# Computational aeroacoustics

Computational aeroacoustics is a branch of aeroacoustics that aims to analyze the generation of noise by turbulent flows through numerical methods.

## History

The origin of Computational Aeroacoustics can only be dated back to the middle of the 1980s, with a publication of Hardin and Lamkin  who claimed, that

” […] the field of computational fluid mechanics has been advancing rapidly in the past few years and now offers hope that” computational aeroacoustics, “where noise is computed directly from a first principles determination of continuous velocity and vorticity fields, might be possible, […] “

Later in a 1986 publication  the same authors introduced the abbreviation CAA. The term was first used for a low Mach number approach as it is described under EIF . Later in the beginning of the 1990s, the authors reported that they have developed a wide range of sources of noise. Such numerical methods can be far field integration methods (eg FW-H   ) as well as Direct numerical methods optimized for the solutions (eg of a mathematical model describing the aerodynamic noise generation and / or propagation. With the rapid development of the computational resources this field has undergone spectacular progress during the last three decades.

## Methods

### Direct numerical simulation (DNS) Approach to CAA

The compressible Navier-Stokes equation describes both the flow field and the aerodynamically generated acoustic field. Thus both can be solved for directly. This requires very high numerical resolution due to the large differences in the length scale between the acoustic variables and the flow variables. It is computationally very demanding and unsuitable for any commercial use.

### Hybrid Approach

In this approach the subject is divided into different regions, such that the governing body or the field can be solved with different equations and numerical techniques. This project involved computational fluid dynamics (CFD) and secondly an acoustic solver. The flow field is then used to calculate the acoustic sources. Both steady state (RANS, SNGR (Stochastic Noise Generation and Radiation), …) and transient (DNS, LES, DES, URANS, …) fluid field solutions can be used. These acoustical sources are provided to the second solver which calculates the acoustical propagation. Acoustic propagation can be calculated using one of the following methods:

1. Integral Methods
1. Lighthill’s analogy
2. Kirchhoff integral
3. FW-H
2. LEE
3. pseudospectral
4. EIF
5. EPA

#### Integral methods

There are multiple methods, which are based on a known solution of the acoustic wave equation to compute the far field of a sound source. Because a general solution for wave propagation in the free space, these solutions are summarized as integral methods. Some of the sources of this problem are known (eg a finite element simulation of a moving mechanical system or a dynamic dynamic CFD simulation of the sources in a moving medium). The integral is taken over (source time), which is the time at which the source is sent to the signal, which arrives at a given position. Common to all integral methods is, that they can not account for the rate of change or the average rate of change. When applying Lighthill’s theory Stokes equations of Fluid mechanics, volumetric sources, and other analogies. Acoustic analogies can be very efficient and fast, as the known solution of the wave equation is used. One far away observe takes a long time to watch. The use of analogies in the context of a large number of analogies (addition / subtraction of many large numbers with result to zero). domain is limited somehow. While in theory the sources have not been zero, the application can not always fulfill this condition. Especially in connection with CFD simulations, this leads to large cut-off errors.

##### Lighthill’s analogy

Also called ‘ Acoustic Analogy ‘. To obtain Lighthill’s aeroacoustic analogy Navier-Stokes equations are rearranged. The operator is a wave operator, which is applied to the disturbance or disturbance respectively. The right hand side is identified in the acoustic sources in a fluid flow, then. As Lighthill’s analogy follows from the Navier-Stokes equations without simplification, all sources are present. Some of the sources are then identified as turbulent or laminar noise. The far-field sound pressure is then given in terms of a volume integral over the domain containing the sound source. The source term term always includes physical sources and such sources, which describes the propagation in an inhomogeneous medium.

The wave operator of Lighthill’s analogy is limited to constant flow conditions outside the source zone. No variation of density, speed of sound and Mach number is allowed. Different ways of analogy, ounce and acoustic wave passes it. Part of the acoustic wave is removed by one source and has a new wave. This often leads to very large volumes with strong sources. Several modifications to Lighthill’s original theory have been proposed for the sound-flow interaction or other effects. To improve Lighthill’s analogy different quantities inside the wave operator. All of them obtain modified source terms, which sometimes allow a clearer view on the “real” sources. The acoustic analogies of Lilley, Pierce,  Howe  and Möhring  are only some examples for aeroacoustic analogies based on Lighthill’s ideas. All acoustic analogies require a volume integration over a source term.

The major difficulty with the acoustic analogy, however, is that the sound source is not compact in supersonic flow. Could be encountered in calculating the sound field, unless the computational domain could be extended in the downstream direction beyond the location where the sound source has completely decayed. Furthermore, an accurate account of the time-effect requires a long record of the time-history of the converged solutions of the sound source, which again represents a storage problem. For realistic problems, the required storage can reach the order of 1 terabyte of data.

##### Kirchhoff integral

Kirchhoff and Helmholtzshown, that the radiation of a surface area – the so-called Kichhoff surface. Then the sound field inside or outside the surface, where can be produced as a superposition of monopolies and dipoles on the surface. The theory follows directly from the wave equation. The source strength of monopolies and dipoles on the surface can be calculated if the normal velocity (for monopolies) and the pressure (for dipoles) are known. A modification of the method allows even to calculate the pressure on the surface based on the normal velocity only. The normal velocity could be given by a FE-simulation of a moving structure for instance. HOWEVER, The present invention relates to the subject of the present invention, wherein the subject of the present invention is described in the following: The Kirchhoff integral method finds for instance application inBoundary element methods (BEM). A non-zero velocity is accounted for by considering a moving frame of reference in which the acoustic wave propagation takes place. Repetitive applications of the method can account for obstacles. The obstacle in the field of obstacle is calculated and then the obstacle is introduced by adding sources on its surface to the normal velocity on the surface of the obstacle. Can be taken into account by a similar method (eg dual reciprocity BEM).

##### FW-H

The integration method of Ffowcs Williams and Hawkings is based on Lighthill’s acoustic analogy. However, by some mathematical modifications under the assumption of a limited source region, which is enclosed by a surface control (FW-H surface), the integral volume is avoided. Surface integrals over monopole and dipole sources remain. Different from the Kirchhoff method, these sources follow directly from the Navier-Stokes equations through Lighthill’s analogy. Sources outside the FW-H can be accounted for by an additional volume integral over quadrupole sources following from the Lighthill Tensor. However, when considering the same assumptions as Kirchhoffs linear theory, the FW-H method equals the Kirchhoff method.

#### Linearized Euler Equations

Considering small disturbances superimposed on a uniform mean flow of density {\ displaystyle \ rho _ {0}}pressure {\ displaystyle p_ {0}} and velocity on x-axis {\ displaystyle u_ {0}}, the Euler equations for a two dimensional model is presented as:

{\ displaystyle {\ frac {\ partial \ mathbf {U}} {\ partial t}} + {\ frac {\ partial \ mathbf {F}} {\ partial x}} + {\ frac {\ partial \ mathbf { G}} {\ partial y}} = \ mathbf {S}},

Where

{\ displaystyle \ mathbf {U} = {\ begin {bmatrix} \ rho \\ u \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ rho _ {0} u + \ rho u_ {0} \\ u_ {0} u + p / \ rho _ {0} \\ u_ {0} v \\ u_ {0} p + \ gamma p_ {0} u \ \\ end {bmatrix} \, \ \ mathbf {G} = {\ begin {bmatrix} \ rho _ {0} v \\ 0 \\ p / \ rho _ {0} \\ gamma p_ {0} v \\\ end {bmatrix}}}

Where {\ displaystyle \ rho}, {\ displaystyle u}, {\ displaystyle v} and {\ displaystyle p} are the acoustic field variables, {\ displaystyle \ gamma} the ratio of specific heats {\ displaystyle c_ {p} / c_ {vb}}for air at 20 ° C {\ displaystyle c_ {p} / c_ {vb} = 1.4}, and the source term {\ displaystyle \ mathbf {S}}on the right-side is distributed unsteady sources. The application of LEE can be found in engine noise studies. 

For high Mach number flows in compressible diagrams, the acoustic propagation may be influenced by non-linearities and the LEE may not be the appropriate mathematical model.

#### Pseudospectral

A pseudospectral Fourier time-domain method can be applied to wave propagation problems relevant to computational aeroacoustics. The original algorithm of the Fourier pseudo spectral A slip wall boundary condition, combined with technical buffer zone to solve some non-periodic aeroacoustic problems has been proposed.  Compared to other computational methods, pseudospectral method is preferred for its high-order accuracy.

#### EIF

Expansion about Incompressible Flow

#### APE

Acoustic Disturbance Equations

Refer to the paper “Acoustic Disturbance Equations Based on Flow Decomposition via Source Filtering” by R.Ewert and W.Schroder.

## See also

• Aeroacoustics
• Acoustic theory

## External links

• Examples in Aeroacoustics from NASA
• Computational Aeroacoustics at the Ecole Centrale de Lyon
• Computational Aeroacoustics at the University of Leuven
• Computational Aeroacoustics at Technische Universität Berlin
• A CAA reading script of Technische Universität Berlin

## References

1. Jump up^ Hardin, JC and Lamkin, SL, “Aeroacoustic Computation of Cylinder Wake Flow,” AIAA Journal, 22 (1): 51-57, 1984
2. Jump up^ Hardin, JC and Lamkin, SL, “Computational aeroacoustics – Present status and future promise,” IN: Aero- and hydro-acoustics; Proceedings of the Symposium, Ecully, France, July 3-6, 1985 (A87-13585 03-71). Berlin and New York, Springer-Verlag, 1986, p. 253-259.
3. Jump up^ Ffowcs Williams, “The Noise from Turbulence Convected at High Speed,”Philosophical Transactions of the Royal Society, Vol. A255, 1963, pp. 496-503
4. Jump up^ Ffowcs Williams, JE, and Hawkings, DL, “Sound Generated by Turbulence and Surfaces in Arbitrary Motion,”Philosophical Transactions of the Royal Society, Vol. A264, 1969, pp. 321-342
5. Jump up^ CKW Tam, and JC Webb, “Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics,”Journal of Computational Physics, Vol. 107, 1993, pp. 262-281
6. Jump up^ Lighthill, MJ, “On Sound Aerodynamically Generated, i”,Proc. Roy. Soc. A, Vol. 211, 1952, pp 564-587
7. Jump up^ Lighthill, MJ, “On Sound Aerodynamically Generated, ii”,Proc. Roy. Soc. A, Vol. 222, 1954, pp 1-32
8. Jump up^ Lilley, GM, “On the noise from air jets”, AGARD CP 131, 13.1-13.12
9. Jump up^ Pierce, AD, “Wave equation for the sound in fluids with unsteady inhomogeneous flow”, J. Acoust. Soc. Am., 87: 2292-2299, 1990
10. Jump up^ Howe, MS, “Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute”, J. Fluid Mech., 71: 625-673, 1975
11. Jump up^ Mohring, W. A well Posed based acoustic analogy was moving acoustic medium. 2010, arXiv preprint arXiv: 1009.3766.
12. Jump up^ Chen XX, X. Huang and X. Zhang, “Sound Radiation from a Bypass Duct with Bifurcations”, AIAA Journal, Vol. 47, No. 2, 2009. pp.429-436.
13. Jump up^ X. Huang and X. Zhang, “A Fourier Pseudospectral Method for Some Computational Aeroacoustics Problems,” International Journal of Aeroacoustics, Vol 5, No. 3, 2006. pp.279-294.