Hydrodynamic-dissipation relation for characterizing flow stagnation

Hydrodynamic stagnation converts flow energy into internal energy. Here we develop a technique to directly analyze this hydrodynamic-dissipation process, which also yields a lengthscale associated with the conversion of flow energy to internal energy. We demonstrate the usefulness of this analysis for finding and comparing the hydrodynamic-stagnation dynamics of implosions theoretically, and in a test application to Z-pinch implosion data.

Figure 1

(a) Inferred kinetic energy dissipation rate ($y=\eta m{v}^3/2R$) at the flow scale ($2R$) vs kinetic energy dissipation rate ($x=-{\dot{E}}_K$) for a region containing different initial proportions of kinetic energy in self-similar cylindrical stagnation flow and nonstagnating (turbulent) flow. When $k_i=0$ (blue, thicker solid line), the region contains only the self-similar stagnation flow. When $k_i=10$ (purple, thinner solid line), the nonstagnating (turbulent) flow has ten times the energy of self-similar flow and dominates the dynamics; intermediate cases are also shown. Time (implicit) progresses clockwise along each curve; in the $k_i=10$ case, it progresses from right to left. For each case, both dissipation values are normalized to the peak $-{\dot{E}}_K$. (b) The fractional hydrodynamic (dissipation) lengthscale $f_l$ vs the kinetic energy dissipation rate for the cases from (a); time is implicit as above. Plotted model parameters $h_i=2.5$, $\overline{\lambda }=0.35$, and $\eta =0.42$; see Sec. 2a.

Figure 2

(a),(b) Scatter plots of $y$ vs $P_R$ for Z-pinch experiment data and synthetic data from matched 2D simulations, respectively. On the right, the continuous simulation data are shown (black, solid) in addition to simulation points matched in time to the experimental measurements (plus one “additional” data point because the last matched time is $P_R\sim 0$ for the simulations). In both cases, time progresses clockwise from the leftmost (green) triangle. (c),(d) Instantaneous values of the fractional hydrodynamic lengthscale, $f_l$, inferred taking $-{\dot{E}}_K\approx P_R$ for the experiments and simulations, respectively, after peak emission as stagnation is approached.