Lifting Surface Methods
Summary
Lifting surface methods are derived either from the Biot-Savart equation for the vortex-induced velocity and the horseshoe model, or from an appropriate expression of Green’s second equation for the velocity potential, when the thickness term is eliminated.
Mathematical details are object of several text books (for example Kats-Plotkin, 1991). Classical methods for lifting surfaces include the theory of Weissinger (1947), Multhopp (1950), Watkins (1955), Giesing (1967), and others.
All the state-of-the art methods describe the lifting surface (wing or blade) as made
up of bound and free vortex rings (lattices), as sketched in the figure below.
The bound vortex rings (or vortex lattice) are placed on the lifting surface (a shift
of one quarter ring in the streamwise direction is needed for several reasons). The
free vortex rings can either be set as a part of the model, or they can be realised
in a time-stepping scheme. The analysis is very similar to the panel method with the
free wake.
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 and an artificial time-step. Two wake relaxation cycles are
generally enough to achieve enough distortion.
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.
The Wake-Relaxation Scheme
The Time-Stepping Scheme
Related Material
Selected References