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METHODS


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.


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 sorces are provided to the second solver which calculates the acoustical propagation. Acoustic propagation can be calculated using one of the following methods :
# Integral Methods
## Lighthill's analogy
## Kirchhoff integral
## FW-H
# LEE
# EIF
# APE


Lighthill's analogy

Also called ' Acoustic Analogy ', the governing equations 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


Kirchhoff integral

See Gustav Kirchhoff


FW-H

See John Ffowcs Williams


Linearized Euler Equations


Considering small disturbances superimposed on a uniform mean flow of density ho_0, pressure p_0 and velocity on x-axis u_0, the Euler equations for a two dimensional model is presented as:

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

where

: \mathbf{U} =
\begin{bmatrix}
ho \
u \
v \
p \
\end{bmatrix} \ , \ \mathbf{F} =
\begin{bmatrix}
ho_0 u + ho u_0\
u_0 u + p/ ho_0 \
u_0 v \
u_0 p + \gamma p_0 u \
\end{bmatrix} \ , \ \mathbf{G} =
\begin{bmatrix}
ho_0 v\
0 \
p/ ho_0 \
\gamma p_0 v \
\end{bmatrix},


where ho, u, v and p are the acoustic field variables, \gamma the reason of specific heats c_p/c_v, for air at 20°C c_p/c_v = 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.


EIF

Expansion about Incompressible Flow


APE

Acoustic Perturbation Equations


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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