Abstract
The attractive fixed-point solution of a nonlinear cascade model is studied for the homogeneous isotropic turbulence containing a parameter C, introduced by Desnyansky and Novikov. With a traditional constant positive external force added on the first shell equation, it can be found that the attractive fixed-point solution of the model depends on both the parameter C and the external force. Thus, an explicit force is introduced to remove the effects of the external force on the attractive fixed-point solution. Furthermore, two groups of attractive fixed-point solutions are derived theoretically and studied numerically. One of the groups has the same scaling behavior of the velocity in the whole inertial range and agrees well with those observed by Bell and Nelkin for the nonnegative parameters. The other is found to have different scaling behaviors of the velocity at the odd and even number shells for the negative parameters. This special characteristic may be used to study the anomalous scaling behavior of the turbulence.
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References
Monin, A. S. and Yaglom, A. Y. Statistical Fluid Mechanics, Pergamon Press, Oxford (1987)
Biferale, L. Shell models of energy cascade in turbulence. Ann. Rev. Fluid Mech., 35, 441–468 (2003)
Kolmogorov, A. N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. R. Soc. Lond. A, 434, 9–13 (1991)
She, Z. S., Fu, Z. T., Chen, J., and Liu, S. D. Hierarchical structure in climate and atmospheric turbulence. Progress in Natural Science, 12(10), 747–752 (2002)
Bohr, T., Jensen, M. H., Paladin, G., and Vulpiani, A. Dynamical Systems Approach to Turbulence, Cambridge University Press, Cambridge (1998)
Gledzer, E. B. Hydrodynamic-type system admitting two quadratic integrals of motion. Dokl. Akad. Nauk SSSR, 209, 1046–1048 (1973)
L’vov, V. S., Podivilov, E., Pomyalov, A., Procaccia, I., and Vandembroucq, D. Improved shell model of turbulence. Phys. Rev. E, 58, 1811–1822 (1998)
Siggia, E. D. High Rayleigh number convection. Ann. Rev. Fluid Mech., 26, 137–168 (1994)
L’vov, V. S. Spectra of velocity and temperature fluctuations with constant entropy flux of fully developed free-convective turbulence. Phys. Rev. Lett., 67, 687–690 (1991)
Bolgiano, R. Turbulent spectra in a stably stratified atmosphere. J. Geophys. Res., 64, 2226–2229 (1959)
Ching, E. S., Guo, H., and Lo, T. S. Refined similarity hypotheses in shell models of homogeneous turbulence and turbulent convection. Phys. Rev. E, 78, 026303 (2008)
Brandenburg, A. Energy spectra in a model for convective turbulence. Phys. Rev. Lett., 69, 605–608 (1992)
Jiang, M. S. and Liu, S. D. Scaling behavior of velocity and temperature in a shell model for thermal convective turbulence. Phys. Rev. E, 56, 441–446 (1997)
Guo, H. Statistics and Structures in Turbulent Thermal Convection, Ph. D. dissertation, The Chinese University of Hong Kong (2007)
Ching, E. S. C., Guo, H., and Cheng, W. C. Understanding the different scaling behavior in various shell models proposed for turbulent thermal convection. Physica D, 237, 2009–2014 (2008)
Bell, T. L. and Nelkin, M. Time dependent scaling relations and a cascade model of turbulence. J. Fluid Mech., 88, 369–391 (1978)
Ching, E. S. C. and Guo, H. Effects of a large-scale mean flow in a shell model of turbulent convection. Int. J. Mod. Phys. B, 21, 4178–4183 (2007)
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Project supported by the National Natural Science Foundation of China (No. 10902007) and the National Basic Research Program of China (973 Program) (No. 2009CB724001)
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Guo, H., Li, C., Qu, Ql. et al. Attractive fixed-point solution study of shell model for homogeneous isotropic turbulence. Appl. Math. Mech.-Engl. Ed. 34, 259–268 (2013). https://doi.org/10.1007/s10483-013-1668-x
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DOI: https://doi.org/10.1007/s10483-013-1668-x