Novel Power Law of Turbulent Spectrum

Affiliation(s)

Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia.

Pan-Asian Center for the Independent Liberal Study of Science Technology and the Humanities, Jusup Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic.

Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia.

Pan-Asian Center for the Independent Liberal Study of Science Technology and the Humanities, Jusup Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic.

ABSTRACT

This paper is concerned with novel power law of turbulent energy spectrum and the relevant experiment in tidal current. The power law in the inertial sub-range has been proposed in such a way that the power of the one-dimensional turbulent energy spectrum varies from 0 to －2 approximately, but it is accompanying the small oscillation with increasing the wave number. The well-known Kolmogorov －5/3 power law is merely one facet, to appear within the present proposed novel power law. The turbulent energy spectra (Su, Sv, Sw) in x-, y-and z-directions, respectively oscillate with the wave number. It is found that the turbulent in the tidal currents is three dimensional, and the intermittence of momentum transport is a predominant and characteristic feature in tidal current.

This paper is concerned with novel power law of turbulent energy spectrum and the relevant experiment in tidal current. The power law in the inertial sub-range has been proposed in such a way that the power of the one-dimensional turbulent energy spectrum varies from 0 to －2 approximately, but it is accompanying the small oscillation with increasing the wave number. The well-known Kolmogorov －5/3 power law is merely one facet, to appear within the present proposed novel power law. The turbulent energy spectra (Su, Sv, Sw) in x-, y-and z-directions, respectively oscillate with the wave number. It is found that the turbulent in the tidal currents is three dimensional, and the intermittence of momentum transport is a predominant and characteristic feature in tidal current.

KEYWORDS

Energy Spectrum, Kinetic Theory, Tidal Current, Inertial Sub-Range, Turbulence Measurement, Power Law

Energy Spectrum, Kinetic Theory, Tidal Current, Inertial Sub-Range, Turbulence Measurement, Power Law

Cite this paper

Osonphasop, C. and Nakagawa, T. (2014) Novel Power Law of Turbulent Spectrum.*Open Journal of Fluid Dynamics*, **4**, 140-153. doi: 10.4236/ojfd.2014.42013.

Osonphasop, C. and Nakagawa, T. (2014) Novel Power Law of Turbulent Spectrum.

References

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http://dx.doi.org/10.1063/1.1694592

[2] Tsugé, S. and Sagara, K. (1975) A New Hierarchy System on the Basis of Master Boltzmann Equation for Microscopic Density. Journal of Statistical Physics, 12, 403-425.

http://dx.doi.org/10.1007/BF01012885

[3] Tsugé, S. and Sagara, K. (1976) Kinetic Theory of Turbulent Compressible Flows and Comparison with Classical Theory. Physics of Fluids, 19, 1478-1485.

http://dx.doi.org/10.1063/1.861350

[4] Tsugé, S. (1979) Separation of Variables in Two-Point Kinetic Equations. Physics Letters, 70A, 266-268.

http://dx.doi.org/10.1016/0375-9601(79)90118-X

[5] Tsugé, S. (1970) On the Divergent Growth of Molecular Fluctuations in Classical Shear Flow. Physics Letters, 33A, 145-146.

http://dx.doi.org/10.1016/0375-9601(70)90699-7

[6] Sagara, K. and Tsugé, S. (1976) A Kinetic Theory of Turbulent and Thermal Fluctuations in Compressible Flows. Nielsen Engineering & Research.

[7] Sagara, K. and Tsugé, S. (1982) A Bimodal Maxwellian Distribution as the Equilibrium Solution of the Two-Particle Regime. Physics of Fluids, 25, 1970-1977.

http://dx.doi.org/10.1063/1.863673

[8] Sagara, K. (1980) Exact Turbulence Correction to Arrhenius Law in the Asymptotic Limit of High Activation Energy. Science & Technology, 21, 191-197.

[9] Tsugé, S. and Ogawa, S. (1993) Molecular and Turbulent Transports Competing in Premixed Flames. In: Takeno, T., Ed., Turbulent and Molecular Processes in Combustion, Elsevier, 35-50.

[10] Bai, B. (1995) Solitary Wave Solution of Turbulent Mixing Layer by the Method of Pseudo-Compressibility. Ph.D. Thesis, Institute of Engineering and Mechanics, University of Tsukuba, Tsukuba.

[11] Nakagawa, T. (1979) A Theory of Decay of Grid-Produced Turbulence. Zeitschrift für Angewandte Mathematik und Mechanik, 59, 648-651.

http://dx.doi.org/10.1002/zamm.19790591111

[12] Nakagawa, T. (1979) On Decay of Grid Produced Turbulence. Department of Mechanical Engineering, Monash University, Clayton.

[13] Nakagawa, T. (1981) The Comparison of a New Theory of Grid-Produced Turbulence with an Experiment. Zeitschrift für Angewandte Mathematik und Mechanik, 62, 6-8.

[14] Nakagawa, T. and Iida, H. (2012) Statistical Theory of Turbulence by the Late Lamented Dr. Shunichi Tsugé—Case Study on Flow through a Grid in Wind Tunnel. Open Journal of Applied Sciences, 2, 18-21.

[15] Ishibashi, K. (1991) Solitary Wave Solution of Turbulent Bénard Convection. Ph.D. Thesis, Institute of Mechanics, University of Tsukuba, Tsukuba.

[16] Ishibashi, K., Tsugé, S. and Nakagawa, T.M.S. (1998) Solitary-Wave Solution of Turbulence with Application to Bénard Convection. In: Bainov, D., Ed., Proceedings of 8th International Colloquium on Differential Equations, Plovdiv, 18-23 August 1997, 227-236.

[17] Tsugé, S. (2013) The Kolmogorov Turbulence Theory in the Light of Six-Dimensional Navier-Stokes’ Equation. In: Nakagawa, T.R.M, Ed., Philosophy of Flow, Vol. 10, Supplement, Royal White Mountains Academia Press, 19.

[18] Tsugé, S. (2004) Scientific Papers on Turbulence by Dr. Shunichi Tsugé. In: Nakagawa, T.M.S., Ed., Philosophy of Flow, Columbus University Press, Hakusan, 10.

[19] Tsugé, S. (2004) Scientific Papers on Combustion by Dr. Shunichi Tsugé. In: Nakagawa, T.M.S., Ed., Philosophy of Flow, Columbus University Press, Hakusan, 11.

[20] Sreenivasan, K.R. (1985) On the Fine-Scale Intermittency of Turbulence. Journal of Fluid Mechanics, 151, 81-103.

http://dx.doi.org/10.1017/S0022112085000878

[21] Antonia, R.A. and Pearson, B. (2000) Effect of Initial Conditions on the Mean Energy Dissipation Rate and the Scaling Exponent. Flow, Turbulence and Combustion, 64, 95-117.

[22] Osonphasop, C. and Hinwood, J.B. (1983) Turbulence Measurements for the Whole Depth of Water in a Tidal Channel. 8th Australasian Conference of Fluid Mechanics, University of Newcastle, 5 September 1983, 10-14.

[23] Osonphasop, C. (1983) The Measurements of Turbulence in Tidal Currents. A Thesis Submitted to the Faculty of Engineering in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy, Department of Mechanical Engineering, Monash University, Clayton.

[24] Osonphasop, C. and Hinwood, J.B. (1984) On Measurement of Turbulence and Shear Stresses in Tidal Currents. APD-IAHR 4th Conference of the International Association for Hydraulic Research, Chiangmai, 25 May 1984, 1-5.

[25] Trevethan, M. and Chanson, H. (2010) Turbulence and Turbulent Flux Events in a Small Estuary. Environmental Fluid Mechanics, 10, 345-368.

http://dx.doi.org/10.1007/s10652-009-9134-7

[26] Chanson, H., Brown, R. and Trevethan, M. (2012) Turbulence Measurements in a Small Subtropical Estuary under King Tide Condition. Environmental Fluid Mechanics, 12, 265-289.

http://dx.doi.org/10.1007/s10652-011-9234-z

[27] Reynolds, O. (1883) An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Philosophical Transactions of the Royal Society of London, 174, 935-982.

http://dx.doi.org/10.1098/rstl.1883.0029

[28] Kolmogorov, A.N. (1941) The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers. Doklady Akademii Nauk SSSR, 30, 301-304.

[29] Bogoliubov, N.N. (1959) Problems of Dynamical Theory in Statistical Physics. AFCRC-TR-59-235.

[30] De Kármán, Th. and Howarth, L. (1938) On the Statistical Theory of Isotropic Turbulence. Proceedings of the Royal Society A, 164, 192-215.

http://dx.doi.org/10.1098/rspa.1938.0013

[31] Chapman, S. (1917) On the Kinetic Theory of a Gas II. Philosophical Transactions of the Royal Society A, 217, 115-197.

[32] Enskog, D. (1917) Kinetische Theorie der Vorgange in Messigverdünnten Gasen. Dissertation, Uppsala University, Uppsala.

[33] Klimontovich, Yu.L. (1967) The Statistical Theory of Non-Equilibrium Processes in a Plasma. MIT Press, Cambridge.

[34] Lamb, H. (1932) Hydrodynamics. 6th Edition, Cambridge University Press, Cambridge, 1-8.

[35] Lorenz, E.N. (1963) Deterministic Nom-Periodic Flow. Journal of the Atmospheric Sciences, 20, 130-141.

http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[36] Ueda, Y. (1992) Strange Attractors and the Origin of Chaos. In: Ueda, Y., Ed., The Road to Chaos, Aerial Press, California, 185-216.

[37] Nakagawa, T. and Hinwood, J.B. (1978) On Measurement of Turbulence in Tidal Currents. Monash University, Clayton, MMEL34.

[1] Tsugé, S. (1974) Approach to the Origin of Turbulence on the Basis of Two-Point Kinetic Theory. Physics of Fluids, 17, 22-33.

http://dx.doi.org/10.1063/1.1694592

[2] Tsugé, S. and Sagara, K. (1975) A New Hierarchy System on the Basis of Master Boltzmann Equation for Microscopic Density. Journal of Statistical Physics, 12, 403-425.

http://dx.doi.org/10.1007/BF01012885

[3] Tsugé, S. and Sagara, K. (1976) Kinetic Theory of Turbulent Compressible Flows and Comparison with Classical Theory. Physics of Fluids, 19, 1478-1485.

http://dx.doi.org/10.1063/1.861350

[4] Tsugé, S. (1979) Separation of Variables in Two-Point Kinetic Equations. Physics Letters, 70A, 266-268.

http://dx.doi.org/10.1016/0375-9601(79)90118-X

[5] Tsugé, S. (1970) On the Divergent Growth of Molecular Fluctuations in Classical Shear Flow. Physics Letters, 33A, 145-146.

http://dx.doi.org/10.1016/0375-9601(70)90699-7

[6] Sagara, K. and Tsugé, S. (1976) A Kinetic Theory of Turbulent and Thermal Fluctuations in Compressible Flows. Nielsen Engineering & Research.

[7] Sagara, K. and Tsugé, S. (1982) A Bimodal Maxwellian Distribution as the Equilibrium Solution of the Two-Particle Regime. Physics of Fluids, 25, 1970-1977.

http://dx.doi.org/10.1063/1.863673

[8] Sagara, K. (1980) Exact Turbulence Correction to Arrhenius Law in the Asymptotic Limit of High Activation Energy. Science & Technology, 21, 191-197.

[9] Tsugé, S. and Ogawa, S. (1993) Molecular and Turbulent Transports Competing in Premixed Flames. In: Takeno, T., Ed., Turbulent and Molecular Processes in Combustion, Elsevier, 35-50.

[10] Bai, B. (1995) Solitary Wave Solution of Turbulent Mixing Layer by the Method of Pseudo-Compressibility. Ph.D. Thesis, Institute of Engineering and Mechanics, University of Tsukuba, Tsukuba.

[11] Nakagawa, T. (1979) A Theory of Decay of Grid-Produced Turbulence. Zeitschrift für Angewandte Mathematik und Mechanik, 59, 648-651.

http://dx.doi.org/10.1002/zamm.19790591111

[12] Nakagawa, T. (1979) On Decay of Grid Produced Turbulence. Department of Mechanical Engineering, Monash University, Clayton.

[13] Nakagawa, T. (1981) The Comparison of a New Theory of Grid-Produced Turbulence with an Experiment. Zeitschrift für Angewandte Mathematik und Mechanik, 62, 6-8.

[14] Nakagawa, T. and Iida, H. (2012) Statistical Theory of Turbulence by the Late Lamented Dr. Shunichi Tsugé—Case Study on Flow through a Grid in Wind Tunnel. Open Journal of Applied Sciences, 2, 18-21.

[15] Ishibashi, K. (1991) Solitary Wave Solution of Turbulent Bénard Convection. Ph.D. Thesis, Institute of Mechanics, University of Tsukuba, Tsukuba.

[16] Ishibashi, K., Tsugé, S. and Nakagawa, T.M.S. (1998) Solitary-Wave Solution of Turbulence with Application to Bénard Convection. In: Bainov, D., Ed., Proceedings of 8th International Colloquium on Differential Equations, Plovdiv, 18-23 August 1997, 227-236.

[17] Tsugé, S. (2013) The Kolmogorov Turbulence Theory in the Light of Six-Dimensional Navier-Stokes’ Equation. In: Nakagawa, T.R.M, Ed., Philosophy of Flow, Vol. 10, Supplement, Royal White Mountains Academia Press, 19.

[18] Tsugé, S. (2004) Scientific Papers on Turbulence by Dr. Shunichi Tsugé. In: Nakagawa, T.M.S., Ed., Philosophy of Flow, Columbus University Press, Hakusan, 10.

[19] Tsugé, S. (2004) Scientific Papers on Combustion by Dr. Shunichi Tsugé. In: Nakagawa, T.M.S., Ed., Philosophy of Flow, Columbus University Press, Hakusan, 11.

[20] Sreenivasan, K.R. (1985) On the Fine-Scale Intermittency of Turbulence. Journal of Fluid Mechanics, 151, 81-103.

http://dx.doi.org/10.1017/S0022112085000878

[21] Antonia, R.A. and Pearson, B. (2000) Effect of Initial Conditions on the Mean Energy Dissipation Rate and the Scaling Exponent. Flow, Turbulence and Combustion, 64, 95-117.

[22] Osonphasop, C. and Hinwood, J.B. (1983) Turbulence Measurements for the Whole Depth of Water in a Tidal Channel. 8th Australasian Conference of Fluid Mechanics, University of Newcastle, 5 September 1983, 10-14.

[23] Osonphasop, C. (1983) The Measurements of Turbulence in Tidal Currents. A Thesis Submitted to the Faculty of Engineering in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy, Department of Mechanical Engineering, Monash University, Clayton.

[24] Osonphasop, C. and Hinwood, J.B. (1984) On Measurement of Turbulence and Shear Stresses in Tidal Currents. APD-IAHR 4th Conference of the International Association for Hydraulic Research, Chiangmai, 25 May 1984, 1-5.

[25] Trevethan, M. and Chanson, H. (2010) Turbulence and Turbulent Flux Events in a Small Estuary. Environmental Fluid Mechanics, 10, 345-368.

http://dx.doi.org/10.1007/s10652-009-9134-7

[26] Chanson, H., Brown, R. and Trevethan, M. (2012) Turbulence Measurements in a Small Subtropical Estuary under King Tide Condition. Environmental Fluid Mechanics, 12, 265-289.

http://dx.doi.org/10.1007/s10652-011-9234-z

[27] Reynolds, O. (1883) An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Philosophical Transactions of the Royal Society of London, 174, 935-982.

http://dx.doi.org/10.1098/rstl.1883.0029

[28] Kolmogorov, A.N. (1941) The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers. Doklady Akademii Nauk SSSR, 30, 301-304.

[29] Bogoliubov, N.N. (1959) Problems of Dynamical Theory in Statistical Physics. AFCRC-TR-59-235.

[30] De Kármán, Th. and Howarth, L. (1938) On the Statistical Theory of Isotropic Turbulence. Proceedings of the Royal Society A, 164, 192-215.

http://dx.doi.org/10.1098/rspa.1938.0013

[31] Chapman, S. (1917) On the Kinetic Theory of a Gas II. Philosophical Transactions of the Royal Society A, 217, 115-197.

[32] Enskog, D. (1917) Kinetische Theorie der Vorgange in Messigverdünnten Gasen. Dissertation, Uppsala University, Uppsala.

[33] Klimontovich, Yu.L. (1967) The Statistical Theory of Non-Equilibrium Processes in a Plasma. MIT Press, Cambridge.

[34] Lamb, H. (1932) Hydrodynamics. 6th Edition, Cambridge University Press, Cambridge, 1-8.

[35] Lorenz, E.N. (1963) Deterministic Nom-Periodic Flow. Journal of the Atmospheric Sciences, 20, 130-141.

http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[36] Ueda, Y. (1992) Strange Attractors and the Origin of Chaos. In: Ueda, Y., Ed., The Road to Chaos, Aerial Press, California, 185-216.

[37] Nakagawa, T. and Hinwood, J.B. (1978) On Measurement of Turbulence in Tidal Currents. Monash University, Clayton, MMEL34.