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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 792 Wave Resistance Computations—A Comparison of Different Approaches S.Gatchell, D.Hafermann, G.Jensen, J.Marzi, M.Vogt (Hamburgische Schiffbau Versuchsanstalt GmbH, Germany) ABSTRACT Steady state free surface computations are still of utmost importance to all model basins. Among the various computational approaches for both, boundary element type as well as volume type methods that have been presented over the past years, none are fully convincing in terms of stability and accuracy for arbitrary ship hulls, being the “principle” business of any model basin. HSVA is developing free surface codes aiming at a most versatile application. Three of these methods are compared in the present paper, showing example computations for a container ship and a fast monohull. 1. INTRODUCTION Steady state free surface flows are still a top priority issue with any model basin. As we do not expect developers of generalised flow codes to provide optimal solutions for this very special Naval Architectural problem, HSVA has dedicated substantial effort and resources to the development of specialised methods suitable for practical applications in ship design and hull form evaluation as well as optimisation. These developments follow three principal lines: 1. The iterative non-linear free surface potential flow code SHALLO (Jensen at al. 1986), which has long been used in hull form design at HSVA and in several other installations, underwent a major renewal, aiming at a more accurate prediction of the wave making resistance of ships. The major changes to the method were the introduction of a new panel type based on desingularised sources and using an integral mass flux boundary condition on the body instead of the usual Neumann condition implied on discrete collocation points used in typical panel codes. Introducing a continuous re-paneling of the hull up to the actual wetted surface, the new code works on arbitrary grid configurations built from triangular and quadrilateral patches. The results obtained with this code are encouraging. 2. The VOF-Code is based on the Euler Equations. The method computes a single phase flow and utilises a Volume-of Fluid (VOF)—approach to describe the free surface. Although it still neglects viscous effects in the flow field, details like breaking waves in the vicinity of the free surface are predicted. Test cases indicate significant improvements in the wave field computed and in the numerical prediction of the wave resistance. The current version of the code is implemented in a multi-block mode. It is parallelised using the message passing paradigm and runs on shared and distributed memory computers. 3. Additionally a commercial RANSE-solver—Comet is applied. This code also allows to compute the free water surface. The method is similar to the one described under 2, but two phases are considered. 4. In another finite volume solver, a level set technique introduced by Osher & Sethian, is used to capture the free surface between two distinguished phases, water and air. In the formulation used here, the governing equations are solved in both the water and the air domains and the two phases are considered as one. A level set function, , defined in both phases is initialised as the distance, with sign, to the undisturbed free surface. Instead of solving the two constitutive equations for density and viscosity, which would cause numerical difficulties, the scalar function , which is continuous even under changes of topology, is convected by the velocity field. Depending on the sign of , the density and the viscosity are given the values of water and air, respectively. Compared to front tracking methods to compute free surface flows, no regridding is necessary and no boundary conditions are applied at the free surface. Further, the position of the free surface need not be explicitly evaluated during the computation, but is carried out in the post-processing. The steep density gradients at the free surface necessitate the interpolation of the physical properties over a few cells around the free surface. However, the interface itself remains sharp. Unfortunately, no stable computational results for the selected test cases were available at the submission deadline for this paper. Hence, the method had to be the authoritative version for attribution. excluded. Important details of the three different approaches, including the mathematical formulation of the free surface conditions used are shown in the following

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 794 In the initial step, an undisturbed free surface is assumed. For each collocation point on the free surface, the flow quantities like velocities and accelerations that are used in the free surface boundary condition are interpolated from the previous step. The first iteration begins by assuming parallel flow with the ship velocity. Point sources are generated above each collocation point on the free surface mesh. They are located at a distance depending on the longitudinal spacing of the undisturbed free surface. To enforce the radiation condition there is no point source above the most upstream collocation point in each row. Instead, there is an extra point source downstream from each row of collocation points (Fig. 2.4). Then a system of linear equations is set-up treating the strengths of the point sources as the unknowns. There are two types of equations: On each collocation point on the free water surface, the combined kinematic and dynamic free surface boundary conditions shown in Jensen, et al. 1986, linearised around an approximate solution for the free surface location and flow potential, is applied. (2.3) On each patch on the submerged body, an equation is set-up requiring the total flow across the patch to be zero (Soding 1993). Thus, a linear system of equation with a full coefficient matrix and relatively weak diagonal is derived. To solve for the unknown source strength a solver combining elimination and iteration steps is used. The potential and its derivatives are then easily determined on each collocation point and panel corner from (2.2). Using the potential and its derivative on the patch corners, the pressure is computed to determine the pressure force. Also the hydrostatic term in Bernoulli's equation is considered in the pressure integration. In addition, the square of the velocity on the body is determined to compute an approximation for a friction form factor: (2.4) This form factor accounts for the change in wetted surface as well as for the inhomogeneous velocity distribution. It should be noted that k must not be confused with the form factor determined as the zero Froude-Number approximation from experiments. Fig. 2.2: “Initial Panel mesh for Trim and sinkage are estimated based on the vertical forces, and the body grid is moved accordingly. potential flow computations—Kriso The wave elevation at the collocation points is computed from Bernoulli's Containership” equation. The wave elevation along the hull is determined by projecting the innermost row of collocation points onto the hull along the local slope of the water surface. This line is used to determine the wetted part of the body grid for the next iteration. Fig. (2.3) shows the development of the body grid during several iteration steps. EXAMPLE COMPUTATIONS Computations have been performed for two well known test cases: i) the KRISO Fig. 2.3: “Wetted surface mesh, start containership as specified for the so called Gothenburg workshop (Gothenburg of iteration—Kriso Containership” 2000) and ii) the Athena hull previously used in a number of experimental and computational case studies (HSVA 1994). For the first test case a panel mesh has been generated based on the IGES data provided for the Gothenburg workshop (Gothenburg 2000) using ICEM/CFD mesh generator. Fig. 2.4: “Wetted surface mesh, end of iteration—Kriso Cotainership” the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 795 The body mesh has 2177 panels. For the water mesh a grid refinement study has been performed using fewer points (V1), a default mesh (V2) and a refined mesh (V3). The factor used to increase grid points per step was The surface meshes are shown in Fig. 2.5. Fig. 2.5: “Grid refinement steps—Kriso Containership” ν-SHALLO results obtained on three different free surface meshes are shown in the following figures 1.6 to 1.8. The conditions used here are (full scale): v=24 kts, FN=0.2599, T=10.8 m. In order to compare with the experimental data given, the results were re-scaled to model conditions. The following figure 1.6 shows resistance coefficients (cW, cR and cT) and a comparison to the experimental data. The two upper curves show the total resistance coefficient taken from measurements (index EFD) and from our computation (index CFD). Fig 2.6. “Grid dependency study—Kriso Containership” Fig. 2.8: “Pressure distribution cP on wetted hull—Kriso containership” Fig. 2.7 shows the wave pattern obtained on the fine mesh (top) and on the coarse mesh (bottom). It can be seen that the fine mesh shows a smoother wave pattern, characterised by smaller wave elevations than the coarse mesh. Comparing those to the authoritative version for attribution. Fig. 2.7: “Kriso containership—computed wave pattern for the fine mesh (top) and for the coarse mesh (bottom)”

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 796 results published in (Gothenburg 2000), it is clear that the finer mesh is closer to experimental data. For the second test case, the Athena hull, a panel mesh was generated based on an existing NAPA model, using NAPA's new panel editor (NAPA 2000). This is presently the predominant input generator for the v-SHALLO code. Fig. 2.9 “Panel models—Athena test case” A study of the sensitivity of results on the discretisation of the hull was also performed. Two different hull models have been created, the first one having 684 panels, the second one with 2300 panels. The water surface has been refined from 564 to 1242 and finally to 2426 panels. Figure 2.9 shows the two different panel meshes on the hull surface. The conditions used for this exercise were v=30 kts, ∇ FN=0.73, T=1.5 m for the full scale ship. The wave pattern obtained on the fine and on the coarse mesh (water surface) are shown in fig. 2.10. Here—in contradiction of the previous exercise—the finer mesh shows substantially higher wave elevations, this in turn being more realistic than the rather “shallow” wave pattern obtained on the coarser mesh. Similar results as for the previous case are plotted in fig. 2.11. Fig. 2.11: “Grid dependency study—Athena” Fig. 2.10: “Athena—Comparison of wave pattern: fine mesh (top), coarse mesh (bottom)” the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 797 3. VOLUME OF FLUID EULER METHOD DESCRIPTION OF METHOD During the past years, methods for solving the Navier-Stokes-Equations have been developed which describe the free surface by adapting the mesh boundaries. These methods, however, only apply to smooth free surfaces. Here the development and application of a Finite-Volume-Code using the Volume of Fluid (VOF) method for the simulation of inviscid, unsteady free surface flows will be described. This method describes the position of the free surface by the void fraction in each computational cell. Two applications will be shown. The transient flow of an inviscid and incompressible fluid is considered. It is described by the conservation equations for mass and momentum, formulated in an inert Cartesian coordinate system. The momentum equations are written as (3.1) where denotes the velocity vector with its components in x, y and z direction, the pressure, ρ the density and the gravity vector. Further variables are the control volume Ω, the velocity of the control volume vb=(ub, vb, wb) T, the surface area S and the normal vector of the control volume surface The pressure is the sum of the dynamic and the hydrostatic pressure (3.2) The vector denotes the position of the free surface, the gravity vector is normal to the undisturbed free surface. For a completely filled control volume the conservation equation for mass is (3.3) To describe the free surface, a function is introduced with a value of either 1 denoting liquid or a value of 0 denoting gas. The function is a discontinuous function. The free surface is the boundary line where changes from 0 to 1. Fluid particles above the free surface will always have a value of 0 and particles below a value of 1. This yields a conservation equation for which is independent of the position of the control volume (3.4) Below the free surface, this leads to a mass conservation equation. The cell fill ratio F is the ratio of the cell volume filled by the liquid to the total cell volume (3.5) By definition this ratio has values between 0 (empty) and 1 (full). At the free surface ambient pressure is assumed: (dynamic boundary condition). Fig. 3.1: The Flowchart of the algorithm the authoritative version for attribution. The figure above shows the algorithmic scheme. At the beginning of every physical time step, the VOF-function F is calculated.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 798 The surface integral needed for the FV formulation is approximated using the product of the flux and the open cell surface ratio. The open cell surface ratio oe of the side e is defined as follows (3.6) Since the cell fill ratio F is a discontinous function, the cell surface ratio cannot be obtained by interpolation and must be calculated iteratively. This leads to a description of the free surface. The velocities close to the free surface are extrapolated using the neighboring values. Afterwards, the SIMPLE Algorithm is applied to solve for the velocities and the pressure. Starting with an approximated solution for the pressure, the linearised momentum equations are solved. To satisfy the continuity equation the pressure and the velocities are corrected by applying the so-called pressure correction. The two steps are repealed until the required accuracy is reached. In addition to the dynamic boundary condition at the free surface, several other boundary conditions must be applied: 1. At the inlet, the velocity is fixed and the pressure is extrapolated, 2. a the outlet, the pressure is fixed and the velocity is extrapolated, 3. at the symmetry planes, fluxes are set to zero said the pressure is mirrored, 4. at the wall, the fluxes are set to zero and the pressure is extrapolated. The waves are damped in direction of gravity by applying an artificial force, which depends on the velocity component w and the distance from the ship. EXAMPLE COMPUTATIONS Computations have been performed for the above mentioned KRISO containership and Athena hull. The conditions used are the same as in the previous computations. The meshes are generated using a HSVA internal mesh generator. For the KRISO test case three meshes with approx. 47000 (coarsest), 160000 and 380000 cells (finest) were generated. The coarsest mesh is shown in figure 3.2. Fig. 3.2: Coarse Grid for the KRISO Containership. The total simulation time is 200 seconds. The flow is accelerated from 0 m/s to u∞ during the first 40 seconds. Figure 3.3 shows the resistance coefficient cw over the simulation time. The finest mesh yields a lower final cW value of 0.0125. The finest mesh calculation also produces smaller oscillations. Fig. 3.3.: Resistance coefficient cW over time—KRISO containership The wave pattern at t=200s lor the KCS test case are shown in figure 3.4 for the coarsest and the finest grid. Both computations show the same overall wave pattern. The finest grid, however, gives a significantly finer resolution of the wave structure. Figure 3.5 compares the wave distribution of the measurements and the simulation on the finest grid. The agreement is very good, although the effect of the artificial damping force can be clearly seen at the outer boundary. The pressure distribution cP for the fine KRISO grid is shown in figure 3.6. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 799 Fig. 3.5: Measured (top) and computed (bottom) wave distribution—KRISO containership Fig. 3.4: Computed wave patterns for the—KRISO containership. (top: coarsest grid, bottom; finest grid) Figure 3.6 Pressure distribution cp on the wetted hull—KRISO containership (fine grid). The grid for the Athena test case consists of approximately 250000 cells. The total simulation time is 80 seconds. The computed wave distribution is shown in figure 3.7. The separation at the stern and the development of the waves can be seen. Fig. 3.7 Computed wave pattern—Athena test case Fig. 3.8 Pressure distribution cp on the wetted hull-Athena test case the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 800 4. RANS-SOLVER COMET DESCRIPTION OF METHOD Comet solves the RANS equations using a well established finite volume method [1]. Free-surface predictions are performed using the front-capturing method. This method requires a kind of a two-phase model. The physical properties (density ρ and viscosity µ ) of the effective fluid depend on physical properties of constituent fluids (e.g. fluid 0 and fluid 1) and a scalar indicator function, known as volume fraction C, according to the following expressions: (4.1) (4.2) where the subscripts 0 and 1 denote the two constituent fluids (e.g. water and air), the function C is used to distinguish between two different fluids. A value of one indicates the presence of fluid 1 and the value of zero indicates the fluid 0. Volume fraction values between these two limits indicate the presence of the interface. The volume fraction C and the mass fraction c of fluid 1 are linked by the expression (4.3) The transport of c is governed by (4.4) It is assumed that for the portion of the solution domain where c has a value between 1 and 0, both fluids share the same velocity and pressure. In this way, this free-surface capturing model reduces to modelling of multi-species flows, where fluid 1 is treated as species and fluid 0 as background fluid. The method is relatively simple on regular grids, but it is difficult for unstructured grids. The free-surface capturing method is based on a convective transport of a scalar quantity which indicates the presence of one of the fluids involved in the free-surface flow. The real or physical interface is sharp and so it should be in the numerical simulation. The use of the UD scheme causes very strong smearing of the interface. The CD preserves the sharpness of the interface but at the same time introduces non-physical oscillations around the interface and produces values of the volume fraction which are beyond the physically meaningful bounds of 0 and 1. The high-resolution interface capturing scheme (HRIC) is designed to overcome these problems and to model accurately the transport of sharp interfaces. At the outlet a pressure boundary is defined. The static pressure at the boundary change due the hydrostatic effects caused by presence of the earth acceleration. EXAMPLE COMPUTATION Computations have been performed for the KRISO-Containership using the same conditions as above. The unstructured grid with approx. 450.000 cells was generated by using ICEM-CFD grid generator. The computations were performed on a PC-cluster with 8 nodes [2]. The wave pattern are shown in figure 5.2 in the following chapter. The agreement with the measurement is good in the neighborhood of the hull. The wave height at the ship is shown in fig. 4.1. Fig. 4.1 Water surface elevation along Ship surface for the KRISO Containership (Comet Calculation) The predicted pressure distribution (cP) is shown in fig. 4.2 Fig. 4.2: “Predicted pressure distribution -(Comet computation)” the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 801 5. COMPARISON OF RESULTS Fig. 5.1: Comparison of VOF-Euler results (above) and Potential flow insults (below) To compare the different approaches we consider the computed wave patterns, the pressure distribution and the computed resistance from both methods for the finest grids used for the Container Vessel and the Athena test cases. Fig. 5.2: “Comparison of wave pattern—Comet—RANSE solution (top), experimental data (bottom)” WAVE PATTERNS For the KRISO containership test case computations were performed for the model fixed in trim and sinkage, as in the measurements. The wave pattern computed with v-SHALLO clearly shows a distinct Kelvin wave pattern, while in the VOF-Euler-solution the waves disappear at some distance from the hull. This is due to the artificial damping force mentioned above. The bow wave is more pronounced in the VOF result. This is expected because the potential flow calculation cannot resolve extremely sleep waves. Also the forward shoulder wave trough is deeper in the VOF solution. The authors believe that this could be some canal effect due to the limited space that is discre the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 802 tised. The details of the stern waves are completely different. In the VOF-RANS solution local bow wave as well as the wave trough at the forward shoulder are even more pronounced than in the previous cases. The far field however shows only limited similarity with the expected Kelvin wave pattern. For the Athena test case the differences are much more significant. The VOF code gives steeper and more pronounced waves. The orientation of the main wave trough is different. The VOF code also predicts the complicated structures typically shad of the transom edge of semi displacement hulls. PRESSURE DISTRIBUTION AND RESISTANCE The overall pressure pattern on the Container-vessel hull is comparable except for the higher pressure and local wave height near the stem. The residual resistance coefficients from the finest grids for the considered Froude-Number of 0.2599 are: 1000 CR Potential Flow Computation 0.47 VOF-Euler Computation 1.25 VOF-RANSE Computation 1.45 Measured at KRISO 0.73 In case of the Athena semi displacement hull at Froude-Number 0.729 the results for the resistance as compared to measurements are as follows: 1000 CR Potential Flow Computation 1.45 VOF-Euler Computation 0.92 Measure at HSVA 2.1 This shows that the quantitative accuracy is still not satisfactory. 6. PRACTICAL CONSIDERATIONS The free surface potential flow code v-SHALLO provides results which describe the global wave pattern and pressure distribution well. Although the computed wave resistance values are much more realistic than from earlier versions of the code, the quantitative accuracy is still not sufficient. Nevertheless it is an efficient tool for hull form development, because grid generation takes typically less than one hour and the computation, even for the finest grid is finished in less than two hours on a Pentium PC. In the practical hull form development cycle, much more time is needed to modify the hull form in the CAD system. For simple geometries the grid generation for the VOF-Euler code is also rather fast and straight forward. For more complex hull forms it is however very time consuming to generate grids with sufficient quality. Although the method obviously can reproduce details of the physical flow much better than the potential flow code, there is no advantage in accuracy for the resistance in the considered examples. The strong dependency of the results on grid resolution indicates that much finer grids are needed. However the computation time for the finest grid was already 30 hours on a single Pentium PC. For the third case, the VOF-RANSE computations the time required to produce the computational grid is certainly the highest in the present comparison. Today's practice using ICEM/CFD as grid generator allows for approx. 1 day for grid generation. The grid size used for this exercise was 450 thousand cells, the computation was performed using 8 nodes of a parallel machine (based on Pentium 600), lasting approx. 25 hours for 12000 iterations (t=120 s). 7. CONCLUSIONS The results presented in this article are promising. This holds less for the accuracy of global results as for the resistance of a ship but certainly for the flow field phenomena such as the wave pattern or pressure distributions. It is obvious that for practical applications potential flow computations are not yet surpassed. Nevertheless, the field type methods solving either Euler or RANS equations are more promising for the future. Today's design process relies heavily on the use of potential flow codes for design optimisation. The present investigation shows that more refined results especially in terms of details predicted in the flow field can be expected from VOF or level set methods. As computation time is still a major concern for these predictions, considerable effort is presently spend on parallelising codes and running them on PC clusters. 8. REFERENCES the authoritative version for attribution. G.Jensen, Z.-X.Mi and H.Söding, 1986: Rankine Methods for the Solution of the Steady Wave Resistance Problem, Sixteenth Symposium on Naval Hydrodynamics, 1986 H.Söding 1993: A Method for Accurate Force Calculation in Potential Flow, Schip Technology Research, Volume 40, 1993: Gothenburg 2000: http://www.iihr.uiowa.edu/gothenburg2000/

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 803 HSVA 1994: Numerische Verfahren zur Antriebsleistungsprognose, HSVA Bericht 1587, 1994 (in German) NAPA 2000: ‘NAPA user meeting 2000', various contributions. C.Schumann, 1999: Berechnung von reibungsfreien Schiffsumströmungen unter Verwendung einer „Volume of Fluid“-Methode zur Beschreibung derfreien Wasseroberfläche, Ph. D. Thesis, Hamburg 1999 C.Schumann, 1998: Berechnung von Schiffsumströmungen mit brechenden Wellen, HSVA Report 1624 I, Dimirdzic; A.D.Gosman, R.I.Isa, M.Peric (1987): A calculation procedure for turbulent flow in complex geometries, Computers & Fluids, 15, 251– 273 (1987) E.Schreck, M.Peric: Computation of fluid flow with a parallel multigrid solver”, Int. J. Numer. Methods in Fluids, 16, 303–327 (1993) the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line WAVE RESISTANCE COMPUTATIONS—A COMPARISON OF DIFFERENT APPROACHES 804 DISCUSSION H.Bingham Technical University of Denmark, Denmark I am surprised that you are able to conclude that Euler codes are not worth pursuing based on un-converged results. Wouldn't you expect an Euler solution to converge using significantly fewer grid elements than a RANS solver? AUTHOR'S REPLY I agree that my conclusion maybe premature. It is however based on the experience that no substantial savings in computational effort or grid generation have been observed. Therefore, I believe that development work should be concentrated on RANSE applications. DISCUSSION M.Ryuji University of Tokyo, Japan This paper's Rankine Source Method used uniform flow at first value as base flow of free surface. But recently, double model flow or exact flow is used to Rankine Source Method as base flow. How about do you think for base flow? AUTHOR'S REPLY According to our experience you save at most one iteration step if you use a double model solution as the start up. This does not seem to make the extra complexity of the code worthwhile. I do not understand what you mean with the term “exact flow” in the context of the start up of a panel method. DISCUSSION L.Raheja Indian Institute of Technology, India You have mentioned that in your first code, you used desingularized panel method in place of conventional panel method. I would like to know, how much advantage you gain by using this method over the conventional one because sometimes you require to make adjustments in the distance from the wall where the source is placed. AUTHOR'S REPLY G.Jensen In using desingularized panel method we used the boundary condition in terms of no-flux across the panel and we did not find any problem with this and we have been using it for last ten years. We even use it for free surface boundary. The computation is very fast because of no singularity being present. the authoritative version for attribution.