Doctors - A Numerical Study of the Resistance of Transom-stern Monohulls

8/13/2019 Doctors - A Numerical Study of the Resistance of Transom-stern Monohulls

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5thInternational Conference

on High Performance Marine Vehicles,

8-10 November, 2006, Australia

A Numerical Study of the Resistance

of Transom-Stern Monohulls

Lawrence J. Doctors

The University of New South Wales, Sydney, NSW 2052, Australia

[email protected] Volker Bertram

Abstract

Previous experimental data for the wave profiles behind a series of transom-stern ship models has

been re-analyzed in this paper. Convenient and accurate regression formulas for the ventilation of the

stern and the length of the transom hollow are provided here.

Predictions of the wave resistance and, hence, the total resistance can now be improved by using this

approach, together with the traditional thin-ship formulation for resistance. However, the accuracy of

these predictions is improved further if one also considers that there must be a minimal effective

hollow length based on the flow behind the well-known backward-facing step.

These theories, together with the use of form factors, are applied to a series of fourteen high-speed

monohull vessels. It is shown that these methods provide a high degree of predictive accuracy.

1. Introduction1.1 Previous Work

A large proportion of the high-speed vessels currently being constructed or being used in passengerservice share the characteristic of possessing a cut-off, or transom, stern. This feature of the vessel

defies simple hydrodynamic analysis because the extent and detailed shape of the hollow cavity in thefree surface created behind the vessel are unknown. They must be part of the mathematical solution tothe problem. In principle, it is known that the pressure acting on the surface of the hollow must be

atmospheric, and that the flow must separate from the transom tangentially. A proper analysis of the

problem would require a fully three-dimensional treatment in which it would be necessary to iterate

the geometric shape of this hollow, until the relevant kinematic and dynamic conditions weresatisfied.

This is no easy task, given that some aspects of the geometry are highly nonlinear (and perhaps even

intractable) within the framework of potential-flow theory. In particular, the closure of the hollow,

which seems always to be accompanied by a considerable amount of spray and unsteadiness in theflow, appears to be a formidable problem in fluid mechanics. This region of the flow is usuallyreferred to as a rooster tail because of the shape of the spray thrown into the air.

A small number of researchers has addressed this question in an attempt to gain some understanding

of the flow. An early paper was written by Milgram (1969), in which the wake behind a vessel wasmodelled as a region filled with deadwater. The flow was assumed to separate in various ways nearthe stern and, in the more refined version of his approach, the separation was taken to occur

tangentially to the hull surface. The traditional thin-ship theory of Michell (1898) was applied to theresulting hypothetical body" encompassing both the hull itself and the wake region. A similar

approach was taken by Doctors and Day (1997) for the prediction of the resistance of transom-sternvessels at high speeds, in which it was assumed that the hollow extended to infinity, and a hydrostatic

correction was employed to the total resistance.

Another study, published by Tulin and Hsu (1986), specifically addressed the stern flow and the

appropriate method of applying the abovementioned condition of constant atmospheric pressure onthe surface of the hollow. The comment was made that the traditional wave- resistance problem may


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Figure 1: Definition of the Problem Figure 1: Definition of the Problem(a) Principal Features (b) Fitting Mesh to the Vessel

not be properly posed and that the resulting computed wave resistance may not be unique in value! In

their theory, the Kutta condition was essentially applied at the girth of the transom stern. The dilemmaof the analysis of the rooster tail was apparently side-stepped by limiting the theory to a very high

Froude number F = gLU/ , in which U is the speed of the vessel, g is the acceleration due to

gravity, and L is the wetted length of the vessel when at rest. In this limiting case, the rooster tailwould be located far behind the vessel. Good agreement with experimental measurements ofresistance of Series 64 vessels was demonstrated.

Molland, Wellicome, and Couser (1994b) tested dierent types of sink models in order to close thewater flow behind the vessel. In this approach, one must ensure that the total source strength, for both

the vessel itself and the hollow region, should be zero. They felt that their results were not aspromising as they had hoped. In particular, they wished to be able to predict the resistance at lower

Froude numbers. Consequently, they next employed in their research what one might call an

engineering model of the transom hollow. In this approach, an experimentally determined length-to-

width ratio for the hollow was assumed and the hollow was then incorporated into the stern of thevessel. Further results on this and related research have been presented by Insel and Molland (1991),Molland, Wellicome, and Couser (1994a and 1995), and Couser (1996).

The empirical correction of linearized wave-resistance theory was studied in a series of collaborative

papers by Doctors, Renilson, Parker, and Hornsby (1991), Doctors and Renilson (1992 and 1993), andDoctors (1995a and 1995b). In these studies, both catamarans and monohulls were tested in a towingtank in water of various depths. Attempts to correlate the experimental results for the resistance with

the linearized theory were then made. It was found that the theory could be used quite accurately to

predict the effects of changes in the water depth or the spacing between the demihulls of a catamaran.In addition, the influences of sloping river banks could also be included in an approximate manner.

However, a different correction factor was required at each speed, limiting the utility of the approach

to some extent.

Mention should be made here of the work of Hanhirova, Rintala, and Karppinen (1995). These

authors also fitted a Michell-type formula to a set of experimental results. They performed a

regression analysis on thedifferencebetween such a prediction and the measurements.

A more sophisticated approach to estimating the length of the transom-stern hollow was proposed byDoctors and Day (1997) using their fire-hose model. In this approach, the length of the hollow was

computed by a technique which allowed it to increase in a realistic manner with the speed, unlike thefixed-length concept of Molland and his coworkers cited above.

However, there was no confirmation that the length of the hollow was indeed correct, despite the fact

that the predictions of resistance were in very good agreement with model experimental data. The


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Table 1: Coefficients of Transom-Ventilation Equation

Number of

Coefficients

Regression CoefficientsType of

Analysis

Nfit C1 C2 C3 C4

Static 2 3 4 0.1570

0.15590.002472

1.835

1.8301.862

0.01580

0.2859

0.3588

Dynamic 2 3 4 0.17670.1795

0.004856

1.7741.786

1.821

-0.035660.1990

0.3126

Table 2: Coefficients of Hollow-Length Equation

Number ofCoefficients

Regression CoefficientsType ofAnalysis

Nfit C1 C2 C3 C4

Static 2 3 4 0.1135

0.094090.6095

3.025

2.8392.733

0.4603

0.3468

-0.1514

Dynamic 2 3 4 0.06209

0.055980.2491

3.276

3.1793.107

0.2507

0.1598

-0.1225

transom was assumed to be fully ventilated at all speeds, which meant that the transom-stern drag,

associated with the missing hydrostatic pressure on the stern, was too high at low speeds.

1.2 Current Work

Subsequent to this initial effort, Robards and Doctors (2003) presented the outcome of a very

extensive set of tests on a series of five ship models possessing a rectangular transom, and ballasted tofive different drafts. This led to 25 combinations of geometry. The transom ventilation and the

centreplane profile of the wave elevation were both measured for a range of speeds. This permitted a

more scientific estimate of the desired parameters | the degree of transom ventilation and the transom-

hollow length. This data has been reanalyzed a number of times; by Doctors (2003), Doctors andBeck (2005), and Doctors (2006a and 2006b).

The last publication forms the basis of the current project. In that paper, the analysis of the transom-stern data yielded two regression formulas. The rst formula is for the ventilated or dry" proportion

of the transom, which is expressed as follows:

4

3

2

1

*C

NT

C

C

Tdry RT

BFC

T

==

(1)

Here, FT = gTU/ is the transom-draft Froude number, B/T is the beam-to-draft ratio, and RNT

= /3gT is the Reynolds number. The other symbols are the transom draft T and the kinematic

viscosity of the water

A second regression formula, of the same type, was derived for the \apparent" dimensionless hollow

length, as follows:


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4

3

2

1

*C

NT

C

C

T RT

BFC

T

L

= (2)

Figure 2: Pictorial View of Ship Models Figure 2: Pictorial View of Ship Models

(a) Model 1 (b) Model 2

Figure 2: Pictorial View of Ship Models Figure 2: Pictorial View of Ship Models(c) Model 3 (d) Model 4

The supposed geometry of the flow is presented in Figure 1(a), in which the symbols are alsoindicated. The adjective apparent" is emphasized above, because the formula is only suited to theshape of the measured free surface. Because of the presence of the stagnant water in the hollow at low

speeds, it is reasonable to assume that a minimum effective wavemaking length should be based onincluding the entire stagnant-water region.

Figure 2: Pictorial View of Ship Models Figure 2: Pictorial View of Ship Models(e) Model 5 (f) Model 6


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The derived coefficients in Equation (1) and Equation (2) are presented in Table 1 and Table 2,

respectively. In both cases, two sets of coefficients are listed, based on two methods of analysis. Thefirst method, referred to as Static uses the at-rest value of the transom-stern draft in order to effect the

calculations and the subsequent prediction of the resistance. The second method, referred to asDynamic, employs the underway transom draft instead. It can be observed that the ventilation varies

approximately with the square of the speed while the apparent hollow length varies approximately

with the cube of the speed.

The purpose of the present research is to apply this approach to a practical series of ship hulls. In this

way, we are continuing the work of Sahoo, Doctors, and Renilson (1999) for monohulls, Sahoo and

Doctors (2003) for the effects of restricted depth, and Sahoo, Doctors, and Pretlove (2006) for astaggered-demihull catamaran.

2. Experiments in Towing Tank

2.1 AMECRC High-Speed Monohulls

The models were tested in the towing tank at the Australian Maritime College (AMC) in

Launceston, Tasmania. The principal dimensions of the towing tank are presented in Table 3.The depth of the water, namely 1.500 m, was sufficient for the depth Froude number

gd

UFd = to be less than unity over most of the speed range of interest.

Table 3: Particulars of Towing Tank

Table 4: AMECRC Systematic Series

The principal geometric characteristics of the fourteen ship models, constituting the

AMECRC high-speed monohull series, appear in Table 4. The table lists the displacement ,

the length-to-beam ratio L/B, the beam-to-draft ratio B/T, the block coefficient CB, the

prismatic coefficient CP, and the slenderness coefficient L/1/3

. The models have been


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described in some detail by Bojovic (1995a and 1995b) and Bojovic and Goetz (1996). The

models all possess a waterline length L of 1.6 m. Six of the fourteen models are presented in

the six parts of Figure 2.

These hulls were chosen as being representative of high-speed patrol vessels. However, it

may be noted that some of the dimensions, such as the beam-to-length ratio are large for this

application.

2.2 Series of Investigations

The fourteen model vessels were all tested in relatively deep water up to a length Froude number ofunity. Additionally, some finite-water-depth tests were conducted. As reported by Bojovic (1996),

some variations in the displacement and the longitudinal centre of gravity LCG were also studied.

The tests were conducted in the towing tank at the Australian Maritime College during the period1994 to 1996.

3 Numerical Approach

3.1 Mesh Refinement

Figure 3 is employed here to illustrate the relevance of the appropriate computational mesh to be usedin the evaluation of the wave-resistance component. As seen in Figure 1(b), the hull is gridded

longitudinally with Nxpanels and the hull is gridded vertically with Nzpanels. For the six vesselspresented here, it can be observed from the plots that a mesh of 60 x 20 is more than adequate to

represent the hull accurately.

Figure 3: Refinement of Mesh (a) Model 1 Figure 3: Refinement of Mesh (b) Model 2

Figure 3: Refinement of Mesh (c) Model 3 Figure 3: Refinement of Mesh (d) Model 4


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While there are some relatively large differences at low Froude numbers, when a coarser mesh is

employed, it must be borne in mind that these are plots of the coefficient of wave resistance, defined

by

SU

RC WW

2

2

1

= , in which RW is the wave resistance, is the water density, and S is the wetted

surface of the vessel. Thus, the absolute error in the wave resistance is seen to be unimportant at theselow Froude numbers.

Figure 3: Refinement of Mesh (e) Model 5 Figure 3: Refinement of Mesh (f) Model 6

The theoretical curves in the plots of Figure 3 all display a step reduction in resistance at a Froude

number of 0.9682. This point in the speed range corresponds to a depth Froude number of unity. Thisdrop is related to the loss of the transverse-wave component at this speed. The phenomenon is less

significant as the width of the channel is increased.

3.2 Resistance Components

The resistance components for the six chosen vessels are given in Figure 4. The total resistance RT,the wave resistanceRW, the hydrostatic (or transom-stern) resistance RH, and the frictional resistance

RFare all rendered dimensionless by the vessel weight W. The frictional resistance is computed usingthe ITTC 1957 correlation line described by Lewis (1988). The experimental data appears as a series

of symbols.

Figure 4: Resistance Components (a) Model 1 Figure 4: Resistance Components (b) Model 2

It is most encouraging to see the excellent correlation between the predictions for the total resistance,which is effected as a simple sum of the components, with the experimental data. This is true, even for

the relatively fat Model 5.


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4. Theoretical Model for Transom-Stern Flow

We now turn to Figure 5, in which different theoretical models for the transom-stern flow arecompared for accuracy. Table 5 should be consulted for the explanation of the two codes describing

the theory for the transom-water-level drop and the transom-hollow length, respectively.

Figure 4: Resistance Components (c) Model 3 Figure 4: Resistance Components (d) Model 4

Figure 4: Resistance Components (e) Model 5 Figure 4: Resistance Components (f) Model 6

In all the cases depicted here, there is an improvement in predictive accuracy, if (a) one uses a fittedregression curve to model the water drop rather than assuming the transom is fully ventilated, (b) the

regression is based on coefficients using the static transom dimensions rather than the dynamictransom dimensions, and (c) the backward-facing-step reattachment length is used to place a lower

limit on the effective transom-hollow length.

Figure 5: Transom-Stern Flow Model

(a) Model 1


(b) Model 2


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In addition to these comments, the application of a wave-resistance form factor fWof 0.8928 and a

frictional-resistance form factor of 1.192, in the equation for total resistance, namely:

RT = fWRW+RH+fFRF, (3)

generally increases the accuracy of the prediction. These two factors were obtained through a root-mean-square regression analysis of all the 28 test conditions for the AMECRC series.


(c) Model 3


(d) Model 4


(e) Model 5


(f) Model 6

5. Variation of Test Conditions

5.1 Change in Displacement

As an additional challenge, we consider the effect of changing the displacement, with zero trim, oftwo of the vessels (Model 6 and Model 9), in the two parts of Figure 6. In these and thesubsequent

graphs, the difference in specific total resistance RT/W is plotted. The difference is calculated relative

to the 100% load case at zero trim.

In a general sense, it can be argued that the methodology works to predict these changes in Figure6(a). The estimates for the effect of load change are slightly overpredicted when the load is increased

beyond 100%. Of course, it must be remembered that the main effect of change in displacement

(either -10%, +10%, or +20%) is accounted for by the form of the chosen plotted parameter. That is,the method of plotting greatly exaggerates any errors. In truth, all of the predictions for the total

resistance are accurate to within one or two percent in an absolute sense.


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The predictions for these small relative changes is slightly worse for Model 9 in Figure 6(b).

Table 5: Theories for the Transom-Stern Hollow

5.2 Change in Trim

Finally, the influence of change in the bow-down trim angle is depicted in the two parts of Figure 7,

for the same two models. This exercise poses an even greater challenge than the previous exercise inwhich the displacement was changed.

Figure 6: Change in Displacement (a) Model 6 Figure 6: Change in Displacement (b) Model 9

Figure 7: Change in Displacement (a) Model 6 Figure 7: Change in Displacement (b) Model 9


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Nevertheless, the numerical values of the predictions are more than acceptable for a Froude numberless than about 0.35. Also, in a general sense, the ordering of the predictions matches that of the

experimental data at high values of the Froude number.

6. Conclusions

This research has demonstrated that the enhanced models for the water flow behind the transom leadto much more accurate estimates of the transom hydrostatic drag and the wave resistance. As a

consequence, the total resistance is predicted with much better accuracy. In particular, it can be notedthat the correlation of theory and experiment is now exceptionally good for the difficult low-speed

region of the resistance curve.A start to determining suitable form factors for the resistance components demonstrates that even

further improvements to the accuracy of the theory are achievable. It is seen that it is advisable to

choose a wave-resistance form factor less than unity and a frictional-resistance form factor greaterthan unity. These statements are in accord with previous, preliminary, work on this matter. The need

for a frictional-resistance form factor greater than unity has also been reported by other researchers.

It is recommended that further effort be invested in clarifying the proper values for the two formfactors. It is suggested that these factors should depend on the actual vessel geometry.

7. Acknowledgments

The tests for the ventilation of the transom stern were performed in the Towing Tank at the Australian

Maritime College by Mr Simon Robards, postgraduate student at The University of New South Wales(UNSW), under the able supervision of Mr Gregor Macfarlane of the AMC. The initial analysis of the

ventilation process behind the transom stern was done by Mr Robards and was reported in the paper

by Robards and Doctors (2003).

The authors would like to acknowledge the assistance of the Australian Research Council (ARC)

Discovery-Project Grant Scheme (via Grant Number DP0209656). This work was also partially

supported by Office of Naval Research (ONR) contract N00014-04-1-0266 as well as by contractN00014-05-1-0875.

The in-kind support of this work from UNSW is also cheerfully acknowledged.

8. References

Bojovic, P.: AMECRC Systematic Series Calm Water Testing Results, Australian Maritime

Engineering Cooperative Research Centre, Report AMECRC IR 95/5, 93+vi pp (1995)

Bojovic, P.: Presentation of the AMECRC Systematic Series Calm Water Testing Results Project

1.1.2, Australian Maritime Engineering Cooperative Research Centre, Report AMECRC IR 95/10,

41+vii pp (1995)

Bojovic, P.: Influence of Displacement and LCG Variation on Calm Water Resistance, Australian

Maritime Engineering Cooperative Research Centre, Report AMECRC IR 96/4, 60+vi pp (1996)

Bojovic, P. and Goetz, G.: Geometry of AMECRC Systematic Series, Australian MaritimeEngineering Cooperative Research Centre, Report AMECRC IR 96/6, 53+v pp (1996)

Couser, P.R.: An Investigation into the Performance of High-Speed Catamarans in Calm Water and

Waves. Chapter 5: Theoretical Calculation of Calm Water Resistance, University of Southampton,Department of Ship Science, Doctoral thesis, pp 91122 (March 1996)


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Doctors, L.J.: The Prediction of Ship Resistance by a Blended Theoretical and ExperimentalMethod, Proc. Seventh International Conference on Computational Methods and Experimental

Measurements (CMEM 95), Capri, Italy, pp 413422 (May 1995)

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of Naval Architects of Korea, Seoul, Korea, Vol. 1, pp 648659 (September 1995)

Doctors, L.J.: Hydrodynamics of the Flow behind a Transom Stern, Proc. Twenty-Ninth Israel

Conference on Mechanical Engineering, Paper 20-1, Technion, Haifa, Israel, 11 pp (May 2003)

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(September 2005)

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Resistance of a Catamaran, Proc. Second International Conference on Fast Sea Transportation(FAST 93), Yokohama, Japan, Vol. 2, pp 12311244 (December 1993)

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Speed River Catamaran, Proc. First International Conference on Fast Sea Transportation (FAST

91), Norwegian Institute of Technology, Trondheim, Norway, Vol. 1, pp 3552 (June 1991)

Hanhirova, K., Rintala, S., and Karppinen, T.: Preliminary Resistance Prediction Method for FastMono- and Multihull Vessels, Proc. International Symposium on HighSpeed Vessels for Transport

and Defence, Royal Institution of Naval Architects, London, England, pp 6-16-17 (November 1995)

Insel, M. and Molland, A.F.: An Investigation into the Resistance Components of High SpeedDisplacement Catamarans, Trans. Royal Institution of Naval Architects, 11 pp, Discussion: 9 pp

(April 1991)

Lewis, E.V. (Ed.): Principles of Naval Architecture: Volume II. Resistance, Propulsion and Vibration,Society of Naval Architects and Marine Engineers, Jersey City, New Jersey, 327+vi pp (1988)

Michell, J.H.: The Wave Resistance of a Ship, Philosophical Magazine, London, Series 5, Vol. 45,pp 106123 (1898)

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No. 1, pp 6971 (March 1969)


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Molland, A.F., Wellicome, J.F., and Couser, P.R.: Resistance Experiments on a Systematic Series of

High Speed Displacement Catamaran Forms: Variation of Length-Displacement Ratio and Breadth-Draught Ratio, University of Southampton, Department of Ship Science, Report 71, 82+i pp (March

1994)

Molland, A.F., Wellicome, J.F., and Couser, P.R.: Theoretical Prediction of the Wave Resistance of

Slender Hull Forms in Catamaran Configurations, University of Southampton, Department of ShipScience, Report 72, 24+i pp (March 1994)Molland, A.F., Wellicome, J.F., and Couser, P.R.: Resistance Experiments on a Systematic Series of

High Speed Displacement Catamaran Forms: Variation of Length-Displacement Ratio and Breadth-

Draught Ratio, Trans. Royal Institution of Naval Architects, 14 pp (November 1995)

Robards, S. and Doctors, L.J.: Transom-Hollow Prediction for High-Speed Displacement Vessels,Proc. Seventh International Conference on Fast Sea Transportation (FAST 03), Ischia, Italy, Vol. 1,

pp A1.19A1.26 (October 2003)

Sahoo, P.K. and Doctors, L.J.: A Study on Wave Resistance of High-Speed Displacement HullForms in Restricted Depth, Proc. Seventh International Conference on Fast Sea Transportation

(FAST 03), Ischia, Italy, Vol. 1, pp A3.25A3.32 (October 2003)

Sahoo, P.K., Doctors, L.J., and Pretlove, L.: CFD Prediction of the Wave Resistance of a Catamaranwith Staggered Demihulls, Proc. International Conference in Marine Hydrodynamics (MAHY 06),

Visakhapatnam, India, pp 867879 (January 2006)

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Resistance of High-Speed Round-Bilge Hull Forms, Proc. Fifth International Conference on FastSea Transportation (FAST 99), Seattle, Washington, pp 803814 (AugustSeptember 1999)

Tulin, M.P. and Hsu, C.C.: Theory of High-Speed Displacement Ships with Transom Sterns,J. ShipResearch, Vol. 30, No. 3, pp 186193 (September 1986)

Doctors - A Numerical Study of the Resistance of Transom-stern Monohulls

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