The document provides a detailed overview of aerodynamic principles, focusing on vector calculus and fluid mechanics for analyzing potential flows. It discusses the Laplace equation, boundary conditions in aerodynamics, and various elementary potential flows, including uniform flow, source/sink flow, and vortex flow. Additionally, it introduces the source-panel method for computational fluid dynamics (CFD) to determine flow around arbitrary shapes, particularly in non-lifting scenarios.

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

2. Review of Vector Calculus
• Gradient of a scalar is a vector
• Curl of a vector is a vector
• Divergence of a vector is a scalar

3. Review of Vector Calculus
• Let A be a vector and p be a scalar. The line, surface and volume
integrals are related by

4. Review of Fluid Mechanics

5. Review of Fluid Mechanics

6. Review of Fluid Mechanics

7. Review of Fluid Mechanics

8. Review of Fluid Mechanics

9. Review of Fluid Mechanics
Note: We have made no assumption
of inviscidity or compressibility

10. Review of Fluid Mechanics Note: We have assumed inviscid flow

11. Incompressible and Inviscid Flows
• Some very important flows can be solved by neglecting
compressibility (𝑀∞ ≤ 0.3) and viscosity (away from body)
• For incompressibility, the condition (derived from continuity
equation) is
• For inviscid flows (aka potential or irrigational flows), the condition is

12. Laplace Equation for Potential Flows
• The equation 𝛻2 𝜙 = 0 is called the Laplace equation.
• It is one of the most famous and extensively equations in math/physics
• It has well known solutions, therefore it is easier to solve potential
(incompressible inviscid) flows analytically
• Since stream functions and potential functions are cousins, we can show
for 2D flows that

13. Notes on the Laplace Equations
• Any irrotational, incompressible flow has a velocity potential and stream
function (for 2D flow) that both satisfy Laplace’s equation.
• Conversely, any solution of Laplace’s equation represents the velocity
potential or stream function (2D) for an irrotational, incompressible flow.
• Laplace’s equation is a second-order linear partial differential equation.
• The fact that it is linear is important, because the sum of any particular
solutions of a linear differential equation is also a solution of the equation!
• Since irrotational, incompressible flow is governed by Laplace’s equation
and Laplace’s equation is linear, we conclude that a complicated flow
pattern for an irrotational, incompressible flow can be synthesized by
adding together a number of elementary flows that are also irrotational
and incompressible.

14. How to solve the Laplace Equations
• Our strategy is to develop flow solutions for several different
elementary flows, which by themselves may not seem to be practical
flows in real life.
• However, we then proceed to add (i.e., superimpose) these
elementary flows in different ways such that the resulting flow fields
do pertain to practical problems.
• All of these flows have the same governing equation, i.e. 𝛻2 𝜙 = 0
• How, then, do we obtain different flows for the different bodies? The
answer is found in the boundary conditions.

15. Boundary Conditions in Aerodynamics
• There are two boundary conditions in all external flows
• Infinity boundary conditions
Far away from the body (toward infinity), in all directions, the
flow approaches the uniform freestream conditions.
• Wall boundary conditions
It is impossible for the flow to penetrate the body surface.

16. Elementary Potential Flow 1: Uniform Flow
• It is clear that
• Integrating these equations
• In polar coordinates
• The flow is obviously irrotational, therefore circulation Γ = 0

17. Elementary Potential Flow 2: Source/Sink Flow
The source strength is defines as
The potentials, streams and velocities can be
calculated as
Source/Sink flow obeys mass
conservation 𝛻. 𝑉 = 0
everywhere except at origin
since 𝑉𝑟 → ∞. But we accept
it as a ‘singularity’

18. Combining Uniform Flow with Source/Sink:
Rankine Halfbody

19. Source and Sink Pair in a Uniform Flow:
Rankine Oval

20. Elementary Potential Flow 3: Doublet
• When source and sink of equal strength are placed at the same point
• 𝜅 is called the doublet strength
• The streamlines are families of circles with diameter
Where C is some constant

21. Non-lifting Flow over a cylinder

22. A bit more on the flow over Cylinder
• We know that pressure coefficient is
• On the surface of the cylinder, we can
show that
• Therefore,
• Cp varies from 1.0 at the stagnation
points to −3.0 at the top/bottom.

23. Elementary Flow 4: Vortex Flow
• All the streamlines are concentric circles about a given point.
• Velocity along any given circular streamline is constant, but vary from
one streamline to another inversely with distance from the center.
• Vortex flow is incompressible 𝛻. 𝑉 = 0 everywhere.
• Vortex flow is irrotational everywhere 𝛻 × 𝑉 = 0 except at origin, but
we except it as a singularity (exception).
• To evaluate the constant, take the circulation around a given circular
streamline of radius r
Therefore, for vortex flow, the
circulation taken about all
streamlines is the same value,
namely, Γ= −2πC.

24. Summary of Elementary Flows

25. Lifting Flow over Cylinder
• If the cylinder is spinning, it will produce finite (measurable) lift.
• We model this as the superimposition of uniform flow+doublet+vortex

28. Lift due to Spinning of Cylinder
• At the surface of the cylinder (r=R), 𝑉𝑟 = 0 and 𝑉𝜃 = −2𝑉∞ sin 𝜃 −
Γ
2𝜋𝑅
• Since, 𝐶 𝑝 = 1 −
𝑉
𝑉∞
2
= 1 − 4 sin2
𝜃 +
2Γ sin 𝜃
𝜋𝑅𝑉∞
+
Γ
2𝜋𝑅𝑉∞
2
• The drag coefficient is given by
• The lift force can be found by

29. The Kutta-Joukowski Theorem
• Lift per unit span is directly proportional to circulation.
• This is a powerful relation in theoretical aerodynamics called the
Kutta-Joukowski theorem, obtained in ~ 1905.
• Rapidly spinning cylinder can produce a much higher lift than an
airplane wing of same planform area
• However, the drag on the cylinder is also much higher than a well-
designed wing. Hence, no rotating cylinders on aircrafts.
• Although 𝐿′ = 𝜌∞ 𝑉∞Γ was derived for a circular cylinder, it applies in
general to bodies of arbitrary cross section.

30. K-J Theorem and the Generation of Lift
• Let curve A be any curve in the flow enclosing the airfoil.
• If airfoil is producing lift, the velocity field around the airfoil will be such
that the line integral of velocity around A will be finite
• The integral of velocity around any curve not including the body is zero
• Hence, the lift produced by the airfoil is given by 𝑳′ = 𝝆∞ 𝑽∞ 𝚪
• The Kutta-Joukowski theorem states that lift per unit span on a two-
dimensional body is directly proportional to the circulation around the
body.
• Just like we synthesized flow over a spinning cylinder by adding a vortex to
the non-lifting flow, we can synthesize flow over an airfoil by distributing
vortices all over and inside the airfoil.
Note: Lift
produces
circulation, not
the other way
round. Think why?

31. Numerical Source Panel Method: Our first
flavor of CFD
• We added elementary flows in certain ways and discovered that resulting
streamlines turned out to fit certain body shapes.
• But it is not practical to randomly add elementary flows and try to match
the body shape we seek to find the flow around!
• We want to specify the shape of an arbitrary body and solve for the
distribution of singularities which, in combination with a uniform stream,
produce the flow over the given body.
• For the moment, we will concentrate on non-lifting flows….for a reason.
• This technique is called the source-panel method, a standard tool in
aerodynamic industry since the 1960s.
• Numerical solution of potential flows by both source and vortex panel
techniques has revolutionized the analysis of low-speed flows. We will
consider vortex methods later for lifting flows.

32. Recall: Source/Sink Flow
The source strength is defines as
The potentials, streams and velocities can be
calculated as
Now, instead of having a
single source, we want to
have a number of sources
placed side by side along a
contour: “Source Sheet”

33. The Source Sheet
• Define λ = λ(s) to be source strength per unit length
along s. For infinitesimal sources, ds acts like a
regular line source of strength λds and depth l
• Recall that the strength Λ of a single line source
was defined as the volume flow rate per unit depth
in the z direction.
• Typical units for Λ are square meters per second,
but for λ are meters per second.
λ may be negative, so it is
really a combination of
sources and sinks

34. SPM- Derivation
• Consider a point P at distance r from portion ds of the source sheet
• This portion of strength 𝜆𝑑𝑠 produces an infinitesimally potential 𝑑𝜙
𝑑𝜙 =
𝜆𝑑𝑠
2𝜋
ln 𝑟
• Complete velocity potential at point P, induced by the entire source
sheet from a to can be found as 𝜙 = 𝑎
𝑏 𝜆 ln 𝑟
2𝜋
𝑑𝑠
• We can now wrap the entire body with this source sheet and
superimpose uniform flow.

35. SPM-Discretizing the Continuous Equations
• Approximate the source sheet by n straight panels j=1,2,…,n
• Let source strength per panel 𝜆𝑗 be constant.
• The panel strengths 𝜆1, 𝜆2, … , 𝜆𝑗, … , 𝜆 𝑛 are unknown
• We want to iterate till the body surface becomes a streamline of the
flow i.e. 𝑉𝑟 = 0 at the surface.
• ↑THIS is the boundary condition we apply at each control point (i.e.
the middle point of each panel).
• Let us now put this in numbers.

36. SPM-Numerical Technique
The integration is because for a given panel ‘r’
varies as we move across the panel

37. The Numerical Recipe (1)
• Since point P is just an arbitrary point, but we are more interested in
what is happening at the surface, let’s move P to the body surface.
• Then the effect of each source panel on a given panel is
• Slope of ith panel is
𝑑𝑦
𝑑𝑥 𝑖
, but its angle with the flow is 𝛽𝑖
• The normal component of free stream flow to each panel is
𝑉𝑖 𝑛∞
= 𝑉∞ cos 𝛽𝑖

38. The Numerical Recipe (2)
• The normal component of velocity induced at (xi , yi ) by the source
panels is
𝑉𝑖 𝑛 𝑠𝑜𝑢𝑟𝑐𝑒𝑠
=
𝜕
𝜕𝑛𝑖
𝜙(𝑥𝑖, 𝑦𝑖) =
1
2𝜋
𝑗=1
𝑛
𝜆𝑗
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑛𝑖
𝑑𝑠𝑗
• We can show that this can be simply expanded as
𝑉𝑖 𝑛 𝑠𝑜𝑢𝑟𝑐𝑒𝑠
=
𝜆𝑖
2
+
1
2𝜋
𝑗=1
𝑗≠𝑖
𝑛
𝜆𝑗
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑛𝑖
𝑑𝑠𝑗

39. The Numerical Recipe (3)
• The velocity normal to the ith panel at (xi , yi ) is
𝑉𝑖 𝑛
= 𝑉∞ cos 𝛽𝑖 +
𝜆𝑖
2
+
1
2𝜋
𝑗=1
𝑗≠𝑖
𝑛
𝜆𝑗
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑛𝑖
𝑑𝑠𝑗
• Therefore, we seek to impose the distribution of sources which gives
us zero normal velocity at each node (i.e. body becomes a streamline)
𝑉∞ cos 𝛽𝑖 +
𝜆𝑖
2
+
1
2𝜋
𝑗=1
𝑗≠𝑖
𝑛
𝜆𝑗
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑛𝑖
𝑑𝑠𝑗 = 0, ∀𝑖 = 1, … , 𝑛
These are n linear equations with n unknowns (𝝀 𝟏, 𝝀 𝟐, … , 𝝀 𝒏)

40. The Numerical Recipe (3)
• Finally, once we find the source distribution (𝜆𝑖, 𝜆2, … , 𝜆 𝑛), we can find the
actual velocity tangent to the surface
𝑉𝑖 = 𝑉∞ sin 𝛽𝑖 +
1
2𝜋
𝑗=1
𝑗≠𝑖
𝑛
𝜆𝑗
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑠𝑖
𝑑𝑠𝑗 = 0, ∀𝑖 = 1, … , 𝑛
• We can then find the pressure coefficient at each node
𝐶 𝑝,𝑖 = 1 −
𝑉𝑖
𝑉∞
2
• The pressure forces can then be easily resolved in lift drag and moment!

41. Source Panel Method Applied to a Cylinder
• Let us begin by dividing the
cylinder into 8 panels.
• Coordinates of the ith panel’s
control point are (xi,yi), and the
edge points of the panel are
(Xi,Yi) and (Xi+1,Yi+1).
• Let 𝑉∞ be in the x-direction.
• Let us first evaluate the integrals
𝐼𝑖𝑗 =
𝑗
𝜕 ln 𝑟𝑖𝑗
𝜕𝑛𝑖
𝑑𝑠𝑗

42. SPM applied to the cylinder (2)
• It is easier to
deal with flow
angles with
tangent to the
surface rather
than normal
to it

43. SPM applied to the cylinder (3)

44. The SPM applied to the cylinder (4)
• To obtain the actual velocity over the cylinder
• The velocity due to upstream flow is 𝑉∞ 𝑠,𝑖
= 𝑉∞ sin 𝛽𝑖
• Velocity induced by sources
• Therefore the total velocity is
• And coefficient of pressure is

45. SPM applied to Cylinder (5)
See Matlab code developed in class