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 are always active as the equilibrium (background) temperature enters the governing energy equation explicitly. From the T0 (SI unit: K) list, select an existing temperature variable (from another physics interface) if available, or select to define a different value or expression. The default is and set to 293.15 K (that is, 20oC). From the p0 (SI unit: Pa) list, select an existing absolute pressure variable (from another physics interface) if available, or select to define a different value or expression. The default is and set to 1 atm.

Thermoviscous acoustics Model

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

•

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.

•

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)

•

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:

•

μ (SI unit: Pa·s).

•

μB (SI unit: Pa·s). The bulk viscosity parameter describes the difference between the mechanical and thermodynamic pressures. It is associated with losses doe to expansion and compression. Its value is difficult to measure and typically require absorption experiments to be determined. Its numerical value is of the same order as the dynamic viscosity. See, for example, Ref. 8 for fluids and Ref. 9 for gases.

•

k (SI unit: W/(m·K)).

•

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

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. When the option (the default) 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 or the material model does not define both a pressure and temperature dependence for ρ0, the coefficient of thermal expansion and the compressibility need to be set manually, or they evaluate to 0.


	For most materials, selected from the material library, it is necessary to set the coefficient of thermal expansion and the compressibility using one of the nondefault options. If the material is air, the From equilibrium density option works well as the equilibrium density ρ0 = ρ0(p0,T0) is a function of both pressure and temperature. For water the coefficient of thermal expansion is well defined as ρ0 = ρ0(T0), while the compressibility can easily be defined using the From speed of sound option.

The section displays if or is selected as the under .

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 lists — (the default), , , or . For , enter a value for βT (SI unit: 1/Pa = Pa-1).

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.

•

c (SI unit: m/s).

•

γ (dimensionless). The default is 1.

•

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:

•

The viscous power dissipation density ta.diss_visc.

•

The thermal power dissipation density ta.diss_therm.

•

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 fluids 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 setting the Isothermal compressibility to User defined and here enter the adiabatic (isotropic) value βs (remember that for fluids βs = γ·βT). Then, in the solver sequence under Solver Configuration>Solver 1> Dependent Variables select Define by study step to User defined and under >Temperature variation (mod1.T) click to clear the Solver for this field box. See also Solver Suggestions for Large Thermoviscous Acoustics Models for suggestions on how to set up the solver for large problems.