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Ultrasound Flowmeter with Generic Time-of-Flight Configuration
Introduction
Knowing the speed of a moving fluid is important in all cases where the fluid is used to transport material or energy. It just so happens that this process is fundamental to such a great part of the tasks we do or interact with in our everyday lives: from industrial applications in production of for instance foods and beverages, drugs, and other chemicals, to personal tasks such as measuring water consumption at your house or determining how much gas was pumped into your car, to medical applications like determining the flow of blood in the veins or the inhalation of air into the lungs. Thus, it should come as no surprise that a large variety of methods have been developed to determine the flow speed.
In the time-of-flight or transit-time method for determining flow speed an ultrasonic signal is transmitted across the main flow in a pipe to non-invasively determine its speed. By transmitting the signal at an angle relative to the main flow, the ultrasound signal will travel faster than the speed of sound if it moves in the direction of the main flow, or slower than the speed of sound if it moves against it. The difference in travel times in the two directions increase linearly with the speed of the main flow. One can therefore infer the speed of the main flow without putting any physical objects into it. Flowmeters of this type find many uses particularly in industrial settings where their non-invasiveness is advantageous and examples include abrasive, corrosive and sanitary flows, and in fact all flows in which immersed objects would degrade or significantly obstruct the flow.
The only major limitation to the use of time-of-flight ultrasound flowmeters is for fluids containing significant amounts of bubbles or suspended particles. The reason is that the ultrasound will be attenuated significantly by the objects resulting in only a very weak signal arriving at the sensor across the main pipe. To measure the flow speed in these cases, a single combined transmitter and receiver emits the ultrasound, which is reflected off the suspended objects, and the flow speed can be calculated from the Doppler shift between the emitted and received signal.
Here we illustrate how to model a generic wetted transient-time ultrasound flowmeter in COMSOL Multiphysics. Wetted ultrasound flowmeters have a dedicated signal tube for the ultrasound signal, and the whole device is therefore mounted into the industrial plant. Conversely, portable clamp-on ultrasound flowmeters also exist, they are mounted on the outside of the main pipe during operation; this type of flowmeter is less accurate due to ultrasound interference resulting from transmitting the signal through the pipe wall.
The model set up here will solve the transient problem of a signal traversing the flow downstream. The signal moving upstream is precalculated and imported as data. The difference in arrival times is used to estimate the speed of the main flow. Due to the high frequency of the employed signal (~ 2.5 MHz), we will use the Convected Wave Equation, Time Explicit physics interface, found under the Ultrasound branch, which is tailored to transient high-frequency situations. The method is very memory efficient and is able to solve the ~7.5 million degrees of freedom (~107 DOFs) of this model using around 7.5 GB of RAM.
Note: This application requires both the Acoustics Module and the CFD Module.
Model Definition
A schematic illustration of the modeled ultrasound flowmeter (using one symmetry plane) with annotations and geometry is shown in Figure 1.
Figure 1: Schematic illustration of ultrasound flowmeter. The smaller signal tube of length L1 is angled relative to the main pipe which contains the flow under investigation.
The flowmeter consists of a main pipe and a smaller signal tube (transducer duct), which is placed at some angle α relative to the main pipe. The diameter D of the main pipe is 5 mm while the signal tube has a diameter of 2 mm and is angled at α = 45° relative to the main pipe. The length of the signal tube is L = 2L1 + L2. The fluid is taken to be water at normal atmospheric conditions, and the mean speed of the background flow is taken to be U0 = 10 m/s.
This model solves the full three-dimensional problem. The axial symmetry is exploited for computational efficiency, so only half the channel is modeled.
Background Flow: CFD
The first step in the modeling procedure is to establish a realistic background flow (p0, u0). This flow is taken to be stationary, and is obtained using a CFD simulation. The background flow is expected to be turbulent with a Reynolds number (Re) of about 5·104 under the operating conditions. The flow is modeled using the Turbulent Flow, k-ω interface of the CFD Module.
Acoustics: Convected Wave Equation, Time Explicit
In the acoustics problem, an ultrasound interrogation signal is prescribed at the transducer (signal) end of the signal tube. The time-of-flight method requires that a signal is sent from both ends so in an actual devices both of these ends act as transmitter as well as receiver. In the current model we only model the signal moving downstream while the signal moving upstream is imported from a data file. The data can easily be generated by simply switching the source and receiver boundaries.
The interrogation signal is a sine wave modulated with a Gaussian pulse. It is modeled by prescribing the normal velocity vn at the source boundary
(1)
where A = 0.1 mm is the signal amplitude, ω0 = 2πf0, f0 = 2.5 MHz, and T0 = 1/f0. The signal in Equation 1 has a bandwidth of roughly f0 which is the same as the carrying signal frequency. Using the Convected Wave Equation, Time Explicit interface, the mesh needs to have a size roughly half the size of the smallest wavelength λ0 (largest frequency) that should be resolved. This is the case since the interface, per default, uses fourth order shape functions. This combination of mesh size and shape functions gives the optimal performance for the discontinuous Galerkin method, on which the interface is based.
Estimating the Flow Speed
One can construct an estimate of the time it takes the two signals to reach their receivers on the opposite side. This estimate can be used to derive an approximate expression for the mean speed of the background flow U0; in fact, this is the working principle behind ultrasound flowmeters. That is, measure the time difference ΔT between the two emitted signals and use this time to derive the mean flow speed, possibly corrected by a correction factor as described below.
The background flow is predominantly in the main tube, with very low speeds in the signal tube, see Figure 1. The distance of the signal tube across the main pipe is denoted by L2, while the part extending on either side has axial length L1 (the side branches). Thus, the total signal tube length satisfies L = 2L1 + L2. The direction of the signal tube is denoted by nL with the angle between the signal tube and the main pipe being the aforementioned α. Assuming that the background flow in the side branches can be neglected, the time T1 it takes to traverse is therefore T1 = L1/c0, independently of whether the ultrasound signal is moving upstream or downstream. Now, assuming that the background flow is constant across the main pipe (this can be a reasonable approximation for very turbulent flows), the velocity field can we written u0 = U0ex. The time T2 it takes the signal to propagate the distance L2 depends on whether it is moving upstream or downstream. The time T2 is, under the assumptions made,
where c0 is the speed of sound. Therefore, the expected time difference ΔT between the two signals is given by
This expression above can be used to estimate the average background mean flow based on a measurement of the time difference ΔT. The expression is
(2)
Note that a critical simplifying assumption for Equation 2 is that the background flow is constant across the tube. In reality, most background flows are not constant across the main pipe. The extreme case is laminar flow where the flow profile will be a parabola, but typical turbulent flow will also exhibit a varying velocity profile.
Results and Discussion
The background mean flow is first simulated using the Turbulent Flow interface from the CFD Module. The velocity amplitude and flow profile are depicted in Figure 2 and Figure 3, respectively.
Figure 2: Background mean flow magnitude in the flowmeter.
Figure 3: Background mean flow profile, plotted through the cross section of the main duct.
The acoustic pressure in the symmetry plane of the flowmeter is depicted for four different times in Figure 4. This sequence (including all stored times) can be animated within COMSOL under the Export node selecting Animation>Player. This gives a very intuitive way to visualize the propagating pulse.
Figure 4: Pressure signal in the symmetry plane of the ultrasound flowmeter at 4 times.
The acoustic particle velocity at a time t = 8·10-6 s just before the pulse reaches the receiver is depicted in Figure 5. The instantaneous intensity amplitude and the intensity vectors (normalized) is depicted in Figure 6 at the same instance as the acoustic velocity in Figure 5. The intensity plot enables a better visualization of the energy flow of the acoustic signal in the flowmeter system.
Figure 5: Acoustic particle velocity in the symmetry plane of the flowmeter.
Figure 6: Acoustic instantaneous intensity amplitude (colors) and normalized intensity vectors as white arrows depicted just before the signal reaches the receiver.
In Figure 7, the pressure signal is depicted at time t = 4·10-6 s along the main axis of the transducer tube (see Figure 1). Finally, in Figure 8, one of the important results of this model. The average pressure as recorded on the receiver for the signal moving downstream and the signal moving upstream. The latter data is imported but can be easily generated by switching the source and receiver selections. Using this figure, it is possible to measure the time difference between the two signals. By zooming in on the figure, by visual inspection, it can be found to be ΔTsimulated = 4.86·10-8 s (this is stored in the parameter DT_simulated).
This simulated time difference can be compared with the expected time difference that can be calculated for the known flow profile u0(x) (the flow profile is, of course, not normally known in measurements). The calculated time ΔTcalc is given by
(3)
This integral is evaluated under Results>Derived Values node and it gives
ΔTcalc  = 5.01·10-8 s which is very close the simulated (visually read) value mentioned above (ΔTcalc is stored in the parameter DT_calc).
Now, using the simulated time difference ΔTsimulated in the expression for the estimated mean flow velocity given in Equation 2 you get U0,est = 10.66 m/s. This value is evaluated (both for the simulated and the calculated time differences) under the Results>Derived Values node.
Flow Profile Correction Factor
The error in estimating U0 from the simplified model above is one of the central problems with data analysis for ultrasound flowmeters, Ref. 1. This is typically described by the flow profile correction factor (FPCF)
The issue in determining FPCF theoretically is that the background flow is in general not known; this is especially true for turbulent flows. Moreover, the flow profile across the signal tube likely depends on both the acoustic medium, the flow speed, the pitch angle α as well as design parameters such as the radii of the main pipe and the signal tube.
In this model we do not make any explicit assumptions about the exact velocity profile in the signal tube. Instead, this is determined by the CFD simulation and therefore explicitly include contributions from all the unknown factors listed above. Therefore, COMSOL Multiphysics can be used to calculate the instrument-specific FPCF, resulting in more accurate instruments across varying flow speeds and geometries. We find for the specific case above
and the model could easily be extended to sweep across, for instance, flow speed and thus obtain FPCF for each situation. Tabulating these values would allow users to obtain the correct result across different working conditions; such tables can easily be included into the data analysis software shipped with the instrument.
Other factors affecting the performance of the flowmeter can also be investigated and simulated. For example, the geometry of the signal tubes and how they taper into the main duct could be of interest. Maybe avoiding unnecessary reflections and diffractions. The modification of the emitted acoustic signal by the flow profile (due to gradients in the flow or even density gradients) can also be studied and used to enhance the detection technique, and allowing better prediction of the flow profile.
Figure 7: Acoustic pressure signal depicted along the central transducer axis.
Figure 8: The average acoustic pressure on the receiver for the signal moving downstream (modeled in this tutorial) and the signal moving upstream (data imported).
Reference
1. J.C. Jung and P.H. Seong, “Estimation of the Flow Profile Correction Factor of a Transit-Time Ultrasonic Flow Meter for the Feedwater Flow Measurement in a Nuclear Power Plant,” IEEE Transactions on Nuclear Science, vol. 52, 2005.
Application Library path: Acoustics_Module/Ultrasound/ultrasound_flow_meter_generic
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Turbulent Flow>Turbulent Flow, k-ω (spf).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ultrasound_flow_meter_generic_parameters.txt.
Geometry 1
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D/2.
4
In the Height text field, type L.
5
Locate the Axis section. From the Axis type list, choose x-axis.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.5*D
7
Clear the Layers on side check box.
8
Select the Layers on bottom check box.
9
Select the Layers on top check box.
Cylinder 2 (cyl2)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/2.
4
In the Height text field, type L_transducer.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type -L_transducer/2.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the object cyl2 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type alpha.
5
Locate the Point on Axis of Rotation section. In the x text field, type L/2.
6
Locate the Rotation section. From the Axis type list, choose y-axis.
7
Click  Build All Objects.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
3
In the Settings window for Union, click  Build All Objects.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose zx-plane.
Partition Objects 1 (par1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object uni1 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
Click  Build All Objects.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
Switch to wireframe rendering to more easily see the domains inside the model, when selecting.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
3
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
4
From the Geometric entity level list, choose Domain.
5
On the object par1, select Domains 1, 3, 5, 7, 9, and 11 only.
6
Click  Build All Objects.
Form Composite Domains 1 (cmd1)
1
In the Geometry toolbar, click  Virtual Operations and choose Form Composite Domains.
2
On the object fin, select Domains 2–5 only.
3
In the Geometry toolbar, click  Build All.
The geometry should look like Figure 1.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Water, liquid.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Definitions
Symmetry
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 2, 6, and 16 only.
5
In the Label text field, type Symmetry.
Flow Inlet
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Flow Inlet in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 1 only.
Flow Outlet
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Flow Outlet in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 19 only.
Source
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Source in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 10 only.
Receiver
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Receiver in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 14 only.
Walls
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Walls in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 3, 4, 7–9, 11–13, 17, and 18 only.
Turbulent Flow, k-ω (spf)
Inlet 1
1
In the Model Builder window, under Component 1 (comp1) right-click Turbulent Flow, k-ω (spf) and choose Inlet.
2
In the Settings window for Inlet, locate the Boundary Selection section.
3
From the Selection list, choose Flow Inlet.
4
Locate the Boundary Condition section. From the list, choose Fully developed flow.
5
Locate the Fully Developed Flow section. In the Uav text field, type Uin.
This boundary condition ensures a fully developed turbulent flow profile at the inlet.
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
In the Settings window for Outlet, locate the Boundary Selection section.
3
From the Selection list, choose Flow Outlet.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
In the Settings window for Symmetry, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Mesh - CFD
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, type Mesh - CFD in the Label text field.
Size
Right-click Component 1 (comp1)>Mesh - CFD and choose Edit Physics-Induced Sequence.
Free Tetrahedral 1
1
In the Settings window for Free Tetrahedral, locate the Domain Selection section.
2
From the Geometric entity level list, choose Domain.
3
Select Domain 2 only.
Swept 1
1
In the Mesh toolbar, click  Swept.
2
Drag and drop below Free Tetrahedral 1.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 12.
4
Click  Build All.
5
Click  Build All.
Study 1 - CFD
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - CFD in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots check box.
Compute the flow solution. This will take about 1 hour on a 4 core machine at 3.6 GHz (in general it depends on the hardware) and require 5 GB of RAM.
4
In the Home toolbar, click  Compute.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Acoustics>Ultrasound>Convected Wave Equation, Time Explicit (cwe).
4
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Study 1 - CFD.
5
Click Add to Component 1 in the window toolbar.
6
In the Home toolbar, click  Add Physics to close the Add Physics window.
Convected Wave Equation, Time Explicit (cwe)
1
Click the  Show More Options button in the Model Builder toolbar.
2
In the Show More Options dialog box, in the tree, select the check box for the node Physics>Advanced Physics Options.
3
Click OK.
Symmetry 1
1
Right-click Component 1 (comp1)>Convected Wave Equation, Time Explicit (cwe) and choose Symmetry.
2
In the Settings window for Symmetry, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Acoustic Impedance 1
1
In the Physics toolbar, click  Boundaries and choose Acoustic Impedance.
2
Select Boundaries 1 and 19 only.
Define the driving signal as given in Equation 1.
Global Definitions
Analytic 1 (an1)
1
In the Home toolbar, click  Functions and choose Global>Analytic.
2
In the Settings window for Analytic, type vn in the Function name text field.
3
Locate the Definition section. In the Expression text field, type A*sin(omega0*t)*exp(-(f0*(t-3*T0))^2).
4
In the Arguments text field, type t.
5
Locate the Units section. In the table, enter the following settings:
Argument
Unit
t
s
6
In the Function text field, type m/s.
Convected Wave Equation, Time Explicit (cwe)
Normal Velocity 1
1
In the Physics toolbar, click  Boundaries and choose Normal Velocity.
2
In the Settings window for Normal Velocity, locate the Boundary Selection section.
3
From the Selection list, choose Source.
4
Locate the Normal Velocity section. In the vn(t) text field, type vn(t).
Create a nonlocal average coupling to be used in postprocessing to calculate the average pressure at the receiver.
Definitions
Average 1 (aveop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Average.
2
In the Settings window for Average, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Receiver.
Now, set up the absorbing layers (sponge layers) used to truncate the computational domain.
Absorbing Layer 1 (ab1)
1
In the Definitions toolbar, click  Absorbing Layer.
2
Select Domains 1 and 3 only.
Create an interpolation function and import data for a signal moving the opposite way (upstream) to the signal modeled here (downstream). You will use that data in postprocessing. The data was generated by switching the source and receiver selections and running the model.
Global Definitions
Interpolation 1 (int1)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file ultrasound_flow_meter_generic_upstream_signal.txt.
If you want to import the data into the model, and not rely on the .txt file, click Import.
6
Find the Functions subsection. In the table, enter the following settings:
Function name
Position in file
p_upstream
1
7
Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Piecewise cubic.
8
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
Column 1
s
9
In the Function table, enter the following settings:
Function
Unit
p_upstream
Pa
Create the acoustics mesh. In contrast to the CFD mesh, this mesh can be relatively coarse.
Mesh - Acoustics
1
In the Mesh toolbar, click Add Mesh and choose Add Mesh.
2
In the Settings window for Mesh, type Mesh - Acoustics in the Label text field.
Free Tetrahedral 1
In the Mesh toolbar, click  Free Tetrahedral.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lam0/1.5.
5
In the Minimum element size text field, type lam0/2.
With this setting you get a bit more than 1.5 elements per wavelength at the frequency f0. This may need to be increased to, for example, lam0/2 in order to well resolve the small frequency content above f0, that exists in the Gaussian pulse.
Free Tetrahedral 1
When modeling using physics interfaces that are based on the DG method, it is important to avoid small mesh elements, since they control the time steps taken by the solver. In order to avoid this use the Element Quality Optimization functionality available for the tetrahedral mesh. This step is very important.
1
In the Model Builder window, click Free Tetrahedral 1.
2
In the Settings window for Free Tetrahedral, click to expand the Element Quality Optimization section.
3
Select the Avoid too small elements check box.
4
From the Optimization level list, choose High.
5
Click  Build All.
An important part of solving aeroacoustic problems is to correctly map the CFD solution onto the acoustics domain. This is done using the built-in Background Fluid Flow Coupling multiphysics feature and the dedicated Mapping study.
Multiphysics
Background Fluid Flow Coupling 1 (bffc1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Domain>Background Fluid Flow Coupling.
2
In the Settings window for Background Fluid Flow Coupling, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Smoothing section. In the Diffusion constant text field, type 1e-4.
Return to the Convected Wave Equation Model node and see that the background mean flow pressure and velocity automatically take the corresponding mapped quantities.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check boxes for Turbulent Flow, k-ω (spf) and Convected Wave Equation, Time Explicit (cwe).
4
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Multiphysics>Mapping.
5
Click Add Study in the window toolbar.
Study 2 - Mapping
1
In the Model Builder window, click Study 2.
2
In the Settings window for Study, type Study 2 - Mapping in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots check box.
Step 1: Mapping
1
In the Model Builder window, under Study 2 - Mapping click Step 1: Mapping.
2
In the Settings window for Mapping, locate the Solution to Map section.
3
From the Study list, choose Study 1 - CFD, Stationary.
Make sure that the acoustic mesh is chosen for mapping the CFD solution upon.
4
Click to expand the Destination Mesh Selection section. In the Home toolbar, click  Compute.
Add Study
1
Go to the Add Study window.
2
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Turbulent Flow, k-ω (spf).
3
Find the Multiphysics couplings in study subsection. In the table, clear the Solve check box for Background Fluid Flow Coupling 1 (bffc1).
4
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study - Acoustics
1
In the Model Builder window, click Study 3.
2
In the Settings window for Study, type Study - Acoustics in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots check box.
Step 1: Time Dependent
1
In the Model Builder window, under Study - Acoustics click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,T0/12,30*T0).
This setting saves the solution 12 times per period and runs the model for 30 periods. The setting only influences when to store the solution (and thus the file size). The internal time steps taken by the solver are automatically controlled by COMSOL to fulfill the appropriate CFL condition.
4
Click to expand the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.
5
From the Method list, choose Solution.
6
From the Study list, choose Study 2 - Mapping, Mapping.
7
Find the Store fields in output subsection. From the Settings list, choose For selections.
8
Under Selections, click  Add.
9
In the Add dialog box, in the Selections list, choose Symmetry, Source, and Receiver.
10
Click OK.
The amount of data generated in this model is large. A good way to restrict the file size is to only store the data on desired selections. Here the solution is only stored on the symmetry plane, and on the source and receiver faces.
Check that the correct mesh is used for the transient acoustic simulation.
11
In the Settings window for Time Dependent, click to expand the Mesh Selection section.
Now solve the convected wave equation (acoustics) model. This will take around 3.5h (depending on the hardware) and will require about 7.5 GB of RAM. Remember that the model contains around 7.5 million degrees of freedom (DOFs).
12
In the Home toolbar, click  Compute.
Before generating any plots and looking at results, process the datasets. It is good practice to add the same selections to the result dataset as the selections chosen in the Store fields in output. For postprocessing purposes we will only select the symmetry plane, source, and receiver in the Acoustics dataset (not selecting the absorbing layer).
Results
In the Model Builder window, expand the Results node.
Study - Acoustics/Solution 3 (sol3)
In the Model Builder window, expand the Results>Datasets node, then click Study - Acoustics/Solution 3 (sol3).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 6, 10, and 14 only.
Finally, create a dataset that is a cut line following the axis of the transducer duct.
Cut Line 3D 1
1
In the Results toolbar, click  Cut Line 3D.
2
In the Settings window for Cut Line 3D, locate the Data section.
3
From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
4
Locate the Line Data section. From the Line entry method list, choose Point and direction.
5
Find the Point subsection. In the x text field, type L/2.
6
Find the Direction subsection. In the x text field, type nLx.
7
In the y text field, type nLy.
8
In the z text field, type nLz.
9
From the Snapping list, choose Snap to closest boundary.
In the following postprocessing the Resolution is increased under the Quality section in all acoustics plots. This is because the shape functions used for the convected wave equation interface are 4th order per default. So in order to properly represent the solution, the resolution needs to be increased.
Background Flow Velocity
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Background Flow Velocity in the Label text field.
3
Locate the Color Legend section. Select the Show units check box.
Surface 1
1
Right-click Background Flow Velocity and choose Surface.
2
In the Background Flow Velocity toolbar, click  Plot.
The background mean flow velocity amplitude plotted on the surface of the geometry should look like Figure 2.
Background Flow Velocity Profile
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Background Flow Velocity Profile in the Label text field.
Line Graph 1
1
Right-click Background Flow Velocity Profile and choose Line Graph.
2
Select Edge 7 only.
3
In the Settings window for Line Graph, locate the x-Axis Data section.
4
From the Parameter list, choose Expression.
5
In the Expression text field, type z.
6
Click to expand the Legends section. Select the Show legends check box.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
Mean flow profile, CFD
Line Graph 2
1
Right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Study 2 - Mapping/Solution 2 (sol2).
4
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Background Fluid Flow Coupling 1>Mapped velocity - m/s>bffc1.u_mapx - Mapped velocity, x component.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Mean flow profile, Mapped
6
In the Background Flow Velocity Profile toolbar, click  Plot.
The CFD and the mapped background mean flow profiles are depicted in Figure 3.
Acoustic Pressure
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Acoustic Pressure in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
4
Locate the Color Legend section. Select the Show units check box.
Surface 1
1
Right-click Acoustic Pressure and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type p2.
4
Locate the Coloring and Style section. From the Color table list, choose Wave.
5
From the Scale list, choose Linear symmetric.
6
Click to expand the Quality section. From the Resolution list, choose Custom.
7
In the Element refinement text field, type 6.
8
In the Acoustic Pressure toolbar, click  Plot.
Add a deformation to better visualize the nature of the waves propagating in the symmetry plane.
Deformation 1
Right-click Surface 1 and choose Deformation.
Deformation 1
1
In the Model Builder window, expand the Results>Acoustic Pressure>Surface 1 node, then click Deformation 1.
2
In the Settings window for Deformation, locate the Expression section.
3
In the x component text field, type 0.
4
In the y component text field, type p2.
5
In the z component text field, type 0.
6
In the Acoustic Pressure toolbar, click  Plot.
The first plot that is shown is for the last stored time, at 30*T0. Change the Time selection to see the solution at other instances. Some are shown in Figure 4. Alternatively, create an animation (under the Export node) that shows the time dependent evolution of the pulse.
Acoustic Pressure
1
In the Model Builder window, under Results click Acoustic Pressure.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Time (s) list, choose 2E-6.
4
In the Acoustic Pressure toolbar, click  Plot.
Acoustic Velocity
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Acoustic Velocity in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
4
From the Time (s) list, choose 8E-6.
5
Locate the Color Legend section. Select the Show units check box.
Surface 1
1
Right-click Acoustic Velocity and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Convected Wave Equation, Time Explicit>Acceleration and velocity>cwe.v_inst - Acoustic velocity amplitude - m/s.
3
Locate the Coloring and Style section. From the Scale list, choose Linear symmetric.
4
Locate the Quality section. From the Resolution list, choose Custom.
5
In the Element refinement text field, type 6.
6
In the Acoustic Velocity toolbar, click  Plot.
The acoustic velocity amplitude is depicted in Figure 5. Once again, change the Time selection to study the wave propagation.
Proceed to plot the instantaneous intensity as depicted in Figure 6.
Acoustic Intensity
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Acoustic Intensity in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
4
From the Time (s) list, choose 8E-6.
5
Locate the Color Legend section. Select the Show units check box.
Surface 1
1
Right-click Acoustic Intensity and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Convected Wave Equation, Time Explicit>Intensity>cwe.Ii_mag - Instantaneous intensity magnitude - W/m².
3
Locate the Coloring and Style section. From the Color table list, choose ThermalDark.
4
Locate the Quality section. From the Resolution list, choose Custom.
5
In the Element refinement text field, type 6.
Arrow Surface 1
1
In the Model Builder window, right-click Acoustic Intensity and choose Arrow Surface.
2
In the Settings window for Arrow Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Convected Wave Equation, Time Explicit>Intensity>cwe.Iix,...,cwe.Iiz - Instantaneous intensity.
3
Locate the Coloring and Style section. From the Arrow length list, choose Logarithmic.
4
In the Range quotient text field, type 400.
5
Locate the Arrow Positioning section. In the Number of arrows text field, type 1000.
6
Locate the Coloring and Style section. From the Color list, choose White.
7
In the Acoustic Intensity toolbar, click  Plot.
Now, proceed to plot the pressure pulse along the central transducer duct axis in a 1D graph. The figure should look like Figure 7.
Pressure Along Transducer Axis
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pressure Along Transducer Axis in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 3D 1.
4
From the Time selection list, choose From list.
5
In the Times (s) list, select 4E-6.
Line Graph 1
1
Right-click Pressure Along Transducer Axis and choose Line Graph.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type p2.
4
Click to expand the Quality section. From the Resolution list, choose Extra fine.
5
In the Pressure Along Transducer Axis toolbar, click  Plot.
The final graph to create shows the average pressure signal at the receiver for the pulse moving downstream (the model simulated here). The figure also includes the pressure from a signal moving upstream, as taken from the imported data. After zooming, the graph should look like Figure 8.
Pressure Signal at Receivers
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pressure Signal at Receivers in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
Global 1
1
Right-click Pressure Signal at Receivers and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
aveop1(p2)
Pa
Pressure moving downstream
p_upstream(t)
Pa
Pressure moving upstream (imported)
4
In the Pressure Signal at Receivers toolbar, click  Plot.
Finally, calculate the integral expression given in Equation 3 under the Derived Values node. Here you can also calculate the predicted mean flow speeds from Equation 2. The results are discussed in the main results section.
Line Integration 1
1
In the Results toolbar, click  More Derived Values and choose Integration>Line Integration.
2
In the Settings window for Line Integration, locate the Data section.
3
From the Dataset list, choose Cut Line 3D 1.
4
From the Time selection list, choose First.
5
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
1/(cwe.c0-(nLx*cwe.u0x+nLy*cwe.u0y+nLz*cwe.u0z))-1/(cwe.c0+(nLx*cwe.u0x+nLy*cwe.u0y+nLz*cwe.u0z))
 
 
6
Click  Evaluate.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study - Acoustics/Solution 3 (sol3).
4
From the Time selection list, choose First.
5
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
L2/(DT_calc*cos(alpha))*(-1+sqrt(1+DT_calc^2*c0^2/L2^2))
 
 
L2/(DT_simulated*cos(alpha))*(-1+sqrt(1+DT_simulated^2*c0^2/L2^2))
 
 
6
Click  Evaluate.