This document summarizes the Hess-Smith panel method for analyzing aerodynamic forces on airfoils. It begins with background on aerodynamics and the different types of air flow. It then describes the 2D Hess-Smith panel method, which involves discretizing an airfoil shape into panels and calculating source strengths to model the air flow. The document provides the theoretical equations for calculating velocity potential and solving for source strengths. It concludes by explaining the Python code used to implement the 2D source panel method on a NACA 0010 airfoil.

1. The Hess Smith Panel Method
Lindsey Marinello
Ilisha Ramachandran
May 15, 2015
1 Background
Aerodynamics is the study of air flow or more general gas dynamics. Principally, the study of aerody-
namics is used to understand interactions between air and solid bodies, and furthermore, to determine the
resultant forces of these interactions. Such analysis holds many applications. Most commonly, it is used
to understand flight and aids the development of higher-efficiency aircraft and spacecraft, but it also has
applications for architectural designers and engineers in the automotive industry, among others.
For flight applications of aerodynamics, there are two main types of flow to be considered: turbulent flow
and laminar flow. Laminar flow describes a smooth streamlines of flow over a body, whereas turbulent flow,
as suggested by its name, is “rougher,” and is caused by disturbances due to the air viscosity and frictional
forces between the air particles and the solid body. These disturbances result in cyclical vortices of air.
Further, there are four main forces to be considered with regarding the flight of an object: backwards drag,
forward thrust, upwards lift, and the downwards weight of the aircraft.
In order to attain flight, the upwards lift force must overcome the downwards force of the weight of
the craft. The downwards forces responsible for flight are a combination of Newtonian forces and pressure
forces. For a wing tilted at some angle of attack, as a downward-directed stream of airflow leaves the
trailing edge, an opposing force pushes back upward on the airfoil; this is the Newtonian contribution to
the lift. Additionally, at this angle, air must travel more quickly over the upper edge of the airfoil than
the lower edge due to the resulting asymmetry in the upper and lower boundary surfaces of the airfoil. By
Bernoulli’s principle, higher-velocity air above the wing has lower pressure than the air below, resulting in
another upward force due to pressure. Throughout this essay, wings will be represented by their airfoil or
cross-section.1
Wing airfoils and the forces discussed are shown in the figures below.
2 The 2D Hess-Smith Panel Method
Our project explores the first simple and practical form of aerodynamic analysis ever developed – a
method developed by Hess and Smith of Douglass Aircraft Company in 1966.2
Panel methods can be used
to determine the lift, drag, pressure, and thrust forces of a given geometry based on the movement of flow
over the object during flight. Such methods are powerful because of their simplicity, generality, and relatively
1”The Angle of Attack for an Airfoil.” HyperPhysics. Georgia State University, n.d. Web. 12 May 2015.
2Alonso, Juan. ”Lecture 5: Hess-Smith Panel Method.” AA210b - Fundamentals of Compressible Flow II. Stanford Univer-
sity, n.d. Web. 12 May 2015
1

2. Figure 1:
Figure 2:
fast computation speed, which makes them effective tools for concept design analysis during the engineering
process.
To implement the panel method, a selected geometry must first be simplified by discretization into panels.
For a 3-D object, a smooth model of an aircraft must be turned into a polyhedron whose surface is broken up
into flat, 2-D panels. Then, through computation, a constant value of a desired characteristic is calculated
for each panel area based on its orientation in an airflow. A 2-D panel method, by comparison, can be
performed by analyzing the cross-section of the body. Starting with a curvature that represents the outline
of the cross-section of a simple body, this involves converting the shape into a polygon with straight-line
panels. For simple aerodynamic geometric shapes such as wings, which do not vary widely in their geometry
about their length, this is a reasonable approximation. When applied in two dimensions, panel methods
are commonly used to compare the predicted aerodynamic properties of different airfoil shapes at varying
angular orientations relative to the direction of travel.3
Panel methods ignore the effects of viscosity and assume that air is incompressible. Therefore, their
predictions cannot capture the development of boundary layers, turbulent flow, wake velocities, and other
conditions. Instead, the streamlines will appear laminar over the entire surface. Therefore, it is a poor
model for solid bodies traveling past subsonic speeds, where these effects are significant. However, the most
common applications of aerodynamics involve streamlined bodies moving at subsonic speeds, whose designs
are made to maximize laminar flow over most of their surfaces. Hence, despite its limitations, the panel
method is still a useful tool due to its computational simplicity.4
To limit the complexity of our project and the computation required by the Jupyter server, our project
analyzes and demonstrates a 2-D panel method. To simplify further, we implement the Source Panel Method,
which assumes that flow is irrotational, in addition to being frictionless and incompressible. Because lift
depends on circulation, we will not be able to calculate lift generated by this method. However, we can
3Smith, Richard. ”Why Use a Panel Method?” Symscape. N.p., Mar. 2007. Web. 12 May 2015
4Ibid.
2

3. calculate the velocity potential over our airfoil and, from the potential, calculate and plot a streamline
around the body that is largely accurate.
Due to the steep learning curve of this method, we have analyzed and demonstrated Python code written
by Dr. Lorena A. Barba, from her online course modules entitled AeroPython. Rather than generate our
own program, we added additional comments to her program code or made minor changes when appropriate.
3 The 2-D Source Panel Method: Theoretical Analysis
Figure 3:
Figure 2, above, demonstrates the basic geometry of the problem. Visualized on the Cartesian coordinate
system is symmetric airfoil with its chord line placed on the x axis. The equation for the velocity potential
of an irrotational, incompressible, and inviscid (i.e. without viscosity) flow is the integral of the velocity
gradient:
φT = φ∞ + φS + φV (1)
φ∞ is the free-stream velocity, representing a potential due to the velocity of an oncoming stream of wind
at angle α in the x direction over the airfoil curvature.
φ∞ = U∞x sin α (2)
φS is the source potential due to sources distributed over the upper and lower airfoil surfaces. Because
the radial velocity from a source is equal to vs = σs
2πr where σs is a constant representing source strength,
when integrated into our potential equation and applied over the curve:
φs =
σ(s)
2πr
· ln(r) · ds (3)
Assuming that flow is irrotational, × φ = 0 so φv = 0. Thus,
φT = U∞(x · cos α) +
σ(s)
2π
· ln(r) · ds (4)
3

4. To solve this equation for a physically realistic scenario, we make use of a flow-tangency boundary
condition.” This corresponds to the idea that a stream, upon meeting the airfoil, should divide and flow over
the surface of the airfoil, rather than penetrating it. Mathematically, this means normal velocity of the flow
along the airfoil surface equal to zero.
un =
∂
∂n
φ(x, y) = 0 (5)
Using this boundary condition, we can solve for the source strengths σ that are sufficient to create a
divided stream around the airfoil 5
After discretizing the airfoil, we are left with a polygon of j flat panels with i midpoints or control
points represented by (xci, yci). The Hess-Smith method approximates the total source velocity potential
by assuming there is a constant source strength over each panel. Sources are placed at each control point,
and the total source contribution to the potential over each panel is computed using a line integral which
depends on each panel point’s distance from the control point as well as the angle βi between the control
point normal and the x axis.This allows us to change our continuous equation to a discrete one:
0 = U∞ cos βi +
σi
2
+
Np
j=1,j=i
σj
2π
(xci
− xj(sj)) cos βi + (yci
− yj(sj)) sin βi
(xci
− xj(s))
2
+ (yci
− yj(s))
2 dsj (6)
The extra σi
2 term is a condition that provides the source strength at the first control point, the leading
edge of the airfoil. Giving this point a source strength of σi
2 , we prevent the the free-stream velocity near
this point from penetrating the airfoil surface, and allow the stream to divide around it.
This gives us 0 = [U∞] + [A][b] where
Aij =



1
2 , if i = j
1
2π
(xci
−xj (sj ))cos βi+(yci
−yj (sj ))sin βi
(xci
−xj (s))
2
+(yci
−yj (s))
2 dsj , if i = j
(7)
We can rewrite the system as follows:
[A][σ] = [b] (8)
The system can be solved for the appropriate source strengths so that we can find the total potential,
φT . Then we can solve for the streamline functions and plot them. Streamline functions, Φ(u, v) are defined
for incompressible flow as:
u (x, y) =
∂
∂x
{φ (x, y)} (9)
v (x, y) =
∂
∂y
{φ (x, y)} (10)
5Barba, Lorena A. “Lesson 11: Vortex Source Panel Method.” AeroPython. George Washington University, Apr. 2015.
Web. 6 May 2015
4

5. When plotted, streamlines represent lines of constant velocity, indicating flow lines that have been dis-
placed around the airfoil. 6
4 Code Explanation
First, we allow our program import any NACA airfoil geometry from the Airfoil Tools database. For
our project, we have chosen to analyze NACA 0010 airfoil. The four digits of the NACA code correspond
to various ratios between the size of the camber, thickness, and chord line of the airfoil. The loadtxt()
function, as implemented, uses a known delimiter value to read and separate the data file into arrays of x
and y coordinates which, when plotted, outline the airfoil.
Once we have our airfoil data separated into arrays of its Cartesian coordinates, we must discretize
the shape. This process is simplified using object-oriented programming. The class for the panels has
six methods - xa, xb, ya, yb, xc, and yc – which correspond to start and end-points of each panel and the
midpoints. These midpoints, xc and yc, are referred to as “control points,” and are the points at which the
flow tangency boundary condition is applied.The orientation of these panels is then calculated in relation
to βi. These panels have the value of source strength, tangential velocity, and the pressure co-efficient built
in. Initially, these values are set to 0, but they will be appended later on. For this project, we will only be
calculating the source strenght and tangential velocity.
Now that we have an easy way to describe the panels, we must generate them using the define panels
function, which has three parameters – x and y, which are the coordinates of the airfoil geometry, and a
quantity N of panels. This function calculates the x.ends[i], and y.ends[i], x.ends[i+1], and y.ends[i+1] which
are the starting and ending points of each panel. Then, it returns an instance of the panel class that is defined
by these start and end points. In order to do so, the function must first define the start and endpoints of
each panel. Rather than make each panel the same length, this function makes a better approximation by
creating smaller panels near the leading and trailing edges of the airfoil, which have much higher curvature
than the body, by using a circle. The function defines a radius R equal to half the chord of our airfoil, which
is obtained by subtracting x.min from x.max and halving the distance. A circle centrepoint is also defined
by averaging the these values. Then, a linspace is created such that it stores N +1 equally-spaced values of
the R cos(θ) value from 0 to 2π about the center of the circle defined. These x values are copied to become
the x-values of the panel’s x-endpoints. The next step uses a loop to search within the airfoil geometry for
two consecutive points, (x[I], y[I]) and (x[I + 1], y[I + 1]), within that are spaced on an interval containing
the x.ends[i] calculated from the circle. With two consecutive points, the y.ends values can be calculated by
linear interpolation.
We can apply this code for the panels of a circle to our airfoil shape. This can be done by translating
each of the points on the circle (endpoints and center points) onto our shape.For accuracy, the more curved
parts of the airfoil should have smaller panels. This can be achieved by forming a circle around the airfoil,
with the two ends of the airfoil touching the circle. The resulting non-uniform distribution is used to store
the x-coordinates of the circle, which correspond to the x-points on the airfoil. The define panels function
returns an array of objects at particular instances, containing information about the panels.
The code described above gives us almost all of the information we need about the airfoil, besides the
source strength, tangential velocity, and the pressure co-efficient. These can only be calculated once the
airfoil encounters a stream of air. The next part of the code creates the freestream. The freestream has a
uniform flow velocity of U∞ and an angle of attack α = 0. We set the freestream conditions by creating a
class and passing in the U∞ and α as inputs. We use equation (6) to solve for the boundary conditions. By
6Ibid.
5

6. imposing these conditions, we can enforce the flow tangency condition on each control point of our pabel.
We input this equation using numpy’s integrate function.
Now that we have the freestream conditions set up, we can calculate the values for the source strength
on each panel. The source strength is calculated by solving the linear system of equations:
[A][σ] = [b]
In the code, this is done by defining two functions, the build matrix and build rhs functions. The
build matrix function builds an N x N matrix, fills the diagonals with 1
2 , which accounts for the source
strength of the ith control point and its effect on each panel, seen in the beginning of equation (7), then
inputs the second half of the equation. The build rhs function creates a vector of free-stream values for
each panel, which are calculated from the alpha and the airfoil geometry, which is known. The equation
[A][σ] = [b] is then solved using numpy’s linalg.solve function to give us the values for source strength.
Finally, we plot the streamlines around our airfoil. We were able to solve the equation for the potential
once we found the source strength on each of our panels. We then find the x and y components of the velocity
by taking the derivative of the velocity potential. The function get velocity field solves these equations
and outputs the velocity field. This can be plotted onto a mesh grid, which is a feature of numpy.
5 Extensions of the Source Panel Method
By solving the velocity potential using the Hess-Smith source panel method, additional steps can be
taken to calculate a pressure distribution map over the surface of our airfoil. Additionally, one could vary
the effect of the angle of attack on the pressure co-effcient over the airfoil, and graph the results to compare
and contrast the stresses experienced by different airfoil shapes.
As mentioned before, the source panel method is limited because it prevents us from estimating the
amount of lift generated by this airfoil by ignoring the effect of vorticity and, by extension, circulation. This
leads to strange behaviour at the trailing edge of the airfoil. If one were to zoom in closer the sharply-
pointed trailing edge, they would see the flow lines penetrating the airfoil near the tip and the upper and
lower streams crossing.
A more accurate model, as written in Python by Dr. Lorena Barba in Lesson 11 of her AeroPython
module, would be an implementation of a Source-Vortex method. By Hess and Smith’s method, this involves
fixing a constant total vorticity strength over the entire body. To solve this extra set of equations, a Kutta
boundary condition is implemented at the trailing edge of the airfoil. Essentially, this condition fixes the
airflow leaving the top and bottom side of the trailing edge to be equal and parallel. As a result, a straight
stream of flow leaves the tip of the airfoil, mimicking the real-world phenomenon of flight. Using the Source-
Vortex method, one can calculate the total lift generated by an airfoil, among other characteristics.
With or without the vortex method, the power of the Hess-Smith Panel Method lies in the fact that, by
applying a few simple aerodynamic equations, one can generate design feedback for any geometry quickly
and with low computational requirements.
6

7. 6 References
6.1 Figures
Jhbdel, ”Airfoil with flow,” Wikimedia Creative Commons.
”The Angle of Attack for an Airfoil.” HyperPhysics. Georgia State University, n.d. Web. 12 May 2015.
¡http://hyperphysics.phy-astr.gsu.edu/hbase/fluids/angatt.html¿.
6.2 Texts
Alonso, Juan. ”Lecture 5: Hess-Smith Panel Method.” AA210b - Fundamentals of Compressible Flow
II. Stanford University, n.d. Web. 12 May 2015.
Barba, Lorena A. ”Lesson 10: Source Panel Method.” AeroPython. George Washington University, Apr.
2015. Web. 6 May 2015.
Barba, Lorena A. “Lesson 11: Vortex Source Panel Method.” AeroPython. George Washington Univer-
sity, Apr. 2015. Web. 6 May 2015.
Dimitriadis, G. ”Lecture 4: Panel Methods.” Aeroelasticity and Experimental Aerodynamics Lab. Uni-
versite de Liege, n.d. Web. 12 May 2015.
http://www.ltas-aea.ulg.ac.be/cms/uploads/Aerodynamics04.pdf
Sengupta, Tapan K. Theoretical and Computational Aerodynamics. Wiley, 2014.
http://adl.stanford.edu/aa210b/LectureN otesf iles/Hess − Smith.pdf
Smith, Richard. "Why Use a Panel Method?" Symscape. N.p., Mar. 2007. Web. 12 May 2015.
http://www.symscape.com/blog/whyusepanelmethod
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