Panel Methods
Summary
Potential flow models have long been used in aerodynamics in a more or less sophisticated form. In the past decade the progress in computer technology has stimulated the use of the panel methods to an increasing scale of complexity.
For most of the methods currently in use the governing equation is based on Green’s second identity, and is solved with the requirement of zero normal flow at the solid boundaries (Neumann boundary condition).
The notes in this chapter are restricted to incompressible flows. Formulations for compressible flows have also been formulated (Morino et. al, 1992), but they are far less advanced from the application point of view.
Approximations
In the meanwhile the formulation of the problem has undergone very little refinement, whereas it was becoming evident that the step from the formulation to the numerical solution of the equations was not a simple one. This step involves approximations of various sorts, among which two are easily identified:
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The grid generation, eg. the subdivision of the geometry into a number of elements of appropriate size and shape;
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The order of variation of the unknown over each element (low- or higher-order panel methods).
Formulations
While most methods work, some work better with thick bodies (using Dirichlet boundary conditions), and others work better with thin bodies (using Neumann boundary conditions). Among the latter ones the velocity potential method is probably the most accurate.
Methods that use potential, source and doublet distributions are one order less singular than the corresponding influence coefficients of a velocity (or vorticity) method. Therefore, potential methods tend to be less sensitive to the paneling details.
The computation of the influence coefficients (a major requirement in terms of CPU) is faster for potential based methods than for corresponding velocity methods. The storage requirement is reduced to one third.
A low order potential method is substantially more accurate than a velocity (or
vorticity) method at given number of panels.
There is a number of boundary conditions that must be satisfied: boundary conditions
on the body; Kutta condition at the trailing edge; conditions on the vortex wake.
On the body the only condition that can be specified is on the velocity normal to the
wall: zero if there is no transpiration, equal to the transpiration velocity
otherwise. In a numerical procedure, depending on the formulation, either Dirichlet
or Neumann boundary conditions are set.
The Kutta condition is one of the fundamental boundary conditions in all
aerodynamics. There are various formulations: in terms of velocity, pressure,
velocity potential, vorticity, doublet, wake exit angle, etc., for both linear and
non-linear steady and unsteady flow (at low reduced frequency).
In simple terms any
of the formulations is aimed at eliminating the infinite velocity at the trailing
edge line, and is an implicit definition of circulation around the body (the
equivalence can be proved with the Kutta-Joukowski theorem).
In a numerical approach to the solution of the potential flow with a panel method one
Kutta condition or another can make a difference. Although equivalence among the
terms specified above can be proved theoretically, in the numerical approach the
state of affairs is somewhat different.
There are linear, non-linear, and iterative
schemes for the definition and improvement of the Kutta condition in a panel method
solution.
The slipstream is technically called wake. The wake geometry is important
because introduces a non-linearity in the problem. Aside from this, boundary
conditions are easily set according to the Kutta condition: trailing edge doublet,
vorticity or potential is constant along a wake line shedding from the trailing
edge. Other (theoretical) conditions include: zero pressure jump, zero normal
velocity jump, and zero potential jump across the wake.
The actual application of the vortex theory has led to more and more
sophisticated models. Models based on linearized theory and empirical evidence
assume rigid wakes, moving uniformly according to the free stream.
There
are models incorporating corrections that account for the deformation of the
wake caused by contraction (propellers, helicopter rotors) or relaxation
(wind turbines). These models are called quasi- linearized. The last step in
this development is the fully-non linear model, based on the concept of mutual
interaction among vortices.
The free wake concept allows the vorticity to evolve in a free motion, and
represents the physically correct approach to the unsteady aerodynamics.
It consists in specifying the wake geometry at the beginning of the solution process,
and solve the problem for the fixed wake. Compute the induced velocity at each wake
control point. The wake points are moved by an amount dx, according to the induced
velocity q and an artificial time-step t. Two wake relaxation cycles are generally
enough.
Modification of the wake relaxation scheme consists in assuming a physical time
step. The wake is shed from the trailing edge line, and its size increases linearly
with the time step.
At each time a new row of wake panels is released), and all the
preceding panels are convected streamwise with the local velocity field. The Kutta
condition is used to fix the vorticity strength to be shed into the wake.
This method is numerically more efficient, in the sense that only the induced
velocity of the actual wake points has to be computed. At fixed number of time steps
the time-stepping method is roughly twice as fast.
Airfoil flows have been solved for a number of years with all possible formulations
and Kutta conditions in both steady and unsteady flow. There are studies that
address the precision of each method.
Three dimensional flows are somewhat more complicated and programming seriously more
elaborate. Nevertheless, there have been several improvements of the Kutta condition
(especially for rotor flows and marine propellers), while the improved hardware has
allowed the development of very sophisticated tools.
VSAERO is one of the codes most widely used and documented, both at the research and
industry levels. Problems that are routinely solved consists of tens of thousands of
panels for configurations as complex as a full aircraft.
Boundary Conditions
Boundary Condition on the Body
Kutta Condition
Numerical Kutta Condition
Boundary Conditions on the Wake
Wake Analysis
Wake-Relaxation Scheme
Time-Stepping Scheme
State-of-the-Art
Related Material
Selected References