Uniform Layer Waveguide

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Uniform Layer Waveguide

Whenever dimensions in waveguides become small compared to the viscous and thermal boundary layers, it is necessary to model acoustics using thermoviscous acoustics. In the present model, the thermoviscous acoustic wave field in a shallow uniform waveguide is modeled and compared with an analytical solution.

Model Definition

An infinitely wide slit of length L and height H is subject to a harmonically varying pressure drop of 1 Pa. In order to reduce the model size, only a section of width L is modeled using symmetry conditions. The waveguide top and bottom are modeled as no slip isothermal walls.

Model parameters are summarized in Table 1.

Table 1: Loaded parameters.
Parameter	Expression	Description
f0	500 Hz	Frequency
T0	293 K	Ambient temperature
p0	1 atm	Atmospheric pressure
H	1 mm	Waveguide height
L	5 mm	Waveguide side lengths
pin	1 Pa	Inlet pressure
dvisc		Approximate expression for the viscous boundary layer thickness at 20oC and 1 atm.

Analytical Theory

A thermoviscous acoustics problem can in principle be thought of as a three wave problem, each with its own wave number: the acoustic k0, the viscous kv, and the thermal kh, with

where c0 is the isentropic speed of sound, ω the angular frequency, ρ0 is the static density, μ is the dynamic viscosity, Cp is the heat capacity, and k is the thermal conductance.

The thermal and viscous waves are rapidly decaying waves normal to a wall. The three waves interact and in the case of geometries with small dimensions, this interaction becomes evident. In simple geometries, analytical solutions of this interaction exist. In the case of a uniform slit, the cross-sectional variation of the temperature T and velocity u is:

where the function Ψ is a complex-valued scalar field

is either v or h. The pressure gradient and the pressure are in this simple geometry given by (disregarding the phase)

Results and Discussion

At 500 Hz, the characteristic size of the viscous boundary layer is about 0.1 mm (for air at 20°C), this length is 1/10 of the waveguide thickness H. This is seen in Figure 1 as the changing colors region near the wall where the velocity is varying to fulfill the no slip condition.

Figure 1: Slice plot of the instantaneous particle velocity U. Outside the boundary layer (red regions) the velocity profile becomes flat as in pressure acoustics.

The velocity and temperature profiles are probed using a 3D cut line and compared to the expressions found in the theory. The velocity profile is shown in Figure 2, while the temperature profile is shown in Figure 3. The results agree very well. When modeling acoustics in small dimensions, it is essential to include the thermal and viscous losses (see Ref. 1).

Figure 2: Comparison of the analytical and numerical velocity profiles.

Figure 3: Comparison of the analytical and the COMSOL generated solution for the amplitude of the acoustic temperature variation T.

1. H. Tijdeman, “On the propagation of sound waves in cylindrical tubes,” J. Sound Vib, vol. 39, no. 1, pp. 1–33, 1975.

Acoustics_Module/Verification_Examples/uniform_layer_waveguide

Modeling Instructions

From the menu, choose .

In the window, click .

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In the window, click .

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In the tree, select .

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In the tree, select .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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Browse to the model’s Application Libraries folder and double-click the file uniform_layer_waveguide_parameters.txt.

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In the window, under right-click and choose .

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In the window for , locate the section.

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Browse to the model’s Application Libraries folder and double-click the file uniform_layer_waveguide_variables.txt.

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type L.

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In the text field, type L.

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In the text field, type H.

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Locate the section. In the text field, type -H/2.

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Click the button in the toolbar.

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In the toolbar, click to open the window.

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Go to the window.

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In the tree, select .

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Click in the window toolbar.

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In the toolbar, click to close the window.

Thermoviscous Acoustics, Frequency Domain (ta)

Thermoviscous Acoustics Model 1

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In the window, under click .

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In the window for , locate the section.

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In the T0 text field, type T0.

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In the p0 text field, type p0.

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In the toolbar, click and choose .

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Select Boundaries 2 and 5 only.

Pressure (Adiabatic) 1

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In the toolbar, click and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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In the pbnd text field, type pin.

Pressure (Adiabatic) 2

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In the toolbar, click and choose .

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Select Boundary 6 only.

When modeling thermoviscous acoustics, the mesh is important as it needs to resolve the boundary layers properly. In the parameters list the quantity d_visc approximates the thickness of the viscous boundary layer, in air, for the frequency f0. This parameter is used in the boundary layer mesh properties in order to get a proper mesh.

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In the window, under right-click and choose .

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Select Boundary 1 only.

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Right-click and choose .

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Select Edges 4 and 6 only.

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In the window for , locate the section.

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In the text field, type 3.

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In the window, right-click and choose .

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In the window, right-click and choose .

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Boundary Layers 1

In the window, right-click and choose .

Boundary Layer Properties

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Select Boundaries 3 and 4 only.

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In the window for , locate the section.

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In the text field, type 3.

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From the list, choose .

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In the text field, type 0.8*d_visc.

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The finished mesh should look like that in the figure below.

Mesh of the uniform layer waveguide including the boundary layer mesh.

Step 1: Frequency Domain

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In the window, under click .

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In the window for , locate the section.

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In the text field, type f0.

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In the toolbar, click .

Acoustic Velocity (ta)

The second default plot shows a slice plot of the instantaneous particle velocity Figure 1.

To compare the analytical and numerical velocity and temperature profiles, as done in Figure 2 and Figure 3, follow the steps given below.

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In the toolbar, click .

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In the window for , locate the section.

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In row , set to L/2, to L/2, and to H/2.

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In row , set to L/2, to L/2, and to -H/2.

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1D Plot Group 4

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In the toolbar, click .

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In the window for , type Velocity Profile in the text field.

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Locate the section. From the list, choose .

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Right-click and choose .

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In the window for , locate the section.

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In the text field, type abs(u).

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Click to expand the section. Select the check box.

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From the list, choose .

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In the table, enter the following settings:


Legends
computed

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In the window, right-click and choose .

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In the window for , locate the section.

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In the text field, type abs(Uana).

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Click to expand the section. Find the subsection. From the list, choose .

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Find the subsection. From the list, choose .

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In the text field, type 20.

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Locate the section. Select the check box.

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From the list, choose .

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In the table, enter the following settings:


Legends
analytical

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In the toolbar, click .

Velocity Profile 1

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Right-click and choose .

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In the window for , type Temperature Profile in the text field.

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type abs(T).

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In the window, click .

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In the window for , locate the section.

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In the text field, type abs(Tana).

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In the toolbar, click .