Transonic flow computations using nonlinear potential methods

Abstract This presentation describes the state of transonic flow simulation using nonlinear potential methods for external-aerodynamic applications. The presentation begins with a review of the various potential equation forms (with emphasis on the full potential equation) and includes a discussion of pertinent mathematical characteristics and all derivation assumptions. Impact of the derivation assumptions on simulation accuracy, especially with respect to shock wave capture, is discussed. Key characteristics of all numerical algorithm types used for solving nonlinear potential equations, including steady, unsteady, space marching, and design methods, are described. Both spatial discretization and iteration scheme characteristics are examined. Numerical results for various aerodynamic applications are included throughout the presentation to highlight key discussion points. The presentation ends with concluding remarks and recommendations for future work. Overall, nonlinear potential solvers are efficient, highly developed and routinely used in the aerodynamic design environment for cruise conditions.

[1]  T. Baker.  A Numerical Method to Compute Inviscid Transonic Flows Around Axisymmetric Ducted Bodies , 1976 .

[2]  Antony Jameson,et al.  A brief description of the Jameson-Caughey NYU transonic swept-wing computer program: FLO 22 , 1976 .

[3]  F. Dejarnette,et al.  INTERACTIVE THREE-DIMENSIONAL BOUNDARY-LAYER METHOD FOR TRANSONIC FLOW OVER SWEPT WINGS , 1989 .

[4]  R. Johnson,et al.  Computational wing design for an advanced trainer , 1983 .

[5]  D. Rizzetta,et al.  Numerical Solution of Three-Dimensional Unsteady Transonic Flow over Swept Wings , 1982 .

[6]  S. Takanashi,et al.  Iterative three-dimensional transonic wing design using integral equations , 1985 .

[7]  R. Houwink,et al.  Results of an improved version of LTRAN2 for computing unsteady airloads on airfoils oscillating in transonic flow , 1979 .

[8]  J. V. Vooren,et al.  Inviscid drag prediction for transonic transport wings using a full-potential method , 1990 .

[9]  Antony Jameson,et al.  Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method , 1979 .

[10]  J. W. Boerstoel.  Review of the Application of Hodograph Theory to Transonic Aerofoil Design and Theoretical and Experimental Analysis of Shock-free Aerofoils , 1976 .

[11]  J. L. Steger,et al.  Shock waves and drag in the numerical calculation of isentropic transonic flow , 1972 .

[12]  W. Mason,et al.  A numerical three-dimensional viscous transonic wing-body analysis and design tool , 1978 .

[13]  B. Arlinger.  Axisymmetric transonic flow computations using a multigrid method , 1981 .

[14]  J. Steinhoff,et al.  Free-wake analysis of compressible rotor flows , 1987 .

[15]  P. J. Drape Henne,et al.  Inverse Transonic Wing Design Method , 1981 .

[16]  V. Shankar,et al.  Computational transonic design procedure for three-dimensional wings and wing-body combinations , 1979 .

[17]  C. Hirsch.  Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows , 1990 .

[18]  B. Grossman,et al.  Nonlinear Supersonic Potential Flow over Delta Wings , 1980 .

[19]  S. Osher,et al.  Treatment of supersonic flows with embedded subsonic regions , 1985 .

[20]  Friedrich L. Bauer,et al.  Supercritical Wing Sections II , 1974 .

[21]  A. Cenko,et al.  Potential flow applications to the prediction of the F-18E aircraft flowfield , 1994 .

[22]  A. R. Seebass,et al.  Finite area method for nonlinear supersonic conical flows , 1984 .

[23]  Guru P. Guruswamy,et al.  Effects of modal symmetry on transonic aeroelastic characteristics of wing-body configurations , 1989 .

[24]  Computation of transonic flow about finite lifting wings. , 1972 .

[26]  R. M. Hicks,et al.  Computational optimization and wind tunnel test of transonic wing designs , 1980 .

[27]  F. R. Bailey.  Numerical calculation of transonic flow about slender bodies of revolution , 1971 .

[28]  H. L. Stone.  ITERATIVE SOLUTION OF IMPLICIT APPROXIMATIONS OF MULTIDIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS , 1968 .

[29]  A. Ecer,et al.  Block-structured solution scheme for analyzing three-dimensional transonic potential flows , 1986 .

[30]  M. Hafez,et al.  A minimal residual method for transonic potential flows , 1982 .

[31]  Y. Wong.  Newton-like minimal residual methods applied to transonic flow calculations , 1985 .

[32]  Guru P. Guruswamy,et al.  Transonic aeroelasticity of wings with tip stores , 1987 .

[33]  A. Brandt.  Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems , 1973 .

[34]  E. M. Murman,et al.  Computation of Transonic Flows past Lifting Airfogs and Slender Bodies , 1972 .

[35]  C. Tung,et al.  Numerical solution of the full-potential equation for rotors and oblique wings using a new wake model , 1985 .

[36]  Viscous-Inviscid Interaction for Transonic Wing-Body Configurations Including Wake Effects , 1982 .

[37]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[38]  S. Velichko,et al.  Analysis of transonic flow by method of viscous-inviscid interaction - Correction for wind tunnel wall interference , 1996 .

[39]  D. Rizzetta,et al.  Unsteady transonic flow over wings including inviscid/viscous interaction , 1983 .

[40]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[41]  D. R. McCarthy,et al.  Multigrid Code for Three-Dimensional Transonic Potential Flow about Inlets , 1982 .

[42]  T. Baker,et al.  A fast algorithm for the calculation of transonic flow over wing/body combinations , 1983 .

[43]  G. P. Guruswamy,et al.  Transonic Aeroelastic Analysis of the B-1 Wing , 1986 .

[44]  R. Houwink,et al.  Improved Version of LTRAN2 for Unsteady Transonic Flow Computations , 1980 .

[45]  J. Douglas,et al.  A general formulation of alternating direction methods , 1964 .

[46]  Lakshmi N. Sankar,et al.  Navier-Stokes/Full Potential/Free-Wake Method for Rotor Flows , 1997 .

[47]  S. R. Bland,et al.  Flow instabilities in transonic small disturbance theory , 1985 .

[48]  C. Lombard,et al.  The Geometric Conservation Law - A link between finite-difference and finite-volume methods of flow computation on moving grids , 1978 .

[49]  Anna Smith,et al.  The Panel Method: Its Original Development , 1990 .

[50]  Numerical design of transonic wings in curvilinear coordinates , 1994 .

[51]  C. Hirsch,et al.  A multigrid method for the transonic full potential equation discretized with finite elements on an arbitrary body fitted mesh , 1982 .

[52]  Shigeru Obayashi,et al.  Multiobjective Genetic Algorithm for Multidisciplinary Design of Transonic Wing Planform , 1997 .

[53]  L. C. Wrobel.  Numerical computation of internal and external flows. Volume 2: Computational methods for inviscid and viscous flows , 1992 .

[54]  M. E. Tauber,et al.  Computation of transonic flow about helicopter rotor blades , 1986 .

[55]  T. Grundy,et al.  Progress in Astronautics and Aeronautics , 2001 .

[56]  Michael B. Bieterman,et al.  An adaptive grid method for analysis of 3D aircraft configurations , 1992 .

[57]  John T. Batina,et al.  Unsteady Transonic Flow Calculations for Realistic Aircraft Configurations , 1989 .

[58]  Dennis F. Fuglsang,et al.  NON-ISENTROPIC UNSTEADY TRANSONIC SMALL DISTURBANCE THEORY , 1985 .

[59]  William H. Mason,et al.  Pressure and force data for a flat wing and a warped conical wing having a shockless recompression at Mach 1.62 , 1981 .

[60]  Jacques Periaux,et al.  On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I) least square formulations and conjugate gradient solution of the continuous problems , 1979 .

[61]  Terry L. Holst,et al.  A new consistent spatial differencing scheme for the transonic full-potential equation , 1984 .

[62]  Antony Jameson.  Transonic potential flow calculations using conservative form , 1975 .

[63]  M. J. Siclari.  Supersonic Nonlinear Potential Flow with Implicit Isentropic Shock Fitting , 1982 .

[64]  Vijaya Shanka,et al.  Conservative Full Potential, Implicit Marching Scheme for Supersonic Flows , 1982 .

[65]  Herman Deconinck,et al.  Finite Element Methods for Transonic Blade-to-Blade Calculation in Turbomachines , 1981 .

[66]  Guru P. Guruswamy,et al.  Interaction of fluids and structures for aircraft applications , 1988 .

[67]  W. F. Ballhaus,et al.  Numerical solution techniques for unsteady transonic aerodynamics problems , 1980 .

[68]  R. M. Traci,et al.  Small Disturbance Transonic Flows About Oscillating Airfoils and Planar Wings. , 1975 .

[69]  Guru P. Guruswamy,et al.  Unsteady transonic flow simulation on a full-span-wing-body configuration , 1987 .

[70]  Hongde Jiang,et al.  An Accurate Spatial Differencing Scheme for a Three-Dimensional Full Potential Equation , 1986 .

[71]  E. Murman,et al.  Analysis of embedded shock waves calculated by relaxation methods. , 1973 .

[72]  T. A. Reyhner.  Transonic Potential Flow around Axisymmetric Inlets and Bodies at Angle of Attack , 1977 .

[73]  A. Hahn,et al.  Comparison of low and high nacelle subsonic transport configurations , 1997 .

[74]  R. Buskirk,et al.  Implicit calculations of transonic flows using monotone methods , 1981 .

[75]  D. Rizzetta,et al.  Numerical solution of three-dimensional unsteady transonic flow overswept wings , 1980 .

[76]  F. Roggero,et al.  Aerodynamic Calculation of Complex Three-Dimensional Configurations , 1993 .

[77]  M. Siclari,et al.  Entropy corrections to supersonic conical nonlinear potential flows , 1984 .

[78]  W. F. Ballhaus,et al.  Numerical calculation of transonic flow about swept wings. , 1972 .

[79]  A. R. Seebass,et al.  A new method for designing shock-free transonic configurations , 1978 .

[80]  B. M. Spee,et al.  Applied Computational Transonic Aerodynamics , 1982 .

[81]  J. Cole,et al.  Calculation of plane steady transonic flows , 1970 .

[82]  J. Steger,et al.  A conservative finite difference algorithm for the unsteady transonic potential equation in generalized coordinates , 1982 .

[83]  G. P. Guruswamy,et al.  Unsteady transonic aerodynamic and aeroelastic calculations about airfoils and wings , 1984 .

[84]  O. J. Rose,et al.  Full-potential analysis of a supersonic delta wing/body , 1989 .

[85]  A. Jameson,et al.  Implicit Approximate-Factorization Schemes for Steady Transonic Flow Problems , 1978 .

[86]  B L Hinson,et al.  Acquisition and Application of Transonic Wing and Far-Field Test Data for Three-Dimensional Computational Method Evaluation. Volume II. Appendix B. Experimental Data. , 1980 .

[87]  Lakshmi N. Sankar,et al.  Unsteady aerodynamic modeling of a fighter wing in transonic flow , 1986 .

[88]  Samson Cheung,et al.  Aerodynamic design - Parallel CFD and optimization routines , 1995 .

[89]  William J Rae,et al.  A Study of Inlet Conditions for Three-Dimensional Transonic Compressor Flows. , 1978 .

[90]  J. B. Malone,et al.  Numerical simulation of two-dimensional unsteady transonic flows using the full-potential equation , 1984 .

[91]  A. Jameson,et al.  A finite volume method for transonic potential flow calculations , 1977 .

[92]  John T. Batina,et al.  Unsteady transonic flow calculations for wing/fuselage configurations , 1986 .

[93]  S. R. Bland,et al.  Linear/nonlinear behavior in unsteady transonic aerodynamics , 1983 .

[94]  R. M. Hicks,et al.  TRANSONIC WING ANALYSIS USING ADVANCED COMPUTATIONAL METHODS , 1978 .

[95]  M. D. Salas,et al.  A comparative study of the nonuniqueness problem of the potential equation. [for transonic flow past airfoils] , 1983 .

[96]  Kenneth B. Walkley,et al.  Application of a full potential method to supersonic aircraft designand analysis , 1989 .

[97]  Joseph L. Steger,et al.  A Finite-Difference Method for Transonic Airfoil Design , 1973 .

[98]  George S. Dulikravich,et al.  Aerodynamic shape design and optimization - Status and trends , 1992 .

[99]  Roger C. Strawn,et al.  Prediction of unsteady transonic rotor loads with a full-potential rotor code , 1987 .

[100]  J. W. Slooff,et al.  Remarks on the suitability of various transonic small perturbation equations to describe three-dimensional transonic flow; examples of computations using a fully-conservative rotated difference scheme , 1976 .

[101]  D. Kwak,et al.  Nonreflecting far-field boundary conditions for unsteady transonic flow computation , 1980 .

[102]  J. L. Steger,et al.  Relaxation Techniques for Three-Dimensional Transonic Flow about Wings , 1972 .

[103]  Antony Jameson,et al.  Relaxation solutions for inviscid axisymmetric transonic flow over blunt or pointed bodies. , 1973 .

[104]  M. Madson.  Sonic Boom Predictions Using a Solution-Adaptive Full-Potential Code , 1994 .

[105]  Oktay Baysal,et al.  Three-dimensional aerodynamic shape optimization of wings using sensitivity analysis , 1994 .

[106]  Ahmed A. Hassan,et al.  Airfoil Design for Helicopter Rotor Blades - A Three-Dimensional Approach , 1994 .

[107]  J. E. Carter,et al.  Interaction of a two-dimensional strip boundary layer with a three-dimensional transonic swept-wing code , 1978 .

[108]  G. Volpe,et al.  The inverse design of closed airfoils in transonic flow , 1983 .

[109]  Conservative full-potential calculations for axisymmetric, transonic flow , 1983 .

[110]  D. P. Young,et al.  A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics , 1990 .

[111]  An Embedded-Mesh Potential Flow Analysis. , 1984 .

[112]  D. Caughey,et al.  Calculation of Transonic Inlet Flowfields Using Generalized Coordinates , 1980 .

[113]  C. J. Borland,et al.  Nonlinear transonic flutter analysis , 1982 .

[114]  P. M. Goorjian,et al.  Computations and aeroelastic applications of unsteady transonic aerodynamics about wings , 1984 .

[115]  J L Steger,et al.  Generation of three-dimensional body-fitted coordinates using hyperbolic partial differential equations , 1984 .

[116]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[117]  George S. Dulikravich,et al.  Aerodynamic shape design and optimization , 1991 .

[118]  Richard H. Burkhart,et al.  A new approach to the solution of boundary value problems involving complex configurations , 1986 .

[119]  Jean-Jacques Chattot,et al.  Calculation of three-dimensional unsteady transonic flows past helicopter blades , 1980 .

[120]  W. F. Ballhaus,et al.  Implicit approximate-factorization schemes for the low-frequency transonic equation , 1975 .

[121]  Charles Hirsch,et al.  FINITE ELEMENT METHODS FOR TRANSONIC FLOW CALCULATIONS , 1980 .

[122]  T. H. Lê.  Transonic potential flow calculation about complex bodies by a technique of overlapping subdomains , 1982 .

[123]  A. Jameson,et al.  Implicit approximate-factorization schemes for the efficient solution of steady transonic flow problems , 1977 .

[124]  S. Osher,et al.  Stable and entropy satisfying approximations for transonic flow calculations , 1980 .

[125]  Effect of Frequency in Unsteady Transonic Flow , 1979 .

[126]  A. Eberle.  A finite volume method for calculating transonic potential flow around wings from the pressure minimum integral , 1978 .

[127]  R. Hindman.  Geometrically induced errors and their relationship to the form of the governing equations and the treatment of generalized mappings , 1981 .

[128]  Terry L. Holst,et al.  Fast, Conservative Algorithm for Solving the Transonic Full-Potential Equation , 1980 .

[129]  R. L. Sorenson,et al.  Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations , 1979 .

[130]  Robert M. Bennett,et al.  Time-marching transonic flutter solutions including angle-of-attack effects , 1983 .

[131]  David Nixon,et al.  Nonisentropic Potential Formulation for Transonic Flows , 1983 .

[132]  P. Gelhausen,et al.  Validation of a new unstructured full potential formulation , 1995 .

[133]  H. J. Cunningham,et al.  Wing-flutter calculations with the CAP-TSD unsteady transonic small-disturbance program , 1989 .

[134]  Jerry C. South,et al.  Vector processor algorithms for transonic flow calculations , 1980 .

[135]  Domenico Quagliarella,et al.  Inverse and Direct Airfoil Design Using a Multiobjective Genetic Algorithm , 1997 .

[136]  Marcel Vinokur,et al.  Conservation equations of gasdynamics in curvilinear coordinate systems , 1974 .

[137]  Lakshmi N. Sankar,et al.  Inverse aerodynamic design method for aircraft components , 1987 .

[138]  Influence of nonconservative differencing on transonic streamline shapes , 1976 .

[139]  David Catheral.  Optimum Approximate-Factorization Schemes for Two-Dimensional Steady Potential Flows , 1982 .

[140]  Leland A. Carlson,et al.  Aerodynamic sensitivity coefficients using the three-dimensional full potential equation , 1994 .

[141]  Michael B. Bieterman,et al.  Practical considerations in aerodynamic design optimization , 1995 .

[142]  J. Zierep,et al.  Symposium Transsonicum III , 1989 .

[143]  W. Rae.  Calculations of Three-Dimensional Transonic Compressor Flowfields by a Relaxation Method , 1977 .

[144]  Andrew J. Majda,et al.  Numerical Radiation Boundary Conditions for Unsteady Transonic Flow , 1981 .

[145]  Timothy J. Baker,et al.  Potential flow calculation by the approximate factorisation method , 1981 .

[146]  Wagdi G. Habashi,et al.  Finite Element Solutions of Transonic Flow Problems , 1982 .

[147]  M. Hafez,et al.  Conjugate Gradient Methods Applied to Transonic Finite Difference and Finite Element Calculations , 1982 .

[148]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[149]  B. Arlinger.  Calculation of Transonic Flow around Axisymmetric Inlets , 1975 .

[150]  Nonunique Solutions to the Transonic Potential Flow Equation , 1984 .

[151]  K. Fung.  Far field boundary conditions for unsteady transonic flows , 1981 .

[152]  D. Quagliarella,et al.  Genetic algorithms applied to the aerodynamic design of transonic airfoils , 1994 .

[153]  N. Sankar.  A Multigrid Strongly Implicit Procedure for Transonic Potential Flow Problems , 1983 .

[154]  B. Wedan,et al.  A method for solving the transonic full-potential equation for general configurations , 1983 .

[155]  Pieter G. Buning,et al.  User's manual for the HYPGEN hyperbolic grid generator and the HGUI graphical user interface , 1993 .

[156]  Richard G. Hindman,et al.  Generalized Coordinate Forms of Governing Fluid Equations and Associated Geometrically Induced Errors , 1982 .

[157]  H. S. Murty.  Nonunique solutions in unsteady transonic flow , 1993 .

[158]  M. Hafez,et al.  Artificial Compressibility Methods for Numerical Solutions of Transonic Full Potential Equation , 1979 .

[159]  Ching S. Chen,et al.  Three-dimensional viscous rotor flow calculations using a viscous-inviscid interaction approach , 1990 .

[160]  N. Sankar,et al.  A Strongly Implicit Procedure for Steady Three-Dimensional Transonic Potential Flows , 1981 .

[161]  Comparisons of Computed and Experimental Pressures for Transonic Flows about Isolated Wings and Wing-Fuselage Configurations , 1975 .

[162]  J. Steger,et al.  Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow , 1980 .

[163]  John T. Batina,et al.  Unsteady transonic algorithm improvements for realistic aircraft applications , 1988 .

[164]  Koji Isogai,et al.  Numerical Study of Transonic Flutter of A Two-dimensional Airfoil , 1980 .

[165]  L. T. Chen,et al.  A transonic wing-body flowfield calculation with improved grid topology , 1985 .

[166]  M. Siclari.  Shock fitting in conical supersonic full potential flows with entropy effects , 1984 .

[167]  J. Benek,et al.  A flexible grid embedding technique with application to the Euler equations , 1983 .

[168]  J. W. Boestoel.  A multigrid algorithm for steady transonic potential flows around aerofoils using Newton iteration , 1981 .

[169]  Terry L. Holst,et al.  Multizone Chimera Algorithm for Solving the Full-Potential Equation , 1998 .

[170]  B. Grossman,et al.  Numerical Procedure for the Computation of Irrotational Conical Flows , 1979 .

[171]  K. Ramachandran,et al.  Rotor hover performance prediction using a free-wake, computational fluid dynamics method , 1989 .

[172]  Gerald Warnecke,et al.  Finite Elements for Transonic Potential Flows , 1990 .

[173]  G. Dulikravich.  Analysis of artificial dissipation models for the transonic full-potential equation , 1988 .

[174]  John T. Batina,et al.  Unsteady transonic small-disturbance theory including entropy and vorticity effects , 1989 .

[175]  D. F. Fuglsang,et al.  Application of XTRAN3S and CAP-TSD to fighter aircraft , 1991 .

[176]  B. Epstein,et al.  Multigrid transonic computations about arbitrary aircraft configurations , 1989 .

[177]  G. Volpe.  Multigrid approximate factorization scheme for two-element airfoil flows , 1987 .

[178]  Woodrow Whitlow.  Application of a nonisentropic full potential method to AGARD standard airfoils , 1988 .

[179]  Antony Jameson,et al.  Numerical Calculation of Transonic Potential Flow about Wing-Body Combinations , 1979 .

[180]  J. J. Douglas.  On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods , 1955 .

[181]  Guru P. Guruswamy.  Integrated approach for active coupling of structures and fluids , 1989 .

[182]  A. Jameson.  Iterative solution of transonic flows over airfoils and wings, including flows at mach 1 , 1974 .

[183]  K. McGrattan.  Comparison of transonic flow models , 1992 .

[184]  W. R. Van Dalsem,et al.  Efficient simulation of separated three-dimensional viscous flows using the boundary-layer equations , 1985 .

[185]  Koji Isogai,et al.  Numerical calculation of unsteady transonic potential flow over three-dimensional wings with oscillating control surfaces , 1984 .

[186]  A. Jameson,et al.  MULTIPLE SOLUTIONS OF THE TRANSONIC POTENTIAL FLOW EQUATION , 1982 .

[187]  F. T. Johnson,et al.  A transonic rectangular grid embedded panel method , 1982 .

[188]  Vijaya Shankar,et al.  Computational Transonic Inverse Procedure for Wing Design , 1982 .

[189]  Terry L. Holst,et al.  Comparison of the full potential and Euler formulations for computing transonic airfoil flows , 1985 .

[190]  T. Tranen,et al.  A rapid computer aided transonic airfoil design method , 1974 .

[191]  Stanley Osher,et al.  Improved finite-difference schemes for transonic potential flow calculations , 1984 .

[192]  T. Holst,et al.  A consistent spatial differencing scheme for the transonic full-potential equation in three dimensions , 1985 .

[193]  T. Yang,et al.  Aeroelastic time response analysis of thin airfoils by transonic code LTRAN2 , 1981 .

[194]  J. B. Malone,et al.  AN IMPLICIT CONSERVATIVE ALGORITHM FOR STEADY AND UNSTEADY THREE-DIMENSIONAL TRANSONIC POTENTIAL FLOWS 81-1016 , 1981 .

[195]  S. Osher,et al.  An efficient, full-potential implicit method based on characteristics for supersonic flows , 1983 .

[196]  A. K. Oppenheim,et al.  Fundamentals of Gas Dynamics , 1964 .

[197]  R. Strawn,et al.  Conservative full-potential model for unsteady transonic rotor flows , 1987 .

[198]  P. R. Garabedian,et al.  Analysis of transonic airfoils. , 1971 .

[199]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[200]  B. Vanleer,et al.  On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe , 1984 .

[202]  Lakshmi N. Sankar,et al.  Three-dimensional Navier-Stokes/full-potential coupled analysis for viscous transonic flow , 1993 .

[203]  R. P. Fedorenko.  The speed of convergence of one iterative process , 1964 .

[204]  William Gropp,et al.  Parallel Newton-Krylov-Schwarz Algorithms for the Transonic Full Potential Equation , 1996, SIAM J. Sci. Comput..

[205]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[206]  Numerical calculation of the transonic potential flow past a cranked wing , 1983 .

[207]  John F. Liutermoza,et al.  Second-Order-Accurate Calculation of Transonic Flow over Turbomachinery Cascades , 1979 .

[208]  H. Elbanna,et al.  Determination of aerodynamic sensitivity coefficients based on the transonic small perturbation formulation , 1990 .

[209]  D. S. Whitehead,et al.  A finite element method for the solution of two‐dimensional transonic flows in cascades , 1985 .

[210]  T. Jayachandran,et al.  Finite-element method applied to transonic flow over a bulbous payload shroud , 1989 .

[211]  Francis X. Caradonna,et al.  Numerical Calculation of Unsteady Transonic Potential Flow over Helicopter Rotor Blades , 1975 .

[212]  John T. Batina,et al.  Supersonic far-field boundary conditions for transonic small-disturbance theory , 1990 .

[213]  J. L. Steger,et al.  Numerical calculation of transonic flow about two-dimensional airfoils by relaxation procedures , 1971 .

[214]  Jerry C. South,et al.  Stability analysis of intermediate boundary conditions in approximate factorization schemes , 1986 .

[215]  Roger C. Strawn,et al.  An experimental and computational study of rotor-vortex interactions , 1988 .

[216]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[217]  C. Boppe,et al.  Calculation of transonic wing flows by grid embedding , 1977 .

[218]  J. Van der Vooren,et al.  Inviscid drag prediction for transonic transport wings using a full-potential method , 1991 .

[219]  Guru P. Guruswamy,et al.  Unsteady transonics of a wing with tip store , 1986 .

[220]  Michael D. Madson.  Transonic analysis of the F-16A with under-wing fuel tanks - An application of the TranAir full-potential code , 1987 .

[223]  Thong Q Dang,et al.  Transonic computational method for an aft-mounted nacelle/pylon withpower effect , 1990 .

[224]  Comparison between computations and experimental data in unsteady three-dimensional transonic aerodynamics, including aeroelastic applications , 1982 .

[225]  R. Nicolaides.  On the ² convergence of an algorithm for solving finite element equations , 1977 .

[226]  W. Hackbusch.  Convergence of multigrid iterations applied to difference equations , 1980 .

[227]  Leland A. Carlson,et al.  Direct-Inverse Transonic Wing-Design Method in Curvilinear Coordinates Including Viscous Interaction , 1989 .

[228]  Achi Brandt,et al.  Application of a multi-level grid method to transonic flow calculations , 1976 .

[229]  Roger C. Strawn,et al.  An entropy and viscosity corrected potential method for rotor performance prediction , 1988 .

[230]  Alan E. Arslan,et al.  Integrated determination of sensitivity derivatives for an aeroelastic transonic wing , 1994 .

[231]  C. W. Boppe,et al.  Simulated transonic flows for aircraft with nacelles, pylons, and winglets , 1980 .

[232]  Terry L. Hoist.  FAST TRANSONIC AERODYNAMIC ANALYSIS SUITABLE FOR THE PRELIMINARY DESIGN ENVIRONMENT , 1999 .

[233]  N. J. Yu,et al.  Efficient Transonic Shock-Free Wing Redesign Procedure Using a Fictitious Gas Method , 1979 .

[234]  Transonic potential flow calculations by two artificial density methods , 1988 .

[235]  Speed-up of the strongly implicit procedure with application to subsonic/transonic potential flows , 1990 .

[236]  S. Luo,et al.  Theoretical and Computational Dynamics of a Compressible Flow , 1991 .

[237]  M. Hafez,et al.  Application of conjugate gradient methods to transonic finite difference and finite element calculations , 1981 .

[238]  R. M. Hicks,et al.  Transonic wing design using potential-flow codes - Successes and failures , 1981 .

[239]  P. Goorjian,et al.  Computation of Unsteady Transonic Flows by the Indicial Method , 1977 .

[240]  John Batina.  An efficient algorithm for solution of the unsteady transonic small-disturbance equation , 1987 .

[241]  Wagdi G. Habashi.  Advances in computational transonics , 1985 .

[242]  M. J. Siclari,et al.  Entropy corrections to supersonic conical nonlinear potential flows , 1985 .

[243]  Zhu Ziqiang,et al.  Numerical Computation of Improved Transonic Potential Method , 1992 .

[244]  S. Cheung.  Parallel computational fluid dynamics design on network-based computer , 1996 .

[245]  Shigeru Obayashi,et al.  Comparison of Optimization Algorithms for Aerodynamic Shape Design , 1996 .

[246]  Michael A. Saunders,et al.  User''s guide for NPSOL (Ver-sion 4.0): A FORTRAN package for nonlinear programming , 1984 .

[247]  T. Holst,et al.  Conservative schemes for the full potential equation applied to transonic flows , 1979 .

[248]  J. Vadyak,et al.  A Grid Overlapping Scheme for Flowfield Computations About Multicomponent Configurations , 1983 .

[249]  Reece E. Neel,et al.  Advances In Computational Fluid Dynamics: Turbulent Separated Flows And Transonic Potential Flows , 1997 .

[250]  M. Hafez,et al.  Frontiers of Computational Fluid Dynamics 2002 , 2001 .

[251]  D. S. Dulikravich,et al.  CAS22 - FORTRAN program for fast design and analysis of shock-free airfoil cascades using fictitious-gas concept , 1982 .

[252]  Arvin Shmilovich.  Calculation of transonic flows with separation past arbitrary inletsat incidence , 1988 .

[253]  Joe F. Thompson,et al.  Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies , 1974 .

[254]  Antony Jameson,et al.  Progress in Finite-Volume Calculations for Wing-Fuselage Combinations , 1980 .

[255]  P. Goorjian,et al.  Implicit Finite-Difference Computations of Unsteady Transonic Flows about Airfoils , 1977 .

[256]  Tuncer Cebeci,et al.  An Interactive Scheme for Three-Dimensional Transonic Flows , 1986 .

[257]  Guru P. Guruswamy,et al.  Efficient algorithm for unsteady transonic aerodynamics of low-aspect-ratio wings , 1985 .

[258]  D. Kinney,et al.  An Unstructured Full Potential - Integral Boundary Layer Formulation , 1997 .

[259]  T. A. Reyhner.  Cartesian Mesh Solution for Axisymmetric Transonic Potential Flow Around Inlets , 1977 .

[261]  F. X. Caradonna,et al.  A conservative implicit finite difference algorithm for the unsteady transonic full potential equation , 1980 .

[262]  N. Yu,et al.  Transonic flow simulations for complex configurations with surface-fitted grids , 1981 .

[263]  Terry L. Holst,et al.  Chimera Donor Cell Search Algorithm Suitable for Solving the Full Potential Equation , 1997 .

[264]  T. A. Egolf,et al.  Hovering rotor airload prediction using a full potential flow analysis with realistic wake geometry , 1985 .

[265]  Edgar G. Waggoner,et al.  Transonic wind-tunnel wall interference prediction code , 1988 .

[266]  Lawrence. Davis,et al.  Handbook Of Genetic Algorithms , 1990 .

[267]  R. M. Hicks,et al.  An assessment of airfoil design by numerical optimization , 1974 .

[268]  H. Sobieczky,et al.  Shock-Free Wing Design , 1980 .

[269]  P. A. Newman,et al.  Conservative versus nonconservative differencing: Transonic streamline shape effects , 1976 .

[270]  P. A. Henne,et al.  Applied Computational Aerodynamics , 1990 .

[271]  E. Atta,et al.  Component-adaptive grid interfacing , 1981 .

[272]  G. B. Cosentino,et al.  Numerical optimization design of advanced transonic wing configurations , 1985 .

[273]  Guru P. Guruswamy,et al.  Effects of viscosity and modes on transonic aerodynamic and aeroelastic characteristics of wings , 1985 .

[274]  John Bridgeman,et al.  Development of an Overset/Hybrid CFD Method for the Prediction of Hovering Performance , 1997 .

[275]  M. Hafez,et al.  Some iterative schemes for transonic potential flows , 1985 .

[276]  Terry L. Holst,et al.  Aerodynamic Shape Optimization Using A Combined Distributed/Shared Memory Paradigm , 1999 .

[277]  Aga M. Goodsell,et al.  TranAir and Euler computations of a generic fighter including comparisons with experimental data. [full-potential equations for transonic flow , 1989 .

[278]  Leland A. Carlson,et al.  Transonic wing design using inverse methods in curvilinear coordinates , 1988 .

[279]  J. L. Steger,et al.  Transonic Flow about Two-Dimensional Airfoils by Relaxation Procedures , 1972 .

[280]  Garret N. Vanderplaats,et al.  CONMIN: A FORTRAN program for constrained function minimization: User's manual , 1973 .

[281]  R. A. Kennelly.  Improved method for transonic airfoil design-by-optimization , 1983 .

[282]  Pradeep Raj,et al.  A Cost-Effective Method for Shock-Free Supercritical Wing Design , 1982 .

[283]  K. Ramachandran,et al.  The free-wake computation of rotor-body flows , 1990 .

[284]  G. B. Cosentino,et al.  Numerical optimization design of advanced transonic wing configurations , 1985 .

[285]  Y. Tassa,et al.  An algorithm for unsteady transonic potential flow past airfoils , 1981 .

[286]  P. M. Goorjian,et al.  Implicit computations of unsteady transonic flow governed by the full-potential equation in conservation form , 1980 .

[287]  A. Jameson.  Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows , 1993 .

[288]  R. M. Hicks,et al.  Wing Design by Numerical Optimization , 1977 .

[289]  R. Buskirk,et al.  Monotone implicit algorithms for the small-disturbance and full potential equations applied to transonic flows , 1983 .

[290]  Michael B. Bieterman,et al.  Recent progress in aerodynamic design optimization , 1999 .

[291]  Woodrow Whitlow,et al.  Entropy Condition Satisfying Approximations for the Full Potential Equation of Transonic Flow , 1985 .

[292]  David A. Caughey,et al.  Application of the Multigrid Method to Calculations of Transonic Potential Flow about Wing-Fuselage Combinations , 1982 .

[293]  John Bridgeman,et al.  The development of a CFD potential method for the analysis of tilt-rotors , 1991 .

[294]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[295]  Roger C. Strawn,et al.  Advanced rotor computations with a corrected potential method , 1989 .

[296]  Edward N. Tinoco,et al.  TRANAIR applications to engine/airframe integration , 1989 .

[297]  John T. Batina,et al.  Calculation of steady and unsteady pressures on wings at supersonic speeds with a transonic small disturbance code , 1991 .

[298]  N. Sankar,et al.  Strongly Implicit Procedure Applied to the Flowfield of Transonic Turbine Cascades , 1982 .

[299]  R. Gregg,et al.  An assessment of mass conservation for potential flow shock simulation , 1982 .

[300]  E. H. Twizell.  Computational methods for partial differential equations , 1984 .

[301]  R. M. Hicks,et al.  Wing analysis using a transonic potential flow computational method , 1978 .

[302]  Analysis of Transonic Cascade Flow Using Conf ormal Mapping and Relaxation Techniques , 1977 .

[303]  T. Holst.  Implicit Algorithm for the Conservative Transonic Full-Potential Equation Using an Arbitrary Mesh , 1979 .

[304]  Scott D. Thomas,et al.  Numerical solution of transonic wing flowfields , 1982 .

[305]  M. Hafez,et al.  Entropy and vorticity corrections for transonic flows , 1983 .

[306]  E. Oran.  Frontiers in Computational Fluid Dynamics , 1997 .

[307]  N. Bakhvalov.  On the convergence of a relaxation method with natural constraints on the elliptic operator , 1966 .

[308]  D. Rizzetta,et al.  Time-Dependent Response of a Two-Dimensional Airfoil in Transonic Flow , 1979 .

[309]  P. F. Bradley,et al.  Vectorized schemes for conical potential flow using the artificial density method , 1984 .

[310]  J. Steger.  Implicit finite difference simulation of flow about arbitrary geometries with application to airfoils , 1977 .

[311]  G. Volpe,et al.  The role of constraints in the inverse design problem for transonic airfoils , 1981 .

[312]  P. Garabedian,et al.  Design of supercritical swept wings , 1982 .

[313]  D. Nixon,et al.  Modifications to Transonic Flow Codes for Unsteady Perturbations Around an Experimental Mean , 1989 .

[314]  Robert M. Bennett,et al.  Application of transonic small disturbance theory to the active flexible wing model , 1995 .

[315]  P. Gill,et al.  Quasi-Newton Methods for Unconstrained Optimization , 1972 .

[316]  A. Jameson,et al.  Accelerated Iterative Calculation of Transonic Nacelle Flowfields , 1976 .

[317]  Leland A. Carlson,et al.  Transonic airfoil analysis and design using Cartesian coordinates , 1976 .

[318]  D. A. Caughey.  Multi-grid calculation of three-dimensional transonic potential flows , 1983 .

[319]  Stanley Osher,et al.  A fast, time-accurate, unsteady full potential scheme , 1987 .