Explicit Finite Difference Methods: Some Selected Applications to Inviscid and Viscous Flows

  • J. D. AndersonJr.


In this chapter we round-out our introductory treatment of computational fluid dynamics by discussing some applications of explicit finite difference methods to selected examples for inviscid and viscous flows. These examples have one thing in common—they are results obtained by either the present author and/or some of his graduate students over the past few years. This is not meant to be chauvinistic; rather this choice is intentionally made to illustrate what can be done by uninitiated students who are new to the ideas of CFD. These examples demonstrate the power and beauty of CFD in the hands of students much like yourselves who may have little or no experience in the field. Moreover, in all cases the applications are carried out with computer programs designed and written completely by each student. This is following the author’s educational philosophy that each student should have the experience of starting with paper and pencil, writing down the governing equations, developing the appropriate numerical solution of these equations, writing the FORTRAN program, punching the program into the computer, and then going through all the trials and tribulations of making the program work properly. This is an important aspect of CFD education. No established computer programs (‘canned’ programs) are used; everything is ‘home-grown’, with the exception of standard graphics packages which are used to plot the results. Therefore, by examining these examples, you should obtain a reasonable feeling for what you can expect to accomplish when you first jump into the world of CFD applications.


AIAA Paper AIAA Journal Nozzle Flow Space Grid Point Governing Flow Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References (for Chapters 1–7)

  1. 1.
    Liepmann, H.W. and Roshko, A., Elements of Gasdynamics, Wiley, New York, 1957.MATHGoogle Scholar
  2. 2.
    Moretti, G. and Abbett, M., ‘A Time-Dependent Computational Method for Blunt Body Flows,’ AIAA Journal, Vol. 4, No. 12, December 1966, pp. 2136–2141.ADSMATHCrossRefGoogle Scholar
  3. 3.
    Anderson, John D., Jr., Fundamentals of Aerodynamics, 2nd Edition McGraw-Hill, New York, 1991.Google Scholar
  4. 4.
    Anderson, John D., Jr., ‘Computational Fluid Dynamics An Engineering Tool?’ in Numerical/ Laboratory Computer Methods in Fluid Dynamics (Ed. A.A. Pouring ), ASME, New York, 1976, pp. 1–12.Google Scholar
  5. 5.
    Graves, Randolph A., ‘Computational Fluid Dynamics: The Coming Revolution,’ Astronautics and AeronauticsVol. 20, No. 3, March 1982, pp. 20–28. Google Scholar
  6. 6.
    Kopal, Z., Tables of Supersonic Flow Around Cones, Dept. of Electrical Engineering, Center of Analysis, Massachusetts Institute of Technology, Cambridge, 1947.Google Scholar
  7. 7.
    Taylor, G.I. and Maccoll, J.W., The Air Pressure on a Cone Moving at High Speed,’ Proceedings of the Royal Society (A), Vol. 139, 1933, p. 278.MATHGoogle Scholar
  8. 8.
    Fay, J.A. and Riddell, F.R., ‘Theory of Stagnation Point Heat Transfer in Dissociated Air,’ Journal of the Aeronautical Sciences, Vol. 25, No. 2, Feb. 1958, pp. 73–85.MathSciNetGoogle Scholar
  9. 9.
    Blottner, F.G., ‘Chemical Nonequilibrium Boundary Layer,’ AIAA Journal, Vol. 2, No. 2, Feb. 1964, pp. 232–239.MATHCrossRefGoogle Scholar
  10. 10.
    Blottner, F.G., Nonequilibrium Laminar Boundary-Layer Flow of Ionized Air,’ AIAA Journal, Vol. 2, No. 11, Nov. 1964, pp. 1921–1927.MATHCrossRefGoogle Scholar
  11. 11.
    Hall, H.G., Eschenroeder, A.Q. and Marrone, P.V., ‘Blunt-Nose Inviscid Airflows with Coupled Nonequilibrium Processes,’ Journal of the Aerospace Sciences, Vol. 29, No. 9, Sept. 1962, pp. 1038–1051.MATHCrossRefGoogle Scholar
  12. 12.
    Chapman, Dean R., ‘Computational Aerodynamics Development and Outlook,’ AIAA Journal, Vol. 17, No. 12, Dec. 1979, pp. 1293–1313.ADSMATHCrossRefGoogle Scholar
  13. 13.
    Advanced Technology Airfoil Research, NASA Conference Publications 2045, March 1978.Google Scholar
  14. 14.
    Anderson, John D., Jr., Modern Compressible Flow: With Historical Perspective, 2 Edition McGraw-Hill, New York, 1990.Google Scholar
  15. 15.
    Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, Wiley, 1960.Google Scholar
  16. 16.
    Kutler, P., ‘Computation of Three-Dimensional, Inviscid Supersonic Flows,’ in H.J. Wirz (ed.), Progress in Numerical Fluid Dynamics, Springer-Verlag, Berlin, 1975, pp. 293–374.CrossRefGoogle Scholar
  17. 17.
    Hildebrand, F.B., Advanced Calculus for Applications, Prentice-Hall, New Jersey, 1976.MATHGoogle Scholar
  18. 18.
    Anderson, Dale A., Tannehill, John C. and Pletcher, Richard H., Computational Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1984.MATHGoogle Scholar
  19. 19.
    Sullins, G.A., Anderson, J.D., Jr. and Drummond, J.P., ‘Numerical Investigation of Supersonic Base Flow with Parallel Injection,’ AIAA Paper No. 82–1001.Google Scholar
  20. 20.
    Sullins, G.A., Numerical Investigation of Supersonic Base Flow with Tangential Injection, M.S. Thesis, Department of Aerospace Engineering, University of Maryland, 1981.Google Scholar
  21. 21.
    Holst, T.L., ‘Numerical Solution of Axisymmetric Boattail Fields with Plume Simulators,’ AIAA Paper No. 77–224, 1977.Google Scholar
  22. 22.
    Roberts, B.O., ‘Computational Meshes for Boundary Layer Problems,’ Lecture Notes in Physics, Springer-Verlag, New York, 1971, pp. 171–177.Google Scholar
  23. 23.
    Thompson, J.F., Thames, F.C. and Mastin, C.W., ‘Automatic Numerical Generation of Body-Fitted Curvilinear Coordinate Systems for Fields Containing Any Number of Arbitrary Two-Dimensional Bodies,’ Journal of Computational Physics, Vol. 15, pp. 299–319, 1974.ADSMATHCrossRefGoogle Scholar
  24. 24.
    Wright, Andrew F., A Numerical Investigation of Low Reynolds Number Flow Over an Airfoil, M.S. Thesis, Department of Aerospace Engineering, University of Maryland, 1982.Google Scholar
  25. 25.
    Corda, Stephen, Numerical Investigation of the Laminar, Supersonic Flow over a Rearward-Facing Step Using an Adaptive Grid Scheme, M.S. Thesis, Department of Aerospace Engineering, University of Maryland, 1982.Google Scholar
  26. 26.
    Dwyer, H.A., Kee, R.J. and Sanders, B.R., ‘An Adaptive Grid Method for Problems in Fluid Mechanics and Heat Transfer,’ AIAA Paper No. 79–1464, 1979.Google Scholar
  27. 27.
    Abbott, I.H. and von Doenhoff, A.E., Theory of Wing Sections, McGraw-Hill Book Company, New York, 1949; also, Dover Publications, Inc., New York, 1959.Google Scholar
  28. 28.
    Hess, J.L. and Smith, A.M.O., ‘Calculation of Potential Flow about Arbitrary Bodies,’ in Progress in Aeronautical Sciences, Vol. 8 (edited by D. Kucheman), Pergamon Press, New York, pp. 1–138.Google Scholar
  29. 29.
    Chow, C.Y., An Introduction to Computational Fluid Dynamics, John Wiley & Sons, Inc., New York, 1979.Google Scholar
  30. 30.
    Winkelmann, A.E. and Tsao, C.P., ‘An Experimental Study of the Flow on a Wing With a Partial Span Dropped Leading Edge,’ AIAA Paper No. 81–1665, 1981.Google Scholar
  31. 31.
    Anderson, John D., Jr., Corda, Stephen and VanWie, David M., ‘Numerical Lifting Line Theory Applied to Drooped Leading-Edge Wings Below and Above Stall,’ Journal of Aircraft, Vol. 17, No. 12, Dec. 1980, pp. 898–904.CrossRefGoogle Scholar
  32. 32.
    Cho, T.H. and Anderson, J.D., Jr., ‘Engineering Analysis of Drooped Leading-Edge Wings Near Stall,’ Journal of Aircraft, Vol. 21, No. 6, June 1984, pp. 446–448.CrossRefGoogle Scholar
  33. 33.
    Johnson, J.L., Jr., Newsom, W.A. and Satran, D.R., ‘Full-Scale Wind Tunnel Investigation of the Effects of Wing Leading-Edge Modifications on the High Angle-of-Attack Aerodynamic Characteristics of a Low-Wing General Aviation Airplane,’ AIAA Paper No. 80, 1844, 1980.Google Scholar
  34. 34.
    Ames Research Staff, ‘Equations, Tables, and Charts for Compressible Flow,’ NACA Report 1135, 1953.Google Scholar
  35. 35.
    Anderson, John D. Jr., ‘A Time-Dependent Analysis for Quasi-One-Dimensional Nozzle Flows with Vibrational and Chemical Nonequilibrium,’ NOLTR 69–52, Naval Ordnance Laboratory, White Oak, MD, 1969.Google Scholar
  36. 36.
    Anderson, John D., Jr., ‘A Time-Dependent Analysis for Vibrational and Chemical Nonequilibrium Nozzle Flows,’ AIAA Journal, Vol. 8, No. 3, March 1970, pp. 545–550.ADSCrossRefGoogle Scholar
  37. 37.
    MacCormack, R.W., ‘The Effect of Viscosity in Hypervelocity Impact Cratering,’ AIAA Paper No. 69–354, 1969.Google Scholar
  38. 38.
    Anderson, John D., Jr., ‘Time-Dependent Solutions of Nonequilibrium Nozzle Flow—A Sequel,’ AIAA Journal, Vol. 5, No. 12, Dec. 1970. pp. 2280–2282.ADSGoogle Scholar
  39. 39.
    Hall, J.G. and Russo, A.L., ‘Studies of Chemical Nonequilibrium in Hypersonic Nozzle Flows,’ AFOSR TN 59–1090, Cornell Aeronautical Laboratory Report AD-1118-A-6, November 1969.Google Scholar
  40. 40.
    Anderson, John D., Jr., ‘On Hypersonic Blunt Body Flow Fields Obtained with a Time-Dependent Technique,’ NOLTR 68–129, Naval Ordnance Laboratory, White Oak, MD, August 1968.Google Scholar
  41. 41.
    Dallospedale, Carlo L., ‘A Numerical Solution for the Two-Dimensional Flowfield in an Internal Combustion Engine with Realistic Valve-Geometry,’ M.S. Thesis, Department of Aerospace Engineering, University of Maryland, College Park, MD, 1978.Google Scholar
  42. 42.
    Berman, H.A., Anderson, J.D., Jr. and Drummond, J.P., ‘A Numerical Solution of the Supersonic Flow Over a Rearward Facing Step with Transverse Non-Reacting Hydrogen Injection,’ AIAA Paper No. 82–1002, 1982.Google Scholar
  43. 43.
    Berman, H.A., Anderson, J.D., Jr., and Drummond, J.P., ‘Supersonic Flow over a Rearward Facing Step with Transverse Nonreacting Hydrogen Injection,’ AIAA Journal, Vol. 21, No. 12, December 1983, pp. 1707–1713.ADSMATHCrossRefGoogle Scholar
  44. 44.
    Baldwin, B.S. and Lomax, H., ‘Thin Layer Approximations and Algebraic Model for Separated Turbulent Flows,’ AIAA Paper 78–257, 1978.Google Scholar
  45. 45.
    Kothari, A.P. and Anderson, J.D., Jr., ‘Flows Over Low Reynolds Number Airfoils—Compressible Navier–Stokes Numerical Solutions’, AIAA Paper No. 85–0107, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • J. D. AndersonJr.

There are no affiliations available

Personalised recommendations