Acta Mechanica

pp 1–17

# An acoustic model of a Helmholtz resonator under a grazing turbulent boundary layer

• Lewin Stein
• Jörn Sesterhenn
Original Paper

## Abstract

Acoustic models of resonant duct systems with turbulent flow depend on fitted constants based on expensive experimental test series. We introduce a new model of a resonant cavity, flush mounted in a duct or flat plate, under grazing turbulent flow. Based on previous work by Goody, Howe and Golliard, we present a more universal model where the constants are replaced by physically significant parameters. This enables the user to understand and to trace back how a modification of design parameters (geometry, fluid condition) will affect acoustic properties. The derivation of the model is supported by a detailed three-dimensional direct numerical simulation as well as an experimental test series. We show that the model is valid for low Mach number flows ($$M=0.01{-}0.14$$) and for low frequencies (below higher transverse cavity modes). Hence, within this range, no expensive simulation or experiment is needed any longer to predict the sound spectrum. In principle, the model is applicable to arbitrary geometries: Just the provided definitions need to be applied to update the significant parameters. Utilizing the lumped-element method, the model consists of exchangeable elements and guarantees a flexible use. Even though the model is linear, resonance conditions between acoustic cavity modes and fluid dynamic unstable modes are correctly predicted.

## List of symbols

$$Re_\tau$$

Friction Reynolds number $$u_{\tau } \delta _{99} / \nu =\delta _{99} / \delta _{\nu }$$.

$$St_\mathrm{TBL}$$

Strouhal number of the TBL $$\omega \,\delta _{99}/u_{0}$$.

$$St_\mathrm{neck}$$

Strouhal number of the neck $$\omega \,L_{x,\mathrm{neck}}/u_{+}$$ (Eq. (12)).

Y

Characteristic impedance $$Y=S/\rho c$$ of a duct with the constant cross section S.

Z

Complex-valued acoustic impedance $$Z=(r+i k l)Y$$.

$$\beta$$

Frequency-dependent tuning of the opening impedance amplitude (Eq. (16)).

$$\delta _\nu$$

Viscous lengthscale $$\nu /u_{\tau }$$.

$$\delta _{99,\mathrm{neck}}$$

Boundary layer thickness $$\delta _{99}$$ at the neck center (Fig. 1).

$$\delta _{99}$$

Boundary layer thickness $$u(y=\delta _{99})=0.99\,u_{0}$$.

$$\tau _w$$

Wall shear stress $$\rho _w \nu \, \mathrm{d}/\mathrm{d}y{<}u_x{>}|_{y=y_w}$$.

l

Acoustic reactance (length correction).

r

Acoustic resistance (excitation or damping).

$$u_{+}$$

Vortex sheet convection velocity (Eq. (15)).

$$u_{0}$$

Free stream velocity.

$$u_{\tau }$$

Wall friction velocity $$\sqrt{\tau _{w}/\rho _w}$$.

v

Acoustic volume flux.

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## Notes

### Acknowledgements

We gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (SE824/29-1) and the provision of computational resources by the High-Performance Computing Center Stuttgart (ACID11700).

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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

## Authors and Affiliations

• Lewin Stein
• 1
• Jörn Sesterhenn
• 1
1. 1.Institut für Strömungsmechanik und Technische Akustik, Technische Universität BerlinBerlinGermany