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The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is ...
The meaning of the turbulence is given below, [2] [3] [4] homogeneous, the statistical properties are invariant under arbitrary translations of the coordinate axes; isotropic, the statistical properties are invariant over a full rotation group, which includes rotations and reflections of the coordinate axes.
Consider a two-point velocity correlation tensor for homogeneous turbulence (,) = (,) (+,) ¯.For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940, [6]
In fluid dynamics, turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real-life scenarios. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows.
Lumley received the 1990 Fluid Dynamics Prize of American Physical Society, "For his outstanding contributions to the understanding of turbulent flow, in particular, the fundamental structure of turbulent shear flows, the effects of drag-reducing additives, and his widely recognized contributions to the statistical theory of turbulence, and for his personal and intellectual leadership in the ...
Science while reviewing this book had observed: "If ever a book on turbulence could be called definitive, it is this book by two of Russia's most eminent and productive scientists in turbulence, oceanography, and atmospheric physics." [5] A. S. Monin, An Introduction to the Theory of Climate, D. Reidel Publishing Company, Netherlands, 1986. [6]
This accounts for the transfer of kinetic energy from the mean flow to the fluctuating velocity field. It is responsible for sustaining the turbulence in the flow through this transfer of energy from the large scale mean motions to the small scale fluctuating motions. This is the only term that is closed in the Reynolds Stress Transport Equations.
The statistical theory of turbulence in viscous liquids describes the fluid flow by a scale-invariant distribution of the velocity field, which means that the typical size of the velocity as a function of wavenumber is a power-law. In steady state, larger scale eddies at long wavelengths disintegrate into smaller ones, dissipating their energy ...