The Effect of Transom on the Wave Resistance of a Ship S.J

Home , Transom (nautical)

The effect of Transom on the wave resistance of a ship

S.J. Watson & A.J. Musker

Defence Research Agency, Haslar, Gosport, f 0)22,4(7, #ampa

NOMENCLATURE

F Froude number =

g acceleration due to gravity L ship length

n' normal vector to hull surface S wetted surface of hull x,y,z Cartesian coordinates f wave elevation

v kinematic viscosity p fluid density velocity potential

D Cw wave resistance coefficient =

Re Reynolds number = RW wave resistance

Uoo ship speed

1. Introduction

Although considerable advance's in the theory of ship wave resistance have been made in recent decades, a good deal of this work is restricted to linearised theory and ideal hull shapes - with the consequence that there is always an element of doubt in the mind of the designer as to the reliability of such methods.

More recently a number of so-called Rankine source methods have

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appeared in the literature, following the pioneering work of Gadd [1] and Dawson [2]. The main advantage of this approach lies in its potential for satisfying more correctly the boundary conditions on the free surface. This is achieved by distributing a source density, of the simple Rankine type, over the whole boundary of the fluid domain rather than just the hull - as is done in methods which use Kelvin (or Michell-Havelock) sources. The source density distribution has to be truncated in practice since an infinite ocean cannot be modelled exactly in this way because of the finite computer memory available. Nevertheless, the numerical analyst is able to estimate such errors and Rankine source density models will soon become a powerful design tool.

An additional source of difficulty lies in the modelling of hull forms with transom sterns. Transom sterns are geometrically discontinuous and are used by the designer to achieve lower wave resistance for a given hull length and speed. The (viscous) flow separates cleanly beneath the transom (where the transom joins the keel - see Fig 1) but the position of the local free surface, in relation to the calm water-line, varies as a function of ship speed. At low speeds, a large recirculating flow is formed downstream of the transom similar in form to the classic 'backward-facing step' problem. At higher speeds, however, the free surface drops to the keel line and the transom becomes dry.

Clearly, in order to model the presence of a transom, some account of the recirculating region must be taken. In this respect, potential flow methods become deficient because of their restriction to irrotational flows. The complete Navier-Stokes problem, including the viscosity-generated turbulence and free-surface, is not considered here because of the complexity and cost. Instead, a hybrid technique is explored whereby the potential flow is established using a panel method under the assumption that the flow separates cleanly at the base of the transom regardless of ship speed. This is accomplished by applying a Kutta condition at the end of the keel. The wave resistance is calculated initially in accordance with the corresponding pressure field by simple integration over the wetted surface. A separate calculation is then performed in which the free surface at the stern is modelled using a viscous field method. For the purpose of this calculation only, the flow is regarded as being 'driven' by the external potential flow which is itself regarded as constant and unaffected by the transom.

In this way, the area of the transom which is wetted can be easily calculated and a correction to the potential flow wave resistance can be made in terms of the local hydrostatic head. In practice, this tends to lower the original estimate of resistance since the hydrostatic head at the transom has the effect of pushing the hull forwards.

This paper describes the calculations required to predict the resistance correction as a function of speed. The method is tested using the ATHENA

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hull as a benchmark [3]. For completeness, however, an overview of the potential flow method is also presented; a complete account of the theory has been presented by Musker [4].

2. Potential Flow Method

The fluid is assumed to be inviscid, incompressible, irrotational, free from surface tension effects and infinitely deep (the effect of finite depth could easily be added if required). A Cartesian coordinate system is used throughout the analysis with the xy plane in the calm free surface and the z axis positive downwards. The x axis is directed upstream.

A scalar velocity potential is defined such that -grad0 is the local fluid velocity vector. Laplace's equation applies everywhere within the fluid domain:

On the hull boundary the following kinematic condition must be satisfied:

(gradQ) -n* = 0 (2)

where n' (n^, n'y, n^) represents the outward normal vector on the hull surface. The hydrodynamic pressure within the fluid is given by Bernoulli's theorem:

p = pgz + -|p (ul - \grad$\*} (3)

where g is the acceleration due to gravity and U= is the ship speed defined such that

- grad cj> = (-U»,o,o) (4)

at oo except where waves are present. Applying equation (3) to the free surface, where the atmospheric pressure is assumed to be zero, leads to:

(5)

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where f represents the wave elevation measured with respect to the calm water plane (z=0). The kinematic condition on the free surface is:

_ dx

In addition, free surface disturbances must be propagated in the downstream direction. Finally, a Kutta condition is applied at the line where the transom joins the hull, such that the flow leaves the hull tangentially with the whole transom totally dry.

We aim to satisfy approximate forms of equations (5) and (6) on the calm water plane, z=0, with the aid of Maclaurin expansions of (5) and (6). A numerical solution to equation (1), subject to the above boundary conditions, can be achieved by distributing a source density on the boundary of the fluid domain.

The source distribution is made discrete by representing the hull and free surface by a large number of panels, each of which is associated with a constant density. Such a distribution automatically satisfies the mass continuity requirement, equation (1), and successive applications of the boundary conditions at a large number of control points leads to a system of non-linear algebraic equations embracing all the unknown source densities and wave elevations. A fuller account of the theory and associated algebra can be found in [4]. For the purpose of the present investigation only one iteration is performed, corresponding to linearised free-surface boundary conditions.

Once the source densities and wave elevations are known, the velocity and pressure fields may be calculated. The wave resistance is then:

where S is the wetted area of the hull. This integration is performed taking due account of both the computed dynamic water-line and the hydrostatic term associated with it (see equation (3)). In general it is impossible to discretise a hull which does not possess fore and aft symmetry such that the resistance is zero for double body flow (without waves). This arises because of the discontinuous nature of both the direction cosines and the computed pressure distribution. This residual numerical resistance is therefore quantified by a separate calculation and subtracted from the calculated wave resistance.

The hull panels are arranged to mesh exactly with the free surface nodes

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along the water-line. Below the water-line, panels are constructed automatically, using a mixture of quadrilateral and triangular panels, such as to allow the number of panels at a particular section to be reduced in the ratio of local section depth to keel depth. This procedure is implemented with the aid of the BLINES computer-aided design system [5].

3. Field Method for Transom Region

The full scale ATHENA hull is 47 m long - which, for a Froude number F = 0.35, corresponds to a full-scale Reynolds number of 2.4 x 10®. This is high by Navier-Stokes standards, however the boundary layer can be expected to be well-behaved, separating abruptly at the transom with little if any thickening. Accordingly, it was felt that although the under-lying mechanics are viscous and turbulent, the simple resistance correction due to the hydrostatic head could be estimated from the Euler or possibly laminar equations of motion.

From indirect experimental evidence, there are clear signs of spanwise variation in the waterline across the transom. However, for the purpose of this initial study, it was felt that an estimate of the resistance correction could be made from the extent to which the waterline is drawn down at the hull centre- plane, and that this draw-down need only be calculated using a two- dimensional calculation based on the hull centre-plane. In terms of a hull length L, the axial extent of the domain was set to 0.1L upstream of the transom and 1.5L downstream. The base of the domain is 0.2L below the mean waterline. The depth of submergence of the transom was 0.012L (see Fig

1).

The FIDAP code [6] was used for this part of the investigation. The equations of motion are reduced to discrete algebraic equations by a Petrov- Galerkin finite element methodology with mass continuity enforced via a pressure correction algorithm.

The motion of free surfaces are tracked using a technique developed by Saito and Scriven [7]. Free surfaces are defined by groups of nodes from the computational mesh which are then constrained to move along fixed pre- defined lines referred to as spines. Although this description of the free surface provides a simple description of free surfaces, it has limitations. Notably, extreme motions such as those leading up to and including wave-breaking cannot captured. Further complications arise when a free surface is intersected by curved hull surfaces. In this context, the computational mesh becomes progressively distorted as the Froude number is increased towards the value at which the transom becomes dry; in fact, so distorted that this value can only be predicted by extrapolation.

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The major parameters influencing the calculations are mesh density, Froude number, Reynolds number and the boundary condition imposed at the hull surface. The hull boundary condition comprised two types. It was possible to impose either zero velocity (termed no-slip) or zero normal velocity (termed free-slip). In this latter case, a Kutta condition in the form of the flow direction at the base of the transom must be specified. It was found that for stable calculations this direction had to be inclined by about 5% to the horizontal.

A structured computational mesh was used, consisting solely of quadrilateral elements with biquadratic and linear basis functions for velocity and pressure respectively. Two mesh densities were used, with 4000 nodes in the basic mesh and 16000 in the fine mesh. Note that both meshes were designed to resolve features local to the transom rather than to capture the wave-making of the hull. Indeed, far downstream of the transom, the elements were highly stretched axially and this tended to suppress disturbances at the outflow boundary.

Experiments suggest that the transom becomes dry at about F=0.35. Therefore, calculations were performed sequentially from Fr = 0.035 to Fr =

0.35 in increments of 0.035 with the results for each case being used as initial conditions for the next. Initial velocity conditions for the first case were generated by replacing the free surface by a free-slip rigid boundary.

The inlet velocity profile was generated by the potential flow method (described in the previous section) applied to the three-dimensional hull for a single Froude number; this profile was then scaled to provide inlet velocities at the other Froude numbers.

4. Results

As expected, the no-slip simulations developed relatively thick boundary layers between the inlet and the transom, irrespective of mesh density and

Reynolds number. For the free-slip condition, the draw-down increases compared with the laminar calculations (see Fig 2).

From the predicted draw-down, a resistance correction was generated by integrating the hydrostatic head over the transom wetted surface area, (analytically vertically, and by trapezoidal rule in the spanwise direction). Note that a linear variation in the free surface height was assumed, from zero draw- down at the beam to a maximum at the centre-plane. Note that the position of zero draw-down at the beam was matched to the local wave elevation - as predicted by the potential flow method described in Section 2. A comparison of non-dimensional wave resistance between prediction and experiment is given in Fig 3. The triangle symbols indicate the potential flow predictions and the solid symbol indicates the effect of including the correction from the field

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calculation. Unfortunately, the region of over-lap between experiment and prediction is confined to a single Froude number, F = 0.28. The correction is substantial and is extremely encouraging considering it was the result of a simple two-dimensional analysis.

5. Conclusions

This paper has described a method of calculating the wave resistance of a ship hull fitted with a transom stern. The basic wave resistance of the hull without the transom is calculated using a potential flow technique. These predictions are then corrected (by superposition) using results from a two- dimensional field calculation which is performed in the region local to the transom.

From these preliminary calculations, it is clear that the hydrostatic head on a wetted transom does lead to a substantial effect on the resistance of the hull at low Froude numbers. Initial results indicate that the approach adopted in this investigation is worthy of further investigation. In particular, more work is required to explore the feasibility of performing fully three-dimensional calculations at realistic Reynolds numbers.

6. Acknowledgements

The authors are indebted to Dr Y Wang (formerly of Liverpool University) for his help in preparing modifications to the computer code described in Reference [4] to take account of the Kutta condition.

7. References

1. Gadd, G.E. A Method of Computing the Flow and Surface Wave Pattern around Full Forms, Transactions of the Royal Institution of Naval Architects, Vol 118, 1976.

2. Dawson, C.W. A Practical Computer Method for Solving Ship-Wave Problems, Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, University of California, Berkeley, September 1977.

3. Bai, K.J. and McCarthy, J.H. ATHENA Model Group Discussion, Proceedings of the workshop on Ship Wave-Resistance Computations, DTMB, Bethesda, Maryland, USA. November 1979.

4. Musker, A.J. A Panel Method for Predicting Ship Wave Resistance, Proceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, August 1988.

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5. Catley, D., Okan, M.B. & Whittle C., Unique Mathematical Definition of a Hull Surface, its Manipulation and Interrogation, WEMT, Paris, July 1984.

6. Engelman, M.S., FIDAP Users Manual Revision 6.0 Volume 1, Fluid Dynamics International Inc. April 1991.

7. Saito, H. and Scriven, L.E., Study of coating flow by the Finite Element Method. Journal of Computational Physics 42 p53. 1981.

Published with the permisiion of the Controller of Her Britannic Majesty's Stationery Office

0.1 L 1.55 L

Transom Calm waterline Hull

0.3L

Figure 1 Computational Domain

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100

a o 80- cd cd M 0 no slip free slip 60-

I- 8

0.0 0.07 0.14 0.21 0.28 0.35

Froude number

Figure 2 Comparison of Draw-down for

Various Hull Boundary Conditions

o 2.4 i — Experiment (fixed) ^ Computed •t srH CQ 2.0- # Corrected computed 2

1 1.6-

cd 1.2-

0.8- 3 0.4-

0.0 T T "1 0.25 0.35 0.45 0.55 0.65 0.75

Froude number

Figure 3 Wave Pattern Resistance