CFD correlation comparison

In the literature review it has been argued that steady - and even unsteady - RANS simulations can not reproduce all flow features for an unsteady bluff body flow with large scale 3D eddy structures in the wake. Nevertheless, in industrial applications it is in general not feasible to construct a grid that is fine enough for LES and / or to run a DES or LES simulation with a small enough time step. Therefore, despite its shortcomings, SRANS methods are still used to predict wheel flow aerodynamics and to get an idea of the resulting force coefficients. The industry standard seems to be to use the SA turbulence model, because of its robustness, whereas academia prefers to apply a version of the k-²model (see table 1.4), because of the claimed better performance in separated flow regions [17, 97]. Due to the sheer extent of the current research, which involves three geometrical parameters to vary (h, gap and overlap), it has been decided to limit the computational research to SRANS simulations. This section will first discuss the grid sensitivity of the results and after this a number of the turbulence models will be compared for the isolated wheel flow.

Grid Sensitivity The simulation for the baseline grid with a SA SRANS solver produces a difference in drag force coefficient between computations and experiments of -6.6%. Table 3.1 shows that coarsening (13.3% in cell dimension per direction) and refinement (13.6%) of the grid do not result in a consistent trend in the changes, since both cases yield a slightly lower drag value. The Fine grid case drag is 1% less than the Baseline case, but moves away from the experimental value. The other characteristics in table 3.1 do however display consistent trends with grid refinement. The wheel lift increases with refinement, the separation from the top of the wheel at the centreline moves downstream and the positive and negative pressure peak near the contact patch grow in magnitude as the number of cells increases. The prediction of the centreline stagnation position is mainly dependent on the local grid resolution2 _{and is in line with the experiments for all}

2_{The grid resolution angle at the stagnation point is reported in table 3.1 for the various cases. The}

dimension of the cell in which the highest stagnation pressure occurs is twice the mentioned variance; so

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cases.

In order to get a better understanding of the dependency of the flow features on grid refinement, the figures 3.1 to 3.5 also show the pressure distributions for the Fine case. The Coarse case shows similarity to both the Baseline and Fine cases and is not included. Comparison of the centreline pressure distributions, figure 3.1, reveals that the largest difference between the Fine and the Baseline cases arise around the separation region and in the recovery from the negative pressure peak at the contact patch towards the wake. The Fine grid captures more suction over the top of the wheel before separation; the local extreme isCP =−0.521 for the Fine case compared to −0.493 for the Baseline and

−0.520 for the coarse grid. A same trend can be found for the other tyre tread locations (P2 and P4), whereas on the wheel sides (P3 and P5) refinement has most influence on the suction experienced around 0◦ _{and 180}◦_{, resulting in a slightly better capturing of the}

local acceleration experienced in the experiments. Intriguingly, the Coarse case predicts these phenomena in the same way as the Fine case, giving the impression that the effect of cell distribution overrides the influence of cell refinement, since refinement repositions the nodes. However the Coarse case predicts a little bit less suction in the wake than the other cases and than the experiments.

In an attempt to promote uniform reporting on the quantification of uncertainty in CFD simulations Roache [100,101] has introduced a Grid Convergence Index (GCI)3. This index makes it possible to estimate the error of the numerical outcomes as a result of the grid size and allows comparison between simulations. For the isolated wheel simulations the GCI values based on the Fine grid are GCICD = 2.8% and GCICL = 4.3%. The Baseline grid will be used in the remainder of this work, because the final outcome seems to be relatively insensitive to grid refinement.

Turbulence models Next, a number of turbulence models have been tried out for the

baseline grid using a SRANS solver. The results for the realizable k-²and k-ω SST model have been included in the tables and figures, whereas none of the other models provided

3_{The Grid Convergence Index is based on the theory of generalized Richardson extrapolation [100] and}

can also be used for non-integer grid refinements, implying that the grid does not have to be doubled or halved exactly. The basic idea is to approximately relate the results from any grid refinement test to

the expected results from a grid doubling using a second-order method. The index is defined asGCI =

3|²|/(rp_{−1), where ²= (f}

2−f1)/f1 is the normalized error of the coarse solutionf2 with respect to the

fine solutionf1,ris the quotient of the fine grid size divided by the coarse grid size andpis the accuracy

order of the code, in this case 2. The factor 3 is added by Roache as a safety factor for the error estimate and the GCI is usually presented in a percentage by multiplying the value with 100%.

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in the software would converge to a solution. The general problem was related to rapid growth of the turbulent viscosity ratio in a number of cells and further analysis revealed that this occurred only in cells in the outer domain. This is a consequence of the design choice to create a fully structured grid, since this means that the aspect ratio of the cells in the outer regions increases excessively in order to limit the total number of cells. A hybrid approach, which uses the same grid around the wheel and a more equally sized tetrahedral grid in the rest of the domain should solve this problem, but could not be implemented within this research due to time constraints.

Both alternative turbulence models predict a lower drag value and a higher lift than the SA model, see table 3.1. With respect to the pressure distributions it can be concluded that the realizable k-²case performs better in predicting the suction before separation at C_P = −0.702 for the centreline, compared to −0.836 in the experiments. This is also true for the other pressure distributions on the tyre tread and for P4 it even matches the experiments. The k-ω SST performance lies between the baseline and the realizable k-². Another area where these two models produce better correlation with the experiments than the SA model is the suction region around the corners of the contact patch (see Z in figure 3.7) aroundθ = 80◦ _{for P2 and P4. On the other hand in the wake region the SA}

model surprisingly produces better correlation in general. Due to the overall discrepancies in pressure distribution correlation on the side of the wheel, it is difficult to evaluate the models in these areas. Also it is expected that some of these differences result from the use of a slightly different wheel geometry for the pressure measurements compared to the CFD geometry. Considering the results, especially the drag coefficient correlation, it has been decided to use the SA model for all other simulations. The robustness of this turbulence model adds an extra advantage to this choice.

General correlation conclusions Finally, to complete this section on CFD correla-

tion, the overall performance of the Baseline SA SRANS simulation will be evaluated. Qualitatively the steady RANS simulation captures most of the time-averaged features shown in the experiments. However a few of them are predicted differently, these are:

• The positive and negative pressure peak in the simulations are larger than those in the experiments (in the order of 75%); the grid refinement study shows that this is primarily resolution dependent, since the peak magnitude increases with refine- ment. The uniform experimental angular resolution is approximately 1◦_{, whereas}

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refinement.

• The local acceleration at θ = 45◦ _{for the centreline pressure distribution is larger}

in the experiments than in the CFD, which leads to CP-values of 0.61 and 0.71

respectively. This can be traced back to the difference in wheel geometries; the experimental wheel with which the pressures have been measured has a 5.8% lower aspect ratio than the 050 geometry of the CFD model. Thus favouring the secondary flow around the sides and creating larger flow acceleration in this area. A similar result has been found by McManus et al. [17] for a comparison of URANS results for a stationary wheel, where the experimental wheel had a 33% higher aspect ratio, resulting in an approximately 0.2 higherCP-value in this region compared to in the

simulations.

• The separation over the top of the wheel is predicted fairly accurately - at most 5◦

too early - for all three sensor locations on the tyre tread, however the suction at this point is underpredicted. This can also be concluded from figure 3.6 (a), which shows higher velocities over the wheel in the experiments than in the computations. Comparison of the other results in this figure reveals that the PIV velocities are genuinely higher than those in the simulations and it is expected that this is a result of the higher blockage in the tunnel due to the wheel support system, compared to the CFD in which only the wheel has been modeled. The underprediction of the suction at θ = 0◦ _{for P3 and P5 can then partly be explained with the same}

reasoning. Adding the integrated CFD pressure distributions over the top half of the wheel fromθ= 180◦_{to 360}◦_{for the five sensor locations yields a 40% underprediction}

in lift compared to in the experiments, whereas the lower half only has a 10% lower value. The fact that the CFD does not resolve the broad extreme from θ = 325◦

to 210◦ _{following the separation (at the centreline) is an additional reason for this}

dicrepancy in lift prediction. The unsteady flow feature responsible for this will be discussed in the following section.

• The differences in pressure distribution P2 between experiments and CFD can be explained by the difference in wheel geometry; in the experiments this sensor is effectively more on the rounded wheel shoulder and no longer contacts the ground. This leads to the generally higher suction, except for near the contact patch where the flow is no longer forced around the wheel but can pass underneath, inducing less suction atθ= 80◦ _{and explaining the lack of the negative pressure peak behind the}

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contact patch. Similarly the experimental sensor P4 is located closer to the edge of the contact patch, though still contacting the ground, considering the presence of the negative pressure peak.

• Finally, the differences in the flow field behind the wheel are not as extreme as indicated in figure 3.6 (c). The CFD flow field at a slightly higher cross section looks more comparable to the PIV results and figure 3.8 shows that the presented CFD result just misses the arch-shaped vortical structure downstream of the top of the wheel, while the PIV plane still cuts through this feature. Further proof for this can be found in figure 3.5, which shows a high acceleration for the CFD atθ= 180◦ _due

to the turning of the flow into the wake, whereas in the PIV this characteristic is replaced by two much smaller suction peaks and a local CP-maximum around the

location of the presentedz-plane (at θ= 187◦).

The limitations mentioned above have to be remembered when analyzing CFD results for the wheel. Furthermore, considering the difference in wheel geometry and test conditions between the experimental and computational pressures, it is suggested to use the pressure data primarily qualitatively. The accuracy is nevertheless high enough to be able to deduce trends and relative changes by comparing different flow situations. Regarding the ACFD case4_{, it is remarkable that the force coefficients are very similar, while the ACFD case does}

not resolve the pressure peaks near the contact patch due to a coarser grid in this region. This shows that such an approach can still result in useful data at much lower labour costs for grid generation and can also be used for comparisons as long as the limitations of the method are kept in mind.

In general it is expected that the computational lift is underpredicted, although no ex- perimental direct lift measurement is available for reference. A comparison of the sectional downforce coefficients in table 3.2 shows that the main differences logically occur on the tyre tread. From the pressure distributions it can be concluded that the underprediction of the suction over the top of the wheel and the lack of the following local suction down- stream are the main contributors to this deficiency. If the sectional downforce coefficients for the tyre tread (P1, P2 and P4) are added and compared then it can be seen that the experimental value is 3.3 times as high. Therefore it is expected that the real lift on the wheel is more likely to be betweenCL=−0.09 and−0.3 than the−0.09 that follows from

4_{This hybrid grid has been created in a similar way as at the industrial sponsor, using wall functions}

and roughness instead of ay+_{= 1 boundary layer resolving grid, a much larger domain. The wheelarm}

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the computations. Even though there is a large uncertainty involved in the isolated wheel lift, it should still be possible to derive trends as a result of the wing presence at a later stage, as long as the same approach is consistently being used. The difference in drag is much smaller (at 6.6%) and the main contribution to this is the underprediction of the suction on the upstream half of the wheel due to the different wheel aspect ratio.