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– Laplace’s Equation

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

freestream flow

• P-G equation can be converted to Laplace’s equation by a transformation

0)1( 2 =+− ∞ yyxxM φφ

0=+ yyxx φφ

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

• Laplace’s Equation

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

02 =∇ φ

• Solution to Governing Equations– Field Method

– Singularity Method

Some Potential Theory

• What is “singularities” ?

These are algebraic functions which satisfy Laplace’s 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:

φ on Σ+ k design problem

– Neuman Problem:

∂φ/ ∂n on Σ+ k analysis problem

• Some other key properties of potential flow theory– If either φ or ∂φ/ ∂n is zero everywhere on Σ+

k then φ = 0 at all interior points.

– φ 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 ⋅=∫∫∫ ∫∫

dSgradgraddVR S

n∫∫∫ ∫∫ ⋅−=∇−∇ )()( 22 ωχχωωχχω

• The derivation

Gauss Divergence Theorem

+Laplace’s equation

The integral expression for the potential

• The integral expression for the potential

• Comments on the integral expressionThe problem is to find the values of the unknown source and doublet strengths σ and μ for a specific geometry and given freestream, φ∞.

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' ∫∫ ∂

∂−−= ∞ μσφφ π

• 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 θ = tan-1(y/x)

' 14

1 13 : ( ) [ ( )]BS

D p dSr n rπφ φ σ μ∞

∂= − −

∂∫∫

' 14

( ) ( )2 : ( ) [ ln ]2 2s

q s sD p r dsπγφ φ θ

π π∞= − −∫Uniform onset flow

cos sinV x V yα α∞ ∞+ q is the 2D source strength

This is a vortex singularity of strength γ

• 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 dSγφ α α θπ π∞

=

= + + −∑ ∫

Definition of Each Panel

– Nodes: ith and i+1th

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

jitjin iiiiii θθθθ sincos,cossin +=+−=

Where:

i

iii

i

iii l

xxl

yy −=

−= ++ 11 cos,sin θθ

• Representation of Boundary Condition (1)– The flow tangency condition

The coordinates of the midpoint of control point

2

21

1

+

+

+=

+=

iii

iii

yyy

xxx

The velocity components at the control point

),(,),( iiiiii yxvvyxuu ==

Niieachforvuorvu

iiii

iiii

,,1,,0cossin0)cossin()(0…==+−

=+−⋅+⇒=⋅θθ

θθ 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 =1

NtVtV ⋅−=⋅ 1

• Representation of Boundary Condition (3)– The Kutta condition

NtVtV ⋅−=⋅ 1

NNNN

NNNN

vuvuor

vuvu

θθθθ

θθθθ

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, γ 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 eqn’s., 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, γ.

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

7. Determine the pressure distribution from Bernoulli’s 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

γα

γα

where qi and γ 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:

πβ

π

2

)ln(21

*

1,*

ijS

ij

jis

ij

ij

v

rr

u

=

−= +

)ln(212

1,*

*

ij

jiv

ijv

rr

v

u

ij

ij

+=

=

π

πβ

• Step 3. Flow tangency conditions to get N equations

Nivu iiii ,,1,0cossin …==+− θθ

∑=

+ ==+N

jiNijij NibAqA

11, ,1…γ

where

• Step 4. Kutta Condition to get equation N+1

NNNN vuvu θθθθ sincossincos 1111 +−=+

∑=

++++ =+N

jNNNjjN bAqA

111,1,1 γ

where

• Step 5. Solve the system for qi, γ

∑=

+ ==+N

jiNijij NibAqA

11, ,1…γ

∑=

++++ =+N

jNNNjjN bAqA

111,1,1 γ

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

• Step 7. Find the surface pressure coefficient

2)(1∞

−=Vu

C i

i

tP

Summary of Classic Panel Method

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

contributions from all the singularities,namely qi, γ.

2.Get N eqn’s 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, γ.