panel method optimzation

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Optimisation of Wing Planform Using 3D Panel Methods

Vinay Kiran C K

Indian Institute Of Technology Madras

May 14, 2010

Vinay Kiran C K (IITM) Optimisation of Wing Planform Using 3D Panel Methods

May 14, 2010 1 / 33
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Overview

Outline

1 Introduction

2 Theory

3 Boundary Conditions and Influence Coefficients

4 Programming Methodology

5 Results


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Introduction

Problem Statement

Aerodynamic Design of Micro-Air Vehicle


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Introduction

Problem Statement


Choose planform type

Vinay Kiran C K (IITM) Optimisation of Wing Planform Using 3D Panel Methods May 14, 2010 3 / 33

I d i
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Introduction

Problem Statement


Choose planform type

Elliptical, Rectangular, Zimmerman


I t d ti
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Introduction

Description of Geometry

Figure: Inverse Zimmerman Figure: Zimmerman


Introduction
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Introduction

Airfoil Profile


Introduction
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Introduction

Airfoil Profile

Airfoil from the NACA 3-Digit Reflex Airfoil Series.


Introduction
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Introduction

Airfoil Profile


y

c=

k1

6

xc

r3

k2

k1(1 r)3

x

c r3

x

c+ r3

, 0

x

c r

yc

= k16

k2k1

xc

r3

k2k1

(1 r)3 xc

r3 xc

+ r3, r < x

c 1


Introduction
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Introduction

Airfoil Profile


y

c=

k1

6

xc

r3

k2

k1(1 r)3

x

c r3

x

c+ r3

, 0

x

c r

yc

= k16

k2k1

xc

r3

k2k1

(1 r)3 xc

r3 xc

+ r3, r < x

c 1

m is Chordwise Location for maximum ordinate of airfoil or camberline

r is chordwise location for zero value of second derivative of 3-digit or3-digit-reflex camber-line equation

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


Introduction
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Camber-line Designation m r k1k2k1

221 0.10 0.1300 51.990 0.000764

231 0.15 0.2170 15.793 0.00677

241 0.20 0.3180 6.520 0.0303251 0.25 0.4410 3.191 0.1355


Introduction
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Figure: Cambers of NACA 3-Digit Reflex Family


Introduction
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Solution Method


Introduction
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Solution Method

Assumption of inviscid, imcompressible flow made


Introduction
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Solution Method


Solving Laplaces Equation

2 = 0


Introduction
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Solution Method



2 = 0

Use of 3-D Panel Methods Vortex Lattice Method (VLM)


Introduction
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Solution Method



2 = 0

Use of 3-D Panel Methods Vortex Lattice Method (VLM)

Low Speed Aerodynamics by Katz & Plotkin


Introduction
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Discretizing The Geometry

Surface Triangulated Using Gmsh


Introduction


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The level of fineness of the mesh can be set.


Introduction
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Export mesh as a vtk file


Introduction
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Export mesh as a vtk file


Introduction
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Figure: A CharacteristicLength of 0.1

Figure: A CharacteristicLength of 0.05


Theory Fundamental Flows
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The Free Vortex


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The Free Vortex

Solution of Laplaces Equation which has non-zero circulation

(rP) =

2

V(r) =

2r


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The Vortex Filament

A Linear Superposition of Point Vortices




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The Vortex Filament


(rP) =

ba

2ds


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The Vortex Filament


(rP) =

ba

2ds

V(r) =1

r


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Figure: A Straight Line Vortex Filament

VP =

4

r1 r2

r1

r2

2r0

r1

r1

r2

r2

where,r0 is the vector ABr1 is the vector APr2 is the vector BP


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The Vortex Ring

Consider a triangular panel PQR



Th V Ri
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The Vortex Ring

Consider a triangular panel PQR

(rP) =

2ds

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Theory Modelling the Wake

M d lli th W k


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Modelling the Wake


Theory Modelling the Wake

M d lli th W k
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Modelling the Wake

Modelled as a series of horse-shoe vortices

Figure: Modelling the Wake Figure: Horse Shoe Vortex

Strength the same as that of the trailing edge panel


Boundary Conditions and Influence Coefficients

B d C diti s
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Boundary Conditions


Boundary Conditions and Influence Coefficients Boundary Conditions

Boundary Conditions


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Boundary Conditions



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Boundary Conditions


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Boundary Conditions

Physically, there can be no flow across a solid boundary.

Mathematically, this can be written as V n = 0



Boundary Conditions
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Boundary Conditions



Vi = V +N

j=0

Vij



Boundary Conditions
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Boundary Conditions



Vi = V +N

j=0

Vij

Vi ni = 0

V

ni +

N

j=0

Vij

ni = 0



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Boundary Conditions



Vi = V +N

j=0

Vij

Vi ni = 0

V

ni +

N

j=0

Vij

ni = 0

Expand the summation

Vi1 n1 + Vi2 n2 + ..... + ViN nN = V ni


Boundary Conditions and Influence Coefficients Influence Coefficients

Influence Coefficients
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Influence Coefficients

V11 n1 V12 n1 V1N n1V21 n2 V22 n2 V2N n2

......

......

......

......

......

VN1 nN VN2 nN VNN nN

=

V n1 V n2

...

...

V nN

(1)

Since was assumed to be constant over each ring, it factors out.

a11 a12 a1Na

21a

22 a

2N......

......

......

......

......

aN1 aN2 aNN

1

2......

N

=

V n1

V

n2......

V nN

(2)


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Now, each of the dot product terms on the LHS is called an InfluenceCoefficient


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A is the influence coefficient matrix


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System of equations Ax = b


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System of equations Ax = b

Solve for


Boundary Conditions and Influence Coefficients Secondary Computations

Lift Production
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Figure: Circulation Causing Lift



Lift Production
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Figure: Circulation Causing Lift

Identify the component of circulation that contributes to thegeneration of lift.

L = V



Secondary Computations
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Velocity is computed using the same subroutine that computed theinfluence coeffs



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Coefficient of Pressure.

Cp = 1

VV

2



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Coefficient of Pressure.

Cp = 1

VV

2


Programming Methodology

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Create mesh in Gmsh. Export as a .vtk file.



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Extract triangles data from vtk file in main program.



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Generate list of trailing edge points



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Generate list of trailing edge triangles



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Generate list of trailing edge trianglesCompute Influence Coefficients.



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Generate list of trailing edge trianglesCompute Influence Coefficients.

outer loop sets up control point



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Compute Influence Coefficients.

outer loop sets up control pointinner loop cycles through all the panels



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outer loop sets up control pointinner loop cycles through all the panelsif panel is a trailing edge triangle:



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add influence of wake



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add influence of wakeSolve for . Gaussian Elimination or SVD


Programming Methodology Code
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f o r i i n ra ng e (N) :p a n e l i = T r i Pa n e l ( i , 1 . 0 , p o i n t s l i s t , t r i a n g l e l i s t )c t r l p t [ i ] = p a n e l i . c t r l p o i n t ( )S [ i ] = p a n e l i . a r ea ( )a r e a o f p a n e l [ i ] = norm ( S [ i ] )n c ap [ i ] = S [ i ] / a r e a o f p a n e l [ i ]r h s [ i ] = np . d o t ( v i n f , n c ap [ i ] )f o r

ji n

ra ng e (N) :i f i s a t e t r i a n g l e ( j , t e t r i a n g l e s ) ==1:w a k e v e l = w a k e i n f l u e n c e ( j , 1 . 0 , c t r l p t [ i ] )

p a n e l j=T r i P a n e l ( j , 1 . 0 , p o i n t s l i s t , t r i a n g l e l i s t )v e l = p a n e l j . v o r i n g ( 1 . 0 , c t r l p t [ i ] )t o t a l v e l = v e l + w a ke v e l

c o e f f [ i ] [ j ] = np . d o t ( t o t a l v e l , n c a p [ i ] )


Results

Results
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Results Effect of Charactersitic Length

Effect of Charactersitic Length
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Figure: Cp CL of 0.075. AoA is 0o Figure: Cp .CL of 0.3. AoA is 0

o


Results Rectangular Planform

Results: Rectangular
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Figure: Cp AoA of 0o

S

Figure: AoA of 0o


Results Rectangular Planform

Results: Rectangular


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Figure: Lift Distrib. AoA of 0o Figure: Cl vs


Results Zimmerman Planform

Results: Zimmerman
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Figure: Cp AoA of 0o Figure: AoA of 0o


Results Zimmerman Planform

Results: Zimmerman
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Results Inverse Zimmerman Planform

Results: Inverse Zimmerman
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Figure: Cp AoA of 0o Figure: AoA of 0o


Results Inverse Zimmerman Planform

Results: Inverse Zimmerman


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THANK YOU


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