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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 941 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 METHOD Analysis of Turbulence Free-Surface Flow around Hulls in Shallow Water Channel by a Level-set Method H.H.Chun, I.R.Park, S.K.Lee (Pusan National University, Korea) ABSTRACT In the present study on turbulence free surface problems in shallow water channel, two fluids Reynods averaged Navier-Stokes equations are solved by using a Finite Volume Method, where SIMPLEC algorithm is used for velocity and pressure coupling, and standard k-ε turbulence model is introduced for modeling Reynolds stresses. A Level-set method is used for capturing the free-surface movement and the influence of the turbulence layer of the free surface is implicitly considered. For the validation of the present numerical scheme, the numerical results for Wigley and Series 60 Cb=0.6 ships in deep water are compared with the experimental results. Computations are made for various depth Froude numbers for the calculations of the shallow water channel flow. In the numerical results, the present solutions show good agreements with the experimental results for the deep water case, and for the case of the shallow water solutions with the viscous effect, present numerical results show reasonable physical phenomena. In addition, it is demonstrated that the level-set method can treat the free surface flows around hulls with a reasonable accuracy together with a simple numerical procedure. INTRODUCTION Shallow-water channel flow near the critical depth Froude number is an unsteady, nonlinear phenomenon and has peculiar flow characteristics, where h is water depth, U is the ship speed and g is the gravitational acceleration. At this critical speed, a ship generates two-dimensional waves propagating in front of the ship, which are faster than the ship speed and show the unsteady flow patterns. These waves are named solitons or solitary waves. The influences of the channel wall and shallow water cause the increase of the resistance and sinkage of the ship at near the critical speed. In the experimental investigations, Thews and Landweber (1935), Helm (1940), Graff et al (1964) and Ertekin (1984) observed these unsteady and nonlinear waves in shallow water towing tanks. Especially, Ertekin (1984) carried out a series of experiments in which certain parameters such as water depth, ship draft and channel width were changed. For the inviscid flow problems, Bai and Kim (1989) used a Finite Element Method for solving a nonlinear free- surface flow for a ship moving in restricted shallow water tank. In case of the numerical works, Choi and Mei (1989) used Kadomtsev-Petviashivili equations and Ertekin & Qian (1989) and Jiang (1998) used Boussinesq equations for solving the nonlinear shallow water waves. Kim and Lee (1996) investigated these phenomena based on Euler equations. For the viscous solutions in shallow water channel, Bertram and Ishikawa (1997) used the hybrid approach computing first squat and potential flow with free-surface calculation and then the viscous flow without free-surface effect at the subcritical depth Froude number. In their numerical results, the pressure on the channel bottom and hull surface are compared with experimental results, where it is explained that the discrepancies are caused by disregard of the deformation of the free-surface. In the present work, the viscous and free-surface effects are considered in the calculation of the shallow water channel flow at the critical and super-critical depth Froude number speeds. For the analysis of turbulence flow, two fluids Reynols averaged Navier-Stokes equations are solved by using a Finite Volume Method, where standard k-ε turbulence model is used for modeling Reynolds stresses. For the free surface treatment, two main approaches (front tracking & front capturing methods) have been used. In front capturing methods, the level-set scheme has been only recently used in free surface problems (Vogt 1998, Dommermuth et al. 1998, Bet et al. 1998 and Park & Chun 1999 (a), (b)). The level-set method is a numerical technique which can follow the evolution of interfaces. These interfaces can develop sharp corners, break apart, and merge together. The level-set method has a wide range of applications, including problems in fluid mechanics, combustion, manufacturing of computer chips, computer animation, image processing, structure of snowflakes, and the shape of soap bubbles. Especially for many complex free surface problems (e.g. breaking wave, spray the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 942 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 METHOD phenomena, slamming problem, water-exit problem and bubble problem) in the naval hydrodynamics, this method is a provocative approach and could be used as a robust numerical scheme. In the present work, this level-set method is introduced for capturing the free-surface movement and implicitly considering the influence of turbulence layer of the free surface. In order to validate the present numerical scheme, the experimental results of Wigley and Series 60 ship at deep water condition are compared with the present numerical results. In the numerical results of the shallow water flow, the wave patterns, pressure distributions on the hull and the friction and pressure resistances computed at different flow conditions are compared with each other and pertinent discussions are included. From these numerical results, the validation of the level-set method can be also checked. MATHEMATICAL FORMULATION Governing Equations In the three-dimension problem, the general integral forms of the time-averaged conservation equations of mass, two- fluid incompressible Navier-Stokes equations and turbulent kinetic energy dissipation rate can be written as (1) where u is the velocity of fluid, q is any conservative quantity, Γq is the associated diffusive coefficient and Sq is the volumetric source term of q. In the present thesis, the standard k-ε turbulence model is used for the turbulence flow. Equation (1) can be written in the vectorized form as (2) where vectors U, Fconv−Fdiff and B are defined as (3) where p: the pressure, k: turbulent kinetic energy, ε: turbulent dissipation rate, µ eff=µ+µt, µ t: turbulent eddy viscosity and Gk: rate of production of turbulence kinetic energy is defined as (4) Turbulent constants of the standard k-ε turbulence model is defined in Table 1. Table 1 Turbulence constants of the standard k-ε turbulence model the authoritative version for attribution. LEVEL-SET FORMULATION As seen in Fig. 1, the immiscible and incompressible two-phase fluids are described by their densities (ρ1, ρ2) and viscosities (µ 1, µ 2), where these physical

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 943 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 METHOD reinitialization procedure, sign(d0)=1 if d0>0, sign(d0)=−1 if d0<0, and sign(0)=0. Second order ENO scheme is used for spatial derivative |∇d| in equation (14) and implicit Euler scheme is used for the time integration. The idea of this equation is that a steady-state solution will be a signed distance function which has |∇d|=1 near the interface with the same zero-level-set as the initial function d0(x). This method generally works well when level-set gradients are neither too flat nor too steep near the interface. The value d(x, t) propagates with speed ±1 along the characteristics that are normal to the interface and converges quickly in a neighborhood of the interface for small time t. The fact that d0(x) is already a good guess for the distance function is effectively used for obtaining the convergence solution after a small time t. Because the interface moves a little, the previous information can be effectively used to update the new level-set distribution by an iterative scheme. The width of the neighborhood around the interface can be defined α∆h, where α>0 and it is sufficient to solve equation (14) up to time α∆h. Since the characteristics are propagating away from the interface with speed of unity, the appropriate time step according to the CFL (Courant Friedriches Lewy) condition is Local Level-Set Method According to the locality property of the level-set method, it is sufficient to calculate the level-set function only in a small narrow band around its zero-level-set for the reinitialization procedure. By doing so, in the case of two-dimensional computation, let N×N be the number of grid point, then computational expense reduces from O(N2) to O(N) (Peng et al. 1999). A PDE-based local level-set method was developed by Peng et al. (1999). The local level-set method described in their paper can be summarized as follows. The level-set function must stay well behaved except for isolated points for numerical accuracy: (16) where c & C are some constants. If the level-set function around the interface is an exact signed distance function, the magnitude of the level-set function gradient at the interface must be unity, namely, (17) The localization makes it only necessary to perform the evolution and reinitialization of the interface within a narrow region around the front. Let 0<β<γ be two constants that will be determined according to the grid size. Around the interface Γ0, a tube with width γ can be defined by (18) Let be a cut-off function: (19) The interface Γ0 is updated by solving the following equation, (20) on T0 with initial data The cut-off function is to prevent numerical oscillations at the tube boundary. Then the new location of the front is given by (21) where is an intermediate level-set function. Let d1(x) be the signed distance function to Γ1. The shifted tube can be defined by (22) A new level-set function would be the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 944 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 METHOD reinitialization procedure, sign(d0)=1 if d0>0, sign(d0)=−1 if d0<0, and sign(0)=0. Second order ENO scheme is used for spatial derivative |∇d| in equation (14) and implicit Euler scheme is used for the time integration. The idea of this equation is that a steady-state solution will be a signed distance function which has |∇d|=1 near the interface with the same zero-level-set as the initial function d0(x). This method generally works well when level-set gradients are neither too flat nor too steep near the interface. The value d(x, t) propagates with speed ±1 along the characteristics that are normal to the interface and converges quickly in a neighborhood of the interface for small time t. The fact that d0(x) is already a good guess for the distance function is effectively used for obtaining the convergence solution after a small time t. Because the interface moves a little, the previous information can be effectively used to update the new level-set distribution by an iterative scheme. The width of the neighborhood around the interface can be defined α∆h, where α>0 and it is sufficient to solve equation (14) up to time α∆h. Since the characteristics are propagating away from the interface with speed of unity, the appropriate time step according to the CFL (Courant Friedriches Lewy) condition is Local Level-Set Method According to the locality property of the level-set method, it is sufficient to calculate the level-set function only in a small narrow band around its zero-level-set for the reinitialization procedure. By doing so, in the case of two-dimensional computation, let N×N be the number of grid point, then computational expense reduces from O(N2) to O(N) (Peng et al. 1999). A PDE-based local level-set method was developed by Peng et al. (1999). The local level-set method described in their paper can be summarized as follows. The level-set function must stay well behaved except for isolated points for numerical accuracy: (16) where c & C are some constants. If the level-set function around the interface is an exact signed distance function, the magnitude of the level-set function gradient at the interface must be unity, namely, (17) The localization makes it only necessary to perform the evolution and reinitialization of the interface within a narrow region around the front. Let 0<β<γ be two constants that will be determined according to the grid size. Around the interface Γ0, a tube with width γ can be defined by (18) Let be a cut-off function: (19) The interface Γ0 is updated by solving the following equation, (20) on T0 with initial data The cut-off function is to prevent numerical oscillations at the tube boundary. Then the new location of the front is given by (21) where is an intermediate level-set function. Let d1(x) be the signed distance function to Γ1. The shifted tube can be defined by (22) A new level-set function would be the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 945 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 METHOD constructed from (23) In the region the motion is modified by the cut-off function. Outside the tube T0, is not updated at all. Since the interface moves less than one grid size, the region for the reinitialization can be defined by (24) Next the reinitialization is performed, and d1(x) is obtained. Conclusively, new level-set function is defined by (25) The numerical values of β and γ depend on the width of the stencil of the schemes used to approximate the spatial derivatives. In the present thesis, β=2∆h and γ=4∆h are taken for a second-order ENO scheme (Harten et al. 1987). LEVEL-SET FORM OF NAVIER-STOKES EQUATIONS The motion of incompressible fluids is described by the conception of the level-set form of Navier-Stokes equations. It is assumed that two fluids are governed by the incompressible Navier-Stokes equations. The stress tensor can be defined as the following equation: (26) where I is the identity matrix and is the rate of deformation tensor. The equations of motion in each fluid domain become (27) (28) where boundary condition on the interface becomes (29) where is the curvature, is the coefficient of surface tension and n is normal vector. Let Ωl and Ωg denote an arbitrary portion of the liquid phase and gas phase. Let Ωl∪g (=Ωl∪Ωg) contains the gas- liquid interface Γl∪g. By integrating the equations of motion in each fluid domain the following equation can be obtained (30) The following relation can be used. (31) Then, equation (30) can be written in the following form: (32) Finally, the level-set form of Navier-Stokes equations for two fluid flows can be written as follows: (33) Conclusively, the level-set form of the two fluid Navier-Stokes equations contains implicitly gas-liquid boundary conditions. NUMERICAL PROCEDURE In the discretization process, all unknown values and physical properties are computed and stored at the center of the control volume. Interpolation and differentiation are necessary to evaluate the convective the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 946 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 METHOD and diffusive fluxes at the cell-face center and each flux can be approximated by using the midpoint rule as follows: (34) (35) (36) The hybrid scheme is used to approximate convective fluxes. For the approximations of the diffusive fluxes, the gradient vector at the cell face can be calculated from the gradients at the cell centers approximated by using midpoint rule based on the Gauss theorem: (37) But this approach may cause oscillatory solutions (Ferziger and Perić 1996). Muzaferija (1994) noted that the following implicit expression is to fit for approximations of diffusive fluxes when the orthogonal regular grid system is used. (38) As seen in Fig. 2, when the grid is irregular, the line connecting points P and E does not pass through the cell face center e. For the case of a non-orthogonal and irregular grid system, Muzaferija (1994) suggested an effective scheme, ‘differed correction method', to prevent the oscillatory solutions: (39) where the unit vector in the ξ-direction and overbar denotes interpolation from neighbor nodal values. (40) (41) (42) (43) If the line connecting points P and E is orthogonal to the cell face, the old-bracket term is zero and usual the central difference approximation of the derivative is recovered. The explicit term at the cell face is used to consider the influence of the cross-derivative. On a non-orthogonal irregular grid system, values of fluid properties at the cell face center can be approximated as (44) The implicit Euler scheme is used for the time integration as follows: the authoritative version for attribution. (45) The discretized linear algebraic equation system can be solved by using the SIP (Stone 1968) solver. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equation Consistent)-algorithm is used for pressure-velocity coupling on cell- centered grids (van Doormal and Rathby 1984). NUMERICAL RESULTS In order to check the accuracy and stableness of the present numerical scheme, the steady waves for two ships (Wigley and series 60 ships) in deep water are

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 947 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 METHOD calculated and compared with experimental results published. In addition, the numerical results published for these two ships by other CFD tools are widely available and the accuracy and effectiveness of the present numerical results can be indirectly compared. The computational conditions for Wigley ship and Series 60(Cb=0.6) ship are as follows: body length L=2m, FN=0.289, RN=3.277×106, 80×26×40CVs and 110×40×60CVs for Wigley ship. FN=0.316, RN=4.0×106 and 110×40×58CVs for Series 60 ship, where H-H type grid is employed for the present calculation. Fig. 3 shows the time history of the side wave profiles along Wigley hull at FN=0.289, RN=3.277×106 with time intervals of 0.7s from 10.5s to 18.9s and the experimental results (Kajitani et al. 1983) are included to check the accuracy. Although there are some discrepancies to notice around the bow, it can be seen that the calculated results have consistency with different times, which means the converged solution with a high stableness. In addition, by comparing the other numerical results published for this wave profile, it can be regarded that the accuracy of the present numerical results is good. Fig. 4 shows the contour of the waves generated by the Wigley ship for two grid densities. It can be also seen from the comparison of the other numerical results published that the present results give a reasonable accuracy. Fig. 5 shows the level-set function contours converged along the hull of Wigley ship at FN=0.289. The distribution of the level-set function shows the uniform thickness along the interface where the thick solid line is zero level-set, namely, the free surface. Fig. 6 shows the wave profiles calculated along the Series 60 ship hull together with the experimental results at FN=0.316 and RN=4.0×106. A good agreement between the two results can be seen. The pressure distribution on the hull surface is also seen in Fig. 6, showing a smooth pressure variation on the hull surface. In Fig. 7, the wake distributions calculated for the Series 60 with the same condition as in Fig. 6 at the two planes (A.P. is x/L=1.0) are shown together with the measured results (Toda et al. 1991). It is rather disappointing to see that present numerical results do not agree well with the experimental results. However, it can be viewed from the comparison with other numerical results published that almost the similar level of numerical accuracy is obtained for the present turbulent model used. In recent published papers, it is recommended that more corrected and appropriate turbulence models should be used for a good prediction of the wake distribution rear the ship hull and grid density dependence of the solution is not so critical (Hino 1994). Fig. 8 shows the turbulent kinetic energy and eddy viscosity distributions around the hull of Series 60 ship at FN=0.316. The free surface turbulence layers in the air and the water region can be seen. The thickness of the turbulence layer developed around the fore part of the hull boundary on the free surface further develops and becomes thicker as it moves backwards. Although the numerical solutions computed by the present scheme near the free surface might not be so exact, the calculated turbulent properties show reasonable physical characteristics. For the shallow channel water calculations, the Wigley hull used previously is taken. In the numerical procedure for shallow water channel problems, blockage coefficient Sb=Ao/(2wh)=0.021, a ratio of water depth to the hull draft h/ T=1.598 and a ratio of channel width to the hull length w/L=2.0 are used, where Ao is the cross-section area of the hull at midship at a given draft. Fig. 1 shows the definition sketch for the shallow water channel problem. For the computational cases of Fh=1 and Fh=1.5, RN=3.6×106 and RN=5.0×106 are used, respectively. The computational channel length x/L=10, where x/L=5 ahead of the hull and x/L=4 behind the hull. In each computation, 210×44×46CVs used and H-H type grid is employed for the present calculation. Computations are started at full speed without flow acceleration. No slip flow condition is used at the hull boundary and the sidewall of the channel, and the symmetry boundary condition is introduced at the bottom and also at the center plane of the channel. The numerical results are shown in Fig. 9. Fig. 9 (1)-(a) shows the steady wave contours at FN=0.316, RN=3.6×106 for deep water which is equal to the critical speed Fh=1 for the shallow channel. The unsteady wave contours at this critical speed with different times for the shallow water channel are plotted in Fig. 9 (2)-(a)~(c). Unlike the steady case, the unsteady wave pattern with a soliton propagating ahead of the ship can be observed. It is known that this the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 948 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 METHOD phenomenon is caused by the effects of blockage and shallow water, and the various values of parameters, h/T, w/L and blockage coefficient, may create a little different flow phenomena. In addition, the results for Fh=1.5, RN=5.0×106 at two different times are included in Fig. 9 (3)-(a)~(b). It is generally known that a soliton develops and propagates forward of the bow of the hull at the critical speed. However, as seen in Fig. 9 (3)-(a)~(b), the wave pattern at the supercritical speed returns to the steady wave pattern with only the divergent wave system. This wave pattern is similar to hypersonic flow, and the angle of the divergent wave is about ±sin−1(1/Fh)≈42°. In Fig. 10, wave profiles calculated along the hull surface at FN=0.316 in the deep water, at the critical speed Fh=1 and at the supercritical speed Fh=1.5 in the shallow water are shown. It can be seen that the difference of the wave heights and patterns between the results for the deep water and shallow channel water is somehow considerable. By comparing the wave profiles for the deep water and the shallow channel of Fh=1 whose two speeds are equivalent, the larger wave near the bow and a deep trough near the stern for the critical speed case, which is of course caused by the big pressure changes due to the shallow water and channel effects as seen later in Fig. 12, creates a large trim by stern. For the three computational cases, the pressure and friction resistances acting on the hull are plotted in Fig. 11. It can be seen that the shallow-water channel effect increases the pressure resistance and the magnitude of the resistance has the maximum value near the critical ship speed in the shallow water channel. In Fig. 12, the pressure distributions on the hull surface are plotted for the three cases. Since the effects of the bottom and sidewalls of the shallow channel are considerable, the magnitude of the pressure difference along the hull is higher for the critical speed case than that for the deep water case. When a ship advances in the shallow water channel at a high speed, the pressure changes on the ship hull caused by the effect of the shallow channel create a large bow wave with a big trough near the stern as seen above and accordingly, cause the severe sinkage and trim of the ship, also resulting a large resistance increase. In Fig. 13, the pressure distributions on the bottom, the centreplane and sidewall of the channel at Fh=1 are shown. Because of the wave propagation ahead of the hull, a high pressure distributions over the upstream region can be seen. In Fig. 14, the velocity distributions and axial velocity contours at x/L=0.75 and x/L=1.0 are shown for the three computational cases. Since the wave heights along the hull surface and the magnitudes of the fluid velocities for the three cases are not all the same, the axial velocity contours show somehow different shapes. Because of the effects of the bottom of the channel, the axial velocity contours at the hull bottom are wider for the channel flow case than that for the deep water case. CONCLUSIONS The level-set approach to solve the turbulence free surface flow around hull in deep water and shallow water channel has been developed. The advantages of the level-set method make stable computation procedure, easy programming and also gives reasonably accurate solutions for the present viscous free-surface flow. It seems that the level of the numerical accuracy obtained by the present method is similar to that of other methods used with the same turbulent model as the present one. Numerical results of the free-surface flow around a hull in shallow water channel with the viscous effects show reasonable physical phenomena. But in the future, it is necessary to compare numerical results with experiments for validation and carry out more systematic study with variations of parameters, h/T, w/L and blockage coefficient. REFERENCE Bai, K.J. and Kim, J.W., “Numerical Computations for a Nonlinear Free Surface Flow Problem,” Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, pp. 403–419, 1989 Bet, F., Hanel, D. and Sharma, S., “Numerical Simulation of Ship Flow by a Method of Artificial Compressibility”, Proceedings of the Twenty-Second Symposium on Naval Hydrodynamics, Washington, pp. 173–182, 1998 Choi, H.S. and Mei, C.C, “Wave Resistance and Squat of a Slender Ship Moving Near the Critical Speed in Restricted Water,” Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, pp. 439–454, 1989 Ertekin, R.C., “Solution Generation by Moving the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 949 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 METHOD Disturbances in Shallow Water: Theory, Computation and Experiment,” Ph.D. Thesis, University of California, Berkeley, 1984 Ertekin, R.C. and Qian, Z.M., “Numerical Grid generation and Upstream Waves for Ships Moving,” Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, pp. 421–437, 1989 Dommermuth, D., Innis, G., Luth, T., Novikov, E., Schlageter, E. and Talcott, J., “Numerical Simulation of Bow Waves,” Proceedings of the Twenty- second Symposium on Naval Hydrodynamics, Washington, D.C., pp. 159–172, 1998 Ferziger, J.H. and Perić, M., Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin, 1996 Graff, W., Kracht, A. and Weinblum, G., “Some Extensions of D.W. Taylor's Standard Series,” Transactions of Society of Naval Architects and Marine Engineers, Vol. 72, 1964, pp. 374–401 Harten, A., and Engquist, B., Osher, S., and Chakravarthy, S., “Uniformly High-Order Accurate Essentially Nonoscillatory Schemes, III,” Journal of Computational Physics, Vol. 71, pp. 231–303, 1987 Helm, K., “Effects of Channel Depth and Width on Ship Resistance,” Hydrodynamische Problem des Schiffsantriebs, Part 2, ed. G.Kempf, Verlag Oldenburg, Munich and Berlin, 1940, pp. 144–171 Hino, T., “A Study of Grid Dependence in Navier-Stokes Solutions for Free Surface Flows around a Ship Hull”, Journal of The Society of Naval Architecs of Japan, Vol. 176, 1994 Jian, T., “Investigation of Waves Generated by Ships in Shallow Water,” Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, pp. 601–612, 1999 Kajitani, H., Miyata, H., Ikehata, M., Tanaka, H., Namimatsu, M., and Ogiwara, S., “The Summary of the Cooperative Experiment on Wigley Parabolic Model In Japan,” Proceedings of The 2nd DTNSTDC Workshop on Ship Wave-Resistance Computations, Bethesda, Maryland, U.S.A., 1983 Kim, S.Y. and Lee, Y.G., “A Study on Upstream Waves for an Arbitrary Hull Shape in Restricted Water Channel,” Proceedings of the Annual Autumn Meeting, SNAK, Inchon, 13–14, November, pp. 107–112, 1998 (in Korean) Muzaferija, S., “Adaptive Finite Volume Method for Flow Predictions Using Unstructured Meshes and Multigrid Approack,” Ph.D. Dissertation, University of London, 1994 Osher, S., and Sethian, J.A., “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” Journal of Computational Physics, Vol. 79, pp. 12–49, 1988 Park, I.R. and Chun H.H., “Analysis of Flow around a Rigid Body in Water-Entry & Exit Problems,” Journal of the Society of Naval Architects of Korea, Vol. 36, No. 4, pp. 37–47, 1999 (a), (in Korean) Park, I.R. and Chun H.H., “A Study on the Level-Set Scheme for the Analysis of the Free Surface Flow by a Finite Volume Method,” Journal of the Society of Naval Architects of Korea, Vol. 36, No. 2, pp. 40–49, 1999 (b), (in Korean) Peng, D.P., Osher, S., Zhao, H.K. and Kang M.G., “A PDE-Based Fast Local Level Set Method,” Journal of Computational Physics, Vol. 155, pp. 410– 438, 1999 Stone, H.L., “Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations,” SIAM J. Numer. Anal., Vol. 5, pp. 530– 558, 1968 Sussman, M., Fatemi, E., Smereka, P., and Osher, S., “An Improved Level Set Method for Incompressible Two-Phase Flows,” Computers and Fluids, Vol. 27, No. 5–6, pp. 663–680, 1997 Thews, J.G. and Landweber, L., “The Influence of Shallow Water on the Resistance of a Cruiser Model,” US Experimental Model Basin, Navy Yard, Washington, DC, Report No. 408, 1935 Toda, Y., Stern, F. and Longo, J., “Mean-flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 Cb=0.6 Ship Model for Froude Numbers 0.16 and 0.316”, IIHR Report No. 352, Iowa Institute of Hydraulic Research, The Univ. of Iowa, Iowa City, Iowa, 1991 van Doormal, J.P. and Rathby, G.D., “Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows,” Numer. Heat Transfer, 7, pp. 147–163, 1984 Vogt, M., “A Numerical Investigation of the Level-set method for Computing Free Surface Waves,” Department of Naval Architecture and Ocean Engineering, Chalmers University of Technology, Report CHA/NAV/R-98/0054, 1998 the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 950 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 METHOD Fig. 1 Definition sketch (shallow water channel) Fig. 2 A non-orthogonal & irregular grid Fig. 3 Wave profiles calculated along Wigley hull; in case of the calculation of 110×40×60CVs, history of wave profiles at intervals of 0.7s from 10.5s to 18.9s shown (FN=0.289, RN=3.277×106, F.P.=1.5 & A.P.=3.5) Fig. 4 Wave contours computed for Wigley ship with two grid densities (FN=0.289, RN=3.277×106) Fig. 5 Converged level-set contours along Wigley ship hull, where the range of levels is from—0.04 to 0.04 (FN=0.289, RN=3.277×106) the authoritative version for attribution. Fig. 6 Wave profiles and pressure distributions along Series 60 ship hull (FN=0.316, RN=4.0×106, F.P.=2 & A.P.=4)

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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 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 the authoritative version for attribution. METHOD results (FN=0.316, RN=4.0×106) Fig. 6 ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET Fig. 7 Axial velocity distributions and contours at x/L=1.0 (A.P.) & 1.2: left side for experimental, right side for numerical 951

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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 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 the authoritative version for attribution. METHOD Fig. 8 Turbulent kinetic energy & eddy viscosity distributions around hull of Series 60 ship (FN=0.316, RN=4.0×106) ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 952

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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 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 the authoritative version for attribution. METHOD supercritical speed Fh=1.5 with h/T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET Fig. 9 Calculated wave patterns of Wigley hull (1) at FN =0.316 in the deep water, (2) at the critical speed Fh=1 and (3) the 953

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 954 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 METHOD Fig. 10 Calculated wave profiles along Wigley hull (1) at FN=0.316 in the deep water, (2) at the critical speed Fh=1 and (3) the supercritical speed Fh=1.5 for h/T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) Fig. 11 History of the resistance of Wigley hull at FN=0.316 in the deep water, and at Fh=1 and Fh=1.5 in the shallow water channel with h/T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) Fig. 12 Calculated pressure distributions on the hull surface at FN=0.316 in the deep water (top), at Fh=1 (middle) and Fh=1.5 (bottom) in shallow water channel with h/T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 955 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 METHOD Fig. 13 Calculated pressure distributions on the bottom, sidewall and center plane of shallow-water channel at Fh=1, h/ T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) Fig. 14 Calculated velocity distributions and axial velocity contours at x/L=0.75 and x/L=1.0 at FN =0.316 in the deep water, at Fh=1 and Fh=1.5 in the shallow water channel with h/T=1.598 and w/L=2.0 (blockage coefficient Sb=0.021) the authoritative version for attribution.

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ANALYSIS OF TURBULENCE FREE-SURFACE FLOW AROUND HULLS IN SHALLOW WATER CHANNEL BY A LEVEL-SET 956 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 METHOD DISCUSSION T.Jiang Versuchsanstalt fur Binnenschiffbau e.V., Germany First, I would like to congratulate the authors to attack this kind of nonlinear and unsteady problems using an unsteady RANS, although is can also be effectively approximated by using shallow-water wave theory, For instance, K-P- equation on Boussinesq equations. The most benefit by using an unsteady RANS could be the better approximation of the wake waves near a transom stern in general and of the bow waves at high supercritical speed, since the bow waves are very sensitive to the bow shape. Turning now to my question. Is we look at your time history of the resistance, it does not seem to be a closed periodic asymptotic solution. However, both our experimental results as well as numerical results have shown these closed periodic procedures. AUTHOR'S REPLY We would like to thank Dr. Jiang for nice and useful comments. In this presentation of our work, the calculated time history of the resistance at the critical speed does not show a periodic behavior. This can be explained by two main reasons: The first reason is that the pressure correction equation could not be calculated accurately in the case of the unsteady critical speed flow condition because of reducing the time consume. The second reason is that our numerical result is that when the second soliton wave just propagates ahead of the bow of a ship after one soliton wave propagates. Therefore, if more accurate calculation of the pressure correction equation is conducted and results of more advanced time steps are obtained, our numerical solution at the critical speed may show a accurate and closed periodic asymptotic behavior. DISCUSSION N. Stuntz University Duisburg, Germany In the simulation the ship model is held fixed during the computation. In the case of the ship flow in restricted wastes in the transcritical range this might have considerable influence on wave patterns and resistance. Have the authors brought about this by having in mind, that the mesh is also fixed during computation using the level-set method? AUTHOR'S REPLY We would like to also thank Mr. Stuntz for the comments on our research. As you referred to your discussion, different conditions of a ship in the restricted shallow water channel might cause different flow patterns (wave height and resistance, etc.) However, it is not easy to simulate the ship at every instant when the ship changes in trim and sinkage at the critical speed due to too much time consumption. Our results in every flow conditions are computed with fixed grid system. More accurate solution can be obtained by using grid adaptation method. the authoritative version for attribution.