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Numerical Eulerian method for linearized gas dynamics in the high frequency regime

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Abstract

We study the propagation of an acoustic wave in a moving fluid in the high frequency regime. We calculate the asymptotic approximation of the solution, around a mean flow, of this problem using an Eulerian method. By introducing the stretching matrix (deformation tensor for the geometrical optics rays) of the linearized Euler system, we deduce the geometrical spreading. This quantity is the key tool for computing the leading order term of the asymptotic expansion thanks to a conservation equation along the group velocity. The main contribution is to construct and implement a numerical scheme in the Eulerian framework for the eikonal equation and for the transport equation on the stretching matrix. We present numerical results for several test cases to study the convergence and validate our approach.

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Notes

  1. The tangent bundle of \(\mathcal {T}\) is \(T\mathcal {T}\). A fiber over \(\varvec{y}\in \mathcal {T}\) of this bundle is the tangent vector space \(T_{\varvec{y}}\mathcal {T}\) of \(\mathcal {T}\) at \(\varvec{y}\). In the same way, the cotangent bundle \(T^*\mathcal {T}\) (or phase space) consists of fibers over \(\varvec{y}\) each of which is the dual vector space to \(T_{\varvec{y}}\mathcal {T}\).

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Acknowledgments

The authors would like to thank the anonymous referee for her/his careful reading and for the helpful comments that have improved substantially the final version of this paper.

Author information

Authors and Affiliations

  1. LAGA, Institut Galilée, Université Paris 13, Villetaneuse, France

    Youness Noumir & Olivier Lafitte

  2. Conservatoire National des Arts et Métiers, Department of Mathematics, and Laboratoire de Mécanique des Structures et des Systèmes Couplés, Paris, France

    François Dubois

  3. Department of Mathematics, University Paris-Sud, Bât. 425, 91405 , Orsay Cedex, France

    François Dubois

  4. CEA Saclay, DM2S, 91191 , Gif sur Yvette Cedex, France

    Olivier Lafitte

Authors
  1. Youness Noumir
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  2. François Dubois
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  3. Olivier Lafitte
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Corresponding author

Correspondence to François Dubois.

Additional information

This work has been supported by EADS-Innovation Works through the funding of the PhD thesis of Y. Noumir.

Appendix: Reduced geometric spreading

Appendix: Reduced geometric spreading

Proposition 2

If the mean flow is independent of the time, and if the solution of the eikonal equation is written as \(\phi (t,\varvec{x})=\phi _1(t)+\phi _2(\varvec{x})\) or \(\phi (t,\varvec{x})={\tilde{\phi }_1}(t)\phi _2(\varvec{x})\) where \(\tilde{\phi }_1\) is a strictly positive function, then the geometrical spreading is reduced to:

$$\begin{aligned} J(s,\varvec{\beta })=-\left| \partial _s\varvec{x}(s,\varvec{\alpha }) ~~\partial _{\varvec{\alpha }}\varvec{x}(s,\varvec{\alpha })\right| , \end{aligned}$$

where \(\varvec{\beta }\equiv (t_0,\varvec{\alpha })\in \mathbb {R}\times \mathbb {R}^{d-1}\) is a parameterization of the incident surface \(\Sigma _{inc}\).

Proof

In these cases, the equation on the ray field is written

$$\begin{aligned} \frac{\partial \varvec{x}}{\partial s}(s,\varvec{\beta })=\varvec{u}_0(\varvec{x}(s,\varvec{\beta }))+c_0(\varvec{x}(s,\varvec{\beta })) \frac{\varvec{\nabla }_{\varvec{x}}\phi _2(\varvec{x}(s,\varvec{\beta }))}{\left| \varvec{\nabla }_{\varvec{x}}\phi _2(\varvec{x}(s,\varvec{\beta }))\right| }, \end{aligned}$$

that can be written also as

$$\begin{aligned} \frac{\partial \varvec{x}}{\partial s}(s,\varvec{\beta })=\varvec{\mathcal {F}}(\varvec{x}(s,\varvec{\beta })). \end{aligned}$$
(35)

By deriving the above equation with respect to \(t_0\), it follows that

$$\begin{aligned} \frac{\partial }{\partial s}\left( \partial _{t_0}\varvec{x}\right) (s,\varvec{\beta })=\nabla _{\varvec{x}} \varvec{\mathcal {F}}\left( \varvec{x}(s,\varvec{\beta })\right) .\partial _{t_0}\varvec{x}(s,\varvec{\beta }). \end{aligned}$$

Given that \(\varvec{x}(0,\varvec{\beta })=(\varvec{\alpha },0)\), it ensues that \(\partial _{t_0}\varvec{x}(0,\varvec{\beta })=\varvec{0}_d\). By uniqueness of the solution of the equation (35), we deduce that \(\partial _{t_0}\varvec{x}(s,\varvec{\beta })=\varvec{0}_d\) for all \(s\).

It remains to note that the geometrical spreading is expressed as the sum of two determinants:

$$\begin{aligned} J(s,\varvec{\beta })=\left| \partial _{t_0}\varvec{x}(s,\varvec{\beta }) ~~ \partial _{\varvec{\alpha }}\varvec{x}(s,\varvec{\beta })\right| -\left| \partial _s \varvec{x}(s,\varvec{\beta }) ~~\partial _{\varvec{\alpha }}\varvec{x} (s,\varvec{\beta })\right| . \end{aligned}$$

This completes the proof of Proposition.\(\square \)

In this case the function \(U\) simplifies to

$$\begin{aligned} U(t,\varvec{x})= \left( \begin{array}{c} \partial _{\alpha }\varvec{x}\left( \mathcal {S}_0(t,\varvec{x}),\mathcal {Y}_0(t,\varvec{x})\right) \\ \partial _{\alpha }\varvec{\xi }\left( \mathcal {S}_0(t,\varvec{x}),\mathcal {Y}_0(t,\varvec{x})\right) \end{array}\right) , \end{aligned}$$

which is solution of the same transport equation (16), but where the second derivatives of the Hamiltonian \(\mathcal {H}^\pm \) are replaced by those of the Hamiltonian \(H^\pm \).

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Noumir, Y., Dubois, F. & Lafitte, O. Numerical Eulerian method for linearized gas dynamics in the high frequency regime. Numer. Math. 127, 641–683 (2014). https://doi.org/10.1007/s00211-013-0598-5

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