Acoustic (poroacoustic) model of the packing

Selects how the acoustic solver represents an absorptive packing: as an equivalent fluid with a complex, frequency-dependent density and bulk modulus. The three empirical models need only the flow resistivity; the JCA and JCAL models need four or five material parameters; "user defined" takes the eight constants of your own fit.

Last updated 2026-11-11

Theory

A rigid-frame fibrous packing behaves acoustically like a lossy fluid. The acoustic finite-element model assigns the packing elements the equivalent density $\rho_{eq}$ and bulk modulus $K_{eq}$ of the selected model, both complex and frequency dependent (time dependence $e^{+i\omega t}$ ). The packing domain is coupled to the gas either directly or through a perforated wall.

Empirical models (Delany-Bazley, Miki, Komatsu). The characteristic impedance $Z_c$ and the propagation constant $k_c$ are power laws in one dimensionless variable $X$ , equations (1)-(3), with eight constants $C_1 \dots C_8$ per model (shown read-only on the card).

Delany-Bazley (Delany and Bazley, 1970) and Miki (Miki, 1990) use $X = \rho_0 f/R$ .
Komatsu (Komatsu, 2008) uses $X = 2 - \log_{10}(f/R)$ .
Miki's modification of the Delany-Bazley fit remains physically realisable (positive real impedance) below its data range; Komatsu's model improves the fit at very low and very high $f/R$ .

The constants are the published coefficients converted to the $X = \rho_0 f/R$ form with the air density of the original measurements, 1.2 kg/m³, as tabulated by Komatsu (2008).

JCA and JCAL (Johnson et al., 1987), (Champoux and Allard, 1991), (Lafarge et al., 1997) are semi-phenomenological models based on the pore geometry: Porosity (JCA / JCAL) $\phi$ , Tortuosity (JCA / JCAL) $\alpha_\infty$ , Viscous characteristic length (JCA / JCAL) $\Lambda$ , Thermal characteristic length (JCA / JCAL) $\Lambda'$ and, for JCAL, Static thermal permeability (JCAL) $k'_0$ , together with the flow resistivity, equations (4)-(5). The gas properties ( $\eta$ , $\gamma$ , $P_0$ , Pr) are those of the local gas, so these models remain valid in hot exhaust gas, unlike the empirical fits, which were measured in room-temperature air. JCA is JCAL with $k'_0 = \phi\Lambda'^2/8$ .

User defined applies equations (1)-(2) with the constants User constant C1 ... C8 and the variable Variable X of the user constants of your own fit; no validity range is checked.

In every model $\rho_0$ and $c_0$ (and, for JCA/JCAL, $\eta$ , $\gamma$ , $P_0$ ) are those of the gas in the packing at its local temperature (the domain mean of a medium mapped from the CFD), and $R$ is the resistivity scaled to that temperature (see Flow resistivity).

Equations

Z_c = \rho_0 c_0\left[1 + C_5 X^{-C_6} - i\,C_7 X^{-C_8}\right]

(1)

k_c = \frac{\omega}{c_0}\left[1 + C_1 X^{-C_2} - i\,C_3 X^{-C_4}\right]

(2)

\rho_{eq} = \frac{Z_c k_c}{\omega}, \qquad K_{eq} = \frac{\omega Z_c}{k_c}, \qquad X = \frac{\rho_0 f}{R}\ \ \text{or}\ \ X = 2 - \log_{10}\frac{f}{R}

(3)

where	meaning	unit
$Z_c$	Characteristic impedance of the packing	Pa·s/m
$k_c$	Complex propagation constant	1/m
$\rho_0, c_0$	Density and sound speed of the gas in the packing	kg/m³, m/s
$f, \omega$	Frequency, angular frequency $2\pi f$	Hz, rad/s
$R$	Flow resistivity at the local temperature	Pa·s/m²
$C_1 \dots C_8$	Model constants: Delany-Bazley 0.0978, 0.7, 0.189, 0.595, 0.0571, 0.754, 0.087, 0.732; Miki 0.122, 0.618, 0.18, 0.618, 0.079, 0.632, 0.12, 0.632; Komatsu 0.0004, -6.2, 0.0069, -4.1, 0.00027, -6.2, 0.0047, -4.1	-

\rho_{eq} = \frac{\alpha_\infty\rho_0}{\phi}\left[1 + \frac{R\,\phi}{i\omega\rho_0\alpha_\infty}\sqrt{1 + i\,\frac{4\alpha_\infty^2\eta\rho_0\omega}{R^2\Lambda^2\phi^2}}\right]

(4)

K_{eq} = \frac{\gamma P_0/\phi}{\gamma - (\gamma - 1)\left[1 - i\,\dfrac{\phi\,\eta/\mathrm{Pr}}{k'_0\rho_0\omega}\sqrt{1 + i\,\dfrac{4 k_0'^2\rho_0\omega}{(\eta/\mathrm{Pr})\Lambda'^2\phi^2}}\right]^{-1}}

(5)

where	meaning	unit
$\phi$	Open porosity	-
$\alpha_\infty$	High-frequency tortuosity	-
$\Lambda, \Lambda'$	Viscous and thermal characteristic lengths	m
$k'_0$	Static thermal permeability (JCA: $\phi\Lambda'^2/8$ )	m²
$\eta$	Dynamic viscosity of the gas	Pa·s
$\gamma$	Ratio of specific heats of the gas, $c^2\rho/p$	-
$P_0$	Absolute pressure of the gas	Pa
$\mathrm{Pr}$	Prandtl number (material setting)	-

Default and rule

Default

Miki

Rule

Miki. It needs only the flow resistivity and, unlike Delany-Bazley, remains physically realisable outside its fitted band.

Source

The constants and validity ranges of the empirical models are the published coefficients as tabulated in Komatsu (2008) (Tables 1-2 and eqs. (15)-(18)), consistent with Delany and Bazley (1970) and Miki (1990). The JCA/JCAL equations follow the formulation of Atalla and Panneton (2000), against which they are checked term by term.

Validation results:

JCA: the measured absorption of the foam of Atalla and Panneton (2000) is reproduced within 0.08 at six frequencies.
A packed annulus in the 3-D acoustic solver agrees within 0.45 dB with a 1-D coupled-line solution using the same $\rho_{eq}$ and $K_{eq}$ .

Valid range Delany-Bazley | Miki | Komatsu | JCA | JCAL | user defined

Validity of the empirical models

Delany-Bazley and Miki: $0.012 < \rho_0 f/R < 1.2$ (the original 0.01-1 m³/kg in air of 1.2 kg/m³); Komatsu: $0.001 < f/R < 1$ m³/kg. The acoustic QA gate lists frequencies outside these ranges as a warning; the models are never extrapolated silently.

Hot gas

The three empirical models are fits to room-temperature air at porosity near 1. In hot gas $\rho_0$ changes $X$ for Delany-Bazley and Miki; Komatsu's $X$ contains no $\rho_0$ . JCA/JCAL use the local gas properties directly.

Rigid frame

All models assume a motionless fibre skeleton; a limp or elastic frame (foams at low frequency) is not modelled.

JCA / JCAL

Checked in a full acoustic run (2026-11-11): every quality check passes; within about 1 dB of Miki below 1 kHz with estimated parameters.

Theory

Equations

Default and rule

Valid range Delany-Bazley | Miki | Komatsu | JCA | JCAL | user defined

Related 10

References 7