TUT 9: Pipe Flow in FluentCFD 814

Adhikar Hariram (18121004)

2013

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ContentsTable of figures......................................................................................................................................3

Introduction...........................................................................................................................................4

Technical Section...................................................................................................................................4

Problem Description..........................................................................................................................4

Problem Setup...................................................................................................................................5

Results...............................................................................................................................................6

Simulation 1...................................................................................................................................7

Simulation 2...................................................................................................................................9

Simulation 3.................................................................................................................................11

Simulation 4.................................................................................................................................13

Conclusion...........................................................................................................................................16

References...........................................................................................................................................17

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Table of figures

Figure 1: Geometrical Description of Problem......................................................................................4Figure 2: Section of Fine Mesh Generated............................................................................................5Figure 3: Dean Vortices..........................................................................................................................7Figure 4: Residuals k-epsilon.................................................................................................................7Figure 5: Pressure Monitor k-epsilon.....................................................................................................8Figure 6: y+ k-epsilon.............................................................................................................................8Figure 7: Velocity Vector Plot k-epsilon.................................................................................................9Figure 8: Residuals k-omega................................................................................................................10Figure 9: Pressure Monitor k-omega...................................................................................................10Figure 10: Velocity Vector Plot k-omega.............................................................................................11Figure 11: Residuals RSM.....................................................................................................................12Figure 12: Pressure Monitor RSM........................................................................................................12Figure 13: Velocity Vector Plot RSM....................................................................................................13Figure 14: Rough Mesh Generated......................................................................................................14Figure 15: Residuals Rough Mesh........................................................................................................14Figure 16: Pressure Monitor Rough Mesh...........................................................................................15Figure 17: Velocity Vector Plot Rough Mesh.......................................................................................15

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Introduction

A highly studied field in fluid mechanics is that of pipe flow due to many industrial processes requiring the transport of fluids through a system via pipes. In the field of pipe flow; information about the pressure drop along the length of the pipe has been studied empirically and thus pipe flow provides an appropriate test case for the validation of CFD results. This pressure drop information is also available for various bends and elbows in a piping system. Experimental information is also available for the flow patterns found beyond such bends. Thus flow through a 900 pipe bend has been investigated for which qualitative and quantitative data is available.

Technical Section

Problem DescriptionThe problem modelled consisted of a simple pipe with a 900 bend section of radius R. A 2-d representation of the problem can be seen in figure 1.

Figure 1: Geometrical Description of Problem

The values for the fluid properties as well as pipe dimensions and flow conditions can be found in table 1, and are all derived from the student number 18121004.

Table 1: Values used in Simulation

Property/Dimension Valueρ 200 kg/m3

V 4 m/sμ 1x10-5 Pa.sD 4 cmR 40 cm

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Problem SetupThe first step of solving the problem involved creating the geometry for the problem and this was done using Autodesk Inventor. This software was used due to the problem needing to be modelled in 3-d and the Autodesk Inventor 3-d modelling being easier to use than the ANSYS design modeller. The need for a 3-d geometry was a result of secondary, swirling flow effects expected close downstream of the pipe bend. Once the pipe geometry was created in Autodesk Inventor it was exported as an iges file and then imported into ANSYS workbench. In ANSYS design modeller the geometry was then frozen and a new solid was created that enclosed a section of the pipe. This would allow for a body of influence to later be used for the mesh sizing in that region. This new solid was also then frozen before the meshing step was undertaken.

The meshing was done using the ANSYS meshing utility and in order to get an appropriate mesh size the maximum cell sizing was set to an appropriately small value. The maximum face size was also set to a value approximately half of the maximum cell size as this is the default ratio in the meshing application. The next step was to apply an inflation layer along the entire pipe surface in order to capture the boundary layer. This was defined using a first layer thickness such that an appropriate y+ value could be achieved along the wall surface. This value needed to lie between 30 and 300 in order for the wall functions to be valid for the various turbulence models. Finally a mesh sizing was used to ensure that the mesh would be appropriately small in the region downstream of the midpoint of the elbow as this region is where the flow was likely to experience swirling. A body of influence was used to do this by using the previously mentioned, newly created solid as the body of influence. A cross section of a portion of the resultant mesh can be seen in figure 2.

Figure 2: Section of Fine Mesh Generated

It should be noted that the flow is from left to right in figure 2. The last step of the meshing procedure was to set the names selections to be later used in FLUENT, namely, 'inlet', 'outflow', and 'pipe'. Once the meshing was completed the mesh could be imported into FLUENT and the problem could be setup and solved. Once the mesh was imported into FLUENT, the first step was to change the viscous model from laminar to turbulent with the specific turbulence model being used being changed for each simulation. The fluid properties were then changed to the values given in table 1 before setting the boundary conditions for the problem. The inlet velocity was set to the value given in table 1 and the turbulence specification was changed to turbulence intensity and hydraulic diameter. The turbulence intensity was left as the default value of 10% and the hydraulic diameter was changed to the pipe diameter as given in table 1. For the pipe, which was recognised as a wall in

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FLUENT, the roughness height was left as the default value of 0m which implies a perfectly smooth wall. The next change that was made from the default settings was to change the pressure velocity coupling to the COUPLED scheme as this was found, with past experience, to converge faster than the SIMPLE method whilst still providing accurate results. The coupled algorithm solves both the momentum and pressure based continuity equation simultaneously (1) . The spatial discretisation scheme was also changed to being second order upwind due to previous experience showing the robustness and effectiveness of this scheme in most cases. With the upwind differencing scheme, cell face values upwind of the cell centroid is taken to be the same as the value of the cell centroid upwind of the cell concerned (2). With the second order upwind scheme, the cell face values are computed using a Taylor series expansion of the cell-centred solution about the cell centroid (1). The next step was to setup an integral surface monitor on the outflow face in order to correctly judge the convergence of the solution. This monitor was setup to measure the integral of the surface pressure on the outlet with convergence being judged at the point where this monitor remain constant. The lower limit on the residuals were also decreased in order to obtain increased accuracy in the solution as well as to fully ensure that convergence could be reached. Once this was done, the solution was then initialised using hybrid initialisation. This solves the Laplace equation to produce a velocity field and smooth pressure distribution in the domain (1). The solution was then calculated and new planes were created downstream of the end of the pipe bend which would allow the secondary flow effects to be investigated.

ResultsWhen conducting the various simulations, the most important effects that were of concern were the following:

1. Effect of various turbulence models2. Mesh independence of the solution

Before the simulated results could be analysed; the theoretical predictions needed to first be determined as they would provide some form of validation of the results. The first type of result that could be used for comparison was the pressure drop along the length of the pipe. This was determined using the equation (3):

H l=flDU 2

2 g−(1 )

Equation (1) gives the head loss for a flow through pipes in meters, and to convert it to Pascals needs to be multiplied by the density and gravity. The friction factor, f, is a function of the flow Reynolds number and the surface roughness and is obtained from the Moody chart (3). This chart can commonly be found in many fluid mechanics books and at the back cover of (3), and manages to account for smooth pipes. The friction factor was found for the pipe at a Reynolds number of 3.2x106

(which clearly indicated turbulent flow) to be 0.014. For the pipe bend an equivalent length was also found using a chart that relates it to the radius of the bend and diameter of the pipe. This chart was, again, found in (3). Using the sum of the equivalent length and the straight sections of the pipe, the predicted head loss in the pipe was found to be 817.6Pa. It should be noted that the equations and factors used to predict this pressure drop are all based on empirical data.

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The second type of result that could be used to validate the results was the flow pattern downstream of the pipe bend. In practice, it was found that two counter rotating vortices form downstream of the pipe bend, known as Dean Vortices (4). These vortices have the form shown in figure 3 and thus provide some qualitative results against which the results could occur.

Figure 3: Dean Vortices

Having information about the expected pressure drop as well as the expected flow patterns within the pipe; the simulated results could be analysed and validated.

Simulation 1The first simulation conducted had been done using the standard k-epsilon (k-ε ) turbulence model with standard wall equations. This standard k-epsilon model is based on model transport equations for the turbulence kinetic energy and its dissipation rate (1). When running the solution it was first vital to correctly judge the convergence of the solution before extracting and analysing any results. This was done by monitoring the residuals and the surface monitor at the outlet. The convergence behaviour of these monitors can be seen in figures 4 and 5.

Figure 4: Residuals k-epsilon

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Figure 5: Pressure Monitor k-epsilon

From the surface monitor it can be seen that convergence occurred within 350 iterations, however from monitoring the residuals it can be seen that full convergence and greatest possible accuracy was only achieved after 475 iterations.

Before analysing the results completely, it was also vital to first check the value of y+ all along the pipe surface as this would validate the use of the standard wall functions. This was checked using a histogram plot of y+ on the pipe's surface. This can be seen in figure 6.

Figure 6: y+ k-epsilon

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From figure 6 it can clearly be seen that the y+ is always within the required range of 30-300 everywhere along the pipe's surface. This shows that the use of the standard wall functions, for the given mesh, was in fact valid. Due to this y+ being mostly dependent on the size of the mesh it can be said that all other simulations run with this mesh would also fall within this range. Once it was established that the y+ values for this mesh size were valid, the results of the simulation could be fully analysed. One of these results can be seen in figure 7, which shows the velocity vector plot on a plane just downstream of the pipe bend.

Figure 7: Velocity Vector Plot k-epsilon

It should be noted that the inside of the pipe bend relative to the plane in figure 7 is at the bottom of the figure, nearer to the blue, slower velocity vectors. From figure 7 it can be seen that the simulation produces the predicted Dean vortices downstream of the plane. It was also found, using a faceted averaged; total pressure report at the inlet and outlet, that the pressure drop along the pipe was 870.01Pa. It was however found that when an area-weighted average was taken on the face that the pressure drop was 551.90Pa. This could be a result of the extremely small face size of the inflation cells which result in a much smaller value for the pressure difference. Thus the facet averaged values have been used for comparison purposes, which shows that the simulation was in the region of 94% accurate with regards to the total pressure drop along the pipe. This, combined with the expected flow patterns being produced, confirms that the solution was indeed accurate.

Simulation 2The second simulation was run using the SST-k-omega (SST-k-ω) model. This model is a blend of the robust and accurate formulation of the k-omega model near the wall with the free-stream independence of the k-epsilon model in the far field (1). More information, as well as the governing equations for this model can be found in (1). As with the first simulation, the convergence behaviour is presented first with the same monitors in place. These can be seen in figures 8, and 9.

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Figure 8: Residuals k-omega

Figure 9: Pressure Monitor k-omega

From figures 9 it can be seen that the k-omega model converges within 300 iterations however due to the decreased limit on the residuals, full convergence is only reached within 475 iterations; much similar to the k-epsilon model. As previously stated, the value of y+ is mesh dependent and hence remained within the required bounds for the wall function to be valid even for this simulation. The velocity vector plot that resulted by using the k-omega model can be seen in figure 10. Note that this plot is taken in the exact same orientation and location as the plot in figure 7.

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Figure 10: Velocity Vector Plot k-omega

From figure 10, the velocity vectors appear to be much the same as those in figure 7 which exhibit the Dean vortices found in reality. With the k-omega model, the pressure drop along the pipe was found to be 832.40Pa, which is in the region of 98% accuracy with respect to the predicted values. This shows that the k-omega model is in fact more accurate than the k-epsilon model and is highly accurate in predicting the correct flow formation as well as pressure drops.

Simulation 3The third simulation that was run The third simulation that was run using the fine mesh was done using the Reynolds Stress Transport model (RSM). The RSM closes the Reynolds-averaged Navier-Stokes equations by solving transport equations for the Reynolds stresses, together with an equation for the dissipation rate (2). More detail can be found in (2). Again, the convergence behaviour of the solution is shown in figures 11 and 12.

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Figure 11: Residuals RSM

Figure 12: Pressure Monitor RSM

From figures 11 and 12 it can be seen that the RSM model takes sufficiently longer to converge than the k-epsilon or k-omega model. This is a result of the added equations that need solving when using the RSM model. The velocity vector plot generated using the RSM model can be seen in figure 13.

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Figure 13: Velocity Vector Plot RSM

From figure 13 it can be seen that the RSM predicts the dean vortices that are expected for this form of pipe flow. Along with this result, the pressure drop along the pipe was found to be in 865Pa which, when compared to the predicted results, was less accurate than the k-omega model but more accurate than the k-epsilon model. This is an unexpected result due to the RSM model being the most complex form of turbulence model. This could be a result of the simulation solving the governing equations directly, thus when comparing to experimental data it may provide different results. Due to the increased accuracy of the RSM model it can be said that, under ideal conditions, the true pressure drop may be closer to 865Pa than to 817Pa as predicted by experimental data.

Simulation 4The final simulation that was conducted was done using a rougher mesh than for the first three simulations. This was done due to the limit in the number of cells when using the academic licence in FLUENT. It should be noted however that the inflation layer parameters were not changed as these provided an acceptable y+ value in the previous simulations and would thus provide an accurate y+ for this simulation. A similar section of the mesh as that in figure 2 can be seen in figure 14.

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Figure 14: Rough Mesh Generated

The simulation run with the rough mesh was done using the SST-k-omega turbulence model alone due to this model having previously provided the most accurate results as well as having the fastest convergence rate. The convergence of the simulation done with this rough mesh was judged by the monitors in figures 15 and 16.

Figure 15: Residuals Rough Mesh

Figure 16: Pressure Monitor Rough Mesh

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From figures 15 and 16 it can be seen that the solution converges at a much faster rate when a rough mesh it used as opposed to a fine mesh. This is due to the decreased number of cells in the domain which requires less time to resolve the governing equations for each cell. The resultant velocity vector plot for the simulation run with the rough mesh can be seen in figure 17.

Figure 17: Velocity Vector Plot Rough Mesh

From figure 17 it can be seen that even with a rough mesh, the Dean vortices are still correctly predicted downstream of the pipe bend. Finally, in order to check the accuracy of the solution, the pressure drop across the pipe was found to be 828.08Pa. This is almost exactly the same as the value found when the simulation was run under the same conditions but with a finer mesh. Thus it can be said that the solution is indeed mesh independent.

Conclusion

From the simulations conducted it can be concluded that when analysing a pipe flow problem, the k-omega turbulence model may prove to be the best turbulence model due to its fast convergence rate and high accuracy. The RSM model may prove to be more accurate over a wide range of applications, however the accuracy gain is not worth the additional computational cost for this problem. It can also be concluded that all tested turbulence models do provide results that are fairly accurate both in terms of qualitative and quantitative results. It can also be concluded that for such simple flow geometries, such as that of a pipe with an elbow, that the solution is indeed mesh independent.

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References

1. Ansys-Help. Ansys FLUENT Theory Guide. Ansys FLUENT Theory Guide. s.l. : ANSYS, 2011.

2. HK, Versteeg and W, Malasekera. 5.6 The upwind differencing scheme. An Introduction to Computational Fluid Dynamics The Finite Volume Method. Essex : Pearson Education Limited, 2007.

3. Fox, R W, Pritchard, P J and T, McDonald A. Introduction to Fluid Mechanics. Massachussets : John Wiley & Sons, 2009. ISBN-13.

4. A, Kalpakli. Experimental study of turbulent flows through. Stockholm : Royal Institute of Technology, 2012. SE-100.

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