Copyright © A. Filippone (1996-2001). All Rights Reserved.

Computational Aerodynamics

In this Chapter

Computational aerodynamics, as a branch of the computational fluid dynamics (CFD), is the latest entry among the tools of the aerodynamic sciences, barely 40 years old. In the 1950s, with matured numerical methods and rudimental computers (by today’s standards !), the first solutions were attempted (ex. simple laminar boundary layers); late in the 1960s the first panel method solutions of practical problems (airfoils and wings) could be produced in relatively short time.

At the present date, computations are routinely performed with multi-block Navier-Stokes solvers for problems as complex are fighter aircraft in complete configurations (Agarwal, 1999), military aircraft with rotor-fuselage interactions, multi-stage turbomachinery, etc.


Importance of Aerodynamic Theory

Experimental aerodynamics is historically the oldest approach to aerodynamics. It was only when the Wright brothers invented the wind tunnel at the beginning of the century that rational data could be gathered and analyzed. Theoretical methods appeared at a later time (late 1910s) to provide information, usually in a closed mathematical form, that could be applied to general cases.

Theoretical research that set outstanding guidelines include Kutta and Joukowsky (airfoil theory), Prandtl (boundary layer), Munk ( thin airfoil, airship theory), Theodorsen (unsteady airfoil theory), Wieselberger-Betz (lifting line theories), Glauert (propeller theory), von Karman (vortex street), Hayes (linearized supersonic theory), Whitcomb (transonic area rule), and many others.

With the invention of the digital computer (also called digital wind tunnel) the range of possibilities has been steadily increasing. The computer itself created a new branch of aerodynamics and applied mathematics: numerical methods.

Importance of Experimental Methods

Experimental methods are still considered the ultimate answer to aerodynamic problems, although they require expensive equipment and can be cumbersome from the point of view of the model accuracy.

Theoretical methods are not very useful unless the equations can be solved in general cases and provide quantitative answers. Back in the days when there was no computing facility calculations were performed by hand. Old papers showed plenty of tables of data (few a few decimal digits) resulting from hand computations.

Numerics is the approach we follow more and more. Since numerical models are also tied the the computer resources available, it has become a field of fast progress in the last twenty years.

One of the fundamental events in the development of numerical methods appeared in a study of the late 1920s. The paper due to Courant, Friedrichs and Lewy set the basis of stability of the solution to certain type of PDEs by numerical methods (CFL condition).

Fluid Dynamic Models

With respect to speed and viscosity, the computational methods are governed by different sets of equations. Incompressible low speed flows are governed by the Laplace equation for the velocity potential, that is an equation of elliptic type. The problem is closed with Dirichlet or Neumann boundary conditions (or both).

Methods of solution are either the boundary element methods (panel methods) or finite element methods. Both can be written in discrete form using the Green identity for the Laplace equation.

The equations for subsonic compressible inviscid flows are still elliptic as long as the entire flow field is subsonic.They switch to hyperbolic in the supersonic pockets defined by the shock waves. Methods of solution have been developed to get around this difficulty.

The equations for inviscid supersonic flows are of the hyperbolic type, therefore they require initial data in the time-like direction. Viscous flows are of a wide variety of their own. The classical boundary layer equations (Prandtl) are of parabolic type and require initial conditions at one end of the integration domain.

Numerical Methods

Numerical methods for CFD are mostly concerned with the solution of system of partial differential equations, but the field is so broad as to include convergence acceleration methods (multi-grid, relaxation, artificial viscosity, etc.), stability control, pre-conditioning.

The equations must be classified prior to attempting their solution. Besides countless articles in archival journals, there are entire books (references below) devoted only to numerical schemes for a wide range of equations, therefore it is not the purpose of this simple note.

All the methods are broadly classified into finite differences, finite volumes, and finite element.

Parallel Computing

Given the sheer size of many CFD problems arising in industrial environments, aerodynamic components and processes, the step from sequential to parallel/vector programming is a necessary one. This requires fundamental changes in the hardware, in the language compiler, besides rational computer programming (the latter one to gain the maximum advantage from both hardware and compilers).


Parallel computers and clusters of sequential (single-processor) computers have been made available. One of the main ideas being pursued is the multiple-instruction multiple data (MIMD) processing.

The parallel CFD consists in distributing grid blocks to N different processors (nodes); performing CFD computations on each node; and finally combining the results from N nodes. The goal is to achieve linear speed up of the computer codes (a code shared by N processors wuould be N times faster.)


On the language side, important imporvements over the old Fortran 77 standard have been proposed recently. They include: Fortran 95, High Performance Fortran (HPI), etc.

Summary of Computational Methods

(avaiable on CD-ROM)

  • Panel Methods
  • Lifting Surface Methods
  • Boudary Layers
  • Viscous-Inviscid Interaction
  • Navier-Stokes Equations
  • Boltzmann Gas Lattice

Related Material

(available on CD-ROM)
  • Grid Generation

Selected References

The books listed below contain a large number of specialized publications (papers and collection of papers).

  • Anderson DA, Tannehill JC, Pletcher RH. Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Bristol, PA 19007, 1984.

  • Patankar SV. Numerical Heat Transfer and Fluid Flow, Hemisphere Publ., 1980.

  • Henne PA (editor), Applied Computational Aerodynamics, Progress in Aeronautics and Astronautics, Vol. 125., AIAA Inc. Washington D.C., 1991.

  • Wendt J (editor), Computational Fluid Dynamics, An Introduction, Springer-Verlag, 1996.

  • Ferziger JH, Peric M. Computational Methods for Fluid Dynamics, Springer, 1997.

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Copyright © A. Filippone (1996-2001). All Rights Reserved.