ME-438 Aerodynamics (week 9)

AerodynamicsME-438

Spring’16ME@DSU

Dr. Bilal A. Siddiqui

Vortex Filament: Basic Tool for Numerical Solution of Low Speed Airfoils• Recall the point vortex from previous lectures. • Imagine a straight line perpendicular to the page, going through point

O, and extending to infinity both ways. This is a straight vortex filament of strength .• Flows in planes perpendicular to the vortex filament at O and O’ are identical to each other and are identical to theflow induced by a point vortex of strength .• Point vortex is simply a section of a straight vortex filament

The Vortex Sheet• We introduced the concept of a source sheet, which is an infinite

number of infinitesimal line sources side by side.

• For vortex flow, consider an analogous situation.• Imagine an infinite number of straight vortex filaments side by side,

where the strength of each filament is infinitesimally small. These side-by-side vortex filaments form a vortex sheet• Let s be the distance measured along the vortex sheet in the edge• view. Define as the strength of the vortex sheet, per unit length along s.• Strength of an infinitesimal portion ds of the sheet is .

Vortex Sheet Cumulative Effect• Recall that for the point vortex and • Strength of an infinitesimal portion ds of the vortex sheet is .• For the vortex sheet, analogously•

• Therefore, for the entire sheet ab,

Velocity Jump across the Vortex Sheet• Consider the rectangular dashed path enclosing a section of the sheet

of length ds.• Velocity components tangential to the top and bottom of this

rectangular path are u1 and u2

• Velocity components tangential to the left/right sides are v1 and v2

• The circulation around the closed path is

• But since , this means • Let , such that the closed path collapses to the segment ds. • Thus,

Local jump in tangential velocity across the vortex sheet is equal to the local sheet strength.

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

Physical Relevance of Vortex Panel Method• In real life, there is a thin boundary layer on the surface, due to

friction between the surface and the airflow. • This boundary layer is a highly viscous region in which the large

velocity gradients produce substantial vorticity; that is, ∇ × V is finite within the boundary layer.• Hence, in real life, there is a distribution of vorticity along the airfoil

surface due to viscous effects, and our philosophy of replacing the airfoil surface with a vortex sheet can be thought of a way of modeling this effect in an inviscid flow.

A dilemma• Even if vortex panel method itself requires computers, for thin airfoils, we

can basically replace the airfoil with a single vortex sheet.

• For this case, Prandtl found closed form analytic solutions.• We are faced with a dilemma though.• There are infinite number of vortex strength distributions which can result in

a streamline flow around the airfoil.• We need another boundary condition to fix for a given airfoil at given angle

of attack.• Kutta to the rescue.

The Kutta Condition• The Kutta condition is a principle in steady flow fluid dynamics,

especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils.• It basically states that “A body with a sharp trailing edge which is

moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.”• In other words, since the streamlines on top and bottom surfaces

should be parallel, the trailing edge point ‘a’ will have two velocities in different directions, which is impossible. Therefore, Va=0.

Kutta Condition further explained• 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

What came before: circulation or lift?• The question is how did we get this circulation in the first place for

the Kutta condition and Kutta-Joukowski lift theorem to hold true?• The answer lies in finding what happened in the beginning of time.

The Starting Vortex• When flow is started, the flow tries to curl around the sharp

trailing edge from the bottom to the top surface.• This results in a large velocity around the corner which is not

sustainable. • As flow develops, the stagnation point on the upper surface

moves toward the trailing edge.• So, in fact some vorticity (circulation) is created at the

beginning of flow…but as we saw, this vorticity is quickly washed down the stream.• So how did circulation came to the airfoil?

Kelvin’s Circulation Theorem

• In an arbitrary curve C1, identify the fluid elements that are on this curve at a given instant in time t1, with circulation

• Now, let these fluid elements move downstream. • At t2 these same fluid elements will form another curve C2, around which circulation is • By conservation of momentum, • In other words, • This is the Kelvin Theorem that the circulation in a closed curve remains the same.• A stream surface which is a vortex sheet at some instant intime remains a vortex sheet for all times.

• The starting vortex is what imparts circulation to the airfoil initially, which creates lift for all times to come (till there is flow)