Poroacoustics
The Poroacoustics feature allow modeling of porous materials in the time domain, using an equivalent fluid model. The properties of the equivalent fluid, representing the porous material, are known (measured or simulated) in term of its real and imaginary part of the (isentropic) compressibility βs(f) and density ρ(f), both functions of frequency. To capture the behavior in the time domain, the Poroacoustics feature has built-in functionality for setting up and defining the compressibility and density expressions through a rational approximation (or rational expansion) of the frequency-dependent data. The approximation has an analytical inverse Fourier transform and can thus be used to define the same frequency dependency in the time domain, by setting up a system of memory equations. These equations are automatically defined by the feature and solved in the porous domain. Fitting of the frequency domain data can be performed using the Partial Fraction Fit function for user-defined data. It can also be done automatically for both the Johnson–Champoux–Allard (JCA) and the Johnson–Champoux–Allard–Lafarge (JCAL) models defining the usual poroacoustic material data.
For more information see Partial Fraction Fit under User-Defined Functions in the COMSOL Multiphysics Reference Manual.
Poroacoustics Model
Select the Poroacoustics model as Johnson–Champoux–Allard (JCA) (the default), Johnson–Champoux–Allard–Lafarge (JCAL), or User defined.
When either the Johnson–Champoux–Allard (JCA) or the Johnson–Champoux–Allard–Lafarge (JCAL) models are selected the underlying partial fraction approximation needs to be computed before solving. Before fitting the data it is important that the Maximum frequency to resolve fmax is set to the desired value in the Transient Solver and Mesh Settings section. This ensures a well posed numerical model (mesh, time stepping, and approximation). Enter the necessary material data (see below). Then, in the Approximation () menu, select Compute partial fraction approximation (). The information message changes from “The selected poroacoustics model has frequency-dependent material properties. Compute their partial fraction approximations before solving” to “Partial fraction approximations of the frequency-dependent material properties successfully computed”. The approximation can be cleared by selecting Clear approximation () in the Approximation () menu.
If changes are made to either the material data or the maximum frequency to resolve, make sure to Clear approximation () and then Compute partial fraction approximation () again.
Fluid Properties
This section is visible if the Poroacoustics model is set to either Johnson–Champoux–Allard (JCA) or Johnson–Champoux–Allard–Lafarge (JCAL).
Select Fluid material from the list of materials found under the Materials node. Note that the materials used here cannot have any space dependent properties. If material properties are user defined, they need to be defined through a parameter.
Then select From material (the default) or User defined for the necessary fluid properties:
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Density (of the fluid) ρf (SI unit: kg/m3)
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Ratio of specific heats γ (SI unit: 1)
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Heat capacity at constant pressure Cp (SI unit: J/(kg·K))
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Thermal conductivity k (SI unit: W/(m·K))
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Dynamic viscosity μ (SI unit: Pa·s)
Porous Matrix Properties
This section is visible if the Poroacoustics model is set to either Johnson–Champoux–Allard (JCA) or Johnson–Champoux–Allard–Lafarge (JCAL).
Select Porous elastic material from the list of materials found under the Materials node. Note that the materials used here cannot have any space dependent properties. If material properties are user defined, they need to be defined through a parameter.
Then select From material (the default) or User defined for the necessary fluid properties:
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Porosity εp (SI unit: 1)
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Flow resistivity Rf (SI unit: Pa·s/m2)
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Viscous characteristic length Lv (SI unit: m). Select how to Specify it, either directly or through the Viscous characteristic length parameter s (SI unit: 1).
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Thermal characteristic length Lth (SI unit: m)
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Tortuosity factor τ∞ (SI unit: 1)
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Static thermal permeability κ'0 (SI unit: m2). Only visible for the JCAL model.
Isentropic Compressibility
This section is visible if the Poroacoustics model is set to User defined. Select the Partial fraction fit as User defined (the default) or From function.
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For the User defined option enter all the necessary parameters that are present in the rational approximation (rational expansion), that is, the Frequency-independent (asymptotic) compressibility β∞ (SI unit: 1/Pa), enter the Real residues and poles (R and ξ) into the table, and enter the Complex residues and poles (Q and ζ) into the table (note that the conjugate terms are automatically added). The quantities can also be loaded from a file (remember to include the number column). Select if the rational approximation function is scaled for frequency data or angular frequency data, by selecting the Equation scaled for frequency data option (selected per default).
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For the From function option select the Reference, which is a Partial Fraction Fit function defined in the model, then click the Import icon () to copy the fitted data to the compressibility field, and the residues and poles tables. To go to the function source click the Go to Source icon ().
Density
This section is visible if the Poroacoustics model is set to User defined. Select the Partial fraction fit as User defined (the default) or From function.
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For the User defined option enter all the necessary parameters that are present in the rational approximation (rational expansion), that is, the Frequency-independent (asymptotic) density ρ∞ (SI unit: kg/m3), enter the Real residues and poles (R and ξ) into the table, and enter the Complex residues and poles (Q and ζ) into the table (note that the conjugate terms are automatically added). The quantities can also be loaded from a file (remember to include the number column). Select if the rational approximation function is scaled for frequency data or angular frequency data, by selecting the Equation scaled for frequency data option (selected per default).
For the From function option select the Reference, which is a Partial Fraction Fit function defined in the model, then click the Import icon () to copy the fitted data to the density field, and the residues and poles tables. To go to the function source click the Go to Source icon ().
The necessary data, the complex-valued and frequency-dependent data for the compressibility and the density, can be generated from a small frequency domain model that uses the Poroacoustics in the The Pressure Acoustics, Frequency Domain Interface.
Define a domain with the desired Poroacoustics model, add the necessary material properties, and solve the model for the range of frequencies to be resolved (no sources are needed). Predefined evaluation groups can be selected from the Results Templates to evaluate the real and imaginary part of the equivalent density and compressibility. The following expressions real(acpr.rho_c) and imag(acpr.rho_c) are evaluated for the density; and real(1/acpr.K_eq) and imag(1/acpr.K_eq) for the compressibility.
The Partial Fraction Fit function can point directly to the Results table or the data can be stored in a .txt file and referred to.