Instability and Mixing
The conditions at time t = 0 are the following: a vessel (twodimensional) contains a gas filling it partly, with a boundary against a vacuum. This boundary is not flat but has an irregularity in it in form of a triangular prominence jutting out with dimensions comparable to the diameter of the vessel (about one-fourth of its width).
We want to follow the behavior of the expansion of this gas, under its own pressure, in the vacuum below it. Two problems were considered:
1) the gas was assumed weightless, i.e., there was no external force acting on it;
2) the gas, in addition to its own pressure, was acted upon by a constant gravitational field.
The hydrodynamical setup used was the following: the gas was represented by 256 material points. These represent centers of mass of regions in the gas. The treatment is Lagrangian; that is, the calculation follows, in time, the position of these masses. The pressure gradients are represented by forces which our points exert on each other. These are repulsive forces, depending only on the distance between points (and thus having a potential "pressure") and are of the form Fij = a/rjI; where the exponent x depends on the adiabatic equation of the state of the gas considered.
We chose oc = I for our first problem. Given a point Pi (xi,yi), one considers all forces exerted on it by other points (Xj, yj) and computes their resultant vector.
Actually, we limit the pj in computing the forces to the "neighbors" of pi; these are defined as the nearest eight points to pi. This is done for two reasons: the economy of the computation-we have to calculate only eight instead of 256 of the total number of forces for each point under consideration; the second, more fundamental reason is, of course, that in the gradient of pressure only the local configuration matters. The 1/r force law gives divergence at infinity. We might mention here parenthetically that in the search or scanning of points for the nearest eight to the given one, the following was adapted for purely practical, economy, reasons. The points really scanned were 50 "candidates" for the nearest neighbor. They were the 50 originally nearest to pi; the problem was not run long enough so that we had to relocate the original candidates but it is, of course, possible to recheck this periodically. In addition, in order to avoid the use of multiplications and operate
only with the much shorter addition times, we used, in the search for the nearest eight points, the non-Euclidean metric p(vi, vj) =I(xi xj) I + I (Yi- yj) Once the eight points were found, however, the true Euclidean distances were computed in order to find correctly the resulting force.
We shall not describe here the special treatment which has to be accorded to points which adjoin the walls of the vessel and the points on the boundary of the gas (with the vacuum).
Among the quantities that were printed as the results of the calculation we shall mention here only these: an interesting functional of the motion is the kinetic energy of the gas divided into two parts: the energy of the motion in the x direction and in the y direction. We study E1 = 1/2 Emi.2 and E2 = 1/2 Emi]2.
The ratio of the two is a function of time and can serve, in a way, as a measure of mixing or irregularity of the motion. One expects due to the initial irregularity of the boundary this ratio to be positive. From the beginning, sidewise motions ensue. Later on one would expect, of course, the motion to be predominantly downward; as the irregularity increases, the ratio of the two quantities should, in Problem I, increase again and approach a constant less than unity.
It is perhaps remarkable that the time behavior found for this ratio was extremely regular; a graph (Fig. 1) for the first 36 cycles is appended.
One word to explain the need to resort to the rather unorthodox procedures outlined above:
It was found impractical to use a "classical" method of calculation for this hydrodynamical problem, involving two independent spatial variables in an essential way (since the gas interface had an irregularity assumed from the beginning). This "classical" procedure, correct for infinitesimal steps in time and space, breaks down for any reasonable (i.e., practical) finite length of step in time. The reason is, of course, that the computation of Jacobians which define the compression assumes that "neighboring" points, determining a "small" area, stay as neighbors for a considerable number of cycles. It is clear that in problems which involve mixing specifically this is not true. Calculations were made just in order to observe the rapidity of change in the "neighbor" relations on the classical pattern and have shown just what was expected to happen: the proximity relations change radically, for points near the boundary, just when the mixing to be studied is starting and the neighboring relation of our points has to be redefined; i.e., the classical way of computing by referring to initial (at time t = 0) ordering of points becomes meaningless.
The problem can be treated, of course, using the Eulerian variables (of a set-up due to von Neumann to be discussed in our next report)* where this difficulty does not occur. This Eulerian treatment
* There was no subsequent report. (Eds.)
is not suitable, however, for the study of the shape of the boundary-a fictitious, i.e., purely calculational, diffusion and mixing obscures the very phenomenon one wants to study. We shall return to this question in our second report.
In the "classical" set-up involving calculation of Jacobians for determining pressure gradients the cycle was approximately two minutes.
In the problem involving the calculation of forces from the eight "neighbors," the cycle time was three minutes. Meaningful results are obtained in about 100 cycles. Appended are graphs showing the results of a few dozen cycles.