# 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.

Figure 1: Lifting Surface Model

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.

#### The Time-Stepping Scheme

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.

#### Selected References

• Katz J, Plotkin J. Low Speed Aerodynamics, McGraw-Hill, Inc., New York, 1991.

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