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