Hostname: page-component-54dcc4c588-wlffp Total loading time: 0 Render date: 2025-09-11T21:50:13.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of numerical techniques in fluid mechanics

Published online by Cambridge University Press:  04 July 2016

Egon Krause
Egon Krause*
Affiliation:
Aerodynamisches Institut, Rheinisch-Westfälische Technische Hochschule Aachen, Germany
Get access Rights & Permissions [Opens in a new window]

Extract

More than sixty years ago Theodore von Kármán showed in a paper entitled Über die Turbulenzreibung verschiedener Flüssigkeiten (On turbulent friction of various fluids) how the pressure drop of various fluid flows through pipes could be correlated in a single curve. The correlation parameter he used is now known as Reynolds number. It had been named after Osborne Reynolds only three years earlier in 1908 by A. Sommerfeld, shordy after he had left Aachen, in a note at the Internationaler Mathematiker Kongress in Rome. Sommerfeld had recognised the importance of Reynolds' similarity law and was deeply persuaded by his experimental and theoretical work.

Information

Type
Research Article
Information
The Aeronautical Journal , Volume 78 , Issue 764 , August 1974 , pp. 337 - 354
Copyright
Copyright © Royal Aeronautical Society 1974 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Kármán, Th., von. Uber die Turbulenzreibung ver- schiedener Flüssigkeiten. Physikalische Zeitschrift, No 12, pp 126–135, 1911.Google Scholar
2. Reynolds, O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Philos Trans Roy Soc, No 174, 1883.Google Scholar
3. Reynolds, O. On the dynamic theory of incompressible viscous fluids and the determination of the criterion. Philos Trans Roy Soc, No 186, 1895.Google Scholar
4. Navier, C. L. M. H. Mémoire sur les lois du mouvement des fluides. Mémoires de I'Académie des Sciences, No 6, pp 389–416 1823.Google Scholar
5. Stokes, G. On the theories of the internal friction of fluids in motion. Transactions of the Cambridge Philo sophical Society, No 8, pp 287–305, 1845.Google Scholar
6. Blasius, H. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z Math Phys, No 56, p 1, 1908.Google Scholar
7. Prandtl, L. The mechanics of viscous fluids. Division G, Aerodynamic Theory (Durand, W. F., ed.), Vol 3, pp 34-203, 1935.Google Scholar
8. Kármán, Th., von. Aerodynamik, Ausgewahlte Themenim Lichte der historischen Entwicklung. Interavia, Genf. 1956.Google Scholar
9. Bradshaw, P. The understanding and prediction of turbulent flow. The Sixth Reynolds-Prandtl Lecture. The Aeronautical Journal of the Royal Aeronautical Society, Vol 76, pp 403-118, 1972.Google Scholar
10. Henrici, P. Elements of numerical analysis. John Wiley and Sons, Inc, Second Printing, 1965.Google Scholar
11. Gericke, H. 50 Jahre GAMM, Ingenieur-Archiv, 41. Band, Beiheft, Springer- Verlag, 1972.Google Scholar
12. Richardson, L. F. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Transactions of the Royal Society of London, Ser A, Vol 210, pp 307–357, 1910.Google Scholar
13. Richardson, L. F. The deterred approach to the limit. Part 1, Single Lattice, Trans Royal Society, Vol 226, A643, pp 299–349, 1927.Google Scholar
14. Sells, C. C. L. Two-dimensional laminar compressible boundary layer programme for a perfect gas. RAE TR 66243, 1966.Google Scholar
15. Keller, H. B., and Cebeci, T. Accurate numerical methods for boundary layer flows. I. Two-dimensional laminar flows. Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics. Lecture in Physics, Vol 8, 1971.Google Scholar
16. Liebmann, H. Die angenäherte Ermittlung hannonischer Funktionen und konformer Abbildungen, Sitzungsber-ichte der Bayr. Akad. Wiss., Math. Phys. Klasse, Vol 47, pp 385–416, 1918.Google Scholar
17. Courant, R., Friedrichs, K. O., and Lewy, H. Uber die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., Vol 100, 32, 1928.Google Scholar
18. Prandtl, L. and Busemann, A. Näherungsverfahren zur zeichnerischen Ermittlung von ebenen Strömungen mit Uberschallgeschwindigkeit. Stodola Festschrift, Zurich, pp 499–509, 1929.Google Scholar
19. Shortley, G. H. and Weller, R. The numerical solution of Laplace's equation. J Appl. Phys., Vol 9 pp 334–348, 1938.Google Scholar
20. Southwell, R. W. Relaxation methods in theoretical physics. Oxford University Press, New York, 1946.Google Scholar
21. Roache, P. J. Computation fluid dynamics. Hermosa publishers, Alburquerque, 1972.Google Scholar
22. Taylor, T. D. Numerical methods for predicting sub sonic, transonic and supersonic flow. AGARDograph No 187. North Atlantic Treaty Organisation, 1974.Google Scholar
23. Krause, E. Numerische Experimente in der Aerodynamik. Vortrag im Strömungstechnischen Seminar der Universitat Karlsruhe. Wintersemester 1968/69.Google Scholar
24. Multhopp, H. Die Berechnung der Auftriebsverteilung von Tragfliigeln. Luftfahrt-Forschung, Bd. 15, pp 153–169, 1938 Google Scholar
25. Trefftz, E. Prandtlsche Tragflächen und Propellertheorie, Vorträge aus dem Gebiet der Hydro- und Aerodynamik. Innsbruck 1922. S. 34-46, Berlin 1924. Vgl. ZAMM 1, pp 206–218, 1921.Google Scholar
26. Glauert, H. Grundlagen der Tragflügel- und Luftschraubentheorie. deutsch von H. Holl, Berlin, Springer- Verlag, 1929.Google Scholar
27. Ferri, A. Some heat transfer problems in hypersonic flow. Aeronautics and Astronautics, Pergamon Press. New York, pp 344–377, 1960.Google Scholar
28. Krause, E. Boundary conditions at the outer edge of the boundary layer on blunted conical bodies. MS Thesis, Polytechnic Institute of Brooklyn. 1962. Also AlAA Journal1, p 1671, 1963.Google Scholar
29. Krause, E. Numerical treatment of boundary-layer problems. AGARD Lecture Series No 64 on Advances in Numerical Fluid Dynamics, North Atlantic Treaty Organisation. 1973.Google Scholar
30. Crocco, L. A suggestion for the numerical solution of the steady Navier-Stokes equations. AlAA Journal 3, pp 1824–1832. 1965.Google Scholar
31. Schönaur, W. Numerical experiments with a difference model for the Navier-Stokes equations (turb. model). Proc. IUTAM Symp. High-Speed Computing in Fluid Dynamics, Phys Fluid Suppl. II, pp 228–232, 1969.Google Scholar
32. Cheng, S. I. A critical review of the numerical solution of Navier-Stokes equations, Progress in numerical fluid dynamics. Von Kármán Institute for Fluid Dynamics, Lecture Series 63. 1974.Google Scholar
33. Rotta, J. C. Turbulente Strömungen, B. G. Teubner, Stuttgart, 1972.Google Scholar
34. Lamb, H. Hydrodynamics, Dover, Sixth Edition, p 256, 1932.Google Scholar
35. Ahmed, S. R. Berechnung des reibuneslosen Strömungsfeldes von dreidimensionalen auftriebsbehafteten Tragfliigeln. Rümnfen und Flügel-Rümf-Kombinationen nach den Panel-Verfahren, DLR-FB 73-102, 1973.Google Scholar
36. Labrujere, Th. E. A survev of current collocation methods in inviscid subsonic lifting surface theory. Part I. Numerical aspects, Numerical Methods in Fluid Dvnamics. Lecture Series 44, von Kdrmdn Institute for Fluid Dynamics. 1972.Google Scholar
37. Bleckrode, A. L. A survev of current collocation methods in inviscid subsonic lifting surface theory. Part II. Calculational asoects of solving large systems of linear aleebraic emiations on a dieital comrtuter. Numerical Methods in Fluid Dynamics. Lecture Series 44, von Karman Institute for Fluid Dvnamics. 1972.Google Scholar
38. Labrujere, Th. E., Loewe, W. and Shoot, J. W. An anproximate method for the calculation of the pressure distribution on wing-bodv combinations at subcritical soeeds. aerodvnamic interference. AGARD Conference Proceedings. 1971.Google Scholar
39. Smith, A. M. O. and Hess, J. L. Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. Douglas Aircraft Corporation, Rep. ES 40622, 1962.Google Scholar
40. Ting, L. Private communication. Aachen. 1974.Google Scholar
41. Chow, F.. Krause, E.. Lui, C. H. and Mao, J. Numerical investigations of an air-foil in a non-uniform stream. 7. of Aircraft. Vol 7. No 6, 1970.Google Scholar
42. Vooren, J.van der and Labrujere, Th. E. Finite element solution of the incompressible flow over an air-foil in a non-uniform stream. National Aerospace Laboratory, The Netherlands, NLR UP 130 13U, revised edition. 1973.Google Scholar
43. Agyris, J. H. The impact of the digital computer on engineering sciences. The Twelfth Lanchester Memorial Lecture. The Aeronautical Journal of the Royal Aeronautical Society, Vol LXXIV, 1970.Google Scholar
44. Napolitano, L. Finite element methods in fluid dynamics. AGARD Lecture Series No 64, On Advances in Fluid Dynamics, North Atlantic Treaty Organisation, 1973.Google Scholar
45. Vries, G. de and Norris, D. H. The application of the finite element technique to potential flow problems. J. Appl. Mech., 1971.Google Scholar
46. Emmons, H. W. Flow over a compressible fluid past a symmetrical airfoil in a wind tunnel and in free air. NACA TN 1746, 1948.Google Scholar
47. Magnus, R. W., Gallacher, and Yoshihora, H. Inviscid supercritical airfoil theory, Transonic Aerodynamics AGARD Conference Proc, Paris, 1968.Google Scholar
48. Singleton, R. E. Lax-Wendroff difference scheme applied to the transonic airfoil problem. Transonic Aerodynamics AGARD Conference Proc, Paris, 1968.Google Scholar
49. Grossman, R. and Moretti, G. Time-dependent calculations for transonic flow. Proceedings, Third International Conference on Numerical Methods in Fluid Mechanics, Paris, Springer-Verlag, 1972.Google Scholar
50. Laval, P. Méthodes instationaires de calcul des effects d'interaction de paroi en dcoulement bidimensionnel supercritique. Poitiers, Congrfes Francais de Mechanique, 1973.Google Scholar
51. Burstein, S. Z. and Mirin, A. A. Time dependent calculations for transonic flow. Proceedings Third International Conference on Numerical Methods in Fluid Mechanics, Paris, Springer-Verlag, 1972.Google Scholar
52. Murman, E. M. and Krupp, J. Solution of the transonic potential equation using a mixed finite difference system. Lecture Notes in Physics. 8, Springer-Verlag, 1971.Google Scholar
53. Bailey, F. R. On the computation of two- and threedimensional steady transonic flows by relaxation methods, von Kármán Institute for Fluid Dynamics, Lecture Series 63, Progress in Numerical Fluid Dynamics, 1974.Google Scholar
54. Krause, E., Hirschel, E. H. and Bothmann, Th. Numerische Stabilität dreidimensionaler Grenzschichten. ZAMM Sonderheft (48), T205, 1968.Google Scholar
55. Dwyer, H. A. Solution of a three-dimensional boundarylayer flow with separation. AlAA Jnl, Vol 6, pp 1336– 1342, 1968.Google Scholar
56. Hall, M. G. A numerical method for calculating steady three-dimensional laminar boundary layers. Royal Aircraft Establishment, Tech Report 67145, 1967.Google Scholar
57. Krause, E. Comment on solution of a three-dimensional boundary-layer flow with separation, AlAA Jnl, Vol 7, p 575, 1969.Google Scholar
58. Richtmyer, D. D. and Morton, K. W. Difference methods for initial-value problems. Second Edition, Interscience Publishers, 1957.Google Scholar
59. Förster, K. Ubersicht tiber das Stumpfkörperproblem. European Research Programme on Viscous Flows. Arbeitsgruppe fur numerische Mefhoden in der Strömungsmechanik, Profil 1972 herausgegeben von E. H. Hirschel, DFVLR Techn. Memor. WT 1/73, 1973.Google Scholar
60. Roesner, K. Private communication, 1974.Google Scholar
61. Isaacson, E. and Keller, H. B. Analysis of numerical methods. John Wiley and Sons, Inc, 1966.Google Scholar
62. Godunov, S. K. and Ryabenkti, V. S. Special stability criteria for boundary condition problems for non selfadjoint finite difference equations. Uspekhi Mat. Mauk., Vol 18, p 3, 1969.Google Scholar
63. Moretti, G. The importance of boundary conditions in the numerical treatment of hyperbolic equations. Polytechnic Institute of Brooklyn, PIBAL Report No 68-34, 1968.Google Scholar
64. Kreiss, H. O. Boundary conditions for difference approximations of hyperbolic differential equations. Lecture Series No 64 on Advances in Numerical Fluid Dynamics, NATO, 1973.Google Scholar
65. Smolderen, J. J. Stability of explicit time dependent treatment of hyperbolic boundary problems. Progress in Numerical Fluid Dynamics, Lecture Series 63. 1974.Google Scholar
66. Korabelar, A. J. and Hanratty, T. J. Finite difference solution for three-dimensional boundary layers with large positive and negative cross-flows. AlAA Jnl, Vol 9, pp 1527–1532. 1971.Google Scholar
67. Watkins, C. B. Numerical solution of the three-dimensional boundary layer on a spinning sharp body at angle of attack. Polytechnic Institute of Brooklyn. Two- Day Symposium on Application of Computers to Fluid Dynamics Analysis and Design, 1973.Google Scholar
68. Blottner, F. G. and Ellis, M. A. Finite-difference solution of the incompressible three-dimensional boundary- layer equations for a blunt body. Computers and Fluids. Vol I. pp 133–158, Pergamon Press, 1973.Google Scholar
69. Krause, E. Mehrstellenverfahren zur Integration der Grenzschichtgleichungen, DLR-Mitt, pp 71-13, 1971.Google Scholar
70. Peters, N. Lösung der Grenzschichtgleichungen für chemisch reagierende Gase mit einem Mehrstellenverfahren. Dissertation TU Berlin, 1971.Google Scholar
71. Wirz, H. J. Eine Erweiterung des Verfahrens der Zwischenschritte auf allgemeinere parabolische und elliptische Differentialgleichungen, ZAMM 52, pp 329–336, 1972.Google Scholar
72. Hocks, W., , Korschelt, D. Küster, H. and Peters, N. Turbulente dreidimensionale Grenzschichten. European Research Programme on Viscous Flows, DFVLR, Techn. Memor. WT 1/73, 1973.Google Scholar
73. Krause, E., Hirschel, E. H. and Kordulla, W. Fourth Order “Mehrstellen“—Integration for three-dimensional turbulent boundary lays. AIAA Computational Fluid Dynamics Conference Proceedings, 1973.Google Scholar
74. Rotta, J. C. Recent attempts to develop a generally applicable calculation method for turbulent shear layers. AGARD Conference Proceedings No 93 on Turbulent Shear Flows, NATO, 1971.Google Scholar
75. Michel, R., Quémard, C. and Eléna, M. P. Distributions de vitesse des couches limites turbulentes en ecoulement compressible uniforme ou accéléré ONERA Tiré a Part, 693, 1969.Google Scholar
76. Morel, P., Atmospheric dynamics and the numerical simulation of atmospheric circulation. Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Vol 1, Lecture Notes in Physics, 1972 Google Scholar
77. Kutler, P., Reinhardt, W. A. and Warming, R. F. Multishocked three-dimensional supersonic flow fields with real gas effects. AIAA Jnl, Vol 11, pp 657–664, 1973.Google Scholar
78. Kutler, P. Computation of three-dimensional inviscid supersonic flows. Progress in Numerical Fluid Dynamics, von Karmdn Institute for Fluid Dynamics, Lecture Series 63, 1974.Google Scholar
79. MacCormack, R. W. The effect of viscosity in hypervelocity impact cratering. AIAA Paper 69-354, 1969.Google Scholar
80. Kutler, P. Numerical solution for the inviscid supersonic flow in the corner formed by two intersecting wedges. AIAA Paper 73-675, 1973.Google Scholar
81. West, J. E. and Korkegi, R. H. Interaction in the corner of intersecting wedges at a Mach number of 3 and high Reynolds numbers. ARL 71-0241, 1971.Google Scholar
82. Merrit, D. L. and Aronson, P. M. Wind tunnel simulation of head- and bow wave-blast wave interactions. NOLTR 67-123, 1967.Google Scholar
83. Rues, D. Der Einfluβ einfallender Stoβwellen auf ebene Uberschallströmungen um stumpfe Körper. DLR-FB 72-68, 1972.Google Scholar
84. Lomax, H., Bailey, F. R. and Ballhaus, W. F. On the numerical simulation of three-dimensional transonic flow with application to the C-141 wing. NASA TND- 6933 1973.Google Scholar
85. Beccer, J. V. Prospects for actively cooled hypersonic transports. Aeronautics and Astronautics, Vol 9, pp 32–39, 1971.Google Scholar
86. Krause, E. Numerical treatment of boundary-layer and Navier-Stokes equations. VKI Lecture Series 44, Numerical Methods in Fluid Dynamics, DFVLR Techn. Memor, WT 1/72, 1972.Google Scholar
87. Pfeiffer, H., Will, E. and Chung, D. H. Oberflächenkühlung in laminaren Hyperschallgrenzschichten. Vortrag auf dem XXI. Internationalen Astronautischen Kongreβ, Konstanz, 1972 DFVLR-Institut filr Angewandte Gasdynamik, Porz-Wahn.Google Scholar
88. Chung, D. H. Theoretische Untersuchung der Wandkühluns in laminaren hypersonischen Grenzschichten, DLR Mitt, pp 71–13, 28-50, 1971.Google Scholar
89. Patankar, S. V. and Spalding, D. B. A finite-difference procedure for solving the equations of the two-dimensional boundary-layer. Int. Journal Heat and Mass Transfer, Vol 10, pp 1389–1411. 1967.Google Scholar
90. Krause, E. Numerical solution of the boundary-layer equations. AIAA Jnl, Vol 5, p 1231, 1967.Google Scholar
91. Krause, E. Berechnung laminarer kompressibler Grenzschichten mittels impliziter Differenzenverfarren. DLR Mitt 68-33, p 122, 1968.Google Scholar
92. Kordulla, W. and Krause, E., Energiezufuhr in hypersonischen Grenzschichten durch Wasserstoffverbrennung. DLR-Mitt 71-13, Sonderheft (GAMM-Tagung), 1971.Google Scholar
93. Krause, E., Maurer, F. and Pfeiffer, H. Some results of investigations of problems relating to supersonic and hypersonic combustion. ICAS Paper No 72-21, Deutsche Fassung DGLR-Jahrbuch 1972.Google Scholar
94. Krause, E., Hirschel, E. H. and Bothmann, Tr., Die numerische Integration der Bewegungsgleichungen dreidimensionaler laminarer kompressible Grenzschichten. DGLR-Fachbuchreihe, Bd. 3, 1969.Google Scholar
95. Krause, E., Hirschel, E. H. and Bothmann, Th. Differenzenformeln zur Berechnung dreidimensionaler Grenzschichten DLR-FB 69-66, 1969.Google Scholar
96. Ting, L. On the initial conditions for boundary layer equations. Journal of Mathematics and Physics, Vol XLIV, pp 353–367, 1965.Google Scholar
97. Krause, E. and Hirschel, E. H. Exact numerical solution for three-dimensional boundary layers. Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics 8, 1970.Google Scholar
98. Krause, E., Hirschel, E. H. and Bothmann, Th. Normal injection in a three-dimensional laminar boundary layer. AIAA Jnl, Vol 7, p 367, 1969.Google Scholar
99. Krause, E. Numerische Lösungen der Navier-Stokes Gleichungen für inkompressible achsensymmetrische Strömungen. Sonderheft (GAMM-Tagung), 1969.Google Scholar
100. Krause, E. Eindimensionale Zentnfugenströmungen kompressibler binarer Gase. DLR-FB 70-04, 1970.Google Scholar
101. Krause, E. Gleitstromungen in eindimensionalen Gaszentrifugen, in Strömungsmechanische Vorgänge in Gaszentrifugen. DFVLR-Kolloquium 1970, Porz-Wahn, Institut für Angewandte Gasdynamik, 1970.Google Scholar
102. Lugt, H. J. and Haussling, H. J. Laminar flows past a flat plate at various angles of attack. Proceedings of Second International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, 8, 1971.Google Scholar
103. Naumann, A. Strömungsfragen der Medizin. Arbeitsgemeinschaft für Forschung des Landes Nordrhein- Westfalen. Natur-Ingenieur- und Gesellschaftswissenschaften, H.203, 1969.Google Scholar
104. Underwood, F. N. and Mueller, T. J. Numerical studies of the steady axisymmetric flow through a disc-tvpe prosthetic heart valve. Proc. 25th ACEMB, Bal Harbour, Florida, 273, 1972.Google Scholar
105. Mueller, T. T. On the separated flow produced by a fully open disc-type prosthetic heart valve. ASME 1973, Biomechanics Symposium Proceedings, AMD, Vol 2, pp 97–98, 1973.Google Scholar
106. Mueller, T. J. Numerical and physical experiments in viscous separated flows. Progress in Numerical Fluid Dynamics, Lecture Series 63, 1974.Google Scholar
107. Fenstel, E. A., Jensen, C. A. and Mcmahon, F. H. Future trends in computer hardware. Proceedings AIAA Computational Fluid Dynamics Conference, 19-20 July, 1973.Google Scholar
108. MacCormack, R. W. Private communication, February 1974.Google Scholar