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Biot (viscous losses), this model is primarily used in earth sciences when the saturating fluid is a liquid like water or oil. The model is based on Biot’s original work and only includes the effects of viscous losses in the pores.
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Biot-Allard (thermal and viscous losses), this model is intended for simulating porous materials where the saturating fluid is air. This is for modeling sound absorbers, liners, foams used in headphones and loudspeakers, cloth and much more.
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Three different waves can propagate in a poroelastic material; fast pressure waves, slow pressure waves, and shear waves. Their speed can be evaluated through the variables pelw.cp_fast, pelw.cp_slow, and pelw.cs_poro, respectively. These should not be confused with the material properties for the constituting solid (pelw.cp and pwel.cs) and fluid (pelw.c). All three poroelastic wave types should be resolved by the computational mesh.
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Young’s modulus and Poisson’s ratio to specify drained Young’s modulus (elastic modulus) Ed (SI unit: Pa) and Poisson’s ratio νd (dimensionless). For an isotropic material, Young’s modulus is the spring stiffness in Hooke’s law, which in 1D form is σ = Edε, where σ is the stress and ε is the strain. Poisson’s ratio defines the normal strain in the perpendicular direction, generated from a normal strain in the other direction and follows the equation ε⊥ = −υε||
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Shear modulus and Poisson’s ratio (the default for the Biot-Allard model) to specify drained shear modulus Gd (SI unit: Pa) and Poisson’s ratio νd (dimensionless).
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Young’s modulus and Shear modulus to specify drained Young’s modulus (elastic modulus) Ed (SI unit: Pa) and drained shear modulus Gd (SI unit: Pa).
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Bulk modulus and shear modulus (the default for the Biot model) to specify the drained bulk modulus Kd (SI unit: Pa) and the drained shear modulus Gd (SI unit: Pa). The bulk drained modulus is a measure of the solid porous matrix’s resistance to volume changes. The shear modulus is a measure of the solid porous matrix’s resistance to shear deformations.
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Pressure-wave and shear-wave speeds to specify the drained pressure-wave speed cp (SI unit: m/s) and the shear-wave speed cs (SI unit: m/s).
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Bulk modulus and shear modulus (the default for the Biot model as described above) to specify the drained bulk modulus Kd (SI unit: Pa) and the drained shear modulus Gd (SI unit: Pa).
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Drained density of porous material to specify the drained density of the porous material in vacuum ρd (SI unit: kg/m3). The drained density ρd is equal to (1 − εp) ρs where ρs is the density of the solid material from which the matrix is made and εp is the porosity.
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Permeability to specify the permeability of the porous material κp (SI unit: m2). The permeability is a measure of the ability of the porous material to let fluid pass through it. It hence gives some measure of the pore size and thus correlates to the viscous damping experienced by pressure waves propagating in the saturating fluid.
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Porosity to specify the porosity of the material εp (dimensionless). It defines the amount of void volume inside the porous matrix and takes values between 0 (no porous material only fluid) and 1 (fully solid material no fluid).
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Biot-Willis coefficient to specify the Biot-Willis coefficient αB (dimensionless). This coefficient relates the bulk modulus (compressibility) of the drained porous matrix to a block of solid material. It is defined as
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Tortuosity factor (high frequency limit) or the structural form factor (dimensionless). This is a purely geometrical factor that depends on the microscopic geometry and distribution of the pores inside the porous material. It is independent of the fluid and solid properties and is normally >1. The default is 2. The more complex the propagation path through the material, the higher is the absorption. The tortuosity partly represents this complexity.
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Shear modulus and Poisson’s ratio (the default for the Biot-Allard model as described above) to specify drained shear modulus Gd (SI unit: Pa) and Poisson’s ratio νd (dimensionless).
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Drained density of porous material to specify the drained density of the porous material in vacuum ρd (SI unit: kg/m3). The drained density ρd is equal to (1 − εp) ρs where ρs is the density of the solid material from which the matrix is made and εp is the porosity.
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Porosity to specify the porosity of the material εp (dimensionless). It defines the amount of void volume inside the porous matrix and takes values between 0 (no porous material only fluid) and 1 (fully solid material no fluid).
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Flow resistivity to specify the (static) flow resistivity of the porous material Rf (SI unit: Pa·s/m2). The flow resistivity is a measure of the ability of the porous material to let fluid pass through it. It hence gives some measure of the pore size and thus correlates to the viscous damping experienced by pressure waves propagating in the saturating fluid. The flow resistivity is also sometimes denoted σ (using the unit N·s/m4) and it is related to the permeability through κp = μ/Rf.
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Isotropic structural loss factor to specify the loss factor of the porous matrix ηs (dimensionless). This value introduces the damping due to losses in the porous structure by transform the elastic moduli into complex valued quantities. This quantity can be frequency dependent if necessary.
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Tortuosity factor (high frequency limit) or the structural form factor τ∞ (dimensionless). This is a purely geometrical factor that depends on the microscopic geometry and distribution of the pores inside the porous material. It is independent of the fluid and solid properties and is normally >1. The default is 2. The more complex the propagation path through the material, the higher the absorption. The tortuosity partly represents this complexity.
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Viscous characteristic length to specify the viscous length scale Lv (SI unit: m). This value is sometimes denoted Λ and replaces the hydraulic radius used in simpler models to account for the viscous losses that appear in the acoustic boundary layer at pore walls.
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Thermal characteristic length to specify the thermal length scale Lth (SI unit: m). This value is sometimes denoted Λ’ and replaces the hydraulic radius used in simpler models to account for the thermal losses that appear in the acoustic boundary layer at pore walls.
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Biot-Willis coefficient how to calculate the Biot-Willis coefficient αB by selecting From material, Rigid assumption (the default), General model, or User defined. This coefficient relates the bulk modulus (compressibility) of the drained porous matrix to a block of solid material.
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From material to pick up the value from the domain material.
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Rigid assumption (the default) and the model defines a rigid porous matrix (Voigt upper bound) where αB = εp.
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User defined enter a value for the Biot-Willis coefficient αB (dimensionless). In the limp limit when αB = 1 it is recommended to use the Poroacoustics feature of The Pressure Acoustics, Frequency Domain Interface instead of the Poroelastic Waves interface. In poroacoustics the limp limit, when the structure is so “fluffy” that it moves with the fluid, has the assumption Kd = 0 and αB = 1 included explicitly.
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Dynamic viscosity to define the dynamic viscosity of the saturating fluid μf (SI unit: Pa·s). The parameter is important for the amount of viscous damping experienced by the acoustic waves.
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Compressibility of the saturating fluid χf (SI unit: 1/Pa). Remember that the fluid compressibility χf is related to the fluid bulk modulus Kf (SI unit: Pa) and the speed of sound c, through the relation
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Biot’s low frequency range models damping at low frequencies where the acoustic boundary layer (the viscous penetration depth) is assumed to span the full width of the pores. This is also the so-called Poiseuille limit.
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For Biot’s high frequency range also select Specify as Reference frequency or Characteristic pore size. Either enter a Reference frequency fr (SI unit: Hz) or a Characteristic pore size a (SI unit: m). This model implements a correction factor to the viscosity that accounts for the relative scale difference between a typical pore diameter and the acoustic boundary layer thickness. The modified viscosity is of the form
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See High Frequency Correction (Biot Model) for more details.
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Dynamic viscosity to define the dynamic viscosity of the saturating fluid μf (SI unit: Pa·s). The parameter is important for the amount of viscous damping experienced by the acoustic waves.
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Ratio of specific heats to define the ratio of specific heats (adiabatic index) of the saturating fluid γ (dimensionless).
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Heat capacity at constant pressure to define the (specific) heat capacity at constant pressure of the saturating fluid Cp (SI unit: J/(kg·K)).
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Thermal conductivity to define the thermal conductivity of the saturating fluid k (SI unit: W/(m·K)). The parameter is important for the amount of thermal damping experienced by the acoustic waves.
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