Simcenter STAR-CCM+ How to calculate the Transmission Loss of a Helmholtz resonator

In the picture below it is shown the classical side branch Helmholtz resonator scheme consisting of an enclosed volume of air (cavity) communicating with the external duct through a small opening neck:

The Helmholtz resonator behaves as an ideal acoustical mass-spring system. The volume of air in the cavity compresses and expands periodically due to the influx and efflux of air through the neck, making the cavity act as a spring with the fluid inertia (mass) concentrated in the neck. Part of the incident sound energy is absorbed by the Helmholtz resonator with maximal sound reduction that occurs at its resonant frequency. This characteristic frequency depends on the volume of the cavity and the geometry of the neck and is calculated using the formula below:

where: 

c = speed of sound

A = cross-sectional area of neck

V = volume of cavity

l = length of neck
 

The example attached to this article reproduces a side branch Helmholtz resonator and is shown schematically in the picture below:
 

 

P1, P2, P3, P4 are the four probe points for calculating the pressure signals placed in the same position of the microphones.

From an unperturbed flow (Ma = 0) we introduce a single sinusoidal acoustic pulse. This is realized by specifying a field function of the Mach number at the entrance of the duct (right-top corner in the plot) corresponding to around 20Pa amplitude pressure pulse.The pressure-time curves at inlet and outlet of the duct are shown below:
 

The pressure signals calculated at locations P1, P2, P3, P4 are dispayed in the plot:
 

Transmission Loss (TS) is an important acoustic parameter that measures the sound reduction performance of the Helmholtz resonator.

 

In this article we propose two methods for calculating the TL, one is based on 2-microphone measurements and the other on 4-microphone measurements. Both methods use a Java macro that elaborates the Fourier Transforms of pressure signals with Data Set Functions and computes the TL per each value of frequency.

Two-Microphone Method
The simplest way to define the Transmission Loss is the difference between the incident and the transmitted acoustic sound pressure levels measured before and after the Helmholtz resonator respectively:
 

In our example we perform the calculation using the two microphones P1 and P4.

The Java macro ComputeTwoPointsTransmissionLoss.java extracts the FFTs of the pressure signals P1 and P4 with Data Set Functions, per each value of frequency subtracts the two FFTs, saves the Transmission Loss data in a table and creates the TL-frequency plot.

F In the java macro you need to specify the start/end time for the FFTs analysis. These parameters depend on the specific case you are simulating.

Four-Microphone Method
A more elaborate formula is proposed by SAE using 4 points (microphones):
 

P1 and P2 are the incident acoustic pressures located before the Helmholtz resonator, P3 and P4 are the transmitted acoustic pressures located after the resonator. It is assumed that each couple of microphones are placed at the same distance s with P2 downstream of P1 and P4 downpstream of P3. 

kr  is the wave number which is a function of frequency f defined as:
 

c    = speed of sound

Ma = Mach number

The Java macro ComputeFourPointsTransmissionLoss.java computes the complex FFTs of the pressure signals P1, P2, P3, P4 with Data Set Functions, per each value of frequency calculates the TL, saves the Transmission Loss data in a table and creates the TL-frequency plot.

F  In the java macro you need to specify the reference value of Mach number, Ma, the distance between the microphones, s, and the start/end time for the FFTs analysis. All these parameters depend on the specific case you are simulating.

Two-microphone Method: Results
In our study we assume that the frequency of interest is within the range [0, 10000] Hz. The plot of Transmission Loss created by the java macro shows that the highest sound absorption occurs at frequency around 5400 Hz corresponding to a noise reduction of 45 dB. 
 

Four-microphone Method: Results
A similar result is achieved with the four-microphone method but we see the the TL plot indicates that the highest sound absorption occurs at frequency around 6000 Hz corresponding always to a noise reduction of 45 dB. 
 

Please notice that other approaches are available for simulating this case, for example the Perturbed Convective Wave Equation (PCWE) but also using Simcenter 3D.