COMSOL 6.4 - Active Flame Validation

In jet engines and gas turbines, an interaction between the heat release from the flame and the acoustic resonances in the engines can lead to unstable modes that can be damaging to the engine. This occurs because oscillations in the fuel supply results in oscillations in the heat release. The oscillations in the heat release can interfere positively or negatively with the acoustic resonances in the engine such that the oscillations are either dampened or amplified.

To model this effect, it is necessary to have a model of the heat release and how it depends on the acoustic field. This model demonstrates the usage of the domain feature in the interface with a simple model that can be compared to an analytical solution.

Model Definition

The model consists of a simple 2D geometry, a rectangle with length L, representing a pipe with one closed end and one open end. At the closed end and the two sides, a hard wall boundary condition is used, while at the open end the acoustic pressure is set to zero. In the middle of the rectangle, there is a small domain with an active compact flame. To the left of the flame, the temperature is 300 K and to the right the flame has heated up the air to 1200 K.

The active flame is modeled with the domain feature , which adds a heat source in the domain that depends on the acoustic velocity at a reference point. The reference point is set to be on the left boundary of the domain. The heat q(x) released by the flame is given by

(1)

where τ(x) is the time-lag, nu(x) is the interaction index, Us is the steady velocity field, and qs is the steady heat release. To ease the analytical modeling of the eigenfrequencies, define the interaction index nu(x) by the parameter n as

(2)

where the specific heat capacity ratio γ and the background pressure p0 are used. Both τ and n are chosen to be independent of the spatial coordinates. The eigenfrequencies can be found analytically as functions of τ and n (see Ref. 1):

(3)

Here, Γ is given as Γ = ρ2c2/ρ1c1, where the subscript 1 refers to the material parameters to the left of the flame and subscript 2 to those to the right of the flame. The first cosine gives solutions where the active flame does not interfere with the eigenfrequency, while the second parenthesis gives the modes where the active flame impacts the eigenfrequency. The active flame will both shift the resonance frequency either up or down in frequency but it will also dampen or amplify the resonance mode. The damping or amplification is represented by the imaginary part of the eigenfrequency. With the used time convention of exp(iωt) a positive imaginary part represents damping of the mode while a mode with a negative imaginary part is amplified by the active flame. Note the time convention is opposite to the convention used in Ref. 1. A mode that is amplified by the active flame results in an unstable mode. Unstable modes in an engine can be damaging to the engine and they are therefore important to avoid during the design.

Results and Discussion

With an eigenfrequency study step, the first four eigenmodes of the system are found, corresponding to modes with 1/4, 3/4, 5/4, and 7/4 wavelengths in the pipe. The four modes are shown in Figure 1 with the closed end to the left and the open end to the right. Because of the difference in temperature on opposite sides of the active flame, the sound speed and therefore the wavelength is different on each side.

Figure 1: The four lowest eigenmodes of the pipe and active flame.

In Figure 2, the eigenfrequencies are plotted with the real part of the eigenfrequency on the x-axis and the imaginary part on the y-axis. The eigenfrequencies are plotted for both n = 0 (inactive flame) with blue plus signs and n = 5 (active flame) with green plus signs. The theoretical predictions are plotted with circles for both n = 0 and n = 5. It can be seen that all the eigenfrequencies for the inactive flame has zero imaginary part, this is because there is no damping in the system when the flame is inactive. For the second eigenfrequency (at around 700 Hz) the eigenfrequency of the active mode is identical to the eigenfrequency of the passive mode. This mode is not influenced by the active flame and is described by the first cosine in Equation 2.

Figure 2: The real and imaginary part of the eigenfrequencies plotted for the lowest four eigenmodes. The analytical solution plotted with blue circles, the simulations with a passive flame with blue plus signs and the active flame with the green plus signs.

For the first and fourth modes, the imaginary part of the eigenfrequency is positive with the active flame. This means that the modes are dampened by the active flame. This interaction between the acoustic field and the heat released from the flame also reduces the real part of the eigenfrequency. For the third mode, the imaginary part of the eigenfrequency is negative and the mode is amplified by the active flame. The real part of the frequency is shifted upward due to the active flame.

Notes About the COMSOL Implementation

The contribution from the flame model depends nonlinearly on the frequency through the exponential function in Equation 1. Therefore, the model uses the ARPACK nonlinear solver. In the nonlinear solver two settings are changed from the default. The is turned off, when it is turned an eigenvalue can be returned twice. Secondly, the is set to 100 because this is the order of the expected interval between the eigenvalues.

Reference

1. F. Nicoud, L. Benoit, C. Sensiau, and T. Poinsot, “Acoustic modes in combustors with complex impedances and multidimensional active flames,” AIAA J., vol. 45, no. 2,pp. 426–441, 2007.

Acoustics_Module/Tutorials,_Pressure_Acoustics/active_flame_validation

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

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

In the parameter file the analytical eigenfrequencies are calculated. They are calculated iteratively due to the form of the differential equation determining the eigenfrequencies.

Definitions

In the window, expand the > node.

Geometry 1

Rectangle 1 (r1)

In the window, expand the > node.

Right-click and choose .

In the window for , locate the section.

In the text field, type L.

In the text field, type L/10.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	L/2
Layer 2	delta_f

Select the checkbox.

Clear the checkbox.

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Add Material

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to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 1 only.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
T_s	300[K]	K
n_u	n/delta_fu_s/Q_s(acpr.rho*acpr.flm1.Cp)/acpr.flm1.alpha_p

Variables 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 2 and 3 only.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
T_s	1200[K]	K
n_u	n/delta_fu_s/Q_s(acpr.rho*acpr.flm1.Cp)/acpr.flm1.alpha_p

Pressure Acoustics, Frequency Domain (acpr)

Pressure Acoustics 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type T_s.

In the pA text field, type P_s.

Pressure 1

In the toolbar, click

and choose .

Select Boundary 10 only.

Flame Model 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

In the nu text field, type n_u.

In the τu text field, type tau_u.

In the Qs text field, type Q_s.

In the Us text field, type u_s.

Locate the section. From the list, choose .

Locate the section. Click to select the

toggle button.

Select Boundary 4 only.

Locate the section. Select the checkbox.

Mesh 1

In the window, under click .

In the window for , locate the section.

In the table, clear the checkbox for .

From the list, choose .

Click

Use the nonlinear eigenvalue solver since the flame model have a nonlinear dependency of the frequency.

Study 1

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type 5.

In the text field, type 10.

Find the subsection. In the text field, type 100.

In the text field, type 2000.

In the text field, type -20*n.

In the text field, type 20*n.

Set the factor to 100 as it is the order of the expected difference between the eigenvalues. is turned off, if it is turned on the solver might return an eigenvalue twice.

In the text field, type 100.

Clear the checkbox.

Parametric Sweep

In the toolbar, click