CAE-Companion-2018-2019

Theory WISSEN CAE

The model The aimof the present model is to determine the impedance Z(ω) using a detailed thermoacousticmodel of a single hole as it is computationally impossible tomodel thewhole perforated plate. Themodel of one holewill give a precise value of Z in- cluding thermal losses and viscous losses as well as all hole-hole interactions (see Figure 1). Moreover, there are no free param- eters here. The so-called end correction is included explicitly when solving this detailed thermoacousticmodel.

Step 4: The pressure acoustics model using the Kirby and Cummings impedance model is solved. The results are shown in Figure 2: The transition loss has been plotted as a function of frequency for the different model versions and for experimental results by Selamet et al, 2003. The model that matches the experimental values best is the one including the thermoacoustic effect.

Outlet

Porous filling

Inlet

Figure 1: The thermoacoustic sub-model of one of the perforated plate holes (left) is used to determine lumped parameters which are then applied as internal impedance boundary condition in the full-scale model of the muffler (right). The effect of having a porous backing on one side of the perforated plates is however not included in the initial model. The semi-analytical Kirby-Cummings impedancemodel, taking a porous backing into account, is implemented for comparison. Now the simulation is performed in four steps: Step 1: The thermoacoustic sub-model of the single hole is solved; Step 2: The pressure acoustics model using the default perforated plate boundary condition is solved; Step 3: The pressure acoustics model using the impedance determined from the sub model is solved;

Figure 2: Transition loss as a function of frequency: Blue: without including the thermoacoustic effect, green: Thermoacoustic effect included, turquois: thermoacoustic and effect of porous backing included. The red dots show experimental results. References: Selamet et al., JSV, 262, p. 509 (2003) COMSOL, Acoustics Module User’s Guide, chapter 6, pp. 327-339 (2014)

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