About the Prospects for Passage to Instability

I. V. Lebed^{*}

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The results of the direct numerical integration
of the Navier-Stokes equations are evaluated against experimental data for problem
on a flow around bluff bodies in an unstable regime. Experiment records several
stable medium states for flow past a body. Evolution of each of these states, after
losing the stability, inevitably goes by periodic vortex shedding modes. Calculations
based on the Navier-Stokes equations satisfactorily reproduced all observed stable
medium states. They were, however, incapable of reproducing any of a vortex shedding
modes recorded experimentally. The solutions to the classic hydrodynamics equations
successfully reach the boundary of instability field. However, classic solutions
are unable to cross this boundary. Most likely, the reason for this is the Navier-Stokes
equations themselves. The classic hydrodynamics equations directly follow from the
Boltzmann equation and naturally contain the error involved in the derivation of
classic kinetic equation. Just the Boltzmann
hypothesis, which closed kinetic equation, allowed us to con- struct classic hydrodynamics on only three lower
principal hydrodynamic values. The use of the Boltzmann hypothesis excludes higher
principal hydrodynamic values from the participation in the formation of classic
hydrodynamics equations. The multimoment hydrodynamics
equations are constructed using seven lower principal hydrodynamic values. The numerical
integration of the multimoment hydrodynamics equations in the problem on flow around
a sphere shows that the solutions to these equations cross the boundary and enter
the instability field. The boundary crossing is accompanied by appearance of very uncommon acts in scenario of system
evolution.

References

[1] I. V. Lebed, “On the Inapplicability of Navier-Stokes Equations to Interpreting the Turbulence,” Physica A, Vol. 315, No. 1-2, 2002, pp. 228-235.
doi:10.1016/S0378-4371(02)01254-2

[2] I. V. Lebed and S. Y. Umanskii, “The Appearance and Development of Turbulence in a Flow Past a Sphere: Problems and the Existing Approaches to Their Solution,” RJ Physical Chemistry B, Vol. 1, No. 1, 2007, pp. 52-73.

[3] I. V. Lebed, “Method of Two-Particle Distribution Functions. Hydrodynamic Equations,” Chemical Physics Reports, Vol. 15, No. 6, 1996, pp. 861-883.

[4] I. V. Lebed, “The Method of Pair Functions as Applied to the Problem of a Flow around a Quiescent Solid Sphere,” Chemical Physics Reports, Vol. 16, No. 7, 1997, pp. 1263-1301.

[5] I. V. Lebed, “About the Behavior of the Entropy of a Gas Flow Losing Its Stability,” Chemical Physics Reports, Vol. 17, No. 3, 1998, pp. 411-439.

[6] S. Taneda, “Experimental Investigation of the Wake behind a Sphere at Low Reynolds Numbers,” Journal of the Physical Society of Japan, Vol. 11, 1956, pp. 1104-1108.
doi:10.1143/JPSJ.11.1104

[7] H. Sakamoto and H. Haniu, “The Formation Mechanism and Shedding Frequency of Vortices from a Sphere in Uniform Shear Flow,” Journal of Fluid Mechanics, Vol. 287, 1995, pp. 151-171.
doi:10.1017/S0022112095000905

[8] R. H. Magarvey and R. L. Bishop, “Transition Ranges for Three-Dimensional Wakes,” Canadian Journal of Physics, Vol. 39, No. 10, 1961, pp. 1418-1422.
doi:10.1139/p61-169

[9] E. Achenbach, “Vortex Shedding from Spheres,” Journal of Fluid Mechanics, Vol. 62, No. 2, 1974, pp. 209-221.
doi:10.1017/S0022112074000644

[10] J. M. Chomaz, P. Bonneton and E. J. Hopfinger, “The Structure of the Near Wake of a Sphere Moving Horizontally in a Stratified Fluid,” Journal of Fluid Mechanics, Vol. 254, 1993, pp. 1-21.
doi:10.1017/S0022112093002009

[11] J. Gerrard, “The Wakes of Cylindrical Bluff Bodies at Low Reynolds Numbers,” Philosophical Transactions of the Royal Society of London A, Vol. 288, No. 1354, 1978, pp. 351-382. doi:10.1098/rsta.1978.0020

[12] M. M. Zdravkovich, “Smoke Observation of the Formation of a Karman Vortex Street,” Journal of Fluid Mechanics, Vol. 37, No. 3, 1969, pp. 491-496.
doi:10.1017/S0022112069000681

[13] B. E. Eaton, “Analysis of Laminar Vortex Shedding behind a Circular Cylinder by Computer-aided Flow Visualization,” Journal of Fluid Mechanics, Vol. 180, 1987, pp. 117-145. doi:10.1017/S0022112087001757

[14] J. Cohen, I. G. Shukhman, M. Karp and J. Philip, “An Analytical-Based Method for Studying the Nonlinear Evolution of Localized Vortices in Planar Homogenous Shear Flows,” Journal of Computational Physics, Vol. 229, No. 20, 2010, pp. 7765-7773.
doi:10.1016/j.jcp.2010.06.035

[15] I. Kim and A. J. Pearlstein, “Stability of the Flow Past a Sphere,” Journal of Fluid Mechanics, Vol. 211, 1990, pp. 73-83. doi:10.1017/S0022112090001501

[16] R. Natarajan and A. Acrivos, “The Instability of the Steady Flow Past Spheres and Disks,” Journal of Fluid Mechanics, Vol. 254, 1993, pp. 323-344.
doi:10.1017/S0022112093002150

[17] H. G. Schuster, “Deterministic Chaos,” Physik-Verlag, Weinheim, 1984.

[18] A. G. Tomboulides, S. A. Orszag and G. E. Karniadakis, “Direct and Large-Eddy Simulations of Axisymmetric Wakes,” AIAA Paper, No. 93-9546, 1993.

[19] R. Mittal, “Planar Symmetry in the Unsteady Wake of a Sphere,” AIAA Journal, Vol. 37, No. 3, 1999, pp. 388-400.
doi:10.2514/2.722

[20] T. A. Johnson and V. S. Patel, “Flow Past a Sphere Up to a Reynolds Number of 300,” Journal of Fluid Mechanics, Vol. 378, 1999, pp. 19-70.
doi:10.1017/S0022112098003206

[21] A. G. Tomboulides and S. A. Orszag, “Numerical Investigation of Transitional and Weak Turbulent Flow Past a Sphere,” Journal of Fluid Mechanics, Vol. 416, 2000, pp. 45-73. doi:10.1017/S0022112000008880

[22] C. P. Jackson, “A Finite-Element Study of the Onset of Vortex Shedding in Flow Past Variously Shaped Bodies,” Journal of Fluid Mechanics, Vol. 182, 1987, pp. 23-45.
doi:10.1017/S0022112087002234

[23] K. Hannemann and H. Oertel Jr., “Numerical Simulation of the Absolutely and Convectively Unstable Wake,” Journal of Fluid Mechanics, Vol. 199, 1989, pp. 55-88.
doi:10.1017/S0022112089000297

[24] P. J. Strykowski and K. Hannemann, “Temporal Simulation of the Wake behind a Circular Cylinder in the Neighborhood of the Critical Reynolds Number,” Acta Mechanica, Vol. 90, No. 1-4, 1991, pp. 1-20.
doi:10.1007/BF01177395

[25] G. K. Batchelor, “An Introduction to Fluid Dynamics,” Cambridge at the University Press, Cambridge, 1970.

[26] V. Gushchin and P. Matyushin, “Vortex Formation Mechanisms in the Wake behind a Sphere for 200 < Re < 380,” Fluid Dynamics, Vol. 41, 2006, pp. 795-809.
doi:10.1007/s10697-006-0096-x

[27] S. Baghery, P. Schlatter, P. Schmid and D. Henningson, “Global Stability of a Jet Cross-Flow,” Journal of Fluid Mechanics, Vol. 624, 2009, pp. 33-44.
doi:10.1017/S0022112009006053

[28] J. Jeong and F. Hussain, “On the Identification of a Vortex,” Journal of Fluid Mechanics, Vol. 285, 1995, pp. 69-94. doi:10.1017/S0022112095000462

[29] L. G. Loitsyanskii, “Mechanics of Liquids and Gases,” Pergamon, Oxford, 1966.

[30] R. L. Liboff, “Introduction to the Theory of Kinetic Equations,” Wiley, New York, 1969.

[31] J. H. Ferziger and H. G. Kaper, “Mathematical Theory of Transport Processes in Gases,” North-Holland Publishing Company, Amsterdam, 1972.

[32] I. V. Lebed and S. Y. Umanskii, “On the Possibility of Improving Classic Hydrodynamics Equations by an Increase in the Number of Hydrodynamic Values,” RJ Physical Chemistry B, Vol. 6, No. 1, 2012, pp. 149-162.

[33] I. V. Lebed, “Derivation of the Equations for Pair Distribution Functions,” Chemical Physics Reports, Vol. 14, No. 5, 1995, pp. 599-615.

[34] I. V. Lebed, “Equations of Pair Distribution Functions,” Chemical Physics Letters, Vol. 165, No. 2-3, 1990, pp. 226-228. doi:10.1016/0009-2614(90)85433-D

[35] I. V. Lebed, “Hydrodynamic Equations Stemming from Two Particle Distributions in the Limit of Weak Non-equilibrium. Analysis of Invertibility of Equations,” Chemical Physics Reports, Vol. 15, No. 12, 1996, pp. 1725-1750.

[36] H. Grad, “About Kinetic Theory of Rarefied Gases,” Communications on Pure and Applied Mathematics, Vol. 2, No. 4, 1949, pp. 331-407. doi:10.1002/cpa.3160020403

[37] I. Prigogine, “From Being to Becoming,” Freeman, San Francisco, 1980.

[38] I. V. Lebed, “The Development of Instability in Problem on Flow around a Sphere,” RJ Physical Chemistry B, Vol 8, No. 1, 2014, in Press.