OPTIMIZATION OF WING PLANFORM USING THREEDIMENSIONAL PANEL METHODS

A Project Report

submitted by

VINAY KIRAN C K

in partial fulfilment of the requirementsfor the award of the degree of

BACHELOR OF TECHNOLOGY

DEPARTMENT OF Aerospace EngineeringINDIAN INSTITUTE OF TECHNOLOGY MADRAS, CHENNAI

April 2010

CERTIFICATE

This is to certify that the project titled Optimisation of Wing Planform Using 3D Panel

Methods, submitted by Mr. Vinay Kiran C K (AE06B022), at the Department of Aerospace

Engineering, to the Indian Institute of Technology, Madras, for the award of the degree of

Bachelor of Technology, is a bonafide record of the project work done by him under my

supervision. The contents of this report, in full or in parts, have not been submitted to any other

Institute or University for the award of any degree or diploma.

Prof. M. Ramakrishna

Professor

Dept. of Aerospace Engineering

IIT-Madras, 600 036

Prof. P. Sriram

Head of the Department

Dept. of Aerospace Engineering

IIT-Madras, 600 036

ACKNOWLEDGEMENTS

I wish to thank my guide Prof. M. Ramakrishna for giving me this opportunity to work under

him. I am grateful to him for guiding me all through, suggesting corrections and improvements

from time to time. I’d like to thank my friends for making my stay at IIT a memorable one.

Special thanks are due to Nitesh Shah and Rajesh Gandham for their inputs and suggestions.

Finally, I’d like to thank my parents and brother for their love and confidence in my abilities.

i

ABSTRACT

KEYWORDS: Vortex Lattice Method, Unstructured Grid, Zimmerman

One of the problems in the design of Micro Air Vehicles is the choice of the shape of the

wing planform. The main objective of the project was arriving at the optimum shape by the

use of the Vortex Lattice Method. In this method, the wing is replaced by the mean surface,

and an unstructured grid is used to represent this mean surface. VLM is discussed in detail

in the book Low Speed Aerodynamics ((1)), which uses rectangular panels on structured grids.

In this project, the VLM is applied on an unstructured grid. Three planform configurations,

rectangular, zimmerman and the inverse zimmerman are investigated. Python was used to

design the code using object oriented principles. The solution to the problem is obtained as a

superimposition of more fundamental flows, like the vortex flow. The Vortex Lattice Method

demands less programming efforts and offers results that are good initial estimates.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

ABSTRACT ii

LIST OF FIGURES 1

ABBREVIATIONS 2

1 INTRODUCTION 3

1.1 Elementary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Source Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Vortex Filament . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Derivation for the Induced Velocity by a Straight Line Filament . . 6

1.3 Vortex Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Panel Methods 8

2.1 Geometry & Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Wing Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Vortex Lattice Method 12

3.1 Choice of Singularity Element . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Boundary Conditions & Influence Coefficients . . . . . . . . . . . . . . . . 12

3.3 Modelling the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Secondary Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.2 Coefficient of Pressure . . . . . . . . . . . . . . . . . . . . . . . . 15

iii

3.4.3 Calculation of Lift . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Results 17

4.1 Rectangular Planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 The Inverse Zimmerman Configuration . . . . . . . . . . . . . . . . . . . . 19

4.3 The Zimmerman Configuration . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Concluding Remarks & Future Work 25

A Code Snippets 26

LIST OF FIGURES

1.1 Source Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 A Free Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 A Vortex Filament . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 A Straight Line Vortex Filament . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 A Triangular Vortex Ring Panel . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 A Gmsh example illustrating the effect of Characteristic Length. The figure onthe left uses a CL of 0.3. The one on the right has a CL of 0.6 . . . . . . . . 9

2.2 Camber Profiles NACA 3-Digit Reflex Airfoil Series . . . . . . . . . . . . 10

3.1 Horse Shoe Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Component of Circulation that Contributes to the Generation of Lift . . . . 16

4.1 Cp distribution over the wing. Rectangular. . . . . . . . . . . . . . . . . . . 17

4.2 Γ distribution over the wing. Rectangular. . . . . . . . . . . . . . . . . . . 18

4.3 Net Circulation per Panel Distribution over the wing. Rectangular. . . . . . 18

4.4 Cp distribution over the wing. Inverse Zimmerman . . . . . . . . . . . . . 19

4.5 Γ distribution over the wing. Inverse Zimmerman . . . . . . . . . . . . . . 20

4.6 Net Circulation per Panel Distribution over the wing. Inverse Zimmerman . 20

4.7 Cp distribution over the wing. Zimmerman . . . . . . . . . . . . . . . . . . 21

4.8 Γ distribution over the wing. Zimmerman . . . . . . . . . . . . . . . . . . 22

4.9 Net Circulation per Panel Distribution over the wing. Zimmerman . . . . . 22

4.10 Cl vs α curve. Rectangular. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.11 Cl vs α curve. Zimmerman. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.12 Cl vs α curve. Inverse Zimmerman. . . . . . . . . . . . . . . . . . . . . . 24

1

ABBREVIATIONS

AoA Angle of Attack

AR Aspect Ratio

BEM Boundary Element Method

CL Characteristic Length

Cl Coefficient of lift

Cp Coefficient of Pressure

MAV Micro Air Vehicle

SVD Singular Value Decomposition

VLM Vortex Lattice Method

2

CHAPTER 1

INTRODUCTION

One of the problems in the design of Micro Air Vehicles is the choice of the shape of the wing

planform. That is, given a numerical value for the magnitude of the area after wing loading

calculations, which shape is best suited for the planform. One could choose between several

options including Elliptical, Rectangular and Zimmerman kind of configuration. This project

aims at solving the problem of choosing the optimum planform shape using Three Dimensional

Vortex Panel Methods, which belong to the category of Boundary Element Methods.

For flows where the characteristic speeds are lesser than the speed of sound and where

viscous effects are negligible, i.e. in inviscid incompressible flows, the equation governing the

flow is simply the Laplace’e Equation.

∇2φ = 0

Since the Laplace’s Equation is a linear differential equation, solutions can be constructed

by linear superposition of other solutions. This property is exploited and one uses the funda-

mental flows, namely the Source Flow and the Vortex Flow to construct the solution to any

given problem. Since the source flow has zero circulation, it is not sufficient to model lifting

flows, in which case vortices are used to construct the solutions.

1.1 Elementary Flows

1.1.1 Source Flow

Source flows are the basic building blocks used in modelling non-lifting flows. The most basic

flow is the point source, using which one can construct more complicated ‘source sheets’, which

can be used in 2D as well as 3D. The equation for the point source can be derived using the

assumption of radial flow. The following equations are quoted from Anderson(2).

Vθ = 0

Vr =Λ

2πr

Figure 1.1: Source Flow

where Λ is the source strength. Physically, Λ is the rate of volume flow from the source,

per unit depth perpendicular to the page.

~V (~r, θ) =Λ

2πrr

1.1.2 Vortex Flow

The other fundamental flow is the Point Vortex. Due to it’s net non-zero circulation, it can be

used to model lifting flows. The following is quoted from Anderson(2).

Vθ = − Γ

2πr

Vr = 0

~V (~r, θ) = − Γ

2πrθ

4

where Γ is the circulation or the strength of the vortex, and θ is the unit vector in the tangential

direction.

Figure 1.2: A Free Vortex

1.2 Vortex Filament

The influence of Vortex Filaments can be computed by treating it as a continuous distribution

of point vortices. And the summation is replaced by an integral. The formula is quoted from

(2)

φ(x, y) =

∫ P2

P1

γds

2πln r

where γ is the vortex strength per unit length.

Figure 1.3: A Vortex Filament

5

1.2.1 Derivation for the Induced Velocity by a Straight Line Filament

Consider the case where we have a vortex filament extending from point A to point B, and

we have to calculate the velocity induced by this segment at a point P in the flow domain.

By segment AB, we mean a segment that is a part of a closed vortex ring ( By Helmholtz’s

Theorem a vortex filament cannot start or end in a flow field). Then the velocity induced at P

due to AB, using the Biot-Savart Law is (from chap. 2 (1)),

Figure 1.4: A Straight Line Vortex Filament

~VP =Γ

4π

~r1 × ~r2‖~r1 × ~r2‖2

~r0 ·(~r1r1− ~r2r2

)where,

~r0 is the vector ~AB

~r1 is the vector ~AP

~r2 is the vector ~BP

6

1.3 Vortex Ring

A vortex ring is a closed circuit vortex filament. The treatment is same as that for a vortex

filament, except that the integral is replaced by a closed loop integral. Hence,

φ =

∮γ

2πrds

In the Vortex Lattice Method, one uses rectangular vortex rings, that is rings composed of

four straight line segments. In the current project, triangular vortex rings are used. This way,

one can benefit from the time and effort saved in modelling complicated geormetries. Using an

unstructured grid helps in achieving uniform mesh characteristic length throughout the domain.

Figure 1.5: A Triangular Vortex Ring Panel

7

CHAPTER 2

Panel Methods

Panel Methods are a class of numerical techniques used to solve inviscid incompressible flows.

Since the equation being solved is the Laplace’s Equation, it is also generally called potential

flow code. They were eventually replaced by Finite Volume Solvers. But the Panel methods

still provide good initial estimates for inviscid flows. The solution generally consists of three

main parts. The fisrt is the discretization of the geometry into panels of the desired size. For this

purpose, a software called Gmsh (5) was used. The second part is to obtain the vortex strengths

required to simulate the flow. This is done by setting up a system of linear equations, which are

obtained from the boundary conditions. The third and final part is the post-processing which

involves performing the secondary computations viz. calculating the pressures, loads, Cl, Cd.

2.1 Geometry & Grid Generation

Gmsh(5) is a GUI based free open source mesh generator which also has pre and post-processing

facilities. Once the geometry is defined in it, a mesh is generated. The fineness of the mesh can

be controlled by setting the ‘Characteristic Length’. The data can be outputted in a number of

formats. In this program, the vtk file format was chosen. The mid-points of the triangles are

chosen to be the control points, i.e. the points where the no normal condition will be applied.

Figure 2.1: A Gmsh example illustrating the effect of Characteristic Length. The figure on the

left uses a CL of 0.3. The one on the right has a CL of 0.6

2.2 Boundary Conditions

The Boundary Condition applied is Dirichlet Boundary Condition. Since the flow is inviscid,

only the normal component of velocity is set to zero. The no normal flow condition is applied

at each of the panel control points. Due to the thin wing approximation that has been made, the

boundary conditions are chosen at the mean camber rather than the actual surface. Since we

have N control points, we are provided with N equations. Also we have N unknowns i.e. the

vortex strengths. Hence, we have a system of equations that can be solved.

2.3 Wing Geometry

Flying wings generally use a reflexed airfoil profile. This is done to compensate for the absence

of the restoring pitching moment which the tail wing would have provided in a conventional

aircraft. The reflex towards the rear end of the airfoil can be thought of as a ‘modified tail’. For

our design course, we used the MH-60 airfoil (7), which has a history of having being used in

tail-less aircraft. For this program however, the NACA-251 was used. It is an airfoil from the

9

NACA 3-Digit Reflex Airfoil family. The camber for this profile is given as,

y

c=k1

6

[(xc− r)3

− k2

k1

(1− r)3x

c− r3x

c+ r3

], 0 ≤ x

c≤ r

y

c=k1

6

[k2

k1

(xc− r)3

− k2

k1

(1− r)3x

c− r3x

c+ r3

], r <

x

c≤ 1

where,

• m is Chordwise Location for maximum ordinate of airfoil or camber line

• r is chordwise location for zero value of second derivative of 3-digit or 3-digit-reflexcamber-line equation

• k1 and k2 are constants that determine the shape of the airfoil.

Figure 2.2: Camber Profiles NACA 3-Digit Reflex Airfoil Series

The values of the constants for airfoils are given below. Data is taken from NASA Technical

Memorandum 4741 (3)

The planform shapes used are the rectangular, Zimmerman and the Inverse Zimmerman

Configurations. (For shapes, see results.4 )

10

Table 2.1: NACA 3-Digit Reflex Airfol ProfilesCamber-line Designation m r k1 k2/k1

221 0.10 0.1300 51.990 0.000764231 0.15 0.2170 15.793 0.00677241 0.20 0.3180 6.520 0.0303251 0.25 0.4410 3.191 0.1355

11

CHAPTER 3

The Vortex Lattice Method

The Vortex Lattice Method is an extension of the Vortex Panel Method to the three dimensional

case. In the vortex panel method, the airfoil is discretized as straight line panels with the

vortex strength varying in a certain way either constant strength, or linearly, or any other higher

order variation. ( The geometry can also be discretized as curved panels. The straight line

approximation is the simplest ). A point, mostly the mid-point, on each of these panels is

chosen as the control point where the no normal flow condition is applied. Finally, the Kutta

Condition requires that the flow leave the trailing edge smoothly. In the vortex panel methods,

this is done by forcing the two panels at the trailing edge to have the same vortex strength.

3.1 Choice of Singularity Element

In the Vortex Lattice Method, the wing, (the mean surface rather) is divided into triangular

panels. The approach follows the one outlined in (1), the difference being the method used in

the book uses rectangular panels. Once the mean surface is divided into triangles, each of the

triangles is now a Vortex Ring, and each of the arms of the ring is a vortex filament. The vortex

strength is constant over each panel, and at this stage is the unknown that has to be computed.

3.2 Boundary Conditions & Influence Coefficients

The induced velocity at any point is the sum of the velocities induced by each of the vortex

panels and the freestream velocity V∞. For the application of the boundary conditions, the

control points of the panels become points of interest.

~Vi =∑N

j=0

~Vij + ~V∞

where i denotes the ith control point and j denotes the jth vortex ring.

Now, since there is no mass flux across the mean surface, we must have

~V · n = 0

Now expanding the ~V term, we get

∑N

j=0

~Vij · n = − ~V∞ · n

~Vi1 · ni + ~Vi2 · ni + ...+ ~ViN · ni = − ~V∞ · ni

The application of the boundary condition at each of the N panels gives us N equations. In

matrix form, the above set of equations can be written as

~V11 · n1 V12 · n1 · · · · · · ~V1N · n1

~V21 · n2 V22 · n2 · · · · · · ~V2N · n2

......

......

......

......

......

~VN1 · nN VN2 · nN · · · · · · ~VNN · nN

=

− ~V∞ · n1

− ~V∞ · n2

...

...

− ~V∞ · nN

(3.2.1)

Since the vortex strength γ is assumed to be constant over each ring, it factors out.

a11 a12 · · · · · · a1N

a21 a22 · · · · · · a2N

......

......

......

......

......

aN1 aN2 · · · · · · aNN

γ1

γ2

...

...

γN

=

− ~V∞ · n1

− ~V∞ · n2

...

...

− ~V∞ · nN

(3.2.2)

In the above matrix equation, each of the terms is called an Influence Coefficient. See the

appendix for the code (A) . An influence coefficient aij is physically the velocity induced by

unit strength jth vortex ring at the ith control point. So, finally we have a system of linear equa-

tions Ax = b, which can be solved by using a standard technique like Gaussian Elimination. In

13

the current case, the influence coefficient matrix was often singular. A best fit solution using the

method of Least Squares was obtained. The function pinv in the SciPy Linear Algebra package

computes the best fit solution using the Singular Value Decomposition method.

Next, we are interested in calculating the tangential velocity distribution at the panel control

points.

3.3 Modelling the Wake

An important aspect here is that of modelling the wake. Following Katz(1), the wake is mod-

elled as a series of horse-shoe vortices. To satisfy the Kutta-Condition at the trailing edge, it

is required that the net circulation at the trailing edge vanish. The procedure followed is thus.

Take a trailing edge panel. Now add a horse-shoe vortex of the same strength to cancel the ef-

fect of the triangular panel. In practice, each arm of the horse-shoe vortex is treated as straight

line vortex filament. 1.2.1

Figure 3.1: Horse Shoe Vortex

14

3.4 Secondary Computations

3.4.1 Velocity

The velocity at a control point is computed by calling the voring (short for Vortex Ring) sub-

routine, this time with the real vortex strengths, to get the induced velocity due to each panel.

The contribution from each panel is summed up to get the total induced velocity at a control

point. The free-stream velocity is added to the give the total velocity.

3.4.2 Coefficient of Pressure

The Coefficient of Pressure is,

Cp =p− p∞12ρV 2∞

The use of Bernoulli’s Equation in the above equation for an inviscid, incompressible, steady

flow leads us to the simplifiation of the equation to

Cp = 1− V 2

V 2∞

3.4.3 Calculation of Lift

Once the circulation (vortex strengths) over each panel have been determined, the lift is com-

puted by summing up the compnents of the circlation that cause lift. This is the component of

the circulation which is perpendicular to the direction of the free-stream velocity. According to

the Kutta-Joukowski theorem, the lift force per unit span is

L = ρV∞Γ∞

where,

• ρ is the free stream density of air

• V∞ is the free-stream velocity

15

• Γ∞ is the net circulation around the wing

Figure 3.2: Component of Circulation that Contributes to the Generation of Lift

16

CHAPTER 4

Results

The code was run for different cases. For each different characteristic lengths were used. For

the same geometry, different characteristic lengths were used. All results were visualized in

Mayavi2 (4). The aspect ratio of the wings in all three cases is 1.73

4.1 Rectangular Planform

Below are plots for Cp, Γ, and net circulation per panel for a rectangular planform. The Char-

acteristic Length used was 0.1. The mesh contained 1100 triangles. The angle of attack is

0◦.

Figure 4.1: Cp distribution over the wing. Rectangular.

Figure 4.2: Γ distribution over the wing. Rectangular.

Figure 4.3: Net Circulation per Panel Distribution over the wing. Rectangular.

18

4.2 The Inverse Zimmerman Configuration

Below are plots for Cp, Γ, and net circulation per panel for the inverse zimmerman configura-

tion. The Characteristic Length used was 0.1. The mesh contained 1100 triangles. The angle

of attack is 0◦.

Figure 4.4: Cp distribution over the wing. Inverse Zimmerman

19

Figure 4.5: Γ distribution over the wing. Inverse Zimmerman

Figure 4.6: Net Circulation per Panel Distribution over the wing. Inverse Zimmerman

20

4.3 The Zimmerman Configuration

Below are plots for Cp, Γ, and net circulation per panel for the zimmerman configuration. The

Characteristic Length used was 0.1. The mesh contained 1110 triangles. The angle of attack is

0◦.

Figure 4.7: Cp distribution over the wing. Zimmerman

21

Figure 4.8: Γ distribution over the wing. Zimmerman

Figure 4.9: Net Circulation per Panel Distribution over the wing. Zimmerman

22

Presented below are the Cl vs α curves for the three respective planforms. The code was

run for angles 0, 2, 4, 6, 8, 10 degrees. The magnitudes are off; this part requiers more work.

Figure 4.10: Cl vs α curve. Rectangular.

Figure 4.11: Cl vs α curve. Zimmerman.

23

Figure 4.12: Cl vs α curve. Inverse Zimmerman.

24

CHAPTER 5

Concluding Remarks & Future Work

The Vortex Lattice Method was implemented using an unstructured mesh on rectangular, Zim-

merman and Inverse Zimmerman configuration planforms and an attempt was made to optimize

the planform shape based on the Cl values. A code was written in Python using NumPy and

Scipy, and Cp, Cl, Γ and velocity distribution over the wing was obtained. VLM is much

faster than the more involved Finite Volume Method, and can be used to obtain quick initial

estimates.(1)

Improvements include using higher order approximations in representing the mean sur-

face and finally, directly modelling the actual surface using higher order elements. For two-

dimensional flows, Ramachandran et. al (6) showed that using cubic panels for representing

the geometry, and using linear vortex strength variation on a panel, improves the accuracy of

the solution and accurate results can be obtained using fewer panels.

APPENDIX A

Code Snippets

Listing A.1: Influence Coefficient Calculationsfor i in range(N):

panel_i = TriPanel(i, 1.0, points_list, triangle_list)ctrl_pt[i] = panel_i.ctrl_point()S[i] = panel_i.area()area_of_panel[i] = norm(S[i])n_cap[i] = S[i]/area_of_panel[i]rhs[i] = np.dot(-v_inf, n_cap[i])for j in range(N):

if is_a_te_triangle(j, te_triangles)==1:wake_vel = wake_influence(j, 1.0, ctrl_pt[i])

panel_j = TriPanel(j, 1.0, points_list, triangle_list)vel = panel_j.voring(1.0, ctrl_pt[i])total_vel = vel + wake_velcoeff[i][j] = np.dot(total_vel, n_cap[i])

Here, in the nested for loop, the influence coefficients are computed and the rhs matrix is

set up. The outer loop cycles through all the control points, whereas the inner loop computes

the influence coefficients. TriPanel is a class that contains method voring (vortex ring). Every

call to TriPanel sets up a triangle based on the input vtk data. voring calls function vortex line

three times, once for each arm of the (triangular) vortex ring. The influence coefficient is the

dot product of the induced velocity due to the jth vortex ring and the normal of the ith panel.

REFERENCES

[1] Joseph Katz and Allen Plotkin, Low-Speed Aerodynamics, pp. 378 412, (1991).

[2] John D. Anderson, Fundamentals of Aerodynamics, pp. 115-173, (1991).

[3] Charles L. Ladson and Darrell W. Sproles, Computer Program To Obtain Ordinates for

NACA Airfoils, NASA Technical Memorandum 4741, 91-1596, (1996).

[4] Prabhu Ramachandran and Gael Varoquaux, The Mayavi data visualizer, http://

code.enthought.com/projects/mayavi, (2005–).

[5] Christophe Geuzaine and Jean-Francois Remacle, Gmsh: a three-dimensional finite ele-

ment mesh generator with built-in pre and post-processing facilities, International Journal

for Numerical Methods in Engineering, Vol. 79, Issue 11, pp 1309-1331, (2009).

[6] Prabhu Ramachandran, S. C. Rajan and M. Ramakrishna, A Fast Multipole Method For

Higher Order Vortex Panels In Two-Dimensions, SIAM Journal on Scientifc Computing,

Vol. 26, no. 5, 1620-1642, (May 2005)

[7] Martin Hepperle, http://www.mh-aerotools.de/airfoils/mh60koo.htm

27