# Acoustics/Flow-induced Oscillations of a Helmholtz Resonator

![](Acoustics_flow_induced.JPG "Acoustics_flow_induced.JPG")

## Introduction

The importance of flow excited acoustic resonance lies in the large
number of applications in which it occurs. Sound production in organ
pipes, compressors, transonic wind tunnels, and open sunroofs are only a
few examples of the many applications in which flow excited resonance of
Helmholtz resonators can be found.\[4\] An instability of the fluid
motion coupled with an acoustic resonance of the cavity produce large
pressure fluctuations that are felt as increased sound pressure levels.

Passengers of road vehicles with open sunroofs often experience
discomfort, fatigue, and dizziness from self-sustained oscillations
inside the car cabin. This phenomenon is caused by the coupling of
acoustic and hydrodynamic flow inside a cavity which creates strong
pressure oscillations in the passenger compartment in the 10 to 50 Hz
frequency range. Some effects experienced by vehicles with open sunroofs
when buffeting include: dizziness, temporary hearing reduction,
discomfort, driver fatigue, and in extreme cases nausea. The importance
of reducing interior noise levels inside the car cabin relies primarily
in reducing driver fatigue and improving sound transmission from
entertainment and communication devices.

This Wikibook page aims to theoretically and graphically explain the
mechanisms involved in the flow-excited acoustic resonance of Helmholtz
resonators. The interaction between fluid motion and acoustic resonance
will be explained to provide a thorough explanation of the behavior of
self-oscillatory Helmholtz resonator systems. As an application example,
a description of the mechanisms involved in sunroof buffeting phenomena
will be developed at the end of the page.

## Feedback loop analysis

As mentioned before, the self-sustained oscillations of a Helmholtz
resonator in many cases is a continuous interaction of hydrodynamic and
acoustic mechanisms. In the frequency domain, the flow excitation and
the acoustic behavior can be represented as transfer functions. The flow
can be decomposed into two volume velocities.

:   qr: flow associated with acoustic response of cavity
:   qo: flow associated with excitation

## Acoustical characteristics of the resonator

### Lumped parameter model

The lumped parameter model of a Helmholtz resonator consists of a
rigid-walled volume open to the environment through a small opening at
one end. The dimensions of the resonator in this model are much less
than the acoustic wavelength, in this way allowing us to model the
system as a lumped system.

Figure 2 shows a sketch of a Helmholtz resonator on the left, the
mechanical analog on the middle section, and the electric-circuit analog
on the right hand side. As shown in the Helmholtz resonator drawing, the
air mass flowing through an inflow of volume velocity includes the mass
inside the neck (Mo) and an end-correction mass (Mend). Viscous losses
at the edges of the neck length are included as well as the radiation
resistance of the tube. The electric-circuit analog shows the resonator
modeled as a forced harmonic oscillator. \[1\] \[2\]\[3\]

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***Figure 2***

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V: cavity volume

$\rho$: ambient density

c: speed of sound

S: cross-section area of orifice

K: stiffness

$M_a$: acoustic mass

$C_a$: acoustic compliance

The equivalent stiffness K is related to the potential energy of the
flow compressed inside the cavity. For a rigid wall cavity it is
approximately:

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$K = \left(\frac{\rho c^2}{V}\right)S^2$

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The equation that describes the Helmholtz resonator is the following:

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$S \hat{P}_e =\frac{\hat{q}_e}{j\omega S}(-\omega ^2 M + j\omega R + K)$

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$\hat{P}_e$: excitation pressure

M: total mass (mass inside neck Mo plus end correction, Mend)

R: total resistance (radiation loss plus viscous loss)

From the electrical-circuit we know the following:

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$M_a = \frac{L \rho}{S}$

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$C_a = \frac{V}{\rho c^2}$

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$L ' = \ L + \ 1.7 \ re$

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The main cavity resonance parameters are resonance frequency and quality
factor which can be estimated using the parameters explained above
(assuming free field radiation, no viscous losses and leaks, and
negligible wall compliance effects)

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$\omega_r^2 = \frac{1}{M_a C_a}$

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$f_r = \frac{c}{2 \pi} \sqrt{\frac{S}{L' V}}$

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The sharpness of the resonance peak is measured by the quality factor Q
of the Helmholtz resonator as follows:

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$Q = 2 \pi \sqrt{V (\frac{L'} {S})^3}$

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$f_r$: resonance frequency in Hz

$\omega_r$: resonance frequency in radians

:   L: length of neck
:   L\': corrected length of neck

From the equations above, the following can be deduced:

-   The greater the volume of the resonator, the lower the resonance
    frequencies.
-   If the length of the neck is increased, the resonance frequency
    decreases.

### Production of self-sustained oscillations

The acoustic field interacts with the unstable hydrodynamic flow above
the open section of the cavity, where the grazing flow is continuous.
The flow in this section separates from the wall at a point where the
acoustic and hydrodynamic flows are strongly coupled. \[5\]

The separation of the boundary layer at the leading edge of the cavity
(front part of opening from incoming flow) produces strong vortices in
the main stream. As observed in Figure 3, a shear layer crosses the
cavity orifice and vortices start to form due to instabilities in the
layer at the leading edge.

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***Figure 3***

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From Figure 3, L is the length of the inner cavity region, d denotes the
diameter or length of the cavity length, D represents the height of the
cavity, and $\delta$ describes the gradient length in the grazing
velocity profile (boundary layer thickness).

The velocity in this region is characterized to be unsteady and the
perturbations in this region will lead to self-sustained oscillations
inside the cavity. Vortices will continually form in the opening region
due to the instability of the shear layer at the leading edge of the
opening.

## Applications to Sunroof Buffeting

### How are vortices formed during buffeting?

In order to understand the generation and convection of vortices from
the shear layer along the sunroof opening, the animation below has been
developed. At a certain range of flow velocities, self-sustained
oscillations inside the open cavity (sunroof) will be predominant.
During this period of time, vortices are shed at the trailing edge of
the opening and continue to be convected along the length of the cavity
opening as pressure inside the cabin decreases and increases. Flow
visualization experimentation is one method that helps obtain a
qualitative understanding of vortex formation and conduction.

The animation below shows, in the middle, a side view of a car cabin
with the sunroof open. As the air starts to flow at a certain mean
velocity Uo, air mass will enter and leave the cabin as the pressure
decreases and increases again. At the right hand side of the animation,
a legend shows a range of colors to determine the pressure magnitude
inside the car cabin. At the top of the animation, a plot of circulation
and acoustic cavity pressure versus time for one period of oscillation
is shown. The symbol x moving along the acoustic cavity pressure plot is
synchronized with pressure fluctuations inside the car cabin and with
the legend on the right. For example, whenever the x symbol is located
at the point where t=0 (when the acoustic cavity pressure is minimum)
the color of the car cabin will match that of the minimum pressure in
the legend (blue).

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![](theplot.gif "theplot.gif")

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The perturbations in the shear layer propagate with a velocity of the
order of 1/2Uo which is half the mean inflow velocity. \[5\] After the
pressure inside the cavity reaches a minimum (blue color) the air mass
position in the neck of the cavity reaches its maximum outward position.
At this point, a vortex is shed at the leading edge of the sunroof
opening (front part of sunroof in the direction of inflow velocity). As
the pressure inside the cavity increases (progressively to red color)
and the air mass at the cavity entrance is moved inwards, the vortex is
displaced into the neck of the cavity. The maximum downward displacement
of the vortex is achieved when the pressure inside the cabin is also
maximum and the air mass in the neck of the Helmholtz resonator (sunroof
opening) reaches its maximum downward displacement. For the rest of the
remaining half cycle, the pressure cavity falls and the air below the
neck of the resonator is moved upwards. The vortex continues displacing
towards the downstream edge of the sunroof where it is convected upwards
and outside the neck of the resonator. At this point the air below the
neck reaches its maximum upwards displacement.\[4\] And the process
starts once again.

### How to identify buffeting

Flow induced tests performed over a range of flow velocities are helpful
to determine the change in sound pressure levels (SPL) inside the car
cabin as inflow velocity is increased. The following animation shows
typical auto spectra results from a car cabin with the sunroof open at
various inflow velocities. At the top right hand corner of the
animation, it is possible to see the inflow velocity and resonance
frequency corresponding to the plot shown at that instant of time.

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![](curve.gif "curve.gif")

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It is observed in the animation that the SPL increases gradually with
increasing inflow velocity. Initially, the levels are below 80 dB and no
major peaks are observed. As velocity is increased, the SPL increases
throughout the frequency range until a definite peak is observed around
a 100 Hz and 120 dB of amplitude. This is the resonance frequency of the
cavity at which buffeting occurs. As it is observed in the animation, as
velocity is further increased, the peak decreases and disappears.

In this way, sound pressure level plots versus frequency are helpful in
determining increased sound pressure levels inside the car cabin to find
ways to minimize them. Some of the methods used to minimize the
increased SPL levels achieved by buffeting include: notched deflectors,
mass injection, and spoilers.

# Useful websites

This link: 1 takes you to the website of EXA
Corporation, a developer of PowerFlow for Computational Fluid Dynamics
(CFD) analysis.

This link:
2 is a
small news article about the current use of(CFD) software to model
sunroof buffeting.

This link:
3
is a small industry brochure that shows the current use of CFD for
sunroof buffeting.

# References

1.  Acoustics: An introduction to its Physical Principles and
    Applications ; Pierce, Allan D., Acoustical Society of America,
    1989.
2.  Prediction and Control of the Interior Pressure Fluctuations in a
    Flow-excited Helmholtz resonator ; Mongeau, Luc, and Hyungseok
    Kook., Ray W. Herrick Laboratories, Purdue University, 1997.
3.  Influence of leakage on the flow-induced response of vehicles with
    open sunroofs ; Mongeau, Luc, and Jin-Seok Hong., Ray W. Herrick
    Laboratories, Purdue University.
4.  Fluid dynamics of a flow excited resonance, part I: Experiment ;
    P.A. Nelson, Halliwell and Doak.; 1991.
5.  An Introduction to Acoustics ; Rienstra, S.W., A. Hirschberg.,
    Report IWDE 99--02, Eindhoven University of Technology, 1999.

