Computational Aeroacoustics

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Computational Aeroacoustics (CAA) as defined by Allan D. Pierce (1992), is the direct simulation of acoustic fields generated by flows and of the interaction of acoustic fields with flows. Direct implies that the computation is only based on fundamental physical principles without reliance on empirical results or heuristic conjectures. With the rapid development in the field of computational resources and Aeroacoustics this field has undergone spectacular progress during the last decade.

Contents

[edit] Methods

[edit] Direct numerical simulation(DNS) Approach to CAA

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

[edit] Hybrid Approach

In this approach the computational domain is split into different regions, such that the governing acoustic or flow field can be solved with different equations and numerical techniques. This would involve using two different numerical solvers, first a dedicated Computational fluid dynamics (CFD) tool and secondly an acoustic solver. The flow field is then used to calculate the acoustical sources. Both stationary (RANS, SNGR (Stochastic Noise Generation and Radiation), ...) and instationary (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. EIF
  4. APE

[edit] Lighthill's analogy

Also called 'Acoustic Analogy', the governing Navier-Stokes equations are rearranged to be in wave-type form. There is some question as to which terms should be identified as part of the sound source and retained in the right-hand side of the equation and which terms should be in the left-hand side as part of the operator. The far-field sound pressure is then given in terms of a volume integral over the domain containing the sound source. Several modifications to Lighthill's original theory have been proposed to account for the sound-flow interaction or other effects. The major difficulty with the acoustic analogy, however, is that the sound source is not compact in supersonic flow. Errors 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 retarded time-effect requires keeping a long record of the time-history of the converged solutions of the sound source, which again represents a storage problem. See Aeroacoustics#Lighthill's equation

[edit] Kirchhoff integral

See Gustav Kirchhoff

[edit] FW-H

See John Ffowcs Williams

[edit] Linearized Euler Equations

Considering small disturbances superimposed on a uniform mean flow of density ρ0, pressure p0 and velocity on x-axis u0, the Euler equations for a two dimensional model is presented as:

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

where

\mathbf{U} = \begin{bmatrix} \rho \\ u \\ v \\ p \\ \end{bmatrix} \ , \ \mathbf{F} = \begin{bmatrix} \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 ρ, u, v and p are the acoustic field variables, γ the reason of specific heats cp / cv, for air at 20°C cp / cv = 1.4, and the source term \mathbf{S} on the right-side represents distributed unsteady sources.

For high Mach number flows in compressible regimes, the acoustic propagation may be influenced by non-linearities and the LEE may no longer be the appropriate mathematical model.

[edit] EIF

Expansion about Incompressible Flow

[edit] APE

Acoustic Perturbation Equations

[edit] References

  • Lighthill, M. J., "On Sound Generated Aerodynamically, i", Proc. Roy. Soc. A, Vol. 211, 1952, pp 564-587
  • Lighthill, M. J., "On Sound Generated Aerodynamically, ii", Proc. Roy. Soc. A, Vol. 222, 1954, pp 1-32
  • Lighthill, M. J., "A General Introduction to Aeroacoustics and Atmospheric Sounds", ICASE Report 92-52, NASA Langley Research Centre, Hampton, VA, 1992
  • Ffowcs Williams, "The Noise from Turbulence Convected at High Speed", Philosophical Transactions of the Royal Society, Vol. A255, 1963, pp. 496-503
  • Ffowcs Williams, J. E., and Hawkings, D. L., "Sound Generated by Turbulence and Surfaces in Arbitrary Motion", Philosophical Transactions of the Royal Society, Vol. A264, 1969, pp. 321-342
  • C. K. W. Tam, and J. C. Webb, "Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics", Journal of Computational Physics, Vol. 107, 1993, pp. 262-281

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