Accounting for viscous and thermal effects in time in computational problems of acoustics

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Abstract

The problem of acoustic wave propagation with thermoviscous boundary conditions is studied. For thermoviscous boundary conditions, a time-dependent formulation is presented based on the concept of a fractional derivative. A weak formulation of the problem is given, which is reduced to a system of Volterra-type integro-differential equations using the finite element method. An implicit finite-difference scheme is constructed for the numerical solution of this system. To verify it, the problem of sound propagation in a thin pipe is modeled, the results of numerical modeling are compared with the analytical solution.

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About the authors

А. I. Korolkov

University of Manchester

Email: laptev97@bk.ru
United Kingdom, Oxford Road, Manchester, M13 9PL

A. Yu. Laptev

Lomonosov Moscow State University

Author for correspondence.
Email: laptev97@bk.ru
Russian Federation, Leninskie Gory, Moscow, GSP-1, 119991

A. V. Shanin

Lomonosov Moscow State University

Email: laptev97@bk.ru
Russian Federation, Leninskie Gory, Moscow, GSP-1, 119991

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Supplementary files

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2. Fig. 1. Approximate view of the region Ω.

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3. Fig. 2. Geometry of the model.

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4. Fig. 3. Comparison of numerical simulation with analytical solution. The dotted line indicates the pulse at a distance of 50 cm from the emitting surface without taking into account thermoviscous effects, the dashed line indicates taking into account thermoviscous effects, and the solid line indicates the analytical solution (12).

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5. Fig. 4. Dependence of the difference between the numerical and analytical solutions on the grid step.

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