This thesis presents a renormalization group method to tackle the problem of reducing the number of degrees of freedom necessary to describe fluid turbulence. Attempts to apply renormalization group methods to fluid turbulence have been in general based on an existing formulation of a multivariate normal model of turbulence. This model is an appropriate zero-order model for perturbation theories of turbulence in the context of the moment-closure problem because of the Gaussian distribution of the zero-older field for which moments of all orders can be expressed in terms of pair correlations. However, this is not a relevant attribute for renormalization group methods. A critical review of renormalization group methods based on the multivariate normal model is presented. An alternative approach was developed by McComb and Watt [Phys. Rev. A, 46, 4797 (1992)], who introduced a formal conditional average. The rescaling two-field theory leads to the derivation of a recursion relation, which eliminates finite blocks of turbulent velocity modes while maintaining the form invariance of the dynamical equation. In the present thesis, it is shown that the theory was heuristic in some respects and there were inadequate explanations in the procedure of that theory. A new formulation of renormalization group method, based on an alternative interpretation of the two-field theory, is presented here. A new zero-order model field is proposed for the Fourier modes in which there is no coupling between the high-wavenumber band of modes being eliminated and the remaining low-wavenumber modes. This model has the property that high-wavenumber modes can be eliminated without the need for a conditional average. The model field is then made the basis of a formal perturbation series which recovers the results of the two-field theory in a way which eliminates certain ambiguities, and allows one to see clear relationship between a turbulent velocity field and the zero order model field. The results are the systematic derivations of an equation for the high wavenumber modes that exhibits form invariance under the renormalization group transformation, and an expression for the effective viscosity for use in the computational simulation of homogeneous and isotropic turbulence. A value for Kolmogorov constant of α = 1:6 is obtained, when the fixed point for the effective viscosity is numerically calculated.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:664103 |
| Date | January 1998 |
| Creators | Yang, Taek-Jin |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/11635 |
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