Airfoil

Airfoil terminology

The various terms related to airfoils are defined below:

• The suction surface (typically upper surface) is generally associated with higher velocity and lower static pressure.
• The pressure surface (typically lower surface) has a comparatively higher static pressure than the suction surface. The pressure gradient between these two surfaces contributes to the lift force generated for a given airfoil.

The geometry of the airfoil is described with a variety of terms :

• The leading edge is the point at the front of the airfoil that has maximum curvature (minimum radius).
• The trailing edge is the point on the airfoil most remote from the leading edge. The angle between the upper and lower surfaces at the trailing edge is the trailing edge angle.
• The chord line is the straight line connecting leading and trailing edges. The chord length, or simply chord, ${\displaystyle c}$, is the length of the chord line. That is the reference dimension of the airfoil section.

The shape of the airfoil is defined using the following geometrical parameters:

• The mean camber line or mean line is the locus of points midway between the upper and lower surfaces. Its shape depends on the thickness distribution along the chord;
• The thickness of an airfoil varies along the chord. It may be measured in either of two ways:
  • Thickness measured perpendicular to the camber line. This is sometimes described as the "American convention";
  • Thickness measured perpendicular to the chord line. This is sometimes described as the "British convention".

Some important parameters to describe an airfoil's shape are its camber and its thickness. For example, an airfoil of the NACA 4-digit series such as the NACA 2415 (to be read as 2 – 4 – 15) describes an airfoil with a camber of 0.02 chord located at 0.40 chord, with 0.15 chord of maximum thickness.

Finally, important concepts used to describe the airfoil's behaviour when moving through a fluid are:

• The aerodynamic center, which is the chord-wise location about which the pitching moment is independent of the lift coefficient and the angle of attack.
• The center of pressure, which is the chord-wise location about which the pitching moment is momentarily zero. On a cambered airfoil, the center of pressure is not a fixed location as it moves in response to changes in angle of attack and lift coefficient.

In two-dimensional flow around a uniform wing of infinite span, the slope of the lift curve is determined primarily by the trailing edge angle. The slope is greatest if the angle is zero; and decreases as the angle increases. For a wing of finite span, the aspect ratio of the wing also significantly influences the slope of the curve. As aspect ratio decreases, the slope also decreases.

Thin airfoil theory

Thin airfoil theory is a simple theory of airfoils that relates angle of attack to lift for incompressible, inviscid flows. It was devised by German mathematician Max Munk and further refined by British aerodynamicist Hermann Glauert and others in the 1920s. The theory idealizes the flow around an airfoil as two-dimensional flow around a thin airfoil. It can be imagined as addressing an airfoil of zero thickness and infinite wingspan.

Thin airfoil theory was particularly notable in its day because it provided a sound theoretical basis for the following important properties of airfoils in two-dimensional inviscid flow:

1. on a symmetric airfoil, the center of pressure and aerodynamic center are coincident and lie exactly one quarter of the chord behind the leading edge.
2. on a cambered airfoil, the aerodynamic center lies exactly one quarter of the chord behind the leading edge, but the position of the center of pressure moves when the angle of attack changes.
3. the slope of the lift coefficient versus angle of attack line is ${\displaystyle 2\pi }$ units per radian.

As a consequence of (3), the section lift coefficient of a thin symmetric airfoil of infinite wingspan is:

${\displaystyle c_l=2\pi \alpha }$

where ${\displaystyle c_l}$ is the section lift coefficient,

${\displaystyle \alpha }$ is the angle of attack in radians, measured relative to the chord line.

(The above expression is also applicable to a cambered airfoil where ${\displaystyle \alpha }$ is the angle of attack measured relative to the zero-lift line instead of the chord line.)

Also as a consequence of (3), the section lift coefficient of a cambered airfoil of infinite wingspan is:

${\displaystyle c_l=c_{l_0}+2\pi \alpha }$

where ${\displaystyle c_{l_0}}$ is the section lift coefficient when the angle of attack is zero.

Thin airfoil theory assumes the air is an inviscid fluid so does not account for the stall of the airfoil, which usually occurs at an angle of attack between 10° and 15° for typical airfoils. In the mid-late 2000s, however, a theory predicting the onset of leading-edge stall was proposed by Wallace J. Morris II in his doctoral thesis. Morris's subsequent refinements contain the details on the current state of theoretical knowledge on the leading-edge stall phenomenon. Morris's theory predicts the critical angle of attack for leading-edge stall onset as the condition at which a global separation zone is predicted in the solution for the inner flow. Morris's theory demonstrates that a subsonic flow about a thin airfoil can be described in terms of an outer region, around most of the airfoil chord, and an inner region, around the nose, that asymptotically match each other. As the flow in the outer region is dominated by classical thin airfoil theory, Morris's equations exhibit many components of thin airfoil theory.

Derivation

In thin airfoil theory, the width of the (2D) airfoil is assumed negligible, and the airfoil itself replaced with a 1D blade along its camber line, oriented at the angle of attack α. Let the position along the blade be x, ranging from 0 at the wing's front to c at the trailing edge; the camber of the airfoil, dy⁄dx, is assumed sufficiently small that one need not distinguish between x and position relative to the fuselage.

The flow across the airfoil generates a circulation around the blade, which can be modeled as a vortex sheet of position-varying strength γ(x). The Kutta condition implies that γ(c)=0, but the strength is singular at the bladefront, with γ(x)∝1⁄√x for x ≈ 0. If the main flow V has density ρ, then the Kutta–Joukowski theorem gives that the total lift force F is proportional to $${\displaystyle \rho V{\int }_0^c\gamma (x)dx}$$and its moment M about the leading edge proportional to $${\displaystyle \rho V{\int }_0^{c}x\gamma (x)dx.}$$

From the Biot–Savart law, the vorticity γ(x) produces a flow field $${\displaystyle w(x)=\frac{1}{2\pi }{\int }_0^c\frac{\gamma (x^{\prime })}{x-x^{\prime }}d{x}^{\prime }\text{,}}$$ oriented normal to the airfoil at x. Since the airfoil is an impermeable surface, the flow ${\displaystyle w(x)}$ must balance an inverse flow from V. By the small-angle approximation, V is inclined at angle α-dy⁄dx relative to the blade at position x, and the normal component is correspondingly (α-dy⁄dx)V. Thus, γ(x) must satisfy the convolution equation $${\displaystyle \left(\alpha -\frac{dy}{dx}\right)V=-w(x)=-\frac{1}{2\pi }{\int }_0^c\frac{\gamma (x^{\prime })}{x-x^{\prime }}d{x}^{\prime }\text{,}}$$ which uniquely determines it in terms of known quantities.

An explicit solution can be obtained through first the change of variables$${\displaystyle x=c\cdot \frac{1+\cos (\theta )}{2},}$$and then expanding both dy⁄dx and γ(x) as a nondimensionalized Fourier series in θ with a modified lead term:$${\displaystyle \begin{array}{cc}& \frac{dy}{dx}=A_0+A_1\cos (\theta )+A_2\cos (2\theta )+\dots \\ & \gamma (x)=2(\alpha +A_0)\left(\frac{\sin \theta }{1+\cos \theta }\right)+2A_1\sin (\theta )+2A_2\sin (2\theta )+\dots \text{.}\end{array}}$$ The resulting lift and moment depend on only the first few terms of this series.

The lift coefficient satisfies $${\displaystyle C_L=2\pi \left(\alpha +A_0+\frac{A_1}{2}\right)=2\pi \alpha +2{\int }_0^{\pi }\frac{dy}{dx}\cdot (1+\cos \theta )d\theta }$$ and the moment coefficient $${\displaystyle C_M=-\frac{\pi }{2}\left(\alpha +A_0+A_1-\frac{A_2}{2}\right)=-\frac{\pi }{2}\alpha -{\int }_0^{\pi }\frac{dy}{dx}\cdot \cos (\theta )(1+\cos \theta )d\theta \text{.}}$$ The moment about the 1/4 chord point will thus be $${\displaystyle C_M(1/4c)=-\pi /4(A_1-A_2)\text{.}}$$ From this it follows that the center of pressure is aft of the 'quarter-chord' point 0.25c, by $${\displaystyle \mathrm{\Delta }x/c=\pi /4((A_1-A_2)/C_L)\text{.}}$$ The aerodynamic center is the position at which the pitching moment M′ does not vary with a change in lift coefficient: $${\displaystyle \frac{\mathrm{\partial }(C_{M^{\prime }})}{\mathrm{\partial }(C_L)}=0\text{.}}$$ Thin-airfoil theory shows that, in two-dimensional inviscid flow, the aerodynamic center is at the quarter-chord position.