4 Computational Fluid Dynamics
301
implicit scheme proposed by Crank–Nicolson (1947), the artificial viscosity
method for calculating shock tube problems proposed by von Neumann
and Richtmyer (1950), conservative scheme of lax (1954), the alternating
direction method and characteristic line assembly method of Peaceman,
Rachford (1955), and Douglas (1956), particle method in the lattice of
Harlow et al. (1957), etc. Theoretical research had made fruitful progress
in the compatibility, convergence, and stability of difference schemes. Lax
equivalence theorem, von Neumann stability analysis method, and the analysis and formality of artificial viscosity term had acquired quite perfect results.
A variety of problems such as the problems of incompressible viscous flow
and shock tube were calculated. However, due to the limitation of computer
function and the difficulty of man–machine conversation, most of the fluid
dynamic models that could be realized were ideal fluid models. During
this period, the main purpose of the work was to lay the foundation of
computational fluid dynamics.
In the 1960s, the wide application of high-speed, large capacity, and multifunctional computers promoted the rapid development of various numerical
methods of fluid dynamics, such as lattice method (MAC, FLIC, CEL, etc.),
shock-capture method and assembly method, fractional step method and
operator splitting method, line method, spectrum method, random selection
method, and finite element and boundary element method. These methods
were more precise and could be adapted to a variety of practical problems.
At the same time, the research of mathematical model and the theoretical analysis of discretization method had been developed deeply, and lots
of analytical, discrete, and statistical hydrodynamic models had been established; the qualitative analysis theory of difference method, from previous
analysis of compatibility, convergence, and stability, to the theoretical analysis of dissipation, dispersion, transmission, monotonicity, and conservation,
and in addition, the analysis method had been developed from Fourier analysis to energy analysis and residual effect analysis. These theoretical studies
had provided the scientific basis for the design, selection, and application of
numerical methods.
In 1965, the American scientists Harlow and Welch put forward the idea
of a staggered grid, in which the velocity component and pressure are stored
on a grid with half step difference. In this way, the problem of the chessboard
unreasonable pressure field when the velocity and pressure are stored on the
same set of grids was effectively solved, and the original variable method (the
method with velocity and pressure as the variables) for solving the N-S equation (the differential equation of incompressible viscous fluid motion) was
established.
301
implicit scheme proposed by Crank–Nicolson (1947), the artificial viscosity
method for calculating shock tube problems proposed by von Neumann
and Richtmyer (1950), conservative scheme of lax (1954), the alternating
direction method and characteristic line assembly method of Peaceman,
Rachford (1955), and Douglas (1956), particle method in the lattice of
Harlow et al. (1957), etc. Theoretical research had made fruitful progress
in the compatibility, convergence, and stability of difference schemes. Lax
equivalence theorem, von Neumann stability analysis method, and the analysis and formality of artificial viscosity term had acquired quite perfect results.
A variety of problems such as the problems of incompressible viscous flow
and shock tube were calculated. However, due to the limitation of computer
function and the difficulty of man–machine conversation, most of the fluid
dynamic models that could be realized were ideal fluid models. During
this period, the main purpose of the work was to lay the foundation of
computational fluid dynamics.
In the 1960s, the wide application of high-speed, large capacity, and multifunctional computers promoted the rapid development of various numerical
methods of fluid dynamics, such as lattice method (MAC, FLIC, CEL, etc.),
shock-capture method and assembly method, fractional step method and
operator splitting method, line method, spectrum method, random selection
method, and finite element and boundary element method. These methods
were more precise and could be adapted to a variety of practical problems.
At the same time, the research of mathematical model and the theoretical analysis of discretization method had been developed deeply, and lots
of analytical, discrete, and statistical hydrodynamic models had been established; the qualitative analysis theory of difference method, from previous
analysis of compatibility, convergence, and stability, to the theoretical analysis of dissipation, dispersion, transmission, monotonicity, and conservation,
and in addition, the analysis method had been developed from Fourier analysis to energy analysis and residual effect analysis. These theoretical studies
had provided the scientific basis for the design, selection, and application of
numerical methods.
In 1965, the American scientists Harlow and Welch put forward the idea
of a staggered grid, in which the velocity component and pressure are stored
on a grid with half step difference. In this way, the problem of the chessboard
unreasonable pressure field when the velocity and pressure are stored on the
same set of grids was effectively solved, and the original variable method (the
method with velocity and pressure as the variables) for solving the N-S equation (the differential equation of incompressible viscous fluid motion) was
established.