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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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 474 Validation of Tab Assisted Control Surface Computation C.-H.Sung, B.Rhee, I.-Y.Koh (Naval Surface Warfare Center, Carderock Division, USA) ABSTRACT A numerical procedure for the prediction of the forces and moments of a tab assisted control surface (TAC) has been developed. The control surface consists of a stern stabilizer, a flap, and a tab. The numerical procedure is based on solving the incompressible Reynolds-averaged Navier-Stokes equations coupled with several two-equation turbulence models. Some features of the numerical method used have been highlighted. In particular, the preconditioning method, multigrid method, and nonreflecting far field boundary condition have been discussed. The wall boundary condition for the specific dissipation rate which is important in obtaining good convergence for the -Ï equation turbulence model have also been briefly discussed. Computed results of the lift and drag coefficients of the control surface, flap torque coefficients and tab torque coefficients at various angles of attack of the stabilizer and of flap angles and tab angles have been predicted within 10 percent from the measured values even at high angles of attack at 15 degrees. The discrepancies of the torque coefficients of flap and tab are somewhat higher particularly at high flap and torque deflections. There are two reasons for these higher discrepancies. The first reason is that the turbulence models are inherently weak in the flow regime where separation is severe, and the second is that the grid solution in both the flap and tab gap is not sufficient. These will be the topics for future investigations. INTRODUCTION A control surface here will be defined as consisting of a stern stabilizer, a flap and a tab. The stabilizer maybe fixed or movable but the flap and tab are always movable. Control surfaces have at least three major functions applicable to both aircraft and marine vehicles. (1) The stabilizer, flap, and tab can be aligned to form a high cambered control surface to increase the lift significantly. In the aerospace industry, this is the so called high-lift multi-element airfoil (or wing). (2) Control surfaces are normally designed to provide adequate lift at lower speed operation, but undesirable excessive control may occur at high speed. A smoother control at high speed may be achieved by keeping the stabilizer fixed and using the flap and/or tab for control. (3) At high speed, an excessively large torque can arise in the stabilizer. This large torque can be reduced by deflecting the flap and/or tab in the direction opposite to the direction of the angle of attack of the incoming flow. The purpose of this paper is to report the progress made in the development of a predictive capability of the forces and moments of the tab assisted control surface (TAC). The design of efficient and desirable control surfaces by applying the predictive capability developed here is left for future work. There is an extensive experimental and computational literature on the high-lift multi-element airfoil (wing). Many references can be found in [1]. There is an extensive set of data of a NACA 0015 airfoil with a flap and a tab measured by Sears et al. [2]. For marine applications, work on relatively low aspect ratio control surfaces (sternplanes or rudders) is mostly experimental. Very little computational work has appeared in the literature. Forces and moments on control surfaces (no flap nor tab) have been measured by Whicker et al. [3] and those on a flapped control surface (no tab) have been measured by Bowers [4]. Water tunnel experiments on a series of 12 rudders with systematic variations of flap area and flap balance have been performed by Kerwin et al. [5]. Three variations of skeg-rudders (i.e., fixed main control surfaces with movable flaps) have been investigated in a wind tunnel [6]. The effect of gaps between the rudder and the skeg has also been investigated. It has been observed that the effect of the gap is insignificant. Unlike the case in aerospace industry, there are not many computational papers on control surfaces. Recently, Soding [7] discussed the application of potential theory in rudder flow predictions. The effects of flaps and tabs were not discussed. Computations for 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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 477 multigrid consisting of the medium and coarse grids is run for about 50 cycles. The solution is then interpolated to the fine grid to start the 3-level multigrid computation. In general, a solution adequate for engineering applications can be achieved in 100â500 multigrid cycles. This efficiency is at least as good as the best Computational Fluid Dynamics (CFD) codes available but is far from the Textbook Multigrid Efficiency (TME, less than 10 cycles) advocated by Achie Brandt [19]. Achieving TME is a noble goal, and will have a significant impact on engineering applications of CFD. Boundary Conditions Only the solid wall and the farfield boundary conditions need to be discussed. At the wall, the three components of velocity and the turbulent kinetic energy are set equal to zero, the pressure p is derived from assumption that the pressure gradient normal to the wall is zero. Finally, the wall boundary condition of specific dissipation rate Ï originally given by Wilcox (p. 148 in [10]) is modified as (13) where â¦w is the vorticity at the wall and ao is a constant varying from a value of 6 given by Wilcox to 20. The choice of ao may vary the convergence rate slightly but once convergence is achieved, the solution is about the same. The motivation in deriving the modified wall boundary condition (13) is to get rid of the requirement that the first grid normal distance must be given. The non-dimensional normal distance y+ requirement creates a difficulty for coarser grids because the first grid normal distances tend to be too large in the coarse grids. With Eqn (13), the normal distance does not appear and the y+ of the first grid normal distance of the finest mesh should be of the order 1 or 2. At the far field, the gradients of the three components of velocity and the gradients of the two turbulence quantities and Ï are set to zero. The pressure is obtained by a non-reflecting condition discussed by Hedstrom [20], Rudy and Strikverda [21] and Sung [22]. This is one of the most important boundary conditions for external flows and will be outlined. The idea is based on the characteristic formulation of hyperbolic equations, such that the outgoing solution modes will not be reflected back into the computational domain to corrupt the solution. To do this, the time derivatives of the characteristic variables corresponding to the positive eigenvalue Î»+ at the left boundary and the negative eigenvalue Î»_ at the right boundary are set equal to zero, i.e., (14) (15) Râ1 Râ1 q is the vector defined in Eqn (7) and is the matrix of the left eigenvectors. The matrix is quite complex for general preconditioning. But for the simpler case of non-symmetric preconditioning, Eqn (14) is quite simple and is given as Eqn (17) is used as the outflow far field boundary (16) (17) condition of P* after the three components of velocity are specified. the authoritative version for attribution. Two-Equation Turbulence Models Several two-equation turbulence models have been implemented in the general-purpose code named IFLOW. It is well known that the convergence of the two-equation turbulence models is rather temperamental. Two techniques have been used in IFLOW and as a result the same convergence rate as in the case of the Baldwin-Lomax turbulence model has been achieved. One of the techniques is the point-implicit method for source terms. Here, the positive part of the source term is treated explicitly, the negative part implicitly. This technique is in fact quite widely used. The other technique is to establish a lower bound for the specific dissipation rate Ï by the Schwartz inequality. To illustrate the method, it is sufficient to consider a linear Reynolds stress model

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 478 (18) By Schwartz inequality, it can be shown that (19) Taking square of the both sides of Eqn (18) gives (20) A lower bound for Ï is then obtained by combining Eqns (19) and (20) as (21) The proportionality factor in Eqn (21) can be taken as a value in the neighborhood of 2. Different values for this factor can affect the convergence rate. But once the convergence is achieved, they all give about the same solution. The value used in this paper is 2.1. DESCRIPTION OF EXPERIMENT As mentioned earlier, the control surface model consists of a stabilizer, a plain flap and a plain tab as show in Figure 1. The control surface has NACA 0018 airfoil sections. Specifically, the tip chordlength is 8.40 in., root chordlenght is 10.66 in., span is 8.44 in.. Both flap and tab gaps are 1/16 in. with the flap gap widened at both ends. At the root section, the flap hinge axis is located at 7.63 in. and the tab hinge axis is at 9.70 in.. The entire control surface model is mounted on a pedestal to place the model outside of any test section boundary layer. The control surface model was tested in the semi-closed jet test section of the 24 in. water tunnel at David Taylor Model Basin. The test section is 21 in. high by 27 in. wide with an area contraction ratio of 8.1. The control surface model was hung vertically from the top. The strain-gaged stabilizer and flap dynomometers measured lift, drag, and torque about their respective hinge axes. The tab dynamometer measured only torque about its hinge. But since the tab dynamometer was fastened to the flap, the flap dynamometer measured the combined loads of the flap and the tab. The nominal test speed was 10.9 to 12.0 ft/s. Fig 1. Grid used in the computation of flow over an NACA 0018 airfoil with flap and tab Based on a mean chordlength of 9.53 in., the Reynolds number is 9.7 x 105. Forces and moments of the TAC model under various combinations of the angles of attack of the stabilizer and the deflections of the flap and the tab were measured. The angle of attack of the stabilizer varied from â15 to +15 degrees, flap deflection from â27 to +27, and the tab deflection from â60 to +60. The values of lift, drag, and torque coefficients are based on the mean chordlength. Corrrections due to blockage, wall, and pedestal were made. The net blockage effects of the model on velocity were about 1.2% for the whole range of angles of attack. The true angle of attack of the model was 3.5% higher than the measured value. The corrected values will be used for comparison with the computed results. DISCUSSION OF RESULTS the authoritative version for attribution. Convergence and Grid-Independent Solution As Computational Fluid Dynamics (CFD) plays an increasingly important role in practical engineering applications, it is important to have some idea about how accurate and reliable the computed solutions are. It is possible to perform meaningful error analysis on a simple problem in a Cartesian computational domain with a uniform grid for inviscid or laminar flows. However, it is not possible to analyze the order of accuracy of a spatial discretization scheme in a highly stretched computational domain in a curvilinear coordinate system. The situation

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 479 becomes much worse with the additional complication from turbulence models. Efforts to quantify the errors in RANS computations for practical problem shave produced dubious results. Nevertheless, attempts must be made to establish confidence in the RANS solutions. The most important thing to do is to assure that the solution is monotonically convergent. But this is not sufficient. It is well known that a nonphysical converged solution can be obtained. For example, one can construct a coarse grid for a turbulent flow about a body with the nondimensional first grid normal distance to the wall y+ on the order of several hundred and obtain a very nicely converged solution. But the solution will not be close to the real physics because a turbulent boundary layer can not be developed for such a large y+ for a conventional two-equation turbulence model. To resolve this difficulty, a sequence of finer grids must be used until the change in the solution due to the grid refinement is small enough to be acceptable to engineering requirement. Thus a careful check of convergence history and mesh refinement to obtain a grid-independent solution are the most effective approach to establish confidence in the results. Other researchers as in e.g., [23] have adopted a similar view. A C-grid with four blocks was used in the computation. The first block wraps around the entire control surface, the second block is on top of the control surface, the third covers the gap between the stabilizer and the flap and the final block covers the gap between the flap and the tab. The water tunnel is not modeled in the computation. A total of three meshes were considered. The coarse mesh consists of 112x28x20, 44x8x8, 8x8x12, and 8x8x12 grid cells for the first, second, third, and fourth block, respectively. This mesh consists of a total of about 65K grid cells. The medium mesh doubles the number of grid cells in each curvilinear coordinate direction of each block and has a total number of grid cells of about half a million. The fine mesh will have the number of grid cells increased by 50 percent in each direction of each block of the medium mesh, giving a total number of grid cells of about 1.6 millions. It will be seen that the solutions obtained by the fine and the medium mesh are almost identical, indicating that a grid independent solution has been achieved. The boundary conditions imposed are the following. The farfield boundary conditions at both the upper and lower wakes are zero gradient for the three components of the Cartesian velocity, the turbulence quantities and Ï, and the nonreflecting boundary condition for the pressure. The nonreflecting boundary condition is important for good convergence and accuracy as mentioned earlier. On the outflow boundary at the top of the computational domain are imposed fixed values for the three components of the Cartesian velocity, the two turbulence quantities and Ï, and zero gradient of the pressure. Because of the presence of the pedestal, the symmetric boundary condition is applied at the bottom of the computational domain. Non-slip boundary condition for the velocity and zero gradient for the pressure are applied at the wall boundary. The turbulent kinetic energy vanishes at the wall and the dissipation rate at the wall has been described earlier. For other boundaries such as between grid blocks and the interface between the upper and the lower wake, exact boundary conditions are applied. Some typical convergence histories of the root-mean-square of pressure for the case without flap and tab deflections and the case with 20 degrees deflections for both flap and tab are shown in Figure 2, where residue is defined as the root- mean-square value of the difference between the current calculated pressure and the last calculated one. It can be seen that flap and tab deflections do not seem to affect convergence rate. The residues for both cases drop more than three orders of magnitude in 200 multigrid cycles. The forces and moments become steady at about 200 multigrid cycles. It should be noted that the drop in the residue due to the multigrid starting procedure has not been included in Figure 2. This explains why logic10 (residue) starts at somewhere between â1 and â2, instead of 0. Fig 2. Root-mean square residue of pressure vs. multigrid cycles the authoritative version for attribution.

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 480 Forces and Moments In the following discussion, the medium grid with a total of about half a million grid cells was used for the computed results. The grid consists of four blocks with grid size of 224x56x40, 88x16x16, 16x16x24, and 16x16x24. The Reynolds number based on the mean chordlength is 9.7x105. The coefficients of the forces and moments are also defined using the mean chordlength as the characteristic length. The lift and drag coefficients will be defined as the total forces applied to the entire control surface. The flap torque coefficient will include both the torque applied to the flap and the tab while the tab torque coefficient will include the torque applied to the tab alone. Figure 3 shows the comparison between measurement and computation of the lift coefficient as the angle of attack of the stabilizer varies from â6 to +15 degrees with no deflections for both flap and tab. The error bars on the data show 10% discrepancy in measurements. The lift coefficient is almost linear indicating insignificant viscous effect in this range of angles of attack. Both the predictions by the fine and the medium grids agree well with the measurement. However, the coarse grid prediction starts to deviate form the measurement by more than 10% after an angle of attack of 9 degrees, indicating insufficient grid resolution. Figure 3. Comparison of calculated and measured lift coefficients with flap and tab at zero deflection Figure 4 shows the comparison of the drag coefficient under the conditions as similar to those in Figure 3. Although a grid independent solution has been achieved between the fine and the medium grids, the drag coefficient is overpredicted by more than 20% in the neighborhood of the zero angle of attack and within 10% for greater than 10 degrees. The coarse grid prediction is even worse, again due to insufficient grid resolution. The effect on lift coefficient of varying the flap deflection from â15 degrees to +15 degrees is shown in Figure 5, where angle of attack for the stern stabilizer remains zero. Figure 4. Comparison of calculated and measured Fig 5. Comparison of calculated and measured lift drag coefficients with flap and tab at zero deflection coefficients with stern stabilizer and tab at zero deflection A grid independent solution has been achieved between the fine and the medium grid up to 10 degrees of flap deflection. At 15 degrees of flap deflection, the fine grid prediction is still within 10% of the measured values but the the authoritative version for attribution. medium grid prediction

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 481 degenerates rapidly. The coarse grid prediction is inadequate beyond 5 degrees of flap deflection. One physical feature is worthy of mentioning. Consider a lift coefficient of 0.2 [Fig 3]. This lift can be achieved by an angle of attack of slightly less than 6 degrees of the entire control surface. It can also be achieved by a flap deflection of about 10 degrees but at a much smaller torque requirement. This is the essence of using a flap and also a tab which would be discussed later. Finally, the effect on lift, flap torque, and tab torque coefficients of varying the tab deflection fromâ60 degrees to +60 degrees are presented in Figures 6 through 8, respectively. Here, the stabilizer is at zero angle of attack and the flap has no deflection. Figure 6. Comparison of calculated and measured lift Figure 7. Comparison of calculated and measured coefficients with stern stabilizer and flap at zero flap torque coefficients with stern stabilizer and flap deflection at zero deflection A grid independent solution has not been obtained in the calculation of the lift coefficient as shown in Figure 6. However, the prediction of the lift from the fine grid is within 10% of measurement even at high tab deflection of 60 degrees. There is one discrepancy when tab deflection is less than 10 degrees. The slope of the measured lift is linear near zero tab deflection but is not zero. The predicted slope is almost zero when tab deflection is less than 10 degrees. If the measurement were correct, the discrepancy could be explained as insufficient grid resolution around the tab. The small increase in the lift due to a small tab deflection has not been picked up even by a grid as large as 1.6 million grid cells. It was mentioned earlier than a lift coefficient of 0.2 can be achieved by either an angle of attack of about 6 degrees of the entire control surface or by a flap deflection of about 10 degrees. This lift can also be obtained by a tab deflection of 40 degrees with even smaller tab torque requirement. The comparison of the flap torque coefficient is shown in Figure 7. It has a similar characteristic as the lift coefficient shown in Figure 6. A grid independent solution has not been achieved at high tab deflection, and the slope near zero tab deflection is much flatter than the measurement. The predicted slope of the tab torque coefficient near zero tab deflection seems to agree better with the measurement but the predicted torque coefficient at high tab deflection deviates from the measurement by more than 10%. It should be noted that the tab torque coefficient is smaller than the flap torque coefficient by approximately one order of magnitude. This is the main reason that the tab assisted control surface is of great practical interest. Figure 8. Comparison of calculated and measured tab torque coefficients with stern stabilizer and flap at zero deflection the authoritative version for attribution.

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 482 CONCLUSIONS A numerical procedure for the prediction of the forces and moments of a tab assisted control surface has been developed. The procedure is based on solving the incompressible Reynolds-averaged Navier-Stokes equations coupled with a -Ï, turbulence model. Computed results of lift, flap, and tab torque coefficients were compared with the measured data at a Reynolds number of 9.7x105 based on the mean chordlength. Three meshes with grid size of 65K, one half million, and 1.6 millions were used to investigate the grid independent solution. A grid independent solution was achieved in most of the cases except for some cases with high flap and tab deflections. The trend of the changes in the forces and moments due to the variations in the angle of attack of the stabilizer and the deflection of the flap and tab has been completely captured. In most cases investigated, the predictions are within 10% of the measurements. Some exceptions are the tab torque coefficients at high tab deflections and the slopes of the lift and flap torque coefficients near zero tab deflection. It is suggested that both the turbulence model and the grid resolution need to be improved. The fact that even with a grid as large as 1.6 million cells, a grid independent solution can only be achieved in most, but not all cases, indicates that more efficient numerical schemes and turbulence models are urgently needed. Despite all these limitations, the predictive procedure presented here is already a useful tool for the design of efficient control surfaces. ACKNOWLEDGMENTS This work is funded by the Office of Naval Research, Code 333, under the Mechanics and Energy Conversion Science and Technology Division (PE0602121). Dr Patrick Purtell is the technical monitor of this program. Dr. Nguyen Thang is the monitor at David Taylor Model Basin. Helpful discussions of experiment and measured data with Mr. David Bochinski at David Taylor Model Basin are gratefully acknowledged. Computer resources provided by the Department of Defense High Performance Computing Modernization Office (DOD-HPCMC) at NAVO and the Arctic Region Supercompting Center in Fairbank, AK are also gratefully acknowledged. REFERENCES 1. AGARD Conference Proceedings 515 on âHigh-Lift System Aerodynamicsâ, September, 1993. 2. Richard I.Sears and Robert B.Liddel, âWind-Tunnel investigation of Conrol-Surface Characteristics, VIâA 3 Percent-Chord Plain Flap On the NACA 0015 Airfoilâ. NACA Wartime Report 454, June 1942. 3. Whicker, L.Folger and Leo F.Fehlner, âFree-Stream Characteristics of a Family of Low-Aspect Ratio, All-Movable Control Surfaces For Application to Ship Designâ, David Taylor Model Basin Report 933, December 1958. 4. Bowers, Allen, âWind Tunnel Investigation of the Characteristics of a Flapped Control Surface Mounted on a Simulated Submarine Hullâ, University of Maryland Wind Tunnel Report No. 259, June, 1959. 5. Kerwin, Justine E., Philip Mandel and S.Dean Lewis, âAn Experimental Study of a Series of Flapped Ruddersâ, Journal of Ship Research, December, 1972 6. Goodrich, G.J. and A.F.Molland, âWind Tunnel Investigation of Semi-Balanced Ship Skeg-Ruddersâ, The Royal Institute of Naval Architects, pp. 285â307, 1979. 7. Soding, H., âLimits of Potential Theory in Rudder Flow Predictionsâ, Twenty-Second Symposium on Naval Hydrodynamics, Washington, D.C., pp. 264â276, August 9â14, 1998. 8. Chau, Shiu-Wu, âComputation of Rudder Force and Moments in Uniform Flowâ, Ship Technology Research Vol. 45, pp. 3â13, 1998. 9. Gowing, Scott, Thang Nguyen and David Bochinski, âT.A.C. Test Static Results in the 24â Water Tunnelâ, NSWC, CD, not yet published, 1999. 10. Wilcox, D.C., Turbulence Modeling for CFD, DCW Industries, Inc. CA, 1993 11. Speziale, Charles G., âComparison of Explicit and Traditional Algebraic Stress Models of Turbulenceâ, AIAA Journal vol. 35, No. 9, September 1997. 12. Chorin, A.J., âA Numerical Method for Solving Incompressible Viscous Flow Problemâ, Journal of Computational Physics, vol. 2, 275, 1967. 13. Turkel, E., âPreconditioned Methods for Solving the Incompressible and Low Speed Compressible Equationsâ, Journal of Computational Physics, vol. 72, 277, 1987. the authoritative version for attribution.

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 483 14. Yee, H.C., âA Class of High-Resolution Explicit and Implicit Shock-Capturing Methodsâ, NASA Technical Memorandum 101088, February, 1989. 15. James on, A., âTime Dependent Calculations Using Multigrid with Applications to Unsteady Flows Past Airfoils and Wingsâ, AIAA 91â1596, June 1991. 16. Liu, C., X.Zheng and C.H.Sung, âPreconditioned Multigrid Methods for Unsteady Incompressible Flowsâ, Journal of Computational Physics, vol. 139, 35â57, 1998. 17. Brandt, A., âMultigrid Techniques: 1984 Guide, with Applications to Fluid Dynamicsâ, 1984, 191 pages, ISBN-3â88457â081â1; GMD-Studien Nr 85; Available from GMD-AIW, Postfach 1316, D-53731, St. Augustin 1, Germany, 1984. 18. Jameson, A., âMultigrid Algorithms for Compressible Flow Calculationsâ, Lecture Notes in Mathematics, No. 1228, Proceedings of the Second European Conference on Multigrid Methods, Cologne, pp. 166â201, October 1â4, 1985. 19. Brandt, A., âBarriers to Achieving Textbook Multigrid Efficiency (TME) in CFDâ, NASA/CR-1998â207647, ICASE Interim Report No. 32, April 1998. 20. Hedstrom, G.W., âNonreflecting Boundary Conditions for Nonlinear Hyperbolic Systemâ, Journal of Computational Physics, vol. 30, pp. 222â237, 1979. 21. Rudy, D.H., and J.C.Strikwerda, âBoundary Conditions for Subsonic Compressible Navier-Stokes Equationsâ, Computers and Fluids, vol. 9, pp. 327â338, 1981. 22. Sung, C.H., âAn Explicit Runge-Kutta Method for 3D Incompressible Turbulent Flowsâ, DTNSRDC/SHâ1244â01, July 1987. 23. Jameson, A. and L.Martinelli, âMesh Refinement and Modeling Errors in Fluid Simulationâ, AIAA Journal vol. 36, No. 5, May 1998. the authoritative version for attribution.

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 VALIDATION OF TAB ASSISTED CONTROL SURFACE COMPUTATION 484 DISCUSSION Y.Tahara Oskaka Prefecture University Japan In your computation, laminar-to-turbulent flow transition was not considered, although your experimental condition (i.e., Reâ106) apparently implies that there exists laminar-flow region on the wing (stabilizer in your definition) surface. Inclusion of the effects is generally essential for accurate prediction of hydrodynamic forces especially for drag component [Tahara et al., 1998, 2000]. In addition, the conventional two equation model used in your work may not be suitable for the purpose. REFERENCES: Tahara, Y., et al., âAn Application of RaNS Equation Method to Strut/Bulb Configuration of America's Cup Sailing Yacht and Comparison with Experiments,â J. Kansai Society of Naval Architects, No. 230, 1998, pp. 163â171. Tahara, Y., et al., âDevelopment of Ballast Bulb for IACC Sailing YachtâEspecially for Investigation on Basic Low Drag Form,â J. Kansai Society of Naval Architects, No. 234, 2000, pp. 51â59 AUTHOR'S REPLY Due to a relatively high turbulence level in a water tunnel, early experimental tests indicated that the flow was turbulent at a Reynolds number of about one million based on a mean chordlength of 9.53 inches. For this reason, computations were made assuming the flow was completely turbulent. Transition from laminar to turbulence was not considered. Admittedly, turbulence models are not perfect for a complex flow such as the one investigated here. However, the âÏ turbulence model used here worked quite satisfactorily in our opinion. It is believed that further improvement of accuracy can be made by increasing the grid size, particularly in the leeward side of the flow region. This will be investigated in the future. the authoritative version for attribution.