Theory

Besides the viscous losses inside the holes, the oscillating flow drawn into and out of each hole rubs on the plate surface. Maa (1998) adds $\tfrac{1}{2}\sqrt{2\omega\rho\eta}$ in total, which is $\alpha = 1/2$ in the notation of Allam and Abom (2011). Allam and Åbom fitted the factor to their measurements: $\alpha = 4$ for sharp-edged holes and $\alpha = 2$ for holes with rounded edges.

The hole impedance itself is the exact solution for the oscillating flow in a short tube Maa (1998) with the mass end correction $0.85\,d$ ; with grazing flow the resistance rises by $K M/\sigma$ and the end correction falls by $F_\delta$ Allam and Abom (2011).

Equations

z = \frac{i\omega t}{\sigma c}\left[1 - \frac{2}{k\sqrt{-i}}\,\frac{J_1(k\sqrt{-i})}{J_0(k\sqrt{-i})}\right]^{-1} + \frac{2\alpha R_s}{\sigma\rho c} + K\frac{M}{\sigma} + i\,\frac{\omega\,0.85\,d\,F_\delta}{\sigma c}

(1)

where	meaning	unit
$z$	Normalised transfer impedance of the plate	-
$k$	Perforate constant, $d\sqrt{\omega/(4\nu)}$	-
$J_0, J_1$	Bessel functions of the first kind	-
$\alpha$	Edge factor (this field)	-
$R_s$	Surface resistance, $\tfrac{1}{2}\sqrt{2\omega\rho\eta}$	Pa s/m
$K$	Grazing-flow factor, 0.15	-
$M$	Grazing Mach number	-
$d$	Hole diameter	m

F_\delta = \left[1 + (12.6\,M)^3\right]^{-1}

(2)

where	meaning	unit
$F_\delta$	Flow factor of the end correction	-

Default and rule

Default: As in Maa 1998 (α = 1/2)
Rule: As in Maa 1998.
Source: Maa (1998), eqs. 2 and 5; Allam and Abom (2011), eqs. 3, 23 and 24 (edge factors and K measured).

Valid range α = 1/2 (Maa 1998), 2 (rounded edges), 4 (sharp edges)

Choices

α = 1/2 (Maa 1998), 2 (rounded edges), 4 (sharp edges).

Hole edges of a micro-perforate

Theory

Equations

Default and rule

Valid range α = 1/2 (Maa 1998), 2 (rounded edges), 4 (sharp edges)

References 2

Theory

Equations

Default and rule

Valid range α = 1/2 (Maa 1998), 2 (rounded edges), 4 (sharp edges)

Related 3

References 2