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.

The passage of a shock wave through a heavy gaseous cylinder is one of the most canonical shock–interface interactions in compressible fluid mechanics.1–4 As the shock passes over the cylinder, baroclinic vorticity is generated from the misalignment of the pressure gradient, which points normal to the shock front, and the density gradient, which is dominated by the interface between the cylinder and the surrounding fluid.3 This vorticity, equal and opposite in magnitude on the top and bottom of the cylinder, drives the flow long after the conclusion of the transient shock dynamics, ultimately resulting in the formation of a vortex dipole visualized in experiments.1,5,6

Although this problem has been studied extensively in two dimensions, little attention has been paid to the three-dimensional dynamics of the resulting vortex cores. In three dimensions, the flow may be subject to perturbation growth via the Crow instability (CI).7 The CI, originally considered in the context of wingtip vortices shed into the wakes of aircraft, stimulates the growth of long-wavelength vortex core perturbations, which emerge as a mirror image of the perturbations on the other core.8,9 Due to this symmetry, the CI causes the cores to grow together at regularly spaced intervals given by a dominant wavelength. At each location where the vortex cores touch, a complex vortex reconnection process triggers the pinch-off of an isolated vortex ring.10–12 Past studies have reported evidence of a short-wavelength vortex core instability related to the CI following the interaction of a shock and a spherical bubble,13 a problem linked to the shocked cylinder given their related two-dimensional physics,1,4 suggesting that the vortex core dynamics driving the CI could affect shocked cylinders.

Furthermore, the CI has recently been proposed as a mechanism for stimulating the formation of clumps in equatorial gas rings surrounding stars,14 including that surrounding the progenitor of Supernova 1987A (SN1987A)15 and protoplanetary disks.16 In such systems, the interaction of solar wind with the equatorial gas induces shock waves and shear flow over the top and bottom of the ring17–19 conducive to the formation of a vortex dipole akin to that generated from the interaction between a shock and a heavy gaseous cylinder. As the CI-unstable perturbations grow, the circumstellar gas cloud thickens where the vortex cores separate and thins where the cores approach, resulting in the formation of the famous clumps along the equatorial ring of SN1987A20,21 and possibly planetesimals in protoplanetary disks.

Although experiments have examined shocked gaseous cylinders,1,2 a number of challenges in stimulating the CI exist with classical methods. First, the shocked cylinder must be long enough to support a wavelength unstable to the CI. For experiments utilizing a cylindrical film suspended between two plates to generate the initial interface between the heavy and surrounding fluids,1 it may not be possible to achieve adequate separation of the plates without bursting the film prematurely. Other experiments utilize a falling column of heavy gas within the driven section of a conventional shock tube as the initial cylinder.2,6 However, a well-characterized perturbation along the length of the falling column that would support perturbation growth via the CI has, to the authors' knowledge, never been achieved. Laser-driven shock wave experiments, however, possess the unique advantage of a high degree of control over the initial perturbation geometry.22–24 In such experiments, a laser deposits energy on the back side of an initially solid target. As material is ablated, a strong shock wave is driven into and immediately vaporizes the target, stimulating the flow of plasma. This technique has long been utilized to study classical fluid instabilities, including the Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz instabilities,25–30 but never the CI. However, experimental evidence of similar vortex core instabilities in high-energy-density flows exists,31–33 suggesting that an experimental study of the CI on a laser-driven platform is possible.

Our objective is to assess the feasibility of an experiment utilizing a laser-driven shock wave to examine the growth of the CI along a heavy cylinder at a facility akin to the Omega EP laser. The target would consist of a perturbed heavy plastic cylinder suspended in a lighter foam material, creating the interfacial density gradient necessary for vorticity deposition. The shock wave, driven perpendicular to the cylinder centerline, would pass through and vaporize the target, stimulating a flow subsequently visualized with x-ray radiography. In Sec. II, the stability theory characterizing the CI is applied to a vortex dipole generated from a shocked cylinder and subsequently verified with three-dimensional simulations. In Sec. III, a scaling law is developed to motivate the choice of experimental parameters enabling the visualization of the CI on time scales relevant to the Omega EP laser and spatial scales compatible with diagnostic resolution limits. Results are summarized and conclusions drawn in Sec. IV.

We consider a shock wave of Mach number M propagating through a fluid of density ρl and impinging on a heavy gaseous cylinder of density ρh and diameter D, as shown in Fig. 1(a). As the shock passes over the cylinder, baroclinic vorticity is generated due to the misalignment of the pressure gradient p, which points normal to the shock front, and the density gradient, which is dominated by and points normal to the cylinder surface.3 The vorticity ω deposited by the shock then governs the evolution of the flow and ultimately stimulates the formation of a vortex pair with core separation distance b, core thickness c, and circulation Γ.1,6

FIG. 1.

(a) Two-dimensional schematic illustrating the formation of a vortex dipole from a heavy gas cylinder before (left), during (center), and after (right) the passage of a shock wave. The transmitted and reflected shocks at the cylinder interface are labeled TS and RS, respectively. (b) Three-dimensional perturbed vortex cores.

FIG. 1.

(a) Two-dimensional schematic illustrating the formation of a vortex dipole from a heavy gas cylinder before (left), during (center), and after (right) the passage of a shock wave. The transmitted and reflected shocks at the cylinder interface are labeled TS and RS, respectively. (b) Three-dimensional perturbed vortex cores.

Close modal
In three spatial dimensions, this vortex pair could be unstable to the CI,6,7 which excites perturbations of wavenumber k in the plane orthogonal to the dipole cross section, as shown in Fig. 1(b), to grow exponentially with growth rate a. The original stability analysis considers perturbed planar vortex cores in an incompressible, inviscid, and irrotational flow governed by the Biot–Savart law, with the dynamics closed by a kinematic constraint requiring perturbations to move at the local flow velocity.7 A normal-mode ansatz is applied to the system, which is then linearized, and first-order terms are collected into a matrix eigenvalue problem from which the nondimensional growth rate of symmetric perturbations (i.e., perturbations on one core emerge as a mirror image of those on the other core) can be directly computed as
α = a 2 π b 2 Γ = [ ( 1 ψ + β 2 ϕ ) ( 1 + χ β 2 ϕ ) ] 1 / 2 ,
(1)
where
ψ = 0 cos β ξ + β ξ sin β ξ ( ξ 2 + 1 ) 3 / 2 d ξ ,
(2)
ϕ = δ cos ξ + ξ sin ξ 1 ξ 3 d ξ ,
(3)
χ = 0 cos β ξ ( ξ 2 + 1 ) 3 / 2 d ξ .
(4)
β = k b is the nondimensional wavenumber, k is the dimensional wavenumber, δ = k d is the nondimensional cutoff parameter, and d = 0.321 c.7 The core separation distance and thickness are, therefore, required to determine the three-dimensional stability of a vortex dipole formed along a laterally shocked heavy cylinder. These parameters are extracted from two-dimensional simulations, enabling the determination of the CI-unstable wavenumber range subsequently explored in three-dimensional simulations, as described next.

Given the large Reynolds numbers in both classical experiments examining shocked cylinders1 and laser-driven shock wave experiments exhibiting vortex core dynamics related to the CI,33 we examine simulations of the multi-species Euler equations for the setup outlined in Fig. 1(a) with M = 1.2 and ρ h / ρ l = 10. Computation is performed with our in-house, finite-volume, hydrodynamics code with a fifth-order weighted essentially non-oscillatory scheme, shown to be efficient and accurate for multi-component flows with shocks and interfaces,34–36 and adaptive mesh refinement.37 An ideal gas equation of state is assumed for both the cylinder and the surrounding gas, with adiabatic index γ = 1.4 throughout the domain ( γ = 5 / 3, as is typical for a plasma, is also examined, with little difference in the results when scaled by the sound speed). The equations are nondimensionalized by the sound speed in the unshocked light fluid us and the diameter of the cylinder. The spatial domain occupies x [ 0 , 44 ] in the direction of shock propagation, y [ 0 , 10 ] in the vertical direction, and z [ 0 , λ / 2 ], where λ = 2 π / k is the perturbation wavelength, in the direction along the cylinder axis. An effective resolution of 32 computational cells across the diameter of the cylinder is utilized. Half of the domain in the xy plane is simulated, with a reflecting boundary at the line of symmetry and outflow elsewhere. Furthermore, half of a perturbation wavelength is simulated in the z-direction with reflecting boundaries. In order to capture both the interface and shock dynamics, the mesh is refined based on a density-gradient threshold ensuring that both the shock and cylinder surface occupy regions of the mesh with the highest resolution.

The interface between the heavy cylinder and the surrounding fluid is initialized with a hyperbolic tangent function with perturbations only in the vertical direction. Specifically, the density is given as
ρ = ρ h + ( ρ l ρ h ) ( 1 + tanh ( 50 ( D pert D / 2 ) ) ) / 2 ,
(5)
where
D pert = 2 ( x 2 + ( y h 0 sin ( k z ) ) 2 ) 1 / 2 ,
(6)
and h 0 = 0.1 D is the initial perturbation size. A passive scalar is advected in the flow to track the fluid originally within the cylinder,
η = η h + ( η l η h ) ( 1 + tanh ( 50 ( D pert D / 2 ) ) ) / 2 ,
(7)
where η l = 0 and η h = 1 are the values of the scalar in the light fluid and heavy cylinder, respectively. This perturbation maintains a constant cylinder thickness in the direction of shock propagation, ensuring that the circulation varies negligibly along the length of the cylinder3 and resulting in a perturbation of only the vortex core position in the vertical direction.

While pointwise convergence can never be achieved when solving the Euler equations, we require that the vortex dipole parameters needed for the three-dimensional stability analysis, b, c, and Γ, are converged to within 5% of their values obtained from a highly resolved two-dimensional simulation with seven levels of refinement. This condition is achieved with as few as three levels of refinement. Although two-dimensional simulations can readily be performed with additional resolution, we restrict our two-dimensional simulations to three levels of refinement to enable a meaningful comparison with our three-dimensional simulations, which would exceed our computational limits with additional refinement.

Figure 2 shows the two-dimensional dynamics of the shock–cylinder interaction at z = 0 (i.e., where the cross section of the cylinder is circular) and illustrates the formation of the vortex dipole with density and vorticity cross sections and mass fraction and Q-criterion contours, where Q > 0 identifies regions of the flow where rotation dominates straining motions.38 When the incident shock wave impinges on the cylinder, shock waves are both transmitted and reflected at the cylinder interface due to the greater impedance of the cylinder compared to the surrounding fluid, as shown at time t u s / D = 1.3, which also illustrates how vorticity is deposited onto the interface as the shock wave passes over it. The greater impedance of the cylinder also causes the shock in the cylinder to travel more slowly compared to the shock in the surrounding fluid, resulting in the highly curved shock seen at time t u s / D = 1.9. This curvature focuses the shock as it breaks out of the back side of the cylinder, resulting in the wave structure visible at time t u s / D = 2.5. The waves generated by shock focusing within the cylinder also deposit vorticity onto the cylinder interface, but these additional wave interactions become progressively weaker with time. Once the wave interactions are complete, the interface is left to evolve governed by the deposited vorticity, resulting in the flattening of the cylinder in the direction of shock propagation and elongation in the perpendicular direction seen at time t u s / D = 4.8. The interface near the back of the cylinder begins to roll up into a reentrant jet that propagates in the direction opposite to shock propagation, as seen at time t u s / D = 12.0. As this process continues, the single surface enclosing the stretching vortex fragments into several isolated vortices, as shown at time t u s / D = 24.5, some of which are shed into the wake of the cylinder, carrying the cylinder fluid with them and entraining the surrounding fluid. These vortices continue to interact in time, ultimately resulting in a steady leading vortex with concentrated vorticity, shown at time t u s / D = 50.5, which is potentially subject to perturbation growth via the CI in three dimensions.

FIG. 2.

The two-dimensional density and mass fraction contour, η = 0.05 (above the symmetry line) and vorticity and Q-criterion contour, Q = 0 (below the symmetry line) at times t u s / D = [ 0.0 , 1.3 , 1.9 , 2.5 , 4.8 , 12.0 , 24.5 , 50.5 ], where the density scale is logarithmic and ω max , t D / u s = [ 0.0 , 5.6 , 6.2 , 6.6 , 5.4 , 2.9 , 1.9 , 1.7 ], respectively.

FIG. 2.

The two-dimensional density and mass fraction contour, η = 0.05 (above the symmetry line) and vorticity and Q-criterion contour, Q = 0 (below the symmetry line) at times t u s / D = [ 0.0 , 1.3 , 1.9 , 2.5 , 4.8 , 12.0 , 24.5 , 50.5 ], where the density scale is logarithmic and ω max , t D / u s = [ 0.0 , 5.6 , 6.2 , 6.6 , 5.4 , 2.9 , 1.9 , 1.7 ], respectively.

Close modal
Anticipating the range of wavenumbers subject to perturbation growth via the CI [Eq. (1)] requires only the ratio of the core separation distance and thickness, b/c. We identify the leading vortex core as the isolated region of positive Q containing the point of maximum vorticity, which we find to be nearly indistinguishable from the λ2 criterion for identifying vortices.39 The area of this region S is nearly circular, so we calculate the core thickness as an equivalent diameter given by
c = ( 2 S / π ) 1 / 2 .
(8)
Furthermore, the core separation distance is calculated as twice the y-position of the centroid,
b = 2 S S y d S .
(9)
Equations (8) and (9) yield b / c = 1.24, which, via Eq. (1), suggests that the CI will excite a range of wavenumbers β ( 0.0 , 2.1 ) with a maximum nondimensional growth rate of α = 0.8 near β = 1.1. Perturbation growth within this unstable range in three-dimensional simulations is discussed next.

Figure 3 shows mass fraction η = 0.05 and Q-criterion Q = 0.05 Q max contours from a three-dimensional simulation with β = 0.8, where Qmax is the maximum value of Q in the simulation. There is no rotation in the initial condition, therefore, Q < 0 everywhere. After the shock passes over the cylinder, however, vortex cores that take on the shape of the perturbation can be seen at time t u s / D = 11.7. In this snapshot, the interfacial vorticity is still collecting in the forming vortex, resulting in a high-vorticity shroud of fluid surrounding the nascent core, which can be visualized as the apostrophe-like shape of the Q-criterion contour in the two-dimensional cross section presented in Fig. 2 at time t u s / D = 12.0. The rotation induced by the forming vortex pair stretches the cylinder in the vertical direction. Both Fig. 2 and time t u s / D = 11.7 in Fig. 3 demonstrate how the fluid initially comprising the cylinder is elongated in the vertical direction due to the motion induced by the forming vortex pair.

FIG. 3.

Mass fraction (left) and Q-criterion (right) contours of η = 0.05 and Q = 0.05 Q max, respectively, at times t u s / D = [ 0.0 , 11.7 , 22.8 , 34.2 , 46.1 , 58.5 ] from top to bottom.

FIG. 3.

Mass fraction (left) and Q-criterion (right) contours of η = 0.05 and Q = 0.05 Q max, respectively, at times t u s / D = [ 0.0 , 11.7 , 22.8 , 34.2 , 46.1 , 58.5 ] from top to bottom.

Close modal

By time t u s / D = 22.8, the vortex pair is fully formed and the initial growth of the imposed perturbation can be seen. Thin vortex filaments oriented in the direction perpendicular to the primary vortex pair can also be seen forming, along with additional vortex filaments that are shed into the wake of the primary vortex pair. The influence of the perpendicular filaments, which result from additional instabilities that excite higher spatial frequencies than the CI,9,40 has augmented by time t u s / D = 34.2. The flow is still primarily dominated by the continued growth of perturbations excited by the CI,41 which continue to increase in amplitude and initiate the formation of the clumps seen in the mass fraction field. Fluid comprising the original cylinder can be seen shed into the wake of the vortex pair in the mass fraction field. Time t u s / D = 46.1 is just before the breakup and reconnection of the primary two vortex cores into isolated vortex rings. High-frequency perturbations of the primary vortex core excited by the elliptic instability can be seen in the region where the CI forces the cores together.8,42 The elliptic instability contributes to the breakup of the original vortex cores43,44 and their ultimate reconnection into the isolated vortex rings shown at time t u s / D = 58.5. The wake behind the clumps continues to stretch and is subject to further instability growth stemming from the weaker vortex filaments present. Clump formation is now complete, and the space between the clumps is occupied entirely by fluid not originally contained in the initial perturbed cylinder.

Additional three-dimensional simulations with β = [ 0.0 , 0.4 , 1.2 , 1.6 , 2.0 ] are performed to examine the behavior of the CI throughout the anticipated unstable band. A number of challenges prevents an accurate determination of the growth rate directly from simulations. Foremost, the onset of perturbation growth via the CI is obfuscated by the transient dynamics of the initial formation of the vortex cores. This transience alone would not preclude a calculation of the growth rate if perturbations continued to grow long after the initial transients subsided. However, soon after the symmetric perturbations start increasing in size, nonlinear effects begin to dominate where the cores come together at a regular series of pinch-off locations. As a result, an adequately long time period over which linear growth occurs, from which a perturbation growth rate could be extracted, is not clearly identifiable. Instead, we adopt the amount of time it takes the original vortex pair to pinch-off into a series of isolated vortex rings, pinch-off time Tpinch, as a proxy to quantify perturbation growth rates.

The pinch-off time for each simulation is given in Fig. 4, which can be compared to the growth rate given as a function of the normalized wavenumber. Unsurprisingly, the unperturbed cylinder, β = 0.0 does not exhibit perturbation growth. However, wavenumbers predicted to be unstable, β = 1.6 and β = 2.0, also yield stable perturbations. This disagreement can likely be attributed to an inconsistency in the vortex core identification. The original stability analysis utilizes potential theory to examine the stability of line vortices (i.e., zero-thickness cores) in an otherwise irrotational flow.7 The vortex core “thickness” is determined by matching the known self-propagation velocity of a slender vortex ring to that implied by the Biot–Savart law integrated along the length of the vortex ring up to a calibrated distance from the singularity in the governing equations. This thickness does not inherently match the vortex definition utilized presently, Q > 0, or any other vortex identification criterion.39 The stability theory is, therefore, subject to the method utilized to identify the vortex cores. If, instead of Q > 0, the contour given by Q = 0.5 Q max is used as a threshold for identifying the vortex core, which results in a separation-thickness ratio b / c = 2.51, the stability analysis predicts a range of unstable wavenumbers in better agreement with the simulations, as demonstrated by the dashed black line in Fig. 4. Although determining the precise growth rates and range of unstable wavenumbers may be elusive, the CI is clearly responsible for initiating the formation of clumps in the simulations.

FIG. 4.

The growth rates calculated from Eq. (1) for b / c = 1.24 (solid black) and b / c = 2.51 (dashed black) and the inverse pinch-off time determined from simulations (orange).

FIG. 4.

The growth rates calculated from Eq. (1) for b / c = 1.24 (solid black) and b / c = 2.51 (dashed black) and the inverse pinch-off time determined from simulations (orange).

Close modal

In this section, we estimate a time scale for achieving CI-induced pinch-off and calculate the relevant laser parameters and target geometry required to visualize clump formation in an experiment on the Omega EP laser facility. Furthermore, we verify that plasma effects are negligible compared to the hydrodynamic development of the CI.

A successful experiment would visualize the formation of clumps from an initially perturbed cylinder on time and spatial scales appropriate for a given experimental facility. On Omega EP, the strength of a laser-driven shock wave will far exceed that of the M = 1.2 shock examined in Sec. II. Therefore, we perform an additional two-dimensional simulation with a shock strength of M = 10 and verify that the net result of the shock interaction is the formation of a vortex dipole similar to that in Fig. 2, with the primary difference being the rate at which the flow evolves. These time dynamics scale proportionally with the circulation deposited by the shock wave onto the cylinder interface, which can be estimated from an analytical scaling law3 as
Γ = S ω d S = ( 1 + π 2 ) u s D γ + 1 ( 1 ρ l ρ h ) ( M + 1 M 2 M 2 ) ,
(10)
where γ = 5 / 3 is the adiabatic index. The amount of time required for the CI to induce pinch-off scales with the factor by which the original perturbation must increase in size in order for the symmetric cores to overlap, which occurs when the perturbation reaches an amplitude on the order of b 0.75 D, determined from the two-dimensional simulation shown in Fig. 2 and using Eq. (9) applied to a vortex core identified with the Q = 0 contour. Therefore, the original perturbation must increase in size by a factor of
e a t = b / h 0 = 0.75 D / h 0
(11)
for pinch-off to occur. We choose an experimental wavelength β exp = 0.8 for which both simulations and theory predict near-optimal perturbation growth, with α ( β exp ) 0.8, as seen Fig. 4. Combining Eqs. (1), (10), and (11) and assuming that the laser-driven shock is strong such that M + M 1 2 M 2 M yields the time required for pinch-off,
T pinch = 2 π ( 0.75 D ) 2 ln ( 0.75 D / h 0 ) 0.8 ( 1 + π 2 ) u shock D γ + 1 ( 1 ρ l ρ h ) ,
(12)
where u shock = M u s is the velocity of the shock in the light fluid.

With this scaling, an experimental target can be designed such that CI-induced clumping can be observed on spatial and temporal scales measurable at the Omega EP laser facility.45 The target would consist of four components: an ablator, a radiation shield, foam, and the perturbed cylinder, as shown in Fig. 5. The purpose of the ablator is to convert the laser energy into fluid kinetic energy in the form of a shock wave. This conversion is achieved by directly irradiating the ablator with the laser, resulting in a shock wave propagating in the direction opposite ablation (i.e., into the rest of the target). This laser–plasma interaction generates intense x rays and electrons that can propagate ahead of the leading shock wave, imparting unwanted preheat. A radiation shield would, therefore, be placed immediately upstream of the ablator.46–48 Past experiments designed to achieve a similar shock wave utilized a 100 μm-thick ablator with a 150 μm-thick shield, which could be employed presently.49 Adjacent to the shield is a foam with relatively low density, which forms the interface with the perturbed cylinder of relatively high density embedded within. The shock wave passes through the foam and cylinder, immediately ionizing both to a plasma state and stimulating gaseous flow. The cylinder would be made of a material opaque to x rays to enable x-ray radiography with a spherical crystal imager29,49–53 for flow diagnosis and clump visualization, as is routinely done in laser-driven experiments of hydrodynamic instabilities.25,49,52–55 The diagnostic line of sight would be along the axis perpendicular to both the direction of laser propagation and the center axis of the perturbed cylinder.

FIG. 5.

A schematic for a laser-driven shock experiment.

FIG. 5.

A schematic for a laser-driven shock experiment.

Close modal

The achievable shock velocities can be estimated from the shock pressure, which is related to the laser intensity as given in the text by Drake.56 A laser pulse with an intensity of 0.4 × 10 14 W cm 2 and wavelength of 351 nm, achievable at the Omega EP laser,57 yields a shock pressure of 4.5 Mbar in the foam. The Saha equation, with an ideal gas equation of state for both electrons and ions, then yields a plasma temperature of T = 35 eV, an ionization of Z = 0.5, and a shock velocity of u shock = 39 km s 1. Furthermore, for a cylinder with diameter D = 100 μm and perturbation amplitude h 0 = 0.1 D = 10 μm and wavelength λ exp = 2 π b / β exp = 2 π ( 0.75 D ) / 0.8  600 μm with density ρ h = 1.42 g cm 3 embedded in a foam with density ρ l = 0.14 g cm 3, clumping is expected to occur at approximately 40 ns after the shock wave passes through the cylinder. This time scale is routine for the Omega EP laser, which can drive a steady shock wave for up to 30 ns and diagnose a flow as late as 400 ns.57 Due to the comparable diagnostic time and the duration of the laser drive, we do not expect significant adverse effects from shock unsteadiness. In addition, we estimate that the cylinder pressure reaches 1% of its equilibrium value 5 ns after the shock first arrives at the cylinder based on the one-dimensional shock physics along the lineout through the diameter of the cylinder in the direction of shock propagation. As a result, unsteady wave reflections are not expected to significantly affect the development of the CI after the incident shock wave passes through the cylinder.

The wavelength of the perturbation must be conducive to visualizing clump formation within the spatial resolution of the spherical crystal imager, which is approximately 20 μm with an anticipated 20 μm of motion blur. Figure 6 shows the mass fraction field at t u s / D = 58.5 (latest time shown in Fig. 3) integrated along the diagnostic line of sight, scaled to the size of the proposed target and blurred to the resolution of the spherical crystal imager. Although additional physics are involved in the execution and diagnosis of a laser experiment, Fig. 6 represents what may be expected from an experimental radiograph, which demonstrates that the formation of isolated clumps can clearly be visualized within a typical 1 mm field of view. This field of view is furthermore expected to capture the desired dynamics even if significant asymmetry in either the incident shock or cylinder cross section causes a vorticity imbalance within the vortex cores that leads to the deflection of the dipole.9 

FIG. 6.

The line-integrated mass fraction at time t u s / D = 58.5 (latest time in Fig. 3) projected onto the diagnostic plane of view (the xz plane), scaled to the dimensions of the proposed experiment and blurred to the resolution of the spherical crystal imager. The dashed red box approximates the field of view.

FIG. 6.

The line-integrated mass fraction at time t u s / D = 58.5 (latest time in Fig. 3) projected onto the diagnostic plane of view (the xz plane), scaled to the dimensions of the proposed experiment and blurred to the resolution of the spherical crystal imager. The dashed red box approximates the field of view.

Close modal

As mentioned previously, laser-driven shock experiments have examined other hydrodynamic instabilities which, presently, could obfuscate the visualization of the CI. Specifically, the shocked cylinder could be subject to the Richtmyer–Meshkov instability.26,27 To estimate the effect of the Richtmyer–Meshkov instability, we integrate the buoyancy-drag equations58 for our proposed experiment over the estimated 40 ns between shock acceleration and diagnosis to determine the growth of perturbations with an initial length scale based on the roughness of targets examined in previous experiments59 of 1 μm, assumed for both the amplitude and wavelength. Such perturbations are expected to grow to an amplitude of 9 μm, an order-of-magnitude less than the initial diameter of the cylinder and the amount by which the intentionally seeded perturbation is expected to grow via the CI. In addition, the elliptic instability8,42 may excite short-wavelength distortions of the vortex cores, as it does in the simulation presented in Fig. 3 where the CI forces the vortex cores together. However, based on the linear growth rates of the elliptic instability and CI9 with initial b / c = 2.54, an unintentional perturbation with an initial amplitude of 1 μm subject to the elliptic instability would not overtake the intentionally seeded perturbation with an initial amplitude of 10 μm subject to the CI before the later increases in magnitude by more than a factor of 20, long after pinch-off is expected. As a result, the CI is expected to dominate any competing instability mechanisms amplifying unintended perturbations along the cylinder.

While the present goal is to visualize the hydrodynamic development of the CI, the plasma state of the flow may introduce effects from magnetic fields and radiation that affect the laser-driven shock experiments. We, therefore, must verify that the flow is hydrodynamically dominated. Because a plasma can carry an electrical current, it can self-generate a magnetic field,60,61 which introduces anisotropy in transport coefficients depending on whether the transport is parallel or perpendicular to the magnetic field.62 Presently, the differential rotation of electrons and ions in the vortex cores produces a magnetic field-generating current. Because the magnetic field results from rotation within the cores, we assume that resistivity is negligible such that the magnetic field strength is B = m i q ( Z + 1 ) ω, where q is the charge of an electron and mi is the ion mass.61 The average vorticity within the cores is estimated from Eq. (10), enabling an order-of-magnitude estimate of the plasma beta, β plasma = p t h p mag, the ratio of the thermal pressure in the plasma, p t h = n ( 1 + Z ) k B T, where n is the ion number density and kB is the Boltzmann constant, and the magnetic pressure p mag = B 2 2 μ 0, where μ0 is the permeability of free space. Assuming a compression ratio in the strong shock limit, we calculate that βplasma exceeds 1010, indicating that magnetic fields would not significantly affect an experiment.

The effect of radiation can be quantified by comparing the radiation ablation pressure pabl to the shock pressure. Assuming a Plankian radiation field at the foam–cylinder interface, the ablation pressure can be conservatively estimated as
p abl = 0.5 ( γ 1 ) σ T 4 m i ( Z + 1 ) T = 0.2 Mbar ,
(13)
where σ is the Stefan–Boltzmann constant.56 Noting that the ablation pressure is highly sensitive to the temperature, and that our temperature calculated from the Saha equation represents an upper bound, Eq. (13) also represents an upper bound estimate of the ablation pressure. Still, the shock pressure is more than 20 times the ablation pressure, and, therefore, radiative effects are not expected to have a significant role in an experiment.

Clumping induced by the growth of three-dimensional perturbations subject to the CI along a vortex dipole generated from a laterally shocked cylinder is examined. The unstable wavenumber range is determined from theory using parameters extracted from two-dimensional simulations, suggesting that long-wavelength perturbations experience growth. This result is consistent with the classical theory and verified by three-dimensional simulations, which additionally demonstrate how this growth leads to the pinch-off of isolated clumps of fluid comprising the original cylinder. Such physics are fundamental to compressible fluid mechanics and may also be relevant to a number of star systems involving the interaction between solar wind and circumstellar material, including the remnant of SN1987A.

Furthermore, we develop a time scale on which clumping is expected in experiments and leverage it to confirm the feasibility of an experiment on the Omega EP laser facility, suggesting a set of target geometry and laser parameters conducive for diagnosing CI-induced clumping with x-ray radiography. Specifically, a shocked cylinder with ten times the density of the foam it is embedded in, with a diameter of 100 μm perturbed in the axial direction and a wavelength of 600 μm, is expected to develop clumps approximately 40 ns after it is accelerated by a shock wave generated from a laser pulse with intensity 0.4 × 10 14 W cm 2 and wavelength 351 nm. Moreover, we estimate that physics involving magnetic fields and ionization would not pollute the visualization of the desired hydrodynamics. These physics, however, constitute an exciting avenue for future work on this topic. In particular, the effect of magnetic fields in suppressing the growth of the CI may enable isolation of the mechanism that suppresses the vortex stretching that leads to turbulence. The effect of radiation can additionally be studied with stronger shock waves driven by for example, the National Ignition Facility.63,64

The work of M. Wadas is supported by the Cecil and Sally Drinkward and the Caltech Presidential Postdoctoral Fellowships, and the work of H. LeFevre is supported by the NSF MPS-Ascend Postdoctoral Research Fellowship under Grant No. 2138109. This work is funded by the U.S. Department of Energy NNSA Center of Excellence under cooperative agreement number DE-NA0004146.

The authors have no conflicts to disclose.

M. Wadas: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Visualization (lead); Writing – original draft (lead). Heath Lefevre: Conceptualization (supporting); Formal analysis (equal); Investigation (equal); Writing – original draft (supporting). Y. Elmore: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting). X. Xie: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting). W. White: Conceptualization (supporting); Software (lead). C. Kuranz: Supervision (equal). E. Johnsen: Conceptualization (equal); Supervision (equal).

The data that support the findings of this study are available within the article.

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