Eric Johnsen

The dynamics of supersonic shock-droplet interactions are important in applications ranging from rotating detonation engines to hypersonic vehicle design. Studying these systems numerically is challenging as the interface between the liquid and surrounding gas must be tracked accurately and conservatively across a wide range of length scales, and different interface regularization techniques can dramatically affect the flow physics. In this work, four interface treatments are compared in the context of a canonical two-dimensional shock-droplet interaction: no regularization (Euler), the tangent-of-hyperbola interface capturing (THINC) reconstruction, and the conservative diffuse-interface (CDI) and accurate conservative diffuse-interface (ACDI) models. A compressible six-equation multiphase model is used to evolve the flow and all four are evaluated at shock Mach numbers $M_s = 2$ and $M_s = 3$. The effects of each technique are assessed qualitatively through droplet morphology and quantitatively through the liquid mass centroid trajectory and integral liquid-gas interface length, which together characterize bulk droplet motion and the development of small-scale interfacial structure. The simulations show that all four methods capture the early-time parent-droplet dynamics with broad agreement, but diverge substantially in the late-time wake structure, the degree of child-droplet development, and the total liquid-gas interface length. The Euler scheme produces continuously diffusing filamentary structures with no well-defined children, while THINC, CDI, and ACDI all produce distinct child droplets, but with morphological differences that depend on numerical parameters and Mach number. These results demonstrate that while bulk droplet translation is largely insensitive to the choice of interface treatment at early time, the resolved wake structure and droplet morphology are strongly model-dependent such that the choice of regularization mechanism and its parameters must be informed by the target observable in predictive simulations.

Prior work has shown that pulsed high-powered microwave exposure can lead to rapid thermoelastic expansion of brain tissue which is subject to a strain-focusing effect. The rapid increase in thermal strain causes pressure waves to form in the brain and reverberate as tensile waves. This study investigates possibly synergies between brain material properties, like viscosity, shear modulus, and cavitation nuclei size, and pressure waveform properties like amplitude and frequency. Cavitation is modeled using the Keller–Miksis equation modified to incorporate viscoelastic effects. A fourth-order variable time step Runge–Kutta scheme calculates bubble radii and wall velocities from the pressure profile. By performing a 6-D parameter sweep, regions of the parameter space where cavitation is more likely to occur are found. This will identify exposure limits to minimize microwave induced brain injury.

The inertial collapse of a cavitation bubble concentrates potential energy within the bubble. This process redistributes energy between the bubble and the surrounding liquid, while also emitting a shock wave. In the incompressible limit, bubble dynamics are governed by the driving pressure, but at high collapse speeds, compressibility effects become critical. We present a theoretical framework that corrects radiated energy estimates, derives closed-form expressions for bubble energy and volume at collapse, and relates shock pressure directly to governing parameters. Our results provide a foundation for describing more complex systems, such as bubble clouds and bubbles near boundaries.

Mechanical loads produced from the impact of droplets on high-speed projectiles are important considerations for their operation. To accurately predict the pressure produced along the surface of the object during impact, it is important to correctly model the air/water interface of the droplet. Novel modeling approaches such as the Phase-Field model enable control of the (numerical) interface thickness. However, the interactions of the grid resolution and interface thickness with the resulting simulation results are not well understood. In this work, the resolution requirements for numerical simulations of a water droplet impacting a rigid wall at Mach 4 are computationally investigated in two dimensions. Numerical simulations are performed using a second-order accurate method with adaptive mesh refinement (AMR) and a consistent and conservative Phase-Field method. The extent to which the maximum wall pressure and its location depend on the choice of numerical interface thickness and resolution parameters (mesh size, levels of mesh refinement) is quantified. The work provides a comprehensive resolution study by investigating the convergence of the Phase-Field parameter, AMR levels, and grid size required for various flow problems to determine suitable numerical parameters for interface control provided by the Phase-Field mechanism.

We present the use of x-ray fluorescence spectroscopy (XFS) to a sensitive temperature diagnostic in shocked foams at temperatures of 30-75 eV. Cobalt-doped foams were shock compressed using a planar drive at the OMEGA laser facility and photo-pumped with a Zn Heα x-ray source. Analysis of the resulting cobalt Kβ x-ray fluorescence spectra using collisional radiative codes allows the temperature to be determined in the shocked foams. This method provides a sensitive and robust technique to determine temperatures in high-energy-density physics experiments in the tens of electronvolts temperature range. In these experiments, we find that radiation hydrodynamic simulations predict a lower temperature in the shocked foams compared to analysis of the XFS data using collisional radiative models. Although additional experiments with an independent temperature diagnostic to absolutely calibrate XFS spectra for these conditions will be required to resolve this discrepancy, these results demonstrate the excellent temperature sensitivity of XFS spectra for high-energy-density physics experiments.

Solids are often assumed to behave as viscous fluids under high-strain rates. This behavior has been studied experimentally in metals but largely unexplored in brittle ceramic materials. In this study, we present a technique for measuring the viscosity of MgO using time-resolved velocimetry to track the growth rate of a perturbation caused by the Richtmyer-Meshkov instability at the OMEGA EP laser facility. To interpret the results, we use an in-house Eulerian hydrocode to simulate our experiments and model the plastic deformation of solid MgO as a viscous fluid. Results indicate that MgO has a surprisingly low upper bound to its effective viscosity of $\ensuremath{\sim}{10}^{2}\phantom{\rule{0.16em}{0ex}}\mathrm{Pa}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ at $175\ifmmode\pm\else\textpm\fi{}15\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}, \ensuremath{\sim}3500\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, and ${10}^{6}\ensuremath{-}{10}^{7}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ strain rate.

Despite the extensive research on bubble collapse near rigid walls, the bubble collapse dynamics in the presence of shear flow near a rigid wall is poorly understood. We conduct direct simulations of the Navier–Stokes equations to explore the bubble dynamics and pressures during bubble collapse near a rigid, flat wall under linear shear flow conditions. We examine the dependence of the bubble collapse morphology and wall pressures on the initial bubble location and shear rate. We find that shear distorts the bubble, generating two re-entrant jets – one developing from the side opposite to the mean flow and the other from the far end toward the wall. Upon impact of the jet on the opposite side of the bubble, water-hammer shocks are produced, which propagate outward and interact with the convoluted bubble shape. The shock stretches the bubble towards the wall, resulting in a closer impact location for the jet originating from the far end compared with the case with no shear flow. The water-hammer pressure location can be approximated as the theoretical distance travelled by a particle initialised at the bubble centre with the corresponding constant shear flow velocity. The maximum wall pressures can thus be predicted by considering the distance between the far jet impingement location and the wall along the wall-normal direction. As the shear rate is increased, the maximum wall pressure increases, although only marginally. We determine the critical initial stand-off distance from the wall at which the bubble morphology is shear dominated, i.e. characterised by converging re-entrant jets.

Bound-preserving analysis is performed for the consistent and conservative Phase-Field method for compressible N -phase flows ( N ⩾ 1 ). A convex admissible set is first proposed, which follows a physical energy constraint and only requires convex equations of state , independent of the multiphase flow models. Under piece-wise polynomial approximation and for the HLL flux, our analysis determines the requirements to preserve physical bounds. A new numerical Phase-Field flux vector is derived to maintain solutions in the admissible set. Our analysis consistently reduces to two- and single-phase flows and is general for any choice of phasic minimum pressures. A parabolic correction for conservation is proposed to develop a numerical approach that not only preserves physical bounds but also maintains consistency, conservation, and equilibria of velocity, pressure, and temperature. Various compressible single-/two-/ N -phase flows are investigated to verify the analysis and numerical approach. The extension of our analysis to general equations of state is also discussed and verified.

Refining and coarsening grids depending on solution needs is a means to address the prohibitive cost of high fidelity turbulence simulations, but nonuniform grids can lead to issues, particularly when combined with high order methods. One issue is that of numerical reflections, which can occur when a wave crosses a change in grid resolution. This issue was first studied for the Discontinous Galerkin method for changes in the grid spacing h in 1D by Hu and Atkins. We extend their eigensolution analysis approach to changes in polynomial order p and simultaneous changes in the grid spacing h and the polynomial order p, and reinterpret the results in the context of an adaptively refined grid. We find that the numerical reflections can be orders of magnitude larger than spatial discretization errors, though they are confined to regions near the interface. We find that for some combinations of changes in h and changes in p, the reflections due to a grid interface decrease in amplitude, but we did not find any combinations of changes in h and p that fully eliminate reflections. We perform numerical tests to verify our analysis and evaluate how well it holds for nonlinear cases. Future work will include extension of the eigensolution analysis to changes in h and p in two dimensions, where there is no general solution that prevents the reflections, and they are not confined to near the interface.

Prior work by Dagro et al. (2021) has shown that high-powered microwave exposure can lead to rapid thermal expansion of brain tissue and has a strain-focusing effect. The rapid increase in thermal strain causes pressure waves to form in the brain and reverberate as tensile waves. This study investigates the possibility of tissue damage from these tensile waves through the growth of intrinsic cavitation nuclei. Cavitation is modeled using the Keller–Miksis equation modified to incorporate viscoelastic effects. A fourth-order variable time step Runge-Kutta scheme calculates bubble radii and wall velocities from the pressure profile. Our simulations show that the maximum hoop stretch caused by high-powered microwave exposure considered in Dagro et al. (2021) is less than the threshold needed to cause morphological degeneration in neural cells, as given in Estrada etal. (2021). By performing a parameter sweep, we find that pressures 1000 times larger than those reported by Dagro et al. (2021) are needed to cause injury. These correlations can help provide safety limits against brain injury due to microwave energy.

Passive flow control devices, such as vortex generators (VGs), can effectively modulate the turbulent boundary layer flow near regions of adverse pressure gradients, but the interactions between the salient flow structures produced by VGs and those of the separated flow are not fully understood. In this study, a spatially evolving turbulent boundary layer interacting with a wall-mounted cube ahead of a backward-facing ramp is investigated using wall-resolved large-eddy simulations for a Reynolds number of 19,600, based on the inlet boundary layer thickness and freestream velocity. Different cube configurations are examined to isolate the effects of cube height and streamwise position. The large counter-rotating flow produced by the larger cubes generated more turbulent kinetic energy, thereby resulting in smaller separated regions over the ramp. Although significantly lower levels of turbulent kinetic energy dissipation than production are observed in the outer flow regions of the horseshoe vortex and in the induced counter-rotating flow, the spatial distribution of dissipation is similar to that of the turbulent kinetic energy. When the upstream position of an isolated single cube, relative to the leading ramp edge, is more than three cube heights, the streamwise decay of the counter-rotating flow results in lower levels of turbulent kinetic energy.

The growth of three-dimensional perturbations subject to the Crow instability along a vortex dipole resulting from the passage of a shock wave through a heavy gaseous cylinder is examined numerically. A linear stability analysis is performed based on geometric parameters extracted from two-dimensional simulations to determine the range of unstable wavenumbers, which is found to extend from 0.0 to 1.3 when normalized by the core separation distance. The analysis is then verified by comparison to three-dimensional simulations, which clearly show the development of the instability and the pinch-off of the vortex dipole into isolated vortex rings, which manifest as clumps of the original cylinder material. A scaling law is developed to determine the relevant spatiotemporal scales of the instability development, which is then used to assess the feasibility of a high-energy-density experiment visualizing clump formation. Specifically, a shocked cylinder with an initial diameter of 100 μm consisting of a perturbation of approximate wavelength and amplitude of 600 and 10 μm, respectively, is expected to form clumps resulting from the Crow instability approximately 40 ns after it is shocked, with dynamics which can be readily visualized on the Omega EP laser facility.

An explanation for the origin and number of clumps along the equatorial ring of Supernova 1987A has eluded decades of research. Our linear analysis and hydrodynamic simulations of the expanding ring prior to the supernova reveal that it is subject to the Crow instability between vortex cores. The dominant wave number is remarkably consistent with the number of clumps, suggesting that the Crow instability stimulates clump formation. Although the present analysis focuses on linear fluid flow, future nonlinear analysis and the incorporation of additional stellar physics may further elucidate the remnant structure and the evolution of the progenitor and other stars.

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

The five-equations model has been used extensively to study compressible multiphase and multi-component flows using lower-order numerical schemes, while the solution to these equations with a high-order scheme has received far less attention. In this work, we present a discontinuous Galerkin finite element scheme with a novel limiting procedure that can efficiently resolve compressible multiphase and multi-component flows with high-order accuracy. The proposed scheme is conservative, preserves velocity, pressure, and temperature equilibria, bounds the phasic masses, the volume fraction, and the pressure for various convex equations of state, and can be used with a variety of limiters. We present several numerical tests in one and two dimensions to verify and validate our methodology, including a multi-material Richtmyer-Meshkov instability, the impingement of a shock in air with an SF6 block, and the interaction of a strong shock in water with an air bubble.

The origin of the clumps along the gaseous circumstellar ring surrounding the remnant of Supernova 1987A has puzzled scientists for decades.Twenty-thousand years prior to the supernova, the interaction between solar wind from the progenitor star and the ring likely generated vorticity conducive to the formation of a circular vortex dipole subject to the cylindrical Crow instability, as shown above.Our analysis predicts a dominant unstable wavenumber consistent with the number of clumps, and simulations reproduce both the clumping behavior and the thin annulus of mass, shed by the vortex dipole, recently observed with the James Webb Space Telescope, as shown below.

Structures evoking vortex rings can be discerned in shock-accelerated flows ranging from astrophysics to inertial confinement fusion. By constructing an analogy between vortex rings produced in conventional propulsion systems and rings generated by a shock impinging upon a high-aspect-ratio protrusion along a material interface, we extend classical, constant-density vortex-ring theory to compressible multifluid flows. We further demonstrate saturation of such vortex rings as the protrusion aspect ratio is increased, thus explaining morphological differences observed in practice.

A numerical model for cavitation in blood is developed based on the Keller-Miksis equation for spherical bubble dynamics with the Carreau model to represent the non-Newtonian behavior of blood. Three different pressure waveforms driving the bubble oscillations are considered: a single-cycle Gaussian waveform causing free growth and collapse, a sinusoidal waveform continuously driving the bubble, and a multi-cycle pulse relevant to contrast-enhanced ultrasound. Parameters in the Carreau model are fit to experimental measurements of blood viscosity. In the Carreau model, the relaxation time constant is 5-6 orders of magnitude larger than the Rayleigh collapse time. As a result, non-Newtonian effects do not significantly modify the bubble dynamics but do give rise to variations in the near-field stresses as non-Newtonian behavior is observed at distances 10-100 initial bubble radii away from the bubble wall. For sinusoidal forcing, a scaling relation is found for the maximum non-Newtonian length, as well as for the shear stress, which is 3 orders of magnitude larger than the maximum bubble radius.

Damage from repeated bubble collapse to neighboring rigid objects is a consequence of cavitation. Few studies exist on the dynamics of bubbles collapsing near a corner, i.e., two flat rigid surfaces making a right angle. Here we solve the three-dimensional compressible Navier-Stokes equations for gas and liquid flows to quantify the pressure fields. The second wall affects the collapse by (i) causing the re-entrant jet to no longer point in the direction normal to the closest wall but at an angle toward the corner and (ii) causing the maximum wall pressure to be observed in the corner when the bubble is sufficiently close to equidistant from each boundary due to shock reflections.

View Video Presentation: https://doi.org/10.2514/6.2023-1400.vid We present in this paper our approach to expand the venerable Von Neumann analysis technique to nonuniform grids in one and two dimensions. This is a con- tinuation, and generalization, of our previous work on triangular grids, both struc- tured and semi-unstructured. The grid’s un-structuredness is introduced into a block of few grid cells, then this block is replicating over the whole computational domain. Resultant grids are essentially non-uniform or semi-unstructured at cell- to-cell level, but still retain a much-needed regularity at block level. We show the viability of this approach by analyzing the scalar linear diffusion equation on grids with varying degrees of non-uniformity. Our observations are then verified via numerical experiments on the same governing equation.