This chapter will look at the generation and control of lift of an aerofoil. All the aerodynamic forces and moments with the accompanied mathematical formulae shall be analysed using the aerofoil. The modification of various geometries of road vehicles will be seen in how they are used to reduce the drag and lift forces generated.
Physical Formulae
The first aerofoil designs arrived from Horatio Philips in 1884 and were marked by their thinness. At that time a plate at an angle of incidence would produce lift but not efficiently. Otto Lilienthal, inspired by the wings of birds, made the realisation that by altering the camber of an aerofoil, lift was produced more efficiently. In the 1930's the NACA [National Advisory Committee for Aeronautics], created the asymmetric aerofoil. This was outlined by its aerodynamic efficiency, creating maximum lift with minimum drag at lower angles of attack.Image:Airfoil.svg
Figure 3. Asymmetric Aerofoil Profile
A symmetric aerofoil with an angle of incidence of O° would provide no lift since no pressure difference would be created between the top and bottom sides. The asymmetric aerofoil, as seen in figure 3.1, works by creating pressure discontinuity between its top and bottom surfaces. A symmetric aerofoil with an angle of incidence of O° would provide no lift, since a pressure discontinuity would not be present. This pressure distribution is created by the air wind velocity. The air travelling on the top side of the aerofoil is allowed to accelerate due to the aerofoils convex shape, creating a region of low pressure. On the underside however, air flows at lower velocities creating a region of high pressure. This pressure difference between top side and bottom side creates lift in the vertical direction as shown in figure 3.2. The parameters that are of high importance when determining the aerodynamic efficiency of the asymmetric aerofoil are the camberline and angle of attack.
The correlation between pressure and speed is analysed mathematically by Bernoullis equation, assuming density does not change with pressure, and is given by the following equation:
eqn: 3.1
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Figure 3.
As the aerofoil travels through an air stream, it is subjected to aerodynamic forces and moments. These forces arise from two main sources, the pressure distribution over the surface of the body that acts normal to the body and the shear stress distribution that acts parallel to the surface of the body. With this in mind it is possible to see the forces acting on the aerofoil and in the direction in which they act. Figure 3.3 shows the four main forces acting on the aerofoil. The lift force [L], as a result of the pressure distribution acting normal to the air flow, the Drag force [D], as a result of the shearing forces acting parallel to the airflow with their resultant force [R]. It is from the lift force that the aerofoil is given the vertical motion and the drag force is the force resisting motion in the direction of travel. The resultant force R is comprised of the Lift and Drag forces but also the Axial [A] and Normal [N] forces which act parallel and perpendicular to the aerofoil.
Figure 3.3http://i47.photobucket.com/albums/f182/leftieman/Aircraft%20Physics%20Discussion%20Images/airfoil.jpg
It is of importance to mention that the total effect on pressure, for attached air flow, contributes only to lift and not to drag and this is known as d'Almberts Paradox []. The drag created by friction is caused by the viscous boundary layer.
Given the angle of attack α and the velocity of the air, the lift and drag forces can be solved algebraically using the following equations.
eq: 3.2
eq: 3.3
In addition to the normal and axial forces, rotational moment forces are also created. The moment force promoting clockwise rotation by increasing angle of inclination is taken to be positive, whereas the moment force promoting anticlockwise motion and hence decreasing the angle of inclination is taken to be negative. In order to fully appreciate the importance of angle of attack and the forces exerted on the aerofoil, it is important to introduce dimensionless coefficients of the force and moments. The reason in defining dimensionless coefficients is that the value of the coefficient is independent of speed and relates strictly to object shape. These coefficients are:
Lift coefficient eqn: 3.4
Drag Coefficient eqn: 3.5
Moment coefficient eqn: 3.6
An important conclusion of the basic aerofoil theory is that this resultant force acts at the quarter chord of a symmetrical aerofoil which is known as the centre of pressure. The point at which the pitching moment is independent of angle of attack is known as the aerodynamic centre and is also located near the quarter chord. The position of the centre of pressure will vary in asymmetrical aerofoils with varying angles of attack where as the aerodynamic centre will remain close to the quarter chord line. In race car application the aerodynamic centre is not as important as the centre of pressure. [Katz]
The characteristics of how an asymmetric aerofoil behaves with varying angles of inclination can be seen in figure 3.4. These results were obtained by wind tunnel testing of a general purpose aerofoil predating the NACA system, known as Clark Y.
Lift and Drag curves for a typical airfoil
Figure 3.4 Clark Y aerofoil with aspect ratio = 6
It can be seen from figure 3.4 that the lift coefficient increases linearly with increasing angle of attack whereas the drag coefficient increases exponentially. At a critical angle of attack α=17° for this specific aerofoil, the lift coefficient gradient becomes negative. At this point the wing is said to stall, the point where its aerodynamic efficiency is terminated and the drag force supersedes the lift, as shown in figure 3.5. The reason for this is due to air flow separation at the leading edge.
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Figure 3. 5 Wing Stall
In order to understand flow separation it is of high importance to understand the theory behind the boundary layer since it is the key in understanding how lift and drag forces are created. Air flowing over a flat plate will have a thin layer slow down due to friction between surface and air resulting in a relative velocity of V=0m/s at the surface of the plate. As you move away from the plate, the air increases speed linearly until the height where air is unaffected by the plate and moves at its free stream properties. This thin layer of affected air is known as the boundary layer and is shown in figure 3.5.
Figure 3. 5 Boundary LayerProfile of a boundary layer. (NASA EP-89, 1971, p. 68)
Within the boundary layer, laminar and turbulent flow may be evident. As the air strikes the front of the aerofoil the flow is laminar, as you move down the length of the aerofoil, the boundary layer thickness increases and the flow becomes turbulent, due to the fluctuating turbulent velocity components. The point at which laminar flow becomes turbulent is known as the transition point. The different types of flow may be flowing at the same average speed but ultimately have different effects on the aerofoil. Laminar flow has the benefit of reducing the skin friction which adds to the total drag whereas the turbulent flow has the potential to maintain flow on the contours of the body inhibiting flow separation due to larger momentum transfer.
Figure 3.6 Boundary Layer movement against Aerofoilhttp://www.zenithair.com/kit-data/images/ht-875a.gif
This transition from laminar to turbulent has an adverse affect on pressure. The air moves from low to high pressure and due to the energy dissipated at point of contact, the air cannot return to its free stream conditions. As the boundary layer moves down this adverse pressure gradient, the speed of the boundary layer drops down to zero, at this point the flow separates. As the flow separates it creates a localised region of low pressure at the rear of the car. Within the region of separated air, the flow will begin to recirculate opposing that of the free stream as is evident from figure 3.7. At this point the lift decreases whilst the drag increases. Therefore for an aerofoil to work efficiently the boundary layer separation must be controlled. [Allen:1986]
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Figure 3.7
An important factor affecting the boundary layer is the Reynolds Number. This is a dimensionless number that provides a ratio of inertial and viscous forces given by the following mathematical equation:
eqn: 3.7
The Reynolds number is an important factor in distinguishing the behaviour of the boundary layer. With a high Reynolds number less drag can be achieved since a thinner boundary layer can be maintained. An increase in the momentum of flow outside the layer is also achieved, both factors delaying flow separation and therefore stall. The opposite applies for lower Reynolds numbers as it will promote flow separation increasing drag.
Road Vehicle Aerodynamics
The aerodynamic forces on road vehicles are the result of complex interactions between flow separations and the dynamic behaviour of the released vortex wake. Successful car design needs to take advantage of these interactions to improve flow control by means of active or passive control devices [A.Sphon]. Examples of these design modifications will be analysed showing how drag can be minimised.
For vehicles aerodynamic drag is the sum of the tangential or skin friction forces and the normal or pressure forces parallel to the vehicles velocity vector but in the opposite direction. For vehicles with mostly attached flows that do not produce much or any lift, the skin friction contribution is often the most important drag component. The other main component of aerodynamic drag is pressure drag. In the absence of viscosity, the pressure forces on the vehicle cancel each other and so the drag is zero but since we always have viscosity, it modifies the pressure distribution which leads to a drag contribution called pressure drag and is the dominant component for vehicles with extended regions of separated flow. The sum of skin friction and pressure drag is called viscous or profile drag. When a vehicle produces lift, a third drag component called induced drag is also applicable. Induced drag comes as a result of the modification in the vehicle pressure distribution caused by the trailing vortex system that accompanies the production of lift. Induced drag tends to be the most important component for vehicles operating at speeds that are lower than their designed speed. At transonic and supersonic speeds another drag-producing mechanism arises from the radiation of energy away from the vehicle in the form of pressure or shock waves, this component is called wave drag. The total pressure drag is the result of three different flow phenomena, these being, boundary-layer displacement effect, the trailing vortex system, and shock waves. The design approaches followed to reduce each of these pressure drag components tend to vary. Each of these drag components is affected according to the Reynolds number and the Mach number [C.P. van Dam: 1999].
Before analysis can be done on the improvement of vehicle geometry and its subsequent effect on the aerodynamic forces experienced, it is important to explain how the air behaves as it flows over the body. There are two types of separation that occur to the air flow around a vehicle due to its bluff body shape. The first type has a quasi-two-dimensional character defined by the line of separation which is perpendicular to the local flow direction. If reattachment occurs downstream then separation bubbles are formed. Since the separation itself is mainly two-dimensional with separation line normal to the flow and vortex axes parallel to the separation line, it is designated quasi-two-dimensional and can be seen in figure 3.8. This type of flow can occur at the leading edge of the front hood, at the sides on the fenders, on the cowl and on the front spoiler, and possibly in the notch of a notchback.
Figure 3. 8
The counter-rotating vortex pair of a notchback, fastback and squareback is shown in figure 3.9. The lower vortex rotates counterclockwise and is responsible for carrying the contamination to the rear of the vehicle. The upper vortex rotates in the opposite direction. After the separation bubble closes, a pair of counter-rotating longitudinal vortices forms in the trailing wake. This produces an upwash in the case of a square back, and induces a downwash in the trailing wake flow on a notchback or fastback.
Figure 3.9 Counter rotating transverse vortices in the wake of cars with three typical rear end
On a square back, the vortex pair rises in the flow direction and wanders toward the plane of symmetry. On fastbacks and notchbacks the vortices approach the road downstream and move to the outside. It can be postulated that these longitudinal vortices are the continuation of the lateral vortices described above. There is a velocity decrease toward the centre of the vortex. The longitudinal vortices are slowly exhausted downstream by dissipation.
The second type of separation is three dimensional in nature and high in energy as displayed in figure 432. Vortex trains are formed at sharp edges where the flow is oblique, as with a delta wing. Such a vortex pair is formed on the two A-pillars and C-Pillars and is bent back towards the roof at the upper end of the Apillars. A strong vortex pair forms at the rear of the vehicle, depending upon the inclination of the rear end. These rear vortices interact with the external flow field and with the quasi-two-dimensional wake.
Fig 2.18 Three dimensional flow separation
When rear modification started taking place at angles of φ=45° it was noticed that flow separation occurred at the trailing edge of the roof, but as the angle was reduced to φ=30° the drag increased by 10% due to flow separation occurring at the bottom slant angle of the rear. Given this a bistable condition was determined between the angles of 28°<φ<32°, meaning that separation could take place anywhere. This can be seen in figure 4567 where angle of inclination φ is plotted again drag coefficient [Cd].
Figure Influence of rear end slope angle φ on drag coefficient, separation line and wake,
The front end of the car, regardless to common misconception is not the most important aerodynamic feature required for efficiency when designing a car. It simply aids the air in flowing smoother over it, and the end result is only as accurate as the changes done to the rest of the body. As air strikes the front a stagnation point is formed, due to the proximity of the road surface, air tends to flow beneath the car. The air flowing underneath flows faster than on top creating a pressure difference with a positive lift force applied to the car.
From the work done by G.W Carr it is known that by increasing the radius of the leading edge the drag of the body is reduced rapidly until a certain value where drag remains constant. Applying this theory to the car geometry, it shows that only with a slight rounding of the leading edge, the flow separation at the front is prevented. To achieve the precise radius for maximum aerodynamic efficiency at the front, it is determined by experimental methods and is highly dependent on Reynolds number. Following the work of A.Gilhaus and V.Renn , it is confirmed that the inclination angle of the hood coupled with the inclination angle of the front face share similar characteristics in the way that they also have a saturation effect and above a certain angle of inclination, there is no further drag decrease. To see the maximum benefit gained by altering front end geometry, an "optimum nose" was designed and tested on. The results indicated a drag improvement of Cd=0.05 and the conclusion was that, as long as flow separation is averted, no more reductions can be achieved by front end geometry modifications. The testing did however prove that the lower the stagnation point at the front the better the aerodynamic efficiency.
Figure Effect of the stagnation point position on drag
Another realisation was that rounding the corners was not essential but similar results could be achieved by chamfering the edges. Front end geometry modifications allow the air flow to remain attached for longer, able to withstand steeper adverse pressure gradients without separating as it flows over the rest of the body contours.
As the air flows over the front end towards the windshield, flow separation can occur at three different locations, these being at the base of the windshield at the junction with the hood, at the junction with the roof and at the A-pillars. As mentioned earlier, the separation occurring at windshield junctions are quasi-two dimensional and weak in strength since their axes are parallel to the flow separation and usually reattach further downstream. The separation occurring at the A-pillar however is highly three dimensional and strong in power. The parameters governing the geometry at this stage are the windshield inclination and the radius of the A-pillars. From diagram 3.4, following the work confirmed by Scibor Rylski, the separation point S and reattachment R are plotted against windshield angle γ. The results obtained are only considered as a trend since the junction between hood and bonnet is rounded and the windshield itself is curved, both attribute inhibit flow separation.
Figure Flow separation on the bonnet and reattachment on the windscreen, as a function of γ
The size of the separation bubble created is determined by the inclination angle δ of the windshield and of the angle between hood and windshield φ and as shown in the figures below the subsequent effect they have on drag.
Figure Effect of Winshield Angle φ on drag
Figure Effect of windshield angle δ on drag
What can be confirmed, by the work of R.Buchheim, is that for angles larger than δ>60° there is no further improvement in drag. Even if an angle greater than δ=60° did bring further improvements it is not a viable option because visibility problems arise from double refraction as well as an increase in temperature within the driving cabin. As shown in figure >>, the effect of the increasing windshield angle is indirect to the drag coefficient. From this it is deduced that for larger windshield angles a lower negative peak pressure is produced at the junction with the roof allow the boundary layer to flow over the adverse pressure gradient with less momentum loss. The large windscreen inclination angle brings about another aerodynamic improvement and that is in relation to the A-pillars. Less air is forced to the A-pillars creating weaker vortices allowing for better pressure recovery downstream. This is coupled with rounding of the A-pillars to aid in wind noise and water flow as shown by figure 3.9.
Figure Development of the A-pillar and C-pillar of the Audi 100 III
As the air flows towards the roof and the attempt to lower drag at this section arises from the longitudinal curvature of the roof. The effect of the roof curvature could have a detrimental effect on drag as shown in figure 4.5, unless the radii junctions between roof and windscreen and roof and rear window are large enough to allow for low negative peak temperatures. As the roof is given its curvature it is of importance not to alter the frontal area of the car because where the drag coefficient could be reduced, the total drag [Cd x A] would increase. This is also shown in figure 4.5 proven by the work of R.Buchheim. The result from the compulsory inclination angles of the hood, windscreen and roof, the windshield and rear window become more spherical increasing costs. From the figure below it is also important to mention that values were taken at the chord length of the roof arch as a reference variable to the curvature.
Figure Effect of roof camber on drag
There are three types of rear ends for production cars, these being notchback, fastback and squareback and for all profiles; a boat-tail could be added. As mentioned earlier there are two types of flow separation and they are determined by individual parameters. In the case of quasi-two dimensional separation, it is boat-tailing and is determined by the angles at the roof, sides and underpanels, whereas the formation of the three dimensional separation is determined by the angle of the rear end. The aim of shape development is to make the base pressure, pressure at the rear of the car, as high as possible and the base itself where the pressure acts as small as possible, and this is accomplished by tapering the rear or boat-tailing as shown in figure 888.
Figure Boat-tailing applied to a notchback car
Figure Effect of underbody diffuser geometry on drag and rear axle liftThe work done by W.A.Mair indicates that a body of revolution had an optimal tapering of 22° compared to the boat-tail added to the rear of the three types of cars, the actual tapering angle being at 10 degrees. Boat-tailing, despite having advantages when taken from the sides of the cars, also has a positive effect when it is applied on the underbody. The work of J.Potthoff demonstrated that for any area ratio [diffuser outlet area/diffuser inlet area], the use of a long diffuser with a small angle is more effective than a short diffuser as is shown in figures 5.7 but only assuming a smooth underbody.
Figure Effect of underbody diffuser length and angle on drag
A huge advantage of the functioning diffuser is that less lift occurs at the rear axle.
As previously mentioned cars are seen as bluff bodies with angles that aid in flow separation. As the air flows along these edges there are four main types of vortices generated. Firstly flow separations occur on edges running perpendicular to the flow, vortices are formed rolling up the car with their axes being parallel to the separation line. These "free" vortices develop into horseshoe vortices and are defined by their weak in energy nature, and observed at the front edge of the hood, on the sides of fenders and on the front windshield and spoiler. These vortices tend to reattach downstream before final separation occurs at the rear of the car leading to a wake of recirculating air called "dead water". These are quasi-two dimensional in nature. The second type of flow separation experienced occurs at the A and C- pillars of the car with their axes being in the streamwise direction. They are heavily three-dimensional and are defined by their strength and high energy values, and are heavily dependent on the geometry from which they separate from. This type of vortex generation causes a powerful vortex formed at the slanted back of the car extending far behind and dissipating their energy due to the friction experienced as they go closer to the road. This formation at the upper rear surface is determined by the slant angle φ. With an increasing angle, the inward rotating vortices increase in strength with the increase in slant angle. As φ increase passed the critical angle of 30° the vortices break up creating maximum drag. The downwash induced by these vortices guarantee that flow does not separate as it comes from the roof, but due to the high negative pressure difference air flow travels against a steep adverse pressure gradient aiding flow separation increasing drag.
Methodology
The following chapter will provide the methodology involved in creating the model with its various geometries using CAD software. It will also give an explanation of the meshing involved in the pre-processing stage along with justification of the processes involved. Finally with the use of Computational Fluid Dynamics, it will indicate the forces exerted on a road vehicle. The aforementioned investigations will be conducted using the following programs, Solid Works, Gambit v2.3.16, Fluent v6.3.26 and Ansys Workbench v12. The model with all its parts shall be created using Solid Works which is a CAD software or computer modelling program, following this, the model shall be imported into Gambit which is a CFD program. Within Gambit, the model surfaces shall be discretized and the domain meshed for the areas under investigation. This makes up the pre-processing stage of the overall process. After the mesh has been completed, the model shall be imported into Fluent where the computational analysis will be undertaken.
For the 3 dimensional analysis of the model, Ansys Workbench shall be used. Within this program, the meshing, discretizing and computational analysis can all be undertaken without the need for the model to be imported into secondary programs such as Gambit.
Aerofoil Investigation
The first investigation shall illustrate how drag and lift forces are affected by angle of inclination on a 2D aerofoil. Due to the simple geometry of the aerofoil, a CAD program is not necessary to create the aerofoil. Within Gambit, the Cartesian coordinate system will be used to plot vertices. These vertices shall be connected with arching or straight lines in order to create the geometry of the aerofoil as shown in figure 6.1 also known as the bottom-up approach. "Top-down approach refers to an approach where the computational domain is created by performing logical operations on primitive shapes such as spheres, cylinders and blocks" [A.Bakker, 2006].
In order for the solver to simulate fluid flow a spatial domain must be created around the aerofoil and then discretized. This is done using the Cartesian geometry system in Gambit to create a rectangular area around the aerofoil with a parabola on the one side of the aerofoil. This region will be discretized using a quadrilateral submap scheme with a spacing of 1 and can be seen in figure 6.2. Quadrilateral mesh was used for the aerofoil's simple geometry because it can be easily stretched to account for varying gradients in varying directions. In the case of the aerofoil, the gradients normal to the aerofoil are greater than the gradients in tangent to the aerofoil. The cells closest to the aerofoil have much higher aspect ratios. This is the case of simple geometry objects, for more complex shapes, a hybrid mesh which is comprised of quadrilateral and triangular cells would be easier to produce with higher accuracy of results.
Once the mesh has been created, the boundary conditions must be applied to the domain and the aerofoil. The aerofoil edges are defined as walls, no fluid flows through them and if required heat transfer could be permissible. At each node point on the aerofoil, the normal pressure and shear stresses will be calculated, their sum providing the drag and lift forces generated by the aerofoil. The curved surface on the left hand side will be defined as a velocity inlet which is where fluid will be flowing from. The remaining three surfaces are defined as pressure far fields so as to allow fluid to flow freely without any interaction distorting the results. The investigation of the aerofoils aerodynamic performance shall be tested against varying angles of attack, 10°, 20° and 30° so as to deduce where the maximum aerodynamic efficiency of the aerofoil is found. Initially the aerofoil be tested against an angle of inclination of 0° to be used as a datum of comparison. It is known from figure angle o atatk that that with increasing angle of attack there is an increase in both lift and drag. Due to this, the value of interest would be the lift to drag ratio. The highest value obtained at a given angle of attack, will be the most efficient position of the aerofoil. Using Fluent Software, the simulation will be done using the 2Dimensional Double precision with a simulated airflow of 100m/s [223 mph]. The drag force and the lift force shall be reported via the Fluent solver.
Audi TT 3 D
