The Thermoviscous Acoustics, Frequency Domain (ta) interface (
), found under the
Thermoviscous Acoustics branch (
) when adding a physics interface, is used to compute the acoustic variations of pressure, velocity, and temperature. This physics interface is required to accurately model acoustics in geometries of small dimensions, often referred to as
microacoustics. Near walls, viscous losses and thermal conduction become important because a boundary layers exists. The thicknesses of these boundary layers are also known as the viscous and thermal penetration depth. For this reason, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations. It is, for example, used when modeling the response of transducers like microphones, miniature loudspeakers, and receivers. Other applications include analyzing feedback in hearing aids and in mobile devices, or studying the damped vibrations of MEMS structures.
The equations defined by the Thermoviscous Acoustics, Frequency Domain interface are the linearized Navier–Stokes equations in quiescent background conditions solving the continuity, momentum, and energy equations (first order perturbation equations). Thermoviscous acoustics is also known as
viscothermal acoustics or sometimes
thermoacoustics (not to be confused with the field discussing heating and cooling using acoustics). Due to the detailed description necessary when modeling thermoviscous acoustics, the model simultaneously solves for the acoustic pressure
p, the acoustic velocity variation
u (particle velocity), and the acoustic temperature variations
T. It is available for 3D, 2D, and 1D Cartesian geometries as well as for 2D and 1D axisymmetric geometries.
The Thermoviscous Acoustics, Frequency Domain interface is formulated in the so-called scattered field formulation where the total field (subscript
t) is the sum of the scattered field (the field solved for,
p,
u, and
T) and a possible background acoustic field (subscript
b), such that
When no Background Acoustic Fields feature is present (the background field values are zero per default) the total field is simply the field solved for,
pt =
p,
ut =
u, and
Tt =
T. All governing equations and boundary conditions are formulated in the total field variables.
When this physics interface is added, these default nodes are also added to the Model Builder —
Thermoviscous Acoustics Model,
Wall, and
Initial Values. Then, from the
Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click
Thermoviscous Acoustics to select physics features from the context menu.
The Thermoviscous Acoustics, Frequency Domain interface solves, as mentioned, the full linearized Navier–Stokes (momentum), continuity, and energy equations. It solves for the propagation of compressible linear waves in a general viscous and thermally conductive fluid. The length scale at which the thermoviscous acoustic description is necessary is given by the thickness of the viscous boundary layer (the viscous penetration depth), which is
where the definition of the symbols f,
μ,
ρ0,
k, and
Cp may be found in
Table 6-1. The thickness of both boundary layers depends on the frequency
f and decreases with increasing frequency. The ratio of the two length scales is related to the nondimensional Prandtl number
Pr, by
which define the relative importance of the thermal and viscous effects for a given material. In air at 20 oC and 1 atm, the viscous boundary layer thickness is 0.22 mm at 100 Hz while it is only 55
μm in water under the same conditions. The Prandtl number is 0.7 in air and 7 in water.
The Label is the default physics interface name.
The Name is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern
<name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the
name string must be unique. Only letters, numbers, and underscores (_) are permitted in the
Name field. The first character must be a letter.
The default Name (for the first physics interface in the model) is
ta.
Expand the Equation section to see the equations solved for with the
Equation form specified. The default selection for
Equation form is set to
Study controlled. The available studies are selected under
Show equations assuming.
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For Study controlled, the scaling of the equations is optimized for the numerical performance of the different solvers and study types.
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For Frequency domain you can manually enter the scaling parameter Δ under the Thermoviscous Acoustics Equation Settings section.
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For Mode analysis you can manually enter the Mode analysis frequency (applicable in 2D and 2D axisymmetry).
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Click to select Adiabatic formulation to use an adiabatic equation of state and disable the temperature degree of freedom for the thermoviscous acoustic equations. This formulation is applicable when the thermal losses can be disregarded, this is often the case in liquids, like water. In gases, like air, on the other hand the full formulation is necessary. When
Adiabatic formulation is selected all temperature conditions and options are disabled in the user interface.
In 2D and 2D axisymmetric models, click to select Out-of-plane mode extension in order to model the third dimension implicitly. When selected, an additional degree of freedom, for the velocity, is solved for in the out-of-plane direction. The behavior is determined by entering the
Out-of-plane wave number kz in 2D models; or by entering the
Azimuthal mode number m in 2D axisymmetric models. The latter defines an azimuthal wave number
km =
m/
r. All out-of-plane gradients of the fields are then defined in terms of the wave number. The solved equations closely follow the ones used in the
Formulation for the Boundary Mode Interface.
For all component dimensions, and if required, click to expand the Equation section, then select
Frequency domain as the
Equation form and enter the settings as described below.
The default Scaling factor Δ is 1/(
iω). This value corresponds to the equations for a
Frequency Domain study when the equations are study controlled. To get the equations corresponding to an
Eigenfrequency study, change the
Scaling factor Δ to 1. Changing the scaling factor influences the coupling to other physics.
See the settings for Sound Pressure Level Settings for the
Pressure Acoustics, Frequency Domain interface.
To display this section, click the Show More Options button (
) and select
Stabilization. Select
No stabilization applied (the default),
Galerkin least-squares (GLS) stabilization, or
Streamline upwind Petrov-Galerkin (SUPG) stabilization. When linear thermoviscous acoustic problems are solved (like in the frequency domain) the numerical problem is stable with the default P1-P2-P2 discretization. Enabling stabilization will ensure stability also for other combinations of discretization orders.
Select to enable the Activate port sweep option (not selected per default). This option is used to compute the full scattering matrix when
Port conditions are used. For more details see
The Port Sweep Functionality subsection. The section only exists for 3D, 2D, and 2D axisymmetry.
Select the Mode shape normalization as
Amplitude normalized (the default) or
Power normalized. This setting controls if the mode shapes are normalized to have a unit maximum pressure amplitude or carry unit power. The selection determines how the scattering matrix is to be interpreted.
From the list select the element order and type (Lagrange or serendipity) for the Pressure, the
Velocity field, and the
Temperature variation, respectively. The default is
Linear for the pressure and
Quadratic serendipity for the velocity and the temperature.
This physics interface defines these dependent variables (fields): the Pressure p, the
Velocity field u and its components, and the
Temperature variation T. The names can be changed but the names of fields and dependent variables must be unique within a model.
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In the COMSOL Multiphysics Reference Manual see Table 2-4 for links to common sections and Table 2-5 to common feature nodes. You can also search for information: press F1 to open the Help window or Ctrl+F1 to open the Documentation window.
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