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.

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:

#### 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.

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).

#### Numerical Kutta Condition

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.

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.

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.

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.

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