The Seventh International Conference on numerical methods in hydrodynamics was held in June 1980 at Stanford, USA, and its proceedings are published in the present volume. It includes 4 survey lectures and 68 papers presented by authors from 15 countries, including the USSR, USA, France, China, Holland, and Japan.
Sixty of the papers are by authors from six of these countries, half being from the USA.
The subjects embrace a wide range of studies on numerical modelling of various fluid and gas flows.
Both the theory of numerical methods, and hydrodynamic applications in different fields of science and engineering, are covered. In some papers topics of computational mathematics predominate, while other papers focus on specific hydrodynamic problems and analysis of results.
We shall first dwell on the survey lectures. M.S. Longuet-Higgins (England) describes the use of polygonal conformal mappings to solve problems of fluid dynamics with a free boundary, in the cases when the latter is wavy with sharp vertices.
The polygonal mapping maps this boundary into a regular polygon, while the latter is mapped into the unit circle by the Schwartz-Christoffel formula, Mass transport in abrupt gravitational waves, surface waves over the ocean shelf, and the appearance of non-stationary waves due to the wind are considered as applications.
H. Viviand (France) gives a general survey of pseudo-non-stationary methods as applied to the numerical study of transonic gas flows.
These methods are desirable for finding the steady-state solution by means of time establishment, when the evolution of the solution is of no interest.
Pseudo-non-stationary methods can then be constructed, in which simplifications are introduced into the non-stationary terms of the equations, into the boundary conditions, and into the difference scheme.
The methods are notable for economy, flexibility, simplicity of the algorithm and programming, and speed of convergence of the solution, which is insensitive to topology and size variations of the mesh.
V.V. Rusanov's lecture deals with the calculation of the discontinuities arising in the field of multi-dimensional supersonic flow.
Difference schemes of the third order of accuracy are promising here; they have higher accuracy in the domain of the smooth solution,while smearing the discontinuity less and generating in its neighborhood weaker oscillations of the solution than do second-order schemes.
On the other hand, these schemes have inherently non-monotonic properties and are more difficult to realize in the case of irregular meshes. D.R. Chapman (USA) analyzes AIAA Journal publications over the last 20 years and traces the development and applications of computational aerodynamics.
Three problems arising in the aerodynamic computation of aircraft are discussed, namely: numerical modelling of the flow in the context of Euler's equations with subsequent allowance for the boundary layer, on the basis of the Navier-Stokes equations, and when considering large-scale turbulent eddies. It is remarked that, when solving these problems, there was 22% utilization of numerical methods in 1979, and this fraction is continuing to increase rapidly.
The papers mainly focussing on computational mathematics cover various topics. New difference schemes (notably, compact or implicit, or schemes with a high degree of solvability) are proposed, their properties are studied, moving computational meshes are constructed, and methods are given for accelerating the convergence of the iterations in iterative methods, etc.
The finite element method has gained wide popularity.
To compute three-dimensional nonstationary gas flows, C.P. Kentzer develops a numerical method of characteristics, based on the use of rays (bi-characteristics), thereby greatly simplifying the algorithm. Some papers consider transonic gas flows inside a nozzle or in the case of external flow past a body.
Of special interest here and three-dimensional cases (flow in the rotor of a turbine and flow past a wing of finite size), and cases of flow when account is taken of viscosity.
Great attention is paid to the numerical study of various (two-or three-dimensional, or non-stationary) motions of viscous fluid, described both by the boundary layer equations and by the Navier-Stokes equations.
The extremely complex computation of the non-stationary three-dimensional flow of viscous fluid through a cube-shaped cavity is performed by J.P. Benque, Y. Coeffe and R. Herleden.
They integrate the Navier-Stokes equations by a method proposed earlier by Chorin, and introduce two different models of turbulence.