L.Larsson,^{1,2} B.Regnström,^{2} L.Broberg,^{2} D.-Q.Li,^{1,3} C.-E.Janson^{2}

(^{1}Chalmers University of Technology, ^{2}FLOWTECH International AB, ^{3}SSPA Maritime Consulting AB, Sweden)

The status of CFD in hydrodynamics is reviewed. After a brief historical survey a theoretical introduction is given to potential flow panel methods and methods based on the Reynolds-Averaged Navier-Stokes equations. Present capabilities are then discussed, and the attainable accuracy in the prediction of waves, wave resistance, local flow, viscous resistance and propeller effects is assessed. Prospects for improvements, based on present research in grid generation, free surface flows, turbulence modelling, propeller flows and full scale predictions are outlined. Finally, the exciting long term possibilities are presented and a number of conclusions drawn.

The modern research in computational hydrodynamics is intimately linked to the development of the computer, and methods based on massive “number crunching” have replaced the more elegant, but less general, analytical methods developed during the first half of this century. One of the first representatives of the new approach is the Hess and Smith method (1), presented in 1962. For the first time it became possible to compute the 3-D potential flow around arbitrarily shaped bodies with an exact boundary condition applied on the hull surface. The method was later generalized to include lift, certainly important for aerodynamic flows, but of more hydrodynamic interest was the introduction of the free surface by Dawson in 1977 (2). Dawson’s method, although based on a linearization of the free surface boundary conditions, soon became an important tool in ship design, and it was the starting point for research at many organizations.

One of the main objectives of the research was to remove the free surface linearization and to introduce exact boundary conditions satisfied at the exact location of the wavy free surface. In a series of PhD projects at Chalmers University of Technology this route was followed, and in 1989 a paper was presented (Larsson, et al (3)), where the major results of the first three projects were presented. Recently, the fourth project was finished by Janson (4) and a robust non-linear method based on this research is now available in the commercial CFD package SHIPFLOW (5). A parallel development was carried out at MARIN, Holland, and in 1996 Raven (6) presented a non-linear method of a similar kind. In the opinion of the authors this development is now close to its limits. Raven’s and Janson’s theses work, based on all previous efforts in the same area, have led to methods which cannot be much improved under the potential flow approximation. To proceed further, viscosity has to be taken into account.

In the sixties the viscous flow research was directed towards 2-D boundary layer theory and by the end of the decade several methods for arbitrary pressure gradients were available. This research continued for 3-D methods during the seventies and an evaluation was made at a workshop organized by one of the authors in Gothenburg in 1980, see Larsson (7). Like in a previous workshop on wave resistance held in the Washington DC in 1979 a number of methods were applied to some well specified test cases and the differences were analysed in view of the differences in the

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Failures, Fantasies, and Feats in the Theoretical/Numerical Prediction of Ship Performance
L.Larsson,1,2 B.Regnström,2 L.Broberg,2 D.-Q.Li,1,3 C.-E.Janson2
(1Chalmers University of Technology, 2FLOWTECH International AB, 3SSPA Maritime Consulting AB, Sweden)
Abstract
The status of CFD in hydrodynamics is reviewed. After a brief historical survey a theoretical introduction is given to potential flow panel methods and methods based on the Reynolds-Averaged Navier-Stokes equations. Present capabilities are then discussed, and the attainable accuracy in the prediction of waves, wave resistance, local flow, viscous resistance and propeller effects is assessed. Prospects for improvements, based on present research in grid generation, free surface flows, turbulence modelling, propeller flows and full scale predictions are outlined. Finally, the exciting long term possibilities are presented and a number of conclusions drawn.
1. Background
The modern research in computational hydrodynamics is intimately linked to the development of the computer, and methods based on massive “number crunching” have replaced the more elegant, but less general, analytical methods developed during the first half of this century. One of the first representatives of the new approach is the Hess and Smith method (1), presented in 1962. For the first time it became possible to compute the 3-D potential flow around arbitrarily shaped bodies with an exact boundary condition applied on the hull surface. The method was later generalized to include lift, certainly important for aerodynamic flows, but of more hydrodynamic interest was the introduction of the free surface by Dawson in 1977 (2). Dawson’s method, although based on a linearization of the free surface boundary conditions, soon became an important tool in ship design, and it was the starting point for research at many organizations.
One of the main objectives of the research was to remove the free surface linearization and to introduce exact boundary conditions satisfied at the exact location of the wavy free surface. In a series of PhD projects at Chalmers University of Technology this route was followed, and in 1989 a paper was presented (Larsson, et al (3)), where the major results of the first three projects were presented. Recently, the fourth project was finished by Janson (4) and a robust non-linear method based on this research is now available in the commercial CFD package SHIPFLOW (5). A parallel development was carried out at MARIN, Holland, and in 1996 Raven (6) presented a non-linear method of a similar kind. In the opinion of the authors this development is now close to its limits. Raven’s and Janson’s theses work, based on all previous efforts in the same area, have led to methods which cannot be much improved under the potential flow approximation. To proceed further, viscosity has to be taken into account.
In the sixties the viscous flow research was directed towards 2-D boundary layer theory and by the end of the decade several methods for arbitrary pressure gradients were available. This research continued for 3-D methods during the seventies and an evaluation was made at a workshop organized by one of the authors in Gothenburg in 1980, see Larsson (7). Like in a previous workshop on wave resistance held in the Washington DC in 1979 a number of methods were applied to some well specified test cases and the differences were analysed in view of the differences in the

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underlying theory. It turned out that most methods produced acceptable results for the major part of the boundary layer along the hull, while all failed completely near the stern. This prompted several groups to start research on less approximate methods based on the Reynolds-Averaged Navier-Stokes (RANS) equations. By the end of the eighties a number of RANS methods for ship flows had been developed and in a new Gothenburg workshop (Larsson et al (8)), a considerable improvement was noted in the capability to predict the stern flow. This improvement was verified at a new workshop organized by Kodama et al (9) in Tokyo in 1994, where, however, the most important breakthrough was the introduction of the free surface in the RANS methods. No less than 10 methods now featured this capability; a result of the research in several groups since the mid-eighties, as will be further discussed below.
There is no doubt that the most important developments to be expected in the future will be in the viscous flow area. With the increasing power of computers more advanced methods will become available, based either on the Large Eddy Simulation (LES) technique, where the large turbulent eddies are computed and only the smaller ones modelled, or on the DNS technique, where all eddies and scales are computed.
In the following, we will first give a brief theoretical introduction to potential flow panel methods and methods of the RANS type. This will be in Section 2. Thereafter, in Section 3 an assessment will be made of current CFD capabilities. Ongoing research and short term prospects for improvements will be addressed in Section 4, and in Section 5 a forecast will be made of the exciting long term possibilities. Finally, a number of conclusions will be drawn. The emphasis will be on resistance and powering, but some reference will be made to research in other areas of ship hydrodynamics as well.
2. Theory
In this Section a short theoretical introduction will be given, covering potential flow panel methods and Reynolds-Averaged Navier-Stokes methods. For obvious reasons the description has to be brief. This holds particularly for the RANS part, which will be further discussed in later Sections.
2.1 Potential Flow Panel Methods
The flow is assumed steady, irrotational and incompressible. A Cartesian coordinate system is employed, where the origin is at the bow and at the undisturbed free surface level, x downstream, y to starboard and z upwards. U and q represent the undisturbed and disturbed velocities, respectively. Defining a velocity potential by
(1)
this potential will satisfy the Laplace equation
(2)
On the hull boundary the normal velocity is zero
(3)
and on the free surface boundary a similar relation holds. This kinematic condition may be written as
(4)
where h=h(x,y) is the equation for the wavy surface.
A dynamic free surface condition may be obtained from the continuity of stresses across the interface. In a potential flow, this condition degenerates to the simple statement that the pressure must be atmospheric at the surface, and without loss of generality this pressure may be set to zero. Neglecting surface tension and applying the Bernoulli equation the dynamic free surface boundary condition may be written
(5)
At infinity the velocity is undisturbed
(6)
g is the acceleration of gravity and R is the distance from the origin.
Finally, the radiation condition states that no below, this condition is enforced numerically. waves may be transmitted upstream. As explained
The free surface boundary conditions are nonlinear and they have to be applied at an initially unknown surface. This may be accomplished as follows. Assume that an approximate solution is known. Introduce small disturbances and δh and expand equations (4) and (5) in these quantities around the known solution. Delete terms of higher order. This yields, (see Janson (4))
(7)

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(8)
where has been replaced by . In thin ship theory the known solution is taken as Φ=Ux, ho=0, i.e. the undisturbed flow and a flat free surface. Dawson used Φ=ΦDΜ, ho=0, where ΦDM stands for the double model solution. In SHIPFLOW (3), (4), (5), and ho,n+1=hn, where n is the iteration number in an iterative sequence, which starts with the double model solution, as in Dawson’s method. When the process converges the neglected quantities in equations (7) and (8) go to zero and the position where the free surface boundary conditions are applied approaches the correct one. In the limit, the exact solution is thus obtained. Note that very non-linear effects, like spray and wave breaking cannot be predicted.
The problem posed by the field equation (2) and the boundary conditions (3), (6), (7) and (8) is solved by a boundary integral technique where sources are distributed on the hull and part of the free surface. Optionally, the free surface sources may be raised a certain distance above the surface (desingularisation, (4) (6)). The potential at any point due to the distribution of sources is
(9)
where σ is the source density, r is the distance from a point on the surface to the field point and S is the hull and free surface (covered part). Note that the sources satisfy both the field equation (2) and the infinity condition (6) for all σ. The solution thus has to be found from the three remaining boundary conditions (3), (7) and (8).
The problem is discretized by representing the hull and the (possibly raised) free surface by quadrilateral panels, each of which having an individual distribution of sources. In the SHIPFLOW code (3), (4), (5) two options exist: either the panels are assumed flat and the source strength constant on each panel (first order method), or the panels are parabolic with a linear variation in the source strength (second order method). In both cases the number of unknowns is equal to the number of panels. Since the boundary condition is applied at one point (the collocation point) on each panel the problem is closed.
A final free surface boundary condition is obtained by solving (8) for δh, differentiating this relation with respect to x and y and introducing all three quantities in (7). Second derivatives of the unknown potential will then appear and these may be handled in two principally different ways. Either (9) may be differentiated twice to give analytical expressions for the derivatives or first order derivatives in (i.e. velocities) may be differentiated numerically by finite differences on the free surface. The latter technique is Dawson’s approach, which has the advantage that the radiation condition is satisfied automatically (2). If the analytical approach is used the radiation condition has to be satisfied by an upstream shift of collocation points, see e.g. Jensen (10), for a discussion.
Upon introduction of the first and second derivatives of the potential in equation (3) and the combined free surface condition these become linear equations in the source strength (source strength at the collocation point in the second order approach). The system of equations is solved either directly by Gaussian elimination, or by an iterative technique. Note that the diagonal dominance of the matrix, known from aerodynamic methods, is lost in the block related to the free surface.
Having solved for the source strength, velocity components may be obtained from derivatives of (9). Pressures may then be computed using the Bernoulli equation. To obtain forces on the hull the pressure is integrated over the surface. In SHIPFLOW, this can be done in two ways: either by assuming the pressure to be constant on each panel and to act in the negative normal direction, or by using a second order technique, where the pressure varies linearly on each panel, the normal has a varying direction and the integration is carried out over the curved panel surface. It should be mentioned that lifting flows may be computed as well, by distributing dipoles on the lifting surfaces. For simplicity this feature has been omitted in the presentation above.
2.2 Viscous Flow
In Cartesian tensor notation the incompressible Reynolds-averaged Navier-Stokes equations may be written as follows
(10)
The continuity equation reads
(11)
Here, Ui represents the velocity components in the Cartesian coordinate system xi, Ρ is the pressure, ν the kin-

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ematic viscosity, ρ the density and τij the Reynolds stresses.
Although a number of turbulence models have been tested in the Chalmers/FLOWTECH group and elsewhere, as will be shown below, there are essentially only two models in practical use, the Baldwin-Lomax zero equation model (11) and the two equation k-ε model, where k is the turbulent kinetic energy and ε its rate of dissipation. Most methods today avoid the use of wall functions, and this calls for a special treatment of the ε-equation near the wall. A common choice is the two-layer model of Chen and Patel (12). The equations for k and ε read
(12)
(13)
where the production term Pk is defined as
(14)
and the eddy viscosity νt is obtained from
(15)
The constants are as follows
(16)
The most common boundary conditions for equations (10), (11), (12) and (13) are:
On the hull: Ui=k=0 (ε not needed)
Inflow, inner part, if there is an upstream boundary layer solution available: Ui, k and ε from the boundary layer solution
Inflow, outer part: Ui from potential flow solution, k and ε=small. If the calculations start in front of the hull: Ui undisturbed.
Outer edge: Ui from potential flow solution, or undisturbed. Sometimes symmetry conditions, k and ε=small or symmetry conditions.
Outflow: Most commonly Neumann condition for Ui, k and ε
Symmetry plane: Symmetry conditions for Ui, k and ε
Free surface: In most standard methods: symmetry conditions for Ui, k and ε. See section 4.2 for better free surface conditions.
A boundary condition for the pressure may be obtained either from the continuity equation or from one momentum equation.
To obtain an approximate solution to the equations, they are discretized in space and time and linearized. Numerous choices exist for solution algorithms and it is out of scope to review them here. The interested reader is referred to the proceedings of the two workshops (8), (9), where the numerical features of all participating methods are listed in detail. In general, an important capability in ship hydrodynamics is to be able to resolve thin boundary layers while taking large time steps to reach steady state, and this favours low order implicit time differencing. Spatial differencing is usually second order accurate with central differences for the diffusion term and some kind of upwind differencing for the advection term. Accuracy higher than second order has advantages, but is also less robust and more difficult to implement, especially when it comes to the boundary conditions.
3. Current capabilities
CFD is becoming an established tool in hydrodynamic design. Several methods are in routine use at shipyards, consultants and universities, and there exists a wealth of literature on various applications for different types of ships. For a review of methods and applications, see Larsson (13). In this Section an assessment will be made of the capabilities of established methods. Strong and weak points will be highlighted, and the need for the research presented in Section 4 will be explained. The discussion will address five important areas: wave pattern, wave resistance, wake/local flow, viscous resistance and propeller/hull interaction.
3.1 Wave pattern
Figure 1 presents predicted potential flow waves from two different hulls, the slender Series 60, CB=0.60 hull and the very bluff Dyne tanker, designed as a test case at the 1990 Gothenburg workshop (8). Its block coefficient is 0.85. The Froude number for the slender hull is 0.316 and for the bluff one 0.165. Wave contours along two longitudinal cuts are shown for each hull. One cut is close to the hull and the other is further out. The bow is at x=0.0 and the stern at x=1.0. Both calculations and measurements are displayed. The number of panels used in the computation was 25

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Figure 1. Wave cuts for two hulls (4)
Circles: measurement, Line: calculation
per wave length in the longitudinal direction on the surface for the Series 60 hull and 15 per wave length for the tanker. Laterally, 20 uniform panels were used for the slender hull and 15 stretched panels for the bluff one. The reason for the worse resolution of the tanker waves is the much smaller wave length, which calls for many more panels to achieve the same resolution. The number of panels followed the recommendations to the users of the SHIPFLOW code, considering available computer capacity, and the results were obtained with this software.
In general, the correspondence between calculations and measurements is quite good. For the Series 60 the wave peaks are slightly underpredicted along the hull, which indicates that the resolution is still not sufficient. Even though as many as 25 panels are used per wave length, systematic grid refinement studies (6) indicate that this is not enough to completely capture the peaks. For the cut closest to the hull an over prediction is noted in the wake. Most likely, this is caused by neglected viscous effects, i.e. the influence of the boundary layer/wake displacement effect.
The wave predictions for the tanker are surprisingly accurate, at least along the forebody, and no systematic under prediction of the peaks is noted. Considering the lower resolution this may seem to contradict the above discussion. However, most likely some breaking of the bow wave is present in this case which somewhat reduces the measured wave height. No such effects are of course considered in the calculation, so the very good correspondence may be attributed to a cancellation of (relatively small) errors. It is interesting to note what happens further aft on the hull, and particularly in the wake. The innermost cut exhibits an increasing over prediction of the wave peaks, and in the wake the predicted waves are several times larger than the measured ones. A much stronger effect of the neglected displacement effect is thus noted for this bluff hull. However, at the second cut there is practically no over prediction. Wave contours for this case show that this is due to the fact that the overpredicted stern waves have not reached the outermost cut at the end of the panelized free surface. This indicates that the waves from the rest of the hull are well predicted even at the downstream edge of the panelized surface.
The two hulls in the example represent extremes in hull bluffness and the good results of the predictions indicate that waves from the major part of the hull may be computed with good accuracy. This is utilized by ship designers in the optimization of ship forebodies and bulbs. By investigating the predicted wave pattern and wave profile, and comparing with the computed pressure distribution on the hull, experienced designers are guided in their optimization process. Good descrip-

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tions of this technique are found in several papers from the Daewoo design team in Korea, see e.g. (14), and according to other users of the SHIPFLOW code, this approach is common also elsewhere. Undoubtedly, this is the best way of utilizing hydrodynamics CFD at present.
Great care is however recommended in the optimization of ship afterbodies using the above approach. As was seen in Figure 1, the real waves are modified at the stern due to viscous effects, not considered in a potential flow. When comparing two similar designs it may be tempting to assume that the ranking is unaffected by this effect, but this is not necessarily true, as was realized in a notable failure by two of the authors in the optimization of the Series 60 hull (15). Applying numerical shape optimization an optimized hull with significantly lower total resistance was developed. In later grid refinement studies this new shape was consistently better than the original hull. However, when the new hull was model tested, its resistance was slightly larger than for the original hull. A careful investigation of the possible reasons for this failure was carried out and it was found that the optimization caused a slight increase in bow wave height, while it minimized a stern wave, whose importance in reality was considerably smaller than in the computations. Further, there was a vertical shift of the boundary layer displacement thickness on the new hull, causing an increase in the wave resistance.
For high speed hulls the situation is different. If the hull has a submerged transom, the boundary layer grows much less than for a cruiser stern, where the girth length is reduced as the stern is approached and where the local convergence of streamlines may be very strong. There is thus a much smaller effect of the displacement thickness for transom stern hulls. This is, of course, provided the transom is dry. If the flow recirculates behind the transom, the problem cannot be handled by potential flow methods. Various proposals have been put forward for the boundary condition to apply at the transom edge, see (6) for review. In SHIPFLOW, the wave height at the upstream edge of each wake panel row is set equal to the transom draft at this position and this produces quite realistic waves, both behind the transom (the rooster tail wave) and laterally (5). There are thus much better prospects for optimizing the stern of a transom hull than of a displacement hull. An accurate prediction of the sinkage and trim is, however, quite important in this case, particularly at the lower range of Froude numbers for which the transom is dry. In this speed range the hydrostatic resistance, caused by the loss of hydrostatic pressure at the transom, is much larger than the hydrodynamic resistance and the former is proportional to the square of the transom draft. An interesting study of the effect of a trim change is presented in (13).
A problem attracting more and more interest is the wake wash of high speed ships. From the above discussion panel methods may appear to be a very suitable for this type of problem, but a difficulty is the large distances at which wave predictions are requested. Normally the interest is in far field waves several boat lengths away from the hull. Obviously, this is very difficult for a method that relies on a panelization of the free surface. Very limited stretching is permitted if the numerical damping of the waves is to be kept within acceptable limits. On the other hand, the non-linear free surface effects are limited to a region rather close to the hull, so predictions by a linear theory may be permitted outside this region. One possibility explored several years ago by Gadd (16) and Aanesland (17) is to match a near field method of the type described here with a far field method using Kelvin sources that automatically satisfy the linear Kelvin condition on the surface. This possibility does not seem to have been explored recently. A simple extension of the wavy surface outside the panelized region based on linear theory has been used by Hughes (18) and others.
The remedy for the stern wave problem in the case of displacement hulls is to abandon the potential flow approach and use viscous methods of the RANS type. As mentioned above, RANS methods with a free surface are becoming more and more common, but they are not at all as established as the panel methods. A major problem is resolution. Due to restrictions in computer capacity the free surface and the water layer just below the surface cannot be resolved well enough to avoid excessive damping, at least not for the low Froude number range. A more thorough discussion on free surface RANS methods will be given in Section 4.2.
3.2 Wave resistance
The wave resistance is normally computed by adding the longitudinal components of the pressure forces on all panels. This means adding of the order of 1000 contributions of different signs, which almost cancel each other. For slow ships the sum may be of the same order as one of the larger contributions. The discretization error for such ships is thus often of the same order as the resistance. In a linear method the problem may be reduced by subtracting the resistance computed for a hull without a free surface, i.e. the so called double model case. According to d’Alemberts paradox the resistance for this case (without lift) is zero, so the computed resistance may be representative of the error associated with a given panelization. Unfortunately, this technique is much more difficult to apply in a non-

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linear method, where the panels normally change in the iterative process. In SHIPFLOW this is entirely true and in principle all panels change in the iterative process, when the free surface converges to its final shape. Other methods, Jensen (19), use a given panelization on the hull, which extends through the free surface and integrate only over the (part of the) panels below the surface. Although the chances of using d’Alemberťs paradox are better for these methods the correction is still approximate due to the differences in the integration close to the water surface in the wave case and the double model case.
Better corrections for the discretization error can thus be devised for linear methods, but they are still inferior to the non-linear ones due to the linearization. As explained in Section 2, the linearization means two types of approximation: a neglect of the non-linear terms in the boundary conditions and a transfer of the conditions to the undisturbed free surface. Both contribute to inaccuracies. As shown by Raven (6) the approximation of the equations leads to a flow of energy through the surface. This may either increase or decrease the resistance. The erroneous location of the conditions normally leads to a reduction in resistance, since the positive contribution to the resistance from the bow wave above the undisturbed free surface is neglected, while a non-existing suction force below this surface is included. At the stern the effects are reversed, but normally much smaller. Very frequently, negative values of the wave resistance are found for linear methods and low Froude numbers.
There are thus several reasons why resistance calculations are inaccurate in the low speed range. A negative wave resistance is seldom found for non-linear methods, but the accuracy in the predicted resistance is still unacceptably low for slow ships, i.e. at least up to a Froude number of 0.20. In the intermediate speed range the prospects are better, since the wave resistance at these speeds is much larger than the discretization error. The most accurate calculations of the wave resistance are found for semi-planing hulls, from the Froude number where the flow clears the transom, around 0.35, up to about 1.0. Above this Froude number non-linear effects like spray, not accounted for, become too important.
One way to improve the wave resistance prediction is to derive an empirical correction to the computed value. This may be accomplished by comparing large sets of measurements and calculations for a specific type of ships. At SSPA this technique is used for round bilge semi-planing hulls and the Technical University of Berlin (20) reports on successful corrections to slender cargo ships. It is very unlikely, however, that a similar correction can be found for slow ships.
A more scientific way of improving the resistance prediction is to abandon the pressure integration technique and obtain the resistance from the computed waves. Since these are rather well predicted a more accurate resistance might be envisaged. Some attempts to follow this track have been made recently. A longitudinal cut method was tested by two Chalmers students in 1995 with promising results, and this idea was pursued by two of the authors in 1997 (15). More work is planned to optimize the technique, but there is no doubt that an improvement relative to the pressure integration technique is possible. A very interesting alternative is presently being investigated at MARIN (21), where a transverse cut technique is being developed. In principle, this technique is less approximate, since no truncation of the cut within the wave system needs to be done. While a transverse cut may cover the whole Kelvin wedge, a longitudinal cut must be truncated somewhere and analytically extended to infinity.
3.3 Wake/local flow
In Figure 2 a comparison is made between the predicted and measured wake contours of a very bluff hull, the Dyne tanker, designed for the 1990 workshop (8). The results are from the workshop, and are thus quite old, but they are still typical of most methods using standard turbulence models.
As can be seen in the figure, the contours in the propeller disk are not well predicted. While the computed contours are quite smooth, the measured ones exhibit an island and a quite pronounced “hook”. During the nineties much of the research in viscous flow CFD has been directed towards improving the ability to predict these features.
On closer inspection it turns out that the wake hooks, which are present for large classes of ships, are caused by the bilge vortices (one on each side), generated at the bilge and hitting the propeller plane inside the propeller disk. An accurate calculation of these vortices is thus important for the prediction of the hooks. This calls for a more advanced turbulence modelling than what was available in the early nineties. At the 1990 workshop the only models used were zero equation models of the Baldwin-Lomax type and the two equation k-ε model. The latter is still the most widely used one and is normally available in commercial RANS codes. Therefore, good wake contours cannot be expected using standard codes, at least for full ship forms. Very slender hulls, on the other hand, do not normally have pronounced bilge vortices, and quite accurate wake contours can be predicted, see for instance Zhang et al (22). It should be pointed out that, although the details of the contours are not well captured for the bluff hull, integrated quantities are better.

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The radial variation of the circumferentially averaged wake, important for designing the pitch distribution, is thus better than the contours but not accurate enough for design purposes (23). The wake fraction, on the other hand, can normally be obtained with sufficient accuracy (22), (23), (24).
Figure 2. HSVA Tanker, wake contours in the propeller plane. Outermost contour: 0.9, interval: 0.1. Top: measurement. Bottom: calculations (8).
Since the wake contours in the propeller plane are insufficiently accurate, local viscous flow predictions may be questioned. Certainly, if predictions are requested as far aft as the propeller great care must be exercised when interpreting the results. However, local flow directions are often requested elsewhere, for the positioning of bilge keels, brackets and fins. Experience shows that such predictions are mostly sufficiently accurate for design purposes. As pointed out above, transom stern ships, have a relatively thin boundary layer even close to the stern, and bilge vortices are seldom a problem. For such hulls very good viscous flow predictions are possible, an important advantage, since these hulls often have brackets, whose direction can be optimized.
To improve the situation, more accurate turbulence models are required and several predictions have been reported where considerably improved results are demonstrated. This development will be discussed in Section 4.3.
3.4 Viscous resistance
The inability to compute the details of the wake flow certainly has some effect on the accuracy of the prediction also of the viscous resistance. However, the resistance is an integrated value, so just as in the case of the wake fraction, the details of the flow do not necessarily have to be very accurate to obtain a value of engineering interest. Therefore, existing RANS solvers should be capable of predicting the viscous resistance accurately enough for ranking purposes, provided certain conditions are satisfied. Where many methods fail is the grid quality. In particular, an accurate representation of the hull shape at the ends is important. Many grid generators produce a staircase-like hull shape at the stern contour, with very distorted numerical cells. Since this is an area of large importance for the resistance, both the erroneous direction of the normal to the hull surface and the inaccurate pressure contribute to an unrealistic pressure drag. For this to be computed accurately it is also important to have a sufficient number of grid points in the direction normal to the hull to resolve the pressure variation across the boundary layer. The frictional resistance, on the other hand, relies on an accurate prediction of the flow in the inner part of the boundary layer. The innermost points of the grid must thus satisfy given requirements for y+, be it a wall function method or a method using the no-slip condition. Grid dependence studies are certainly important in all CFD work, but especially in resistance prediction.
The viscous resistance was provided by eight methods at the 1994 workshop (9). Three were seriously in error and three predicted the resistance, as well as the split between the pressure and frictional contributions, quite well. There was no discernable advantage of the k-ε model as compared to the Baldwin-Lomax model and the methods were in most respects quite similar, so the different performance must be due to numerical details, most probably related to the grid. This conjecture is substantiated by the fact that careful grid refinement studies have been reported elsewhere for two of the three best methods, see Ju and Patel (25), and Ishikawa (26). In both papers successful rankings of hull afterbodies are reported. Streckwall (27) also reports on successful rankings and in a recent paper Masuda and Kasahara (24) present interesting calculations of the form factor for 20 hulls using the NICE code (28). The latter introduce a correlation between the predicted and measured form factors, and using this correlation, the error in the predicted values are within 2% for 18 out of the 20 hulls. As mentioned above, this is an good way of increasing the engineering value of CFD. It should be mentioned that a similar correlation was developed for the wake fraction.

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3.5 Propeller/hull interaction
Already in 1977 Schetz and Favin proposed an actuator disk approach to introduce propeller effects in RANS methods. By applying body forces to the numerical cells in the propeller disk the flow may be accelerated in the same way as by a propeller with an infinite number of blades, producing the same thrust and torque. This approach was further refined and improved by Stern and his co-workers in Iowa (29), and in the Chalmers group by Zhang (22). More recently the method has been applied to design problems at the University of Athens (30). A somewhat simpler approach is to apply a pressure jump in the propeller disk, see Streckwall (27).
An actuator disk is now included in the SHIPFLOW code, where the body forces are computed by a lifting line propeller analysis method, run interactively with the RANS solver. Using the velocity computed in each numerical cell within the propeller disk, the lifting line program computes the circulation and thereby also the axial and tangential body forces, which are returned to the flow solver in an iterative process. Strong under relaxation is required to introduce the body forces, but the iterations are included in the normal pressure/ velocity iterations, so the computer effort is not much increased Since the code can be run with a propeller only, corresponding to an open water test, the propulsive factors can be computed.
Although the actuator disk has been available in SHIPFLOW for a couple of years it does not seem to have been much used. It is therefore difficult to assess the accuracy of the computed propulsive factors. Zhang’s own calculations (22), as well as more recent work in a Master’s thesis indicate that the thrust deduction can be obtained with good accuracy even for high speed hulls, where a large part of the thrust deduction comes from trim changes. The effective wake and the relative rotative efficiency also seem to be reasonable. The actuator disk approach should be further evaluated and used by the designers. It seems to be a useful tool for power prediction (of course with the limitations of the resistance calculation). For more accurate predictions of blade flows and pressure fluctuations the more advanced methods presented in Section 4.4 will be required.
4. Present research
As explained above, the most advanced potential flow methods have now reached a state where the major obstacle for further improvement is the inherent neglect of viscosity, and the efforts to fine tune available methods have been covered already in Section 3. Therefore, the present section will only cover research in the RANS area. The work presented may be expected to yield improvements in existing solvers in the relatively near future, say five years, while the more long range perspectives will be discussed in Section 5. Although reference will be given to research elsewhere, the emphasis will be on the work in the Chalmers/FLOWTECH group. Four areas will be covered: grid generation, free surface boundary conditions, turbulence modelling and propeller blade flows.
4.1 Grid generation
The ship hull is normally a smooth surface and the flow domain surrounding the hull may rather easily be transformed into a rectangular box, in which the computations are performed. Ship flows are therefore good candidates for single block structured grids, and so far the vast majority of ship flow methods employ grids of this kind. In the 1990 and 1994 workshops (8), (9) all methods used single block structured grids. There are, however, several reasons for introducing more advanced gridding. Although the hull itself is smooth, it always has appendages of different kinds. These are normally neglected, but for more advanced calculations the rudder, possible shafts and shaft brackets, bilge keels, fins, etc. have to be taken into account. A single block grid necessarily has to include singular points or lines in front of and behind the hull and this calls for special treatment in the solver. Further, a large number of grid points are normally wasted in regions where they are not needed in structured single block grids and, as pointed out above, it is difficult to obtain a high grid quality close to the ends of the hull.
For these reasons there is a growing interest in developing methods based on less restrictive gridding techniques. Little interest has however, been shown in completely unstructured grids, common in structural mechanics. Such techniques offer great flexibility, but impose more work on the solver, which has to take care of the connectivity information and has to deal with large sparse matrices. It is also more complicated to develop higher order schemes and multi-grid convergence acceleration techniques. Completely unstructured grids are also unsuitable for boundary layers at high Reynolds numbers, where extremely large gradients are experienced in the direction normal to the surface. These disadvantages are often offset by the advantage in flexibility for general purpose CFD applications, where unstructured grids may be an alternative, but in hydrodynamics this does not seem to be the case and only a few calculations with unstructured grids have been reported, see Hino (31) and a recent paper by Yang and Loehner (32).
Instead, the recent interest among hydrodynamicists has been directed towards multi-block methods,

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where the structure is maintained within each block. Not surprisingly, this development started in submarine hydrodynamics, where appendages play an important part. Standard multi-block techniques for submarines are reported by Sung et al (33), McDonald and Whitfield (34) and Bull (35). Other applications are presented by Cowles and Martinelli (36) and Beddhu et al (37). The traditional multiblock techniques offer more flexibility than single block grids, but they are still restricted due to the need for matching at the common boundaries of adjacent blocks. A more modern approach is to let the grids overlap without any restrictions on continuity of grid lines. This technique, known in the United States as the Chimera technique, has become a useful tool in many engineering branches, see the 1st–4th Symposia on Overset Composite Grid & Solution Technology. (The 4th one to be held at the Army Research Laboratory, Aberdeen, Maryland, USA 23–25 September 1998). Recent applications of overlapping grids in hydrodynamics include Weems et al (38), Lin et al (39) and Masuko (40).
In the Chalmers/FLOWTECH group a new 3-D overlapping grid generator CHALMESH has been developed by Dr. Anders Petersson, based on his earlier 2-D code XCOG (41). The new code has an efficient user interface and seems more robust than other grid generators of this kind. Based on overlapping surface grids, which have to be provided for general cases, overlapping volume grids are grown hyperbolically out from the surface. The body-fitted grids may be embedded in one or more background grids, which may be curvi-linear or Cartesian. Holes are automatically cut out and the necessary overlap is ensured. Interpolation weights are computed and saved for all interpolation points at the edge of each grid. To avoid mismatch, special techniques are used to handle the interpolation between the extremely thin cells close to boundaries where the no-slip condition is to be applied. CHALMESH is reported in (42), and the code is available free of charge for research purposes on the Internet under the address: http://www.na.chalmers.se/%7Eandersp/chalmesh/chalmesh.html. It is presently used at different departments at several universities in Sweden and the USA.
Figure 3 shows the grid generated around a three-bladed propeller. Only surface grids are shown for two of the blades, while a cut through the body-fitted grids is shown for the third blade. It is seen that there are three grids on each blade, one on each side and one wrapped around the edge. These grids are embedded in a background cylindrical grid. By arranging the grids in this way quite orthogonal grid lines may be obtained. On ship hulls typically a few grid patches are wrapped around the stern, after which a
Figure 3. The grid around a 3-bladed propeller. Body fitted volume grid shown on one blade only.
couple of rectangular grids are attached. All these are embedded into a background cylindrical grid. The rectangular grids are added to remove the line singularity of the background grid.
A new RANS method for overlapping grids has been developed within the group, mainly by Björn Regnström. Finite difference discretization is used and the discretized equations are solved implicitly for all blocks, i.e. the matrix contains equations for all points including interpolation points. To keep the discretization stencil as small as possible, central differencing is used for all terms. Alternatively, a mix of second order central and first order upwind discretization may be used for the convective terms. To stabilize the solution if central differences are used, artificial dissipation is added. The variables are collocated and Rhie-Chow interpolation is used to avoid pressure fluctuations in the SIMPLE pressure/velocity updating scheme. As will be seen below, the free surface is treated in a special way and a range of turbulence models are available. By the time of writing, laminar flow calculations have been carried out for the propeller of Figure 3, and the first turbulent results have been obtained for a ship hull. The results indicate that the overlapping algorithm works as expected, but some other fine tuning of the method is required before it can be used on a regular basis.
4.2 Free surface boundary conditions
As mentioned above, the most important result of the 1994 workshop (9) was the breakthrough of the

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free surface RANS methods. No less than 10 methods featured this capability. The technique was not new, however. The Marker and Cell (MAC) method had been available for 30 years and the Volume of Fluid (VOF) method for more than 10, but they had mainly been applied to internal flow problems, such as sloshing. In ship hydrodynamics the first references are from the mid-eighties, when Miyata et al introduced their version of the MAC method called TUMMAC (43). A large number of references to subsequent developments in different groups are available and may be found in Larsson (13). They will not be repeated here. See however Miyata (44) for an overview of the impressive developments at the Tokyo University. Recent references, not in (13), include Yang and Loehner (Euler solution) (32), Beddhu et al (37) and Haussling et al (45). The latter is especially interesting, since it deals with the difficult transom stern problem.
Extensions into other areas of hydrodynamics are presented in the papers by Rhee and Stern (46), Campana et al (47) and Ohmori et al (48). While Rhee and Stern compute the wave diffraction from a fixed hull subjected to an approaching wave train, the others consider the side force and turning moment of a hull at a yaw angle.
In general, predicted wave profiles along the surface of the hull agree well with measurements, while the wave height away from the hull is consistently underpredicted. This is true especially for the diverging waves, which are not well captured even at high Froude numbers (above 0.3). In the low Froude number range (below 0.2) all waves are damped out at a relatively short distance from the hull. The reason for this is obviously bad resolution. To investigate the requirements on grid resolution a Korean visiting scientist in the Chalmers/FLOWTECH group, Dr. Kang, carried out systematic grid refinement studies for the waves generated by a submerged airfoil (49). A 2-D free surface RANS code was used and great care was taken to reduce the numerical damping on the surface. It was found that approximately 50 points per wave length were needed to preserve the waves.
In a 3-D case the diverging waves have the shortest wave lengths, and as pointed out by Mori and Hinatsu (50), about five times denser grids are required in the transverse direction as compared to the longitudinal direction to really capture these waves. This means that the requirements on the number of grid points become prohibitive on today’s computers. For instance, at a Froude number of 0.15, where the fundamental wave length is about 1/7 of the hull length, approximately 106 grid points would be required on the wavy surface. Now, the dense grid could probably be concentrated in a layer relatively close to the surface, but to maintain the density in the vertical direction very near the surface the total number of points would be somewhere in the range 107–108.
Since the wave length is proportional to the square of the Froude number, the number of points on the surface would be inversely proportional to the fourth power of the Froude number, if the extension of the gridded part of the surface was kept constant. Now, the gridded part normally increases somewhat with wave length, but normally much less than linearly. Assuming that the required number of points in the vertical direction is independent of the wave length and the free surface is unchanged, the total number of points at a Froude number of 0.3 would be one order of magnitude smaller than at 0.15, i.e. in the range 106–107. No predictions presented so far have been anywhere near this number. The largest cases presented have had a total number of grid points around 106, while the number of points on the surface has been of the order of 104. This is why the Kelvin wave pattern is not obtained. It should be pointed out that panel methods seem to be able to predict very detailed wave patterns even at the lowest Froude numbers using less than 104 panels on the surface. The inability of the RANS methods to capture the waves is, of course, only a temporary problem. As will be seen below there are good prospects for increasing the number of points on the surface considerably within the not too distant future. Further, a good Kelvin wave system might not be needed for other predictions of interest, such as wave resistance, sinkage and trim, side forces and other local phenomena.
In the first free surface RANS methods of the MAC and VOF types the grid was kept fixed and the free surface tracked in the grid at every time step. This technique has been abandoned in most recent methods (a notable exception is TUMMAC), where the grid is fitted to the free surface and deformed as the waves grow with time. As pointed out in Section 2, there are two boundary conditions on the surface, the kinematic one (4) and the dynamic one. (5) represents an inviscid approximation of the latter. The exact dynamic condition expresses continuity of stresses across the surface, i.e
(17)
where the star represents values in the air.
This condition is, however rarely used. With few exceptions (Alessandrini and Delommeau (51)) the inviscid approximation is adopted, i.e. the pressure is assumed constant and set to zero. Bernoulli’s theorem cannot be used, however, so (17) may now be written

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nonlinear loads. Whereas the former method is well established and proven to be valid in a variety of cases, the latter one allows much stronger compression of the simulation time, but it requires further development and validation.
[1] Söding, H. and Tongue, E., “Computing Capsizing Frequencies of Ships in Seaway,” Third Stability Conference, Gdansk 1986
[2] Söding, H. “Current Problems in Ship Loads” (in German), Jahrbuch Schiffbaut. Ges. (STG) 1991.
AUTHORS’ REPLY
You are correct that different responses require different simulations when using the presented wave conditioning technique. However, we do not agree that the methods which you mentioned are more efficient with respect to both the required simulation time and the accuracy of the results. A principal difference between the approach followed in [1] and our approach is that we are not looking for the probability of occurrence of a non-linear event rather than the magnitude of the non-linear event. The probability of this event is defined in an earlier stage as the most probable largest value according to linear theory. The Most Likely Extreme Response method was presented to calculate the non-linear response in a well-defined condition, which is known to give the largest response in a linear calculation. The required simulation time depends on the shape of the auto-correlation function of the response. For vertical bending, this is typically 4 to 5 cycles whereas the length can be more than 10 cycles for roll motions. The procedure as described in [1] required the simulation of 30 events and 400 cycles to predict the probability of capsizing. Spending a similar simulation time using the MLER-method, it is possible to calculate the non-linear extremes for about 40 to 80 responses. The method mentioned in your second reference [2] shows more similarity with our method although the wave-conditioning process is much easier to understand and to realise in the Most Likely Extreme Response method. In addition, the Most Likely Extreme Response method is very flexible to slight modifications of the mean period at the instant of occurrence of the extreme response. This allows the user to control the instantaneous wave steepness to be within physical limits.
[1] Söding, H. and Toguc, E. “Computing Capsizing Frequencies of Ships in a Seaway,” Third Stability Conference, Gdansk 1986.
[2] Söding, H. “Recent Problems in Ship Loads (in German),” Jahrbuch Schiffbaut. Ges. (STG) 1991.

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Investigation into Ventilated Hydrofoils for Ride Control of High-Speed Craft
K.Thiagarajan, R.Shock (Australian Maritime Engineering CRC, Ltd., Australia)
Abstract
This paper investigates the feasibility of using an air-ventilated foil as an alternative to active-incidence-fins for ride control of high speed craft. Numerical simulations using FLUENT software demonstrates the features of the air cavity, and the reduction in lift simulated compares well with theory and published experimental results for a NACA 0010 foil at zero angle of attack. The ventilation number of the cavity was found to be inversely proportional to the cavity length. Further work involves testing foils with different vent locations at different angles of attack.
1. Introduction
Research into high speed craft offer navies of countries, e.g. Australia and the US, a potential for developing unique capabilities and increasing effectiveness. Increasing speed without compromising comfort and safety is an attractive strategy for ferry operators. As ships increase speed, their dynamic response in a seaway become increasingly complex. Ride control devices for high speed crafts need to be effective in responding rapidly to a dynamic environment, while minimizing penalties e.g. appendage drag.
The pitching motion of a vessel in a seaway is controlled by active anti-pitch fins, e.g. a T-foil fitted to a hull at the bow. In the case of a catamaran, a T-foil is fitted under each demihull. Even keel during motion is then maintained by incidence and effective camber control of the foil surface, in effect controlling the resultant lift force generated. Each foil typically sits on a rocker arm that is pivoted by a hydraulic ram to change incidence, and produce the required force. The hydraulics required to actuate the foil include motors, pressure cylinders, oil filters, and regulators, all of which can weigh up to 3 tons per demi-hull for an 85 m catamaran. The control machinery associated with these T-foils is thus large. Further, experience of operators indicates that moving parts underwater pose heavy maintenance problems. It would thus be desirable to develop a ride control device whose lift force could be altered without the need for moveable parts underwater, as well as reduce machinery needed to accomplish the required extent of ride control.
The research presented in this paper focuses on the use of air injection over a hydrofoil as a means to control the lift force of a fixed system of anti-pitch fins. Compressed air is introduced to the suction side of foils to destroy a significant portion of the hydrodynamic lift. Since ventilated foils do not require changes in angle of attack, underwater moving parts are minimized. The ventilated foil system also allows the use of high lift, low drag cambered foils for the foil geometry. Thus the overall drag penalty due to the ride control operation can be lowered.
The advantages of ventilated foil systems stem from the fact that they require no mechanical movement to adjust the lift forces generated. Current ride control devices adjust the angle of attack of the foil, as well as trim tabs to control the lift, requiring multiple through-hulls and various support machinery such as hydraulics and control rods. The ventilated foil approach instead relies solely on air injection regulated from a compressor, or the atmosphere. This limits the through hulls to one for the air pipe and a mounted strut for the foil. If the ventilated foil system can generate the same lift control as conventional systems, savings in cost can be

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realized in initial outlay, maintenance, and reduced operating cost.
The use of air entrainment to control hydrofoil boats has been attempted with some degree of success by the German design company Supramar in the 1960’s (Gunston, 1970). The highly successful Supramar PT.50 hydrofoil boat used air entrainment to control the lift of both the forward and aft lifting foils. Supramar also used this concept effectively for a number of other hydrofoil boats, from the smaller PT.20 to the larger PT.150. Through venting air over the suction side of the foils, the lift on the hydrofoils was adjusted to control the motions of the vessel. A fully submerged rear foil fitted with separate air-stabilization systems on the port and starboard sides showed that the stabilization reduced the roll response of the ship by 75% in roll (Gunston 1970). This same concept can conceivably be adapted to high speed catamaran ferries.
Previous investigations into ventilated foils dates back to the 1960s and earlier, where the concept was explored by model tests (Lang 1959; Lang et al. 1959). These tests conducted at the US Naval Ordinance Test Station’s (NAVORD) water tunnel on a NACA 0010 (10% thickness) foil indicated significant reduction in lift by varying vent location and air flow rate at a given Reynolds number. The air cavity development was found to be unstable initially, and cavity length varied until the air flow was increased to a critical value. At this critical ventilation number, the cavity closure point shifted to two chord lengths downstream, and the foil was fully vented. Increasing the air flow rate beyond this point was found to produce marginal changes to the foil performance. Reducing the airflow after reaching the critical point also did not appreciably change the lift force, pointing to a hysteresis effect The slope of the lift coefficient—angle of attack curve for an infinite span hydrofoil was found to change from the unvented theoretical value of 2π, to a value close to π, when the vents were successively moved from the trailing edge to about 3% of chord from the leading edge. It was noted that the drag coefficient in a fully vented flow increased to about 150% of the fully wetted value. While a portion of the increase was attributed to tunnel interference effect, it was found that the drag force decreased once the critical ventilation number was reached.
The topic of foil cavitation has been studied extensively in the past, and is not reviewed in this paper. While the mechanisms of occurrence are different for cavitation and ventilation, some flow similarities may be expected, particularly when a long cavity exists behind the foil. Based on this flow similarity, theoretical treatment of cavitating flows may be extended for ventilated foil flows. A linear theory for cambered foils with long trailing cavity at zero incidence and zero cavitation number was developed by Tulin and Burkart (1955). The solution strategy was based on reducing the flow characteristics of the foil and the trailing cavity to an equivalent problem of the classical thin airfoil. Observation of cavitating flows showed that the slenderness ratio of the cavity (ratio of diameter to length of cavity) approached a value of σ/2 (σ −cavitation number) as σ→0 and the trailing cavity shape was elliptic (Tulin, and Burkart, 1955). Comparison with experiments showed that linear theory predicted the length of the cavity to within 5% of the measured value. Tulin and Hsu (1980) developed a theory for foils with short cavity, and were able to include effects of foil thickness in the formulation. Kinnas (1991) extended the linear cavity theory further to incorporate leading edge correction, for foils with cavity initiating close to the leading edge. Results from this theory were found to correlate well with linear theory for cases where cavity starts further from the leading edge.
A theory for potential flow past thin airfoils with ventilation was developed by Fabula (1962), using the open and closed cavity termination models of Tulin (1956). A conformal mapping technique was employed to transform the foil surface to a semi-circle, and the problem solved as a function of cavity inception point and cavity length. Results of this theory applied to vented flat plates are used in this paper. As will be shown later, these results show remarkably close correlation with the experimental results of Lang et al. (1959) and those of FLUENT.
2. Aspects of flow past a ventilated foil
If a foil is ventilated on its suction (low pressure) side, the overall pressure distribution is altered.

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Figure 1: Ventilated foil parameters
With sufficient injection, a cavity is formed on the surface of the foil, leading to an increase in pressure on the suction side, resulting in a decrease in lift force. By varying the airflow rate, the cavity length and hence the resultant lift force can be adjusted.
There are several parameters influencing the ventilated flow problem. The geometrical parameters are shown in Figure 1. The angle of attack, α, is fixed for this problem and hence can be considered as a geometrical parameter. Operational parameters of importance include the ambient pressure and oncoming velocity, and the inlet air pressure and air velocity expressed together as the air flow rate Q. The dimensionless air flow rate per unit length is defined as (Lang et al. 1959):
(1)
The cavity pressure is given in terms of the ventilation number or the cavity number,
(2)
and is similar in definition to the cavitation number. However the relationship between K and CQ does not appear to be straightforward. Lang et al. (1959) noted that there appeared to be a relationship between the air flow rate and the cavity length and pressure that depended on the angle of attack. It was observed that beyond a certain CQ value nominally around 0.004, the
Figure 2: Lift coefficient derivative versus vent location
ventilation number was unaltered, and the increased airflow resulted in increased length of the cavity. A similar trend is noticeable from Fabula’s (1962) theory when cavity length is an order of magnitude larger than the chord length.
The three forces of importance for the foil are obviously the lift, drag and overturning moment However, for the purposes of this paper, we will concentrate primarily on the lift coefficient, with the proviso that drag penalties due to ventilation be left as future work within this project. The lift coefficient is given traditionally for two-dimensional sections as
(3)
It is of interest to know the variation in lift coefficient with respect to the following two parameters:
vent location, given by the ratio a/c.
flow rate CQ, or the ventilation number K.
It is fairly obvious that maximum lift change will occur when the vent is located close to the leading edge, and this change will decrease as the vent moves leeward. The lift coefficient derivative should approach 2π as the vent approaches the trailing edge. These are indeed

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Figure 3: Lift coefficient derivative versus ventilation number/2α; experimental values of Lang et al. and theory of Fabula.
found to be the case, Figure 2, both from experimental results of Lang et al. (1959) and the theoretical results of Fabula (1962) using the closed cavity model. The open cavity model appeared to give unrealistic results for the case of cavity length slightly larger than 1, and for long cavity lengths, it gave results identical to the closed cavity model. So, only the closed cavity results are shown in Figure 2. The figure shows some interesting trends. The experiments closely agree with theory, even though the theoretical results pertain to a flat plate, while the experiments were conducted on a NACA 0010 foil. The figure indicates that unless thicker foils are used, results may not differ appreciably from the flat plate case. The foil case tends towards a value of π as the vent approaches the leading edge, indicating about 50% of lift can be dumped by ventilating the entire upper surface of a foil. It is interesting that much greater lift can be lost by using a flat plate, where the curve approaches a value of π/2, indicating a loss of 75% of the lift. It would be useful to evaluate how a thicker foil performs against these results.
Correlation of experimental and theoretical results for the case of a/c=0.3, and various Κ numbers are shown in Figure 3. There are undoubtedly few difficulties in precisely estimating the ventilation number in the experiments. While the trends between theory and experiment appear to be similar, albeit at different offsets, the experimental cases include data for partially vented conditions, while the theoretical curve is for the fully vented condition for different cavity lengths. Due to its limitations, the theory cannot be applied for partially vented conditions. Further, it is seen that much higher ventilation is required for the smaller angle of attack case (about 130% higher at Cl/α=4), which is not intuitively obvious. No clear conclusions may be drawn from this analysis. It is hoped that FLUENT simulations may show a clearer trend between Κ and Cl.
3. Numerical Simulation
Numerical simulations of ventilated foil flows are currently ongoing using FLUENT Ver 4.4, a pressure-based segregated finite-volume method solver. This software has a capability for solving multi-phase flows, with options for grid refinement around large pressure gradient locations. The simulations are expected to provide a means of obtaining fairly accurate results for ventilated foils that can be compared against corresponding results from experiments and theory.
The multi-phase model used in the simulations is the Volume Of Fluid (VOF) model included in FLUENT. In this model, a single set of momentum equations is used to describe both fluids, as given below:
(4)
where the density ρ, and dynamic viscosity μ are determined by tracking the volume fraction of each cell. The volume fraction (εk) of a cell is determined by solving the continuity equation for the fluids,
(5)
where Sεk denotes any source terms in the problem. A standard RNG k-ε turbulence model is used for closure. The momentum equations are solved using an explicit time stepping method, with a user-defined time step and maximum

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Figure 4: Computational grid in FLUENT
allowable iterations per time step. The volume fraction equation can be solved either at every iteration of the momentum equation, or for every time step. To speed up processing time, the latter option is used for the present simulations. Surface tension effects are included in the simulation, and the standard value of surface tension of water at 20° C is used.
The geometry and the mesh are created using a program called preBFC Ver. 3.0. The grid is a structured mesh with quadrilateral cells as required by FLUENT. The computational domain consists of 301 cells in the i direction and 151 cells in the j direction. The grid needs to be highly resolved near the foil for accurate modeling of the flow. Further, there is a requirement for significant resolution downstream to visualize the cavity, as the latter grows and moves downstream of the foil. This is accomplished with an O-H type grid, which consists of a cyclic boundary at the leading edge of the foil. A rectangular grid is used behind the foil to visualize the downstream cavity and any shed air bubbles. The O-grid at the foil surface is highly resolved, while aft of the foil, the Η-grid is coarser as x approaches xmax. The transition from circular to rectangular grid takes place half a chord length behind the foil. The two interior regions are each 100 cells by 101 cells giving a total of 45451 computational cells. The grid was smoothed using the “Spacing control” option in preBFC over the entire domain.
Results reported in this paper are obtained from simulations conducted on a NACA 0010 foil with 1 m chord length, and an air orifice width of 0.01 m. The flow conditions, similar to Lang et al. (1959), are as follows.
Air vent location
3% chord
Water inlet velocity
10 m/s
Air inlet velocity
5 m/s
Dimensionless air flow rate
0.005
Ambient pressure
1.3 E05 Pa
Using a time step of 0.01 sec, the simulations were run up to t=1.9 sec. The simulation was started with the foil in the fully wetted condition, and air injection was initiated 0.3 sec into the simulation. It was observed that if air injection was started at t=0, the resulting air cavity showed a ballooning effect at the trailing edge due to the presence of the starting vortex, and this balloon was later swung around as the vortex moved downstream. The simulation results were unrealistic in this case. If the fully wetted flow was allowed to develop initially and then air injection introduced, the cavity was much better developed and simulation results were found to be reliable. With a total of 190 time steps and a maximum of 120 iterations per time step, the vented simulations took approximately 24 hours per case.
4. Results and discussion
The results presented and discussed in this section pertain to time evolution of the ventilated flow and foil characteristics for a NACA 0010 foil at zero angle of attack. A parametric study in terms of K and a/c are beyond the scope of this paper, but will be part of proposed research. Figures 5 a–d show the volume-averaged density contours as the flow develops over time. At t=0.54 sec (about 0.2 sec after air injection), the cavity shows signs of development, with the trailing wake showing a mixture of air and water. The cavity is well developed beyond 0.84 sec, and has extended downstream of trailing edge. With passage of time, the cavity continues to grow, albeit gradually, and the boundary layer comprising a mixture of water and air is evident all around the cavity. Patches of air-water mixture can also be seen in the trailing end of the cavity, which is due to entrainment of the slower flow into the cavity. The pressure contour plots (Figure 5 e–h) show that the trailing edge pressure point moves further

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Fig 5: Simulation results for NACA 0010 foil at α=0°, U∞=10 m/s, air flow=5 m/s at 3% of chord from leading edge. (A–D) Density contours of mixing phases, and (E–H) Relative static pressure contours, at various time steps.

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Fig 5: Simulation results for NACA 0010 foil at α=0°, U∞=10 m/s, air flow=5 m/s at 3% of chord from leading edge. (A–D) Density contours of mixing phases, and (E–H) Relative static pressure contours, at various time steps.

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Figure 6: Variation of ventilation number (K) and cavity length (l/c) with time.
downstream as the cavity develops. The leading edge stagnation pressure values remain unaffected by the cavity development.
From the simulations, the cavity pressure and cavity length were estimated, and are plotted in Figure 6. The cavity number peaks initially, and then decreases gradually to level out at a value of about 0.22. The corresponding trend for the cavity length is a steep increase initially, and then a more gradual growth with time.
The cavity pressure and length can be expected to be interrelated. If at two instants of time, the cavity pressures and volumes are known, they can be related by the ideal gas law (assuming constant temperature):
(6)
The cavity volume depends on the net volume flow rate, i.e. the inflow from the vent minus the outflow through the boundary layer and the wake. The cavity length is proportional to the cavity volume, and the relationship may be assumed linear if perturbations in cavity thickness are small. Then, based on Eq. (6) the following relationship can be expected:
(7)
From the data in Figure 6, it is seen that this relationship is satisfied to within 10% over the time length specified, i.e.
Figure 7: Change in lift coefficient with time for the NACA 0010 foil at zero angle of attack.
The ventilation number may thus be expected to vary as the inverse of cavity length during the primary stage of cavity development Experimental observations of Lang et al. (1959) showed unclear trends with respect to the ventilation number, but it was felt that K tends to remain a constant at later stages of the cavity development.
Figure 7 shows the change in lift coefficient with time for the NACA 0010 foil. The lift coefficient is originally at its fully wetted value of 0.2, gradually decreasing until the steady state cavity is developed at about 1.2 sec (0.9 sec after air injection). Cl is noticed to stabilize at around a value of −0.21±5%. This agrees well with Lang et al’s (1959) results where the least squared straight line fit to experimental Cl data intersects at −0.22 for zero angle attack. The data also shows that 90% of steady state lift value is achieved in less that 1 sec after air injection. This response time information is an important consideration when developing a ride control system. It is of interest to know how this time varies as the operational parameters are changed. This will be part of the future research in this project. This simulation has shown convincing evidence as to the utilization of FLUENT for simulation of the flow of interest.
5. Conclusion and future work
A reasonably comprehensive set of experimental data on lift coefficient for a NACA 0010 foil has shown good agreement with

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theoretical results with respect to vent location, but unclear trends with respect to the ventilation number. Simulations conducted on the same foil with the commercial software FLUENT have shown good correlation with experimental data for the lift coefficient at zero degree angle of attack. The product of ventilation number and cavity length is found to be roughly a constant for the duration of cavity development.
Further work with FLUENT includes the following:
NACA 0010 foil at different angles of attack
Same foil with different vent locations.
Other foil sections of interest, and flow conditions of relevance to fast craft.
Studies on drag induced by ventilation.
Variations to the basic configuration of ventilated foils.
It is also proposed to use the leading-edge-corrected linear theory of Kinnas (1991) to compare with results of FLUENT.
The present work was initiated by the Australian fast ferry industry with a view to assess the feasibility and viability of using ventilated foils for ride control of high speed crafts. Ongoing simulations will provide farther information that would enable ventilated foils to be a proven alternative to incidence control as means of pitch control on a number of vessels, particularly fast ferries.
6. Acknowledgments
This work forms part of Australian Maritime Engineering CRC’s Program B: Performance of Surface Marine Vehicles. The work is supported by the industry participants of the research Task, Incat Designs Sydney through Mr. Todd Maybury, and Austal Ships through Mr. Tony Elms. The second author’s Masters work is supported by an Overseas Postgraduate Research Scholarship of the Australian Government, and an ΑΜΕ CRC Postgraduate Scholarship. The authors would like to acknowledge the help of Dr. Liang Cheng and Mr. Xiaojang Wang, Department of Civil Engineering, University of Western Australia for the use of their computer facilities and the FLUENT software for this project. Suggestions offered by Mr. Kim Klaka and Dr. Patrick Couser of ΑΜΕ Perth Research Core are appreciated.
7. References
Elms, Τ., “Hydrofoil Lift Control Using Air Injection” Bachelor of Engineering Thesis, University of New South Wales, 1992.
Fabula, Α., “Thin Air-foil Theory applied to Hydrofoils with a Finite Cavity and Arbitrary Free-Streamline Detachment” Fluid Mechanics, Vol. 12, Part 2, 1962
Fluent Inc, “Fluent User’s Guide Vol. 2” Fluent Inc, Lebenon, N.H, 1995.
Gunston, B., “Hydrofoils & Hovercraft, New Vehicles for Sea and Land”, New York, Doubleday, 1970.
Kinnas, S.A., “Leading-edge corrections to the linear theory of partially cavitating hydrofoils”, J. Ship Res. Vol. 35, No. 1, 1991, pp. 15–27.
Lang, T.G., “Base Vented Hydrofoils” NAVORD Report 6606, October 19, 1959.
Lang, T.G., Daybell, D.A., Smith, K.E. “Water-Tunnel Tests of Hydrofoils with Forced Ventilation”, NAVORD Report 7008, 10 November 1959.
Tulin, M. and Burkhart, M.P., “Linearized Theory for Flows About Lifting Foils at Zero Cavitation Number” Technical Report C-638, David Taylor Model Basin, Bethesda, MD, 1955.
Tulin, M., and Hsu, C.C., “New Applications of Cavity Flow Theory” Hydronautics, Inc Report, 1980.
Tulin, M., “Supercavitating Flow Past Foils and Struts” NPL Symposium on Cavitation in Hydrodynamics, Her Majesty’s Stationary Office, London, England, 1956.

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DISCUSSION
T.Sarpkaya
Naval Postgraduate School, USA
If the physics of the desired control is to separate the boundary layer on the upper surface, why is the air introduction advantageous? Why not use water injection and avoid undesirable lift fluctuations? Even better, why not induce the boundary-layer separation point, and thereby control its angular motion (a few millimeters), the length of the separation zone, and thus the lift as in the case of thrust controls for missiles? Your comments will be appreciated.
AUTHORS’ REPLY
One of the reasons for using air introduction on the ride control foils is to overcome the force limitation cavitation puts on the foils. At the current speeds of high speed craft, most notably catamaran ferries, ride control foils are required to be quite large, thus heavy, to produce the required force without cavitating. Air injection on the surface of the foil gives the ability to greatly adjust the lift of the foil as well as delaying the onset of cavitation. While a movable plate along the foil would give us the ability to control the lift of the foil to a certain extent, it would invariably induce cavitation along the foil. Air injection allows greater control of the cavity along the entire foil with variable flow rates of injection and the ability to use multiple vents along the surface of the foil. By using a large flow rate injected at the leading edge, a large force can be generated to dampen motions in a heavy sea-state, whereas in relatively calm seas, short bursts can be initiated toward the trailing edge, thereby reducing drag when large damping forces are not needed.
The choice of air entrainment instead of using water is being pursued due to the ease of implementing such a system. While it is possible to use the surrounding water as the injection fluid, this would require a more robust support system than that of air injection. One of the aims of this project is to reduce the number of through-hulls below the waterline. Whereas atmospheric pressure can easily be supplied within the strut, water would require a port in the strut or along the hull. It was therefore decided that air would be a more advantageous medium.