Punekar, Jyothika Narasimha
(1996)
Numerical simulation of nonlinear random noise.
*University of Southampton, Doctoral Thesis*.

## Abstract

Nonlinear acoustics is an important subject area in the aeronautical sciences. Two such areas are (i) the generation of the sonic boom from aircraft travelling at supersonic speeds and (ii) in the generation and propagation of noise from jet propulsion engines. The present work focuses on the problem of the propagation of plant finite amplitude sound waves which evolve from an initial random noise signal. The work is concerned with the numerical solution of Burgers equation, the standard equation of nonlinear acoustics, for the case of high Reynolds number using a new accurate numerical algorithm which we refer to as the Convolution method. The direct numerical method uses the exact solution of Burgers equation based on its reduction to standard linear diffusion equation using Cole-Hopf transformation and where the attenuation is by thermoviscous diffusion only. Comparison of the numerical solutions with known exact inviscid solutions for simple waveforms shows convergence towards the inviscid waveform in each case as the Reynolds number is increased.

The present work is also concerned with waveforms containing a fundamental wave and a superimposed second harmonic. It is shown that for such waveforms, nonlinear interaction give rise to zero crossings in the plots of change in the wavenumber spectrum with time. These zero crossings do not occur after the shock formation time.

Comparisons are also made with Lighthill's analytical model for the one-dimensional propagation of randomly spaced sawtooth shock waves in the inviscid limit. In Lighthill (1994) it was shown that the number of shocks in a long period taken from a given noise sample decreases owing to the phenomenon of 'bunching' whereby the weaker are engulfed in time by the stronger shock waves. From a numerical example Lighthill showed that the average shock strength in the assemblage of sawtoothed waves decreases as 1√*t* and the same result was found in the present work for large but finite Reynolds number. The result differs from the decay of a regular constant amplitude sawtooth waves which decreases as 1√*t*.

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