A Method for Predicting the Influence of an Additive Bulb on Ship Resistance

7/24/2019 A Method for Predicting the Influence of an Additive Bulb on Ship Resistance.

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1 INTRODUCTION

The hydrodynamic design of bulbs, being of major importance for many types of ships, is a fre-quent activity in towing tanks. However, the selection of the proper bulb is not a trivial task,since it is related to the complicated flow phenomena around the bow that are more or less diffi-cult to predict. Usually, such a design is based on previous experience or on systematic series ofexperiments, e.g. Kracht (1978), but there are many cases where both practices may fail. There-fore, new configurations have to be tested in order to achieve an effective reduction of the shipresistance. In these cases, the drawing and the manufacturing of various bulbs that are adjustedto the original hull is not only expensive but also time-consuming.An alternative, promising way to accelerate the whole procedure and reduce the required cost

is to apply CFD tools that may provide useful information about the effectiveness of a new bulbdesign. The aim of the present work is to investigate the possibility of relevant methods to pre-dict the hydrodynamic behavior of such configurations. Although the application of advanced RANS methods that solve the real free-surface problem are still very demanding, it is noticeable

that the flow around the bow can be effectively simulated regarding the fluid as inviscid (unlesswave-breaking or spray occur). In this respect, a potential flow code has been developed to cal-culate the free-surface geometry and next, an existing RAN S solver is applied to calculate the re-sistance. This is not a new concept and various methods have been developed to face the prob-lem, each one with its own advantages and shortcomings. In addition, a fast representation(based on the conformal mapping technique) is applied to generate an additive bulb with the purpose of comparing its behavior with an implicit one having the same general characteristics.

2. THE NUMERICAL METHOD

2.1 The potential flow solver

The numerical method that has been developed to solve the potential flow around ships is basedon an iterative procedure which, basically, decouples the two free-surface boundary conditions.

met o or pre ct ng t e n uence o an a t ve u on s presistance.

G.D. TzabirasSchool of Naval Architecture and Marine Engineering

National Technical University of Athens

9 Heroon Polytechniou str., Zografos 15773, Athens, Greece

ABSTRACT: The present work is concerned with the numerical prediction of the ship resis-

tance by combining two CFD approaches: an inviscid code and a RANS solver. The main purpose is to decide effectively and rapidly about optimized bulb configurations for performingmodel experiments. A potential flow method has been developed to solve the non-linear free-surface problem that calculates the geometry of the free-surface and, in a first approximation,the wave resistance. Then, by applying the RANS code beneath the predetermined surface, thetotal resistance is calculated. The method is applied for a model of a passenger-ferry that has been tested in the towing tank of NTUA. In order to accelerate the whole procedure in the de-sign stage, the effect of an additive bulb, that is “equivalent” to the original implicit, is studied.



Unlike most of the non-linear codes (e.g. Janson, 1997, Raven, 1993) which employ potentialformulations that involve implicitly these conditions, primitive variables are introduced herethat is, a velocity-pressure coupling solution is applied. To solve for the kinematic boundaryvalue problem (Laplace equation) the ship hull and the real free-surface are covered with quad-rilateral panels, Figure 1.

Figure1. Free-surface panels.

Two co-ordinate systems (x, y, z) are introduced, i.e. the ship reference system which is employed to construct the panels and the absolute system with z=0 on the undisturbed free-surfacewhere all the flow equations refer to. The co-ordinates are transformed successively between thetwo systems taking also into account the possible sinkage and trim corrections. Both the kine-matic and the dynamic boundary conditions are satisfied on the free-surface at the end of the it-erative algorithm which follows the three main steps:

1. For a known free surface geometry solve the Laplace problem which satisfies the kine-

matic condition on both the hull and the free-surface2. Solve the vertical momentum equation by introducing the normal pressure gradient on

the free-surface which is calculated from step 1 using the Bernoulli equation and calcu-late the corresponding velocity component.

3. Update the free surface by a two-step Lagrangian-Eulerian procedure using the verticalvelocity components calculated in step 2.

The above steps are repeated until convergence, which is effectively obtained when the dynamic boundary condition is satisfied under a specified criterion.

The first step follows the classical methodology of Hess & Smith (1968), that is a constantsource distribution is assumed on each quadrilateral panel. This approach has also been used in previous works that adopt different formulations. The unknown sources have to satisfy the ki-nematic condition, i.e, the normal velocity at the “null” point of each panel equals zero. Theyare calculated by solving the corresponding linear system by the Gauss-Seidel iterative algo-rithm which is quite efficient, since the relevant matrix is diagonally dominant.

After the calculation of sources, the velocity components (u x , u y , u z ) are calculated on the null points of the surface elements and the total pressure p* is derived from the Bernoulli equation.The total pressure is the sum of the static pS and the hydrostatic term ρgz . Obviously, for an arbitrary surface, pS is different than the ambient pressure (which is assumed equal to zero). Thisdifference is introduced as a source term to calculate the vertical velocity u z * on the free surface by solving the corresponding inviscid momentum equation:

2 y z x z z

u uu u u p

y z z

(1)



Equation (1) is solved numerically after integrating on the surface control volumes, as shownin Figure 2. The result of this integration is a set of non-linear algebraic equations correspondingto the surface panels having the form

*( )( ) P zP E zE W zW U zU D zD P P

A u A u A u A u A u x y p gz (2)

Figure 2. Surface control volume

The vertical side δz * of the control volume in Figure 2 is an arbitrary parameter that controlsthe convergence of the procedure. It appears in the convective terms Ai of equation (2), but es-sentially determines the influence of the pressure gradient. By performing numerical experi-ments on various ship forms and Froude numbers, it has been found that efficient values for δz

*

should be of the order (δxδy). The convective coefficients can be approximated by any higherorder upstream difference scheme. However, in the present investigation, the zero-order up-stream scheme is applied in the stream-wise UD direction, in order to avoid as far as possibledownstream reflections. Then, only one sweep of the computational domain is needed to solvefor the vertical velocity component, but fine discretization in the longitudinal direction is re-quired to achieve satisfactory results for the free-surface geometry. In addition, to avoid nu-merical diffusion in the lateral direction, an alternative option can also bee applied which is based on the integration along local flow-lines. The latter can be easily determined around the

surface control points using the potential flow results. In this case, equation (2) reduces to thesimple form:

*( )( ) P zP U zU P P

A u A u x p gz (3)

where u zU * is calculated by linear interpolation on the upstream face of the control volumealong the local streamline passing through point P.

After the vertical velocity components have been calculated, the new free surface is computed by performing two corrective steps as described by Tzabiras (2004). First, following thelocal flow-lines, points on a transverse surface line define corresponding points on the next,downstream line and a new transverse profile is generated. Next, this new cut is corrected sothat the flow rate through the corresponding surface panel vanishes. This way, the surface flow-

lines associate the dynamic with the kinematic condition.The application of the method to multiple-hull forms or transom sterns (without “dead” wa-

ter) is straightforward. Besides, since accuracy depends strongly on the number of panels, to re-duce the computational effort the solution may start with a coarse grid which is successively re-fined to the maximum number of panels that are defined in the input data. Except thegeometrical interpolations, there is no difficulty to pass from the one grid resolution to anotherdue to the decomposition which is followed. The wave resistance as well as vertical forces andmoments are calculated by integrating the static pressures on the hull panels. The latter are usedto calculate the dynamic sinkage and trim whenever this is required and the center of gravity isknown. This procedure is carried out by changing sinkage and trim values after a certain numberof iterations is completed, until final convergence is achieved.



2.2 Panel construction

The number of panels that are required to make accurate predictions for the free-surface ge-ometry and the wave resistance component depends on the examined ship form as well as on theFroude number. Low Froude number flows or vessels with bulbous bows at high speeds requirefine longitudinal and transverse discretization which needs a fast method for generating the pan-

els. The conformal mapping technique, e.g. Tzabiras (1997), has been employed in the presentinvestigation which transforms ship or bulbous bow sections onto the unit circle through thegeneral transformation:

1

1

2

N n

n

n

z a a

(5)

In transformation (5) z represents the complex plane on the unit circle, ζ the arbitrary hull or bulb section, while coefficients an are calculated iteratively for an initial set of data sections.The intermediate sections that are needed for fine longitudinal grids are generated straight away by calculating the corresponding coefficients applying cubic interpolation. Then, the panels on

the hull or the bulb are directly constructed by finding the intersection of the free surface withthe analytical representation of the sections and using any desired distribution of grid pointsgirth-wise (equal arcs or equal angles on the circle plane). The points on a transverse cut of theupdated free surface are found by interpolation following an exponential arrangement.

The geometry of an additive bulb can be defined by introducing the six geometrical parame-ters of Figures 3a, b, e.g. Kracht (1978), Tzabiras (1997). Three of them are linear, i.e. thelength L B of the bulb up to FP, its maximum breadth B B and the vertical distance Z B of the lead-ing edge from the base line. The remaining are: the transverse area A BT of the bulb section at FP,the area A LB of the longitudinal projection of the bulb contour and the volume of the bulb V B.Evidently, there are many ways to design a bulb which satisfies these parameters. In the presentwork the geometry of an additive bulb is produced using a on a three-parametric Lewis repre-sentation defined by the two principal dimensions and the area of any section, longitudinal ortransverse (Tzabiras, 1997). Assuming a subdivision of A LB in the upper (U) and the lower (D)

regions, the longitudinal bulb profile can be directly generated. The same procedure holds forthe basic section A BT . The other transverse sections along the bulb vary proportionally to their principal dimensions T(x)B(x) where B(x) is calculated by an iterative Newton-Raphson methodso that the total volume of the bulb equals V B. To fit the additive bulb after FP it is assumed thatits basic section remains constant downstream. Then, the panel construction on the hull is basedsimply on keeping the maximum breadth of the bulb or the hull, at a certain depth. To obtain asmooth distribution of points girth-wise, the intersection of the bulb with the corresponding hullsection is firstly defined and, then, points are redistributed in both directions. This procedure isshown in Figures 4a, b where an original hull is transformed by applying an additive bulb.

Figure 3a,b. Left (a): transverse section at FP of an additive bulb . Right (b): Longitudinal profile



Figure 4a,b: Left (a): original bow and additive bulb Right (b): Transformed bow lines

2.4 The RANS solver

The RANS multi-block code for ship flows developed at NTUA (e.g. Tzabiras, 2004) is used to

solve the viscous flow under the specified free surface. The numerical solution is based on thefinite volume approach, employs curvilinear co-ordinate systems and staggered meshes andsolves the velocity and the pressure fields by a pressure correction technique. Two turbulencemodels can be applied that are based on k-ε, and k-ω-SST variants. The code may also be applied to solve independently the free surface flow following a surface-tracking process.

3 TEST CASES

In order to test the developed potential flow method, a first application for the well documentedSeries-60, c B=0.6 hull is presented in the sequel. Computations were carried out for two Froudenumbers, i.e. Fr=0.25 and 0.35. In both cases the grid was refined sequentially every 100 steps,

starting from 5000 panels to 25000 panels at the finest mesh. The computation domain extendedfrom – L PP to 3 L PP in the longitudinal dimension and 1.5 L PP on both sides of the model, where L PP =3.048m and x=0 coincides with FP. The calculated results for the wave profile on the hullare compared to the experimental data by Garofallidis (1996) in Figures 5a, b. In general, thecalculated profile follows the experimental one, except the region close to AP, where the poten-tial theory predicts higher values. The convergence history of the wave resistance coefficient C W

is presented in Figure 6, showing that after 300 steps C W has practically converged, while the procedure ended in 450 iterations where the dynamic condition is satisfied. The results for C W ,or (C R), presented in Tables 1&2, show that the potential flow predicts lower values than themeasured, which is an expected result since the viscous effects are ignored.

Figure 5a, b. Left (a): Comparison of calculated vs. experimental wave profiles at Fr=0.25 Right (a): Comparison of profiles at Fr=0.35 (Series-60 model)



Figure 6. Convergence history of C W

Table 1. Comparison between calculated and experimental values (S-60, Fr=0.25) ________________________________________________________________

C Rx103 sinkage(m) trim(deg)

_____________________________________________________________________________________________________

Experimental 0.8 0.0061 -0.12Calculated 0.45 0.0079 -0.07 ______________________________________________________________________________________________________

Table 2. Comparison between calculated and experimental values (S-60, Fr=0.35) _________________________________________________________________

C Rx103 sinkage(m) trim(deg)

______________________________________________________________________________________________________

Experimental 1.26 0.0142 -0.07Calculated 1.23 0.0148 -0.05 ______________________________________________________________________________________________________

Next, the method was applied for a 1:30 model of a passenger-ferry ship with transom stern

and L PP =115.5m. The model had initially a conventional bow and an implicit bulb was designedto improve its resistance. According to Figure 3, the main characteristics of the bulb at full scalewere: L B=4.9m, B B=1.62m, Z B=2.75m, T B=5m. The ship’s draft was equal to 5.2m and the de-sign speed 19 knots, corresponding to Fr=0.29. The resistance experiments were carried out atthe Laboratory for Ship and Marine Hydrodynamics ( LSMH ) of NTUA. Potential flow results

were obtained by applying the aforementioned grid refinement procedure and a total of 5000(hull)+22000(free-surface) elements were applied. Besides, computations were carried out with

an additive bulb that had equivalent parameters to the implicit. The integrated wave resistancecomponents are compared to the experimental C R in Table 3. Calculations do not include sink-

age and trim corrections. A first conclusion, derived from the experimental results, is that thedesigned bulb will not improve drastically the resistance of the ship, provided also that the non-dimensional coefficients are divided by the wetted surface which is higher when the bulb is

added. The potential flow predicts higher values in both cases, which shows a rather unexpected behaviour because the viscous pressure is not taken into account. Apart from any numerical

problems, probably due to the adopted quadrilateral panels, it should be noted here that a seriouswave-breaking and spray was observed during experiments, which was more intense with theconventional bow. Since the present method cannot take into account these phenomena, suchdifferences may be expected. However, it is noteworthy that the calculated results follow thecorrect trends, while the additive bulb shows better behaviour with respect to the implicit. Thewave profiles on the hull, plotted in Figure 7, show that both bulbs produce essentially the sameshape, which shifts the crest of the original hull towards FP. There are no substantial differences

at the stern region, where the flow separates at the transom, as shown ib Figure 8.



Table 3. Comparisons of CR between experiments and potential flow results ______________________________________________________________ C Rx10

3conventional bow implicit bulb additive bulb

__________________________________________________________________________________________________

Experimental 1.32 1.26Calculated 1.38 1.27 1.21 __________________________________________________________________________________________________

Figure 7. Calculated wave profiles on the hull. Figure 8. Free-surface around transom

The computed free-surface in all cases was, then, taken as a known boundary to apply the

RANS code underneath. The computations domain was divided in three blocks covering respec-tively the bow, the stern and the transom and wake regions. The k-ε model with wall functionswas adopted as the turbulence model, taking into account that ship had a very low block coeffi-cient. The external boundary conditions for the flow variables were calculated from the potentialflow solution as described by Tzabiras & Garofallidis (2005) and, therefore, a restricted domaincould be used for viscous calculations that require very fine grids to obtain reliable solutions. Amesh-refinement technique was applied (e.g. Tzabiras 1997) to perform grid dependence tests.Typical results are presented in Table 4 for the case of the implicit bulb, where the first numberdenotes the number of nodes in the longitudinal direction, the second refers to nodes girth-wiseand the last, nodes normal to a ship’s section. As observed, the results obtained with the finestgrid can be considered as reliable for practical conclusions.

Table 4. Grid dependence tests with the RANS solver ______________________________________________________________ Bow Grid 111x27x31 221x54x81 327x81x91C T x10

23.658 3.625 3.628

__________________________________________________________________________________________________

Stern & Transom 126x27x31 251x54x62 371x81x91C T x10

34.534 4.437 4.389

___________________________________________________________________________________________________

The calculated total resistance coefficient C T (non-dimensionialized by the wetted surface instill water in each case) is compared to the experimental values in Table 5. Although the resultsshow a surprisingly good agreement, it should be considered that the input free-surface in calcu-lations has two main drawbacks: first, it cannot take into account the wave breaking and, sec-

ond, the “inviscid” stern waves are normally higher than the real. Nevertheless, the comparisons



demonstrate that RANS computations are substantially more reliable than the initial potentialflow predictions of C R because they include all the viscous influences on the velocity and pres-sure fields. The benefit from the potential solution relies on the prediction of the free-surfacegeometry which seems to be satisfactory for taking decisions about the proper design of bulbous bows. In addition, a significant conclusion is that the additive bulb behaves just as the implicitone and may be a strong alternative due to the easier adjustment of bow lines. From both the

experiments and calculations, it is also obvious that the employed bulb does not reduce signifi-cantly the total resistance and, consequently, is not suggested for the real ship.

A final run was carried out for the full scale ship with the implicit bulb under the calculatedfree-surface. The computed total resistance coefficient was C T =2.485x10-3, while the viscous pressure coefficient C P =0.962x103. As expected, the latter is lower than the model scaleC P =1.23x10

3, owing to the reduction of the viscous pressure component. This result verifies that

the Froude hypothesis is not valid for the particular hull.

Table 5. Comparisons of C T between experiments and RANS calculations ______________________________________________________________ C Rx10

3conventional bow implicit bulb additive bulb

__________________________________________________________________________________________________

Experimental 4.529 4.444Calculated 4.475 4.470 4.451 __________________________________________________________________________________________________

CONCLUSIONS

The combined application of the potential and the viscous codes to predict the total resistance ofmodels or ships with bulbous bows seems to be a reliable tool in the design stage. The additive bulb produces the same results with an implicit bulb with the same geometrical parameters. RANS computations are needed to obtain reliable results, while they can be applied to full scalewithout any problem. Further comparisons with experimental data are needed to ascertain theefficiency of the particular procedure.

REFERENCES

Garofallidis, D.A. 1996. Experimental and numerical investigation of the flow around a shipmodel at various Froude numbers. Ph.D. Thesis, Department of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece.

Hess, J.L. and Smith A.M.O. 1968. Calculation of potential flow about arbitrary bodies, Prog. Aeraunaut. Sci., vol 8, 1-136

Janson, C.E. 1997. Potential flow panel methods for the calculation of free-surface flows withlift. Ph.D. Thesis, Department of Naval Architecture and Ocean Engineering, ChalmersTechnical University, Gothemburg, Sweden.

Kracht, A.M. 1978. Design of bulbous bows. SNAME TransactionsRaven, H.C. 1992. Nonlinear ship wave calculations with the RAPID method. 6th International

Conference on Numerical Ship Hydrodynamics, Iowa City, USATzabiras, G.D. 1997. A numerical study of additive bulb effects on the resistance and self-

propulsion characteristics of a full ship form. Ship Technology Research, vol. 44(1), 98-108Tzabiras, G.D. 2004. Resistance and self-propusion calculations for a Series-60, cB-0.6 hull at

model and full scale. Ship Technology Research, vol. 51(1), 21-34Tzabiras, G. And Garofallidis, D. 2005. Computation of the resistance of a Series-60, cB-0.6

model under a measured free-surface. Proc. Marine Transportation and Exploitation ofOcean and Coastal Resources, Lisbon, Portugal, 295-300

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A Method for Predicting the Influence of an Additive Bulb on Ship Resistance