Acoustic analogy
Encyclopedia
Acoustic analogies are applied mostly in numerical aeroacoustics
to reduce aeroacoustic sound sources to simple emitter types. They are therefore often also referred to as aeroacoustic analogies.
In general, aeroacoustic analogies
are derived from the compressible Navier-Stokes equations
(NSE). The compressible NSE are rearranged into various forms of the inhomogeneous acoustic wave equation
. Within these equations, source terms describe the acoustic sources. They consist of pressure and speed fluctuation as well as stress
tensor
and force terms.
Approximations are introduced to make the source terms independent of the acoustic variables
. In this way, linearized equations are derive which describe the propagation of the acoustic waves in a homogeneous, resting medium. The latter is excited by the acoustic source terms, which are determined from the turbulent fluctuations. Since the aeroacoustics are described by the equations of classical acoustics, the methods are called aeroacoustic analogies.
The Lighthill-Analogy considers a free flow, as for example with an engine jet. The nonstationary fluctuations of the stream are represented by a distribution of quadrupole
sources in the same volume.
The Curle-Analogy is a formal solution of the Lighthill Analogy, which takes hard surfaces into consideration.
The Ffowcs Williams
-Hawkings-Analogy is valid for aeroacoustic sources in relative motion with respect to a hard surface, as is the case in many technical applications for example in the automotive industry or in air travel. The calculation involves quadrupole
, dipole
and monopole
terms.
Computational Aeroacoustics
While the discipline of Aeroacoustics is definitely dated back to the first publication of Sir James Lighthill in the early 1950s, the origin of Computational Aeroacoustics can only very likely be dated back to the middle of the 1980s...
to reduce aeroacoustic sound sources to simple emitter types. They are therefore often also referred to as aeroacoustic analogies.
In general, aeroacoustic analogies
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
are derived from the compressible Navier-Stokes equations
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
(NSE). The compressible NSE are rearranged into various forms of the inhomogeneous acoustic wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
. Within these equations, source terms describe the acoustic sources. They consist of pressure and speed fluctuation as well as stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
and force terms.
Approximations are introduced to make the source terms independent of the acoustic variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. In this way, linearized equations are derive which describe the propagation of the acoustic waves in a homogeneous, resting medium. The latter is excited by the acoustic source terms, which are determined from the turbulent fluctuations. Since the aeroacoustics are described by the equations of classical acoustics, the methods are called aeroacoustic analogies.
The Lighthill-Analogy considers a free flow, as for example with an engine jet. The nonstationary fluctuations of the stream are represented by a distribution of quadrupole
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
sources in the same volume.
The Curle-Analogy is a formal solution of the Lighthill Analogy, which takes hard surfaces into consideration.
The Ffowcs Williams
John Ffowcs Williams
Professor John Eirwyn Ffowcs Williams BSc PhD Hon DSc Soton MA ScD Cantab FREng is Emeritus Rank Professor of Engineering at the University of Cambridge and a former Master of Emmanuel College, Cambridge ....
-Hawkings-Analogy is valid for aeroacoustic sources in relative motion with respect to a hard surface, as is the case in many technical applications for example in the automotive industry or in air travel. The calculation involves quadrupole
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
, dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
and monopole
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
terms.
Literature
- Blumrich, R.: Berechnungsmethoden für die Aeroakustik von Fahrzeugen. Tagungsband der ATZ/MTZ-Konferenz Akustik 2006, Stuttgart, 17-18.5.2006..
- Lighthill, M. J.: On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. 211(A) S. 564-587, London, 1952.
- Lighthill, M. J.: On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. 222(A) S. 1-34, London, 1954.
- Ffowcs Williams, J. E. and Hawkings, D. L.: Sound Generation by Turbulence and Surfaces in Arbitrary Motion. Philosophical Transactions of the Royal Society, Vol. A264, Nr. 1151, 1969, S. 321-342
- Curle, N.: The Influence of solid boundaries upon aerodynamic sound. Proc. Roy. Soc. Lond., A23, 505-514. 1955