Aeroacoustics is a branch of
acousticsAcoustics is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound . A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering...
that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. Although no complete scientific theory of the generation of noise by aerodynamic flows has been established, most practical aeroacoustic analysis relies upon the so-called
Acoustic AnalogyAcoustic 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....
, whereby the governing equations of motion of the fluid are coerced into a form reminiscent of the
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
of "classical" (i.e. linear) acoustics.
Aeroacoustics is that branch of one,which deals with the study of noise generation through,either turbulent fluid motion or aerodynamic forces interacting with surfcaces.
The most common and a widely-used of the latter is
Lighthill's aeroacoustic analogy. It was proposed by
James LighthillSir Michael James Lighthill, FRS was a British applied mathematician, known for his pioneering work in the field of aeroacoustics.-Biography:...
in the 1950s when noise generation associated with the
jet engineA jet engine is a reaction engine that discharges a fast moving jet of fluid to generate thrust in accordance with Newton's laws of motion. This broad definition of jet engines includes turbojets, turbofans, rockets, ramjets, pulse jets and pump-jets...
was beginning to be placed under scientific scrutiny.
Computational AeroacousticsWhile 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...
(CAA) is the application of numerical methods and computers to find approximate solutions of the governing equations for specific (and likely complicated) aeroacoustic problems.
Lighthill's equation
Lighthill rearranged the Navier–Stokes equations, which govern the
flow-Relating to the movement of material:*Environmental flow, the amount of water necessary in a watercourse to maintain a healthy ecosystem*Flow chemistry, a chemical reaction run in a continuous stream...
of a compressible viscous
fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
, into an inhomogeneous
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
, thereby making a connection between
fluid mechanicsFluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the...
and
acousticsAcoustics is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound . A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering...
. This is often called "Lighthill's analogy" because it presents a model for the acoustic field that is not, strictly speaking, based on the physics of flow-induced/generated noise, but rather on the analogy of how they might be represented through the governing equations of a compressible fluid.
The first equation of interest is the
conservation of massThe law of conservation of mass/matter, also known as principle of mass/matter conservation is that the mass of a closed system will remain constant over time, regardless of the processes acting inside the system. A similar statement is that mass cannot be created/destroyed, although it may be...
equation, which reads
where and represent the density and velocity of the fluid, which depend on space and time, and is the substantial derivative.
Next is the conservation of momentum equation, which is given by
where is the thermodynamic
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, and is the viscous (or traceless) part of the stress tensor.
Now, multiplying the conservation of mass equation by and adding it to the conservation of momentum equation gives
Note that is a
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
(see also
tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this...
). Differentiating the conservation of mass equation with respect to time, taking the
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of the conservation of momentum equation and subtracting the latter from the former, we arrive at
Subtracting , where is the
speed of soundSound is a vibration that travels through an elastic medium as a wave. The speed of sound describes how far this wave travels in a given amount of time. In dry air at , the speed of sound is . This equates to , or about one mile in five seconds...
in the medium in its equilibrium (or quiescent) state, from both sides of the last equation and rearranging it results in
which is equivalent to
where is the
identityIn linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
tensor, and denotes the (double)
tensor contractionIn multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...
operator.
The above equation is the celebrated Lighthill equation of aeroacoustics. It is a
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
with a source term on the right-hand side, i.e. an inhomogeneous wave equation. The argument of the "double-divergence operator" on the right-hand side of last equation, i.e. , is the so-called
Lighthill turbulence stress tensor for the acoustic field, and it is commonly denoted by .
Using
Einstein notationIn mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas...
, Lighthill’s equation can be written as
where
and is the
Kronecker deltaIn mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...
. Each of the acoustic source terms, i.e. terms in , may play a significant role in the generation of noise depending upon flow conditions considered. describes unsteady convection of flow (or Reynold's Stress), describes sound generated by shear, and describes non-linear acoustic generation processes.
In practice, it is customary to neglect the effects
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
of the fluid, i.e. one takes , because it is generally accepted that the effects of the latter on noise generation, in most situations, are orders of magnitude smaller than those due to the other terms. Lighthill provides an in-depth discussion of this matter.
In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present.
Finally, it is important to realize that Lighthill's equation is
exact in the sense that no approximations of any kind have been made in its derivation.
Related model equations
In their classical text on
fluid mechanicsFluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the...
, Landau and Lifshitz derive an aeroacoustic equation analogous to Lighthill's (i.e., an equation for sound generated by "turbulent" fluid motion) but for the
incompressible flowIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some...
of an inviscid fluid. The inhomogeneous wave equation that they obtain is for the
pressure rather than for the density of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation is
not exact; it is an approximation.
If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid is incompressible) to obtain an approximation to Lighthill's equation is to assume that , where and are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into we obtain the equation
And for the case when the fluid is indeed incompressible, i.e. (for some positive constant ) everywhere, then we obtain exactly the equation given in Landau and Lifshitz , namely
A similar approximation [in the context of equation ], namely , is suggested by Lighthill [see Eq. (7) in the latter paper].
Of course, one might wonder whether we are justified in assuming that . The answer is in affirmative, if the flow satisfies certain basic assumptions. In particular, if and , then the assumed relation follows directly from the
linear theory of sound waves (see, e.g., the linearized Euler equations and the
acoustic wave equationIn physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure or particle velocity as a function of space and time...
). In fact, the approximate relation between and that we assumed is just a
linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
to the generic
barotropicIn meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...
equation of stateIn physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
of the fluid.
However, even after the above deliberations, it is still not clear whether one is justified in using an inherently
linear relation to simplify a
nonlinear wave equation. Nevertheless, it is a very common practice in
nonlinear acousticsThis article is about sound waves being distorted as they travel.Non-linear acoustics is that branch of physics,which deals with the study of sound waves being distorted as they travel.-Introduction:...
as the textbooks on the subject show: e.g., Naugolnykh and Ostrovsky and Hamilton and Morfey.
External links
- M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory," Proc. R. Soc. Lond. A 211 (1952) pp. 564-587. This article on JSTOR.
- M. J. Lighthill, "On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound," Proc. R. Soc. Lond. A 222 (1954) pp. 1-32. This article on JSTOR.
- L. D. Landau and E. M. Lifshitz, Fluid Mechanics 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. ISBN 0750627670, Preview from Amazon.
- K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1. ISBN 052139984X, Preview from Google.
- M. F. Hamilton and C. L. Morfey, "Model Equations," Nonlinear Acoustics, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3. ISBN 0123218608, Preview from Google.
- Aeroacoustics at the University of Mississippi
- Aeroacoustics at the University of Leuven
- International Journal of Aeroacoustics
- Examples in Aeroacoustics from NASA