Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
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Session 3- Wavy/Free Surface Flow: Panel Methods 3
Session 3- Wavy/Free Surface Flow: Panel Methods 3
Pages 93-152
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From page 95... ...
This method, developed at MARIN, uses an iterative procedure based on a Rankine panel method similar to that of Dawson. No convergence problems are usually met in the speed and fullness range of practical ships, and even for cases showing extensive wave breaking in reality.
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From page 96... ...
Specifically: the neglect of nonlinear terms and transfer terms in the free surface boundary condition introduces inaccuracies; the hull form above the undisturbed waterline cannot be taken into account in a linearized method; this eliminates the effects of bow flare, flat sterns, surface-piercing bulbous bows and so on; doubts exist on the modeling of the flow off an immersed transom stern . All these inaccuracies and limitations can in principle be removed by solving the fully nonlinear problem, rather than applying the slow-ship linearization underlying Dawson's method.
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From page 97... ...
Thus we obtain a linear fixed-domain problem to be solved in the current iteration. Having found the solution we update He base flow field and free surface, and start Specifically, defining r1 = H + rl ' Vat = Vat + V9' and substituting this into the dynamic and kinematic boundary conditions, we obtain linearized kinematic and dynamic conditions: (5)
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From page 98... ...
It is important to note that checking these residuals is the only safe procedure; visual inspection of the changes in a hull wave profile, as is often done, does not guarantee that the solution has converged ! 2.3 Implementation To implement the mathematical model outlined above, singularity distributions are specified for all boundaries where non-trivial boundary conditions are to be imposed, i.e.
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From page 99... ...
2, in which the logarithms of the maximum residual errors in the kinematic and dynamic boundary conditions are plotted against the iteration level. numerical damping.
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From page 100... ...
Therefore wave breaking does not necessarily lead to divergence of the iteration process, and nonlinear methods like the one presented here can be a useful tool also in cases with wave breaking. But for completing the flow picture a separate criterion is required to determine whether a potential flow solution in practice will give rise to wave breaking, and some model of the breaking should eventually be incorporated.
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From page 101... ...
Dependent on the hull form the last part of the actual waterline in potential flow may be either a streamline or an envelope of streamlines that leave the hull tangentially and continue on the free surface. At the smooth detachment of the free surface from the hull surface' both the hull boundary condition and the two free surface boundary conditions must be satisfied.
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From page 102... ...
At the transom edge however, the confluence of boundary conditions poses some problems. The two kinematic conditions can be reconciled if the flow leaves the hull tangentially.
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From page 103... ...
However, at a transom edge the hull curvature is infinite, and we want to have a particular hull - free surface intersection. Nevertheless a quite reasonable representation of transom flows appears to be possible in linearized methods as well.
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From page 104... ...
Moreover, since the nonlinear method imposes the exact free surface boundary conditions, we can be more confident that the physical mechanisms governing the transom flow are fully included. 3.4 Results As the actual behaviour at the transom edge in a 3D nonlinear case is not known, I have chosen the following path for verifying the method.
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From page 105... ...
0.65 X/{, 0.85 1.05 nonlinear method Figure 8. Wave profile for varying transom edge height; linearized and nonlinear method ,500 ,200 4'900 ._ ~ 600 a> 3 300 o .
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From page 106... ...
Similarly, all RAPID calculations display the correct free surface slope at the edge, corresponding to a tangential flow off the hull; while in the DAWSON predictions only the cases with the transom edge on or below the still water level have the correct slope. The same tendencies are found in the predicted wave resistance (Fig.
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From page 107... ...
This may be an acceptable approximation for cases with small buttock angle near the transom, but not for the present case where the vertical velocity should be around 0.16. Summarizing, for the cases studied the results with the linearized method appear to be quite reasonable for all immersed transoms, with a moderate deviation due to nonlinear effects; but for transoms above the still water level the representation of the flow off the transom edge is relatively poor.
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From page 108... ...
Hull wave profile; Wigley hull, Fn = 0.316 Pig. 12 compares the wave profiles on the line of collocation points adjacent to the hull and centerline as predicted by DAWSON, RAPID, and the first RAPID-iteration which is a Neumann-Kelvin linearization.
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From page 109... ...
Also the hull wave profile (Fig.
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From page 110... ...
1 . I 0.3 0.5 0.7 0.9 X/L t.1 ~ Figure 16 Hull wave profile, series 60 CB = 0.60, Fn = 0.316 4.3 Container ship Of course further validations for more up-to-date hull forms are desired and will be carried out.
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From page 111... ...
This cannot be studied using a linearized method. Inherent to the linearization is the transfer of the boundary condition towards the undisturbed free surface.
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From page 112... ...
· A converged solution is also obtained for cases in which extensive wave breaking will occur in real flows, such as full hull forms at high speed. A separate criterion for the inception of wave breaking and a model for its effect on the flow field would be required to complete the flow picture in such cases.
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From page 113... ...
[7] Ni, S.-Y., "A Method for Calculating Nonlinear Free Surface Potential Flows using Higher-Order Panels", in: Ph.D.Thesis, Chalmers Univ., Gothenburg, Sweden, 1987.
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From page 114... ...
Have you experienced improved predictions in the wave resistance which result from improved wave profile calculations? Author's Reply It has been found that slow-ship linearized methods: - generally predict the resistance reasonably well for relatively slender vessels at fairly high speed, with substantial wave making, but often underestimate the resistance other wlse; - generally predict a negative wave resistance for full hull forms at low Froude numbers.
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From page 115... ...
To what extent these will affect the location of smooth detachment, and the flow off an immersed transom stern as well, will depend on the hull form and the thickness of the boundary layer or wake. One expects little effect for slender high-speed vessels, but much more for slower, fuller hull forms.
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From page 116... ...
and ED31. Concerning the use of linearized methods I would like to add that, although as a matter of fact the predicted wave resistance is not useful for full ships at low Froude numbers, the wave pattern predicted by our DAWSON code is qualitatively correct, as comparison with R~PID predictions has shown.
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From page 117... ...
Haussling, David Taylor Model Basin The author is to be commended on the development of a powerful method for the prediction of the flow about general hull geometries including nonlinear free-surface effects. I found the treatment of transom sterns interesting, as apparently did much of the audience, judging by the discussion that followed the talk.
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From page 118... ...
In the discussion in Section 2.5 on wave breaking in more general 3D cases, wave breaking is again not defined as the nonexistence of a potential flow solution, but as the physical phenomenon. The wave pattern in Fig.
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From page 119... ...
The hull form is approximated by means of flat triangular panels within which the source strength is assumed piecewise constant. Convergence of the computed velocity potential, wave profile, and lift, moment and drag with respect to the number of panels is evaluated.
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From page 120... ...
The parameter ~ is equal to 0 along the stem profile and 1 along the stern profile, and the parameter ~ is equal to 0 at the static wa terline and 1 at the keel. The hull form is di vided into three regions, which are defined by means of different sets of parametric equations.
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From page 121... ...
The panel configurations listed in Table 1 were not all used in all the convergence studies. These panel arrangements represent a reasonable, although not optimal, panel distribution.
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From page 122... ...
VELOCITY POTENTIAL The vertices of the 72 panels corresponding to the coarsest panel arrangement shown in the upper part of Figure 1 define 52 points. The velocity potential is evaluated at these 52 calculation points, which are panel vertices for all the panel arrangements considered in the study as was already noted.
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From page 123... ...
The upper and lower parts of Figure 3 show that errors in the calculation of the potentials ',bo and ~,600 decrease approximately in proportion to 1/N as the panel-number N increases. Potentials ¢, ~,6w and (SN The velocity potential ~6 is considered in Figures 4 and 5 for three values of the Froude number F equal to 0.1, 0.25, and 0.5, which correspond to the top, center, and bottom rows of these two figures.
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From page 124... ...
o lo 11 ~ ~ i E ~ / ~ ~ hi o o o so C o o ~ on o o ~o o o _ 11 lo D_2 :~ _ ........
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From page 125... ...
00 _ To _ ~ 2 a) Figure 5: Relative error in nonoscillatory and wave potentials at three Froude numbers.
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From page 126... ...
For a given panel distribution, the wave profile is determined using linear interpolation between the values of the free-surface elevation computed at the centers of the waterline segments. Linear extrapolation is used to determine the 126
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From page 127... ...
The fluid velocity Ax is determined, using the relations given previ ously for the wave profile, at the centroid of ev ery hull panel and regarded as piecewise constant within the corresponding panel. The derivatives B~6/ds and b¢/8t for a given panel are evaluated at the centers of the two shortest sides of the panel 127 10j NO rms error in wave profile ~uu ~' 1 NO rms error in wave profile 10 _ .
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From page 128... ...
-fexact ~ /fe~act I ~ corresponding to the lift, drag, and moment are depicted in the top, center, and bottom grids of 128 MOMENT 5 O ' .~...~ 1 o2 103 104 Number of Panels \v ~ \~\ Figure 8: Percent relative error in lift, drag, and moment defined in terms of values at moderate Froude numbers. Figure 7.
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From page 129... ...
Higher-order boundary element methods may therefore offer significant advantages in comparison to constant-panel methods. Table 2 also suggests that conclusions with respect to the benefits of nonlinear, or otherwise more refined, mathematical models could be questionable unless a sufficiently large number of panels is used.
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From page 130... ...
The magnitude of the errors associated with the prediction of wave drag using pressure integration however is not always fully appreciated. As we already noted in reply to Professor Landweber's comments, the primary aim of our study is to investigate panel convergence for conditions representing typical computational methods.
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From page 131... ...
A main finding of this study is that convergence for flat panels is quite slow, as was also shown by Professor Landweber, and that a very large number of panels is required for accurate calculations. We agree with Professor Landweber that a correction for curvature, or the use of a higher order boundary-element method, would significantly reduce the required number of panels and represents a desirable improvement over existing constant-panel methods, as is noted .
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From page 133... ...
Both the collocation method and the Galerkin method are presented to solve the system of equations. NomencIature number of panels; number of intervals between knots order of polynomials number of B-spline vertices number of collocation points on each panel degree of polynomials, N = It-1 maximum number of degrees of the polynomials expansions inflow velocity coordinates of the geometry = x+iy velocity potential XV' Yv B-spline control polygon vertices of the geometry B-spline control polygon vertices of the potential B-spline basis function of order k nth degree influence functions of the source and dipole distributions.
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From page 134... ...
We first present a collocation method, in which an overdetermined linear system is solved by least squares for the unknown potentials. A Galerkin method is then presented which uses the B-spline basis functions as the test functions.
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From page 135... ...
These are sometimes referred to in the literature as control points, but this terminology will be avoided here since this term is commonly used to designate points on a body where boundary conditions of the hydrodynamic problem are to be satisfied. While we started the discussion of B-spline curves by introducing the concept of knots, in most cases one generates a curve by selecting a set of vertices.
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From page 136... ...
along the surface of the body. With the widespread use of potential based panel methods for both two and three dimensional flow problems, it is well known that the potential at a point on the body surface can be expressed in the form of integrals of source and dipole distributions over the body surface.
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From page 137... ...
) The source strength B-spline coefficients, aj~k, are known from the boundary condition, and the dipole moments are unknowns to be solved.
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From page 138... ...
~ Introduction Both the panel geometry and the basis functions for the singularity distributions are defined by polynomials of the general form N f(t) = ~ futn (16)
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From page 139... ...
-f In the derivations of the multipole expansions (22)
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From page 140... ...
, and nX + ins = i ,d~ /~ dt = i At / dt (40) From these relations it follows that the dipole integral can be evaluated in the form In 5.5 Multipole expansions For sufficiently large values of jaw-zoo the logarithmic function in (38)
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From page 141... ...
(59) The local multipole expansions for the subdivided panel are then evaluated in the forms N N In = log (zO-w)
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From page 142... ...
(75) The normal dipole integral is then given as 142
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From page 143... ...
M-1 in = i As, (m + 1)
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From page 144... ...
6. ~ The Collocation Method In this system of equations, the boundary condition is satisfied at collocation points, and the unknowns to be solved are the B-spline polygon vertices of the dipole moments.
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From page 145... ...
where jv,m is the mth B-spline vertex of the dipole moment, and the coefficients /3k can be obtained by expanding the basis functions N JV-m (t)
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From page 146... ...
The Galerkin method used here is the Bubnow-Galerkin method, in which the test functions are selected to be the same as the interpolation functions.
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From page 147... ...
However, the left-hand-side matrix size of the Galerkin method will be smaller than that of the collocation method. For a cubic higher order panel method, with L panels and NC collocation points, the collocation methods will give an L*
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From page 148... ...
the maximum error of the solutions of different order panel methods for the case of a uniform inflow past a square. _` us c' 1 0 z', ~q ~ A_ z .~ 10 En a .° z ~ 1 ~4 o V · ~ ~ ~ ~ Constant Potentials o o o o ~ Cubic Potentials 2 L/4=32 ~L/4= 128 ~ \ \~64 \\2 ~6 1% relative error 8 8 2 i B ~ 2 4 B ~ 1 2 10-3 lo-2 10 Absolute Error Figure 5: The absolute error (maximum error of the solutions)
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From page 149... ...
vs. the computational time of different order panel methods for the hydrofoil case.
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From page 150... ...
That is, for the same error, the matrix size that the constant potential method needs is about 600 times larger than that of the higher order collocation method, and 1400 times larger than that of the higher order Galerkin method! Therefore, the presented method also has the advantage of using less computer resources.
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From page 151... ...
It is hoped that this method will not only be able to provide more accurate solutions than the low order panel method, but also be more effective of solving problems such as wave body interactions, and lifting problems. Acknowledgements This work was performed as part of the Joint Industry Project "Wave effects on offshore structures", and also under the MIT Sea Grant College Program with support provided by the David Taylor Model Basin.
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