^{1,a)}, J. Stuart Bolton

^{2}, Nicholas N. Kim

^{2}, Jonathan H. Alexander

^{3}and Ronald W. Gerdes

^{4}

### Abstract

It has been previously shown that relatively simple computational fluid dynamics (CFD) models can be used to calculate the transfer impedances, including the associated end corrections, of microperforated panels. The impedance is estimated by first calculating the pressure difference across a single hole when a transient input velocity is imposed, and then Fourier transforming the result to obtain the impedance as a function of frequency. Since the size of the hole and the dimensions of the inlet and outlet channels are very small compared to a wavelength, the flow through the hole can be modeled as incompressible. By using those procedures, Bolton and Kim extended Maa's classical theory to include a resistive end correction for sharp-edged cylindrical holes which differs from those previously proposed by the inclusion of a static component. Here it is shown that CFD models can also be used to compute end corrections for tapered holes. Since practical experimental characterization of perforated materials often involves measurement of the static flow resistance, a closed form empirical equation for that quantity has been developed. Finally, it is shown that configurations having equivalent static flow resistances can yield different acoustic absorptions.

I. INTRODUCTION

II. REVIEW OF THEORY

A. Maa model

B. Guo model

C. Bolton and Kim model

III. CFD MODEL

A. Geometry

B. CFD parameters

IV. END CORRECTIONS FOR STATIC FLOW RESISTANCE

A. Straight holes

B. Tapered holes

C. Equal cases

V. DYNAMIC FLOW RESISTANCE

A. Tapered holes without end corrections

B. Straight hole dynamic end corrections

C. Tapered hole dynamic end corrections

VI. CASES HAVING EQUIVALENT STATIC FLOW RESISTANCE

A. Dynamic flow resistance

B. Absorption coefficient

VII. CONCLUSIONS

### Key Topics

- Flow simulations
- 33.0
- Poiseuille flow
- 20.0
- Rheology and fluid dynamics
- 10.0
- Collisional energy loss
- 6.0
- Sound pressure
- 5.0

##### F16S1/00

## Figures

(Color online) Sketch of the CFD geometry. The axis for this axisymmetric geometry is at the bottom of the sketch, and the upper boundaries are slip-surfaces (symmetry).

(Color online) Sketch of the CFD geometry. The axis for this axisymmetric geometry is at the bottom of the sketch, and the upper boundaries are slip-surfaces (symmetry).

Typical computational mesh in the region of the perforation.

Typical computational mesh in the region of the perforation.

Inlet velocity as a function of time (a) and the corresponding Fourier spectrum as a function of frequency (b).

Inlet velocity as a function of time (a) and the corresponding Fourier spectrum as a function of frequency (b).

(Color online) Straight hole steady-state CFD results: (a) pressure field, Pa, (b) velocity magnitude, mm/s, (c) shear rate, 1/s, and (d) energy loss rate, W/m^{3}. In these images, the flow is from the bottom to the top.

(Color online) Straight hole steady-state CFD results: (a) pressure field, Pa, (b) velocity magnitude, mm/s, (c) shear rate, 1/s, and (d) energy loss rate, W/m^{3}. In these images, the flow is from the bottom to the top.

(Color online) Sketch of relevant geometry for understanding the taper angle terms in Eq. (16) . The ratio of the entrance (and exit) angle to 90° (i.e., π/2) enters into the equation.

(Color online) Sketch of relevant geometry for understanding the taper angle terms in Eq. (16) . The ratio of the entrance (and exit) angle to 90° (i.e., π/2) enters into the equation.

(Color online) Tapered hole steady-state CFD results: (a) pressure field, Pa, (b) velocity magnitude, mm/s, (c) shear rate, 1/s, and (d) energy loss rate, W/m^{3}. In these images, the flow is from the bottom to the top.

(Color online) Tapered hole steady-state CFD results: (a) pressure field, Pa, (b) velocity magnitude, mm/s, (c) shear rate, 1/s, and (d) energy loss rate, W/m^{3}. In these images, the flow is from the bottom to the top.

(Color online) Comparison of one hundred and forty steady-state CFD pressure drop results with the three formulas discussed in the text.

(Color online) Comparison of one hundred and forty steady-state CFD pressure drop results with the three formulas discussed in the text.

(Color online) This series of images shows the shear rate (1/s) for the eight cases listed in Table II .

(Color online) This series of images shows the shear rate (1/s) for the eight cases listed in Table II .

(Color online) Inlet pressure from the CFD calculations for cases 2 (“straight”) and 5 (“tapered”) of Table II . The tapered hole was run with flow in both directions, and it can be seen that the two curves are practically on top of each other. The velocity is shown simply for timing reference.

(Color online) Inlet pressure from the CFD calculations for cases 2 (“straight”) and 5 (“tapered”) of Table II . The tapered hole was run with flow in both directions, and it can be seen that the two curves are practically on top of each other. The velocity is shown simply for timing reference.

(Color online) Dynamic flow resistances for 4 cases with straight holes (a) and 4 cases with tapered holes (b). CFD results (solid) are compared with results from Eq. (23) (dashed).

(Color online) Dynamic flow resistances for 4 cases with straight holes (a) and 4 cases with tapered holes (b). CFD results (solid) are compared with results from Eq. (23) (dashed).

(Color online) For a straight case (#2 in Table II ) the velocity magnitude (m/s) is shown in the upper row and the corresponding shear rate (1/s) is shown in the lower row. The times from left to right are at 30, 50, 70, and 90 μs.

(Color online) For a straight case (#2 in Table II ) the velocity magnitude (m/s) is shown in the upper row and the corresponding shear rate (1/s) is shown in the lower row. The times from left to right are at 30, 50, 70, and 90 μs.

(Color online) Resistance (a) and reactance (b) plots for a straight case (#2 from Table II ). Corresponding formula results using Eq. (22) , Eq. (1) (Maa), and Eq. (6) (Guo) with α = 2 and α = 4 are shown as well.

(Color online) For the tapered case (#5 in Table II ) the velocity field (m/s) is shown in the upper row and the corresponding shear rate (1/s) is shown in the lower row. The times from left to right are at 30, 50, 70, and 90 μs.

(Color online) For the tapered case (#5 in Table II ) the velocity field (m/s) is shown in the upper row and the corresponding shear rate (1/s) is shown in the lower row. The times from left to right are at 30, 50, 70, and 90 μs.

(Color online) Resistance (a) and reactance (b) plots for straight and tapered cases (#2 and #5 from Table). Corresponding formula results using Eq. (23) are shown as well. Only a slight difference between forward and reverse flow can be seen in the resistance plot.

(Color online) Resistance (a) and reactance (b) plots for straight and tapered cases (#2 and #5 from Table). Corresponding formula results using Eq. (23) are shown as well. Only a slight difference between forward and reverse flow can be seen in the resistance plot.

(Color online) Dynamic flow resistances for cases with equal static flow resistance, from Table II . Sketches of the eight hole designs, all on equal scales, are shown so the length and diameter of the holes can better be compared; additionally, the horizontal extent of the film (shown shaded) gives an indication of the hole density.

(Color online) Dynamic flow resistances for cases with equal static flow resistance, from Table II . Sketches of the eight hole designs, all on equal scales, are shown so the length and diameter of the holes can better be compared; additionally, the horizontal extent of the film (shown shaded) gives an indication of the hole density.

(Color online) Absorption coefficients for cases with equal static flow resistance, from Table II , and a rigidly terminated air space of 25 mm. Graph (a) shows results using the impedance from the CFD calculations, and graph (b) shows the results using the impedance from Eq. (23) .

## Tables

Parameters defining the steady-state cases.

Parameters defining the steady-state cases.

Parameters defining eight cases expected to have equal static flow resistance.

Parameters defining eight cases expected to have equal static flow resistance.

Parameters defining the time-dependent cases.

Parameters defining the time-dependent cases.

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