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Axially Rotating Pipe and Swirling Jet Turbulent Flows
Introduction
The Single-Phase Flow, SSG-LRR interface is employed to investigate fully developed turbulent flow in a rotating pipe, as well as subsequent development of the swirling jet, emanating from the pipe outlet, into a drastically expanded nonrotating section. Properties of the flow regimes are thoroughly analyzed and illustrated.
Model Definition
The influence of rotation on turbulent flows is not naturally accounted for by eddy-viscosity models. Partially, this drawback can be healed by activating Include rotation-curvature correction. For the case of an axially rotating pipe this would shift axial velocity profile in a qualitatively correct direction. However, tangential (swirl) velocity would retain linear (solid-body) profile, which contradicts experimental observations, direct numerical simulations (DNS) and large-eddy simulations (LES), see Ref. 1 and Ref. 2. In contrast to RANS-EVM, differential Reynolds stress models (RANS-RSM) of turbulence self-consistently respond both to the in-frame flow vorticity and to the frame rotation.
Here, turbulent rotating pipe flow is investigated first. The pipe Reynolds number,
is Re = 20,000. Here, Ub is the streamwise bulk velocity and Dp is the pipe diameter. The flow is investigated at rotation number (or swirl number),
being Ns = 0 and Ns = 0.5. Ω is the angular rotation rate.
The development of a fully developed swirling state, starting from an isotropic turbulent state with solid-body rotation, requires the streamwise length of order L = 30Dp. Periodic Flow Condition in a streamwise-short geometry is employed to save computational resources.
Subsequently, the rotating pipe flow emanates into a large nonrotating region of very slow flow (Ub/30). The incoming angular momentum of the swirling jet fades away quite quickly, but nevertheless has profound effect on the jet development. Again, Ns = 0 and Ns = 0.5 cases are being considered.
Figure 1: The model geometry of the rotating pipe (left) and swirling jet (right). The converged fully developed state of the rotating pipe study is used as an inlet condition for the swirling jet study.
Implementation in COMSOL Multiphysics
Only axisymmetric flow behavior is in the focus of this investigation, thus a 2D-axisymmetric component is used. Swirl flow has to be activated. Figure 1 shows geometries, along with boundary conditions, for both studies, while Figure 2 demonstrates the corresponding meshes. The rotating pipe study is performed using a moving wall condition; the same results (accounting for the shift in tangential velocity) would be obtained if Rotating frame (instead of Wall movement) was used instead. The swirling jet flow is originated via inlet conditions taken from the converged state of the rotating pipe flow analysis. The near-wall cells of the pipe should be meshed exceptionally well, and the development region of the jet (so called near field) should have good meshing. In contrast, the far-field region is used only to relieve the influence of the outlet and slip wall boundary conditions, and the results there are not very reliable. An Auxiliary sweep with continuation is employed for efficient computation of cases with consequent rotation numbers.
Figure 2: Meshes used for the studies of the rotating pipe (left) and swirling jet (right). Extremely thin cells at the rotating wall of the rotating pipe are used to ensure accurate evaluation of traction. The near field of the jet features really fine mesh cells, while the poorly meshed far field serves mostly to ensure that the development region of the jet is not much affected by the distant boundary conditions.
The details of the implementation of the Single-Phase Flow, SSG-LRR interface can be found in the CFD Module User’s Guide; see the section “Theory for the Turbulent Flow Interfaces”.
Results and Discussion
Figure 3 illustrates how the axial velocity profile becomes modified due to rotation with Ns = 0.5. Apparently, the profile appearance shifts from the typical “round nose bullet-like” turbulent form to an “inverted paraboloid” laminar form, which is consistent with the general laminarizing effect of rotation. The swirl velocity profile clearly resembles a paraboloid form. Figure 4 details these observations. It can be shown that the axial velocity in the Ns = 0.5 case has a clearly distinguishable logarithmic region, although it is shorter than in the Ns = 0 case. The swirl velocity curve is placed somewhat below pure parabola, the latter being considered as a canonical (experimentally confirmed) form in the rotating pipe.
Figure 3: The pipe flow axial (nonrotating versus rotating) and tangential (rotating) velocity profiles.
Figure 4: The pipe flow axial and swirl-velocity profiles.
Figure 5: Rotating pipe flow in absolute system: streamlines on a cylindrical surface (v/(NsUb) = 0.76) and arrow surfaces on horizontal planes (v/(NsUb) according to the color legend).
The contrast between Figure 5 and Figure 6 is due to slower-than-linear growth of the swirl velocity v with radius: in rotating system the streamlines are spiraling in clockwise (negative) direction, and the maximum in-frame swirl velocity (ωr-v)/(NsUb) = 0.38 at r/Rp = 0.587. Figure 6 can be obtained directly if using the Rotating frame feature for computation.
Figure 7 compares Ns = 0 and Ns = 0.5 cases, and demonstrates that rotation reduces the shear stress, although profiles normalized by the friction velocity are almost the same. The friction coefficient, defined as
falls from 6.7187·10−3 (Ns = 0) to 5.342·10−3 (Ns = 0.5). Meanwhile, the turbulence kinetic energy level grows just a bit and its maximum shifts away the wall. Profiles normalized by show significant growth due to the rotation.
Figure 6: Rotating pipe flow in rotating frame: streamlines on a cylindrical surface ((ωr-v)/(NsUb) = 0.15) and arrow surfaces on horizontal planes ((ωr-v)/(NsUb) according to the color legend).
Figure 7: The pipe flow: shear stress and turbulence kinetic energy, normalized both by the friction velocity and by the bulk velocity.
Figure 8: Streamlines of the swirling jet.
Figure 8 demonstrates the streamlines of the swirling jet, emanating from the outlet of the rotating pipe. Apparently, swirl velocity falls rapidly in the axial direction and already at 3 Dp (Dp = 0.1 m) maximum swirl velocity is no more than 0.25 of its original maximum.
Figure 9 confirms the intuitive expectation that the swirl leads to faster loss of axial momentum on the axis due to increased mixing with the surrounding fluid (upper left). The form of the curve and characteristic values have good correspondence to the experimental data, although do not reproduce it exactly. At the same time, in the rotating case the axial velocity falls less rapidly with radius at all axial cross sections (upper right). In a wide range of axial coordinates, maximum positive radial velocity happens at approximately the same radius and has approximately the same value (lower left). Although far away from the source the swirl velocity is small, it falls off very slowly with radius (lower right). Notice that negative swirl velocities near the jet axis are not observed.
Figure 9: Top row: axial velocity (without and with swirl) at the jet axis (left) and at various axial cross sections (right). Bottom row — various axial cross sections in swirling case: radial (left) and tangential (right) velocities.
Figure 10 shows that shear stress uw is consistently larger in the Ns = 0.5 case (upper left), except at the central portion of the jet at quite high z/Dp (approximately 10). Axial velocity rms ww demonstrates the same trend (upper right). Axial-tangential stress vw attains strong positive values at small z/Dp (say, 2), but quickly decreases further in the axial direction (lower left). Radial-tangential stress uv exhibits extremely high values at the same locations as vw, and rapidly becomes attenuated with increasing z as well.
[
Figure 10: Top row: shear stress (left) and axial velocity rms (right) in Ns = 0 and Ns = 0.5 cases. Bottom row: axial-tangential stress (left) and radial-tangential stress (right) in the swirling case.
Summary and Outlook
Major consequences of turbulence anisotropy induced by rotation in rotating pipe flow and swirling jet flow can be described by the SSG-LRR Reynolds stress model implemented in COMSOL Multiphysics. Most of the characteristics obtained by the RSM, like friction reduction with growing rotation rate, are compatible with more precise experimental, DNS and LES studies, see Ref. 1 and Ref. 2.
However, in analysis based on SSG-LRR, tangential velocity deviates from the parabolic profile in the rotating pipe, as well as counter-rotating region is absent in the swirling jet. Thus, although correct trends of turbulence response to swirl have been obtained both in rotating pipe and swirling jet, it is desirable to use more reliable tools of analysis in case exact predictions are required.
References
1. L. Facciolo “A Study on Axially Rotating Pipe and Swirling Jet Flows,” Doctoral thesis, Department of Mechanics, Royal Institute of Technology, 2006.
2. N.D. Castro “Large-Eddy Simulation of Axially-Rotating, Turbulent Pipe and Particle-laden Swirling Jet Flows,” Doctoral thesis, Mechanical Engineering, Old Dominion University, 2012; doi.org/10.25777/mpdk-zd57.
Application Library path: CFD_Module/Single-Phase_Flow/rotating_pipe_swirling_jet
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  2D Axisymmetric.
2
In the Select Physics tree, select Fluid Flow > Single-Phase Flow > Turbulent Flow > Turbulent Flow, SSG-LRR (spf).
3
Click Add.
4
Click Add.
5
Click  Study.
6
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Stationary with Initialization.
7
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 the Load button. From the menu, choose Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file rotating_pipe_swirling_jet_parameters_1.txt.
Parameters 2
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click the Load button. From the menu, choose Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file rotating_pipe_swirling_jet_parameters_2.txt.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
u_tau
sqrt(spf.pfc1.delta_p*R_p/(2*rho_w*dL))
 
Friction velocity in pipe
Add Material from Library
In the Home toolbar, click  Windows and choose Add Material from Library.
Add Material
1
Go to the Add Material window.
2
In the tree, select Built-in > Water, liquid.
3
Click the Add to Component button in the window toolbar.
4
In the Home toolbar, click  Add Material to close the Add Material window.
Geometry 1
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type R_p.
4
In the Height text field, type dL.
5
Locate the Position section. In the z text field, type -dL.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
dL/2
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type Ej*R_p.
4
In the Height text field, type Lj.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type (Ej-1)*R_p.
4
In the Height text field, type Lp.
5
Locate the Position section. In the r text field, type R_p.
6
In the z text field, type -Lp.
Rectangle 4 (r4)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 1.25*R_p.
4
In the Height text field, type 0.75*R_p.
5
Locate the Position section. In the z text field, type -0.25*R_p.
Rectangle 5 (r5)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type R_p.
4
In the Height text field, type Lp.
5
Locate the Position section. In the z text field, type -Lp.
Rectangle 6 (r6)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 2*Lj.
4
In the Height text field, type 3*Lj+Lp.
5
Locate the Position section. In the z text field, type -Lp.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object r4 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object r5 only.
6
Select the Keep objects to subtract checkbox.
Difference 2 (dif2)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object r6 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object r5 only.
Polygon 1 (pol1)
1
In the Geometry toolbar, click  Polygon.
2
In the Settings window for Polygon, locate the Coordinates section.
3
In the table, enter the following settings:
r (m)
z (m)
R_p
0
1.01*R_p
0
1.03*R_p
-dL/2
R_p
-dL/2
4
Click  Build All Objects.
Definitions
Explicit 1
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 1 and 2 only.
Complement 1
1
In the Definitions toolbar, click  Complement.
2
In the Settings window for Complement, locate the Input Entities section.
3
Under Selections to invert, click  Add.
4
In the Add dialog, select Explicit 1 in the Selections to invert list.
5
Click OK.
Turbulent Flow, SSG-LRR (spf)
1
In the Model Builder window, under Component 1 (comp1) click Turbulent Flow, SSG-LRR (spf).
2
In the Settings window for Turbulent Flow, SSG-LRR, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domains 1 and 2 only.
5
Locate the Physical Model section. Select the Swirl flow checkbox.
Fluid Properties 1
1
In the Model Builder window, under Component 1 (comp1) > Turbulent Flow, SSG-LRR (spf) click Fluid Properties 1.
2
In the Settings window for Fluid Properties, locate the Fluid Properties section.
3
From the ρ list, choose User defined. In the associated text field, type rho_w.
4
From the μ list, choose User defined. In the associated text field, type mu_w.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the u vector as
0*omega_p*r
phi
Ub
z
Wall 1
1
In the Model Builder window, click Wall 1.
2
In the Settings window for Wall, click to expand the Wall Movement section.
3
From the Translational velocity list, choose Manual.
4
Specify the utr vector as
omega_p*r
phi
Periodic Flow Condition 1
1
In the Physics toolbar, click  Boundaries and choose Periodic Flow Condition.
2
Select Boundaries 2 and 6 only.
3
In the Settings window for Periodic Flow Condition, locate the Flow Condition section.
4
From the Flow condition list, choose Mass flow.
5
In the text field, type Mf.
Pressure Point Constraint 1
1
In the Physics toolbar, click  Points and choose Pressure Point Constraint.
2
Select Point 10 only.
Mesh 1
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 1 and 2 only.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundaries 2, 4, and 6 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 200.
6
In the Element ratio text field, type 10.
7
From the Growth rate list, choose Exponential.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundaries 16 and 17 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 1.
5
Click  Build All.
Study 1
Step 1: Wall Distance Initialization
1
In the Model Builder window, under Study 1 click Step 1: Wall Distance Initialization.
2
In the Settings window for Wall Distance Initialization, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Turbulent Flow, SSG-LRR 2 (spf2).
Step 2: Stationary
1
In the Model Builder window, click Step 2: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Turbulent Flow, SSG-LRR 2 (spf2).
4
Click to expand the Study Extensions section. Select the Auxiliary sweep checkbox.
5
Click  Add.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
Ns (Swirl number)
0 0.5
 
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Compile Equations: Wall Distance Initialization.
3
In the Settings window for Compile Equations, locate the Geometric Entity Selection section.
4
From the Use entities list, choose Selected.
5
Under Selections, click  Add.
6
In the Add dialog, select Explicit 1 in the Selections list.
7
Click OK.
8
In the Study toolbar, click  Compute.
Turbulent Flow, SSG-LRR 2 (spf2)
1
In the Model Builder window, under Component 1 (comp1) click Turbulent Flow, SSG-LRR 2 (spf2).
2
Select Domains 3–8 only.
3
In the Settings window for Turbulent Flow, SSG-LRR, locate the Physical Model section.
4
Select the Swirl flow checkbox.
Fluid Properties 1
1
In the Model Builder window, under Component 1 (comp1) > Turbulent Flow, SSG-LRR 2 (spf2) click Fluid Properties 1.
2
In the Settings window for Fluid Properties, locate the Fluid Properties section.
3
From the ρ list, choose User defined. In the associated text field, type rho_w.
4
From the μ list, choose User defined. In the associated text field, type mu_w.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the u vector as
Ub/10
z
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundary 6 only.
3
In the Settings window for Inlet, locate the Velocity section.
4
Click the Velocity field button.
5
Specify the u0 vector as
withsol('sol1',u,setval(Ns,Ns))
r
withsol('sol1',v,setval(Ns,Ns))
phi
withsol('sol1',w,setval(Ns,Ns))
z
6
Locate the Turbulence Conditions section. Click the Specify turbulence variables button.
7
From the list, choose Symmetric.
8
Specify the R0 matrix as
withsol('sol1',uu,setval(Ns,Ns))
withsol('sol1',uv,setval(Ns,Ns))
withsol('sol1',uw,setval(Ns,Ns))
withsol(’sol1’,uv,setval(Ns,Ns))
withsol('sol1',vv,setval(Ns,Ns))
withsol('sol1',vw,setval(Ns,Ns))
withsol(’sol1’,uw,setval(Ns,Ns))
withsol(’sol1’,vw,setval(Ns,Ns))
withsol('sol1',ww,setval(Ns,Ns))
9
In the ω0 text field, type withsol('sol1',spf.om_global,setval(Ns,Ns)).
Inlet 2
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundaries 13 and 26 only.
3
In the Settings window for Inlet, locate the Velocity section.
4
In the U0 text field, type Ub/30.
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundaries 16 and 17 only.
3
In the Settings window for Wall, locate the Wall Movement section.
4
From the Translational velocity list, choose Manual.
5
Specify the utr vector as
(1+z/dL)*omega_p*r
phi
Wall 3
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundary 28 only.
3
In the Settings window for Wall, locate the Boundary Condition section.
4
From the Wall condition list, choose Slip.
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
Select Boundary 11 only.
3
In the Settings window for Outlet, locate the Pressure Conditions section.
4
Clear the Suppress backflow checkbox.
Mesh 2
In the Mesh toolbar, click Add Mesh and choose Add Mesh.
Distribution 1
Right-click Mesh 2 and choose Distribution.
Size
1
In the Settings window for Size, locate the Element Size section.
2
From the Calibrate for list, choose Fluid dynamics.
Distribution 1
1
In the Model Builder window, click Distribution 1.
2
Select Boundary 6 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 200.
6
In the Element ratio text field, type 10.
7
From the Growth rate list, choose Exponential.
8
Select the Reverse direction checkbox.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 8 only.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundary 19 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 20.
6
In the Element ratio text field, type 40.
7
From the Growth rate list, choose Exponential.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundary 18 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 20.
6
In the Element ratio text field, type 5.
7
From the Growth rate list, choose Exponential.
8
Select the Reverse direction checkbox.
Distribution 3
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundaries 17 and 20 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 20.
6
In the Element ratio text field, type 2.
7
From the Growth rate list, choose Exponential.
Free Triangular 1
1
In the Mesh toolbar, click  Free Triangular.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 3 and 7 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Select Domain 3 only.
5
Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.
6
From the Predefined list, choose Extremely fine.
Size 2
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 16 only.
5
Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.
6
From the Predefined list, choose Extremely fine.
Size 3
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Select Domain 7 only.
5
Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.
6
From the Predefined list, choose Extra fine.
Free Triangular 2
1
In the Mesh toolbar, click  Free Triangular.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 4 only.
Size 1
1
Right-click Free Triangular 2 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Calibrate for list, choose Fluid dynamics.
4
From the Predefined list, choose Finer.
5
Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element growth rate checkbox. In the associated text field, type 1.03.
Size 2
1
In the Model Builder window, right-click Free Triangular 2 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 7 only.
5
Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.
6
Click the Custom button.
7
Click the Predefined button.
8
From the Predefined list, choose Extremely fine.
9
Click the Custom button.
10
Locate the Element Size Parameters section.
11
Select the Maximum element size checkbox. In the associated text field, type 0.0075.
Free Triangular 3
1
In the Mesh toolbar, click  Free Triangular.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 5 and 6 only.
Size 1
1
Right-click Free Triangular 3 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Select Domain 6 only.
5
Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.
6
From the Predefined list, choose Finer.
Size 2
1
In the Model Builder window, right-click Free Triangular 3 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 12, 13, and 25 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.01.
Size 3
1
Right-click Free Triangular 3 and choose Size.
2
Select Domain 5 only.
3
In the Settings window for Size, locate the Element Size section.
4
From the Calibrate for list, choose Fluid dynamics.
5
From the Predefined list, choose Extremely coarse.
6
Click the Custom button.
7
Locate the Element Size Parameters section.
8
Select the Maximum element growth rate checkbox. In the associated text field, type 1.1.
Size 4
1
Right-click Free Triangular 3 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 9 only.
5
Locate the Element Size section. From the Predefined list, choose Coarse.
6
Click the Custom button.
7
Locate the Element Size Parameters section.
8
Select the Maximum element size checkbox. In the associated text field, type 0.075.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 6 and 7 only.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
Select Boundaries 12 and 14 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
In the Number of layers text field, type 7.
5
Click  Build All.
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 Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Stationary with Initialization.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Wall Distance Initialization
1
In the Settings window for Wall Distance Initialization, locate the Physics and Variables Selection section.
2
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Turbulent Flow, SSG-LRR (spf).
Step 2: Stationary
1
In the Model Builder window, click Step 2: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Turbulent Flow, SSG-LRR (spf).
4
Locate the Study Extensions section. Select the Auxiliary sweep checkbox.
5
Click  Add.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
Ns (Swirl number)
0 0.5
 
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node, then click Compile Equations: Wall Distance Initialization.
3
In the Settings window for Compile Equations, locate the Geometric Entity Selection section.
4
From the Use entities list, choose Selected.
5
Under Selections, click  Add.
6
In the Add dialog, select Complement 1 in the Selections list.
7
Click OK.
8
In the Study toolbar, click  Compute.
Results
1
In the Model Builder window, click Results.
2
In the Settings window for Results, locate the Update of Results section.
3
Select the Only plot when requested checkbox.
Evaluation Group 1
In the Results toolbar, click  Evaluation Group.
Global Evaluation 1
1
Right-click Evaluation Group 1 and choose Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
spf.pfc1.delta_p*R_p/(rho_w*Ub^2*dL)
1
 
Surface Average 1
1
In the Model Builder window, right-click Evaluation Group 1 and choose Average > Surface Average.
2
Select Domains 3 and 4 only.
3
In the Settings window for Surface Average, locate the Data section.
4
From the Dataset list, choose Study 2/Solution 3 (sol3).
5
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
w2
m/s
Velocity field, z-component
Surface Average 2
1
Right-click Surface Average 1 and choose Duplicate.
2
In the Settings window for Surface Average, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
v2
m/s
Velocity field, phi-component
Surface Average 3
1
Right-click Surface Average 2 and choose Duplicate.
2
In the Settings window for Surface Average, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
u2
m/s
Velocity field, r-component
Surface Average 4
1
Right-click Surface Average 3 and choose Duplicate.
2
In the Settings window for Surface Average, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
spf2.muT/spf2.mu
1
 
4
In the Evaluation Group 1 toolbar, click  Evaluate.
Evaluation Group 1
1
Go to the Evaluation Group 1 window.
2
Click the Full Precision button in the window toolbar.
Results
1
In the Model Builder window, click Surface Average 4.
2
In the Evaluation Group 1 toolbar, click  Evaluate.
Revolution 2D
1
In the Model Builder window, expand the Results > Datasets node, then click Revolution 2D.
2
In the Settings window for Revolution 2D, click to expand the Revolution Layers section.
3
In the Start angle text field, type 0.
4
In the Revolution angle text field, type 360.
Edge 2D 1
1
In the Results toolbar, click  More Datasets and choose Edge 2D.
2
Select Boundaries 16 and 17 only.
Revolution 2D 3
1
In the Results toolbar, click  More Datasets and choose Revolution 2D.
2
In the Settings window for Revolution 2D, locate the Data section.
3
From the Dataset list, choose Edge 2D 1.
4
Locate the Revolution Layers section. In the Revolution angle text field, type 225.
5
In the Model Builder window, expand the Results node.
Revolution 2D 4
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Revolution 2D and choose Duplicate.
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 Boundary 4 only.
Parametric Surface 1
1
In the Results toolbar, click  More Datasets and choose Parametric Surface.
2
In the Settings window for Parametric Surface, locate the Parameters section.
3
Find the First parameter subsection. In the Maximum text field, type R_p.
4
Find the Second parameter subsection. In the Maximum text field, type 2*pi.
5
Locate the Expressions section. In the x text field, type s1*cos(s2).
6
In the y text field, type s1*sin(s2).
7
In the z text field, type -dL.
Parametric Curve 3D 1
1
In the Results toolbar, click  More Datasets and choose Parametric Curve 3D.
2
In the Settings window for Parametric Curve 3D, locate the Parameter section.
3
In the Maximum text field, type 2*pi.
4
Locate the Expressions section. In the x text field, type 0.9*R_p*cos(s).
5
In the y text field, type 0.9*R_p*sin(s).
6
In the z text field, type -dL.
Transformation 3D 1
1
In the Results toolbar, click  More Datasets and choose Transformation 3D.
2
In the Settings window for Transformation 3D, locate the Transformation section.
3
Select the Scale checkbox.
4
In the z text field, type pipe_plot_z_scale.
Cut Line 2D 1
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 3 (sol3).
4
Locate the Line Data section. In row Point 2, set r to 3*R_p.
5
Select the Additional parallel lines checkbox.
6
In the Distances text field, type 2*D_p 6*D_p 10*D_p.
Cut Line 2D 2
1
Right-click Cut Line 2D 1 and choose Duplicate.
2
In the Settings window for Cut Line 2D, locate the Line Data section.
3
In the Distances text field, type range(D_p,D_p,8*D_p).
Revolution 2D 5
1
In the Model Builder window, under Results > Datasets right-click Revolution 2D 1 and choose Duplicate.
2
In the Settings window for Revolution 2D, click to expand the Revolution Layers section.
3
In the Start angle text field, type 0.
4
In the Revolution angle text field, type 360.
Pipe Axial Velocity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pipe Axial Velocity in the Label text field.
3
Locate the Data section. From the Parameter selection (Ns) list, choose Manual.
4
In the Parameter indices (1-2) text field, type 2,1.
5
Click to expand the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Axial velocity - w/Ub.
7
Locate the Plot Settings section.
8
Select the x-axis label checkbox. In the associated text field, type r/R_p.
9
Select the y-axis label checkbox. Clear the associated text field.
10
Locate the Legend section. From the Position list, choose Center.
Line Graph 1
1
Right-click Pipe Axial Velocity and choose Line Graph.
2
Select Boundary 4 only.
3
In the Settings window for Line Graph, locate the y-Axis Data section.
4
In the Expression text field, type w/Ub.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type r/R_p.
7
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.
8
From the Color list, choose Black.
9
Click to expand the Legends section. Select the Show legends checkbox.
10
From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
Ns=0.5
Ns=0
12
In the Pipe Axial Velocity toolbar, click  Plot.
Rotating Pipe Tangential Velocity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Rotating Pipe Tangential Velocity in the Label text field.
3
Locate the Data section. From the Parameter selection (Ns) list, choose From list.
4
In the Parameter values (Ns) list box, select 0.5.
5
Locate the Legend section. From the Position list, choose Upper left.
6
Locate the Title section. From the Title type list, choose Manual.
7
In the Title text area, type Swirl velocity - Solid: v/(Ns*Ub) Dotted: r/R_p Dashed: (r/R_p)<sup>2</sup>.
8
Locate the Plot Settings section.
9
Select the x-axis label checkbox. In the associated text field, type r/R_p.
Line Graph 1
1
Right-click Rotating Pipe Tangential Velocity and choose Line Graph.
2
Select Boundary 4 only.
3
In the Settings window for Line Graph, locate the y-Axis Data section.
4
In the Expression text field, type v/(Ns*Ub).
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type r/R_p.
7
Locate the Legends section. Select the Show legends checkbox.
8
From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Ns=0.5
10
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.
11
From the Color list, choose Black.
Line Graph 2
1
Right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type r/R_p.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Line
Line Graph 3
1
Right-click Line Graph 2 and choose Duplicate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type (r/R_p)^2.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Parabola
Pipe Shear Stress
1
In the Model Builder window, right-click Pipe Axial Velocity and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Pipe Shear Stress in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Shear stress - Black: uw/u_tau<sup>2</sup> Blue: 200*uw/Ub<sup>2</sup>.
5
Locate the Plot Settings section.
6
Select the x-axis label checkbox. In the associated text field, type r/R_p.
Line Graph 1
1
In the Model Builder window, expand the Pipe Shear Stress node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type uw/u_tau^2.
Line Graph 2
1
Right-click Results > Pipe Shear Stress > Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type 200*uw/Ub^2.
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle (reset).
5
From the Color list, choose Blue.
Pipe Shear Stress
1
In the Model Builder window, click Pipe Shear Stress.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Upper left.
Pipe Turbulence Kinetic Energy
1
Right-click Pipe Shear Stress and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Pipe Turbulence Kinetic Energy in the Label text field.
3
Locate the Title section. In the Title text area, type TKE - Black: spf.tke/u_tau<sup>2</sup> Blue: 200*spf.tke/Ub<sup>2</sup>.
Line Graph 1
1
In the Model Builder window, expand the Pipe Turbulence Kinetic Energy node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type spf.tke/u_tau^2.
Line Graph 2
1
In the Model Builder window, click Line Graph 2.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type 200*spf.tke/Ub^2.
Rotating Pipe 3D Velocity Profiles
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Rotating Pipe 3D Velocity Profiles in the Label text field.
3
Locate the Data section. From the Dataset list, choose None.
4
Click to expand the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Left to right: w/Ub at Ns=0; w/Ub, v/(Ns*Ub) at Ns=0.5.
6
Click to expand the Plot Array section. From the Array type list, choose Linear.
7
In the Relative padding text field, type 1.
Surface 1
1
Right-click Rotating Pipe 3D Velocity Profiles and choose Surface.
2
In the Model Builder window, expand the Results > Views node, then click Surface 1.
3
In the Settings window for Surface, locate the Data section.
4
From the Dataset list, choose Revolution 2D 4.
5
From the Parameter value (Ns) list, choose 0.
6
Locate the Expression section. In the Expression text field, type w/Ub.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the z-component text field, type w/Ub.
4
Locate the Scale section.
5
Select the Scale factor checkbox. In the associated text field, type 0.15.
Surface 2
1
In the Model Builder window, under Results > Rotating Pipe 3D Velocity Profiles right-click Surface 1 and choose Duplicate.
2
In the Settings window for Surface, locate the Data section.
3
From the Parameter value (Ns) list, choose 0.5.
4
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Surface 3
Right-click Surface 2 and choose Duplicate.
View 3D 3
1
In the Settings window for View 3D, locate the View section.
2
Clear the Show axis orientation checkbox.
Surface 3
1
In the Model Builder window, expand the Results > Rotating Pipe 3D Velocity Profiles > Surface 3 node, then click Surface 3.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type v/(Ns*Ub).
Deformation 1
1
In the Model Builder window, click Deformation 1.
2
In the Settings window for Deformation, locate the Expression section.
3
In the z-component text field, type v/(Ns*Ub).
Rotating Pipe Streamlines and Surface Arrows
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Rotating Pipe Streamlines and Surface Arrows in the Label text field.
3
Locate the Data section. From the Dataset list, choose Transformation 3D 1.
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
Click to expand the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Absolute system. Streamlines at r=0.9 R_p and arrow surfaces. Color: v/(Ns*Ub).
Streamline 1
1
Right-click Rotating Pipe Streamlines and Surface Arrows and choose Streamline.
2
In the Settings window for Streamline, locate the Expression section.
3
In the r-component text field, type u.
4
In the phi-component text field, type v.
5
In the z-component text field, type w/pipe_plot_z_scale.
6
Locate the Streamline Positioning section. From the Along curve or surface list, choose Parametric Curve 3D 1.
7
Locate the Coloring and Style section. Find the Line style subsection. From the Type list, choose Tube.
8
Find the Point style subsection. From the Type list, choose Arrow.
9
From the Color list, choose Cyan.
Arrow Surface 1
1
In the Model Builder window, right-click Rotating Pipe Streamlines and Surface Arrows and choose Arrow Surface.
2
In the Settings window for Arrow Surface, locate the Expression section.
3
In the r-component text field, type u.
4
In the phi-component text field, type v.
5
In the z-component text field, type 0.
6
Locate the Data section. From the Dataset list, choose Revolution 2D 4.
7
Locate the Coloring and Style section.
8
Select the Scale factor checkbox. In the associated text field, type 0.2.
Color Expression 1
1
Right-click Arrow Surface 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type v/(Ns*Ub).
Arrow Surface 2
1
In the Model Builder window, under Results > Rotating Pipe Streamlines and Surface Arrows right-click Arrow Surface 1 and choose Duplicate.
2
In the Settings window for Arrow Surface, click to expand the Inherit Style section.
3
From the Plot list, choose Arrow Surface 1.
Transformation 1
1
Right-click Arrow Surface 2 and choose Transformation.
2
In the Settings window for Transformation, locate the Transformation section.
3
In the z text field, type -dL*pipe_plot_z_scale.
Surface 1
1
In the Model Builder window, right-click Rotating Pipe Streamlines and Surface Arrows and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Revolution 2D 3.
4
Locate the Expression section. In the Expression text field, type 1.
5
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
6
From the Color list, choose Gray.
Transformation 1
1
Right-click Surface 1 and choose Transformation.
2
In the Settings window for Transformation, locate the Transformation section.
3
Select the Scale checkbox.
4
In the z text field, type pipe_plot_z_scale.
Transparency 1
1
In the Model Builder window, right-click Surface 1 and choose Transparency.
2
In the Settings window for Transparency, locate the Transparency section.
3
Find the Transparency subsection. In the Transparency text field, type 0.25.
View 3D 4
1
In the Model Builder window, under Results > Views click View 3D 4.
2
In the Settings window for View 3D, locate the View section.
3
Clear the Show axis orientation checkbox.
Rotating Pipe Streamlines and Surface Arrows, Rotating Frame
1
In the Model Builder window, right-click Rotating Pipe Streamlines and Surface Arrows and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Rotating Pipe Streamlines and Surface Arrows, Rotating Frame in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Rotating system. Streamlines at r=0.9 R_p and arrow surfaces. Color: (\omega r-v)/(Ns*Ub).
Streamline 1
1
In the Model Builder window, expand the Rotating Pipe Streamlines and Surface Arrows, Rotating Frame node, then click Streamline 1.
2
In the Settings window for Streamline, locate the Expression section.
3
In the r-component text field, type u.
4
In the phi-component text field, type v-omega_p*r.
Arrow Surface 1
1
In the Model Builder window, click Arrow Surface 1.
2
In the Settings window for Arrow Surface, locate the Expression section.
3
In the r-component text field, type u.
4
In the phi-component text field, type v-omega_p*r.
5
Locate the Coloring and Style section. In the Scale factor text field, type 0.4.
Color Expression 1
1
In the Model Builder window, expand the Arrow Surface 1 node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type (omega_p*r-v)/(Ns*Ub).
Arrow Surface 2
1
In the Model Builder window, expand the Results > Rotating Pipe Streamlines and Surface Arrows, Rotating Frame > Arrow Surface 2 node, then click Arrow Surface 2.
2
In the Settings window for Arrow Surface, locate the Expression section.
3
In the r-component text field, type u.
4
In the phi-component text field, type v-omega_p*r.
Color Expression 1
1
In the Model Builder window, click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type (omega_p*r-v)/(Ns*Ub).
Swirling Jet Axial Velocity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Swirling Jet Axial Velocity in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2/Solution 3 (sol3).
4
Click to expand the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Axial velocity: w2/Ub - Dashed: Ns=0 Solid: Ns=0.5.
6
Locate the Plot Settings section. Select the x-axis label checkbox.
7
Select the y-axis label checkbox.
8
In the x-axis label text field, type r/R_p.
9
Clear the y-axis label text field.
Line Graph 1
1
Right-click Swirling Jet Axial Velocity and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Cut Line 2D 1.
4
From the Parameter selection (Ns) list, choose From list.
5
In the Parameter values (Ns) list box, select 0.
6
Locate the y-Axis Data section. In the Expression text field, type w2/Ub.
7
Locate the x-Axis Data section. From the Parameter list, choose Expression.
8
In the Expression text field, type r/R_p.
9
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
10
Locate the Legends section. Select the Show legends checkbox.
11
From the Legends list, choose Manual.
12
In the table, enter the following settings:
Legends
Ns=0, z=0
Ns=0, z=0.2
Ns=0, z=0.6
Ns=0, z=1
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
In the Parameter values (Ns) list box, select 0.5.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Solid.
5
From the Color list, choose Cycle (reset).
6
Locate the Legends section. In the table, enter the following settings:
Legends
Ns=0.5, z=0
Ns=0.5, z=0.2
Ns=0.5, z=0.6
Ns=0.5, z=1
Swirling Jet Axial Velocity Fluctuations
1
In the Model Builder window, right-click Swirling Jet Axial Velocity and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Swirling Jet Axial Velocity Fluctuations in the Label text field.
3
Locate the Title section. In the Title text area, type sqrt(ww2)/Ub - Dashed: Ns=0 Solid: Ns=0.5.
Line Graph 1
1
In the Model Builder window, expand the Swirling Jet Axial Velocity Fluctuations node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type sqrt(ww2)/Ub.
Line Graph 2
1
In the Model Builder window, click Line Graph 2.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type sqrt(ww2)/Ub.
Swirling Jet Axial Velocity on the 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 Swirling Jet Axial Velocity on the Axis in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Axial velocity: w2/Ub - Dashed: Ns=0 Solid: Ns=0.5.
5
Locate the Plot Settings section.
6
Select the x-axis label checkbox. In the associated text field, type z/D_p.
7
Select the y-axis label checkbox. Clear the associated text field.
8
Locate the Data section. From the Dataset list, choose Study 2/Solution 3 (sol3).
Line Graph 1
1
Right-click Swirling Jet Axial Velocity on the Axis and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 3 (sol3).
4
From the Parameter selection (Ns) list, choose From list.
5
In the Parameter values (Ns) list box, select 0.
6
Select Boundaries 5 and 7 only.
7
Locate the y-Axis Data section. In the Expression text field, type w2/Ub.
8
Locate the x-Axis Data section. From the Parameter list, choose Expression.
9
In the Expression text field, type z/D_p.
10
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
11
From the Color list, choose Black.
12
Locate the Legends section. Select the Show legends checkbox.
13
From the Legends list, choose Manual.
14
In the table, enter the following settings:
Legends
Ns=0
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
In the Parameter values (Ns) list box, select 0.5.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Solid.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Ns=0.5
Swirling Jet Radial Velocity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Swirling Jet Radial Velocity in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Radial velocity: u2/Ub.
5
Locate the Plot Settings section. Select the x-axis label checkbox.
6
Select the y-axis label checkbox.
7
In the x-axis label text field, type r/R_p.
8
Clear the y-axis label text field.
9
Locate the Data section. From the Dataset list, choose Cut Line 2D 2.
10
From the Parameter selection (Ns) list, choose From list.
11
In the Parameter values (Ns) list box, select 0.5.
Line Graph 1
1
Right-click Swirling Jet Radial Velocity 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 u2/Ub.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type r/R_p.
6
Locate the Coloring and Style section. From the Color cycle list, choose Long.
7
Locate the Legends section. Select the Show legends checkbox.
8
From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Ns=0.5, z=0
Ns=0.5, z=0.1
Ns=0.5, z=0.2
Ns=0.5, z=0.3
Ns=0.5, z=0.4
Ns=0.5, z=0.5
Ns=0.5, z=0.6
Ns=0.5, z=0.7
Ns=0.5, z=0.8
Swirling Jet Radial Velocity
1
In the Model Builder window, click Swirling Jet Radial Velocity.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Lower left.
Swirling Jet Swirl Velocity
1
Right-click Swirling Jet Radial Velocity and choose Duplicate.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Upper right.
4
In the Label text field, type Swirling Jet Swirl Velocity.
5
Locate the Title section. In the Title text area, type Swirl velocity: v2/(Ns*Ub).
6
Locate the Axis section. Select the Manual axis limits checkbox.
7
In the x minimum text field, type 0.
8
In the x maximum text field, type 3.
9
In the y minimum text field, type 0.
10
In the y maximum text field, type 0.6.
Line Graph 1
1
In the Model Builder window, expand the Swirling Jet Swirl Velocity node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type v2/(Ns*Ub).
Swirling Jet Shear Stress
1
In the Model Builder window, right-click Swirling Jet Axial Velocity and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Swirling Jet Shear Stress in the Label text field.
3
Locate the Title section. In the Title text area, type Shear stress: uw2/Ub<sup>2</sup> Dashed: Ns=0 Solid: Ns=0.5.
Line Graph 1
1
In the Model Builder window, expand the Swirling Jet Shear Stress node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type uw2/Ub^2.
Line Graph 2
1
In the Model Builder window, click Line Graph 2.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type uw2/Ub^2.
Swirling Jet Radial-Tangential Stress
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Swirling Jet Radial-Tangential Stress in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Radial-tangential stress: uv2/Ub<sup>2</sup>.
5
Locate the Plot Settings section. Select the x-axis label checkbox.
6
Select the y-axis label checkbox.
7
In the x-axis label text field, type r/R_p.
8
Clear the y-axis label text field.
9
Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
10
From the Parameter selection (Ns) list, choose From list.
11
In the Parameter values (Ns) list box, select 0.5.
Line Graph 1
1
Right-click Swirling Jet Radial-Tangential Stress 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 uv2/Ub^2.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type r/R_p.
6
Locate the Legends section. Select the Show legends checkbox.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
Ns=0.5, z=0
Ns=0.5, z=0.2
Ns=0.5, z=0.6
Ns=0.5, z=1
Swirling Jet Axial-Tangential Stress
1
In the Model Builder window, right-click Swirling Jet Radial-Tangential Stress and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Swirling Jet Axial-Tangential Stress in the Label text field.
3
Locate the Title section. In the Title text area, type Axial-tangential stress: vw2/Ub<sup>2</sup>.
Line Graph 1
1
In the Model Builder window, expand the Swirling Jet Axial-Tangential Stress node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type vw2/Ub^2.
Jet streamlines
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Jet streamlines in the Label text field.
3
Locate the Data section. From the Dataset list, choose Revolution 2D 5.
4
Locate the Plot Settings section. From the View list, choose New view.
5
In the Jet streamlines toolbar, click  Plot.
6
Click to expand the Title section. From the Title type list, choose Manual.
7
In the Title text area, type Streamlines starting at a diameter y=0. Color: v/(Ns*Ub).
8
Clear the Parameter indicator text field.
9
Click to collapse the Title section. Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
Streamline 1
Right-click Jet streamlines and choose Streamline.
View 3D 5
1
In the Settings window for View 3D, locate the View section.
2
Clear the Show grid checkbox.
3
Clear the Show axis orientation checkbox.
Streamline 1
1
In the Model Builder window, under Results > Jet streamlines click Streamline 1.
2
In the Settings window for Streamline, locate the Expression section.
3
In the r-component text field, type u2.
4
In the phi-component text field, type v2.
5
In the z-component text field, type w2.
6
Locate the Streamline Positioning section. From the Entry method list, choose Coordinates.
7
In the x text field, type range(-R_p,R_p/40,-3*R_p/4) range(-3*R_p/4,R_p/20,-R_p/2) range(-R_p/2,R_p/10,R_p/2) range(R_p/2,R_p/20,3*R_p/4) range(3*R_p/4,R_p/40,R_p).
8
In the y text field, type 0.
9
In the z text field, type 0.
10
Locate the Coloring and Style section. Find the Line style subsection. From the Type list, choose Tube.
11
In the Tube radius expression text field, type 1.
12
Select the Radius scale factor checkbox. In the associated text field, type 0.001.
13
Click to expand the Advanced section. In the Maximum streamline length text field, type 0.07.
14
Clear the Allow backward time integration checkbox.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type v2/(Ub*Ns).