COMSOL 6.4 - Nonlinear Transfer Impedance of a Tapered Orifice

Nonlinear Transfer Impedance of a Tapered Orifice

This model analyzes the nonlinear transfer impedance of a tapered orifice that can be part of a perforate or microperforated plate (MPP). The analysis is carried out for various degrees of tapering of the perforate and for a frequency range.

A linear analysis is first set up in the frequency domain using the Thermoviscous Acoustics, Frequency Domain interface as well as in the time domain using the Thermoviscous Acoustics, Transient interface (for the time domain, using only a few selected frequencies for comparison). A full nonlinear analysis is finally carried out in the time domain using the Thermoviscous Acoustics, Transient interface and the Nonlinear Thermoviscous Acoustics Contributions feature.

The results are compared with analytical and semi-analytical models in the simple straight cylindrical perforate configuration (circular orifice).

Figure 1: Schematic representation of the tapered orifice in the microperforated plate.

Model Definition

In this 2D axisymmetric model, a plane wave is incident normally on a microperforated plate (MPP) with a tapered orifice as depicted schematically in Figure 1. The wave is propagating in the positive axial z direction, and has the form

(1)

where k is the wavenumber and f0 the excitation frequency. The top radius is b, the orifice center radius is ac (0.4 mm in the model), the bottom radius is a, the taper ratio is a/b (the parameter ab in the model ranging from 0.5 to 1), and tp is the plate thickness (1 mm). In the model, the corners of the orifice include a small fillet of radius ac/50 in order to avoid any singularities at sharp edges.

An in-depth experimental and analytical analysis of the loss mechanisms and loss regimes in a microperforated plate is given in Temiz and others (Ref. 1), for the nontapered case when the taper ratio is 1. Some details or the modeling background are given in Ref. 2. In the linear regime, all losses are due to viscous dissipation in the Stokes layer (viscous boundary layer), which has the characteristic thickness

(2)

where μ is the dynamic viscosity, ω is the angular frequency, and ρ0 is the density. At 100 Hz in air, this characteristic viscous length is 0.22 mm. The ratio of the Stokes layer to the characteristic dimensions of the hole (the diameter d = 2a) is given by the Shear number

(3)

In an MPP, the Stokes layer spans almost the entire perforate such that Sh = O(1). Two cases can the be defined (Ref. 1):

•

The excitation is low and the transfer impedance is linear and only depends on the excitation frequency and the perforation geometry.

•

The excitation amplitude is larger than a given critical value and vortices start to form at sharp edges (local nonlinear effects). This leads to additional resistive losses.

To determine when vortices start to appear, the nondimensional Strouhal number Sr is introduced. It is defined as

(4)

where |ûp| is the absolute value of the averaged acoustic particle velocity in the center of the orifice. A large Strouhal number defines a regime where the particle displacement is small compared to the orifice diameter and no vortices will appear. For very small Strouhal numbers vortices are generated and convected away from the perforate. In between these two limits when Sr = O(1), local vortices appear near the sharp edges of the orifice (Ref. 1). In Temiz and others (equations 13 to 15 in Ref. 1), semi-analytical nonlinear corrections to the linear transfer impedance are given. These expressions are implemented as variables in the model under the node. An analytical expression for the linear transfer impedance is also set up; more details can be found in the Application Library model Transfer Impedance of a Perforate, as well as in the documentation for the feature in the Acoustics Module User’s Guide. The necessary time-averaged quantities that enter the analytical expressions are computed using the timeint() operator for integration in time, as well as spatial integration coupling operators.

Figure 2: Acoustic temperature variation (left half) and acoustic particle velocity (right half) for the linear frequency domain model at 100 Hz. Showing the three tapering configurations with a/b set to 1, 0.75, and 0.5.

Results and Discussion

The temporal evolution of the acoustic (perturbation) fields are best visualized using an in the user interface. As an example, acoustic temperature variation and the acoustic particle velocity is shown side by side in Figure 2 and Figure 3. In the first figure the (instantaneous) values are shown for the frequency domain simulation at 100 Hz for the three tapering configurations a/b equal to 1, 0.75, and 0.5. The time dependency is here given by the usual exp(iωt) and no nonlinear effects can be seen. In Figure 3 the time evolution from t = 0.045 s to 0.048 s is illustrated for the same 100 Hz excitation and for a/b equal to 0.5. The sequence of 4 images clearly shows local vortex shedding. Note that the vortex is highly damped as it moves away from the orifice, this is due to the chosen coarser mesh and the numerical stabilization. It is the dissipated energy in the orifice that dictates the transfer impedance magnitude.

Figure 3: Time evolution (from t = 0.045 s to 0.048 s) for the acoustic temperature variation (left half) and acoustic particle velocity (right half) for the nonlinear time domain simulation for the 100 Hz excitation and for a/b equal to 0.5.

The model implements and compares the COMSOL simulation results with an analytical linear and a semi-analytical nonlinear transfer impedance model. Some of the important results are given in the following Figure 4 to Figure 7.

The linear analytical model (see ) is first compared to the (linear) frequency domain results for the straight pipe configuration
(a/b = 1). Figure 4 shows the real, imaginary, and absolute values of the transfer impedance as function of frequency. The results show good correlation.

Figure 4: Comparison of the simulated and analytical linear transfer impedance as function of frequency. Image showing the real, imaginary, and absolute value for the straight orifice configuration.

A comparison between all the transfer impedance models is shown in Figure 5 and Figure 6, as function of frequency and Strouhal Sr number, respectively. The figures show the analytical linear and analytical nonlinear (see ) models for the straight pipe configuration (no analytical models exist for the tapered configurations). The figure also shows simulation results for the linear frequency domain and time domain, as well as the nonlinear time domain simulation results, all for the three tapering cases (a/b = 1, 0.75, and 0.5). The comparison between the linear frequency domain and time domain results is carried out to validate the computation procedure for the time domain transfer impedance. The results agree perfectly. The nonlinear time domain simulations results are seen to agree well with the semi analytical model from Ref. 1 (straight pipe configuration), validating the simulation method based on the Nonlinear Thermoviscous Acoustic setup. The method can be used to predict the linear and nonlinear acoustic behavior of any perforate configuration and geometry. As described above the nonlinear results deviate from the linear results for small Strouhal numbers (see Figure 6) where the dynamics exhibit stronger (local) nonlinear behavior, here local vortices are generated and shed from the orifice edges.

Finally, Figure 7 shows the relation between the Shear number Sh and Strouhal number Sr, for the range of studied frequencies (from 10 Hz to 500 Hz), for the frequency domain model.

Figure 5: Comparison of all the analytical and numerical simulation results. Analytical linear and nonlinear models for a/b = 1, simulation results showing the linear frequency and time domain (for a/b = 1, 0.75, and 0.5) and the nonlinear time-domain results (for a/b = 1, 0.75, and 0.5). Shown as function of frequency.

Figure 6: Comparison of all the analytical and numerical simulation results. Analytical linear and nonlinear models for a/b = 1, simulation results showing the linear frequency and time domain (for a/b = 1, 0.75, and 0.5) and the nonlinear time-domain results (for a/b = 1, 0.75, and 0.5). Shown as a function of the Strouhal number.

Figure 7: Shear number Sh versus Strouhal number Sr for all the studied frequencies from 10 Hz to 500 Hz, for the linear frequency-domain model.

References

1. M.A. Temiz, J. Tournadre, I.L. Arteaga, and A. Hirschberg, “Non-linear acoustic transfer impedance of micro-perforated plates with circular orifices,” J. Sound Vib., vol. 366, pp. 418–428, 2016.

2. M. Herring Jensen, “Simulation of Perforates: An Application of Nonlinear Thermoviscous Acoustics,” Acoustics Today, vol. 17, Summer 2021.

Acoustics_Module/Nonlinear_Acoustics/nonlinear_transfer_impedance

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