models.aco.flow duct (1)

Solved with COMSOL Multiphysics 4.2

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F l ow Du c t

Introduction

The modeling of aircraft-engine noise is a central problem in the field of computational aeroacoustics. In this example you simulate the harmonically time-varying acoustic field from a turbofan engine under various conditions and calculate the attenuation of the acoustic noise made possible by introducing a layer of lining inside the engine duct.

Model Definition

Assume that the flow in the axisymmetric duct is compressible, inviscid, perfectly isentropic, and irrotational. In terms of variables made dimensionless by division by suitable combinations of a reference duct radius, R, a reference speed of sound, c, and a reference density, , it is described by

Here is the density, v equals the velocity, p denotes the pressure, c equals the speed of sound, and is the constant ratio of the specific heats at constant pressure and volume.

Because the flow is irrotational, you can describe the velocity field, v v r z t = , in terms of a potential, , defined by the equation v = . The basic time- and space-dependent variables describing the flow are then the velocity potential and the density, . These variables (and the velocity field itself) are split into a stationary mean-flow part and a harmonically time-varying acoustic part:

Also assume that the amplitudes of the acoustic variables are small compared to the corresponding mean-flow quantities. This allows for a linearization of the equations of

t------ v + 0=

vt------ v v+ p+ 0=

p

= c2 pdd-------

1–= =

eit+= v V v eit

+= aeit+=

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motion and the equation of state, and it extends the mean-flow + acoustic split to the pressure:

In particular, the linearized equations for the acoustic variables are

The duct geometry used in this model, depicted in Figure 1, is taken from Ref. 1. It is an approximate model of the inlet section of a turbofan engine in the very common CFM56 series.

Figure 1: The duct geometry.

The spinner and duct-wall profiles are given, respectively, by the equations

where 0zzL1, and L1.86393 is the duct length. A noise source is imposed at z0, henceforth referred to as the source plane. This is where the fan would be located in the actual engine geometry. The plane zL corresponds to the fore end of the engine and is referred to as the inlet plane.

p P p eit+=

ia – aV+ + 0=

i V + p=

p C2a=

R1 z max 0 0,64212 0,04777 0,98234 z2+ 1 2

– =

R2 z 1 0,18453 z2– 0,10158 e 11 1 z – – e 11––

1 e 11––

------------------------------------------+=

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For the reference quantities in this model, choose the duct radius, the mean-flow speed of sound, and the mean-flow density at the source plane. Hence, all three of these quantities take the value 1.

To facilitate the COMSOL Multiphysics modeling, add a set of auxiliary domains to the geometry:

• A cylindrical domain—adjoined at the inlet plane and extending to the terminal plane, z2.86393—extends the modeling domain into a region where you can consider the mean flow as being uniform. This allows you to impose the simple boundary condition of a constant velocity potential and a vanishing tangential velocity for the background flow at the terminal plane.

• PML domains, adjoined at the source and terminal planes, allow you to conveniently implement nonreflecting boundary conditions for the aeroacoustic field. At the source plane, the PML domain is split into three annular sections with the innermost and outermost sections damping both in the axial and radial directions, while the central one is damping only in the axial direction.

The remaining boundary conditions for the mean flow consist of a natural boundary condition specifying the mass-flow rate through the source plane via the normal velocity and the density; slip conditions (vanishing tangential velocity) at the duct wall and at the spinner; and axial symmetry at r0.

For the aeroacoustic field, the model considers two different boundary conditions at the duct wall:

• Sound hard—the normal component of the acoustic particle velocity vanishes at the boundary.

• Impedance—the normal component of the acoustic particle velocity is related to the particle displacement through the equation

where Z is the impedance. This boundary condition, first derived by Myers (Ref. 2), was later recast in a weak form by Eversman (Ref. 3); it is this weak version, which is directly suitable for finite element modeling, that is implemented in the Acoustics Module’s Aeroacoustics interface. The impedance boundary condition represents a lined duct wall. In this model, following Ref. 1, the impedance is taken to be Z2 i.

The spinner, in contrast, is always assumed to be acoustically hard.

i v n i V n n V –+ pZ---- =



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This study examines two cases for the mean-flow normal velocity component at the source plane, Vz, which (owing to the choice of reference speed) alternatively can be referred to as the source-plane axial Mach number, M: M0.5, approximately representative of a passenger aircraft at cruising speed, and M0.

The dimensionless angular frequency (nondimensionalized through division by Rc) is 16, and the circumferential mode number is m10. If you want to obtain a deeper understanding of the duct’s aeroacoustic characteristics, you can, of course, perform a systematic exploration of parameter space by varying these quantities independently.

Results and Discussion

T H E M E A N - F L OW F I E L D

For the nontrivial case of a source-plane axial Mach number of M0.5, the resulting mean-flow field appears in Figure 2. Note, in particular, that the velocity potential is uniform well beyond the terminal plane, thus justifying the boundary condition imposed there. Furthermore, as could be expected, deviations from the mean density value appear primarily near the nonuniformities of the duct geometry, such as at the tip of the spinner.

As a complement, a more quantitative picture of the variations of the mean-flow velocity and density profiles along the axial direction appear in the cross-section plots in Figure 3.

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Figure 2: Mean-flow velocity potential and density for source-plane Mach number M = 0.5.

Figure 3: Mean-flow cross-section plot at a sample radius of 0.8.



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T H E N O I S E S O U R C E

With the solution for the mean-flow field at hand, it is possible to calculate the corresponding eigenmodes for the acoustic field at the source plane. Figure 4 shows the resulting velocity-potential profile for the lowest mode. This is the boundary mode used as the source of the acoustic noise field in the duct for the case M 0.5.

Figure 4: The first axial boundary mode at the source plane (z = 0) for the case of a background flow with Mach number M = 0.5.

T H E A E R O A C O U S T I C F I E L D

The pressure fields for the case without a background mean flow, depicted in Figure 5, very closely match those for the corresponding FEM solutions presented in Figure 6 of Ref. 1. Similarly, the results for the attenuation between the source and inlet planes in the lined-wall case are in good agreement: 50.6 dB for the COMSOL Multiphysics solution versus 51.6 dB for the FEM solution in Ref. 1.

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Figure 5: Acoustic pressure field for the cases of hard (top) and lined (bottom) duct wall with no mean flow and at circumferential mode number m = 10 and angular frequency = 16.



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Turning to the case with a mean flow, the pressure field for the hard-wall case in the upper image of Figure 6 closely resembles the FEM solution obtained by Rienstra and Eversman in Ref. 1. For the lined-wall case in the lower image, although the agreement is still quite good, you can note some differences, especially near the source plane. This observation extends to the attenuation, for which the calculated value of 25.2 dB differs slightly more from the value of 27.2 dB obtained in Ref. 1.

However, these discrepancies have a natural explanation: the source mode in the COMSOL Multiphysics calculation was derived for the case of a hard duct wall, whereas Rienstra and Eversman used a noise source adapted to the acoustic lining. The lowest mode for the lined-wall case is a linear combination of the two forward-propagating hard-wall modes. Thus, the noise source term used to obtain the FEM solution visualized in the lower plot of Figure 6 is not optimally adapted to the duct, and it is consequently not maximally attenuated.

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Figure 6: Acoustic pressure distribution for the cases of hard (top) and lined (bottom) duct wall with mean flow (M = 0.5) and at circumferential mode number m = 10 and angular frequency = 16.



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Notes About the COMSOL Implementation

The model involves three physics interfaces, the last of which is used twice:

• Compressible Potential Flow (cpf)—for modeling the mean-flow velocity field

• Boundary Mode Aeroacoustics (aebm)—for calculating the boundary eigenmode to be used as the source of the acoustic noise in the mean-flow background

• Aeroacoustics (ae, ae2)—for modeling the time-harmonic acoustic field above and below the source plane

After an initial modeling stage—consisting of creating the geometry and the mesh, then defining parameters, variables, and model couplings—you proceed with three consecutive stages corresponding to the items in the bullet list above.

As explained in theModel Definition section, this model uses nondimensional variables obtained by dividing each variable by a suitable reference quantity of the same dimension. The reference length is the duct radius at the source plane (which is why it has the value 1). The mean-flow density and speed of sound at the source plane (z0) complete the set of reference variables.

References

1. S.W. Rienstra and W. Eversman, “A Numerical Comparison Between the Multiple-Scales and Finite-Element Solution for Sound Propagation in Lined Flow Ducts,” J. Fluid Mech., vol. 437, pp. 367–384, 2001.

2. M.K. Myers, “On the Acoustic Boundary Condition in the Presence of Flow,” J. Sound Vib., vol. 71, pp. 429–434, 1980.

3. W. Eversman, “The Boundary Condition at an Impedance Wall in a Non-Uniform Duct with Potential Mean Flow,” J. Sound Vib., vol. 246, pp. 63–69, 2001. Errata: ibid, vol. 258, pp. 791–792, 2002.

Model Library path: Acoustics_Module/Industrial_Models/flow_duct

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Modeling Instructions

Initial Stage—Geometry, Mesh, and Common Definitions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click the 2D axisymmetric button.

3 Click Next.

4 In the Add physics tree, select Acoustics>Aeroacoustics>Compressible Potential Flow

(cpf).

5 Click Add Selected.

6 In the Add physics tree, select Acoustics>Aeroacoustics>Boundary Mode Aeroacoustics

(aebm).

7 Click Add Selected.

8 In the Velocity potential edit field, type phi_b.

9 In the Add physics tree, select Acoustics>Aeroacoustics>Aeroacoustics (ae).

10 Click Add Selected twice.

11 Click Next.

12 In the Studies tree, select Custom Studies>Preset Studies for Some Physics>Stationary.

13 Click Finish.

R O O T

1 In the Model Builder window, click the root node.

2 Go to the Settings window for Root.

3 Locate the Unit System section. From the Unit system list, select None.

This setting turns off all unit support in the model.

G L O B A L D E F I N I T I O N S

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 Go to the Settings window for Parameters.



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

Alternatively, you can load the parameters from file. To do this, click the Load from File button, browse to the model’s Model Library folder, and then double-click the file flow_duct_parameters.txt.

G E O M E T R Y 1

First import the duct geometry, which is supplied in the form of an MPHBIN-file.

Import 11 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import.

2 Go to the Settings window for Import.

3 Locate the Import section. Click the Browse button.

4 Browse to the model’s Model Library folder and double-click the file flow_duct.mphbin.

5 Click the Import button.

Next, add the auxiliary cylindrical domain between the inlet plane at z = 1.86393 and the terminal plane at z = 2.86393.

Rectangle 11 In the Model Builder window, right-click Geometry 1 and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type 0.917050.

NAME EXPRESSION DESCRIPTION

gamma 1.4 Ratio of specific heats

M -0.5 Mean flow Mach number

m 10 Circumferential mode number

omega 16 Angular frequency

f omega/(2*pi) Frequency

rho0 1 Reference density

C0 1 Reference mean speed of sound

A 0.01 Acoustic source strength

Z 2-i Duct wall impedance

b 0.01 Impedance onset rate

zi 1.86393 Axial coordinate, inlet plane

zt 2.86393 Axial coordinate, terminal plane

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4 In the Height edit field, type 1.

5 Locate the Position section. In the z edit field, type zi.

6 Click the Build Selected button.

7 Click the Zoom Extents button on the Graphics toolbar.

Finally, attach cylindrical PML domains at both ends.




4 In the Height edit field, type 0.2.

5 Locate the Position section. In the z edit field, type zt.




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3 Locate the Size section. In the Width edit field, type 1.


5 Locate the Position section. In the r edit field, type 0.2.

6 In the z edit field, type -0.2.







5 Locate the Position section. In the r edit field, type 0.423557.

6 In the z edit field, type -0.2.


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Form Union1 In the Model Builder window, right-click Form Union and choose Build Selected.

2 The model geometry is now complete.

The axisymmetric model geometry including duct domain and auxiliary domains.

M E S H 1

Create a mapped mesh that is sufficiently fine to resolve the small-scale acoustic perturbations by following the instructions below.

Mapped 1In the Model Builder window, right-click Mesh 1 and choose Mapped.

Distribution 11 In the Model Builder window, right-click Mapped 1 and choose Distribution.

2 Select Boundary 1 only. This is the duct domain’s boundary along the symmetry axis.

3 Go to the Settings window for Distribution.

4 Locate the Distribution section. In the Number of elements edit field, type 39.


2 Select Boundary 3 only. This is the symmetry-axis boundary segment for the auxiliary domain above the duct.



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2 Select Boundaries 6 and 43 only. These are the source-plane and terminal-plane boundaries. Note that you can make the selection by clicking the Paste Selection button and typing the indices in the dialog box that opens.




2 Select Boundaries 5, 19, 20, 96, and 97 only.




2 Select Boundary 2 only.




2 Select Boundaries 8–18, 22–39, and 66–94 only.

You can do this most easily by copying the text ‘8–18, 22–39, and 66–94’ and then clicking in the Selection box and pressing Ctrl+V or by using the Paste Selection dialog box.




2 Select Boundaries 40 and 95 only.


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2 Select Boundaries 44–59 and 62–65 only.



5 Click the Build All button.

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

The meshed geometry.

D E F I N I T I O N S

Variables 11 In the Model Builder window, right-click Model 1>Definitions and choose Variables.

2 Go to the Settings window for Variables.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Domain.

4 Select Domains 1 and 2 only.



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

Next, define an expression for the source mode’s intensity component normal to the source boundary.

Variables 21 In the Model Builder window, right-click Definitions and choose Variables.


3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Boundary.



Proceed by defining integration operators for the source and inlet planes for use in computing the attenuation.

Integration 11 In the Model Builder window, right-click Definitions and choose Model

Couplings>Integration.

2 Go to the Settings window for Integration.

3 Locate the Operator Name section. In the Operator name edit field, type intop_src.

4 Locate the Source Selection section. From the Geometric entity level list, select Boundary.


Integration 21 In the Model Builder window, right-click Definitions and choose Model

Couplings>Integration.


C sqrt(gamma*cpf.pref*rho^(gamma-1)/cpf.rhoref^gamma)

Mean flow speed of sound

Mz -cpf.Vz Axial Mach number


Iz_src A^2*0.5*real((aebm.p/cpf.rhoref+aebm.Vr*aebm.vr-aebm.Vz*aebm.ikz*phi_b)*conj(rho*aebm.Vz-cpf.rhoref*aebm.ikz*phi_b))

Source mode intensity, normal component

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2 Go to the Settings window for Integration.

3 Locate the Operator Name section. In the Operator name edit field, type intop_inl.



Using these operators, define variables for the power through the source and inlet planes.

Variables 31 In the Model Builder window, right-click Definitions and choose Variables.



The factor 2*pi*r in these expressions account for the azimuthal direction.

Because the variables you just defined cannot be evaluated for the same solution data sets, it is not possible to define a variable for the attenuation. Instead, create an analytic function that returns the attenuation when supplied with two power values.

Analytic 11 In the Model Builder window, right-click Definitions and choose Functions>Analytic.

2 Go to the Settings window for Analytic.

3 Locate the Function Name section. In the Function name edit field, type dw.

4 Locate the Parameters section. In the Expression edit field, type 10*log10(w_src/w_in).

5 In the Arguments edit field, type w_src w_in.

6 Right-click Analytic 1 and choose Rename.

7 Go to the Rename Analytic dialog box and type Attenuation in the New name edit field. Click OK.

Stage I—The Background FlowIn this modeling stage you derive the stationary background on which propagate the time-harmonic acoustic perturbations that you model in Stage III. You calculate this


W_src intop_src(2*pi*r*Iz_src) Power through source plane

W_inl intop_inl(2*pi*r*ae.Iz) Power through inlet plane



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flow field using the Compressible Potential Flow interface defined on the duct geometry proper (Domain 1) and on the auxiliary region (Domain 2) appended at the inlet plane (z = 1.86393). Subsequently, you extend the flow to the PML domains by continuity. Impose a mass-flow boundary condition at the source plane and a normal-flow condition at the terminal plane (z = 2.86393). The duct wall and the spinner are both impervious to the flow.

C O M P R E S S I B L E PO T E N T I A L F L O W

1 In the Model Builder window, click Model 1>Compressible Potential Flow.


3 Go to the Settings window for Compressible Potential Flow.

4 Locate the Reference Values section. In the pref edit field, type cpf.rhoref^gamma/gamma.

5 In the ref edit field, type rho0.

6 In the vref edit field, type M.

Compressible Potential Flow Model 11 In the Model Builder window, expand the Compressible Potential Flow node, then click

Compressible Potential Flow Model 1.

2 Go to the Settings window for Compressible Potential Flow Model.

3 Locate the Compressible Potential Flow Model section. In the edit field, type gamma.

Normal Flow 11 In the Model Builder window, right-click Compressible Potential Flow and choose

Normal Flow.


Mass Flow 11 In the Model Builder window, right-click Compressible Potential Flow and choose Mass

Flow.


3 In the Model Builder window, collapse the Compressible Potential Flow node.

S T U D Y 1

Step 1: Stationary1 In the Model Builder window, click Study 1>Step 1: Stationary.

2 Go to the Settings window for Stationary.

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3 Locate the Physics Selection section. In the associated table, enter the following settings:

4 In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

Mean Flow Velocity (cpf)The first default plot group contains a surface plot of the mean flow velocity magnitude. To reproduce the plot shown in Figure 2, begin by plotting the density instead.

1 In the Model Builder window, expand the Mean Flow Velocity (cpf) node, then click Surface 1.

2 Go to the Settings window for Surface.

3 In the upper-right corner of the Expression section, click Replace Expression.

4 From the menu, choose Compressible Potential Flow>Density (rho).

5 Click the Plot button.


Next, add a contour plot of the mean-flow velocity potential by following these instructions.

7 In the Model Builder window, right-click Mean Flow Velocity (cpf) and choose Contour.

8 In the Model Builder window, click Mean Flow Velocity (cpf).

9 Go to the Settings window for 2D Plot Group.

10 Locate the Plot Settings section. Select the Title check box.

11 In the associated edit field, type Surface: Density Contour: Mean-flow velocity potential.

12 Right-click Mean Flow Velocity (cpf) and choose Rename.

13 Go to the Rename 2D Plot Group dialog box and type rho, Phi in the New name edit field. Click OK.

PHYSICS INTERFACE USE

Boundary Mode Aeroacoustics (aebm) ×

Aeroacoustics (ae) ×

Aeroacoustics (ae2) ×



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Compare the result with Figure 2.

To get a detailed view of the density and velocity profiles along the length of the duct, proceed as follows:

Data Sets1 In the Model Builder window, right-click Results>Data Sets and choose Cut Line 2D.

2 Go to the Settings window for Cut Line 2D.

3 Locate the Line Data section. In row Point 1, set r to 0.8.

4 In row Point 2, set r to 0.8.

5 In row Point 2, set z to 1.86393.



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Mean Flow Velocity, 3D (cpf)The second default plot group is a 225-degree revolution plot of the velocity potential.

Remove this 3D Plot Group node and replace it with a 1D Plot Group.

1 In the Model Builder window, right-click Results>Mean Flow Velocity, 3D (cpf) and choose Delete.

2 Click Yes to confirm.

1D Plot Group 21 Right-click Results and choose 1D Plot Group.


3 Locate the Data section. From the Data set list, select Cut Line 2D 1.

4 Right-click Results>1D Plot Group 2 and choose Line Graph.

5 Go to the Settings window for Line Graph.

6 In the upper-right corner of the y-Axis Data section, click Replace Expression.

7 From the menu, choose Density (rho).

8 In the upper-right corner of the x-Axis Data section, click Replace Expression.

9 From the menu, choose Geometry and Mesh>Coordinate>z-coordinate (z).

10 Click to expand the Coloring and Style section.

11 Find the Line style subsection. From the Color list, select Blue.



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13 Right-click Line Graph 1 and choose Duplicate.



16 From the menu, choose Definitions>Axial Mach number (Mz).

17 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, select Dashed.

18 From the Color list, select Red.


Finish the plot by updating the title and axis labels.

20 In the Model Builder window, click 1D Plot Group 2.



23 In the associated edit field, type Mean flow cross-section plots at r=0.8.

24 Locate the Plot Settings section. Select the x-axis label check box.

25 In the associated edit field, type z-Coordinate.

26 Select the y-axis label check box.

27 In the associated edit field, type Density (solid), axial Mach number (dashed).

28 Right-click 1D Plot Group 2 and choose Rename.

29 Go to the Rename 1D Plot Group dialog box and type rho, Mz in the New name edit field. Click OK.


The resulting plot should closely resemble that in Figure 3.

Stage II—The Boundary Source Mode

P H Y S I C S

As the source generating the acoustic field in the duct, use a single boundary mode imposed at z = 0. More specifically, take this mode to be the lowest propagating axial mode in the duct computed in the background flow field from the previous stage of the modeling process. The subsequent instructions demonstrate how to derive this boundary mode.

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B O U N D A R Y M O D E A E R O A C O U S T I C S

1 In the Model Builder window, click Model 1>Boundary Mode Aeroacoustics.

2 Go to the Settings window for Boundary Mode Aeroacoustics.

3 Locate the Boundary Selection section. Click Clear Selection.


5 Locate the Equation section. In the m edit field, type m.

Aeroacoustics Model 1The following settings couple the aeroacoustics boundary mode to the background flow:

1 In the Model Builder window, expand the Boundary Mode Aeroacoustics node, then click Aeroacoustics Model 1.

2 Go to the Settings window for Aeroacoustics Model.

3 Locate the Aeroacoustics Model section. From the list, select Density (cpf/cpf1).

4 From the cmf list, select Mean flow speed of sound (cpf/cpf1).

5 Specify the V vector as

6 In the Model Builder window, collapse the Boundary Mode Aeroacoustics node.

S T U D Y 1

Step 2: Mode Analysis1 In the Model Builder window, right-click Study 1 and choose Study Steps>Mode

Analysis.

2 Go to the Settings window for Mode Analysis.

3 Locate the Physics Selection section. In the associated table, enter the following settings:

4 Locate the Study Settings section. In the Desired number of modes edit field, type 12.

cpf.Vr r

cpf.Vz z

PHYSICS INTERFACE USE

Compressible potential flow (cpf) ×

Aeroacoustics (ae) ×

Aeroacoustics (ae2) ×



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5 In the Mode analysis frequency edit field, type f.

Before solving, add a parametric sweep over the axial Mach number in preparation for studying the acoustic field in the duct both in the absence and presence of a background flow.

Parametric Sweep1 In the Model Builder window, right-click Study 1 and choose Parametric Sweep.

2 Go to the Settings window for Parametric Sweep.

3 Locate the Study Settings section. Under Parameter names, click Add.

4 Go to the Add dialog box.

5 In the Parameter names list, select M (Mean flow Mach number).

6 Click the OK button.

7 Go to the Settings window for Parametric Sweep.

8 Locate the Study Settings section. In the Parameter values edit field, type 0 -0.5.

9 In the Model Builder window, collapse the Study 1 node.

10 In the Model Builder window, click Study 1.

11 Go to the Settings window for Study.

12 Locate the Study Settings section. Clear the Generate default plots check box.

This disables the generation of additional default plots.

13 Click the Compute button.

R E S U L T S

Reproduce the plot of the desired boundary mode shown in Figure 4 as follows:

1D Plot Group 31 In the Model Builder window, right-click Results and choose 1D Plot Group.


3 Locate the Data section. From the Data set list, select Solution 1.

4 From the Out-of-plane wave number selection list, select From list.

Inspecting the Out-of-plane wave number list you find four solutions with a purely real wave number, three of them positive and one negative. In other words, there are four propagating waves, three of which propagate in the positive z direction and one in the opposite direction. The strong background flow has shifted the wave

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numbers, which in the absence of a mean flow would be symmetrically distributed around zero.

Select the wave propagating in the positive z direction:

5 In the Out-of-plane wave number list, select 5.778448.

6 Right-click Results>1D Plot Group 3 and choose Line Graph.


8 In the upper-right corner of the Selection section, click Activate Selection.



11 From the menu, choose Boundary Mode Aeroacoustics>Velocity potential (phi_b).

12 In the upper-right corner of the x-Axis Data section, click Replace Expression.

13 From the menu, choose Geometry and Mesh>Coordinate>r-coordinate (r).




17 In the associated edit field, type Line Graph: Velocity potential.

18 Locate the Plot Settings section. Select the x-axis label check box.

19 In the associated edit field, type r-Coordinate.

20 Select the y-axis label check box.

21 In the associated edit field, type Velocity potential.


23 Go to the Rename 1D Plot Group dialog box and type phi_b in the New name edit field. Click OK.


Data SetsNext, add two solution data sets for use when computing the attenuation and give them suitable names.

1 In the Model Builder window, right-click Results>Data Sets and choose Solution.

2 Go to the Settings window for Solution.

3 Locate the Solution section. From the Solution list, select M=0.

4 Right-click Solution 4 and choose Rename.



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5 Go to the Rename Solution dialog box and type aebm (M=0) in the New name edit field. Click OK.

6 Right-click Data Sets and choose Solution.

7 Go to the Settings window for Solution.

8 Locate the Solution section. From the Solution list, select M=-0.5.


10 Go to the Rename Solution dialog box and type aebm (M=-0.5) in the New name edit field. Click OK.

Similarly, give the Solution 2 data set a more descriptive name; this data set contains the stored solution for the compressible potential flow that you computed first. Because it does not involve any eigenmode index, it is the natural one to use for the first two plot groups.


12 Go to the Rename Solution dialog box and type cpf in the New name edit field. Click OK.

13 In the Model Builder window, click Cut Line 2D 1.

14 Go to the Settings window for Cut Line 2D.

15 Locate the Data section. From the Data set list, select cpf.

rho, Phi1 In the Model Builder window, click Results>rho, Phi.


3 Locate the Data section. From the Data set list, select cpf.

Derived ValuesNow use the data sets you added to compute the power through the source plane for the cases without and with flow.

1 In the Model Builder window, right-click Results>Derived Values and choose Global

Evaluation.

2 Go to the Settings window for Global Evaluation.

3 Locate the Data section. From the Data set list, select aebm (M=0).




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7 From the menu, choose Definitions>Power through source plane (W_src).

8 Click the Evaluate button.

9 Right-click Global Evaluation 1 and choose Rename.

10 Go to the Rename Global Evaluation dialog box and type W_src (M=0) in the New

name edit field. Click OK.

11 In the Model Builder window, expand the Results>Tables node.

12 Right-click Results>Tables>Table 1 and choose Rename.

13 Go to the Rename Table dialog box and type W_src (M=0) in the New name edit field. Click OK.

14 Right-click Derived Values and choose Global Evaluation.


16 Locate the Data section. From the Data set list, select aebm (M=-0.5).




20 From the menu, choose Definitions>Power through source plane (W_src).

21 Click the small triangle in the Settings window toolbar and choose New Table from the menu.


23 Go to the Rename Global Evaluation dialog box and type W_src (M=-0.5) in the New



25 Go to the Rename Table dialog box and type W_src (M=-0.5) in the New name edit field. Click OK.

Stage III—The Acoustic FieldEquipped with the solution derived in the two previous stages, you can now go on to simulate the acoustic field. Model the noise source through judicious choices of boundary conditions at the source plane (z = 0) for the two time-harmonic Aeroacoustics interfaces. Furthermore, implement non-reflecting boundary conditions at both ends of the duct geometry by using the auxiliary PML domains that you added to the model earlier in the geometry creation steps.

Here are the detailed instructions for the procedure.



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D E F I N I T I O N S

First, add an extrusion coupling that allows you to extend the compressible potential flow quantities rho and C to the PML domain outside the terminal plane (z = 2.86393).

General Extrusion 11 In the Model Builder window, right-click Model 1>Definitions and choose General

Extrusion.

2 Go to the Settings window for General Extrusion.



5 Locate the Destination Map section. Clear the y-expression edit field.

6 Locate the Source section. Select the Use source map check box.

7 Clear the y-expression edit field.

Clearing these edit fields gives a one-dimensional map that only depends on the radial coordinate, r.

Also, create a maximum coupling operator for the duct domain and use it to define a normalized absolute pressure variable.

Maximum 11 In the Model Builder window, right-click Definitions and choose Model

Couplings>Maximum.

2 Select Domain 1 only.

Variables 11 In the Model Builder window, click Variables 1.



A E R O A C O U S T I C S

1 In the Model Builder window, click Model 1>Aeroacoustics.

2 Select Domains 1–3 only.

3 Go to the Settings window for Aeroacoustics.


pabsn abs(ae.p)/mod1.maxop1(ae.p) Normalized pressure

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4 Click to expand the Equation section.

5 In the m edit field, type m.

Aeroacoustics Model 1Couple the aeroacoustics field to the mean-flow field just like you did for the boundary mode.

1 In the Model Builder window, expand the Aeroacoustics node, then click Aeroacoustics

Model 1.


3 Locate the Aeroacoustics Model section. From the list, select Density (cpf/cpf1).

4 From the cmf list, select Mean flow speed of sound (cpf/cpf1).


Aeroacoustics Model 21 In the Model Builder window, right-click Aeroacoustics and choose Aeroacoustics

Model.



4 Locate the Aeroacoustics Model section. From the list, select User defined. In the associated edit field, type genext1(rho).

5 From the cmf list, select User defined. In the associated edit field, type genext1(C).


Perfectly Matched Layers 11 In the Model Builder window, right-click Aeroacoustics and choose Perfectly Matched

Layers.


3 Go to the Settings window for Perfectly Matched Layers.

4 Locate the Geometric Settings section. From the Type list, select Cylindrical.

cpf.Vr r

cpf.Vz z

0 r

genext1(cpf.Vz) z



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Aeroacoustics Model 11 In the Model Builder window, expand the Perfectly Matched Layers 1 node, then click

Aeroacoustics Model 1.


3 Locate the Aeroacoustics Model section. From the list, select User defined. In the associated edit field, type genext1(rho).

4 From the cmf list, select User defined. In the associated edit field, type genext1(C).


Normal Mass Flow 11 Right-click Aeroacoustics and choose Normal Mass Flow.


3 Go to the Settings window for Normal Mass Flow.

4 Locate the Normal Mass Flow section. In the mn edit field, type rho*(-aebm.ikz*A*phi_b).

Impedance 11 In the Model Builder window, right-click Aeroacoustics and choose Impedance.

2 Select Boundaries 44–59 and 62–95 only.

3 Go to the Settings window for Impedance.

4 Locate the Impedance section. In the Zi edit field, type Z/flc2hs(z/zi,b).

The reason behind using the smoothed Heaviside function flc2hs is to make the impedance a continuous (albeit abruptly changing) function across the interfaces between regions with and without an acoustic lining. This is a condition required for the equivalence of Myers’s original impedance boundary condition and its weak reformulation due to Eversman used here to hold (see Ref. 3).

Impedance 21 In the Model Builder window, right-click Aeroacoustics and choose Impedance.


3 Go to the Settings window for Impedance.

4 Locate the Impedance section. In the Zi edit field, type Z/flc2hs((zt-z)/(zt-zi),b).

0 r

genext1(cpf.Vz) z

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A E R O A C O U S T I C S 2

1 In the Model Builder window, click Model 1>Aeroacoustics 2.

2 Select Domains 1 and 4–6 only.

Here, the inclusion of Domain 1—which partly overrides the first Aeroacoustics interface—is just temporary. Because the true geometric scope for the second Aeroacoustics interface, Domains 4–6, is in full a PML, the automatic scaling algorithm for the Perfectly Matched Layers node would fail if you set the selection to Domains 4–6 before setting up the PML.

3 Go to the Settings window for Aeroacoustics.

4 Click to expand the Equation section.

5 In the m edit field, type m.

Aeroacoustics Model 11 In the Model Builder window, expand the Aeroacoustics 2 node, then click


You do not need to make any settings for this node because it will be overridden by the corresponding nodes for the Perfectly Matched Layers that you set up next.

Perfectly Matched Layers 11 Right-click Aeroacoustics 2 and choose Perfectly Matched Layers.

2 Select Domains 4–6 only.


4 Locate the Geometric Settings section. From the Type list, select Cylindrical.

Aeroacoustics Model 11 In the Model Builder window, expand the Perfectly Matched Layers 1 node, then click



3 Locate the Aeroacoustics Model section. From the list, select User defined. In the associated edit field, type rho0.

4 From the cmf list, select User defined. In the associated edit field, type C0.


0 r

M z



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Aeroacoustics Model 21 In the Model Builder window, right-click Perfectly Matched Layers 1 and choose

Aeroacoustics Model.



4 Locate the Aeroacoustics Model section. From the list, select User defined. In the associated edit field, type rho0.

5 From the cmf list, select User defined. In the associated edit field, type C0.

Perfectly Matched Layers 11 In the Model Builder window, click Perfectly Matched Layers 1.


3 Locate the Geometric Settings section. From the Type list, select General.

Note that three Manual Scaling nodes appear under the Perfectly Matched Layers node. You can now remove Domain 1 from the selection for this feature node.

4 In the Model Builder window, click Aeroacoustics 2.

5 Select Domain 1 only, then click the Remove from Selection button.

Velocity Potential 11 In the Model Builder window, right-click Aeroacoustics 2 and choose Velocity

Potential.


3 Go to the Settings window for Velocity Potential.

4 Locate the Velocity Potential section. In the edit field, type phi-A*phi_b.

5 In the Model Builder window, collapse the Aeroacoustics 2 node.

M O D E L W I Z A R D

1 In the Model Builder window, right-click the root node and choose Add Study.

2 Go to the Model Wizard window.

3 In the Studies tree, select Custom Studies>Preset Studies for Some Physics>Frequency

Domain.

4 Click Finish.

0

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S T U D Y 2

Step 1: Frequency Domain1 In the Model Builder window, click Study 2>Step 1: Frequency Domain.

2 Go to the Settings window for Frequency Domain.

3 Locate the Study Settings section. In the Frequencies edit field, type f.

4 In the Model Builder window, click Study 2.

5 Go to the Settings window for Study.

6 Locate the Study Settings section. Clear the Generate default plots check box.

For this study, you will create and set up the plots manually.

7 Right-click Study 2 and choose Show Default Solver.

Solver 61 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 6

node, then click Dependent Variables 1.

2 Go to the Settings window for Dependent Variables.

3 Locate the General section. From the Defined by study step list, select User defined.

4 Locate the Values of Variables Not Solved For section. From the Method list, select Solution.

5 From the Solution list, select Parametric 3.

6 From the Use list, select M=0.

7 From the Out-of-plane wave number list, select 10.837405.

8 In the Model Builder window, right-click Solver 6 and choose Rename.

9 Go to the Rename Solver dialog box and type M=0, lined in the New name edit field. Click OK.

10 Right-click Solver 6 and choose Compute.

R E S U L T S

Derived ValuesBefore setting up a plot of the normalized pressure distribution in the duct, apply a Selection node to the data set for the solution you just computed that restricts the data set to the duct domain. Also, rename the data set for easy identification.

1 In the Model Builder window, expand the Results>Derived Values node.



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Data Sets1 In the Model Builder window, expand the Results>Data Sets node.


3 Go to the Rename Solution dialog box and type ae (M=0, lined) in the New name edit field. Click OK.

4 Right-click Solution 6 and choose Add Selection.

5 Go to the Settings window for Selection.



2D Plot Group 41 In the Model Builder window, right-click Results and choose 2D Plot Group.


3 Locate the Data section. From the Data set list, select ae (M=0, lined).

4 Right-click Results>2D Plot Group 4 and choose Contour.

5 Go to the Settings window for Contour.


7 From the menu, choose Definitions>Normalized pressure (pabsn).

8 Locate the Levels section. From the Entry method list, select Levels.

9 In the Levels edit field, type 0.0001 0.001 0.01 0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9.

10 Locate the Coloring and Style section. From the Contour type list, select Filled.

11 Right-click Contour 1 and choose Duplicate.

12 Go to the Settings window for Contour.

13 Locate the Coloring and Style section. From the Contour type list, select Lines.

14 From the Coloring list, select Uniform.




18 In the associated edit field, type Normalized pressure (lined wall, no flow).


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20 Go to the Rename 2D Plot Group dialog box and type p (M=0, lined) in the New


p (M=0, lined)1 In the Model Builder window, collapse the Results>p (M=0, lined) node.



Compare the plot with the lower plot in Figure 5.

S T U D Y 2

Next compute the solution for the case with a background flow with axial Mach number M = 0.5. To keep the solution for the case M = 0, disable the corresponding solver sequence before generating a new one.

M=0, lined1 In the Model Builder window, right-click Study 2>Solver Configurations>M=0, lined

and choose Rename.

2 Go to the Rename Solver dialog box and type M=0, lined in the New name edit field. Click OK.

3 Right-click Study 2>Solver Configurations>M=0, lined and choose Disable.

Solver 71 Right-click Study 2 and choose Show Default Solver.

2 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 7 node, then click Dependent Variables 1.





7 From the Use list, select M=-0.5.



10 Go to the Rename Solver dialog box and type M=-0.5, lined in the New name edit field. Click OK.




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R E S U L T S

Data SetsAgain, restrict the new solution data set to the duct domain and give it a descriptive name.

1 In the Model Builder window, right-click Solution 7 and choose Rename.

2 Go to the Rename Solution dialog box and type ae (M=-0.5, lined) in the New


3 Right-click Solution 7 and choose Add Selection.




Now use this data set for a filled contour plot of the same kind as the one you created for the case M = 0. Start by duplicating the latter plot.

p (M=0, lined) 11 In the Model Builder window, right-click Results>p (M=0, lined) and choose Duplicate.


3 Locate the Data section. From the Data set list, select ae (M=-0.5, lined).

4 Locate the Title section.

5 In the associated edit field, type Normalized pressure (lined wall, flow).

6 Right-click 1 and choose Rename.

7 Go to the Rename 2D Plot Group dialog box and type p (M=-0.5, lined) in the New name edit field. Click OK.


The plot in the Graphics window should look like the lower plot in Figure 6.

Derived ValuesProceed to compute the attenuation.

1 In the Model Builder window, right-click Results>Derived Values and choose Global

Evaluation.




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5 From the menu, choose Definitions>Power through inlet plane (W_inl).



8 Go to the Rename Global Evaluation dialog box and type W_inl (M=0) in the New




11 Locate the Data section. From the Data set list, select ae (M=-0.5, lined).


13 From the menu, choose Power through inlet plane (W_inl).



16 Go to the Rename Global Evaluation dialog box and type W_inl (M=-0.5) in the New


17 In the Model Builder window, expand the Results>Tables node.


19 Go to the Rename Table dialog box and type W_inl (M=0) in the New name edit field. Click OK.


21 Go to the Rename Table dialog box and type W_inl (M=-0.5) in the New name edit field. Click OK.



24 Go to the Rename Global Evaluation dialog box and type Attenuation in the New




Use the function mod1.dw(W_src, W_inl) that you defined earlier with the computed power values as argument. Begin with the case M = 0:



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27 Locate the Expression section. In the Expression edit field, type mod1.dw(0.027961, 2.420846e-7).

Here, the mod1. prefix is necessary to specify the function’s scope; recall that you added it under Model 1>Definitions.

28 Select the Description check box.

29 In the associated edit field, type Attenuation (M=0, dB).


The result should be approximately 50.6 dB, which is quite close to that in Ref. 1 (51.6 dB). The reason is that both use the same source mode as a result of the fact that there is a single forward-propagating mode in the flow-free case, thus making the two calculations directly comparable.

31 In the Expression edit field, type mod1.dw(9.519e-3, 2.88782e-5).

32 In the Description edit field, type Attenuation (M=-0.5, dB).

33 Click the small triangle in the Settings window toolbar and choose Table 5 -

Attenuation (mod1.dw(0.027961, 2.420846e-7)).

The result should now be roughly to 25.2 dB. Compare this result to the value of approximately 27 dB obtained for the corresponding quantity in Ref. 1. In contrast to that paper, the source mode used in these calculations was derived for the case of a hard duct wall, whereas the lowest mode for the lined-wall case would be a linear combination of the two forward-propagating hard-wall modes. For this reason, the noise source is not an eigenmode and is, consequently, not maximally attenuated.


35 Go to the Rename Table dialog box and type Attenuation in the New name edit field. Click OK.

A E R O A C O U S T I C S

Now compute the acoustic pressure fields for the case of a hard duct wall. To do this, simply disable the Impedance nodes so that the default condition applies.

Impedance 21 In the Model Builder window, right-click Impedance 1 and choose Disable.

2 Right-click Impedance 2 and choose Disable.

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S T U D Y 2

First consider the case without a background mean-flow field. As before, disable the currently enabled solver sequence and then generate a new one.

M=-0.5, linedIn the Model Builder window, right-click Study 2>Solver Configurations>M=-0.5, lined and choose Disable.







7 From the Use list, select M=0.



10 Go to the Rename Solver dialog box and type M=0, hard in the New name edit field. Click OK.


R E S U L T S

Follow the instructions below to reproduce the upper plot in Figure 5.

Data Sets1 In the Model Builder window, right-click Results>Data Sets>Solution 8 and choose

Rename.

2 Go to the Rename Solution dialog box and type ae (M=0, hard wall) in the New


3 Right-click Results>Data Sets>Solution 8 and choose Add Selection.





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p (M=-0.5, lined) 11 In the Model Builder window, right-click Results>p (M=-0.5, lined) and choose

Duplicate.


3 Locate the Data section. From the Data set list, select ae (M=0, hard wall).


5 In the associated edit field, type Normalized pressure (hard wall, no flow).


7 Go to the Rename 2D Plot Group dialog box and type p (M=0, hard) in the New



S T U D Y 2

Finally, consider the case of a hard duct wall and a background mean-flow field.

M=0, hardIn the Model Builder window, right-click Study 2>Solver Configurations>M=0, hard and choose Disable.







7 From the Use list, select M=-0.5.



10 Go to the Rename Solver dialog box and type M=-0.5, hard in the New name edit field. Click OK.


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R E S U L T S

The following steps reproduce the upper plot in Figure 6.

Data Sets1 In the Model Builder window, right-click Results>Data Sets>Solution 9 and choose

Rename.

2 Go to the Rename Solution dialog box and type ae (M=-0.5, hard wall) in the New name edit field. Click OK.

3 Right-click Results>Data Sets>Solution 9 and choose Add Selection.




p (M=0, hard) 11 In the Model Builder window, right-click Results>p (M=0, hard) and choose Duplicate.


3 Locate the Data section. From the Data set list, select ae (M=-0.5, hard wall).


5 In the associated edit field, type Normalized pressure (hard wall, flow).


7 Go to the Rename 2D Plot Group dialog box and type p (M=-0.5, hard) in the New




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