In this Chapter
- Overview
- Numerical Methods
- Parallel Computing
- Summary of Computational Methods (available on CD-ROM)
- Lifting Surface Methods
- Panel Methods
- Boundary Layers
- Viscous-Inviscid Interaction
- Navier-Stokes Equations
- Grid Generation
- Selected References
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.
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).
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 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.
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.
The books listed below contain a large number of specialized publications (papers and
collection of papers).
Importance of Experimental Methods
Fluid Dynamic Models
Numerical Methods
Parallel Computing
Hardware/Software
Languages
Summary of Computational Methods
(avaiable on CD-ROM)
Related Material
(available on CD-ROM)
Selected References