Abstract
The paper concerns the problem on statistical description of the turbulent velocity pulsations by using the method of characteristic functional. The equations for velocity covariance and Green’s function, which describes an average velocity response to external force action, have been obtained. For the nonlinear term in the equation for velocity covariance, it has been obtained an exact representation in the form of two terms, which can be treated as describing a momentum transport due to turbulent viscosity and action of effective random forces (within the framework of traditional phenomenological description, the turbulent viscosity is only accounted for). Using a low perturbation theory approximation for high statistical moments, a scheme of closuring the chain of equations for statistical moments is proposed. As the result, we come to a closed set of equations for velocity covariance and Green’s function, the solution to which corresponds to summing up a certain infinite subsequence of total perturbation series.
Highlights
The problem on statistical description of turbulence, even at the simplest assumptions of stationary state and homogeneity of the velocity field, so far remaines to be unsolved in spite of its scientific and practical importance
For the nonlinear term in the equation for velocity covariance, it has been obtained an exact representation in the form of two terms, which can be treated as describing a momentum transport due to turbulent viscosity and action of effective random forces
Statistical description of turbulent flows means a knowledge of probability to find a given realization of the velocity field u r,t in the elementary volume d u r,t of the space of velocity field realizations that are solutions to hydrodynamic equations [1,2]
Summary
The problem on statistical description of turbulence, even at the simplest assumptions of stationary state and homogeneity of the velocity field, so far remaines to be unsolved in spite of its scientific and practical importance. Note that the effective Reynolds number defined as a ratio of energy input to given mode from large-scale modes to the value of viscous absorbtion caused by interaction with smale-scale modes appears to be approximately unit under the conditions of stationary developed turbulence This provides more rapid convergence of perturbation series. One more peculiarity of developed turbulence, which enables one to reveal the key features of the energy spectrum of turbulent velocity pulsations in some wavenumber range, is a conjecture that the turbulent spectrum is formatted due to nonlinear interactions between modes of close scales, whereas interactions between modes with significantly different scales are performed through the cascade sequence of interactions between modes of all intermediate scales (Richardson’s cascade), in other words, one has to tell about “a locality of intermode interactions in the wave-number space” [11] This is explained by the fact that the interaction between modes of significantly different scales reduces to a simple mechanical translation of small-scale modes by large-scale ones without energy redistribution [12].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
