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Aerodynamics of Road Vehicles – a Challenge for Computational Fluid Dynamics Wolf-Heinrich Hucho Germany, huchowhh@t-online.de ABSTRACT During the development of a new car, information on the aerodynamic properties is needed very early in the process. In the design (styling) phase, three to six different models may be considered. Time is too short to have them all built and tested in a wind tunnel. However Computational Fluid Dynamics (CFD) may be used to obtain the data required - and throughout the remainder of the engineering process. The flow past a car is shown to be governed by separations of different types. Generally separated flow is non-stationary (i.e. varying with time). However, within the time constraints of the development process, present-day computers are not fast enough to compute the flow past a car by Direct Numerical Simulation (DNS) or by Large Eddy Simulation (LES). Hence most present-day CFD codes use Reynolds Averaged Navier Stokes equations (RANS). These are steady state, and all turbulence - including the fluid motion inside the near wake - is modeled as stationary (i.e. constant with time). Whether this simplification is permissible remains open to question. One way out of this trap might be to model the near wake, and confine the use of the turbulence model to those zones where the flow is turbulent in the “classic” sense. Such a code might suffice until Large Eddy Simulation can be applied routinely.

Keywords: aerodynamics, drag predictions, flow separation, CFD validation 1. SETTING THE SCENE Mechanical engineering is increasingly being penetrated by numerical methods, generally called “codes”. Many of these have achieved a high level of sophistication. Applying them is no longer described as “computing”, rather it is called “simulating” the reality, be it static or dynamic. For a long time aircraft engineering held the lead in making use of numerical methods. The following key problems were attacked numerically: • structural analysis: finite element method (FEM); elastic materials; • aerodynamics, first with incompressible air, later compressible;

o ideal (non-viscous) flow: integral boundary layer methods; attached flows; o real (viscous) flow, attached; o partly separated flow;

• flight dynamics: multi-degrees of freedom, more or less coupled. While in aircraft engineering the numerical methods addressing the different problems were all at similar levels of maturity, this was – and perhaps still is – not the case in vehicle engineering. For vehicles, Numerical Aerodynamics has lagged behind by a decade or more. The reason was – and still is – the difficulty of computing separated flows. In the vehicle industry, the

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aerodynamicists had to decide when to invest in CFD. In the author’s opinion, many started far too early. Meanwhile, times have changed. Several CFD codes are now marketed, that give results which seem “pretty close” to reality. However, the codes have yet to prove themselves in the course of daily engineering work. What this means for vehicle aerodynamics will be illustrated briefly in the following section. 2. OBJECTIVES AND REQUIREMENTS Even considering only passenger cars, the objectives of vehicle aerodynamics are many and varied. A compilation is given with Figure 1. This shows that vehicle aerodynamics is concerned with much more than drag and the related performance properties like top speed, and fuel consumption.

Figure 1: The objectives of vehicle aerodynamics

The vehicle aerodynamics objectives in Figure 1 may be translated into the following capabilities required of numerical fluid mechanics1:

• Aerodynamics • flow around the vehicle • flows in compartments or systems, especially as they affect temperatures

o radiator, other heat exchangers, engine compartment o ducts for combustion air, and brakes o passenger compartment

• Fluid mechanics

• engine o scavenging, and combustion, reacting fuel and air o oil flow o piping for liquid cooling

• ducts for heating, ventilation, and air conditioning (HVAC) • torque converter

1 The classification into aerodynamics and fluid mechanics is not strict.

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• fuel piping 3. STATE OF THE ART The easiest way to demonstrate the status and progress of vehicle aerodynamics is to follow the decrease of the drag coefficient cD with time. With D meaning drag, V the driving speed, ρ the density of air, and A the frontal area of the vehicle, the definition of the drag coefficient is as follows:

AV2

Dc2

D ρ=

How this quantity has evolved over the years is shown in Figure 2.

Figure 2: History of drag coefficient since 1960. Emphasized circles are cars with very low drag.

Data from Volkswagen AG and from ATZ. The lower the value of cD, the better the aerodynamicist has been able to influence the car’s shape. The broad scatter of the data suggests that aerodynamics is not given the same priority by every car manufacturer. The steep descent followed the first oil crisis of 1973/74. cD = 0.25 seems to be the asymptotic limit for current passenger cars, and has been achieved by only a few very ambitious projects. Now, aerodynamicists are confronted with a new problem. While for passenger cars the frontal area A remained almost constant within each car class2, a new type of vehicle, the Sports Utility Vehicle (SUV), is coming with much larger frontal areas. Along with their very bluff shape, this imposes new demands on the aerodynamicist, who has to extend their field of experience in order to achieve an acceptably low drag in ever-shorter time. 2 Actually the frontal area is growing with time for two reasons: the demand for more comfort and the fact that human beings are growing taller.

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4. HOW LOW DRAG IS CURRENTLY ACHIEVED 4.1 Evaluation of Aerodynamic Potential The shapes that designers currently prefer are influenced by aerodynamics much more than they realize and are ready to admit. When at the beginning of a new car project several competing designers create new shapes, these are usually quite good, and generally don’t differ very much, from an aerodynamics point of view. The geometry of these “early” models is not worked out to the last detail; only the main proportions are modeled - usually in clay. In this phase of development, the aerodynamicist tries to establish a ranking among those models with regard to their drag coefficient cD. The traditional method would be to take all the different models in a scale 1:4 or 1:5 to a wind tunnel, and to try to find out their (low) drag limits.

Figure 3: Pilgrim-step aerodynamic refinement in the early phase of a development project.

Example Audi A4, from Dietz et al. (2000) The drag coefficient of each model is measured, and its shape is modified here and there to reduce drag. Subsequently, the models go back to the styling studio, and each designer then has to decide how to respond to the findings from the wind tunnel. Generally they do not accept all the changes suggested by aerodynamics, and they also propose new ones. To check the effect on drag, the models have to go to a wind tunnel for a second time, and are again improved by the aerodynamicist. Often this sequence is repeated several times, as shown in Figure 3. In a kind of pilgrim step3, low drag and a good appearance come together – or they don’t. When one or two candidate designs are selected for further development, drag is only one criterion out of many which has to be considered. By this phase of the project, the new car’s dimensions and performance are defined numerically, and the same is expected for its aerodynamics. If CFD is used, the results must be as reliable as the data from small-scale wind tunnel testing. They should be exact to within say ± 5% - and the probable uncertainty should be quoted together with the result. In the example in Figure 3, the uncertainty of ±5% means ∆cD = ± 0.01. Hence the drag coefficient should always be quoted as cD average = 0.25 ± 0.01.

3 Three steps forward, two steps back.

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Although all too rarely done, in this case it means that - with regard to their optimized drag limits - there is no certain difference between all the candidates A to G. 4.2 Investigating Minor but Perceptible Shape Changes Later in the course of development, the drag must be predicted again - and for very small changes in shape. At this stage, the major proportions of the car must remain unchanged. The process - which is called detail optimization4 - is carried out at full scale - first with clay models, and finally with metal prototypes. The accuracy (repeatability) of measurement should now be say ± 2%, or even less. An example is given in Figure 4.

Figure 4: Detail optimization of the Audi A4. Measurements were done with full-scale model; after

Dietz et al. (2000). All the accepted modifications of the car’s body can add up to a noticeable improvement. In this case, the drag coefficient was “pulled down” from 0.300 to 0.250 - or by 17% of the initial value. In context of this conference, the message of this example is that the tool applied during detail optimization - be it a wind tunnel or a computer - must be able to discriminate at least ∆cD = ± 0.25 x 0.02 = ± 0.005. 4.3 Drag and Lift Reduction by Changing Many Small Details An example from this phase is presented in Figure 5, which originates from the development of the Audi A2 model (capable of 3l/100 km) reported by Dietz (2000). More than 80 small and very small details were investigated, mostly tiny little protrusions or recesses, with dimensions comparable to the thickness of the boundary layer. To give an explicit example: even the sides of the tires were made smooth by removing the brand and the numbers by which its type is identified.

4 See Hucho (1998)

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Figure 5: Detail optimization of the Audi A2, 3l/100 km, full-scale model; after Dietz (2000)

All these modifications were necessary to arrive at cD = 0.25 - a value which had to be achieved in order to meet the target fuel consumption of 3.0 l/100 km. A further objective was a low lift at the rear axle, which is important for good road holding. To sum up: vehicle aerodynamics deals with three kinds of modification: • overall proportions; • perceptible shape details; • small and very small changes of detail. To serve as an engineering tool, a numerical code should be able to handle all three. 5. WHAT MAKES A VEHICLE AERODYNAMICABILITY SO CHALLENGING? Computational aerodynamics has certainly reached a high level of maturity, and is able to handle almost all the problems presented by aircraft. These are due to the flows past the complex geometry of a commercial aircraft - fuselage and engines, and wings and tail (each with control surfaces) - all interacting. Moreover, the flows are at high Reynolds numbers, and at Mach numbers close to 1, which means that the compressibility of air has to be taken into account. Local supersonic regions are common, followed by shocks which interact with the boundary layer. Given the severity of these problems, the question arises: what makes vehicle aerodynamics so difficult? In comparison to aircraft, their shapes look simple and they cruise at low speeds, where the compressibility of air is negligible. The answer is as follows: In fluid mechanical terms, road vehicles are bluff bodies in very close proximity to the ground. Their detailed geometry is extremely complex. Internal and recessed cavities which communicate freely with the external flow (i.e. engine compartment, and wheel wells, respectively) and rotating wheels add to their geometrical and fluid mechanical complexity. Moreover, the flow past a

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vehicle is fully three-dimensional and unsteady, with turbulent boundary layers. The Reynolds numbers are high, and the flow field is dominated by steep pressure gradients causing separation - which may be followed by reattachment. Large unsteady near wakes are formed at the rear, followed by a turbulent far wake. Depending on geometrical peculiarities, long trailing vortices may be shed and interact with the surrounding flow, primarily the near wake. Two types of separation may be distinguished:

• those where the line of separation is predominantly perpendicular to the direction of local flow, • and those where a shear layer is shed along a line oblique to the local flow and then rolls up. Both kinds of separation are sketched in Figure 6.

Figure 6: The two types of flow separation; a) line of separation primarily perpendicular to the

local flow direction; b) line of separation oblique to the local flow. A separation of type a) is shown in more detail in Figure 7a. The case considered is the so called “backward-facing step”, a simple looking model which, however, contains all major features of this kind of separation and the formation of a near wake. With some imagination, one may recognize its similarity to the notch of a notchback car. The flow is coming from the left. At point S a shear layer is generated which reattaches at point R. The region below the line from S to R is called the “near wake”5. The fluid therein rotates, and in some cases a small counter rotating vortex may be generated close to the inner corner, as shown. As drawn, the flow pattern appears to be two-dimensional and steady state. However, these would be very careless assumptions! On the contrary, because the shear layer tends to bend inwards, towards the centerline of the vehicle, the re-circulating flow becomes three-dimensional, and is extremely unsteady.

5 In German literature the near wake frequently is called „dead water“ (Totwasser). This indication goes back to Helmholz & Kirchhoff (1869); in their flow model the flow in this region was assumed to be at rest (dead).

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Figure 7: Separation at a line perpendicular to the oncoming flow: a) flow pattern; b) pressure distribution cp(x).

The pressure distribution cp(x) shown in Figure 7b is typical for such near wakes. The base pressure is negative, with the lowest pressure being slightly lower, while to the rear of the near wake, the pressure increases. Following a slight overshoot, the pressure tends asymptotically back to ambient. If considered as two-dimensional, the flow at the rear of a bluff body generally separates from both the top and the bottom. The resulting eddies or recirculations may be periodic or non-periodic. The best known case of the former is the circular cylinder, which over a specific range of Reynolds numbers, gives rise to the famous Karman vortex street (i.e. periodic alternating trailing eddies). Taking a blunt two-dimensional wedge as an example, the two different types of near wake are compared in Figure 8. The periodic near wake, Figure 8 a, can be changed into a non-periodic mode by introducing a splitter plate, Figure 8 b. One notable consequence is that drag is reduced drastically, and a general rule can be derived: periodic near wakes should be avoided. Luckily they rarely occur on cars – with the exception of antennae, where they may cause whistling. The near wake behind a three-dimensional cone is shown in Figure 8 c. Although cars are not bodies of revolution, separation takes place at a line all around their rear body. The near wake therefore shows some similarity to that behind a cone, which again is non-steady. In a case similar to 8 c, a “pumping effect” has been observed by Duell & George (1999). Vortices were expelled periodically from the near wake into the far wake, causing the near wake to expand and contract with a specific frequency. A similar effect has been observed for the rear end of a notchback car by Gilhome et al. (2001). This caused an oscillating drag, and much worse, an oscillating lift at the rear axle with a frequency close to the resonance frequency of the rear suspension.

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Figure 8: Flow separation at the rear of a bluff body; schematic; a) two-dimensional periodic; b) two-dimensional non periodic; c) rotational.

In the literature,6 the unsteady motion of comparatively large eddies inside the near wake is often described as turbulence; 30 to 40% are quoted for the x-direction. However, the peak in the spectrum7 of these large eddies is at a frequency of about 5 to10 Hz, while the frequency of small scale turbulence in the classical sense is of the order of 1 kHz. The motion of these comparatively large eddies therefore seems to be distinctly different from turbulence. The type of separation shown in Figure 6 b can be seen in more detail in Figure 9. The shear layers shed obliquely from both sides of the slant roll up into a pair of trailing vortices, much like the vortex lines in an ideal flow. As long as the shear layer detaches from the body at a sharp edge, its shedding is steady. However, if the separation starts from a smoothly rounded surface, the separation line is not well defined by geometry, and the shedding may be unsteady. Exactly this occurs at the C-pillars of a fastback or a notchback car if they are rounded. It also occurs at the rounded “nose” of a modern high-speed train (like the ICE) when at the rear and running backwards. In this case, the shedding seems to be periodic, and is suspected of causing some discomfort to the passengers in the last carriage at very high speeds. Figure 9 b shows the roll-up in a vertical cross section through the slant, and Figure 9 c depicts the velocity distribution in the same plane. This latter is similar to a classical potential vortex: v ≅ 1/r, however, with a viscous core where v → 0. Finally, in Fig. 9 d the pressure induced on the slant is sketched; the low-pressure peaks are characteristic of a vortex line. Under specific situations (e.g. a large adverse pressure gradient), the longitudinal vortices burst, and the separation turns into a near wake - as shown in Figure 6 b and 7. On cars, both kinds of separation may be present simultaneously, and interact with each other.

6 See Tanner (1967). 7 See Leder (1992).

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Figure 9: Shedding of a shear layer and its rolling up to a longitudinal vortex, line of separation oblique;

a) flow pattern b) shear layer, cross section c) velocity distribution d) pressure distribution

schematic.

The dimensions of the tiny details mentioned when discussing Figure 5 are so small that they are partly or fully immersed in the boundary layer. Hence their effect on the flow past a car seems likely to be very difficult to predict with CFD. 6. BENCHMARKING FOR VALIDATION AND FURTHER REFINEMENT OF CODES 6.1 Elementary Test Cases The preceding three figures show how complex the flow patterns are past bluff bodies like cars. A code designed to predict the flow around a car must be able to reproduce these patterns. Whether this is within the ability of a given code must be determined with geometrically simple benchmark models which allow the investigation of the major typical flow patterns one by one, rather than all together. A selection of elementary geometries is sketched in Figure 10. It is far from being complete. For instance, the effect of rounding an edge is mentioned only in 2-D. However, it is believed to be strong also on slanted front- and rear-ends, i.e. in 3-D. Three categories are distinguished: 2-D, rotational, and 3-D flows. Only a very few of these geometries have been investigated in detail.

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Figure 10: Some elementary benchmark geometries for the validation of CFD-codes, after Hucho

(2000). 6.2 Benchmark Cases Published So Far Two elementary test cases have been investigated in great detail by Hupertz (1998): the flat plate, and the hump. Examples of the latter are shown inverted in the upper left of Figure 10. The experimental data are from the literature, as set out in Hupertz’ thesis. The code applied was VW-Ikarus, a RANS-code with a k-τ turbulence model. This predicted the properties characteristic of the turbulent boundary layer quite well.

Figure 11: A two-dimensional hump, an example for separation and reattachment; after Hupertz

(1998) The results for the hump are shown in Figure 11, where the pressure coefficient is plotted in the form cp(x/l); l is the length of the model considered. Two effects are prominent: separation and reattachment. Both are predicted qualitatively by VW-Ikarus, but separation occurs too late, and reattachment too early. A very similar result is given by Lattice-Boltzmann codes. The flow past a body of revolution with a slanted back end – a model from the middle column in Figure 10 - was computed by Tsuboi et al. (1988). The experimental data go back to Morel (1978) and Bearman (1979). The results are displayed in Figure 12. The code applied was quoted as direct numerical simulation (DNS). However, after analyzing the conditions, Lumley (1982) classified it an “unintentional large eddy simulation (LES)”. The drag coefficient, plotted versus the slant angle on the left diagram, agrees pretty well with the experimental data. Even the “critical”

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slant angle ϕkrit, at which cD falls abruptly, is predicted accurately. This event is caused by the collapse of the vortices and the subsequent change of flow pattern from “fastback” (with a pair of trailing vortices, see Figure 9) to “square back” (with a near wake as sketched in Figure 8).

Figure 12: Effect of base slanting on a body of revolution in free air; left side) drag coefficient cD

(cW) versus slant angle ϕ; right side) crosswise pressure distribution, cp(y/d), on the slant at a slant angle ϕ = 40°. Measurements left diagram Morel (1978), right plot Bearman (1979); computation

Tsuboi et al. (1988). However, when measured and computed pressure distributions along the y-axis are compared, remarkable differences become evident, as can be seen on the right diagram in Figure 12. The distinctive pressure peak, which is typical for an oblique surface with vortex roll up on both sides8, is not well reproduced by the computation. Surprisingly, a force may sometimes be predicted correctly even though the pressure pattern shows major differences between computation and experiment. This apparent contradiction may be explained as follows: While experimental forces and moments on a body are measured directly, computed forces and moments are the results of integrating pressures and shear stresses. Generally, integrating is a good-natured operation; if lucky, differences in the integrand may cancel out. The third example to be discussed here is shown in Figure 13: a generic automobile model, well known from the name of its creator, and called “Ahmed’s body”. This 3-D model was investigated in a position close to a ground plane. The computations were done with a Lattice-Boltzmann code that, in essence, solves the time-dependent Navier Stokes equations.

8 A good example is the vortex roll up at the oblique leading edges of a delta wing (like Concorde)

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Figure 13: Effect of base slanting with „Ahmed’s body“, close to ground; left side drag coefficient versus slant angle, cD(ϕ); right side lift coefficient versus slant angle, cL(ϕ); measurement Morel

(1978), computation Anagnost et al. (1997) The measured and computed forces are pretty close to each other, not only drag but also lift! Here again, the critical angle ϕkrit is predicted exactly9. The pressure distribution on the slant of the same model, halfway down, is shown in Figure 14.

Figure 14: Pressure distribution cp(y) on a slant for the two typical flow regimes: ϕ = 30, a pair of longitudinal vortices; ϕ = 35, near wake; Measurements by Morel (1978), Computation – Lattice Boltzmann CFD code.

For a slant angle ϕ = 30° there is a significant difference between experiment and computation. For a slanted surface at ϕ < 30°, as in Figure 12, the typical low pressure peaks are not

9 The difference in critical angle between Figure 12 and Figure 13 is due to the ground proximity of the body in Figure 13.

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reproduced by CFD. Nevertheless, the computed drag and lift agree well with the experimental data. Good fortune? 7. CONCLUSIONS The flow past bluff bodies like cars is characterized by two kinds of separation: • the near-wake type (for square back shapes and, partly, notchback shapes). • shear layers rolling up to form longitudinal vortices (for fastback shapes). The flow inside the former is unsteady, and the latter will also be unsteady if the shear-layer roll-up is not fixed by sharp edges. Under certain conditions, the flow inside the near wake is periodic; otherwise it is random. The motion of the eddies inside the near wake is distinctly different from turbulence. The size of the eddies is comparatively large, and the maximum in their power spectrum is at low frequencies10. With turbulence it is the other way around. The size of eddies is very small, and the frequency of maximum power is high. Hence one can hardly expect both kinds of random motion to be described well by the same mathematical model. Accordingly, the computation of the flow past bluff bodies like cars must ultimately start from solving the time-dependent Navier Stokes equations. However, direct numerical simulation (DNS) is out of reach for the near future. Large eddy simulation (LES) would be the second best approach. However, this scheme still requires long CPU times, and so is not suitable for day to day vehicle aerodynamics. But CFD is urgently needed for the aerodynamic development of vehicles. So what options remain? One possibility may be to model a near wake. Roshko & Lau (1965) discovered that the pressure distribution in the near wake could be made independent of the body from which it develops, provided that the pressure is suitably standardized11. They derived a universal near-wake model that in essence converts the near wake into a solid body with a constant shape. This is not known beforehand, but is found by iteration. This model has been integrated into inverse panel-codes12,13 with good success. Hence the same approach could be tried with a RANS-code. This would allow the aerodynamicist to stay with a steady-state code - hopefully yielding results comparable with the wind tunnel - until computers are fast enough to enable regular use of Large Eddy Simulation. After having uttered so much critique, this paper should not be ended without summing up the major positive aspects from today’s CFD. Already today's CFD codes:

• do not need a hardware model or prototype; • do not need a wind tunnel. • can cope with relative motion between car and road; • can cope with rotating wheels. • can model at full-scale Reynolds numbers (?). • offer a very deep insight into the flow past a car.

Moreover, the results need no wind tunnel corrections, i.e.:

• no blockage, provided the computation volume is large enough; • no effect from the solid walls of nozzle and collector; • no “horizontal buoyancy”.

10 Close to one of the eigen-frequency of a car (oscillating up and down motion of the rear). 11 For details see Hucho (2002). 12 See Gersten et al. (1988), and in a slightly different version, Geropp & Kim (1995). 13 An overview has been published by Papenfuß (1997).

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Hence today's CFD codes can take over a notable part of the aerodynamic development of vehicles, and seem worth the effort of further improvement. ACKNOWLEDGEMENT The author owes many thanks to Gordon Taylor who was so kind to review this paper. REFERENCES [1] Anagnost; A., Alajbegovic, A., Chen, H., Hill; D., Teixeira, C., Molvig, K.

(1997):Digital Physics – Analysis of the Morel Body in Ground Proximity. SAE- Paper 970 139. Warrendale, Pa.: SAE.

[2] Baxendale, A. (2002): Visions of the future: Automotive CFD. Interview in Fluent news, Vol. XI, Issue 2, 2002.

[3] Bearman, P.W. (1979): Bluff Body Flows Applicable to Vehicle Aerodynamics. In Morel, T., Dalton, C. (Eds.): Aerodynamics of Transportation, New York: ASME, 1–11.

[4] Dietz, S. (2000): A2 – ein Meilenstein in der Fahrzeugaerodynamik. Sonderausgabe ATZ und MTZ, März 2000, 80–94.

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[6] Duell, E.G., George, A.R. (1999): Experimental Study of a Ground Vehicle Body Unsteady Near Wake. Warrendale, Pa.: SAE-SP 1441, 197–208.

[7] Geropp, D., Kim, M.S. (1995): Zonenmethode zur Berechnungebener Körper- umströmungen mit Totwasser und Bodeneinfluss. Archive of Applied Mechanics, 65, 270–278.

[8] Gersten, K. Papenfuß, H.-D., Becker, A., Bauhaus, F.-J., Kronast, M. (1998): Experimentelle und numerische Untersuchungen der Aerodynamik bodengebundener Fahrzeuge. Bochum: Abschlussbericht Teil I des Forschungsvorhabens der Stiftung Volkswagenwerk.

[9] Gilhome, B.R., Saunders, J.W., Sheridan, J. (2001): Time Averaged and Unsteady Near-Wake Analysis of Cars. Warrendale, Pa.: SAE-SP 1600, 191 – 208.

[10] Hucho, W.-H. (1998): Aerodynamics of Road Vehicles. 4th Edition. Warrendale, Pa.: SAE Publication.

[11] Hucho, W.-H.(2000): Two Comments on the Application of Computational Fluid Dynamics to Vehicles. In Kobayashi, T., and Ahmed, S.R. (Eds.): Proceedings of Workshop on CFD in Automobile Engineering, SAE Japan, 1 - 4.

[12] Hucho, W.-H. (2002): Aerodynamik der stumpfen Körper. Wiesbaden: Vieweg Verlag.

[13] Hupertz, B. (1998): Einsatz der numerischen Simulation der Fahrzeugumströmung

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im industriellen Umfeld. Braunschweig: ZLR-Forschungsbericht (Dissertation TU Braunschweig).

[14] Leder, A. (1992): Abgelöste Strömungen – physikalische Grundlagen. Wiesbaden: Vieweg Verlag.

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Innovation and Reliability in Automobile Design and Testing, Florence, April 8 – 10. 1992.

[16] Morel, T. (1978): The Effect of Base Slant on the Flow Pattern and Drag of Three- dimensional Bodies with Blunt Ends. In Sovran, G., Morel, T., and Mason, W.T. (Eds.): Aerodynamic Drag Mechanisms of Bluff Bodies and Road Vehicles. New York: Plenum Press, 191 - 226.

[17] Papenfuß, H.-D. (1997): Theoretische Kraftfahrzeug-Aerodynamik – die Struktur des Strömungsfeldes bestimmt das Konzept. ATZ,99, 100 – 107.

[18] Tanner, M. (1967): Ein Verfahren zur Berechnung des Totwasserdruckes und Widerstandes von stumpfen Körpern bei inkompressibler, nichtperiodischer Totwasserströmung. Dissertation, Göttingen: Mitteilungen aus dem Max-Planck- Institut für Strömungsforschung und der Aerodynamischen Versuchsanstalt, Nr.39.

[19] Tsuboi, K., Shirayama, S., Oana, M., Kuwahara, K. (1988): Computational Study of the Effect of Base Slant. In: Marino, C. (Ed.): Supercomputer Applications in Automotive Research and Engineering Development. Minneapolis, Ma.: Cray Research Inc. book, 257 – 272.