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On jet instability modes of a subsonic Hartmann whistle

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Abstract

Numerical experiments to understand the resonant acoustic response of a subsonic jet impinging on the mouth of a tube, known as the Hartmann whistle configuration, were performed as large-eddy simulations. The tube length was chosen so that its fundamental duct mode, for one end closed and one end open, would match the dominant mode in the exciting jet. When the tube mouth was placed in the path of a regular stream of vortex rings, formed by the instability of the jet’s bounding shear layer, a strong resonant, tonal response (whistling) was obtained. At three diameters from the jet, OASPL was 150–160 dB. A tube with a thicker lip generated a louder response. When the tube was held closer to the nozzle exit, the impinging unsteady shear layer could not provoke any significant resonance. The simulations reveal that the tonal response of a Hartmann whistle operating in subsonic mode is significant.

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Abbreviations

A :

area of the tube inlet (\(\hbox {m}^2\))

a :

speed of sound (\(\hbox {ms}^{-1}\))

f :

frequency (Hz)

L :

length of the tube (m)

M :

Mach number

p :

static pressure (\(\hbox {N}/\hbox {m}^2\))

\(p_0\) :

total pressure (\(\hbox {N}/\hbox {m}^2\))

R :

nozzle pressure ratio (\(p_0/p_{\infty }\))

Re:

Reynolds number

\(r_j\) :

jet radius (m)

S :

stand-off distance (m)

St:

Strouhal number (\(\textit{fU}_j/2r_0\))

\(U_j\) :

jet velocity at nozzle exit (\(\hbox {ms}^{-1}\))

x :

reduced length (no unit)

\(x_{in}\) :

value of variable x at nozzle exit

\(x_\infty \) :

value of variable x at ambient conditions

\(\rho \) :

density (\(\hbox {kg}/\hbox {m}^3\))

\(\theta \) :

polar angle (degrees)

\(\phi _{ij}(f)\) :

cross-power spectral density (\(\hbox {Pa}^2/\hbox {Hz}\))

\(\phi _{ii}(f)\) :

auto-power spectral density (\(\hbox {Pa}^2/\hbox {Hz}\))

\(\tau _{ij}\) :

shear stress tensor

\(\mu \) :

dynamic viscosity (\(\hbox {Nsm}^{-2}\))

PSD:

power spectral density (\(\hbox {Pa}^2/\hbox {Hz}\))

SPL:

sound pressure level (dB), \(\hbox {p}_{ref} = 20 \times 10^{-6}\ \hbox {Pa}\)

OASPL:

overall sound pressure level (dB)

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Acknowledgements

We thank Dr Santosh Hemachandra, Aerospace Engineering, IISc, for helpful discussions of this work.

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Author notes

  1. Ramanathan Varadharajan

    Present address: Physical Chemistry and Soft Matter, Wageningen University and Research Center, 6708 WE, Wageningen, The Netherlands

  2. Manu Kamin

    Present address: Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH, 45221, USA

Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Science, Bangalore, 560012, India

    Ramanathan Varadharajan, Manu Kamin, Subramanian Ganesh & Joseph Mathew

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  1. Ramanathan Varadharajan
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Correspondence to Ramanathan Varadharajan.

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Varadharajan, R., Kamin, M., Ganesh, S. et al. On jet instability modes of a subsonic Hartmann whistle. Sādhanā 43, 140 (2018). https://doi.org/10.1007/s12046-018-0921-z

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