TUT 8: Aerofoil Analysis in

FluentCFD 814

Adhikar Hariram (18121004)

2013

1 | P a g e

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

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

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

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

Problem Setup...................................................................................................................................4

Results...............................................................................................................................................7

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

Simulation 2.................................................................................................................................11

Simulation 3.................................................................................................................................15

Simulation 4.................................................................................................................................19

Conclusion...........................................................................................................................................24

References...........................................................................................................................................25

2 | P a g e

Table of figures

Figure 1: Geometrical Description of Problem......................................................................................4Figure 2: Fine Mesh Generated.............................................................................................................6Figure 3: Residuals k-epsilon.................................................................................................................8Figure 4: Drag Coefficient k-epsilon.......................................................................................................8Figure 5: Lift Coefficient k-epsilon.........................................................................................................9Figure 6: Axis Transformation for Drag and Lift.....................................................................................9Figure 7: y+ k-epsilon...........................................................................................................................10Figure 8: Velocity Contour Plot k-epsilon............................................................................................11Figure 9: Residuals k-omega................................................................................................................12Figure 10: Drag Coefficient k-omega...................................................................................................12Figure 11: Lift Coefficient k-omega......................................................................................................13Figure 12: y+ k-omega.........................................................................................................................14Figure 13: Velocity Contour Plot k-omega...........................................................................................15Figure 14: Residuals RSM.....................................................................................................................16Figure 15: Drag Coefficient RSM..........................................................................................................16Figure 16: Lift Coefficient RSM.............................................................................................................17Figure 17: y+ RSM................................................................................................................................18Figure 18: Velocity Contour Plot RSM..................................................................................................19Figure 19: Rough Mesh Generated......................................................................................................20Figure 20: Residuals Rough Mesh........................................................................................................21Figure 21: Drag Coefficient Rough Mesh.............................................................................................21Figure 22: Lift Coefficient Rough Mesh................................................................................................22Figure 23: y+ Rough Mesh...................................................................................................................23Figure 24: Velocity Contour Plot Rough Mesh.....................................................................................24

3 | P a g e

Introduction

One of the most common applications in the field of CFD is in the aeronautics industry with a large amount of work having being done on the analysis of aerofoils. Amongst the many types and series of aerofoils, the Joukowski aerofoil is one who's shape is steeped in mathematics and as a result the streamlines around such a profile are well understood. There also exists empirical data on the lift and drag characteristics for this series of aerofoil and hence it provides an ideal shape for CFD analysis and verification of results. Thus, the CFD analysis of a Joukowski aerofoil profile with known lift and drag characteristics has been undertaken.

Technical Section

Problem DescriptionThe problem modelled consisted a Joukowski 558 aerofoil profile at an angle of attack of α that is travelling at a velocity equivalent to Mach 0.4. A geometrical description of the problem is shown in figure 1.

Figure 1: Geometrical Description of Problem

The value for α , as well as the other variables used in the simulation can be found in table 1. All of the values used were derived from the student number 18121004.

Table 1: Values used in Simulation

Property/Dimension Valueα 3+40

T 00CH (altitude) 4000m

It should be noted that the fluid properties were left as the database values for air in FLUENT.

Problem SetupThe first step of the problem setup was to create the geometry for the domain, which was done using the ANSYS Design modeller. The first step was to create a sketch for the flow domain that would accurately represent the flow being simulated. As was found in both the FLUENT Tutorial

4 | P a g e

guide, as well as in other online tutorials for the analysis of an aerofoil, the domain was made to be curved at the inlet with a semi-circle with the rest of the downstream domain being made up of a simple rectangle. It should also be noted that the trailing edge of the aerofoil was made to be in-line with the end points of the semi-circle. This is illustrated further on in figure 2, which shows the meshed domain. The domain was also made sufficiently large to ensure that the boundary conditions could be accurately prescribed. Once the domain was sketched; a 2-D surface was created from it as the analysis to be undertaken was a pure 2-D simulation. In order to accurately capture flow phenomenon close to the aerofoil body, the coordinates of the aerofoil were scaled up twice and imported into the Design modeller. From these scaled coordinates, line bodies were created and projected onto the created domain. This split the domain into three parts which allowed for much greater mesh control especially near to the aerofoil body. The next step then required the domain to be meshed using the ANSYS Meshing facility.

Once the domain was in the meshing facility, all previous line bodies were suppressed as only the domain needed to be meshed and not these line bodies. The next step was to apply separate face sizing's onto the two separate regions closest to the aerofoil body as this would allow the flow to be accurately captured in this region. Finally, inflation was applied to the surface of the aerofoil in order to capture the boundary layer effects along its surface. When inserting the inflation layer, the specification was changed to first layer thickness as this would allow for an adequate y+ value to be achieved at the wall. This y+ value was achieved using a flat plate approximation and the total number of layers was determined from the boundary layer thickness, δ , also determined from a flat plate approximation. The details of the equations used to calculate these values for a turbulent flow can be found in (1). The velocity used in the equations given in (1) was determined from the mach number and the speed of sound. For flow in air, this is given as :

M= c

√γRT−(1)

where c is the speed of the fluid concerned,γ is the ratio of specific heats, is the gas constant divided by the fluid's molecular weight and T is in Kelvin. From this equation, the speed of the fluid was found to be 146.02 m/s which resulted in a y of 1 mm at the end of 2 m long plate for a y+ of 60, and a δ value of 26.48mm at a length of 1.8m along a 2m long plate. The y+ of 60 was chosen as this is within the range of 30<y+<300 for which the wall treatment equations used in FLUENT are valid. The length of 1.8 m was also chosen due to flow separation being expected to occur before the trailing edge of the aerofoil. This meant that the maximum boundary layer thickness that was required occurred at a point before 2m along the flat plate approximation. Using the mentioned sizing's and inflation specifications, the mesh that had been generated can be seen in figure 2. It should be noted that the named selections for the boundaries were also created in the meshing application. These selections were namely: 'inlet', 'outlet', 'symmetry' and 'wing'.

5 | P a g e

Figure 2: Fine Mesh Generated

Once the mesh had been generated; the problem then needed to be setup and solved in FLUENT. The first step of setting up the problem was to change the viscosity model from laminar to turbulent, with the specific turbulence model being changed for each simulation in order to investigate the effects of the turbulence model on the results. Next, the operating pressure for the domain needed to be set, and this was a function of altitude as described as follows:

P=P0e−hh0 − (2 )

where P0 is the pressure at ground level, h0 is the height of the atmosphere, which is taken as 7km and h is the altitude of concern which, in this specific simulation, was 4km. This resulted in the operating pressure being 57220 Pa. Next, the inlet conditions needed to be set. When setting the velocity; it was specified using components by splitting the total velocity into x and y components where the total velocity was at an angle of 70 to the x axis. At the inlet the turbulence had to also be specified and this was done using turbulence intensity along with hydraulic diameter. This method was chosen as the hydraulic diameter can be approximated more accurately than the other specification methods. The turbulence intensity was set to 7% as common values used in simulations are between 5% and 10% with values over 10% being highly turbulent and fairly rare in simulations. The hydraulic diameter was set to that of the flat plate approximation of 2 m. The same turbulence conditions were also set at the pressure outlet whilst the gauge pressure at the outlet was set to 0 Pa. Next, the reference values were computed from the inlet and the temperature and pressure was changed accordingly to the values previously mentioned. The next step was to select the solver methods. Due to the effect of the turbulence model being the only variable of concern, the solution methods selected were used in every simulation and thus had to be chosen correctly.

6 | P a g e

For the pressure-velocity coupling, the Coupled solver was used. The coupled algorithm solves both the momentum and pressure based continuity equation simultaneously (2). This scheme was chosen as it was found, from past experience, to give results similar to that of the SIMPLE solver whilst having greatly reduced convergence times. This meant that the solution would converge fairly quickly whilst providing as accurate as possible results. For the spatial discretisation, the Second Order Upwind scheme was chosen. With the upwind differencing scheme, cell face values upwind of the centroid is taken to be the same as the value of the cell centroid upwind of the cell concerned (3). 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 (2). The Second Order Upwind scheme was chosen as it was found, in previous simulations, to provide the most accurate results for the widest range of simulations. It also has increased accuracy over the First Order Upwind scheme, especially in cases where flow separation is expected as was found in (4). The QUICK scheme was also found to provide accurate results, however was not as robust as the Second Order Upwind scheme in terms of the range of problems that the QUICK scheme could be applied to. This was especially evident for natural convection problems as found in (5). Thus, for these reasons the Second Order Upwind scheme was determined to be the most appropriate spatial discretisation scheme for this problem. Next, in order to accurately judge convergence, both a monitor for the drag and lift coefficient on the wing was created as these plots would become constant when the solution has converged. The lower limit on the residuals were then set to 0.0001 in order to obtain greater accuracy in the results. The next step was to then initialise the solution and this was done using hybrid initialisation which solves the Laplace equation to produce a velocity field and smooth pressure distribution in the domain (2). Once initialised, the solution was run and the results were extracted and analysed.

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 analysing the results of each simulation, it was important to establish the results as predicted by the empirical data provided. Using this data, for an aerofoil with an angle of attack of 70, the coefficient of lift was approximately 1.18 and the coefficient of drag was approximately 0.175. The equation governing the drag force and lift force for an aerofoil flow is given as follows:

Fd ,l=12Cd ,lU

2 Aρ−(3)

where the area A is the plan area of the aerofoil which is the projected area of the aerofoil as seen from above or below. This equation applies to both lift and drag and the values predicted for a 1 m wing span with chord length 2 m was: Drag force = 4537.02 N, Lift Force = 30592.49 N.

Simulation 1The first simulation that had been conducted using the aforementioned setup information was done using the standard k-epsilon (k-ε ) turbulence model with standard wall functions. This standard k-epsilon model is based on model transport equations for the turbulence kinetic energy and its

7 | P a g e

dissipation rate (2). These equations and further detail on this model can be found in (2). The results of the simulations run using this model are presented here. First, the convergence nature of the solution can be seen in figures 3, 4 and 5.

Figure 3: Residuals k-epsilon

Figure 4: Drag Coefficient k-epsilon

8 | P a g e

Figure 5: Lift Coefficient k-epsilon

From figures 4 and 5 it can be seen that convergence may have occurred in the region of 1600 iterations. It should also be noted that these plots were used simply as a guide to judge convergence and do not represent the true drag and lift coefficients for the given aerofoil. This is due to the vector direction in which FLUENT measures them being different to the actual direction in which they should be measured. This is illustrated in figure 6 where the direction of measurement is the x and y axis', whereas the actual direction they should be measured in is the x1, y1 axis' due to the manner of the flow direction relative to the domain.

Figure 6: Axis Transformation for Drag and Lift

Using the new direction vectors, as represented in figure 6, it was found that with the standard k-epsilon model, the total lift force per 1 m wingspan was found to be 22702.50N. Similarly the total drag force was found to be 2310.18N. Both these values are found to be lower than the predicted values and this could be a result of the different conditions under which the coefficients were

9 | P a g e

obtained for the predicted results. Due to no information being available concerning the conditions under which the coefficients; used to predict the results; were obtained, it is likely a cause for the difference in the simulated and predicted results. As an example, the viscosity of the fluid would have an effect on the total drag force on the aerofoil thus differences in the values used in the simulation and prediction could result in differences in the total drag force. It could also be that the coefficients may have been obtained using a different working fluid in which case there may be inaccuracies when converting the values to air. Another contributing factor is that for flows over a mach number of 0.3, the fluid would generally need to be modelled as compressible, which was not done in this case. As a result of these arguments, it can be said that for this specific flow scenario; the simulated values are more accurate as a result of the direct nature in which they solve the governing equations. This is especially true for slightly more complex geometries, such as that of an aerofoil. This being said, in order to check that the solution was in fact accurate, the y+ value along the surface of the wing was plotted in order to ensure that it fell between 30 and 300 as this would validate the use of the wall functions. These wall functions in term determine the wall shear stress which are indicative of the drag force on the cylinder. This y+ plot can be seen in figure 7.

Figure 7: y+ k-epsilon

From figure 7 it can be seen that majority of the surface of the aerofoil contains a y+ value in the range mentioned, which validates the use of the wall functions. As previously mentioned, flow separation was also expected to occur at some point along the aerofoil's surface. In order to judge where this separation occurs; the velocity contour plot was investigated and at a point where the boundary layer along the aerofoil stops growing as expected; it was determined that separation occurs at that point. This contour plot can be seen in figure 8.

10 | P a g e

Figure 8: Velocity Contour Plot k-epsilon

From figure 8, the point of separation of the flow was judged to be in the region of 1.9m downstream of the leading edge of the aerofoil. This was found to be fairly close to the predicted value of 1.8m that was used to determine the boundary layer thickness.

Simulation 2The second simulation that was run was done 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 (2). More information, as well as the governing equations for this model can be found in (2). As with the first simulation, the convergence behaviour is presented first with the same monitors in place. These can be seen in figures 9, 10 and 11.

11 | P a g e

Figure 9: Residuals k-omega

Figure 10: Drag Coefficient k-omega

12 | P a g e

Figure 11: Lift Coefficient k-omega

From figures 10 and 11, it can be seen that the k-omega model converges much faster than the k-epsilon model with convergence occurring in the region of 500 iterations. Using this turbulence model the lift and drag force was found to be: Drag force = 1617.26 N, Lift force = 24466.13 N. Again, these values were found to be greatly different to the predicted values. The arguments presented for the first simulation however still apply and provide an explanation for the differences in predicted and simulated results. It should be noted that the SST-k-omega turbulence model is the recommended model for analysis of aerofoils and this model achieves greater accuracy in a greater class of flows (2). Thus it can be said that the results found with this turbulence model is of greater accuracy than those achieved with the k-epsilon model. Again, however, in order to validate the use of the wall functions for this model, the y+ plot was used and can be seen in figure 12.

13 | P a g e

Figure 12: y+ k-omega

From figure 12 it can be seen that y+ value is within the required range for a larger portion of the aerofoil when using the k-omega turbulence model. This shows that the use of the wall functions are valid. It can also be seen that y+ falls within the required range for a greater proportion of the wing when using the k-omega model as compared with the k-epsilon model. This proves that, for this specific flow scenario, the k-omega method is the more accurate of the two. Finally, in order to judge flow separation; the velocity contour plot was once again used. This is seen in figure 13.

14 | P a g e

Figure 13: Velocity Contour Plot k-omega

Investigation of the velocity contour plot revealed that flow separation again only occurs in a region approximately 1.9 downstream of the leading edge of the aerofoil.

Simulation 3The 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). As with the previous simulations, the convergence behaviour has been presented in figures 14, 15 and 16.

15 | P a g e

Figure 14: Residuals RSM

Figure 15: Drag Coefficient RSM

16 | P a g e

Figure 16: Lift Coefficient RSM

From figures 14, 15 and 16, it can be seen that convergence only occurs in the region of 900 iterations. Using the RSM, the drag and lift forces were found to be: Drag force = 1513.09 N, Lift force = 23669.56 N. Once again these values are found to be different to the predicted values with the differences already having been explained for the previous simulations. In comparison to the other turbulence models, the RSM is the most elaborate model and has greater potential to provide more accurate solutions. However this model does employ certain assumptions to close the transport equations and still relies on scale equations (epsilon; omega), both of which are a source of inaccuracy for this scheme. The results for this simulation can be said to be more accurate than the other two, simpler models, however not so much so to warrant the additional computational cost. In order to confirm the accuracy of the solution, again the wall y+ value was checked along the surface of the aerofoil. This can be seen in figure 17 which shows the wall y+ value along the length of the aerofoil.

17 | P a g e

Figure 17: y+ RSM

From figure 17 it can be seen that y+ falls within the appropriate range; for the RSM model; over as much of the aerofoil as with the k-omega model. Due to this being the large majority of the aerofoil surface, it can again be said that the use of wall functions for this model are indeed accurate. Once again, in order to judge the separation of the flow the velocity contour plot was investigated and this can be seen in figure 18.

18 | P a g e

Figure 18: Velocity Contour Plot RSM

From investigation of figure 18 it was again found that the flow separation occurred in the region of 1.9m downstream of the leading edge of the aerofoil. This shows that all three turbulence models produce the same velocity profiles.

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 first mesh having in the region of 60000 cells and thus much further refinement could not be done due to the cell limit in the FLUENT academic licence. The rough mesh was in the region of half the mesh density of the first three simulations in all the regions of the domain and can be seen in figure 19.

19 | P a g e

Figure 19: Rough Mesh Generated

The simulation run with the rough mesh was done using the SST-k-omega turbulence model alone due to this model being recommended for aerofoil analysis. This model was also found to converge much faster than the other two and thus provided the best platform to compare the results from the fine and rough mesh simulations.

As with the previous simulations, the convergence nature of the simulation run with the rough mesh has been represented in figures 20, 21 and 22.

20 | P a g e

Figure 20: Residuals Rough Mesh

Figure 21: Drag Coefficient Rough Mesh

21 | P a g e

Figure 22: Lift Coefficient Rough Mesh

From the convergence plots presented it can be seen that the rough mesh converged much faster than any previous simulation. This is a direct result of having fewer cells and thus faster solving times. It can be seen however that the residuals do not converge as smoothly as with the fine mesh. This is most likely due to the larger cells resulting in the solver having to overcompensate to satisfy the governing equations and this continues to loop over itself around the converged value. It may have been possible to overcome this fluctuation by setting the under-relaxation factors even smaller in order to avoid the overcompensation of the solver. With the rough mesh, the resultant forces on the aerofoil were found to be: Drag force = 1616.54 N, Lift force = 24371.26 N. From these results it can be said the result is fairly mesh independent as these results are extremely close to those achieved with the fine mesh. However it can be seen that due to the way in which the mesh was constructed, the mesh density close to the aerofoil was still fairly high which still results in an accurate solution. Again, to check this accuracy the y+ value along the wall surface has been plotted in figure 23 as this would validate the wall functions.

22 | P a g e

Figure 23: y+ Rough Mesh

From figure 23 it can be seen that the y+ value along the wall falls within the required range almost everywhere along the aerofoils surface. This validates that the wall functions are indeed valid, even for this lowered mesh density. Finally, for the rougher mesh the flow separation again needed to be checked against the previous results, and this was done with the use of the velocity contour plot as before. This is seen in figure 24.

23 | P a g e

Figure 24: Velocity Contour Plot Rough Mesh

For the rough mesh it was found that the flow separation occurred slightly closer to the leading edge of the aerofoil at a distance of around 1.8m. Thus, overall the only major difference between the fine and rough mesh results are the rates of convergence and the velocity profiles that are developed.

Conclusion

From the simulations conducted it can be concluded that when analysing an aerofoil profile that the most appropriate turbulence model is the SST-k-omega model. This is due to it having the cheapest computational cost whilst having fairly accurate results compared to the expensive RSM model. Another conclusion that can be drawn is that the different turbulence models result in similar velocity profiles yet due to differences in the stress modelling of the flow; each turbulence model produces different values of lift and drag. It can also be concluded that when comparing the results of a simulation and empirical data, it is vital that the conditions under which the simulation is run matches the conditions used when obtaining the empirical data. It can also be concluded that for flows with Mach numbers greater than 0.3, compressibility effects likely need to be taken into account else the results of the simulation will be inaccurate. It can also be concluded that for the specific aerofoil shape analysed at the specific angle of attack analysed; the flow remains fairly well structured over majority of the length of the aerofoil as the flow only separates around 0.1m upstream of the trailing edge. Finally it can be concluded that the result is fairly mesh independent as long as the mesh remains fairly fine in the regions close to the aerofoil.

24 | P a g e

References

1. Hariram, A V. Tut 7: Flow Over a Flat Plate. Stellenbosch : University of Stellenbosch, 2013.

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

3. 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.

4. Hariram, A V. Tut 6: Vortex Shedding in Fluent. Stellenbosch : University of Stellenbosch, 2013.

5. —. Tut 5: Natural Convection Heat Transfer in Fluent. Stellenbosch : University of Stellenbosch, 2013.

25 | P a g e