Abstract
Spectral and cumulative spectral transfer, as well as energy and dissipation spectra, have been computed by FFT from exact solutions to Burgers’ equation for Thomas [1] random initial conditions at initial turbulence Reynolds’ numbers of Re 0 = 400 and 1000. Accurate, efficient numerical evaluation of the exact solution on a fine spatial mesh enables spectral computations to be carried out at high wave numbers for large ensembles, thereby confirming the exponential viscous cutoff. The low wave-number white narrow band of the energy spectrum gives way to the well-known k −2-subrange, independently of Rep in the high Rep range. The k −2-subrange, in turn, gives way with increasing wave-number k to an exp(−αk) high wave-number viscous cutoff at a wave number depending on Re 0. Spectral transfer and cumulative spectral transfer functions look very much like their counterparts for three-dimensional isotropic turbulence.
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References
J.H. Thomas, Phys. Fluids, 11, 1245 (1968).
W.C. Meecham and A. Siegel, Phys. Fluids, 7, 1178 (1964).
R.H. Kraichnan, Phys. Fluids, 11, 265 (1968).
W.P.M. Malfliet, Physica, 45, 257 (1969).
T. Tatsumi, Phys. Fluids Suppl. II, 12, II - 258 (1969).
H. Tanaka, J. Meteor. Soc. Jpn., 47, 373 (1969).
J.J. Walton, Phys. Fluids, 13, 1634 (1970).
H. Tanaka, J. Phys. Soc. Jpn., 34, 1390 (1973).
W.C. Meecham, P Iyer, and W.C. Clever, Phys. Fluids, 18, 1610 (1975).
T. Tatsumi and S. Kida, J. Phys. Soc. Jpn., 49, 2014 (1980).
J. Mizushima and T. Tatsumi, J. Phys. Soc. Jpn., 50, 1765 (1981).
G.K. Batchelor, The Theory of Homogeneous Turbulence,Cambridge University Press, Cambridge (1953) (see esp. p 195 note added in 1956 and subsequent printings).
A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, Vol. 2 (ed. J.L. Lumley) MIT Press, Cambridge, MA (1975).
D.T. Jeng, A Study of Burgers’ Model Equation of Turbulence, Magistral thesis, Univ. of Minnesota (1965).
D.T. Jeng, R. Foerster, S. Haaland, and W.C. Meecham, Phys. Fluids, 9, 2114 (1966).
Y.-C. Shih and XB Reed, Jr., Symp. of Fluid Dynamics (eds. A.L. Addy, W.L. Chow, N.S. Vlachos and R.A. White) Dept. Mech. and Ind. Engg, U. Ill., pp. 301–308 (1984).
Y.-C. Shih and XB Reed, Jr., Phys. Fluids, 28, 2088 (1985).
Y.-C. Shih and XB Reed, Jr., Physico-Chemical Hydrodynamics, 6, 853 (1985).
Y.-C. Shih and XB Reed, Jr., Sixth International Conference on Physico-Chemical Hydrodynamics, Oxford Univ. (1987).
XB Reed, Jr. and Y.-C. Shih, unpublished (1988).
S. Keleti and XB Reed, Jr., Developments in Mechanics, 14a, 63 (1987).
S. Keleti and XB Reed, Jr., 5th Symposium on Turbulent Shear Flows, Cornell Univ. (1985) (abstract available from authors).
M. Martell, Checkpoint, 2, 5 ( Oct. 1970, Floating Point Systems, Inc.).
S.A. Orszag, in Fluid Dynamics/Dynamique des Fluids 1973 (eds. R. Balian and J.-L. Peube) Gordon and Breach Science Publishers, London (1977).
J.M. Burgers, Advances in Applied Mechanics, vol. I (eds. R. von Mises and T. von Karman) Academic, New York, p. 171 (1948).
W.H. Reid, Appl. Sci. Res. A 6, 85 (1957).
A.M. Oboukhov, J. Fluid Mech., 13, 77 (1962).
W. Heisenberg, Z. Physik 124, 628 (1948) (German)
L.S.G. Kovasznay, J. Aero. Sci. 15, 745 (1948).
J.O. Hinze, Turbulence, 2nd ed. McGraw-Hill, New York (1975).
M.S. Uberoi, Phys. Fluids, 6, 1048 (1963).
C.W. Van Atta and W.Y. Chen, J. Fluid Mech., 38, 743 (1969).
F.H. Champagne, J. Fluid Mech., 86, 67 (1978).
J. Mizushima and Y. Saito, Phys. Fluids, 28, 1294 (1985).
H. Tòkunaga, J. Phys. Soc. Jpn., 52, 827 (1983).
M.D. Love, J. Fluid Mech., 100, 87 (1980).
H. Tokunaga, J. Phys. Soc. Jpn., 41, 328 (1976).
R.D. Fay, J. Acoust. Soc. Amer., 3, 222 (1931).
J.M. Burgers, Proc. Acad. Sci. Amst.,,53, 247,393, 718, 732 (1950).
E.Y. Rodin, J. Math. Anal. and Appl. 30, 401 (1970).
E.R. Benton, Phys. Fluids, 10, 2113 (1967).
P.G. Saffman, in Topics in Nonlinear Physics (ed. N.J. Zabusky) Springer-Verlag, New York, pp. 485–614 (1968).
J. Mizushima, Phys. Fluids, 21, 512 (1978).
H. Tennekes and J.L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, MA (1972).
S. Kida, Phys. Fluids, 24, 604 (1981).
E. Parzen, Stochastic Processes, Holden-Day, San Francisco (1962).
S. Kida, J. Fluid Mech., 93, 337 (1979).
S.C. Crow and G.H. Canavan, J. Fluid Mech., 41, 387 (1970).
J. Mizushima and A. Segami, Phys. Fluids, 23, 2559 (1980).
T. Tatsumi, Advances in Applied Mechanics, vol. 20 (ed. Chia-Shun Yih) Academic Press, New York, pp. 39–133 (1980).
H.L. Dryden, Quart. J. Appl. Math., 1, 7 (1943).
G.K. Batchelor and I. Proudman, Philos. Trans. A 248, 369 (1956).
R.H. Kraichnan, J. Fluid Mech., 5, 497 (1959).
S.A. Orszag, Theory of Turbulence, Princeton University Ph.D., University Microfilms, Ann Arbor, MI (1966).
U. Frisch, M. Lesieur, and D. Schertzer, J. Fluid Mech., 97, 181 (1980).
S.A. Orszag, Handbook of Turbulence, vol. 1: Fundamentals and Applications (eds. W Frost and T.H. Moulden) Plenum Press, New York, p. 281 (1977).
A.C. Hearn, Ed., REDUCE User’s Manual (version 3.2), Rand Corp., Santa Monica, CA (1985).
VAX UNIX MACSYMA TM Reference Manual,Symbolics Inc., Cambridge, MA (1985).
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Keleti, S., Reed, X.B. (1996). Evaluation of Spectral Behavior for Large Ensembles of Exact Solutions to Burgers’ Equation for Thomas Initial Conditions. In: Funaki, T., Woyczynski, W.A. (eds) Nonlinear Stochastic PDEs. The IMA Volumes in Mathematics and its Applications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8468-7_12
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DOI: https://doi.org/10.1007/978-1-4613-8468-7_12
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