Theory

The designer follows the Cremer-impedance method of Kabral et al. (2014) and Kabral (2015): a locally reacting wall gives the largest damping of a duct's least-attenuated mode when its impedance equals the optimum found by Cremer for a rectangular duct and by Tester for a circular duct with mean flow. A micro-perforated plate gives the resistance and a cavity behind it the (negative) reactance, so the design is made in three steps at one design frequency $f_0$ :

1. the target impedance $Z_C$ from the duct size, $f_0$ and the Mach number; 2. the open-area ratio $\sigma$ that gives its resistance: the plate impedance is the slit (or hole) impedance divided by $\sigma$ , and the micro-perforate models of the program do not depend on $\sigma$ otherwise, so $\sigma = \mathrm{Re}\,z_h / \mathrm{Re}\,Z_C$ ; 3. the reactance that is left for the cavity, $x_{cav} = \mathrm{Im}\,Z_C - \mathrm{Im}\,z_h/\sigma$ , and the smallest annular cavity (outer radius $R$ , rigid end wall) that gives it.

The slit and hole impedances are the program's own micro-perforate models (the same functions the acoustic solve uses). Kabral et al. (2014) draw the Cremer resistance of their M = 0.05 case through 0.34 at 2 kHz (their Fig. 6), which the circular formula gives with $M = -0.05$ : the Mach number is signed, negative when the flow goes against the sound (an intake silencer, where the compressor noise travels upstream). The grazing term of the plate uses $|M|$ .

With the paper's optimised panel (slit width 0.21 mm, thickness 1 mm, duct radius 30 mm, 2 kHz, M = −0.05, air at 20 °C) the designer gives $\sigma$ = 0.0619 and $R$ = 49.1 mm; the paper gives 0.062 (Kabral 2015 Table 2.2) and 49 mm.

The preview is the attenuation of the lined circular duct's axisymmetric mode that starts as the plane wave, from $i k J_0(\kappa r)/Z = \kappa J_1(\kappa r)$ and $k_z = \sqrt{k^2 - \kappa^2}$ , in dB per metre and times the lined length. It leaves out the reflections at the two ends of the lining and the flow in the wave equation, so it is only an indication: the transmission loss comes from the acoustic solve. For a rectangular duct the cavity is a flat air layer of depth $D$ behind the plate, $Z_{cav} = -i\cot(\omega D/c)$ , the air-backed absorber of Maa (1998) (eq. 9); the thin annular cavity of Kabral (2015) (eq. 2.11) tends to it, and the designer returns the smallest depth that gives $x_{cav}$ .

Equations

Z_{Cc} = (0.88 - 0.38\,i)\,\frac{k r}{\pi (1 + M)^2}, \qquad Z_{Cr} = (0.91 - 0.76\,i)\,\frac{k a}{\pi}

(1)

\sigma = \frac{\mathrm{Re}\,z_h(f_0)}{\mathrm{Re}\,Z_C}, \qquad x_{cav} = \mathrm{Im}\,Z_C - \frac{\mathrm{Im}\,z_h(f_0)}{\sigma}

(2)

Z_{cav} = \frac{i\left[H_0^{(1)}(k r) - \dfrac{H_1^{(1)}(k R)}{H_1^{(2)}(k R)} H_0^{(2)}(k r)\right]}{H_1^{(1)}(k r) - \dfrac{H_1^{(1)}(k R)}{H_1^{(2)}(k R)} H_1^{(2)}(k r)}

(3)

where	meaning	unit
$Z_C$	Optimum wall impedance, normalised by $\rho c$ (circular $Z_{Cc}$ , rectangular $Z_{Cr}$ )	-
$k$	Wavenumber $2\pi f_0 / c$	1/m
$r, a$	Duct radius (circular) or height (rectangular)	m
$M$	Mach number, negative against the sound	-
$z_h$	Slit or hole impedance of the micro-perforate model, normalised	-
$\sigma$	Open-area ratio	-
$R$	Outer radius of the annular cavity	m
$H_m^{(n)}$	Hankel function of kind $n$ and order $m$	-

Default and rule

Default: auto (0.08)
Rule: auto (0.08).
Source: Kabral et al. (2014) eqs. 6 and 14; Kabral (2015) eqs. 2.11-2.13 and Table 2.2; Maa (1998) eq. 9 (flat cavity). The micro-perforate impedances are those of the Boundary Conditions card.

Valid range 0 - 1

Design aid

0 - 1. The design is a starting point at one frequency; check it with the acoustic solve.

Liner designer: grazing factor

Theory

Equations

Default and rule

Valid range 0 - 1

References 3

Theory

Equations

Default and rule

Valid range 0 - 1

Related 13

References 3