Numerical Methods
Summary
Methods fow Potential Flows
A fundamental development for transonic potential flows is the upwind differencing, to overcome the symmetry of the solution obtained by central schemes (this scheme in fact is unable to distinguish a flow reversal). Central differencing can be used in subsonic flows and the resulting system solved by a line relaxation process.
The switching, however, may violate the conservation of mass in presence of shock waves. The problem is solved by introducing appropriate artificial viscosity.
Due to the slow convergence characteristics of the numerical method, several
acceleration techniques have been proposed and applied: These include factorization
of the difference operators, and multi-grid techniques.
Most numerical techniques for the Euler equations rely on time-dependent
formulations to reach a steady-state solution (but there are also non-linear
Newton-Raphson methods). This strategy allows the treatment of steady and
unsteady flows with the same formulation and non-oscillatory schock-capturing
schemes.
Time-stepping techniques of the early days include the McCormack scheme.
For steady state solutions the way the calculation procedes is inessential, therefore
any fast time-stepping will do it. Current methods are classified as explicit
(Lax-Wendroff, multi-stage methods, such as leap-frog, Adams-Bashfort and
Runge-Kutta) and implicit (LU decomposition, alternating-direction ADI,
relaxation).
The space discretization needs dissipation terms, in order to damp numerical
oscillations and to have more effective shock-capturing at transonic speeds.
Dissipative schemes come in a wide variety, that include total variation diminishing
(TVD), with or without flux limiting, flux-vector splitting, etc.
Methods for the full Navier-Stokes equations are of the same variety as for
the Euler equation. The additional viscous terms can be approximated by one
adequate numerical differentiation technique.
The NSE that are usually solved
are in fact Reynolds-averaged equations, due to the large disparity of
length scales between turbulence and integration domain. At the Reynolds
numbers of interest in aerodynamics flows become easily turbulent and closure
equations are needed. This is presently one of the great difficulties in CFD.
The number of publications on numerical methods is staggering. Please look at
the references below to get started, and ask if you need more.
Methods for the Euler Equations
Methods for the Navier-Stokes Equations
Selected References