance. For Fn > 0.17, discrepancies due to different grids appear. For F„ < 0.17, results appear to be rather grid independent. The wave resistance computed by RSKFM, especially for the grid combination fp-2-Hhp-l, agrees well w i t h measured data, Fig. 4. Calculations were also carried out for 3 ships w i t h different protruding bow length, Fig. 5. Fig. 6 compares the wave resistance for f u l l load condition. The calculated and experimental wave resistance is almost the same for all 3 ships. Fig. 7 compares the wave resistance i n bafiast condition. Here the longer the bulb, the lower the resistance was computed. This relation agrees with the experiments.

References

BABA, E. (1976), Wave resistance of ships in low speed, Mitsubishi Technical Bulletin No.109

DAWSON, C. W. (1977), A practical computer method for solving ship-wave problems, 2nd Int. Conf Num. Ship Hydrodyn., Berkeley

(15)

KASHIWAGI, M. (1994), A new Green-function method for the 3-D unsteady problem of a ship with forward speed, 9th Int. Workshop Water Waves and Floating Bodies, Kuju

RAVEN, H. (1991): Adequacy of free surface conditions for the wave resistance problem, 18th Symp. Naval Hydrodyn., Washington

SCLAVOUNOS, P.D.; NAKOS, D. E. (1988), Stability analysis of panel methods for free-surface flows with forward speed, 17th Symp. Naval Hydrodyn., The Hague

XIA, F.; LARSSON, L. (1986), A calculation method for the lifting potential flow around yawed surface piercing 3-D bodies, 16th Symp. Naval Hydrodyn., Berkeley

YASUKAWA, H. (1993): A Rankine panel method to calculate steady wave resistance of a ship taking the effect of trim and sinkage into account, Trans. West-Japan Soc. Naval Arch. 86

Table I Principal dimensions of SR107 hull form Full scale Model Ship Length(Lpp) 302.00m 8.000m

Beam(5) 44.20m 1.171m

Draft (d) 17.00m 0.450m Block Coeff. (Cb) 0.826 0.826 Displacement (V) 192,200ton 3,486kg

hp-1 (445 panels for a half body)

hp-2 (768 panels)

fp-1 (900 panels)

fp-2 (1998 panels)

Fig. 2. Free surface panel arrangements

0.02 0.015 0.01 > ^ 0.005 -0.005 -0.0 o fp-1 + hp-1 fp-2 + hp-1 • fp-2 + hp-2 • • 0 . . . Pressure • • Q . . 0.15 Fnpp 0.2 0.015 'h 0.01 Q. 0.005 o 0 0.005 -0.0 , r . . , . CaL(Kochin) — CaL(Pressure) . Exp. 0.15 0.2

Fig. 3. Calculated wave resistance coefficient Fig. 4. Wave resistance coefficient curves for curves for SR107 hull form i n fuU load SR107 huh form i n f u l l load

CO Ü a. cn L.W.L.

<

Fig. 5. Comparison of stem profiles

0.01 C w 0 . 0 0 5 0.01 C w 0 . 0 0 5 • C a l . —e— 1 1 1 1 1 S h i p A 1 1 -_ - i - - _{S h i p B} - — B - - S h i p C -] O -a P 1 -.1 0 . 1 5 0

8

Fig. 6. Calculated (top) and measured (bottom) wave resistance coefficient curves for 3 f u l l ships with different bow shapes i n

f u l l load 0.2 0.01 C w 0 . 0 0 5 G a l . S h i p A - - A - - S h i p B - B - - S h i p C

Fig. 7. Calculated (top) and measured (bottom) wave resistance coefficient curves for 3 f u l l ships w i t h different bow shapes i n ballast condition

A K u t t a Condition for Ship Seakeeping Computations

w i t h a Rankine Panel M e t h o d

Volker B e r t r a m , T U Hamburg-Harburg^ G e r h a r d D . T h i a r t , Univ. Stellenbosch^

1. Introduction

The ship (especially including the rudder) can be considered as a vertical foil of very short span. For a steady yaw angle, i.e. a typical manoeuvring application, one would certainly enforce some kind of K u t t a condition i n a potential flow computation. However, for the corre-sponding periodic motion i n seakeeping, even mentioning of the K u t t a condition is rarely found for 3-d methods. Wu (1994) includes the K u t t a condition to compute the radiation problem for an oscillating surface-piercing plate at forward speed using a Green function method. Zou

(1995) solves the same problem using a Rankine panel method. The only applications to ships

at forward speed are an extended abstract by Zou (1994) for a Rankine panel method and

Schellin and Rathje (1995) for a Green function method. However, Zou solves only the

ra-diation problem and neglects the steady disturbance potential, and Schellin and Rathje have some inconsistency i n fulfllling the free-surface condition. For the related problem of fluttering vibrations of airfoils i n aerospace engineering, enforcing a K u t t a condition appears to be stan-dard procedure. I t is unclear i f omitting the K u t t a condition is based on some physical insight about the negligible effects or oversight. I t is of no importance as long as the applications are limited to head or following waves, i.e. the most frequent test cases. But of course, for prac-tical requirements a method must be applicable for all wave directions. We shall numerically investigate the effect of the K u t t a condition and hope to contribute this way to a clariflcation of its necessity.

2. P h y s i c a l model

We consider a ship w i t h average speed C i n a regular wave of small amplitude h. Bertram

(1996) , Bertram and Thiart (1998) presented a hnearized 3-d Rankine panel method including

all forward-speed effects. The same method is now extended to include the K u t t a condition. We will therefore omit other details of the method and focus here only on the K u t t a condition and the associated dipole elements applied to compute the motions i n oblique waves.

The problem is formulated in right-handed Cartesian coordinate systems. The inertial

Oxyz system moves uniformly with velocity U. x points forward, z downwards. The angle of

encounter / i between body and incident wave is defined such that jj, = 180° denotes head sea and n = 90° sea from starboard.

The ship has 6 degrees of freedom for rigid body motion expressed i n the motion vector ü =

{ui,U2,U3}'^ and the rotational motion vector a = {^4,lis,ue}"^. A perturbation formulation for the potential is used omitting higher-order terms:

^ ^ ( 0 ) ^ ^ ( 1 ) (1)

(f)^'^'> is the part of the potential independent of the wave amplitude h. I t is the solution of

the steady wave-resistance problem, (p^^^ is proportional to h and accounts (linearized) for the contributions of the seaway.

</>(i)=i?e(0(i)(x,y,.2)e-^*) (2) The symbol " denotes a complex amplitude. tOg is the encounter frequency. The harmonic

potential cj)^^^ is decomposed into the potential o f t h e incident wave 0^, the diffraction potential

'Lammersieth 90, D-22305 Hamburg, bertram@schifFbau.uiii-hamburg.d400.de ^Dept. Mech. Eng., 7600 Stellenbosch, South Africa

0*^, and 6 radiation potentials:

<i>^'^ = <p'+r + j 2 f u i (3) 1=1

I t is convenient to divide cf)'^ and 0^ into symmetrical and antisymmetrical parts to take ad-vantage of the (usual) geometrical symmetry:

4^ = cj)<^''+ d)'^'^ (5)

The K u t t a condition requires that at the trailing edge the pressures are equal on both sides. I f the pressure is decomposed into a symmetric and an antisymmetric part, then this is equivalent to requiring zero antisymmetric pressure:

^''"^ = -piiuJeH'" + V0(°)V0^'") = 0 (6) where 0*'" is an antisymmetric unsteady pressure, i = 1,3 and 5 for the three antisymmetric

radiation modes and i = d for the antisymmetric diffraction part.

The four antisymmetric potentials are solved i n a Rankine panel method which uses first-order panels (plane and constant strength) on the hull (up to a height above the average wetted surface) and around the ship. I n addition, a dipole distribution inside the hull and trailing behind the hull is used for the antisymmetric cases. The theoretical details of the dipole dis-tribution are discussed i n the appendix. The K u t t a condition (6) is fulfilled at the last column of collocation points at the ship stern. A corresponding number of semi-infinite dipole strips is used on the centerplane o f t h e ship. The dipole strips start amidships and have the same height as the corresponding last panel on the stern. The simultaneous fulfillment of no-penetration condition and K u t t a condition on the collocation points poses no numerical problem, as already demonstrated by Zou (1994). The four systems of equations for the antisymmetrical potentials share the same coefficient matrix w i t h only the r.h.s. being different. A l l four cases are solved simultaneously using Gauss ehmination. The subsequent computations leading to the motions are then straight-forward.

3. Results for S-175 containership

The S-175 containership using the same discretisation as i n previous computations, Bertram and Thiart (1998), was computed for Proude number Fn = 0.275. The results for the symmetric motions are of course not affected by the K u t t a condition.

Fig. 1 shows the result for p, = 150°, Fig. 2 ïoi p = 60°. The results for heave and pitch show excellent agreement w i t h experiments. This is typical for all wave lengths and angles of encounter investigated. No experimental results are available for surge for these angles of encounter, but the tendency is similar as for head waves i n previous computations, Bertram

and Thiart (1998).

For /X = 150° the K u t t a condition has only significant effects for long waves i n the region where resonance occurs. Here the K u t t a condition simulates to some extent the effect of viscous damping and reduces drastically (by factors between 2 and 4) the antisymmetric motions. However, for roll additional viscous effects (those that would be apparent also at zero speed) reduce i n reality the motions even more. For yaw and sway, no experimental data are here available, but we expect that autopilots i n experiments will also prevent the very large predicted motion amplitudes of the computations.

For fj, = 60° some short-comings of the present method become apparent. Yaw is predicted with good accuracy over most of the range of frequencies, but for low frequencies (long waves) the computations overpredict motions strongly, even i f a K u t t a condition is enforced. We suspect that this is due to an autopilot used i n experiments which restores motions for low frequencies, but this assumption would require further investigation. Roll motions show only satisfactory agreement between computations and experiments. However, the measured roll motions appear less plausible for high frequencies than the computations. Sway shows strong scatter of results. Apparently the K u t t a condition makes things worse here.

A conclusive improvement of results due to the K u t t a condition could not be demonstrated. The improvements for the narrow region of resonance are drastic, but ad hoc solutions hmiting the response amplitude operators to maximum values taking from experiments for similar ships may work almost as well i n practice, while being a lot simpler. Sway motions for low encounter frequencies (in following oblique waves) require special treatment. This w i h be subject to fur-ther research.

References

BERTRAM, V. (1996), A 3-d Rankine panel method to compute added resistance of ships, IfS-Report 566, Univ. Hamburg

BERTRAM, V.; THIART, G.D. (1998), Fully three-dimensional ship seakeeping computations with a surge-corrected Rankine panel method, J. Marine Science Teclmology 3/2

SCHELLIN, T.E.; RATHJE, H. (1995), A panel method applied to hydrodynamic analysis of twin-hull ships, FAST'95, Travemünde

WU, G.X. (1994), Wave radiation by an oscillating surface-piercing plate at forward speed. Int. Shipb. Progr. 41, pp.179-190

ZOU, Z.J. (1995), A 3-d numerical solution for a surface-piercing plate oscillating at forward speed, 10th Workshop Water Waves and Floating Bodies, Oxford

ZOU, Z.J. (1994), A 3-d panel method for the radiation problem with forward speed, 9th Workshop Water Waves and Floating Bodies, Kuju

SÖDING, H. (1975), Springing of ships, ESS-Report 7, Univ. of Hannover

\üMh 1.0 H 90-- 1 ^ .O— e—9—e-3 4 uj^jL/g -90-1.0 \Ü2\lh 0.5-è O O O + . + + —1 [ ¬ 2 3 4 uJLJ^ 90 H 0¬ -90- _{* * * +} O O \üz\lh 1.0 0.5 1 1 0.5 -3 4 w/zVfl 90^ -90 H 2.0 1.0 M l l 2 i3 + •*••» 6 't./ip „ . a 90 O -90 3 4 o i y ^ O é » O O , + + O 1.0- 0.5-90-1 0¬ -90-O» n r ^ ^ ^ 3 4 a-v^i/s 1.0 H 0.5 J M ^4 _ ^ - j b — ^ — l p — 0 _ 90 O -90 1 2 3 4 u./L/g e 4- • j, ^ *

Fig. 1: Response amplitude operators for S-175, p = 150°, F„ = 0.275; • experiment, -|- RPM without Kutta condition, o RPM with Kutta condition

1.0 \Üi\lh 90 O -90-1 -90-1 -90-1 r 1 2 e 3 4 ui^JTTg O r, O O O 1.0 0.5 • 8 O ~1 1 ~ 90 O -90 2 3 4 w\ / l 7 ^ 1.0 0.5 kh 1.0 0.51 \u2\lh 90 O -90 6.0 5.0 4.0 3.0 2.0 i.oH 90 0 -901 1.0 0.5 kh 90¬ 0¬ -90-0 0 ° 90 O -90 * + ^ f

Fig. 2: Response amplitude operators for S-175, p = 60°, F„ = 0.275; experiment, -I- RPM without Kutta condition, o RPM with Kutta condition

—1 1 1 — ^ 2 3 4 uJ^/L/^ + * O + O O e- 4> t t iLIg ^ ? ^ V 1

Appendix: Velocity Potential for a Rectangular Vortex Panel with Harmonically Oscillat-ing Strength

Tiie oscillating ship creates a vorticity. The problem is similar to that of an oscillating airfoil. The circulation is assumed constant within the ship. Behind the ship, vorticity is shed downstreams with ship speed U. Then:

| - f / A ) 7 ( . , . , ^ ) = 0. ( A l ) 7 is the vortex density, i.e. the strength distribution for a continuous vortex sheet. The following

distribution fulfills condition ( A l ) :

7(a;, z, t) = Re {%{z) • e^(-'=/f^)(^-^<.) . e^"=') for x<Xa (A2) where % is the vorticity density at the trailing edge Xa (stern of the ship). We continue the vortex sheet

inside the ship at the symmetry plane y = 0, assuming like Zou (1994) a constant vorticity density:

7(0;, z,t) = Re {% {z) • e'"'*) for aj^ < a; < a;/ (A3) a;ƒ is the leading edge (forward stem of the ship). This vorticity density is spatially constant within the

ship.

A vortex distribution is equivalent to a dipole distribution if the vortex density 7 and the dipole density m are coupled by:

^ " ^ / A n

^ = ^ ( ^ ^ )

The potential of an equivalent semi-infinite strip of dipoles is then obtained by integration. This potential is given (except for a so far arbitrary 'strength' constant) by:

^{x,y,z) = Re\ j f m{i)^ é^'' =Re {^{x,y,z) • é^^') (A5)

with r = yj{x - ^ ) 2 -\-{z- C)= and

~ { (1 - e^(-'/^)«-^»)) + {xf - Xa) for - 0 0 < ^ < a:, ^^^> It is convenient to write <p as:

^{x, y, z) = [yh + f2l2 + hh + fiVh] (A7) where

= [ ^^^^

11^

The velocity components and higher derivatives are then derived by differentiation of $, which reduced to differentiation of ip:

V $ = Re (-7a • V(/5 • e'"'*) with

= yhx - h + f2hx + hhx + fiUhx

= -^1 + yhy + }2hy + h h y + f i { h + Vhy) (Pz = yhz + hhz + h h z + IAVUZ

^xx = yhxx - "^hx + f2hxx + fzhxx + fiVhxx f y y = "^hy + y h y y + hhyy + hhyy + hi'^hy + y h y y ) f x y = Ilx + yhxy - hy + hhxy + hhxy + hihx + yhxy) ^yz = hz + yhyz + hhyz + hhyz + f i { h z + Vhyz) f x z = yhxz - hz + f2hxz + fzhxz + fiyhxz

Hjz - ZQ) + rao){{z - Zu) + Tfu) [{{z - Zo) + rfo){{z - Zu) + rau)

The integral h and its derivatives are:

h

where rao = Vi^ " ^o.)'^ +y^ + { z - ZoY, r f u = \ / { x - Xf)'^ + + _ Zu)'^, etc X — Xa X — Xf X — Xf

hx =

I

Taoiz - Zo+ rao) Tfuiz - Zu + Tfu) rfo{z

y , y X Xr, can be (A15) (A16) (A17) (A18) (A19) (A20) (A21) (A22) (A23) (A24) 'ly hz - Zo + r f o ) y {z - Zu + rau) y ' a a

•aoiz - Zo + Tao) rfu{z - Zu + rfu) rfo{z - Zo + rfo) rau{z - Zu + rau)

1 1 1 1 — + ' ao Tfu rfo nyy llxy -nyz llxz

-rloi^ Zo + rao) -{X - Xa)'^{z - Zp + 2rao)

,.3 _ ^ _ L „ 12 ; ; - ! T- Tao)'^

rjui^ - Zu + rfu) - j x - Xf)^{z -Zu + 2rfu)

rjui^- Zu+rfu)^

rj^jz -Zo + r f o ) - { x - X f ) ^ { z - Zo + 2rfo)

rjoiz - Zo + rfo)^

rjuiz - Zu+ rau) - { x - Xa)'^{z - Zu + 2ra„)

TIU{Z - Zu+Tau)'^ Zo + rao) -y'^jz - Zo + 2rao)

rloi^

+

rlo{z - Zo+rao)"^

rjuiz - Zu + Tfu) -yHz-Zu + 2rfu)

Zu +rfu)'^

rjoiz - Zo + rfo) - y'^{z - Zo + 2rfo)

Zo + rfo)"^

^aui^ ~ Zu ~^ rau) - y ' ^ { z - Zu + 2rau)

+

(g; - Xa)y{z - Zp + 2rao) {x - X f ) y { z - Zu + 2rfu)

rlo{z- Zp+rao)"^ r)u{^- Zu + rfu)^

{ x - X f ) y { z - Zo + 2rfo) ^ {x - Xa)y{z - Zu + 2rau) r%{z- Zo+rfo)^ rluiz - Zu + rau)^

y 'ao X - X X fo Xf + au X — x i ^ x _ ' f o

The integral I2 and its derivatives are: '{z~Zo){x-Xf) arctan -arctan yrfo [Z - Zu){x - X f ) arctan + arctan {z - Zo){x - Xa) i2x yrju

y{z - Zo) y{z - Zo)

yrao

{Z - Zu){x - Xa)

(A25)

yra

Tfoiix - Xf)'^ +y'^) raoiix - Xa)^ + y"^) y{z - Zu) y{z - Zu)

+ '2y

rfuiix - Xf)^ +y^) rau{(x - Xa)'^ + y'^) ~{X - Xf){z - Zo) '

+ {X - Xa){z - Zo) + {X - Xf){z - Zu)

Tfoiix + rfoiiz - Zo)'+y')\ 1

+ 1

rao{{x - Xa)"^ + 2/2) + _{rao{{z - Zo)'+y^)\} 1

+ 1

Tfuiix -Xf)'+y^) + rfu{{z -- Zu)'^ + 2/2)

1

+ 1

'22

- ( a ; - Xa)iz - Zu)

y j x - X f )

rfo{{z-Zo)^+y^) rao{{z-Zo)^ + y^) y j x - X f ) ^ y{x - Xa)

y{x

-

+

r / „ ( ( ^ - ^ „ ) 2 + 2 / 2 ) rau{{z-zu)^+y^)

{x-Xf)y{z-Zo){2>r)^-{z-Zo)^) {x - Xa)y{z - Zo){2>rlo - {z - Zo)^)

+

3rlo - j z - Zo)' , 3rl-{x-xa)

+ {x - Xf)y{z - Zu) {x - Xa)y{z - Zu) ( ( a ; - a : a ) 2 + 2/2)2 ( ( ^ _ ^ ^ ) 2 + ^^2)2 •3r}^-(z-zu)' , 3 r 2 „ - ( a ; - a : / ) ^ + ( ( a ; - a ; / ) 2 + 2 / 2 ) 2 ( ( ^ - ^ „ ) 2 + y 2 ) 2

3rlu-{z-zu)' 3rtu - { x - Xa)'

i2xy L ( ( a ; - a : a ) 2 + + 2 / 2 ) 2 ( ( ^ _ ^ ^ 2 + ^ 2 ) 2 y^i3r%-iz-Zo)') l2yz -rfo Z - Zp ^ao Z - Zy rfu ^Z-Zu rau X — Xf rfo X — X, ( a ; - a : / ) 2 + 2 / 2 r 2 ^ ( ( , ; _ , ; / ) 2 + y 2 ) 2 1 2 / ' ( 3 r L - { z - Z p ) ' )

[x-xa)' + y' rlo{{x-Xa)' + y')'

1 y'{?,r%-{z-zu)') ( a ; - a ; / ) 2 + 2 / 2 r)^{{x - xf)"^ + y'^)'^ 1 y'{?,rlu-{z-zu)') ix-xa)'+y' r 2 J ( a ; - a ; a ) 2 + 2 / 2 ) 2 J 1 2 y ^ ( 3 r 2 ^ - ( a ; - a : ; ) 2 ) ^z-Zo)'+y' r%{{z-zp)^+y^)^ 1 y'i3rlo - { x - Xa)') {z-Zo)'+y' rlo{{z-ZpY+y-^y

X — Xf rfu ^X-Xg rau y y y^[3r}^-ix-Xfr) ' fo ao fu au

The integral Js and its derivatives are:

J3 = arctan

— arctan

(z-zur + y' r%{{z-zur + y'r

(z-z^y+y^ rlMz-Zur + y'Y y y r l . r3 {Z - Zo){x - Xg) yrao > - Zu){x - Xg) yra arctan + arctan {z - Zo) y {z - Zu) y (A26) hx hy l3z

y{z - Zo) y{z - Zu)

rao{{x - Xg)'^ + y"^) rguiix - Xg)'^ +y'^ = -{X - Xg){z - Zo) + {X - Xg){z - Zu) + = 1

+

+

rau{{x - Xa)'+ y') rgu{{z - Zu)'+y') Z - Zu

(Z - ZoY +2/2 (Z - Zu)' + y'

y{x-Xa) y{x-Xg)

+

rgoi{z-Zo)'+y') rgui{z-zu)'+y') {z-Zo)'+y' {z-Zu)'+y

hxx —

hyy =

hxy —

hyz

-[x - Xa)y{z - Zo){3r'go - { z - Zp)') {x - Xg)y{z - Zu){3rlu - jz - Zu)')

rL((a;-a:a)2+2/2)2 " + - ^3J(^ _ J2 + ^2)2 (a: - Xg)y{z - Zp) Ï 3rg„ - {z - Zo)' 3rlp - {x - Xg)' '

rlo [{ix-xg)^+y^)^ + ((^_^„)2+y2)2_

(a; - Xa)y{z - Zu) \ 3rlu ~ {z - Zu)' 3rlu - jx - Xa)'

rlu [{{x-xg)^+y^)^^ {{z-zu)'+y')' 2y{z - Zp) 2y{z

-

{{z-Zo)'+y')' ^ iiz-zu)'+y')' Z - Zp ' ao Z — Z 1 y'jSr'gp-iz-Zo)' {x-Xg)'^+y' rlp{{x - Xg)'^ + y')' 1 y2^3rlu-{z-zu)') I ao X — X

x~Xg)^+y^ rluiix - Xg)'+ y')'

1 y2^:ir'go-{x-Xg)')-J-Zxz — ( ^ _ ^ „ ) 2 + j ^ 2 ,2^((^_^^)2+y2)2 1 y2(3,2^_(^_^^)2) {z-zu)'^-y' rlu{{z-zu)'+y')' {z - Zo)' - y' {z - Zu)' - y'

[{z-Zo)' + y')' ^ {{z-zu)'+y')' y

y

' ao 'au

The integral I4 and its derivatives are:

Xa

-I

h = : ' — 0 0

Z - Zo z - z.

roikx-iY^y') ru{(x-iY^y') (A27)

where r,, = A / { X - i)^ + y' + {z - ZoY and r„ =

{z - zu){x - aZrl - { z - Zu?) {z - Zo){x - Oi^r^ - {z - z^) 2 M

r l i ^ x - i Y + y ' ? r l { { x - i Y + y ' ^ Y {z - Zu)y{2,Tl - { z - z u f ) {z - Zo)y{3rl - {z - Zpf)^ r l { { x - i Y + y ' ? di r l { { x - i Y + y ^ Y di rl rl lAxx Uyy Hyz l-Axz Xa r l { { x - i Y + y ' ) r l { { x - i Y + y - ^ f r u { { x - i Y + y ^ f 8(x - 0 ' f r l - 2 . { x - i f 2 { r l - 2 { x - i f ) r l { { x - i Y + y ' ) r l i i x - i Y + y ' ^ Y r„((a; - 0^ + 2/^)3 di \z-Zo) {z - Zu) r l - i y ' ₊ 2irl - 22/2)

rliix - i ) ' + 2/2) rliix - i)^ + y^)^ r^Hx - i)^ + 2/2)3

Sy' rl - 32/2 ^ 2(7-2 _ 2y2) hxy = y / e ' " ' i / " { x - i ) r l { i x - i ) ' + y ' ) • r3((a;-02+2/2)2 ,^((3. _ ^)2 + ^2)3 3 di r l i i x - i ) ' + y ' ) ^ r U { x - i ) ' + y ' ) \Z-Zu) /4(3r2 - (z - z u f ) V l i k x - i Y + y - ^ Y ' r l i i x - i Y + y - ^ ) + di = -32/ / e

I

The integral with respect to i in the expressions for I4 and its derivatives are evaluated numerically following the modified Simpson's method of Söding (1975):

xi+2h

I

h-The integration interval is truncated 0.1 x {xf - Xa) up- and downstream of the the field point x. h-The wake "panel" is subdivided into a suitable number of "sub-panels" depending on the minimum distance of the field point to the wake.

Viscous Flow Simulations for Conventional

and High-Skew Marine Propellers

Moustafa Abdel-Maksoud, Potsdam Model Basin ^ F l o r i a n Menter, A E A Technology GmbH ^

H a n s Wuttke, Potsdam Model Basin ^

1 Introduction

The design of marine propellers is largely based on potential flow theory in combination with experimental testing. While potential flow theory offers a quick way to obtain insight into the global performance characteristics of a propeller, i t has substantial shortcomings resulting f r o m the assump-tion of inviscid flow. Important effects like boundary layer separaassump-tion due to pressure gradients at off design conditions cannot consistently be included in the method. Empirical assumptions about the drag have to be introduced i n order to include some of the effects of viscosity. The influence of turbulence on the performance cannot be modelled in the inviscid methodology.

Furthermore, the optimization of marine propellers will eventually have to take into account the interaction between the hull and the propeller, as the wake of the hull has a strong effect on the flow conditions at the propeller and vice versa. The interaction is largely influenced by viscous effects, and only viscous flow computations will be able to successfully treat these problems. As a result, the numerical simulation of propeller flows based on the solution of the Reynolds averaged Navier-Stokes (RANS) equations is receiving increased attention. Computations for marine propellers

(or simplifled generic models) have been reported by Kim and Stern (1990), Park and Sankar (1993), Oh and Kang (1992), Abdel-Maksoud et al. (1995, 1996), Sanchez-Caja (1996) and Chen (1996). Encouraging results have been obtained, especially w i t h flue grids, by Sanchez-Caja (1996).

The main obstacle i n the integration of RANS methods into the propeller design process is the complexity involved in the generation of high quality numerical grids. Little information is given on the details of the grid generation in the aforementioned studies. Furthermore, the reported conflgurations were limited to conventional propeller shapes which are easier to discretize than modern skew and high-skew propellers (Figs. 1-3). I n order to have a useful design tool, it is necessary to develop strategies for the efficient generation of grids around a wide range of propeller shapes. Efficient numerical methods must be available to allow a propeller flow computation within hours on a workstation. As the convergence properties of numerical codes deteriorate on grids of low quality (poor grid angles and cell aspect ratios), fast convergence rates require high quality grids.

Grid generation around marine propellers is a difficult problem. The main sources of complexity are the strong twisting of the blades, the large geometric angle of attack against the propeller axis, the periodicity of the grid in circumferential direction and the complex shape of modern propellers. I n addition, the hub has to be included in the discretization. The blunt end of the hub can have a strong infiuence on the grid generation and the grid topology and can severely complicate the entire procedure (Sanchez-Caja 1996). To make RANS computations an attractive alternative to design tools based on potential fiow theory, time spent on grid generation must be limited to a few hours.

A procedure for generating high quality numerical grids around marine propellers will be de-scribed. Grids have been generated for three different propellers and numerical simulations based on these grids will be presented. The commercial computational fiuid dynamics (CFD) package CFX-TASCflow of A E A Technology (Anon. 1995) was used for the grid generation and the numerical simulations.

'Marquardter Chaussee 100, D-14469 Potsdam ^Staudenfeldweg 12, D-83624 Otterfing