# 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:

• The grid generation, eg. the subdivision of the geometry into a number of elements of appropriate size and shape;

• 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.

### Boundary Conditions

#### Kutta Condition

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.

### Wake Analysis

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.

#### 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.

### State-of-the-Art

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.

#### Selected References

• Hess JL, Smith, AMO. Calculation of Potential Flow About Arbitrary Bodies, in Progress in Aeronautical Sciences, Vol. 8, 1967, pages 1-138.

• Maskew B. Program VSAERO. A Computer Program for Calculating the non-Linear Aerodynamic Characteristics of Arbitrary Bodies, NASA CR-166476, 1982.

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

• Berger SA. Laminar Wakes, American Elsevier Publ., Ner York, 1971.

• Kellog OD. Foundations of Potential Flow. Dover Ed. New York, 1953.

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