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. 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 physics interface solves the equations in the frequency domain assuming all fields and sources to be harmonic. The harmonic variation of all fields and sources is given by using the +iω convention. Linear acoustics is assumed. Nonlinear effects can be included when modeling in the time domain using The Thermoviscous Acoustics, Transient Interface and the Nonlinear Thermoviscous Acoustics Contributions feature.

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 physical quantities commonly used in the thermoviscous acoustics interfaces are defined in Table 6-1 below.

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.

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.

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.

Enter a value or expression for the typical wave speed for perfectly matched layers cref (SI unit: m/s). The default is 343 m/s.

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 Lagrange 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.