Appendix F: Solutions to Problems

  • Wolfgang KollmannEmail author


The solutions to the problems set at the end of each chapter are presented here. In some of the solutions, literature references are provided and additional material has been added to aid the interpretation of the results.


  1. 1.
    Bennett, A.: Lagrangian Fluid Dynamics. Cambridge University Press, Cambridge, U.K. (2006)Google Scholar
  2. 2.
    Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)Google Scholar
  3. 3.
    Truesdell, C.A.: The Kinematics of Vorticity. Indiana University Press, Bloomington, Indiana (1954)Google Scholar
  4. 4.
    Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)Google Scholar
  5. 5.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, U.K. (2002)Google Scholar
  6. 6.
    Hamman, C.W., Klewicki, J.C., Kirby, R.M.: On the Lamb vector divergence in Navier-Stokes flows. JFM 610, 261–284 (2008)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Oberlack, M.: Symmetrie. Invarianz und Selbstähnlichkeit in der Turbulenz. Shaker, Aachen, Germany (2000)Google Scholar
  8. 8.
    Oberlack, M., Waclawczyk, M., Rosteck, A., Avsarkisov, V.: Symmetries and their importance for statistical turbulence theory. Bull. J. SME 2, 1–72 (2015)Google Scholar
  9. 9.
    Janocha, D.D., Waclawcyk, M., Oberlack, M.: Lie symmetry analysis of the Hopf functional-differential equation. Symmetry 7, 1536–1566 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: Mathematical Foundation of Turbulent Viscous Flows. Lecture Notes in Mathematics, vol. 1871, pp. 1–43 (2006)Google Scholar
  11. 11.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)Google Scholar
  12. 12.
    Tsinober, A.: An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer, Dordrecht (2009)Google Scholar
  13. 13.
    Gibbon, J.D., Galanti, B., Kerr, R.M.: Stretching and compression of vorticity in the 3D Euler equations. In: Hunt, J.C.R., Vassilicos, J.C. (eds.) Turbulence Structure and Vortex Dynamics, Cambridge University Press (2000)Google Scholar
  14. 14.
    Wilczek, M., Meneveau, C.: Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. JFM 756, 191–225 (2014)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, U.K. (2010)Google Scholar
  16. 16.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)Google Scholar
  17. 17.
    Leonard, A., Wray, A.: A new numerical method for the simulation of three-dimensional flow in a pipe. In: Krause E. (ed.) Lecture Notes in Physics 170. Springer, p. 335 (1982)Google Scholar
  18. 18.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publ. Inc., Mineola, New York (2001)Google Scholar
  19. 19.
    Kollmann, W.: Simulation of vorticity dominated flows using a hybrid approach: I formulation. Comput. Fluids 36, 1638–1647 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Canuto, C., Hussaini, M.Y., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)CrossRefGoogle Scholar
  21. 21.
    Gelfand, I.M., Yaglom, A.M.: Integration in functional spaces and its application in quantum physics. J. Math. Physics 1, 48–69 (1960)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Cameron, R.H., Martin, W.T.: Evaluation of various Wiener integrals by use of certain Sturm-Liouville differential equations. Bull. Am. Math. Soc. 51, 73–90 (1945)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Montroll, E.W.: Markoff chains, Wiener integrals and quantum theory. Comm. Pure Appl. Math. 5, 415–453 (1952)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cameron, R.H., Martin, W.T.: Transformations of Wiener integrals under a general class of linear transformations. Trans. Am. Math. Soc. 58, 184–219 (1945)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Klauder, J.R.: A Modern approach to Functional Integration. Birkhaeuser/Springer, New York (2010)Google Scholar
  26. 26.
    Woyczynski, W.A.: Burgers-KPZ Turbulence: Goettingen Lectures. Lecture Notes in Math, vol. 1700. Springer, Berlin (1999)Google Scholar
  27. 27.
    Egorov, A.D., Sobolevsky, P.I., Yanovich, L.A.: Functional Integrals: Approximate Evaluation and Applications. Kluwer Academic Publication, Dordrecht (1993)CrossRefGoogle Scholar
  28. 28.
    Shen, H.H., Wray, A.A.: Stationary turbulent closure via the Hopf functional equation. J. Stat. Phys. 65, 33–52 (1991)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Novikov, E.A.: Random-force method in turbulence theory. Sov. Phys. JETP 17, 1449–1454 (1963)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Novikov, E.A.: Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290–1294 (1965)ADSMathSciNetGoogle Scholar
  31. 31.
    Panton, R.L.: Incompressible Flow, 2nd edn. Wiley, New York (1996)Google Scholar
  32. 32.
    Lukacs, E.: Characteristic Functions. Hafner Publication, New York (1970)Google Scholar
  33. 33.
    Davidson, P.A.: Turbulence. Oxford University Press, Oxford, U.K. (2004)Google Scholar
  34. 34.
    Kolmogorov, A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous, incompressible fluid at high Reynolds number. JFM 13, 82–85 (1962)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Curl, R.L.: Dispersed phase mixing: part i. theory and effects in simple reactors. AIChE J. 9, 175–181 (1963)CrossRefGoogle Scholar
  36. 36.
    Chen, H., Chen, S., Kraichnan, R.H.: Probability distribution of a stochastically advected scalar field. Phys. Rev. Lett. 62, 2657 (1989)ADSCrossRefGoogle Scholar
  37. 37.
    Duffy, D.G.: Greenś Functions with Applications. Chapman & Hall/CRC, Boca Raton, Florida (2001)Google Scholar
  38. 38.
    Pope, S.B.: Mapping closures for turbulent mixing and reaction. Theoret. Computat. Fluid Dyn. 2, 255–270 (1991)ADSCrossRefGoogle Scholar
  39. 39.
    Miller, K.S.: Multidimensional Gaussian Distributions. Wiley, New York (1963)Google Scholar
  40. 40.
    Wilczek, M., Xu, H., Ouellette, N.T., Friedrich, R., Bodenschatz, E.: Generation of Lagrangian intermittency in turbulence by a self-similar mechanism. New J. Phys. 15, 055015 (2013)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Sagaut, P., Cambron, C.: Homogeneous Turbulence Dynamics. Cambridge University Press, New York (2008)Google Scholar
  42. 42.
    Kuksin, S., Shirikyan, A.: Mathematics of Two-dimensional turbulence, Cambridge Tracts in Mathematics, vol. 194. Cambridge University Press, U.K. (2012)Google Scholar
  43. 43.
    Meleshko, V.V., van Heijst, G.J.F.: On Chaplygin’s investigations of two-dimensional vortex structures in an invscid fluid. JFM 272, 157–182 (1994)ADSCrossRefGoogle Scholar
  44. 44.
    Nickels, T.B.: Turbulent Coflowing Jets and Vortex Ring Collisions. University of Melbourne, Australia (1993). PhD thesisGoogle Scholar
  45. 45.
    Shelley, M.J., Meiron, D.I., Orszag, S.A.: Dynamical aspects of vortex reconnection in perturbed anti-parallel vortex tubes. JFM 246, 613–652 (1993)ADSCrossRefGoogle Scholar
  46. 46.
    Nickels, T.B.: Inner scaling for wall-bounded flows subject to large pressure gradients. JFM 521, 217–239 (2004)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Vieillefosse, P.: Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. France 43, 837–842 (1982)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Vieillefosse, P.: Internal motion of a small element of fluid in an inviscid flow. Physica A 125, 150–162 (1984)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Cantwell, B.J.: Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4, 782–793 (1992)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Meneveau, C.: Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annua. Rev. Fluid Mech. 43, 219–245 (2011)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Mullin, T., Juel, A., Peacock, T.: Sil’nikov Chaos in Fluid Flows. In: Vassilicos, J.C. (ed.) Intermittency in Turbulent Flows, Cambridge University Press. J.C.R. Hunt and J.C. Vassilicos, eds., Cambridge University Press (2001)Google Scholar
  52. 52.
    Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V. Chua, L.O.: Methods of Qualitative Theory in Nonlinear Dynamics. vol.I, World Scientific, London (1998)Google Scholar
  53. 53.
    Rektorys, K. (ed.): Survey of Applicable Mathematics. MIT Press, Cambridge Mass (1969)Google Scholar
  54. 54.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)Google Scholar
  55. 55.
    Kusnetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (2004)Google Scholar
  56. 56.
    Barkley, D.: Theoretical perspective on the route to turbulence in a pipe. JFM 803, P1–1 (2016)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Kollmann, W., Prieto, M.I.: DNS of the collision of co-axial vortex rings. Comput. Fluids 73, 47–64 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Surana, A., Grunberg, O., Haller, G.: Exact theory of three-dimensional flow separation. Part 1. Steady separation. JFM 564, 57–103 (2006)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Chong, M.S., Perry, A.E., Cantwell, B.J.: A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777 (1990)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Vlaykov, D.G., Wilczek, M.: On the small-scale structure of turbulence and its impact on the pressure field. JFM 861, 422–446 (2018)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    Townsend, A.A.: The Structure of Turbulent Shear Flow. Cambridge University Press (1956)Google Scholar
  62. 62.
    Orlandi, P., Carnevale, G.F.: Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles. Phys. Fluids 19, 057106 (2007)ADSCrossRefGoogle Scholar
  63. 63.
    Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press (1995)Google Scholar
  64. 64.
    Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13, 249–255 (2000)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    Lerner, N.: A Course on Integration Theory. Birkh/"auser/Springer Basel (2014)Google Scholar
  66. 66.
    Wang, X.: Volumes of Generalized Unit Balls. Math. Mag. 78, 390–395 (2005)CrossRefGoogle Scholar
  67. 67.
    Folland, G.B.: How to integrate a polynomial over a sphere. Am. Math. Mont. 108, 446–448 (2001)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

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