# Numerical simulation of multi-dimensional acoustic propagation in air including the effects of molecular relaxation

Abstract (Summary)

A computational acoustic propagation model based upon the Navier-Stokes equations
is created that is able to simulate the effects of absorption and dispersion
due to shear viscosity, bulk viscosity, thermal conductivity and molecular relaxation
of nitrogen and oxygen in one or two dimensions. The model uses a fully
nonlinear constitutive equation set that is closed using a thermodynamic entropy
relation and a van der Waals equation of state. The use of the total variables in
the equations rather than the perturbed (acoustical) variables allow for the extension
of the model to include wind, temperature profiles, and other frequency
independent conditions. The method of including sources in the model also allow
for the incorporation of multiple spatially and temporally complex sources.
Two numerical methods are used for the solution of the constitutive equations:
a dispersion relation preserving scheme, which is shown to be efficient and accurate
but unsuitable for shock propagation; and a weighted essentially non-oscillatory
scheme which is shown to be able to stably propagate shocks but at considerable
computational cost. Both of these algorithms are utilized in this investigation
because their individual strengths are appropriate for different situations.
It is shown that these models are able to accurately recreate many acoustical
phenomena. Wave steepening in a lossless and thermoviscous medium is compared
to the Fubini solution and Mendousse’s solution to the Burgers equation, respectively,
and the Fourier component amplitudes of the first harmonics is shown to
differ from these solutions by at most 0.21 %. Nonlinear amplification factors upon
rigid boundaries for high incident pressures and its comparisons to the Pfriem solution
is shown to differ by at most 0.015 %. Modified classical absorption, nitrogen
relaxation absorption, and oxygen relaxation absorption is shown to differ from
the analytical solutions by at most 1 %. Finally, the dispersion due to nitrogen
relaxation and oxygen relaxation are also shown to differ from the analytical soiii
lutions by at most 1 %. It is believed that higher resolution grids would decrease
the error in all of these simulations.
A number of simulations that do not have explicit analytical solutions are then
discussed. To demonstrate the model’s ability to propagate multi-dimensional
shocks in two dimensions, the formation of a Mach stem is simulated. The amplification
factors determined in the test demonstrate a qualitative similarity with
discussions in the literature and explosion data. The ability of the algorithm to
propagate jet noise is then investigated using full scale jet noise as the input into
the algorithm. The waveforms predicted by the model are compared to a Burgers
equation algorithm and using a weak shock theory analysis of the shock propagation
speeds, an under-prediction of shock coalescence is noted in the Burgers
equation algorithm. To establish if the under-prediction of shock coalescence by
the Burgers equation algorithm is the cause of the discrepancy between its predictions
and recently measured scale model data, the WENO scheme is also used
to propagate the scale model jet noise. The predictions by the two models agree
very well for both cold and heat simulated jet cases which is due in part to the
relatively small amplitudes and propagation distances. The two models, however,
do not agree very well with the experimental data and it is concluded that more
work is needed to determine the precise reasons for this discrepancy.
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Bibliographical Information:

Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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