Panel Methods: Theory and Method

A Solution for Incompressible Potential Flow

Introduction Incompressible Potential Flow

The viscous effects are small in the flowfield

The speed of the flow must be low everywhere (M < 0.4)

The flow must be irrotational

Governing Equations Laplaces Equation

Prandtl-Glauert Equation For higher subsonic Mach numbers with small disturbances to the

freestream flow

P-G equation can be converted to Laplaces equation by a transformation

0)1( 2 =+ yyxxM

0=+ yyxx

Introduction The advantages of Panel Method

Flexibility Be capable of treating the range of geometries

Economy Get results within a relative short time

A Story about the creation of Panel Method A.M.O.Smith, The initial development of panel

methods in Applied Computational Aerodynamics, P.A. Henne, ed., AIAA, Washington, 1990.

Outline Some Potential Theory

Derivation of the Integral Equation for the Potential

Classic Panel Method

Program PANEL

Subsonic Airfoil Aerodynamics

Issues in the Problem formulation for 3D flow over aircraft

Example applications of panel methods

Using Panel Methods

Advanced panel methods

Some Potential Theory

Laplaces Equation

Since the equation is linear, superposition of solutions can be used.

02 =

Solution to Governing Equations Field Method

Singularity Method

Some Potential Theory

What is singularities ?

These are algebraic functions which satisfy Laplaces equation, and can be combined to construct flow-fields.

The most familiar singularities are the point source, doublet and vortex

Review on the singularities Point source Vortex Doublet

Producing a streamline pattern using a uniform flow and a point source

+

We could superimpose many sources and sinks to get nearly any flow pattern we desired.

What are the singularity methods ? The solution is found by distributing singularities of

unknown strength over discretized portions of the surface: panels.

The unknown strengths of the singularities is found by solving a linear set of algebraic equations to determine.

The equation governing the flow-field is converted from a 3D problem throughout the field to a 2D problem for finding the potential on the surface.

Boundary Conditions Dirichlet Problem:

on + k design problem Neuman Problem:

/ n on + k analysis problem

Some other key properties of potential flow theory If either or / n is zero everywhere on +

k then = 0 at all interior points. cannot have a maximum or minimum at

any interior point. Its maximum value can only occur on the surface

boundary, and therefore the minimum pressure (and maximum velocity) occurs on the surface.

Derivation of the Integral Equation for the Potential

Motivation To obtain an expression for the potential anywhere in the flowfield

in terms of values on the surface bounding the flowfield.

Gauss Divergence Theorem The relation between a volume integral and a surface integral

dSdVdivR S

nAA = dSgradgraddV

R S

n = )()( 22

The derivation

Gauss Divergence Theorem

+Laplaces equation

The integral expression for the potential

The integral expression for the potential

Comments on the integral expression The problem is to find the values of the unknown

source and doublet strengths and for a specific geometry and given freestream, .

The requirement to find the solution over the entire flowfield (a 3D problem) is replaced with the problem of finding the solution for the singularity distribution over a surface (a 2D problem).

dSrnr

pBS

)]1(1[)( 41' =

More comments on the integral expressionAn integral equation to solve for the unknown surface

singularity distributions instead of a partial differential equation.The problem is linear, allowing us to use superposition

to construct solutions.We have the freedom to pick whether to represent the

solution as a distribution of sources or doublets distributed over the surface.The theory can be extended to include other

singularities.

The basic idea of panel method Approximating the surface by a series of line segments (2D) or

panels (3D)

Placing distributions of sources and vortices or doublets on each panel.

Possible differences in approaches to the implementation

various singularities

various distributions of the singularity strength over each panel

panel geometry

Advantage No need to define a grid

throughout the flowfield

The Classic Hess and Smith Method

Starting with the 2D version and using a vortex singularity in place of the doublet singularity

Where = tan-1(y/x)

' 14

1 13 : ( ) [ ( )]BS

D p dSr n r

= w' 1

4( ) ( )2 : ( ) [ ln ]

2 2s

q s sD p r ds = v

Uniform onset flow

cos sinV x V y + q is the 2D source strength This is a vortex singularity of strength

The idea of Approach Break up the surface into straight line segments

Assume the source strength is constant over each line segment (panel) but has a different value for each panel

The vortex strength is constant and equal over each panel.

The potential equation become

1

( )( cos sin ) [ ln ]2 2

N

j panel j

q sV x y r dS == + +

Definition of Each Panel

Nodes: ith and i+1th

Inclination to the x axis: Normal and tangential unit vectors:

jitjin iiiiii sincos,cossin +=+=Where:

i

iii

i

iii l

xxl

yy == ++ 11 cos,sin

Representation of Boundary Condition (1) The flow tangency condition

The coordinates of the midpoint of control point

2

21

1

+

+

+=

+=ii

i

iii

yyy

xxx

The velocity components at the control point

),(,),( iiiiii yxvvyxuu ==

Niieachforvuorvu

iiii

iiii

,,1,,0cossin0)cossin()(0==+

=++=

jijinV

Representation of Boundary Condition (2)

The Kutta condition The flow must leave the trailing edge smoothly.

Here we satisfy the Kutta condition approximately by equating velocity components tangential to the panels adjacent to the trailing edge on the upper and lower surface.

The solution is extremely sensitive to the flow details at the trailing edge.

Make sure that the last panels on the top and bottom are small and of equal length.

tNt uu =1NtVtV = 1

Representation of Boundary Condition (3) The Kutta condition

NtVtV = 1

NNNN

NNNN

vuvuor

vuvu

sincossincos

)sincos()()sincos()(

1111

1111

+=+

++=++ jijijiji

The boundary conditions derived above are used to construct a system of linear algebraic equations for the strengths of the sources and the vortex.

Steps to determine the solution1. Write down the velocities, ui, vi, in terms of contributions from all the

singularities.

includes qi, from each panel and the influence coefficients which are a function of the geometry only.

2. Find the algebraic equations defining the influence coefficients.

3. Write down flow tangency conditions in terms of the velocities (N eqns., N+1 unknowns).

4. Write down the Kutta condition equation to get the N+1 equation.

5. Solve the resulting linear algebraic system of equations for the qi, .6. Given qi, , write down the equations for uti, the tangential velocity

at each panel control point.

7. Determine the pressure distribution from Bernoullis equation using the tangential velocity on each panel.

Step 1. Velocities The velocity components at any point i are given by

contributions from the velocities induced by the source and vortex distributions over each panel.

= =

= =

++=

++=N

j

N

jvsji

N

j

N

jvsji

ijij

ijij

vvqVv

uuqVu

1 1

1 1

sin

cos

where qi and are the singularity strengths, and the usij, vsij, uvij, and vvij are the influence coefficients.

As an example, the influence coefficient usij is the x-component of velocity at xi due to a unit source distribution over the jth panel.

Step 2. Influence coefficientsLocal panel coordinate system The influence coefficients

due to the sources:

The influence coefficients due to the vortex distribution:

2

)ln(21

*

1,*

ijS

ij

jis

ij

ij

v

rr

u

=

= +

)ln(212

1,*

*

ij

jiv

ijv

rr

v

u

ij

ij

+=

=

Step 3. Flow tangency conditions to get N equations

Nivu iiii ,,1,0cossin ==+

=

+ ==+N

jiNijij NibAqA

11, ,1

where

Step 4. Kutta Condition to get equation N+1

NNNN vuvu sincossincos 1111 +=+

=

++++ =+N

jNNNjjN bAqA

111,1,1

where

Step 5. Solve the system for qi,

=

+ ==+N

jiNijij NibAqA

11, ,1

=

++++ =+N

jNNNjjN bAqA

111,1,1

Step 6. Given qi, and , write down the equations for the tangential velocity at each panel control point.

Step 7. Find the surface pressure coefficient

2)(1

=Vu

C ii

tP

Summary of Classic Panel Method

Key points1.Write down the velocities, ui, vi, in terms of

contributions from all the singularities,namely qi, .2.Get N eqns using flow tangency conditions in

terms of the velocities.

3.Get the N+1 equation using the Kutta condition.

4.Solve the resulting linear algebraic system of equations for the qi, .

Panel Methods: Theory and MethodIntroductionIntroductionOutlineSome Potential TheorySome Potential TheoryDerivation of the Integral Equation for the PotentialThe Classic Hess and Smith MethodSummary of Classic Panel Method