ME438 Aerodynamics is offered by Dr. Bilal Siddiqui to senior mechanical engineeing undergraduates at DHA Suffa University. This lecture set is an introduction to vortex lattice method (VLM) through the Kutta condition and circulation.

1. Aerodynamics
ME-438
Spring’16
ME@DSU
Dr. Bilal A. Siddiqui

2. 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 Γ.
• Flows in planes perpendicular to the vortex filament at O
and O’ are identical to each other and are identical to the
flow induced by a point vortex of strength Γ.
• Point vortex is simply a section of a straight vortex filament

3. 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 𝛾 = 𝛾(𝑠) as the strength of the vortex sheet, per unit
length along s.
• Strength of an infinitesimal portion ds of the sheet is 𝛾 𝑑𝑠.

4. Vortex Sheet Cumulative Effect
• Recall that for the point vortex V = 𝑉𝜃 = −
Γ
2𝜋𝑟
and 𝜙 = −
Γ
2𝜋
𝜃
• Strength of an infinitesimal portion ds of the vortex sheet is 𝛾 𝑑𝑠.
• For the vortex sheet, analogously
• 𝑑𝑉 = 𝑑𝑉𝜃 = −
𝛾𝑑𝑠
2𝜋𝑟
• 𝑑𝜙 = −
𝛾𝑑𝑠
2𝜋
𝜃
• Therefore, for the entire sheet ab,

6. 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
Γ = − 𝑣2 𝑑𝑛 − 𝑢1 𝑑𝑠 − 𝑣1 𝑑𝑛 + 𝑢2 𝑑𝑠 = 𝑢1 − 𝑢2 𝑑𝑠 + 𝑣1 − 𝑣2 𝑑𝑛
• But since Γ = 𝛾𝑑𝑠, this means 𝛾𝑑𝑠 = 𝑢1 − 𝑢2 𝑑𝑠 + 𝑣1 − 𝑣2 𝑑𝑛
• Let 𝑑𝑛 → 0, such that the closed path collapses to the segment ds.
• Thus,
𝜸 = 𝒖 𝟏 − 𝒖 𝟐Local jump in tangential velocity across the
vortex sheet is equal to the local sheet strength.

7. Philosophy of the Vortex Panel Method
• Replace the airfoil surface with a vortex sheet of strength 𝛾(𝑠)
• Find the distribution of 𝛾 along s such that the induced velocity field
from the vortex sheet when added to the uniform velocity 𝑉∞ will
make the vortex sheet (airfoil surface) a streamline of the flow.
• The circulation around the airfoil will be given by Γ = ∫ 𝛾𝑑𝑠
• The lift per unit span can then be calculates by the K-J theorem
𝐿′ = 𝜌∞ 𝑉∞Γ
Developed by Ludwig Prandtl during 1912–1922

8. 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.

9. 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 Γ for a given airfoil at
given angle of attack.
• Kutta to the rescue.

10. 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.

11. 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 Γ 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 𝛾 = 𝑢1 − 𝑢2, this means the Kutta condition is
𝜸 𝑻𝑬 = 𝟎

12. 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.

13. 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?

14. 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 Γ1 = − ∫𝐶1
𝑉. 𝑑𝑠
• Now, let these fluid elements move downstream.
• At t2 these same fluid elements will form another curve C2, around which circulation is
Γ2 = − ∫𝐶2
𝑉. 𝑑𝑠
• By conservation of momentum, Γ1 = Γ2
• In other words,
𝐷Γ
𝐷𝑡
=
𝜕Γ
𝜕𝑡
+
𝜕Γ
𝜕𝑥
+
𝜕Γ
𝜕𝑦
+
𝜕Γ
𝜕𝑧
= 0
• 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 in
time remains a vortex sheet for all times.

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