Hydrodynamical Problems; Heuristic Considerations
In general for problems involving two or more spatial variables, there is at present little hope for obtaining solutions analytically-in closed form. In the several sections that follow, heuristic considerations are set forth with exploratory calculations on the high speed computing machines performed in several cases. The purpose of these was primarily to establish the feasibility of such calculations, to estimate the sizes of problems which can be handled in a reasonable time, and in general to gain experience in the new methods and new fields which, in our opinion, are now open for investigation.
Our approach to the problem of dynamics of continua can be called perhaps "kinetic"--the continuum is treated, in an approximation, as a collection of a finite number of elements of "points;" these "points" can represent actual points of the fluid, or centers of mass of zones, i.e., globules of the fluid, or, more abstractly, coefficients of functions, representing the fluid, developed into a series, e.g., Fourier or Rademacher series. This corresponds to the use of general Lagrangian coordinates in classical mechanics; their use can be rigorously justified in problems where entropy is constant-i.e., holonomic systems.
They are always functions of time, which proceeds by discrete intervals.
We thus replace partial differential equations or integer-differential equations by systems of total differential equations. The number of
elements which we can at present handle is always less than 1,000, the limitation being primarily in the "memory" of the machines, the number of time intervals ("cycles") of the order of 100.
The problems which we study are characterized by lack of symmetry. The positions of our points become, as a rule, more and more irregular as time goes on. This has the consequence that the meaningful results of the calculations are not so much the precise positions of our elements themselves as the behavior, in time, of a few functionals of the motion of the continuum.
Thus in the problem relating to the mixing of two fluids, it is not the exact position of each globule that is of interest but quantities such as the degree of mixing (suitably defined); in problems of turbulence, not the shapes of each portion of the fluid, but the overall rate at which energy goes from simple modes of motion to higher frequencies; in problems involving dynamics of a star cluster, not the individual positions, but quantities like angular momenta of subsystems of the whole system, of size smaller but still comparable to the whole system, etc.
Needless to say, our investigations are in a most preliminary and rudimentary state; we have not made rigorous estimates of the disturbing or smoothing action of the roundoff errors which accumulate, nor the effects of finiteness of the time interval (i.e., the errors due to replacement of differential by differences expressions). In the individual cases that follow the reader will be able to judge for himself how far elementary common sense permits to estimate these effects.
We hope in the future to multiply considerably the number of calculations performed for each of our problems to assay the influence of changes in initial conditions on our conclusions. We repeat that so far the value, if any, of this work is only heuristic.
The first problem considered concerns the behavior of a gas confined in a vessel, expanding into vacuum under its own pressure and weight. The surface is not plane but has an irregularity of a finite size. In other words we consider the problem of instability in the compressible case.