# Engineering Acoustics/Analogies in aeroacoustics

## Acoustic Analogies

!Test facility and testing of supersonic jet engine to assess noise
emissions at NASA Langley Research
Center
A direct way to predict Aerodynamic noise would be to solve the
Navier-Stokes Equations in a general three-dimensional unsteady
compressible numerical simulations. Unfortunately, it is hardly
achievable except for very simple academic configurations within a
delimited region. The idea of an acoustic analogy is to restate the full
equations of gas dynamics providing an equivalent wave equation in
homogeneous medium in uniform motion from the point of view of a distant
observer. This condition leads to simplifications of usual linear
acoustics problems\[1\]. The most systematically used formalisms are
Lighthill\'s analogy and the extensions made by Curle and Ffowcs
Williams & Hawkings, because they offer of wide range of applicability.

It must be started clearly that the aim of an analogy is not essentially
to deduce exacts results or numerical coefficients, but to infer general
laws from the standard procedures associated with the classical wave
equation. A preliminary knowledge of the main flow features coming from
either experiments, CFD or analytical methods is needed to apply these
analogies. Moreover, the degree of accuracy in the flow variables to
extract acoustic results is crucial to ensure the relevance of the
prediction method.

## Governing gas dynamics equations to Lighthill\'s equation

References \[2\] \[3\], development below originates from Aeroacoustics
wikipage.

$$\frac{\partial \rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{v}\right)=\frac{D\rho}{D t} + \rho\nabla\cdot\mathbf{v}= 0,$$\...\...\...\...\...\...\...\.....*conservation
of mass equation (E1)*

where $\rho$ and $\mathbf{v}$ represent the density and velocity of the
fluid, which depend on space and time, and $D/Dt$ is the substantial
derivative.

Next is the conservation of momentum equation, which is given by

$${\rho}\frac{\partial \mathbf{v}}{\partial t}+{\rho(\mathbf{v}\cdot\nabla)\mathbf{v}} = -\nabla p+\nabla\cdot\sigma,$$\...\...\...\...\...\...\...\.....*conservation
of momentum equation (E2)*

where $p$ is the thermodynamic pressure, and $\sigma$ is the viscous (or
traceless) part of the Cauchy stress tensor from the Navier--Stokes
equations.

_Step 1:_ Multiplying (E1) by $\mathbf{v}$ and
adding it to (E2) yields

$$\frac{\partial}{\partial t}\left(\rho\mathbf{v}\right) + \nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}) = -\nabla p + \nabla\cdot\sigma.$$

_Step 2:_ Differentiating (E1) with respect to
time, taking the divergence of (E2) and subtracting the latter from the
former, we get

$$\frac{\partial^2\rho}{\partial t^2} - \nabla^2 p + \nabla\cdot\nabla\cdot\sigma = \nabla\cdot\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}).$$

_Step 3:_ Subtracting $c_0^2\nabla^2\rho$, where
$c_0$ is the speed of sound in the medium in its equilibrium (or
quiescent) state, from both sides of the last equation and rearranging
it results in:

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho = \nabla\cdot\left[\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v})-\nabla\cdot\sigma +\nabla p-c^2_0\nabla\rho\right],$$

which is equivalent to

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho=(\nabla\otimes\nabla) :\left[\rho\mathbf{v}\otimes\mathbf{v} - \sigma + (p-c^2_0\rho)\mathbb{I}\right],$$
where $\mathbb{I}$ is the identity tensor, and $:$ denotes the (double)
tensor contraction operator.

Using *Einstein notation*, Lighthill's equation can be written as

$$\frac{\partial^2\rho}{\partial t^2}-c^2_0\nabla^2\rho=\frac{\partial^2\sigma_{ij}}{\partial x_i \partial x_j},\quad$$

where $\sigma_{ij}$ is the so-called Lighthill\'s stress tensor
$\sigma_{ij} = \rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} - \tau_{ij}$,
further details are provided in the next section. with: $\rho V_i V_j$
is the
inertial
stress tensor of Reynolds $\tau_{ij}$ is the viscous stress tensor of
Reynolds $(P-c_0^2 \rho)$ represents all effects due to entropy
non-homogeneities (important for hot jets with high temperature
gradients)

## Lighthill\'s Acoustic Analogy

Aeroacoustic engineers need to predict noise arising with turboengines
id est localized unsteady flow in a propagating medium, at a first step
of approximation. The basic idea of Lighthill (1952) is to reformulate
the general equations of gas dynamics, in order to derive a wave
equation. The featured variable is the density fluctuation, naturally
preferred because acoustic waves in a gas are due to compressibility. No
special assumption is made nor linearisation introduced.

$$\ \frac{\partial^2 \rho}{\partial t^2} - \frac{\partial^2 (\rho V_i V_j)}{\partial x_i \partial x_j} = - \frac{\partial^2 \sigma_{ij}}{\partial x_i \partial x_j}$$\...\...\...\...\...\...\...\...\...\....*Lighthill\'s
equation*

with
$$\ \sigma_{ij} = \rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} -\tau_{ij}$$
called *Lighthill\'s tensor*

This equation remains true if the same quantity is added on both sides.
Add the $$\ c_0^2 \frac{\partial^2 \rho}{\partial^2 x_j^2}$$ with
$\ c_0$ being the characteristic speed of sound of the undisturbed gas
(precisely the speed of sound in the medium surrounding the flow region,
in the applications; this is different from the local speed of sound
$\ c^2$ in the flow). Then, forming a wave operator on the left-hand
side and removing all other terms on the right-hand side leads to:

$$\ \frac{\partial^2 \rho}{\partial t^2} - c_0^2 \frac{\partial^2 \rho}{\partial x_j^2} = \frac{\partial^2}{\partial x_i \partial x_j} [\rho V_i V_j + (P-c_0^2 \rho)\delta_{ij} -\tau_{ij} ]$$

This result is the well-known Lighthill\'s equation. When applied to a
true problem of acoustics, it reduces to the homogeneous wave equation
at large distances from the flow, since all terms in the right-hand side
can be considered negligible (according to the reasonable assumptions
related to the propagation of acoustic waves, as a small-amplitude,
isentropic motion\[4\]). An alternative form of Lighthill\'s equation
can be written with the pressure instead of the density, as follows:

$$\ \frac{1}{c_0^2} \frac{\partial^2 P}{\partial t^2} - \frac{\partial^2 P}{\partial x_j^2} = \frac{\partial^2}{\partial x_i \partial x_j} (\rho V_i V_j  + \frac{1}{c_0^2} \frac{\partial^2}{\partial t^2}(P-c_0^2 \rho-\tau_{ij}))$$\...\...\...\...\...\...\...*LightHill
equation on the fluctuating pressure*

Maybe this one is less common because density fluctuations are directly
related to compressible effects, whereas pressure fluctuations can exist
to compensate inertial accelerations in the fluid. When temperature
non-homogeneities are involved, however, the fluctuating pressure is
well suited.

Explanation of each term of Lighthill\'s tensor:

Aeoracoustic sources can be separate in three distinct categories,

!Monopole acoustic
source{width="100"}

-   Monopole sources: $\frac{\partial^2}{\partial t^2}(P)$ Spherical
    sources or discrete source delivering flow rate $Q(t)$ fluctuating
    over the time. Can appear only if solid surfaces are encountered.

!Dipole acoustic
source{width="100"}

-   Dipole sources: $-\frac{\partial^2}{\partial t^2}(c_0^2 \rho)$
    Similar to two monopoles placed side by side with opposition phase
    $+Q(t)$ and $-Q(t)$. Dipoles are associated to the force $F(t)$
    according to the axe created by our joined monopoles. Like Monopoles
    they only appear when solid surfaces are involved in the domain.

!Quadrupole acoustic
source{width="100"}

-   Quadrupole sources:
    $\frac{\partial^2}{\partial x_i \partial x_j} (\rho V_i V_j)$
    Constituted of two dipoles side by side in phase opposition, these
    sources cames from turbulent vortices and are usually neglected at
    low velocity flow. They originate from shear-terms in Navier-Stokes
    equations.

As a consequence of the general equations of gas dynamics, Lighthill\'s
equation is exact. All aeroacoustic processes, including generation of
sound by the flow non-homogeneities, sound propagation through the flow
and sound dissipation by viscosity or heat conduction, are accounted
for. Hence, this equation is not tractable as a pure wave equation using
linear acoustics since the right-hand side contains the acoustic field
to be determined and cannot be considered as a true source term. Thus we
need to approximate this term independently of the acoustic variables
which corresponds to neglect some of the mechanisms. To remove this
fundamental difficulty, Lighthill proposed some simplifications
motivated by the thinking that **sound generation by the mixing of fluid
is the dominant mechanism** , especially at the high Reynolds numbers of
interest in aeronautics. This is equivalent to privilege the mechanical
effects related to fluid inertia and discard thermodynamic effects as
secondary ones.

## Lighthill\'s Approximation

Lighthill\'s equation is well posed because it assumes that sources are
contained within the flow and not outside of it. I reduces to the
homogeneous wave equation in the propagation region. But in order to
solve near the sources approximations using comparison between phenomena
has to be done.

Lighthill\'s analogy is often use to calculate jet noise, is this type
of application we have specific conditions leading to simplifications
\[5\]:

!Explain classical approximation made using Lighthill\'s
approximation{width="600"}

Practical approximation used in industrial context:

!Explain classical approximation made using Lighthill\'s
approximation{width="500"}

The approximation makes the equation explicit in the sense of the wave
equation in linear acoustics, to be solved formally by the Green\'s
function technique. When numerical means are used to described the flow,
some assumptions can be removed for a more accurate evaluation and the
equation used to post-process the flow data.

!Description of Pseudo-sound phenomenon and influence on near-field and
far-field assumptions to calculate sound
pressure{width="600"}

If entropic non-homogeneities dominate in a disturbed flow, sources
appears as equivalent monopoles:

Using Ribner\'s splitting and uncompressible fluid laws:

:   ```{=html}
    <center>
    ```
    $\ \Delta p'_t = -\rho_0 \frac{\partial^2(U_i U_j)} {\partial x_i \partial x_j} = -\rho_0 \frac{\partial U_i} {\partial x_j} \frac{\partial U_j} {\partial x_i}$

    ```{=html}
    </center>
    ```

Leading to the final approximate Lighthill equation:

:   ```{=html}
    <center>
    ```
    $\ \Delta p'_a - \frac{1}{c_0^2} \frac{\partial^2p'_a}{\partial t^2} = \frac{1}{c_0^2} \frac{\partial^2p'_t}{\partial t^2}$
    \...\...\...\...\...\...\...*approximate Lighthill equation*

    ```{=html}
    </center>
    ```

## Ffowcs Williams & Hawkings formulation

### History

In 1969, Ffowcs Williams & Hawkings, are the first scientists to express
a fundamental equation to predict noise generate by blades in a flow
\[6\].

### Ffowcs Williams & Hawkings general equation

(They are several formulations of FWH analogy, here is one adapted to
blade noise description phenomena)

Deriving from Lighthill\'s equation, it reveal that fluctuating pressure
generating acoustic sources in a rotor is solution of a specific
inhomogeneous wave equation:

$\frac{\partial^2 p(z,t_z)}{\partial z_i^2} - \frac{1}{c_0^2} \frac{\partial^2 p(z,t_z)}{\partial t_z^2} = S(z,t_z)$

where, $z$ the vector coordinates from a source point and $t_z$ the time
in sources domain.

The source term $S(z,t_z)$ can be written as the following sum:

$S(z,t_z) = Q(z,t_z) + \frac{\partial F_i(z,t_z)}{\partial z_i} + \frac{\partial^2 T_{ij}(z,t_z)}{\partial z_i \partial z_j}$

-   First term $Q(z,t_z)$ represents *thickness noise* generated by
    volume displacement of fluid. A fan blade has its thickness and
    volume. As the rotor rotates, the volume of each blade displaces
    fluid volume, then they consequently fluctuate pressure of near
    field, and noise is generated. This noise is tonal at the running
    frequency and generally very weak for cooling fans, because their
    RPM is relatively low. Therefore, thickness of fan blades hardly
    affects to electronic cooling fan noise. (This kind of noise can
    become severe for high speed turbomachines like helicopter rotors.)
-   Second term is called *loading noise*, comes from fluctuation of the
    force-field $F_i(z,t_z)$ on moving surfaces. In a rotor, It
    originates from nonstationary aerodynamic forces between the fluid
    and the blades. In computational models, this term is represented by
    surface-distributed dipoles. (Dominant for fans)
-   Final term is *shear noise*, which is composed of quadrupoles on the
    surface of the blades \[7\].

The Ffowcs Williams & Hawkings theory allows to solve this equation
knowing the source terms in Green\'s functions.

### Ffowcs Williams & Hawkings extended analogy using permeable control surface

In all applications where the quadrupole term is significant and must be
calculated, which preferentially occurs at high speeds, the computations
can be made cumbersome because the sources are distributed inside a
volume, the boundaries of which are not precisely defined. In contrast,
the surface source terms are much simpler to compute and clearly
delimited. If CFD must be used in a limited domain surrounding the
surfaces, and if the computations are able to reproduce the acoustic
near-field, a more convenient way of solving the acoustic problem can be
proposed by taking the information not on the physical surfaces but on a
delocalised control surface that can be user-defined \[8\]. Double layer
potential description can be apply, to solve the Helmotz equation using
Bessel functions assumption. This generalised form of Ffowcs Williams &
Hawkings' analogy is widely used in recent Computational Aero-Acoustics
(CAA).

!Aeroacoustic computational simulations using FWH analogy and permeable
surface around aerodynamic
profile{width="650"}

N.B: The CFD Domain inside the control surface must be implemented until
turbulence systems generated are fully developed (using for example
k-$\epsilon$criterion).

The formal advantage of an analogy is to state a problem of
aeroacoustics as a usual problem of linear acoustics, by defining
equivalent sources that would produce in a uniform medium the same sound
as what is heard at observer's location from the flow-and-surface
configuration. The difficulty of the initial gas-dynamics equations is
transposed to the description of the source terms. The formal solution
is derived using the theoretical background of linear acoustics but it
may be useless if the equivalent source terms are not determined
elsewhere. Using Lighthill and FWH analogies allow engineers to
calculate aircraft engines noise for example at a lower time-calculation
cost.

Intensive research at NASA and CERFACS facilities try to develop more
efficient calculation schemes to provide improved design tools to
motorists engineer to gain confidence in noise prediction and develop
aero-mechanical-acoustic design for future generation products.

Making approximations means discarding phenomena that are expected
negligible and retaining the dominant features. This is just proposing
an interpretation.

## References

\[1\]Goldstein, M. E. (1976). Aeroacoustics. New York, McGraw-Hill
International Book Co., 1976. 305 p., 1.

\[2\]Tam, C. K. (1995). Computational aeroacoustics-Issues and methods.
AIAA journal, 33(10), 1788-1796.

\[3\]Wang, M., Freund, J. B., & Lele, S. K. (2006). Computational
prediction of flow-generated sound. Annu. Rev. Fluid Mech., 38, 483-512.

\[4\]Colonius, T., Lele, S. K., & Moin, P. (1993). Boundary conditions
for direct computation of aerodynamic sound generation. AIAA journal,
31(9), 1574-1582.

\[5\] Williams, J. F. (1969). Hydrodynamic noise. Annual Review of Fluid
Mechanics, 1(1), 197-222.

\[6\] Williams, J. F., & Hawkings, D. L. (1969). Sound generation by
turbulence and surfaces in arbitrary motion. Philosophical Transactions
of the Royal Society of London. Series A, Mathematical and Physical
Sciences, 264(1151), 321-342.

\[7\] Ianniello, S. (1999). Quadrupole noise predictions through the
Ffowcs Williams-Hawkings equation. AIAA journal, 37(9), 1048-1054.

\[8\]Di Francescantonio, P. (1997). A new boundary integral formulation
for the prediction of sound radiation. Journal of Sound and Vibration,
202(4), 491-509. 
