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Comparisons between CFD and experimental results for a simplified car model in wall proximity

R. Strachan*, K. Knowles‡ and N. Lawson

Department of Aerospace, Power and Sensors Cranfield University, RMCS, Shrivenham,

Swindon, Wiltshire, SN6 8LA

r.strachan@cranfield.ac.uk

Abstract

This paper presents Computational Fluid Dynamic (CFD) and experimental data for the Ahmed reference model in near side wall proximity. During the experiments Laser Doppler Anemometry (LDA), force and moment data was recorded. The aim of the study was to investigate the ability of both the k-ε and Reynolds Stress viscous model to predict the alterations in the flow around the model when in wall proximity in comparison with the isolated case, and to use the CFD data to assist in explaining the trends exhibited in the experimental force results.

Keywords: Ahmed model, wall proximity, low-speed, CFD, LDA

Introduction The Ahmed reference model was originally developed for a time-averaged vehicle wake investigation (Ahmed et al. 1984). It is a car-like bluff body with a curved fore body, straight centre section and an angled rear end, representing a highly simplified ¼ scale lower medium size hatchback vehicle. The specific angle of the back end can be altered between 0° and 40°, in 5° increments. The model’s major dimensions are 1044mm x 389mm x 288mm. A diagram of the Ahmed reference model is shown in Fig 1. The Ahmed body was designed to have a fully attached flow over the front, and to exhibit many of the flow features of an automobile with its 9 interchangeable rear ends. This variation in backlight geometry provides a range of flow characteristics over the back end of the model. The geometry of this bluff body was designed to be such that an experiment could be conducted with reference to only one significant aerodynamic feature, namely the flow over the slanted rear end, as flow was expected to remain attached over the other sections. The Ahmed model has also lent itself well to the validation of CFD codes, as the accuracy of these codes can be determined once again with only one significant aerodynamic factor varying between cases. This allows validation to be conducted without the common difficulty of sources of errors from different regions of the flow field cancelling each other out. Numerous previous experimental investigations have been report on this model (Ahmed et al 1984; Graysmith et al. 1994; Aider et al. 2000; Lienhart and Becker 2000; Strachan et al. 2004). These were conducted in various wind tunnels over a range of airspeeds. In general the model was supported during testing from underneath by four cylindrical struts attached to a fitted ground plane. The present tests were, however, conducted over a moving ground plane with the model being supported from above by an aerodynamic sting.

*Research Student, Aeromechanical Systems Group ‡Head, Aeromechanical Systems Group

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It has been shown that the flow over the rear of the model depends greatly on the angle of backlight being investigated (Ahmed et al, 1984). There have been found to be two critical angles at which the flow structure changes significantly. Below 12.5° (1st critical angle) airflow over the angled back end remains fully attached. Flow then separates from the model when it reaches the vertical back end. The flow from the angled section and the side walls produces a pair of counter-rotating vortices which continue downstream. For backlight angles between 12.5° and 30° (2nd critical angle) the flow over the angled section becomes highly three dimensional. Two counter-rotating lateral vortices are again shed from the sides of the angled back section, and a separation bubble is formed over this part of the body. Above 30°, flow over the angled section is fully separated. Again a set of counter rotating vortices form over the back end, but in this case the vortices are formed from the separation from the top of the model, instead of from the top of the vertical back section. When the flow is in this state a near constant pressure is found across the backlight. Due to these changing flow structures at these critical angles, it was decided to investigate only the 10°, 25° and 40° back angles in wall proximity. In order to more effectively ascertain the validity of the CFD model however, all 9 back angles were tested for the isolated Ahmed model case. A number of computational investigations into the Ahmed model have also been conducted including that of Gillieron and Chometon (1999); Krajnovic and Davidson (2002 and 2004); Gaylard et al. (1998) and Makowski (1997). Gillieron and Chometon used a k-ε turbulence model with the commercial CFD code Fluent version 4.2 for the simulation and compared their results with those of Ahmed et al. (1984). Here Reynolds numbers were approximately Re = 4.3x106. This investigation highlighted the shortcomings of the standard k-ε turbulence model when attempting to predict the drag coefficient of the body over the whole range of backlight angles. Large Eddy Simulation (LES) was performed by Krajnovic and Davidson (2002) on an Ahmed model with 0° backlight angle. This simulation was performed at a lower Reynolds number of Re = 7.25x105 to facilitate the use of LES. This study highlighted the capabilities of LES for qualitative understanding of the flow around the Ahmed model. This work was extended when similar LES studies were performed on both the 25° and 35° Ahmed model (Krajnovic and Davidson, 2004). This study highlighted the advantages of LES turbulence modeling to predict both the flow structure over the back end of the Ahmed model, and the prediction of the drag force, albeit at a lower Re. To the authors’ knowledge, there has been only one experimental [Wallis and Quinlan, 1988] and one computational study [Advantage CFD, 2001] published on the aerodynamic effects of wall proximity. Wallis and Quinlan investigated the forces and moments on a NASCAR model at various distances from a model wall, however, no attempt was made to ascertain the reasons for the changes in aerodynamic forces which were

0º - 40º

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Fig. 1. Schematic diagram of Ahmed reference model (dimensions in mm). Flow is from left to right and the co-ordinate system used for the present study is shown

Ground Plane

a) Side Elevation

a) Plan View

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observed. Advantage CFD’s study modeled a NASCAR at a single distance from the wall. As such no computational data concerning the effects of altering this wall distance were presented. The current work aims to add to the existing literature in both these areas.

Aims and objectives

The aim of the study was to investigate the ability of CFD to predict the alterations in the flow around the Ahmed model when in side wall proximity in comparison with the isolated case and to use the CFD data to assist in explaining the trends exhibited in the experimental force results.

Methodology Experimental Set-up

The model was mounted 50mm (z/L = 0.048) above a moving ground plane and supported from above by an aerodynamic strut (t/c = 4.06, positioned at x/L=0.25 downstream of the leading edge). This configuration allowed analysis of the effect of both the overhead strut and the moving ground plane on the flow around the model. In addition, the flow on the underside of the model was not altered by inclusion of supporting cylindrical struts, which have been commonly used in previous Ahmed model investigations.

Wall Model

The wall was mounted from the side of the rolling road as can be seen in Figs 2 and 3. The wall was made from Perspex so as to minimise any reflection issues arising during the acquisition of the LDA data. The model wall extended 0.5L upstream and 0.5L downstream of the leading and trailing edge of the Ahmed model respectively, and extended to a height of 0.32L above the top edge of the model. For each test case, the gap needed between the rolling road and the wall was kept at 2mm, in order to allow for the travel of the rolling road, whilst minimising any flow escaping at this junction. The model was tested both with and without the inclusion of the wall. Direct comparison of the isolated case results with experimental force results from Graysmith et al. (1994) was possible owing to the similar model and strut configurations employed. This comparison was employed to ensure similarity between current and previous experiments. Wind Tunnel The model was tested in the “D.S. Houghton” open-jet, closed-return wind tunnel at Shrivenham. This has nozzle dimensions of 2.74m by 1.66m high. A continuous belt rolling road (1m wide by 3.5m long) provided moving ground simulation. Throughout the testing the freestream velocity was constant at 25m/s, corresponding to a Reynolds number of approximately Re = 1.7x106 (based on the model length). The rolling road was synchronized with the tunnel freestream velocity and boundary layer control was provided by upstream suction and a knife edge transition. The system achieved an onset boundary layer thickness of 1mm (to 99% freestream dynamic pressure), and a freestream turbulence intensity below 0.25%. LDA System A two-component laser-Doppler anemometer was installed on a three-dimensional computer-controlled traversing system. The 2D LDA system consisted of DANTEC FibreFlow optics with two BSA Enhanced signal processors, Burstware software and a 2.5m focal length DANTEC FibreFlow probe. A JEM Hydrosonic 2000 fog generator was used to seed the flow with a water/glycerol seeding mixture. Using previous error analysis, LDA velocity measurement error is predicted to be better than 1% of full-scale measurement (Lawson and Davidson 1999).

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Time-averaged LDA measurements were made for the u and v components in a number of planes both around and downstream of the model. These included transverse planes at 80mm (0.076L), 500mm (0.48L) and 1044mm (1L) downstream and planes at the trailing edge and x=-50mm (0.048L) at the top of model. All back angles (0°-40°) were tested for the isolated model cases, with the majority of LDA results taken from the 25° case. For the near wall experiments, however, only the 10°, 25° and 40° cases were tested. These were chosen as they cover the three backlight flow structures created by the Ahmed model’s variable geometry. In the current paper, however, only results from the 25° case are presented. Force Measurements A PC-controlled six-component force balance was mounted inside the model. The force balance was calibrated with a known vertical and horizontal force prior to testing. RMS error in force was found to be better than 1% of full-scale measurement. A schematic diagram of the experimental set-up is shown in Figure 2, with a photo of the set-up with the wall included is shown in Figure 3. Numerical Model The commercial CFD code Fluent version 6 was used for the numerical modeling. In order to resolve the boundary layer close to the Ahmed body, a viscous-hybrid mesh scheme was used. The triangular surface mesh for the computations was produced in GAMBIT. Finer triangulation was employed over areas experiencing separation and the final surface mesh was similar to that used by Gillieron and

Ar Ion Laser + LDA optics LDA x-y-z traverse

Rolling Road

LDA

Ahmed Model Supporting Strut

Nozzle Collector

Fig. 2. Schematic diagram of experimental set-up.

Model Wall

Fig. 3. Experimental wind tunnel set-up showing Ahmed model and Perspex wall.

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Chometon (1999). Prisms were then grown from the surface of the Ahmed model in TGrid, with this prismatic region used to resolve the boundary layer. Following the production of these prisms the remainder of the domain was meshed automatically with tetrahedra using TGrid’s automatic mesh generation function. This initial mesh had approximately one million cells for each of the cases, and was exported to Fluent for solving. The computational domain for both the isolated and near wall cases was 8L in length, 2L in height, and 2L in width, with the model leading edge positioned 2L downstream of the fluid inlet and 50mm above the ground plane. For the isolated cases the model was positioned centrally between the side walls. These dimensions give a blockage ratio of 2.8%. As the wind tunnel was of an open jet type, no attempt was made to accurately model the wind tunnel in the CFD model. The dimensions chosen for the simulated wall boundaries should ensure insignificant effect on the airflow around the model (Gillieron and Chometon, 1999). The boundary conditions employed in this computation were as follows: • Uniform freestream velocity of 25 m/s at inlet. • Constant pressure condition at outlet. • No-slip condition on Ahmed body. • Symmetry conditions on the walls, roof and floor of computational domain. The symmetry condition on the floor simulates the rolling road in the wind tunnel. Both the k-ε RNG and Reynolds Stress viscous models were used throughout the computation. It has been shown previously that the RNG k-ε model is superior to the standard k-ε model in both its prediction of aerodynamic coefficients and the prediction of pressure distribution over the Ahmed model (Graysmith et al. 1994). In addition, it has been shown [Makowski and Kim, 2000] that for the 30° case the Reynolds Stress model provides a more accurate prediction of the separation over the backlight. As all back angles were to be tested in the current investigation however, both viscous models were tested to ascertain which would be the most accurate. As the grid is not aligned with the flow due to the unstructured mesh, it was necessary to use second-order discretisation, as first-order discretisation greatly increases the numerical diffusion. Although first-order iterations are more likely to converge quickly, the solution is likely to be less accurate. In order to ensure convergence, however, a number of first-order iterations were performed in each calculation before switching to second-order iterations. Successive mesh refinements were performed in FLUENT using the program’s mesh refinement function. This refines each cell with a static pressure gradient across it greater than a user-defined threshold value. After a number of these refinements the mesh size had increased to between 1.7 and 1.9 million cells (depending on the specific angle being investigated), with at each stage the lift and drag coefficients on each section of the model being used as convergence criteria.

Results and discussion

CFD models comparison

For the isolated Ahmed model case CFD models were run with and without the inclusion of the supporting strut, with both standard and non-equilibrium wall functions, and with both the RNG k-ε and Reynolds Stress viscous models. From the results of these the model with most accurately simulated the flow over the model was chosen to model the near-wall cases. For each of the cases a number of mesh refinements were conducted after convergence of the aerodynamic forces, until around two million cells were reached. This was the maximum number possible due to the available computational power. Figures 4(a) – (c) show contour plots of v/U∞ for both the with and without strut CFD cases, in addition to the current LDA data for a plane at the model trailing edge

Various other sections of the flow around and downstream of the model were analysed in order to determine the merits of including the strut in the simulations, and the trailing edge planes shown above are chosen simply to demonstrate the relative abilities of both CFD cases to model the flow over this particular section of the

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backlight. As can be seen from Figure 4, the with-strut CFD case does not model the flow over the centre section of the model (z/L=±0.1) as closely as the standard CFD model (with not strut). This initially seems peculiar; however, it can be explained by the fact that due to the number of cells required to model the strut, fewer cells were available to model the back end. As such the flow over this section, and subsequently the wake produced by the model is more accurately represented by the standard CFD case. The slight increase in v/U∞ evident in the LDA data between z/L=±0.05 is almost certainly an effect of the strut in the experiments, and is therefore not recreated in the standard CFD case. Despite this, the flow is more accurately predicted by this CFD case than the with-strut CFD, and as such the strut was not included in further CFD modeling. As was expected, the k-ε model over-predicted the pressure drag over the front end for all back angles [Makowski and Kim, 2000]. For the 25° case, this over-prediction was approximately 320%. This compared to the Reynolds Stress model which although also over estimated the pressure drag in this area, did so by 170%. It is therefore clear that although the pressure drag is more closely modelled in the stagnation region by the Reynolds Stress model, neither it nor the RNG k-ε model could be considered accurate. As accurate quantitative drag force data was therefore not expected from any of the CFD simulations run, the relative performance of the two viscous models was also compared on the prediction of the near wake and the flow over the backlight. Figure 5 shows a plot of u/U∞ along the model centreline (z/L=0), downstream of the model trailing edge for both the k-ε and Reynolds Stress models.

Again it must be stressed that a number of planes of data were analysed in order to determine the best viscous

x/L

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Fig 5.(a) centerline plane of near wake - 25° case – LDA data

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Fig 5.(b) centerline plane of near wake - 25° case – CFD data - Reynolds Stress viscous model

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Fig 5.(b) centerline plane of near wake - 25° case – CFD data - k-ε RNG viscous model

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Fig. 4 (a) 25° Ahmed model, x = 0mm plane, LDA data z/L

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Fig. 4 (b) 25° Ahmed model, x = 0mm plane, CFD data –without inclusion of strut

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Fig. 4 (c) 25° Ahmed model, x = 0mm plane, CFD data –strut included

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v/U∞

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model to employ for further CFD modelling, and the planes shown in Figure 5 are chosen simply to highlight one particular area investigated. It can be seen from Figure 5 that the wake formation predicted by the Reynolds Stress model (Fig 5(b)) more closely resembles that measured by experiment (Fig 5(a)). This is particularly true for the reversed flow region, the shape of which is better predicted by the Reynolds Stress model. Because of this it was decided to employ a Reynolds Stress model throughout the near wall computational investigations.

Front End Pressure Changes - CFD Figure 6 shows the change in pressure coefficient between the isolated Ahmed model case and near wall cases, at z/L (distance from side of model to wall) = 0.19, 0.14, 0.1, 0.05 and 0.04. From Fig 6 it can be seen that as the model is moved closer to the wall, a large pressure increase is predicted on the near-wall side of the model front end, skewed slightly to the bottom of the model. In addition, it can be seen that due to the reduction in air passing over the near wall side of the model, more flow is forced over the three other sides, resulting in a higher local velocity and a resultant drop in Cp. This is most evident at the top, near-wall section of the front end, where the largest pressure drop is visible. Figure 7 plots ∆Cp against z/L for a line through y/L=0.185 (the model centre) for the cases shown in Fig 6. It can be seen from Figure 7 that the change in maximum ∆Cp increases far more quickly as the model gets closer to the wall; there is an increase in Cp of 0.1 between 200mm and 100mm from the wall, and an increase of 0.15 between 50mm and 40mm. In addition, there is a similar relationship between wall separation and the drop in Cp evident over the top of the model front end, as shown in Figure 8. This plots ∆Cp against y/L for a line through z/L=0.16 for the cases shown in Fig 6. Again the rapid increase in pressure over the bottom and centre of the front end can be seen, in addition to the rapid drop in pressure over the near-wall top-side of the front end. This drop is 30% lower in magnitude than the maximum predicted Cp increase. As the pressure changes on this section of the model are predicted by the CFD to be greater than those on any other section, they should have the greatest effect on the overall aerodynamic forces.

z/L-0.100.1

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y/L = 0.185 line - 40mm from wally/L = 0.185 line - 50mm from wally/L = 0.185 line - 100mm from wally/L = 0.185 line - 150mm from wally/L = 0.185 line - 200mm from wall

∆ Cp

Fig 7. ∆Cp at y/L=0.185 on the Ahmed model front end at various distances from the wall

Fig 8. ∆Cp at z/L=0.16 on the Ahmed model front end at various distances from the wall

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∆Cp

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Fig 6 (e).Change in Pressure coefficient on 25° Ahmed model front end with wall at z/L=0.22 (40mm)

Fig 6 (d).Change in Pressure coefficient on 25° Ahmed model front end with wall at z/L=0.23 (50mm) Fig 6 (a). Change in Pressure coefficient on 25° Ahmed

model front end with wall at z/L=0.38 (200mm)

Fig 6 (c).Change in Pressure coefficient on 25° Ahmed model front end with wall at z/L=0.28 (100mm)

Fig 6 (b).Change in Pressure coefficient on 25° Ahmed model front end with wall at z/L=0.33 (150mm)

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Back-end pressure and vortex structure results - CFD

Shown in Figure 9 is the pressure changes experienced over the backlight with the inclusion of the wall, when compared to the isolated case.

It can be seen from the scale used in Figure 9 that the changes in Cp are of a smaller magnitude than those experienced at the model front end, but are nonetheless significant. Particularly evident is the overall drop in Cp over the centre section of the backlight as the model approaches the wall, and the change in the strength of the vortices being shed from the sides. This is shown by the drop in Cp over the far wall section, revealing an increase in local velocity and thus vortex strength, with the reverse effect evident over the near wall section. In order to better quantify these changes in vortex structure, Figure 10 plots v/U∞ at y/L=0.28, 50mm upstream of the model trailing edge for the isolated, 200mm from wall and 50mm from wall cases.

It can be seen that maximum v velocity on the far wall side only changes very slightly (0.02U∞) between the most extreme cases plotted, the minimum velocity is significantly altered (0.1U∞). It is also interesting to note that the CFD predicts only a very small change in this minimum velocity between the 200mm and 50mm from wall cases, suggesting that, unlike the pressure changes analysed over the front end of the model, the far wall side vortex structure is not significantly altered between these cases. The downwash over the central section of the backlight is then found to be increased by the presence of the wall on the far wall side of the model, and decreased on the near wall side. The vortex structure changes predicted by the CFD at the near wall side are more significant. The point of minimum velocity is moved toward the side of the model as the wall is moved closer, shifting by approximately z/L=0.2 between the isolated and 50mm from wall cases. In addition, it must be noted that although the

Fig 9 (b). Change in Pressure coefficient on 25° Ahmed model backlight with wall at z/L=0.22 (40mm)

Fig 9 (a). Change in Pressure coefficient on 25° Ahmed model backlight with wall at z/L=0.38 (200mm)

z/L

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Fig 10. v/ U∞ at y/L=0.28 on the Ahmed model backlight, 50mm upstream of the trailing edge – CFD data

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minimum velocity is again increased for the 200mm from wall case, this velocity decreases significantly (0.1U∞) for the 50mm from wall case, with a corresponding increase in maximum v velocity. It is clear therefore that wall proximity has a far greater influence on the near wall vortex structure than on its counterpart.

Experimental force results

Figure 11 shows the experimental lift and drag coefficient changes measured for the near wall cases. Both are plotted as differences from the isolated case.

It can be seen that the lift actually decreases slightly at 200mm and 300mm from the wall, before increasing rapidly as the wall separation becomes smaller. This initial decrease is difficult to explain from the CFD predictions, as it was expected that the pressure drop over the top of the model would have a greater effect on the lift coefficient than any increase in velocity (and subsequent pressure drop) on the bottom of the model front end or underside. Further experimental quantification of the effects of wall proximity on the pressures in these areas will be the subject of further experimental work, in order that this drop in lift can be explained. The rapid increase in CL between 200mm and 5mm wall separation though was expected from the CFD simulations. The large pressure drop over the top of the front end and its tendency to increase sharply as wall separation decreased (see Figure 8) predicted this lift trend. In addition, the pitching moment was found to follow an almost identical pattern to the lift plot shown in Figure11. This again concurs with the CFD which predicted that the pressure drop over the front end would have the greatest effect on the lift coefficient, in turn producing a large positive pitching moment as the wall separation was decreased. The drag coefficient in Figure 11 however does not increase sharply closer to the wall, instead maintaining an almost linear pattern. Also interesting to note is that for distances from the wall greater than 50mm, there was a drop in overall drag coefficient compared with the isolated case. This again agrees with the CFD model, which predicts that this almost linear increase in drag coefficient is mostly a product of an overall pressure increase over the model front end.

Experimentally measured vortex structure changes

In order to compare both the CFD predictions and experimental results of the alterations to the vortices formed over the backlight, Figure 12 plots v/U∞ at y/L=0.28, 50mm upstream of the model trailing edge for the isolated and 200mm from wall cases. This can be compared with Figure 10, which plots CFD results for the same points around the model.

Fig 11. Changes in CL and CD for the 25° Ahmed model in wall proximity.

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The LDA data shows no increase in maximum v velocity in the far wall vortex, which contrasts to the CFD which predicted a slight increase with wall proximity. Both LDA and CFD data however show an increase in minimum velocity magnitude, in addition to an overall increase in downwash over the far wall half of the model. The CFD predicted increase in minimum velocity magnitude is, however, approximately double that of the experimental value. In addition, although the increase in downwash is of approximately equal magnitude for both the LDA and CFD, the CFD predicts that this increase will only be evident up to z/L=0.2, with the velocity returning to isolated case values between z/L=±0.02 before exhibiting an reduction in downwash over the region z/L>0.2. The LDA by comparison, shows this increased downwash continuing to z/L=0.04. It is of course possible that this discrepancy is due to an interaction between the supporting strut and the wall, but unless further testing is performed without it this is impossible to determine.

The minimum v velocity measured experimentally for the near wall side vortex proved to be higher for the 200mm from wall case by approximately 0.1U∞ when compared with the isolated case. This contrasts with the simulation which predicted a very slightly lower velocity for the 200mm from wall case by approximately 0.015U∞. The simulation did predict a drop in velocity magnitude at this point of approximately 0.06U∞ for the 50mm from wall case, but this trend was not continued over all the calculated wall distances. Once further LDA testing has been completed it will be useful to see if any near wall cases do indeed exhibit a rise in v velocity magnitude in the near wall vortex region, but from the current data it must be concluded that for the 200mm from wall case the current simulation does not accurately predict the variations measured experimentally.

Conclusions ● It can firstly be concluded that the Reynolds Stress viscous model provides a more accurate simulation of the

Ahmed body than the k-ε RNG model. Despite this, both models significantly over predict the pressure drag over the front end, and thus can not be used for accurate pressure drag force predictions.

● The simulations predicted that the changes in pressure over the front end would have the most significant effect on the lift and drag force, with the predicted trends in these forces with decreasing wall separation being mirrored by experimental measurements. This suggests that despite the absence of accurate drag predictions from the CFD, the flow structure and how it is altered by the inclusion of a wall is well modelled in this region.

● The simulation of the changes to the vortices shed from the backlight was also, in general, well predicted. Although there were again discrepancies in the magnitude of these changes, the increased downwash over the far wall section and the corresponding decrease on the near wall section was simulated, as was the increase in minimum flow velocity magnitude in the far wall vortex. Further experimental testing is required to ascertain whether the trends in flow velocity predicted by the CFD for the near wall vortex are mirrored by experimental data.

● From the results shown it can be seen that the CFD model employed was an extremely useful tool in analysing and explaining force and flow visualisation measurements.

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0.525° 25ms isolated case - y/L=0.28 line - LDA data25° 25ms - 200mm from wall - y/L=0.28 line - LDA data

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Fig 12. v/ U∞ at y/L=0.28 on the Ahmed model backlight, 50mm upstream of the trailing edge – LDA data

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S.B. Wallis, W.J. Quinlan, “A discussion of aerodynamic interference effects between a race car and a race track retaining wall (a wind tunnel NASCAR case study)” SAE paper 880458, 1988

Definitions, Acronyms, Abbreviations

CFD: Computational Fluid Dynamics

LDA: Laser Doppler Anemometry

CD: Drag Coefficient = SUDrag 2

21

∞ρ

CL: Lift Coefficient = SULift 2

21

∞ρ

U∞: Free-stream velocity

13

u,v,w: velocity components in X, Y and Z directions respectively

X: Streamwise co-ordinate (see Fig 1)

Y: Vertical co-ordinate (see Fig 1)

Z: Transverse co-ordinate (see Fig 1)

L: Model length = 1.044m

ρ: air density

S: Frontal area of Ahmed model = 0.112032m2