Pictoral diagram of electrothermal instabilities on a cylindrical liner with an axial flow of electrical current. The filamentation form (a) with kparallel to B, occurs when . The striation form (b) with wavevector k perpendicular to B, occurs when .
Schematic diagrams of experimental hardware configuration. (a) Half section of final power feed and load, which depicts the load (rod), VISAR, and B-dot current diagnostics used in these experiments. (b) Photograph of installed rod and current return can prior to the installation of VISAR diagnostic fibers. (c) Schematic diagram of the 2-frame x-ray radiography configuration which shows the 3 degree angular separation between each frame.
Surface map of an Al 5052 rod (a) and a Cu rod (b) as measured with laser interferometric metrology at 50× magnification. (c) Three dimensional surface map of the same data for one of the Al rods.
Azimuthally averaged power spectrum for the Al and Cu rods. The dominant wavelength of that appears is a result of the diamond turning process, which creates a small helical groove on the surface as it turns.
1D radial profiles of density (blue), magnetic pressure (red), material pressure (green), and total effective pressure (black) at times (a) (t = −2 ns and 2 MA), (b) (t = 13 ns and 6 MA), and (c) (t = 23 ns and 8 MA) from 1D-HYDRA RMHD simulations.
Identification of regions of density and temperature phase space for Al that are unstable to the striation form of the electrothermal instability. Superimposed on this plot is the parametric evolution of several Lagrangian points located at various initial depths from a 1D simulation. The initial Lagrangian point at the surface of the rod is the lowest trajectory in the plot. Progressively, greater initial depths are listed by their value in microns.
Log density contours from a 2D Al rod LASNEX simulation which show (a) the initial development of electrothermal instabilities immediately after the outer surface layers melt, (b) majority of electrothermal instability development saturating near peak expansion, (c) development of MRT instabilities seeded by previous electrothermal growth as the rod begins to compress, (d) and MRT instabilities entering a stage of non-linear growth.
RMS perturbation in areal density as a function of time from post processed 2D LASNEX calculations of a nominal Al rod simulation (red), a Al simulation with 10× enhanced thermal conduction (green), and a Al simulation with enforced constant electrical resistivity (blue).
Resistivity of Al as a function of temperature for several densities.
(a) Log density contours of an Al rod simulation with an enforced constant electrical conductivity throughout the simulation. (b) Nominal Al rod simulation (SESAME 3719, 29373). (c) Identical simulation with a 10× enhancement in Al thermal conductivity. Both the constant electrical conductivity and enhanced thermal conduction cases exhibit substantially less instability growth, as predicted by electrothermal instability theory.
(a) Experimental and simulated 6151 eV radiographs of the Cu rods at the measured times. The red horizontal lines in the experimental images represent the original radius of the rod. All of the images were taken at a time when MRT instabilities are dominant. In the last two frames, which were taken on the same shot (Z1802), a large azimuthally symmetric inclusion in the target fabrication was present. This inclusion which is evident as the single largest perturbation allowed axial alignment of these frames, and a direct measure of the instability growth that occurs over 14 ns. (b) Current as a function of time measured by the load and magnetically insulated transmission line (MITL) B-dots for the Cu rod experiments. Also shown is the VISAR inferred load current on Z1802 (blue). The MITL B-dots show excellent reproducibility for all of these experiments. We, therefore, assumed the VISAR load current to be identical for all of the experiments despite the fact that the load B-dot measurements show significant discrepancies. Blue squares on the VISAR inferred load current represent the relative time and load current each of the radiographs were taken at.
(a) Experimental and simulated 6151 eV radiographs of the Al rods at the measured times. The initial expansion of the outer surface of the rod can be seen relative to the red horizontal line, which shows the position of the initial rod radius. Instability growth at this time is too small to be resolved. By frame (2), 14 ns later, instabilities have grown sufficiently to be marginally resolved. (b) Current as a function of time measured by the load and magnetically insulated transmission line (MITL) B-dots for both Al rod experiments. Also shown is the VISAR inferred load current on Z1801 (blue). The MITL B-dots show excellent reproducibility for both experiments up to approximately 15 MA and varied somewhat thereafter. However, the load B-dots showed relatively consistent load currents through the time at which the final radiograph was taken. We, therefore, assumed the VISAR load current on Z1801 to be the same for the other Al rod experiment, Z1913. Blue squares on the VISAR inferred load current represent the relative time and load current each of the radiographs were taken at.
(a) Radial surface trajectory for the Cu rod determined from the 50% transmission contour of the simulated radiographs with experimental points overplotted for comparison. Data points with large error bars are the result of being unable to obtain a preshot radiograph. Uncertainty in the relative timing is +/− 1 ns, which makes the error bars related to timing smaller than the width of the data points. (b) Simulated radial trajectories for various EOS and conductivity model combinations.
Radial surface trajectory for the Al rod determined from the 50% transmission contour of the simulated radiographs. Experimental data points are shown in black for comparison. In contrast with Cu, the simulated trajectory for Al using the VISAR measured current (solid curve) lies within experimental error bars.
Comparison of the Al (a) and Cu (b) rod perturbation amplitudes as determined from the 50% transmission contour from radiographs. Solid lines represent amplitudes from simulated radiographs while the discrete points represent experimental data. Time t = 0 starts in the bottom right hand side of each plot and proceeds up and to the left as the rod is compressed. In the simulated curves, it is easy to see the fastest growing instabilities occur during the initial expansion. Cu shots that did not have an associated preshot radiograph have significantly greater uncertainty in the radial position, represented by error bars with dashed lines, and do not provide a meaningful comparison with simulation.
Comparison of the simulated and experimental instability power spectrums as determined from the 50% transmission contour from radiographs. Simulated Al spectra (a) and (b) are represented by blue curves while experimental data is represented by the black curves at t = 23.57 ns and t = 83.04 ns. Simulated Cu spectra (c) and (d) are represented by green curves while experimental data are represented by the black curves at t = 129.45 ns and t = 143.85 ns. Both Al and Cu simulations do a reasonable job of capturing the dominant wavelengths despite highly nonlinear instability growth that has evolved several orders of magnitude in amplitude.
Two color radiography image of the Cu rod taken at t = 95 ns. Image (a) is the image taken at 6151 eV while (b) was taken 2 ns later (effectively the same time) at 1865 eV.
2D composite Abel inverted log density contours formed from two color radiographs obtained at 1865 eV and 6151 eV for both experiment and simulation. Note the maximum density that can be inferred with this technique is , which occurs when the transmission at 6151 eV falls to zero. At this radius, the radial density is artificially set to the maximum density for illustrative purposes.
Axially averaged Cu Abel inverted density profile from both simulation (black curve) and experiment (red curve). The shaded region indicates where the radiograph transmission approaches zero. The outermost edge of this region indicates the maximum density that can be inferred from the Abel inversion. The lower limit of density plotted approximately represents the lower limit of density that can be inferred with this technique, which occurs when radiograph transmission at 1865 eV approaches unity.
Axially averaged Abel inverted density profiles from both simulation and experiment for the last two frames of Al radiographic data at (a) times t = 37.6 ns and t = 83 ns (b). Black curves represent simulated Abel inverted density profiles and the red curves represent Abel inverted density profiles from the experimental radiographs. The shaded region indicates where the radiograph transmission approaches zero. The outermost edge of this region indicates the maximum density that can be inferred from the Abel inversion. The lower limit of density plotted approximately represents the lower limit of density that can be inferred with this technique, which occurs when radiograph transmission approaches unity. However, it should be noted that our uncertainty in transmission of a few percent results in larger uncertainties at the lower inferred densities.
(a) 6151 eV unprocessed radiograph of one of the Al rod experiments, Z1801. The darker shaded region on the left hand side of the rod due to transmission through a partially opaque Be return current post. (b) Same radiograph magnified, rotated 90°, and cropped to show only the right hand side of the rod. The 50% contour trace of this image of the rod is overlayed in white while an additional contour trace of the transmission on the left hand side of the rod is overlayed in red. Only the largest perturbations show complete azimuthal correlation.
Summary of Al and Cu rod experiments.
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