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Design of High-Lift Airfoil for Formula Student Race Car.

I. Introduction

THROUGH the past decades-especially since the end of the 1960s [1]-the importance of aerodynamics for motorsports has been increasing rapidly. Since then more and more motorsports vehicles started using the concept of adding inverted wings to add more downforce to further increase cornering speeds, as generating downforce should increase the adhesion friction between the tires and the ground and, therefore, the vehicle should resist the skidding or rolling forces at corners finally pushing up the skid-speed limit. In addition the generated downforce would enhance braking performance in terms of braking distance, and it will reduce slip effect through increasing traction between the tire and the ground.

Formula student is a student competition that takes place in many countries across the world; Ain Shams University's ASU racing team participates in the Silverstone, United Kingdom, competition organized by IMechE.

The competition is about getting students to design, build, present, and compete with scaled formula race vehicles into static and dynamic events. Static events include presenting the technical design aspects of the vehicle and presenting a business logic case of the vehicle along with the cost analysis report. On the other hand, the dynamic event involves time attack events of skid pad, acceleration, autocross (sprint), and endurance races.

It is arguable that most teams focus on increasing down-force rather than reducing drag because although drag should reduce the vehicle's acceleration; however, the autocross and endurance tracks consist of hard and successive corners starting from 5 to 20 meter radii, whereas straight lines which are not long enough for a vehicle to reach its top speed are intended for safety considerations. Thus the nominal vehicle speed is estimated to be about 50 km/h, and therefore the performance of the vehicle onto these corners appears to be more important and more effective in terms of lap time than that on straight lines.

The problem that appears to every motorsports aerodynamicist is the limiting space rules for aerodynamics devices, thus creating the need for an efficient wing. This efficiency is interpreted into generating as much lift force with the limited available space and keeping in mind to reduce drag as possible.

This article presents the design and optimization of a 2D three-element wing through finding the optimum combination of rigging parameters. The wing consists of three elements: the main element with the longest chord length, in addition to two flaps (Figure 2).

A. High-Lift Aerodynamics Introduction

a. High-Lift Airfoil Gaps Effect Understanding the gaps and slots effects was discussed by Prandtl [2] and attributed mainly to energizing the boundary layer. This concept was then better justified by the work of A.M.O Smith [3] according to the following effects:

1. The slat effect: The slat vortices protect the leading edge of the main element by decreasing the energy of air impacting the main element's leading edge. Thus at higher angles of attack, it reshapes the flow to be smooth over the leading edge and hence delays leading edge separation stall.

2. The circulation effect: The downstream element causes the upstream element to be in high-velocity area. Thus the circulation has to increase in order to meet the Kutta condition. So the downstream increases circulation and thus lift of the upstream element.

3. The dumping effect: The trailing edge of the downstream element has higher velocity than the free stream. So the boundary layer will come off the forward element with high velocity where a reduction in pressure recovery occurs.

4. Off-surface pressure recovery: The boundary layer accelerates at the trailing edge with speed more than free-stream speed and then converts into a wake; the recovery to free-stream velocity will be efficient away from the walls of the airfoil, where the wake should not merge with the walls' boundary layer.

5. Fresh boundary layer: The purpose is to start a fresh boundary layer at each element in order to have three thin boundary layers on three elements that is better than having a thick one on a single airfoil.

b. High-Lift Airfoil Geometry and Rigging as Shown in Figure 2

1. Main element chord C: To which all coefficients and dimensionless parameters are related. Defined as the distance between the main element's leading and trailing edges, although some other references might instead use a mean chord starting from the main element's leading edge down to the last downstream element's trailing edge; however, such definition is problematic as the chord length changes with each iteration and displacement of the elements. Hence a comparison using the dimensionless coefficients of lift and drag will fail; therefore here it is set to be the fixed value of the main element's chord.

2. Flap gap [G.sub.f1]: The diameter of the circle drawn from the upstream element's trailing edge, tangential to the flap, suffix 1 or 2 indicates the specified flap 1 or 2.

3. Flap overhang [O.sub.f1]: The horizontal distance between the upstream element's leading edge and the flap's leading edge.

4. Flap deflection [[delta].sub.f1]: The minimum angle between the flap's chord and the main element's chord.

c. Trailing Edge Devices

1. Plain flap: It is just a part of the main element, but it is given a degree of freedom in order to take the required deflection without slotting.

2. Slotted flap: As Smith explained the effects of slotting, the slotted flap increases the maximum achievable lift through making the boundary layer withstand further camber and higher pressure differential.

3. Fowler flap: Improved flap performance could be concluded when the flap produces a Fowler action, where the Fowler action is defined as "the measure of change in position of the leading-edge of the flap in the plane of the chord of the fore element." Also the extended chord could be defined as the cruise airfoil chord plus the Fowler action; this increase in airfoil area increases the achievable lift without significant increase in drag.

In conclusion, adding a downstream flap to the rear of a race car airfoil should not change the slope of the lift curve, but rather the curve itself will shift upward, thus increasing maximum lift [4]. From the previous and according to the requirements of the application, it was chosen to design a double-slotted flap setting.

B. Computational Fluid Dynamics Introduction

Computational fluid dynamics (CFD) techniques are used to numerically solve the governing equations for a flow field to predict flow variables such as pressure, velocity, and temperature. Thus it is important to define two equation sets related to the focus of this article: firstly the Navier-Stokes mass, momentum, and energy conservation equations and secondly introducing the problem of turbulence modeling. Once these models are defined, they are solved by the finite volume method (FVM) used in ANSYS FLUENT.

a. Continuity and Navier-Stokes Equations These

equations express the conservation of mass and momentum; such equations are then applied to the control volumes through the domain to be solved numerically according to the boundary conditions:

[partial derivative][rho]/[partial derivative]t++[nabla].([rho]V) = 0 Eq. (1)

[mathematical expression not reproducible] Eq. (2)

[mathematical expression not reproducible] Eq. (3)

[mathematical expression not reproducible] Eq. (4)


[rho] = Density

p = Pressure

[g.sub.i] = Gravitational acceleration component in the


u, v, w = Velocity components in the x-, y-, and


[micro] = Fluid dynamics viscosity

b. Turbulence Modeling Turbulent flow is a type of flow characterized by irregular local fluctuations in both magnitude and direction of properties with respect to time. Due to the excessive kinetic energy, the fluid particles overcome the damping effect of viscosity. The problem of modeling the fluctuating irregularities does not only arise in the very short time step among which these fluctuations occur but also that the solution for such problem would be much affected with the initial conditions and boundary conditions; thus the simplest error would result in huge variation of the solution. Thus the problem of turbulence is either solved by the following methods.

1. Direct numerical solution (DNS):

* This simulation resolves the irregular fluctuations completely from the smallest dissipative eddy up to the integral scale ones.

* Such simulation requires significant computational resources; thus it is mainly used for academic applications.

2. Large eddy simulation (LES):

* The large unsteady turbulent eddies are resolved, whereas the small ones are modeled.

* Its idea is to decrease the required computational resources of the DNS using a low-pass filtering method.

3. Reynolds-averaged Navier-Stokes (RANS) equations:

* The fluctuation is decomposed statistically into two components: an average component and a peak component where the fluctuation equals the average plus or minus the peak.

* Reynolds stresses are either all modeled and solved using the six-equation Reynolds stress models, or an analogy using the Boussinesq hypothesis is made to relate the RANS unknowns to a new term called turbulent viscosity. And then zero-, two-, and four-equation models are used to solve this term.

RANS Eddy Viscosity Flow Model. Using Reynolds decomposition:

u = u + u Eq. (5)

where u is a local average x velocity and u is the fluctuation above the average value. Averaging the continuity equation:

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 Eq. (6)

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 Eq.(7)

In the x-direction, we have:

[rho] [partial derivative]u/[partial derivative]t + [rho](u [partial derivative]u/[partial derivative]x + v [partial derivative]u/[partial derivative]y) = -[partial derivative]p/[partial derivative]x + [micro][[nabla].sup.2]u Eq. (8)

Using the decomposition equation:

[mathematical expression not reproducible] Eq. (9)

Thus according to the Boussinesq hypothesis:

[rho] [partial derivative]u/[partial derivative]t + [rho](u [partial derivative]u/[partial derivative]x + v [partial derivative]u/[partial derivative]y) = -[partial derivative]p/[partial derivative]x + [micro][[nabla].sup.2]u + [f.sub.turb,x] Eq. (10)

[f.sub.turb,x] is the force due to turbulent stress in the x-direction.

Now through more derivation, turbulent shear stress is function in turbulent viscosity; this is derived as follows through analogy with normal shear stress and normal dynamic viscosity:

[[tau].sup.turb.sub.xy] = [partial derivative][[tau].sup.turb.sub.xy]/[partial derivative]y dy Eq. (11)

[[tau].sup.turb.sub.xy] = [[micro].sub.turb] ([partial derivative]v/[partial derivative]x + [partial derivative]u/[partial derivative]y) Eq. (12)

The term [[micro].sub.turb] is the coefficient of turbulent viscosity, which the RANS EVM turbulence models are intended to solve as function in a set of partial differential equations, so in other words, turbulence models are equations added to the RANS flow model, in order to solve the turbulence unknown terms.

II. Procedure

A. Optimization Method

Brute force optimization would require giving the model ranges for all design parameters which are the gaps, overlaps, and deflections for both flaps ([G.sub.f1], [O.sub.f1], [[delta].sub.f1], [G.sub.f2], [O.sub.f2], [[delta].sub.f2]) and then solving all possible outcomes and finding the best possible results in terms of lift and drag. If each of the six parameters takes only four values, this produces the required [4.sup.6] = 4096 simulations to cover the whole variations. This method could get a well-optimized wing with good understanding of the parameters' sensitivity; however, it needs a lot of effort, time, and computational resources.

Alternatively to solve the previous problem, the optimization was started with a baseline design using deep understanding of high-lift aerodynamics and the relation between slotting and the boundary layer of each element. Now having a baseline design of acceptable lift and drag coefficients, a parameter is varied while all other parameters are kept constant till reaching lift coefficient maxima; subsequently this parameter value producing the maxima is kept constant, and the next parameter is varied and so on in one direction as follows:

[G.sub.f1] [right arrow] [O.sub.f1] [right arrow] [[delta].sub.f1] [right arrow] [G.sub.f2] [right arrow] [O.sub.f2] [right arrow] [[delta].sub.f2]

Reaching the last parameter ([[delta].sub.f2]) should end the first pass. To find an even better combination, a second pass should be performed using the acquired parameters of the first pass and again starting to vary parameters in the same direction and order until a maximum lift coefficient is.

The optimization procedure is shown in Figure 3.

B. Turbulence Model

The objective is to accurately calculate the pressure field in order to obtain both lift and drag coefficients, keeping in mind the crucial importance of capturing the boundary layer separation accurately, as this is the point before which the maximum lift coefficient is determined.

It was found that both LES and DNS models would not go along with the available resources; thus the choice was made for the RANS model. It is crucial to choose an appropriate RANS model for the problem in hand as each one of them is flawed at a point and accurate at another.

One other modeling challenge was that the vehicle operates at rather low Re numbers-typically below [10.sup.6]; thus the flow lies into the transition region between a fully turbulent and fully laminar flow. So a model that wouldn't predict such transition would detect false stall.

Hence based on the relative advantages and disadvantages of various RANS model options, it was decided to use the transition shear stress transport (SST) model.

The transition SST [5] model is based on the two-equation k--[omega] SST model but further adds two more equations to detect transition; one for the intermittency and the other for the transition onset criteria. Thus two more variables-intermittency and transition momentum thickness Reynolds number [Re.sub.[theta]]-are introduced and shown below.

It is considerable to note that the k - [omega] SST model has the advantage of using a k - [epsilon] model at the far stream [6] and using the k - [omega] model at the wall. Sensitivity is an inherent property of the model and requires some fine tuning. It has been found that for practical applications, the K-omega SST model behaves well with adverse pressure gradients and separation detection. The four-equation Langtry-Menter transition SST model is given by the following four equations:

[mathematical expression not reproducible] Eq. (13)

[mathematical expression not reproducible] Eq. (14)

[mathematical expression not reproducible] Eq. (15)

[mathematical expression not reproducible] Eq. (16)


k = Turbulent kinetic energy

[omega] = Specific turbulence dissipation rate

[gamma] = Intermittency

[Re.sub.[theta]t] = Transition momentum thickness Reynolds number

C. Turbulence Model Validation

Narsipur, Pomeroy, and Selig validated the transition SST model for a two-element high-lift airfoil [7], where the airfoil NLR 7301 was analyzed using this model. The result lift and drag curves were compared against experimental data collected in the low-speed wind tunnel at the National Research Labs, the Netherlands. "The two-element system consisted of the NLR 7301 airfoil section and a flap element of 32% chord at a deflection angle of 20[degrees], a gap of 2.6% chord, and an overlap of 5.3%."

According to their work and experiments, the transition SST model underpredicts [C.sub.1] by nearly 3%, whereas it overpredicts [C.sub.d] by nearly 4% for all angles of attack. The accuracy of the model could be attributed to the well-resolved laminar and turbulent boundary layer.

D. Mesh Properties

In order to fulfill the requirements of the transition SST model, the boundary layer must be resolved by keeping [y.sup.+] value around 1 for the airfoil walls. But in order to make the process more robust, the far-field mesh was set to be unstructured triangular elements, whereas the quad elements are layered only at the airfoil walls to reach the required [y.sup.+] value. Thus a hybrid mesh setting was used.

The domain is a C-domain mesh which is advised by NASA. For refinement around the airfoils to capture the turbulence wakes, a sphere of influence was defined at such region. Figure 4 and Table 1 sum up the final mesh properties for the model.

In order to make sure that the mesh sizing does not affect the final solution, a mesh independency study was carried on where the refinement zone was further refined and the effect on lift and drag coefficients was recorded and displayed in Figure 5.

For mesh independency study, the sphere of influence refinement sizing was varied and plotted against lift and drag coefficients. Figure 5 shows that further refinement after 501,000 cells would result in variation of only 0.32% for [C.sub.1] and 3.67% for [C.sub.d], thus pointing that mesh independency is achieved as target criterion is less than 5%. This convergence was satisfactory for the model and fidelity of the solutions obtained.

III. Results and Discussion

Figure 6 shows the relationship between [O.sub.f1] and [C.sub.1], where a decrease of [O.sub.f1] by 20% increased [C.sub.1] by 0.13% which is a small margin. The next step was varying [G.sub.f1] which appeared to be the most sensitive parameter in comparison to the rest. Increasing [G.sub.f1] by five times increased [C.sub.1] by 7.1%. These results are shown in Figure 7. Then varying [[delta].sub.f1] by 5.71% yielded into further increasing [C.sub.1] by 0.89%, whereas any further attempt to increase deflection was met by boundary layer separation stall. Thus optimizing flap 1 settings yielded into a total [C.sub.1] increase of 8.2%.

Now keeping the acquired flap 1 rigging and proceeding with flap 2 optimization. Reducing [O.sub.f2] by 47.5% increased [C.sub.1] by 0.16% as shown in Figure 8. Then increasing [G.sub.f2] by 30% further increased [C.sub.1] by 0.52% as shown in Figure 9. Thus optimizing flap 2 yielded into further increase of 0.63% in lift coefficient.

Figure 10 describes how optimizing the variables changed lift and drag coefficient. It is noted that flap 1 gap had the highest contribution to the lift coefficient increase, where the final L/D ratio is 41.9.

One can note that nearly all lift and drag curves behave in a similar way before stall limit, which could be attributed to the fact that as pressure differential across the airfoil increases, lift would increase in expense to additional drag; such behavior could be noted too in a single-element airfoil.

Also one idea that would explain the behavior of lift and drag vs. gaps and overlaps is that the intent of a gap is to supply air with sufficient energy to overcome the adverse pressure gradient, thus delaying separation and enhancing overall circulation. Increasing the gap much beyond limit would make air enter with insufficient velocity and unguided direction which might lead to separation of the boundary layer onto the following element. Reducing the gap too much would result in very high throttling losses for air entering the gap; thus it would have insufficient energy to withstand the adverse pressure gradient. As a result, the design point should lie between these two conditions.

Figures 11 and 12 show a comparison between the velocity contours of the baseline vs. the optimized setting; it is clear that the optimized rigging provides more air flow with proper direction to energizing the boundary layer and generating a favorable wake. It is also clear that the wake generated by the baseline design prevented adhesion between air and the second flap due to separation. A problem overcame as shown by the optimized design.

Figures 13 and 14 show how the optimized rigging increased the pressure differential through increasing suction pressure above the wing. In baseline rigging, the maximum suction pressure coefficients of elements were -11.52, -3.55, and -0.5 for the main element, flap 1, and flap 2, respectively, whereas for the optimized rigging, these values increase to -12.67, -8.36, and -1.03 consequently.

Keep in mind the rough number stated by Smith that the maximum achieved value of e is about 70%.

It is important to note that the optimized design did stall on trying to further increase the flap 2 deflection, as well as in attempt to further optimize and do a second-pass variation; thus the boundary layer is on the verge of separation as recommended by Smith.

The following figures show the plot of pressure coefficient for both upper and lower surfaces for each airfoil against horizontal distance measured downstream from the main element's leading edge.

IV. Conclusion and Future Work

This article has shown the effect of varying rigging parameters against lift and drag coefficients for a 2D airfoil. Such airfoil could be used for race or even commercial car applications for both speed and stability performance enhancements. However, if used for commercial vehicles that do not only follow the formula student track, it is advised to use an active wing design so as to use the wing when required at sharp corners, whereas at straight lines where acceleration is required, the wing should not be used.

Clearly the challenge was to keep air attached until the last downstream element through finding the right rigging combination that delays separation due to adverse pressure gradient.

Although 2D design was used to generate the best profile; however, one of the aerodynamics challenges for a formula student vehicle is the short available wingspan due to competition rules; therefore the trailing vortex effect will surely affect both lift and drag. Thus future work should include the 3D simulation of the previous airfoil and should show how to overcome losses generated by the trailing vortex. Some ideas might be using a twisted airfoil in addition to an optimized endplate to minimize air spilling from the upper high-pressure to the lower low-pressure surfaces at the ends.

Afterward the 3D optimized wing's lift and drag coefficients should be inserted to a track simulation code and find out how the vehicle would perform with and without the wing. Then finally such results should be compared to real on-track results for final validation.

V. Acknowledgments

This research article was carried out by the help, support, and facilitations of ASU racing team; we would like to thank all of this entity's leaders and members who have helped us through this research, and we would like to thank the teaching assistants at the automotive department, Ain Shams University, who voluntarily give much time and effort in helping ASURT students, mentioning Eng. Mohamed Abdelwahab, Eng. Ahmed Abdelqader, Eng. Mohamed Essam, and Eng. Mohammed Abdelshakour.

We would like to thank all of our professors, teaching assistants, and staff at Ain Shams University's mechanical engineering department who guided us to the knowledge and mindset we have.

We are also grateful to Eng. Mohammed El-Beheiry (technical analyst at Optumatics) and Eng. Ibrahim Gad El-Haq (teaching assistant, mechanical power department, Ain Shams University) for their help and comments for the sake of this research.

And most importantly, we thank our family, friends, and colleagues who stood by us all along.

CFD -               Computational fluid dynamics
RANS -              Reynolds-averaged Navier-Stokes
LES -               Large eddy simulation
SST -               Shear stress transport
DNS -               Direct numerical simulation
[C.sub.1] -         Lift coefficient
[C.sub.d] -         Drag coefficient
[C.sub.p] -         Pressure coefficient
C -                 Main element fixed chord length
[y.sup.+] -         Non-dimensional wall distance
[G.sub.f1] -        Flap 1 gap
[G.sub.f2] -        Flap 2 gap
[O.sub.f1] -        Flap 1 overlap
[O.sub.f2] -        Flap 2 overlap
[[delta].sub.f1] -  Flap 1 deflection
[[delta].sub.f2] -  Flap 2 deflection

VI. References

[1.] Katz, J., Race Car Aerodynamics: Designing for Speed (Robert Bentley, Incorporated, 1996).

[2.] Prandtl, L., "Uber Fltissigkeitsbewegung bei sehr kleiner Reibung," III International Mathematiker-Kongress, Heidelberg, 1904.

[3.] Smith, A.M.O., "High-Lift Aerodynamics," AIAA Journal of Aircraft 37th Wright Brothers Lecture 12(6), 1975.

[4.] Mason, W.H., "Configuration Aerodynamics," Virginia Tech, 2018.

[5.] Langtry, R.B. and Menter, F.R., "Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes," AIAA Journal 47(12):2894-2906, Dec. 2009.

[6.] Menter, F.R., "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications," AIAA Journal 32(8):1598-1605, Aug. 1994.

[7.] Narsipur, S., Pomeroy, B.W., and Selig, M.S., "CFD Analysis of Multielement Airfoils for Wind Turbines," 30th AIAA Applied Aerodynamics Conference 2012-2781, New Orleans, LA.

Abdelrahman Ibrahim Mahgoub, Hashim El-Zaabalawy, Walid Aboelsoud, and Mohamed Abdelaziz, Ain Shams University, Egypt


Received: 25 Jun 2018

Revised: 04 Oct 2018

Accepted: 14 Oct 2018

e-Available: 05 Dec 2018

TABLE 1 Meshing properties' summary.

Used meshing software  ANSYS WB meshing
Domain description     C-domain mesh with sphere of
Meshing method         Hybrid of tetrahedral mesh and
                       inflating prisms around the airfoils
Maximum skewness       0.51
Size functions         Proximity and curvature
Total mesh count       5E+5
Inflation method       First layer thickness calculated from
                       desired y+ value of 1
Free-stream velocity   14.7 m/s
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Author:Mahgoub, Abdelrahman Ibrahim; El-Zaabalawy, Hashim; Aboelsoud, Walid; Abdelaziz, Mohamed
Publication:SAE International Journal of Commercial Vehicles
Article Type:Report
Date:Mar 1, 2019
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