Thermoviscous Acoustics Model

Use the node to define the model inputs (the background equilibrium temperature and pressure) and the material properties of the fluid (dynamic viscosity, bulk viscosity, thermal conductivity, heat capacity at constant pressure, and equilibrium density) necessary to model the propagation of acoustic compressible waves in a thermoviscous acoustic context. Extended inputs are available for the coefficient of thermal expansion and the compressibility, which enables modeling of any constitutive relation for the fluid.

Model Inputs

This section contains field variables that appear as model inputs. The fields for the p0 and the T0 are always active as they enter the governing equations explicitly. If material properties depend on other model inputs they will automatically appear in this section.

Select (the default), , or an existing variable from another physics interface.

The p0 (SI unit: Pa) has the default value set to 1 atm. The T0 (SI unit: K) has the default value set to 293.15 K (that is, 20oC).


	Details about the Model Input and the Default Model Inputs are found in the Global and Local Definitions chapter of the COMSOL Multiphysics Reference Manual.

Fluid Properties

Define the material properties of the fluid by selecting an — , , or .

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If is selected (the default), the equilibrium density, and its dependence on the equilibrium pressure p0 and temperature T0, is taken from the defined material. Make sure that the Thermal Expansion and Compressibility settings are correct.

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For also select the Gas constant type — select Specific gas constant Rs (SI unit: J/(kg·K) or Mean molar mass Mn (SI unit: kg/mol)

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For enter a value or expression for the ρ0(p0, T0) (SI unit: kg/m3). The default is ta.p0/(287[J/kg/K]*ta.T0), which is the ideal gas law.

The other thermoviscous acoustic model parameters defaults use values For enter another value or expression for:

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μ (SI unit: Pa·s).

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μB (SI unit: Pa·s). The bulk viscosity parameter describes the difference between the mechanical and thermodynamic pressures. It is associated with losses due to expansion and compression.

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k (SI unit: W/(m·K)).

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Cp (SI unit: J/(kg·K)). This is the specific heat capacity or heat capacity per unit mass.


	For a discussion about the air and water materials as commonly used in acoustics and other material properties, such as the bulk viscosity, see the Acoustic Properties of Fluids chapter of this manual.

Thermal Expansion and Compressibility

One of the main characteristics of an acoustic wave is that it is a compressional wave. In the detailed thermoviscous acoustic description, this property is closely related to the constitutive relation between the density, the pressure, and the temperature. This results in the important (linear) relation for the acoustic density variation

where ρt is the total density variation, pt is the total acoustic pressure, Tt is the total acoustic temperature variations, βT is the (isothermal) compressibility of the fluid, and αp the (isobaric) coefficient of thermal expansion (sometimes named α0). If this constitutive relation is not correct, then no waves propagate or possibly they propagate at an erroneous speed of sound. The default behavior is to define both quantities from the speed of sound and the ratio of specific heats (using the option) which are material properties often more readily available.

Note that, when the is selected option under the Thermoviscous Acoustics Equation Settings section the equation of stated reduces to

When the option is selected for the coefficient of thermal expansion and the compressibility, both values are derived from the equilibrium density ρ0(p0,T0) using their defining relations

If the equilibrium density ρ0 is a user-defined constant value, is picked up from another physics interface, or the material model does not define both a pressure and temperature dependence for ρ0, the coefficient of thermal expansion and/or the compressibility will evaluate to 0. Then the default option or the options should be used.


	If the material is air, the From equilibrium density option works well as the equilibrium density ρ0 = ρ0(p0,T0) is defined as a function of both pressure and temperature. For the water material the coefficient of thermal expansion is well defined as ρ0 = ρ0(T0), while the compressibility should be defined using the default From speed of sound option.

The section is displayed if or is selected as the under . For the option the parameters are readily defined.

Select an option from the αp list — , , (the default)or . For enter a value for αp (SI unit: 1/K = K-1).

Select an option from the βT list — , (the default), or . For , enter a value for βT (SI unit: 1/Pa = Pa-1).

The different options for defining the coefficient of thermal expansion and the compressibility stem from their thermodynamic definitions:

For each of the following, and based on the above selection, the default is taken . For enter another value or expression in the text field.

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c (SI unit: m/s).

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γ (dimensionless). The default is 1.

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βs (SI unit: 1/Pa = Pa-1) .


	See the Theory Background for the Thermoviscous Acoustics Branch section for a detailed description of the governing equations and the constitutive relations.

To visualize the dissipated energy due to viscosity and thermal conduction in postprocessing. Three postprocessing variables exist:

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The viscous power dissipation density ta.diss_visc.

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The thermal power dissipation density ta.diss_therm.

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The total thermo-viscous power dissipation density ta.diss_tot.


	In certain cases it can be interesting not to include thermal conduction in the model and treat all processes as adiabatic (isentropic). This is, for example, relevant for most liquids where the thermal boundary layer is much thinner than the viscous. Not solving for the temperature field T also saves some degrees of freedom (DOFs). This is achieved by selecting the Adiabatic formulation option under the Thermoviscous Acoustics Equation Settings section. See also Solver Suggestions for Large Thermoviscous Acoustics Models for suggestions on how to select an iterative solver for large problems.