Investigation of renormalization group methods for the numerical simulation of isotropic turbulence

• Main Author: Young, Alistair John
• Published: University of Edinburgh 1999
• Subjects: 530.44
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.664143

Numerical simulations are becoming increasingly important and useful tools in the study of turbulent flows. Direct numerical simulation (DNS) is a method of computation in which all the important scales of motion are fully-resolved. However, flows with the sort of complexity and turbulent intensity that one might find in the laboratory, or in nature, lie well beyond our current computational reach. The problem lies with the large number of degrees of freedom. In large eddy simulations (LES) this number is reduced by simulating only the large scales of motion, while the effects of the small scales are modelled in some way. In Fourier space, the problem becomes one of eliminating high-wavenumber modes, in some statistical sense, in order to bring the reduced number of degrees of freedom within the capacity of current (or even future) computers. In this thesis, two methods by which such a reduction in the number of modes may be obtained are explored. A conventional approach to the problem is to model the effects of the absent modes by way of an increased viscosity. A number of theoretical ways of obtaining such an <i>eddy-viscosity</i> have been proposed. In this work, we look first at the closely-related renormalization group (RG) theories of McComb. [W. D. McComb and A. G. Watt, Phys. Rev. A 46, 4797 (1992)] and Yang [T.-J. Yang, PhD thesis, University of Edinburgh (1998)]. In these theories, a <i>conditional average </i>is introduced, allowing the high-wavenumber modes to be averaged out, whilst the low-wavenumber modes remain unaffected. The theory of Yang also introduces a model field which is used to aid in the evaluation of the conditional averages. This averaging technique is applied iteratively, in order to eliminate a series of narrow bands until a fixed point is reached. Utilising the results from a DNS, the underlying feasibility of using these theories to obtain an eddy-viscosity is assessed in a number of ways. Firstly, we make a study of the properties of the nonlinear term in the Navier-Stokes equation. This term has been partitioned in the same manner as the RG theories and the results obtained are discussed in both general and RG contexts. Secondly, we give an averaging method which may be regarded as a <i>computable</i> approximation to the conditional average. Results are given and the degree to which this technique actually models the conditional average is considered. Finally, a brief examination of Yang's model field - and its use in the evaluation of the conditional average - is made. The results of these studies suggest that there may be problems with the way in which the RG eddy-viscosity is obtained. However, we note that the majority of our tests have necessarily been limited to relatively low Reynolds number cases.