The first set of figures show the temperature field at times t=1.0, 2.0 and 4.0. After the initial linear mixing phase, bubbles (rising from below) and spikes (falling from above) begin to form. Afterward, the horizontal fluctuation scales grow in size and begin to merge and coalesce. For this problem, the system eventually will return to rest in equilibrium with the acceleration field. The next figure shows the spectra of vertical velocity (in terms of squared amplitude per mode) at the midplane as a function of horizontal wave number for various grid resolutions. The 512-cubed case appears to be converged (with respect to grid resolution), with the middle portion of the curve representing a possible inertial range.
The first figure below shows the passive scalar at t=3, which is in the aftermath of the interaction of the shock with the contact discontinuity. The development of fine-scale, non-chaotic features is evident. The next two figures depict the result of passing an additional Mach 6 shock through the interface from the same and opposite sides, respectively. There is a distinct change in character, as much of the fine-scale structure is smeared out. The next figure shows the width of the mixing layer (measured by the departure of the passive scalar from its initial values) as a function of time. For the second shock phase (rightmost curve), the diamonds indicate the case where the second shock is in the same direction as the first, and the crosses indicate the case where the second shock is coming from the opposite side. For the case in which the second shock is in the same direction as the first, there is an initial compression followed by a significant acceleration of the mixed layer width. As noted earlier, the character of the solution changes significantly, as smaller-scale structures evolve and the fine structure becomes blurred. When the second shock is applied in the opposite direction, there is inversion followed by similar behavior. In fact the forward and reverse shock cases differ by only a slight delay, perhaps due to the time required for the inversion.
The calculation was carried out in three dimensions on a 2048 X 2048 X 1920 mesh, using 960 nodes, arranged in an 8 X 8 X 15 domain decomposition. The simulation ran for just over a week, and attained roughly 0.6 teraflop per second of performance. Over 3 Tbytes of graphics data were produced, distributed over close to 300,000 files.
A volume rendering of the entropy at the conclusion of the calculation is shown in the figure below. The shock had moved from top to bottom and then partially reflected off the bottom wall. We are working in a reference frame in which the contact discontinuity, after being encountered by the shock, is relatively stationary.
We have created several videos. One video shows the characteristic development of bubbles and spikes and their subsequent merger and break-up. The 3-D simulation shows marked contrasts from coresponding 2-D simulations, which indicate a preferred progression to larger, rather than smaller, scales as the simulation progresses. The high resolution allows elucidation of fine scale physics; in particular, when compared with coarser resolution cases, we observe a possible transition from a coherent to a turbulent state with increasing Reynolds number. A second video shows the end of the simulation, observing it from a variety of positions and angles.
A more comprehensive video, containing background music, shows time evolution followed by observations of the final state from various angles, distances, and from within.
A parallelized compressible PPM code has been used to study the interaction of shock waves with a pre-existing three-dimensional turbulent field in a compressible fluid. Such a situation can arise, for example, when multiple shocks pass through an interface of different-density materials. The first shock can produce a Richtmyer-Meshkov instability; the resultant turbulence interacts with subsequent shocks. Of interest is how the shock affects the turbulence --- in particular the turbulence strength, spectrum, anisotropy, and rate of shock propagation.
We first run a 3-D decay problem with triply periodic boundary conditions, starting from an initial k**2*exp(-k**2) spectrum of 8 modes per direction with a width of 2 modes. The initial turbulent Mach number is 0.7, and the initial pressure and density perturbations are zero. This initial state is allowed to decay to a turbulent Mach number of about 0.2, at which time inflow boundary conditions are invoked. At one end (z=0), we specify a density, pressure, and z velocity consistent with downstream conditions for a shock of specified Mach number moving into a fluid at rest (the final state of the decay portion of the simulation). At the other end (z=1), reflecting boundary conditions are imposed. The boundary conditions in the x and y directions remain periodic. As the simulation proceeds, the shock forms, moves from z=0 to z=1, and reflects. At any instant prior to reflection, there is an unshocked turbulent region between the shock and z=1, a shocked turbulent region extending downstream from the shock, and finally a region filled with quiescent fluid that moves in from the inflow boundary. All results described below are for pure PPM Euler (no explicit dissipation) simulations and were run at various resolutions, up to 512 points in each direction.
The first figure shows a 3-D rendering of the v-z field (with the x-y average subtracted off), for a 512-cubed grid. The shock location and the post-shock amplification are evident. The next two figures show z-profiles of the root-mean-square x-y averages of each vorticity component, for Mach 2 and Mach 6 shocks, at 256-cubed resolution. From these figures we note that (a) the x and y vorticities increase immediately behind the shock, (b) the z vorticity initially decreases behind the shock, but subsequently recovers and increases, and (c) the Mach 6 effects are appreciably stronger. The last figure shows v-z spectra for several mesh resolutions (the cases marked 95, 191 and 363 have 128, 256 and 512 points, respectively, in each direction). These spectra suggest that a meaningful inertial range is present in the 256-cubed and 512-cubed simulations, and that the results at 256-cubed and 512-cubed are converged in the energy containing range and the captured portion of the inertial range. Finally, there is the question of the influence of the turbulence on the shock; we find that the shock is (slightly) sped up by the presence of the turbulence, and the shock front is broadened by the turbulence.
Last modified: August, 1999
UCRL-MI-126772 Rev. 2