Like the Ffowcs Williams and Hawkings (FW-H) integral model, this model based on the finite volume solver of the wave equation, obtains information about the sound sources from the calculation of a background flow, which generates sound. Different from the FW-H model, which calculates sound signals at selected receiver locations, the wave equation model can deliver a sound field in the entire simulation domain. In this sense, this model is similar to the direct simulation approach (see Direct Method ). However, unlike the direct simulation, the wave equation model allows you to see a distribution of the sound pressure separately from the flow pressure.
The following limitations apply:
The background flow must be a single phase at low Mach number without significant variations of density or speed of sound.
The effect of convection is neglected by the wave equation.
The 2-D axisymmetric version of Fluent is not supported.
The transient scheme must be the implicit scheme with either the second order or bounded second order enabled (the explicit scheme and the first order implicit time scheme for the flow solver are not supported).
The domain and mesh must be stationary. Mesh and frame motion is not supported.
Overset meshes are not supported.
The User Defined Function of the type
DEFINE_INITDEFINE_INITcannot be used to initialize the wave equation model. Please use instead theDEFINE_ON_DEMANDfunctionDEFINE_ON_DEMAND.
Written for the sound potential,
, the wave equation has the following form:
 | (11–11) |
Where
is the local instant value of the flow static pressure,
is the constant fluid density, and
is the constant speed of sound.
At low Mach numbers, the convective transport of sound may be neglected, so the sound
pressure
is computed from the sound potential using a simplified relation:
 | (11–12) |
The following boundary condition types for the wave equation are currently available in Fluent:
Walls: all walls are considered acoustically hard, ideally reflecting sound.
Permeable boundaries (inlets, outlets, far-field boundaries): these boundaries are handled using non-reflective conditions.
Symmetry and periodic boundaries.
Impedance boundary conditions are not yet available for the wave equation model.
Non-conformal stationary mesh interfaces can be used with the acoustics models. However, local wall zones, which may be created on such interfaces to handle significant geometrical misalignments, reflect sound and may strongly influence the acoustics solution. Therefore it is recommended to avoid such non-physical walls, unless they are covered by a sponge region.
The Laplacian term in Equation 11–11 is discretized in space using the same finite volume procedure that is applied in Fluent for the diffusion terms of the flow equations.
The solution is propagated in time using the
-method by Hilber, Hughes, and Taylor, which is also used for transient
analysis by Mechanical APDL and is documented in the Ansys Mechanical APDL Theory Reference,
section 15.2.2.1: Time Integration Scheme for Linear Systems. This method is an advanced
variant of the popular algorithm by Newmark, which is used for numerical integration of
evolutionary equations with the leading transient term represented by a second order time
derivative, like the wave equation in
Equation 11–11
. It combines
second order accuracy with good stability.
The following sections detail some strategies and considerations to prevent non-physical reflections of sound waves, which have a large influence on solution accuracy.
The numerical solution of the wave equation is known to be very sensitive to any non-uniformities in the mesh that have a size of the order of the sound wave length or larger. Preventing such non-uniformities is commonly recommended and it is easier to estimate the size of a dangerous non-uniformity in the mesh when performing a harmonic analysis of acoustics in the frequency domain. For the time-domain aeroacoustics simulation, estimation of the mesh quality criteria is not as easy because of the broad-band nature of the vortex sound in a turbulent flow. A scale-resolving simulation normally resolves fluctuations down to the mesh cell size, and it may not be obvious how to estimate the dangerous size of mesh non-uniformities. Results of aeroacoustics simulations in the time domain are often less accurate in the high-frequency part of spectrum due to the reflection and dispersion of short waves at such non-uniformities. These inaccuracies can be reduced by the use of the numerical procedures described below in this section. Still, the mesh quality requirements for aeroacoustics simulations are higher than those for incompressible flow simulations in similar configurations. It is recommended to estimate in advance the frequency range of interest, and allocate 12 or more cells per wavelength within this range.
Two kinds of filters are applied to smooth the source term of Equation 11–11 in space and time: a second order explicit linear filter in space suppresses odd-even spatial oscillations, and a high-order explicit filter in time damps the high frequency part of the source spectrum. The space filter is always active, and can be switched off only by changing an rp-variable. As for the time filter, there is a graphical and text user interface to activate or de-activate it. It is recommended to keep the time filter active, which is the default setting.
The acoustics wave equation model should be activated after the time-dependent flow solution has been well established. This means a flow simulation needs to be stopped and then continued with the wave equation solver switched on. However, a sudden activation of the sound source term in the wave equation can create non-physical solution disturbances of high magnitude, which may remain in the domain for a very long time. To avoid this, a slow ramping of the source is applied in the wave equation solver, with the duration specified by the user in terms of number of timesteps. A common rule to estimate this duration is to make it equal to one or few periods of the low frequency sound which is significant and intended to be well-resolved. During the ramping period, the source term is pre-multiplied by a factor, which grows smoothly in time from zero to one.
Another measure to improve the solution quality is limiting (masking) the sound source region. It serves two purposes:
allows only the desired part of the turbulent flow region to generate sound, where the solution is known to be well resolved.
smoothly reduce the sound sources at the borders of the masking region to further improve the solution quality. This is achieved by a smooth transition of the masking marker between zero and one through the user-specified transition thickness. A value of zero means that the model source term is locally active, and the value of one means that the source is completely masked out.
Application of a sponge region helps to avoid any non-physical sound reflection in the following situations:
At open boundaries, there may still be some limited reflection despite the use of a non-reflective boundary condition, if the sound wave propagation direction is not normal to the boundary surface.
In large domains, mesh cell size expansion in the directions away from sound sources is often necessary. Reflections from expanding mesh layers, and especially from internal mesh interfaces, may negatively affect the acoustics solution quality. It is recommended you avoid such mesh expansions and interfaces, which create overly coarse meshes within the acoustics region of interest for important sound frequencies. Outside this region, mesh irregularities should be covered by a sponge region. Similar to the source masking region, the border of the sponge region is smeared with a user-specified transition thickness.
The damping of the wave equation solution is realized by adding an artificial
viscous term to
Equation 11–11
, which has the same form as a
physical viscous term. The artificial viscosity value is selected depending on the local
mesh resolution. Inside the user-specified sponge region, a viscous term with the
default non-dimensional artificial viscosity coefficient,
, is optimized to most efficiently damp the shortest resolved waves
with the wavelength equal to two cell sizes. If there is a need for stronger damping of
longer waves, you can increase the artificial viscosity using the two parameters,
and
:
 | (11–13) |
where
is the location-dependent sponge layer marker, which varies between
zero and one.
The parameter
is a base level for the artificial viscosity, which is used to smooth
the solution everywhere in the acoustics region, meaning regions outside of the sponge
region are also damped by this base level. The default values for the damping parameters
are
= 1 and
= 0.1.
The acoustics wave equation model in Fluent is mainly intended to simulate sound propagation in the mid-field range. This range covers sound sources and the non-uniform flow field, where also sound-reflecting solid obstacles may exist. In many practical applications, there is a need to calculate the perceived sound signals at remote target locations much further away. Propagation of sound through open space, where fluid is either stagnant or moves uniformly, can be calculated using the Kirchhoff surface integral.
The Kirchhoff integral model is a complementary model to the wave equation model and can be applied whenever the wave equation model is used. The current wave equation model can be applied to transient single-phase flow simulations, when the background flow does not cause density changes (constant density fluid model is recommended).
Limitations of the Kirchhoff model, which correspond to the limitations of the wave equation model in Fluent, are:
Effect of convection is not included, meaning that the far field fluid must be stagnant.
Motion of the integration surface (in particular its rotation) is not supported.
Additional limitations of the current implementation of the Kirchhoff model are:
Only a 3-D variant is implemented.
Only a single surface can be used for the integration. A complex surface can be built as an iso-surface of a correspondingly defined scalar field.
Periodicity is not supported.
Moving receivers are not supported.
Model is available only for the “on-the-fly” mode, which means that every next set of signal samples is computed during a transient simulation after each time step. Currently there is no “read-and-compute” mode such as for the Ffowcs Williams and Hawkings model, where the transient data are exported during the simulation, and Fluent performs a single post-processing action after the simulation end to compute the sound propagation to remote receivers.
Due to the linearity of the wave equation model and its integral solution by Kirchhoff, the surface integral may be written for any acoustics field variable. Since the currently implemented wave equation model uses the acoustic potential as the primary variable, the Kirchhoff integral is directly applied to potential:
 | (11–14) |
Where
and
are the receiver coordinates and the reception time,
is the emission (retarded) time,
is the distance from a location on the integration surface to the
receiver,
is the unit vector pointing to the receiver,
is the outward surface unit normal vector, and
is the speed of sound. Each elementary facet of the integration
surface contributes to the integral at its own retarded time
. Integration is performed in
Equation 11–14
over a user-specified surface, which theoretically must be closed. In practice, however,
non-closed integration surfaces are applied in situations, where the Kirchhoff sound
source (a function under the integral in
Equation 11–14
) is
either not known or may be neglected on a portion of a surface. One example is a long
cylindrical surface without the end faces, which is used to compute sound, generated by
a long cylinder in a cross flow. Another example is a spherical surface placed around
the outlet of a duct or pipe in order to calculate sound, radiated from the orifice. In
the latter case a part of a sphere, which is covered by a duct or pipe, is omitted in
the integral.
After the acoustic potential samples are accumulated for the intended duration at
all receivers, the sound pressure signal
is calculated at each receiver as:
 | (11–15) |