Acoustics/Flow-induced Oscillations of a Helmholtz Resonator

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

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]

Figure 2

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:

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

The equation that describes the Helmholtz resonator is the following:

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

$$\hat{P}_e$$: 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:

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

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

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

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)

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

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

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

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

$$ f_r$$: 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.

Figure 3

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.

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).



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.



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

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

This link: 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.