I. Interactions of Fast and Slow Waves in Problems with Two Time Scales. II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow

Citation

Barker, John Wilson (1982) I. Interactions of Fast and Slow Waves in Problems with Two Time Scales. II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ynsy-nh46. https://resolver.caltech.edu/CaltechETD:etd-09182006-090057

Abstract

I. Interaction of Fast and Slow Waves in Problems with Two Time Scales

We consider certain symmetric, hyperbolic systems of nonlinear first-order partial differential equations whose solutions vary on two time scales, a 'slow' scale t and a 'fast' scale t/ε. The large (0(ε^-1)) part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives (a 'symmetriser') is allowed.

In physical applications, it is often the case that the fast scale motion is of little interest, and it is desired to calculate only the slow scale motion accurately. It is known that solutions with arbitrarily small amounts of fast scale motion can be obtained by careful choice of the initial data, and that an error of amplitude 0(ε^p), where p = 2 for one space dimension or p = 3 for two or three space dimensions, in this choice is allowable, resulting in fast scale waves of amplitude 0(ε^p) in the solution.

We investigate what happens when the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude 0(ε). We show that then the perturbation in the solution will also be of amplitude 0(ε). Further, we show that if the large part of the spatial operator is nonsingular in the sense that the number of large eigenvalues of the symbol, P(iω), of the spatial operator is independent of ω, then the error introduced in the slow scale motion will be of amplitude 0(ε²), even though fast scale waves of amplitude 0(ε) will be present in the solution. If the symmetriser is a constant, this holds even if the spatial operator is singular, and further if an error 0(ε^μ) is made in the initial conditions, for any µ > 0, the resulting error in the slow scale motion will be 0(ε^2μ).

Our proofs are based on energy estimates which show that spatial derivatives of the solutions are 0(1), even if time derivatives are not, and the development of the solutions in asymptotic expansions.

II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow

Some previous theories and numerical calculations pertaining to the problem of two-dimensional turburlence are reviewed, and a new numerical experiment is proposed. The purpose of the experiment is to test the hypothesis that narrow regions of concentrated vorticity are produced in two-dimensional flows by advection of vorticity towards dividing streamlines in regions where the local flow is convergent.

The numerical method to be used is described in detail. It integrates the inviscid Euler equations using a Fourier (pseudo-spectral) method for the space derivatives, and a predictor-corrector method due to Hyman (1979) for time stepping. Dissipation is included, following Fornberg (1977), by a chopping of the amplitudes of the higher Fourier modes every few time-steps. This acts as a high-wavenumber energy sink, allowing very high Reynolds number flows to be simulated with relatively little computational effort.

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