Applications of Boundary Element Methods to Fluid Mechanics

  • J. A. Liggett
  • P. L.-F. Liu
Part of the Topics in Boundary Element Research book series (TBOU, volume 1)

Abstract

In this chapter we make an effort to review the applications of boundary methods to fluid mechanics. At the outset we wish to give our definition of boundary methods. First, we exclude such techniques from the general class of finite element methods. Although some of the language and a few of the numerical techniques of the two methods have merged, they are historically quite separate.

Keywords

Permeability Vortex Porosity Beach Vorticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Trefftz, E., Über die Kontraktion kreisförmiger Flüssigkeitsstrahlen. Z. Math. Phys., 64, 34, 1917Google Scholar
  2. 2.
    Jaswon, M. A., Symm, G. T., Integral Equation Methods in Potential Theory and Elastostatics. Academic Press, 1977Google Scholar
  3. 3.
    Brebbia, C. A., The Boundary Element Method for Engineers. John Wiley and Sons, 1978Google Scholar
  4. 4.
    Brebbia, C. A., Walker, S., Boundary Element Techniques in Engineering. Newnes-Butterworths 1980Google Scholar
  5. 5.
    Banerjee, P. K., Butterfield, R., Boundary Element Methods in Engineering Science. McGraw-Hill 1981Google Scholar
  6. 6.
    Liggett, J. A., Liu, P. L.-F., The Boundary Integral Equation Method for Porous Media Flow. George Allen Unwin, 1983Google Scholar
  7. 7.
    Crouch, S. L., Starfield, A. M., Boundary Methods in Solid Mechanics. George Allen and Unwin, 1983Google Scholar
  8. 8.
    Mukherjee, S., Boundary element methods in creep and fracture. Applied Science Publishers, Ltd, Essex, 1982, Also Elsevier Science Publishing, Co, N.Y.Google Scholar
  9. 9.
    Green, G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham 1828Google Scholar
  10. 10.
    Hunt, B. W., Numerical solutions of an integral equation for flow from a circular orifice. Jour. Fluid Mech., 31, 361–377, 1968CrossRefMATHGoogle Scholar
  11. 11.
    Prager, W., Die Druckverteilung an Körpern in ebener Potentialströmung. Physik. Zeitschr., 29, 865–869, 1928Google Scholar
  12. 12.
    Nyström, E. J., Über die praktische Auflösung von linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie. Soc. Sci. Fennica, Comment. Physico-Math., 4, 15, 1–52, 1928Google Scholar
  13. 13.
    Fredholm, I., Sur une classe d’equations fonctionelles. Acta Math., 27, 365–390, 1903CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lotz, I., Calculation of potential past airship bodies in yaw. NACA TM 675, 1932 [also Ingenieur-Archiv, Vol. I I, 1931 ]Google Scholar
  15. 15.
    Vandrey, F., A direct iteration method for the calculation of velocity distribution of bodies of revolution and symmetrical profiles. Admiralty Research Lab. Rept. R 1/G/HY/12/2, 1951Google Scholar
  16. 16.
    Weinstein, A., On axially symmetric flows. Quarterly of Applied Math., 5, No. 4, 1948Google Scholar
  17. 17.
    Van Tuyl, A., On the axially symmetric flow around a new family of half bodies. Quarterly of Applied Math., 7, No. 4, 1950Google Scholar
  18. 18.
    Sadowsky, M. A., Sternberg, E., Elliptic integral representation of axially symmetric flows. Quarterly of Applied Math. 8, No. 2, 1950Google Scholar
  19. 19.
    Landweber, L., The axially symmetric potential flow about elongated bodies of revolution. David Taylor Model Basin Rep., 761, 1951Google Scholar
  20. 20.
    Vandrey, F., On the calculation of the transverse potential flow past a body of revolution with the aid of the method of Mrs. Flügge-Lotz. Astia AD-40089, 1951Google Scholar
  21. 21.
    Smith, A. M. O., Pierce, J., Exact solution of the Neumann problem. Calculation of plane and axially symmetric flows about or within arbitrary boundaries. Douglas Aircraft Company Report No. 26988, 1958 [Summary in Proc. of the 3rd Int. Congress of Applied Math., Brown University, 1958 ]Google Scholar
  22. 22.
    Hess, J. L., Smith, A. M. O., Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. ES 40622, Douglas Aircraft Corp., Long Beach, Calif. 1962 (Also in Jour. of Ship Research 8, No. 2, Sept. 1964 )Google Scholar
  23. 23.
    Davenport, F. J., Singularity solutions to general potential flow airfoil problem. D6–7207, Boeing Airplane Co., Seattle, Wash. 1963Google Scholar
  24. 24.
    Jaswon, M. A., Integral equation methods in potential theory: I. Proc. Royal Soc. A, 275, 23–32, 1963MATHMathSciNetGoogle Scholar
  25. 25.
    Symm, G. T., Integral equation methods in potential theory; II. Proc. Royal Soc. A, 275, 33–46, 1963MATHMathSciNetGoogle Scholar
  26. 26.
    Chaplin, H. R., A method for numerical calculation of slip stream contraction of a shrouded impulse disk in the static case with application to other axisymmetric potential flow problems. David Taylor Model Basin Report No. 1857, 1964Google Scholar
  27. 27.
    Gallagher, R. H., Finite Element Analysis Fundamentals, Prentice Hall 1975Google Scholar
  28. 28.
    Hess, J. L., Review of integral-equation techniques for solving potential-flow problems with complicated boundaries. Innovative Numerical Analysis for the Applied Engineering Sciences. University Press of Virginia, Charlottesville, 131–143, 1980Google Scholar
  29. 29.
    Hunt, B., The mathematical basis and numerical principles of the boundary integral equation method for incompressible potential flow over 3-D aerodynamic configurations. Numerical Methods in Applied Fluid Dynamics (B. Hunt, ed.), Academic Press, 49–135, 1980Google Scholar
  30. 30.
    Carey, G. F., Extension of boundary elements to lifting compressible aerodynamics. Finite Element Flow Analysis (T. Kawai, ed.), Univ. of Tokyo Press, 939–943, 1982Google Scholar
  31. 31.
    Carey, G. F., Kim, S. W., Extension of boundary element method to lifting subcritical flows. 19th Annual Meeting, Society of Engineering Science, University of Missouri-Rolla, 1982Google Scholar
  32. 32.
    Inamuro, T., Adachi, T., Sakata, H., A numerical analysis of unsteady separated flow by discrete vortex model using boundary element method. Finite Element Flow Analysis (T. Kawai, ed.), University of Tokyo Press, 931–938, 1982Google Scholar
  33. 33.
    White, J. W., Kline, S. J., A calculation method for incompressible axisymmetric flows, including unseparated, fully separated, and free surface flows. Report MD-35, U.S. Air Force Office of Scientific Research Mechanics Divison, Contract AF-F44620–74-C-0016; Thermosciences Division, Department of Mechanical Engineering, Stanford University, 1975Google Scholar
  34. 34.
    Wu, J. C., Problems of general visous flow. In: Developments in Boundary Element Methods — 2 (P. K Banerjee and R. P. Shaw, eds.). Applied Science Publishers, 69–109, 1982Google Scholar
  35. 35.
    Wu, J. C., Thompson, J. F., Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation. Computers and Fluids 1, 197–215, 1973CrossRefMATHGoogle Scholar
  36. 36.
    Lennon, G. P., Liu, P. L-F., Liggett, J. A., Boundary integral equation solution to axisymmetric potential flows, 1, Basic formulation. Water Resources Research, 15 (5), 1102–1106, 1979CrossRefGoogle Scholar
  37. 37.
    Lennon, G. P., Liu, P. L-F., Liggett, J. A., Boundary integral equation solution to axisymmetric flows, 2, Recharge and well problems in porous media. Water Resources Research, 15 (5), 1107–1115, 1979CrossRefGoogle Scholar
  38. 38.
    Lennon, G. P., Liu, P. L-F., Liggett, J. A., Boundary integral solutions to three-dimensional unconfined Darcy’s flow. Water Resources Research, 16 (4), 651–658, 1980CrossRefGoogle Scholar
  39. 39.
    Cheng, A H-D., Liu, P. L-F., Liggett, J. A., Boundary calculations of sluice and spillway flows. J. of the Hydraulics Division, ASCE, 107, (HY 10), 1163–1178, 1981Google Scholar
  40. 40.
    Liggett, J. A, Salmon, J. R., Cubic spline boundary elements. Int. J. for Numerical Methods in Engineering, 17, 543–556, 1981CrossRefMATHGoogle Scholar
  41. 41.
    Liu, P. L-F., Liggett, J. A., Applications of boundary element methods to problems of water waves. In: Developments in Boundary Element Methods–2 (P. K. Banerjee and R. P. Shaw, eds.), Elsevier’s Applied Science Publishers, Ltd., 37–67, 1982Google Scholar
  42. 42.
    Nakayama, T., Washizu, K., The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems. Int. J. Num. Meth. Engrg., 17, No. 11, 1631–1646, 1981CrossRefMATHGoogle Scholar
  43. 43.
    Washizu, K, Some applications of finite element techniques to nonlinear free surface fluid flow problems. Finite Element Flow Analysis (T. Kawai, ed.). Univ. of Tokyo Press, 3–15, 1982Google Scholar
  44. 44.
    Faltinsen, O. M., A numerical nonlinear method of sloshing in tanks with two-dimensional flow. J. Ship Research, 23 (3), 193–202, 1978Google Scholar
  45. 45.
    Longuet-Higgins, M. S., Cokelet, E. D., The deformation of steep surface waves on water. I. A numerical method of computation. Proc. of the Royal Society, A 350, 1–25, 1976Google Scholar
  46. 46.
    Cokelet, E. D., Breaking waves - the plunging jet and interior flow-field. Proc. Sym. on Mechanics of Wave-induced Forces on Cylinders, Bristol, 1978Google Scholar
  47. 47.
    Vinje, T., Brevig, P., Numerical solution of breaking waves. Adv. Water Resources, 4, 77–82, 1981CrossRefGoogle Scholar
  48. 48.
    Vinje, T., Brevig, P., Numerical calculations of forces from breaking waves. Preprints, Int. Sym. on Hydrodynamics in Ocean Engineering, Trondheim, Norway, 547–566, 1981Google Scholar
  49. 49.
    Issacson, M. de St. Q., Nonlinear-wave effects on fixed and floating bodies. J. Fluid Mechanics, 120, 267–281, 1982CrossRefGoogle Scholar
  50. 50.
    Engquist, B., Majda, A., Absorbing boundary conditions for the numerical simulation of waves. Math. and Computers, 31, 629–651, 1977CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Smith, W. D., A non-reflecting boundary for wave propagation problems. J. Computational Physics, 15, 492–503, 1974CrossRefMATHGoogle Scholar
  52. 52.
    Betts, P. L., Mohamad, T. T., Water waves: A time-varying unlinearized boundary element approach. Finite Element Flow Analysis (T. Kawai, ed.), University of Tokyo Press, 923–929, 1982Google Scholar
  53. 53.
    LeMéhauté, B., Progressive wave absorber. J. Hyd. Res., 10 (2), 153–169, 1972CrossRefGoogle Scholar
  54. 54.
    Salmon, J. R., Liu, P. L-F., Liggett, J. A., Integral equation method for linear water waves. J. Hydraulics Division, ASCE, 106, (HY 12), 1995–2010, 1980Google Scholar
  55. 55.
    Lennon, G. P., Liu, P. L-F., Liggett, J. A., Boundary integral solutions of water wave problems. J. Hydraulics Division, ASCE, 108 (HY8), 921–931, 1982Google Scholar
  56. 56.
    Mills, R. D., Computing internal viscous flow problems for the circle by integral methods. J. Fluid Mech., 79 (3), 609–624, 1977CrossRefMATHGoogle Scholar
  57. 57.
    Okabe, M., A boundary element approach in the incompressible viscous flow. Finite Element Flow Analysis (T. Kawai, ed.). Univ. of Tokyo Press, 915–922, 1982Google Scholar
  58. 58.
    Biot, M. A, General theory of three-dimensional consolidation. J. of Applied Physics, 12, 155–164, 1941CrossRefMATHGoogle Scholar
  59. 59.
    Biot, M. A, Theory of elasticity and consolidation for a porous anisotropic solid. J. of Applied Physics, 26, 182–185, 1955CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Cleary, M. P., Fundamental solutions for a fluid-saturated porous solid. Int. J. of Solids and Structures, 13, 785–806, 1977CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Cleary, M. P., Moving singularities in elasto-diffusive solids with applications to fracture propagation. Int. J. of Solids and Structures, 14, 81–97, 1978CrossRefMATHGoogle Scholar
  62. 62.
    Cleary, M. P., Fundamental solutions for fluid-saturated porous media and applications to local rupture phenomena. Thesis submitted in partial fulfillment of requirements for for the Ph.D. degree, Brown University, 1976Google Scholar
  63. 63.
    Cheng, A. H-D., Liggett, J. A., Boundary integral equation method for linear porous-elasticity with applications to fracture propagation.Int. J. for Numerical Methods in Engineering, (In press), 1983Google Scholar
  64. 64.
    Cheng, A. H-D., Liggett, J. A., Boundary integral equation method for linear porous-consolidation. Int. J. for Numerical Methods in Engineering, (In press), 1983Google Scholar
  65. 65.
    Liggett, J. A., Hydrodynamic calculations using boundary elements. Finite Element Flow Analysis (T. Kawai, ed.), Univ. of Tokyo Press, 889–896, 1982Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1984

Authors and Affiliations

  • J. A. Liggett
  • P. L.-F. Liu

There are no affiliations available

Personalised recommendations