AerodynamicsME-438

Spring’16ME@DSU

Dr. Bilal A. Siddiqui

Philosophy of the Vortex Panel Method• Replace the airfoil surface with a vortex sheet of strength • Find the distribution of along s such that the induced velocity field

from the vortex sheet when added to the uniform velocity will make the vortex sheet (airfoil surface) a streamline of the flow.

• The circulation around the airfoil will be given by • The lift per unit span can then be calculates by the K-J theorem

Developed by Ludwig Prandtl during 1912–1922

Kutta Condition

• For a cusped trailing edge, the edge angle is almost zero, therefore we can have nonzero speed at the edge. However, since only one pressure can exist at ‘a’, V1=V2

• We can summarize the statement of the Kutta condition as follows:1. For a given airfoil at a given angle of attack, the value of around the airfoil is

such that the flow leaves the trailing edge smoothly.2. If the trailing-edge angle is finite, then the trailing edge is a stagnation point.3. If the trailing edge is cusped, then the velocities leaving the top and bottom

surfaces at the trailing edge are finite and equal in magnitude and direction.

• Since, we have , this means the Kutta condition is

Thin Airfoil Theory• Most airfoils have negligible thickness compared to the chord length

(less that 20%)• The camber is also rather small mostly (less than 5% of the chord)• This allows for some assumptions to be valid for many airfoils• More importantly, it allows exact analytic close formed solution for

the calculation of lift, drag and pitching moment of• “thin” airfoils ()• at low angles of attack ()• and low speeds (incompressible; )• in inviscid flow (irrotational)

Thin Airfoil Theory - 2

• For thin airfoils, we can basically replace the airfoil with a single vortex sheet. For this case, Prandtl found closed form analytic solutions.• Looking at the airfoil from far, one can neglect

the thickness and consider the airfoil as just the camber line. The airfoil camber is z(x).• If we neglect the camber also, we can

basically place all the vortices on the chord line for the same effect.

Thin Airfoil Theory - 3• For the vortex sheet placed on the camber line, • Normal velocity induced by the sheet is .• For the camber line to be a stream line • For another sheet placed on the chord line, • Normal velocity induced by the sheet is . For

small camber (x)• For the camber line to be a stream line • Kutta condition:

Thin Airfoil Theory-4• From geometry

• For small angles • Both camber and angle of attack

are small, so

Thin Airfoil Theory-5• Induced normal velocity at point x due to the vortex filament at

Total induced normal velocity at x is

Thin Airfoil Theory - 6• Therefore, the ‘no penetration’ (abstinence?) boundary condition is

• This is the fundamental equation of the thin airfoil theory.• For a given airfoil, both and are known.• We need to find which • makes the camber line a streamline of the flow• and satisfied the Kutta condition

Specializing Thin Airfoil Theory for Symmetric Airfoils• For a symmetric airfoil, the fundamental equation simplifies to

• Let us transform the variable to another variable

For any particular value of , there is a corresponding particular [ and ]

• The fundamental equation can then be written equivalently as

Thin Airfoil Theory for Symmetric Airfoils-2• The solution to this integral equation can be shown to be

• It can be easily shown to satisfy the fundamental equation as well as the Kutta condition .• Total circulation is found by • This can be evaluation as

Thin Airfoil Theory for Symmetric Airfoils-3• The lift can now be calculated by the K-J theorem

Thus,

These is the fundamental result of thin airfoil theory. The lift curve slope is 0.11 per degree angle of attack! It fits well with experiment.

Thin Airfoil Theory for Symmetric Airfoils-4• For calculating moment about leading edge, the incremental lift due

to circulation caused by a small portion of the vortex sheet is multiplied by its moments arm .• The total moment is the integration of these small moments

Therefore the moment coefficient is

Thin Airfoil Theory for Symmetric Airfoils-5• Shifting this moment to the quarter chord point

• Now recall that:• Center of pressure is point on the chord about which there is zero moment• Aerodynamic center is the point on the chord about which the moment about

the chord does not change with the angle of attack.

• This means the quarter chord point on a symmetric airfoil is both the center of pressure and the aerodynamic center! • This is only true for thin, symmetric airfoils at low speeds!• This too holds up very well in experiments.

Experimental Validation of Thin Airfoil Theory For Symmetric Airfoils

An exampleConsider a thin flat plate at 5 deg. angle of attack. Calculate

• lift coefficient• moment coefficient about the leading edge• moment coefficient about the quarter-chord point• moment coefficient about the trailing edge.