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Transition of fractal dimension in a latticed dynamical system

Description: We study a recursion relation that manifests two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigenbaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns are reproduced in this new route. In the oscillatory instability regime the fracal dimension saturates at D/sub F/ approx. = 2.6 with imbedding dimensions while in the turbulent regime D/sub F/ saturates at 6.0. 19 refs., 3 figs.
Date: March 1, 1986
Creator: Duong-van, M.
Partner: UNT Libraries Government Documents Department

Dimension densities for turbulent systems with spatially decaying correlation functions

Description: The introduction of nonlinear, deterministic, and low-dimensional dynamical systems with chaotic solutions led to many conjectures about how these chaotic systems might be related to fluid turbulence. It appears that the time series, produced by a chaotic solution can be as complex as experimentally observed signals from turbulent hydrodynamics. Furthermore, certain transitions from laminar to turbulent flow have their analog in the transition from ordered to chaotic behavior of deterministic chaotic systems. A basic problem in this context is how far methods from nonlinear dynamical systems can be used to describe experimental turbulence. A frequent objection to the approach of using simple dynamical systems as models for turbulence is that these models might reproduce some temporal chaos but would not correspond to real turbulence, for which the spatial structure also is very irregular and chaotic. The dynamics of a turbulent flow are not expected to be spatially coherent and therefore cannot be described by a few global modes. In the following we use the simplest of coupled maps in order to test the applicability of some ideas which should make it possible to estimate the number of degrees of freedom per unit length of a system which is spatially incoherent. This is done by computing the dimension density of the lattice system through a series of two-point measurements at separated lattice points. Then we compare these results with the spatial decay of the correlation function and also of the mutual information content. We find a qualitative agreement with the expected dependence. For precise quantitative measurements the general problem of accuracy and data limitations appear to become dominant.
Date: January 1, 1986
Creator: Mayer-Kress, G. & Kurz, T.
Partner: UNT Libraries Government Documents Department

A method for treating hourglass patterns

Description: Hourglassing is a problem frequently encountered in numerical simulations of fluid and solid dynamics. The problem arises because certain volume-preserving distortions of cell shape produce no restoring forces. The result is an unrestricted drifting mode in the velocity field that leads to severe distortions of the computational mesh. These distortions cause large errors in the numerical approximations of the equations of motion. The drift may also allow adjacent vertices to get very close to each other. This results in the computational time step based on a Courant stability condition to become very small, effectively halting the calculation. We describe a mathematical formalism that identifies and selectively damps the hourglass patterns. The damping is constructed to preserve the physical aspects of the solution while maintaining a reasonable computational mesh. We further describe the implementation of our scheme in a 2D hydro code, and show the relative improvement in the results of six different test problems that we calculated.
Date: January 1, 1987
Creator: Margolin, L.G. & Pyun, J.J.
Partner: UNT Libraries Government Documents Department

Analysis of the effect of operator splitting and of the sampling procedure on the accuracy of Glimm's method

Description: A study is made of Glimm's method, a method for constructing approximate solutions to systems of hyperbolic conservation laws in one space variable by sampling explicit wave solutions. It is extended to several space variables by operator splitting. Two fundamental problems are addressed. A highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence is proposed. Error bounds are derived for Glimm's method, with van der Corput sampling, as applied to the inviscid Burgers' equation: for sufficiently small times the error in shock locations, speeds, and strengths is no greater than O(h/sup 1/2/abs. value (log h)), and the error in the continuous part of the solution, away from shocks, is O(h abs. value (log h)). Here h is the spatial increment of the grid, with the estimates holding in the limit of h ..-->.. 0. The improved sampling procedure is tested numerically in the case of inviscid compressible flow in one space dimension; it gives high-resolution results both in the smooth parts of the solution and at discontinuities. The operator splitting procedure by means of which the multidimensional method is constructed is investigated. An O(1) error stemming from the use of this procedure near shocks oblique to the spatial grid is analyzed numerically in the case of the equations for inviscid compressible flow in two space dimensions, and a method for eliminating this error, by the use of suitable artificial viscosity, is proposed and tested. 33 figures.
Date: December 1, 1978
Creator: Colella, P.
Partner: UNT Libraries Government Documents Department