Renormalization-Group Analysis of Turbulence

Abstract

ObjectiveThe objective is to understand and extend a recent theory of turbulence based on dynamic renormalization group (RNG) techniques. The application of RNG methods to hydrodynamic turbulence has been explored most extensively by Yakhot and Orszag (1986). They calculate an eddy viscosity consistent with the Kolmogorov inertial range by systematic elimination of the small scales in the flow. Further, assumed smallness of the nonlinear terms in the redefined equations for the large scales results in predictions for important flow constants such as the Kolmogorov constant. The authors emphasize that no adjustable parameters are needed. The parameterization of the small scales in a self-consistent manner has important implications for sub-grid modeling. The RNG TransformationRenormalization group methods were first developed for quantum field theories. They were later applied to the theory of critical points in materials that undergo phase transitions (Ma, 1976). Predictions for the universal exponents characterizing the behavior of thermodynamic quantities near critical points are quite accurate. The common feature of the physical phenomena amenable to RNG analysis is a lack of characteristic length and time scales.The lack of characteristic length and time scales in turbulence makes RNG methods attractive. The universality of the inertial range spectrum in widely varying turbulent flows is also suggestive.The RNG transformation consists of two steps. First, small scales are eliminated by an averaging procedure. Second, space is rescaled. New independent variables are defined on the original intervals by the rescaling. In most cases, the dependent variables must also be rescaled.A set of equations is renormalizable if it is unchanged by the RNG transformation. Renormalizability implies scale invariance. Usually a set of equations is renormalizable only for specific values of its coefficients and the scaling parameters. These points are called fixed points. However, the physics of more general cases is often well described by the physics at a b e d point.The method of attack is to iterate the RNG transformation of the equations. With each transformation the coefficients in the equations change. One looks for a situation in which this iteration procedure converges.In addition to redefining coefficients of existing terms, the scale elimination often generates terms of different form than those in the original equations. PRECEDiNG PAGE BLANK NOT FilMEDhttps://ntrs.nasa.gov/search.jsp?R=19890013450 2018-05-12T00:35:56+00:00Z

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