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On Lighthill’s Acoustic Analogy for Low Mach Number Flows

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New Directions in Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Most predictions of the noise generated by a turbulent flow are done using a model due to Lighthill from the 1950’s (the Lighthill analogy). In a large region of a fluid at rest surrounding a small region containing a small Mach number, high Reynolds number turbulent flow, this is Step I: Solve the incompressible Navier-Stokes equations with the constant density ϱ∞ for the velocity u. Step 2: Compute div(div(ρ∞ u ⊗ u)) and solve the inhomogeneous acoustic equation in both regions for the acoustic density fluctuations R:

$$ \partial _t^2 R - \omega \Delta R = div(div(\rho _\infty u \otimes u)), $$

where \( \sqrt \omega \) is the speed of sound. Current understanding of the derivation of the Lighthill analogy seems to be a variation on Lighthill’s original reasoning and has resisted elaboration by the tools of both formal asymptotics and rigorous mathematics. In this report we give a rigorous derivation of Lighthill’s acoustic analogy (including the sound source div(div(ρ∞ u ⊗ u)) being derived from an incompressible flow simulation. from the compressible Navier-Stokes and energy equation as Ma → 0.

W.L. was partially supported by NSF grants DMS 0508260 and 0810385

Work was completed during the stay of A.N. at the Department of Mathematics and the Department of Mechanical Engineering of the University of Pittsburgh under the financial support of the exchange visitor programm number P-1-00048. The authors thank to G.P. Galdi for inspiring discussions.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15 260, USA

    William Layton

  2. Institut Mathématiques de Toulon, Université du Sud Toulon-Var, BP 132, F-83957, La Garde, France

    Antonín Novotný

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  1. William Layton
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  2. Antonín Novotný
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Editor information

Editors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Vorob’evy Gory, 119991, Moscow, Russia

    Andrei V. Fursikov

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 630 Benedum Engineering Hall, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

  3. Lavrentyev Institute of Hydrodynamics, Lavrentyev prospect 15, 630090, Novosibirsk, Russia

    Vladislav V. Pukhnachev

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Dedicated to A.V. Kazhikov

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Layton, W., Novotný, A. (2009). On Lighthill’s Acoustic Analogy for Low Mach Number Flows. In: Fursikov, A.V., Galdi, G.P., Pukhnachev, V.V. (eds) New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0152-8_14

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