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Applied Mathematics and Mechanics (English Edition) ›› 2013, Vol. 34 ›› Issue (3): 259-268.doi: https://doi.org/10.1007/s10483-013-1668-x

• •    下一篇

Attractive fixed-point solution study of shell model for homogeneous isotropic turbulence

郭昊1,2 李超1,2 屈秋林1,2 刘沛清1,2   

  1. 1. Institute of Fluid Mechanics, Beihang University, Beijing 100191, P. R. China;
    2. Key Laboratory of Fluid Mechanics Under the Ministry of Education, Beihang University, Beijing 100191, P. R. China
  • 收稿日期:2012-01-06 修回日期:2012-10-08 出版日期:2013-03-03 发布日期:2013-02-06
  • 通讯作者: Hao GUO, Ph.D., E-mail: guohao@pku.org.cn E-mail:guohao@pku.org.cn

Attractive fixed-point solution study of shell model for homogeneous isotropic turbulence

Hao GUO1,2, Chao LI1,2, Qiu-lin QU1,2, Pei-qing LIU1,2   

  1. 1. Institute of Fluid Mechanics, Beihang University, Beijing 100191, P. R. China;
    2. Key Laboratory of Fluid Mechanics Under the Ministry of Education, Beihang University, Beijing 100191, P. R. China
  • Received:2012-01-06 Revised:2012-10-08 Online:2013-03-03 Published:2013-02-06
  • Contact: Hao GUO, Ph.D., E-mail: guohao@pku.org.cn E-mail:guohao@pku.org.cn

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摘要： 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.

关键词: topological vector space, condensing preference map, maximal element

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.

Key words: topological vector space, condensing preference map, maximal element

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