^{1,a)}and Véronique Fortuné

^{2}

### Abstract

An iterative algorithm is developed for the computation of aeroacoustic integrals in the time domain. It is specially designed for the generation of acoustic images, thus giving access to the wavefront pattern radiated by an unsteady flow when large size source fields are considered. It is based on an iterative selection of source-observer pairs involved in the radiation process at a given time-step. It is written as an advanced-time approach, allowing easy connection with flow simulation tools. Its efficiency is related to the fraction of an observer grid step that a sound-wave covers during one time step. Test computations were performed, showing the CPU-time to be 30 to 50 times smaller than with a classical non-iterative procedure. The algorithm is applied to compute the sound radiated by a spatially evolving mixing-layer flow: it is used to compute and visualize contributions to the acoustic field from the different terms obtained by a decomposition of the Lighthill source term.

The algorithm development was supported by the French National Research Agency (ANR) via the ‘BruitAero’ program (Grant No. 05-BLAN-0208). Both authors thank Dr. P. Jordan for helpful discussions and for valuable advice in English language.

I. INTRODUCTION

II. IMPLEMENTING AEROACOUSTIC INTEGRALS: CURRENT ISSUES

A. Basic integration method

B. Discussion on the required optimization

III. A RING-GUIDED PROCEDURE

A. Principle

B. Algorithm efficiency

IV. APPLICATION TO THE ANALYSIS OF THE SOUND GENERATED BY A MIXING-LAYER

A. Decomposition of the Lighthill source term

B. Flow computation

C. Acoustic computation

D. Contributions of subterms to the sound field

V. CONCLUSIONS

### Key Topics

- Acoustics
- 18.0
- Acoustic waves
- 14.0
- Acoustic noise
- 8.0
- Spatial analysis
- 8.0
- Acoustic analogies
- 7.0

## Figures

Principle of the ring-guided procedure. Case . For one source time [(a) and (b)] and the following [(c) and (d)], at high emission angle [(a) and (c)] a low emission angle [(b) and (d)]. The procedure tracks observer grid points located between the two full-line circles (bold crosses) while minimizing the number of useless tested points (squares). The dotted-line circles correspond to other source times, and are plotted for visual aid.

Principle of the ring-guided procedure. Case . For one source time [(a) and (b)] and the following [(c) and (d)], at high emission angle [(a) and (c)] a low emission angle [(b) and (d)]. The procedure tracks observer grid points located between the two full-line circles (bold crosses) while minimizing the number of useless tested points (squares). The dotted-line circles correspond to other source times, and are plotted for visual aid.

CPU-time of test simulations with respect to the number of source-observer pairs, for different grid combination. (a) scanning emission-time procedure; (b) ring-guided emission-time procedure [solid line from estimation (9)].

CPU-time of test simulations with respect to the number of source-observer pairs, for different grid combination. (a) scanning emission-time procedure; (b) ring-guided emission-time procedure [solid line from estimation (9)].

Snapshots of the fluctuating pressure, density, dilatation, vorticity and Lighthill’s source term (from top to bottom). The high speed flow is on top.

Snapshots of the fluctuating pressure, density, dilatation, vorticity and Lighthill’s source term (from top to bottom). The high speed flow is on top.

Respective amplitudes (black boxes) and acoustic powers (gray boxes) of the subterms of Eq. (10), in the source domain. The value corresponds to the full Lighthill source term.

Respective amplitudes (black boxes) and acoustic powers (gray boxes) of the subterms of Eq. (10), in the source domain. The value corresponds to the full Lighthill source term.

Fluctuating pressure fields for a mixing layer. (a) direct computation (reference solution); (b) result from Lighthill’s analogy.

Fluctuating pressure fields for a mixing layer. (a) direct computation (reference solution); (b) result from Lighthill’s analogy.

Radial profile of acoustic pressure obtained from the acoustic analogy (symbols), source position , solid and dashed lines represent and decay respectively.

Radial profile of acoustic pressure obtained from the acoustic analogy (symbols), source position , solid and dashed lines represent and decay respectively.

Fluctuating pressure fields resulting from (a) summed contributions of driving source subterms , (b) summed contributions of subterms and 7, (c) summed contributions of subterms and subterms 5 and 7. (d) the subterm 1 alone; (e) summed contributions of subterms , (f) summed contributions of subterms .

Fluctuating pressure fields resulting from (a) summed contributions of driving source subterms , (b) summed contributions of subterms and 7, (c) summed contributions of subterms and subterms 5 and 7. (d) the subterm 1 alone; (e) summed contributions of subterms , (f) summed contributions of subterms .

Single source point test case. (a) Account of observer points concerned by the procedure marching as a function of the ring radius . (b) Dependency on of proportionality ratios between the number of points located on the ring, or outside it, and the ring radius .

Single source point test case. (a) Account of observer points concerned by the procedure marching as a function of the ring radius . (b) Dependency on of proportionality ratios between the number of points located on the ring, or outside it, and the ring radius .

(a) CPU-time and accounts of observer points concerned by the procedure marching as a function of the ring radius, for real source and observer domains, with different source grids. (b) CPU-time by ring step as a function of the number of [source-observer] pairs of which distance is between and . Each symbol corresponds to a source file time step, that is an entire value of .

(a) CPU-time and accounts of observer points concerned by the procedure marching as a function of the ring radius, for real source and observer domains, with different source grids. (b) CPU-time by ring step as a function of the number of [source-observer] pairs of which distance is between and . Each symbol corresponds to a source file time step, that is an entire value of .

Evolution of coefficient with . Symbols: different observer grids for ; straight line: linear regression.

Evolution of coefficient with . Symbols: different observer grids for ; straight line: linear regression.

## Tables

**Algorithm 1.** Sketch of a reception time algorithm (instructions 1 and 9 are optional).

**Algorithm 1.** Sketch of a reception time algorithm (instructions 1 and 9 are optional).

**Algorithm 2.** Sketch of an emission time algorithm.

**Algorithm 2.** Sketch of an emission time algorithm.

**Algorithm 3.** Sketch of a ring-guided emission time algorithm (instructions 4 and 7 are optional).

**Algorithm 3.** Sketch of a ring-guided emission time algorithm (instructions 4 and 7 are optional).

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