Calculating turbulent flows based on a stochastic model

Czibere, Tibor (2006) Calculating turbulent flows based on a stochastic model. JOURNAL OF COMPUTATIONAL AND APPLIED MECHANICS, 7 (2). pp. 155-188. ISSN 1586-2070

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Abstract

Relying on his theoretical and experimental examinations, O. Reynolds arrived at the conclusion that the Navier-Stokes equation of motion describing laminar flow continues to remain valid in terms of velocity fields interpreted by instantaneous values when the fluid is in turbulent motion. Th. von Kármán also used experimental experience to construct his similarity hypothesis [1], according to which, on the one hand, outside the viscous layer close to the wall, the turbulent velocity distribution does not depend on the viscosity of the medium, and on the other, the local (turbulent) flow patterns show mechanical similarities in points of the fully developed turbulent flow field; in other words each of them can be transferred into a common (turbulent) flow pattern by means of a suitably chosen transformation. The turbulence model used in this paper also relies on the same hypothesis. The model's fundamental principle can be summed up in the following words: in any point of the flow field the Helmholz-Thomson vortex theorem valid in the relative coordinate system - that is a coordinate system moving steadily at a velocity equal to the average Reynolds velocity in the given point - is suitable for describing the turbulent velocity fluctuation, thus it can be considered to be the equation of motion of turbulence, which then can be transformed, on the basis of von Kármán's similarity hypothesis, into the coordinate system of the common flow pattern mentioned. And a particular solution to the partial differential equation obtained can be used to represent the stochastic flow of the turbulence while the optional coefficients and phase constants appearing in it as integration constants are considered to be probability variables. By using the scalar components of the turbulent velocity fluctuation obtained in this way -- in this special relative coordinate system -- it is possible to produce the scalar elements of Reynolds' turbulent stress tensor, which can be re-transformed into the physical space on the basis of mechanical similarity. Thus, by using the stochastic turbulence model it becomes possible to produce Reynolds' turbulent stress tensor in a specific way, which in the transport equations of turbulent motion leads to formal changes that can be used in the numerical solutions with advantage.

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