Gregor Seljak

April 8, 2008

1

Introduction

First racing cars were primarily designed to achieve high top speeds and the main goal was to minimize the air drag. But at high speeds, cars developed lift forces, which aﬀected their stability. In order to improve their stability and handling, engineers mounted inverted wings proﬁles1 generating negative lift. First such cars were Opel’s rocket powered RAK1 and RAK2 in 1928. However, in Formula, wings were not used for another 30 years. Racing in this era 1930’s to 1960’s occured on tracks where the maximum speed could be attained over signiﬁcant distance, so development aimed on reducing drag and potencial of downforce had not been discovered until the late 1960’s. But since then, Formula 1 has led the way in innovative methods of generating downforce within ever more restrictive regulations.

Figure 1: Opel’s rocket powered RAK2, with large side wings

2

Airfoils

Airfoil can be deﬁnead as a shape of wing, as seen in cross-section. In order to describe an airfoil, we must deﬁne the following terms(Figure 2) • The mean camber line is a line drawn midway between the upper and lower surfaces.

• The leading and trailing edge are the most forward an rearward of the mean camber line.

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Compared to an aircraft

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• The chord line is a line connecing leading an trailing edge. • The chord length is the distance from the leading to the trailing edge, measured along the chord line.

• The camber is the maximum distance between mean camber line and chord line.

• The thickness is the distance between the upper and lower surfaces.

Figure 2: Airfoil nomenclature

The amount of lift L produced by the airfoil, can be expressed in term of lift coeﬃcient CL

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2

(1)

L = ρ∞ V∞ SCL

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where V∞ denotes the freestrem velocity, ρ∞ ﬂuid density and S the airfoil area.

2.1

Flow over an airfoil

Properties of an airfoil can be measured in a wind tunnel, where constantchord wing spannes the entire test section, from one sidewall to the other. In this conditions, the ﬂow sees a wing without wing tips. Such wing is called inﬁnite wing and streches to inﬁnity along the span. Because the airfoil section is identical along the wing, the properties of the airfoil and the inﬁnite wing are identical. Therefore the ﬂow over an airfoil can be described as a 2D incompressible inviscid ﬂow over an inﬁnite wing. Lift per unit span L′ generated by an arbitrary airfoil(or any other body) moving at speed V∞ through the ﬂiud with density ρ∞ and circulation Γ is

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given by Kutta-Joukowsky theorem

L′ = ρ∞ V∞ Γ .

(2)

Circulation around an airfoil, can be calculated with the concept of a vortex sheet, which was ﬁrst introduced by Prandtl an his colleagues. Consider an airfoil of arbitrary shape and thickness as shown in Figure 3. Circulation can be distributed over the whole airfoil area with surface density(vortex sheet strength) dΓ/ds = γ (s), where γ (s) must satisfy Kutta condition γ (trailing edge) = 0

(3)

Entire circulation is then given by

Γ=

γ (s)ds ,

(4)

where the integral is taken around the complete surface of the airfoil. However, there is no general solution for γ (s) for an airfoil of arbitrary shape and it must be found numericaly, but analytical solutions can be found with some aproximations.

Figure 3: Simulation of an arbitrary airfoil by distributing a vortex sheet over the airfoil surface.

2.2

Thin airfoil theory

Here we discuss thin airfoil in freestream of velocity V∞ under small angle of attack α. Camber and thickness are small in relation with chord length c. In such case, airfoil can be described with a single vortex sheet distributed over the camber line(Figure 4). Our goal is to calculate the variation of γ (s), such that the chamber line becomes streamline and Kutta condition at trailing edge, γ (c) = 0, is satisﬁed.

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Figure 4: Thin airfoil approximation. Vortex sheet is distributed...