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Vortex-merger statistical-mechanics model for the late time self-similar evolution of the Kelvin-Helmholtz instability
Physics of Fluids
Abstract
The nonlinear growth, of the multimode incompressible Kelvin-Helmholtz shear flow instability at all density ratios is treated by a large-scale statistical-mechanics eddy-pairing model that is based on the behavior of a single eddy and on the two eddy pairing process. From the model, a linear time growth of the mixing zone is obtained and the linear growth coefficient is derived for several density ratios. Furthermore, the asymptotic eddy size distribution and the average eddy life time probability are calculated. Very good agreement with experimental results and full numerical simulations is achieved.
... The vortices add more windings with time but their height reaches a finite asymptotic value proportional to the wavelength, as predicted in Equation (3.1). In particular, for an incompressible flow, the evolution of a row of vortices can be solved using methods of complex potential, yielding an asymptotic value of h(t → ∞) = [arcsin (1)/π] λ 0.56λ for the height of an individual eddy (Corcos & Sherman 1976;Rikanati et al. 2003). This analysis assumes a perfect single-wavelength perturbation in an infinite or periodic domain. ...
... This was first observed in the pioneering experiments by Brown & Roshko (1974) and Winant & Browand (1974). The result of a merger between two vortices with heights h 1 and h 2 is the formation of a larger vortex with its height approximately equal to the sum of the component heights, h h 1 + h 2 (Winant & Browand 1974;Rikanati et al. 2003). These interactions cause the vortex population to increase in size 9 The peak-to-valley amplitude refers to the distance along the x-axis between the top (peak) point of the perturbed interface to the bottom (valley) point. ...
... In Figure 4, we plot the average vorticity along the x axis, Ω ≡ ∫ ω y (x, z, t)dx ∫ dx, as a function of the zcoordinate and of time. By tracing the trajectory of highvorticity peaks along the z axis, we can follow the evolution of the vortex population in a "vortex merger tree", similar to the plots extracted from experiments by Brown & Roshko (1974) and from models by Rikanati et al. (2003). This representation highlights the decrease in the number of vortices with time. ...
As part of our long-term campaign to understand how cold streams feed massive galaxies at high redshift, we study the Kelvin-Helmholtz instability (KHI) of a supersonic, cold, dense gas stream as it penetrates through a hot, dilute circumgalactic medium (CGM). A linear analysis (Paper I) showed that, for realistic conditions, KHI may produce nonlinear perturbations to the stream during infall. Therefore, we proceed here to study the nonlinear stage of KHI, still limited to a two-dimensional slab with no radiative cooling or gravity. Using analytic models and numerical simulations, we examine stream breakup, deceleration and heating via surface modes and body modes. The relevant parameters are the density contrast between stream and CGM (), the Mach number of the stream velocity with respect to the CGM () and the stream radius relative to the halo virial radius (). We find that sufficiently thin streams disintegrate prior to reaching the central galaxy. The condition for breakup ranges from for to for . However, due to the large stream inertia, KHI has only a small effect on the stream inflow rate and a small contribution to heating and subsequent Lyman- cooling emission.
... It will lead to separated fluidic layers, near and away from the surface, with different hydrodynamic behaviors in which incompressible and inviscid fluids are in irrotational motion. 21,22 Flow field analysis has demonstrated the existence of vortices at the interface. If the linkage system moves along the same direction as the flow, vortices appear away from the substrate with a wider coverage (Supporting Information Figure S2). ...
... Despite that, the interface between fluidic layers on both occasions acts as an unstable vortex sheet that rolls up into a spiral under the perturbation, which actually addresses the process of KHI in classical hydrodynamics. 21,23 In each fluidic layer, the velocity and density profiles are uniform but discontinuous at the interface. It can be regarded as flows with a tangential velocity difference, at which small perturbations along the interface evolve into a number of large vortices. ...
... It will lead to a sinuous and symmetric geometry (Figure 2b), as the vortices are kept in place due to the zero net velocities. 21 The distance between neighboring peaks λ multiplies when the moving velocity of linkage system increases at equal steps (Text S2). The horizontal displacement ω(λ) will exponentially increase with the moving velocity. ...
The nature of chaos is in that elusive flow that is an advanced order out of our vision. It is wise to take advantage of chaos after recognizing or modifying its unique fractal properties. Here, a magnetron weaving strategy was developed for producing chaotic but monochromatic carbon nanotube tangles (CNT-Ts) under Kelvin-Helmholtz instability (KHI). The self-similarity characteristic facilitated individual ultralong CNTs to manipulate their entropy-driven fractal geometry, resulting in ∼104 μm2 CNT-Ts with variable curvature radius. In addition, based on the rate-selected mechanism, 85% metallic and ∼100% semiconducting CNT-Ts were synthesized and separated simultaneously at different length positions. After ex situ modifying their fractal into aligned CNTs with hydrogel, these CNT-Ts delivered a current of 10 μA μm-1 in transistors with an on/off ratio >107. It has provided the third route as a paradigm of applying one-dimensional nanomaterials by switching between chaos and fractal, in parallel with that of direct synthesis and postseparation.
... The fundamental physics of KH instability, having multiple vortices in the incompressible regime, is well understood. [25][26][27][28] Basic models produce good agreement with experimental data and numerical simulations for this case. In contrast, the physics of compressible TMZ evolution has not been well understood theoretically, for either the single-mode or the multimode vortex regime. ...
... In the incompressible limit, Rikanati et al. 25 presented a statistical model, describing the evolution of incompressible KH TMZ. This model followed the theoretical approach developed by Alon et al. [37][38][39] for describing the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities. ...
... In their modeling of incompressible KH, Rikanati et al. 25 derived the two nonlinear building blocks (single vortex evolution and interaction between vortices) using a simple standard potential flow theory. In the selfsimilar incompressible regime, the initial conditions are forgotten, and the growth of the mixing zone is dominated by merger of saturated vortices. ...
A new model for the evolution of compressible, multimode Kelvin-Helmholtz (KH) instability is presented. The model is built upon compressible single vortex evolution and two-vortex interaction, resulting in a statistical description of the compressible KH mixing zone evolution. These two building blocks, which, due to complicated compressibility effects and the presence of shock waves, cannot be derived using simple flow models, are validated by novel supersonic high-energy-density physics experiments. The model was validated against numerical simulations, experimental results, and previous phenomenological models, confirming the compressible KH scaling law in the self-similar regime in good agreement with simulations and a compilation of experimental data. Moreover, the model extends and confirms the logical validity of previous work, done in the incompressible regime. Therefore, it sheds new light on the evolution of compressible shear layers up to the self-similar regime.
... The vortices add more windings with time but their height reaches a finite asymptotic value proportional to the wavelength, as predicted in Equation (3.1). In particular, for an incompressible flow, the evolution of a row of vortices can be solved using methods of complex potential, yielding an asymptotic value of h(t → ∞) = [arcsin (1)/π] λ 0.56λ for the height of an individual eddy (Corcos & Sherman 1976;Rikanati et al. 2003). This analysis assumes a perfect single-wavelength perturbation in an infinite or periodic domain. ...
... This was first observed in the pioneering experiments by Brown & Roshko (1974) and Winant & Browand (1974). The result of a merger between two vortices with heights h 1 and h 2 is the formation of a larger vortex with its height approximately equal to the sum of the component heights, h h 1 + h 2 (Winant & Browand 1974;Rikanati et al. 2003). These interactions cause the vortex population to increase in size 9 The peak-to-valley amplitude refers to the distance along the x-axis between the top (peak) point of the perturbed interface to the bottom (valley) point. ...
... In Figure 4, we plot the average vorticity along the x axis, Ω ≡ ∫ ω y (x, z, t)dx ∫ dx, as a function of the zcoordinate and of time. By tracing the trajectory of highvorticity peaks along the z axis, we can follow the evolution of the vortex population in a "vortex merger tree", similar to the plots extracted from experiments by Brown & Roshko (1974) and from models by Rikanati et al. (2003). This representation highlights the decrease in the number of vortices with time. ...
As part of our long-term campaign to understand how cold streams feed massive galaxies at high redshift, we study the Kelvin-Helmholtz instability (KHI) of a supersonic, cold, dense gas stream as it penetrates through a hot, dilute circumgalactic medium (CGM). A linear analysis (Paper I) showed that, for realistic conditions, KHI may produce nonlinear perturbations to the stream during infall. Therefore, we proceed here to study the nonlinear stage of KHI, still limited to a two-dimensional slab with no radiative cooling or gravity. Using analytic models and numerical simulations, we examine stream breakup, deceleration and heating via surface modes and body modes. The relevant parameters are the density contrast between stream and CGM (), the Mach number of the stream velocity with respect to the CGM () and the stream radius relative to the halo virial radius (). We find that sufficiently thin streams disintegrate prior to reaching the central galaxy. The condition for breakup ranges from for to for . However, due to the large stream inertia, KHI has only a small effect on the stream inflow rate and a small contribution to heating and subsequent Lyman- cooling emission.
... The evolution in the presence of vortex merger would be expected to differ from the growth of independent vortices, as follows. Independent vortices would saturate at a peak-to-valley amplitude that is roughly half their wavelength [37,77,17]. In the present experiment this would produce no growth beyond that already present ahead of the shock. ...
... That experiment was performed using an intentional initial perturbation consisting of a single sinusoidal mode of much larger wavelength and amplitude than the surface features used in the present experiment, and the tooling was also more precise, producing a smoother surface. The resulting system, consequently, exhibits KH behavior much more closely resembling that of a classical, single-mode vortex sheet [17,77,37]. Indeed, Harding's experimentally-measured growth can be fully accounted for by such a single-vortex model [32]. ...
... Further, if the behavior is dominated by vortex merger, then <λ> will increase in time. (In the limit that the behavior becomes self-similar, <λ> will grow in time approximately as <λ> ∝ t, with a constant height-to-characteristic-wavelength ratio h/<λ> [77].) The baroclinic circulation has a more complex dependence upon the pressure change across the blast wave ∆P , the density change across the interface ∆ρ, as well as the spectral properties, including the initial interface perturbation wavelength <λ 0 > and initial amplitude a. Finally, it is worth stressing that this particular analysis requires the assumption of a broad initial interface spectrum, such that the dynamics will be strongly influenced by vortex merging relatively early in time, and hence will not apply to a system with a dominant initial interface wavelength, such as the one presented by Harding [32]. ...
This dissertation presents a series of experiments on various aspects of shock-driven hydrodynamic instability at high energy density (HED). This is an aspect of physics with ramifications in many important applications, for example in the confinement of fusion fuel and in many astrophysical phenomena. The common theme in this research lies in the experimental technique. These experiments, and others like them, are typically performed using a system of initially-solid plastic and carbon foam, where the surface of the plastic in contact with the foam can be easily machined with a seed perturbation, allowing for precise control of the unstable interface growth under well-characterized initial conditions. A high-powered, pulsed laser is then used to irradiate the system, driving a shock wave into it. This shock ionizes and accelerates the system, converting it into an HED plasma. The acceleration and/or the subsequent motion of the shocked plasma provides the impetus that drives the instability, where the particular mechanisms at work are controlled by the direction of incidence of the shock upon the material interface, as well as by appropriate choice of an initial interface perturbation.
The first three experiments explore various details of three important interface processes: Rayleigh-Taylor and Kelvin-Helmholtz instability, as well as Richtmyer-Meshkov physics. The final experiment studies the generation of fast electrons by the interaction of a laser with a material. These electrons are produced in virtually any HED system involving a laser, and can affect the system's dynamics significantly. They are of particular interest for the fast-ignition concept in inertial-confinement fusion, and also can have an effect on imaging-based diagnostics, such as the X-ray radiography techniques that are the primary method for diagnosing the instability experiments that are the focus of this dissertation.
... In this stage of the growth (i.e., the nonlinear stage), each vortex continues to grow in a decaying fashion. In the final stage of a pure single-mode evolution with A ¼ 0, the vortex saturates to an asymptotic height of ðasinhð1Þ=pÞk % 0:28k [13], resulting in W % 0:56k. We note that in the case of A 6 ¼ 0, the evolution of the vortices is asymmetric with respect to the mean interface. ...
... This way of estimation is reasonable, since the vortex evolution of the KH instability is self-similar (the hydrodynamics equations scale as long as the viscosity and the surface tension effects can be neglected). From Ref. [13], the normalized saturation time scale of a vortex having a wavelength of k is ðDu Á t=kÞ ! 1. For our case, where Du % 20 lm=ns and k ¼ 16 lm, one gets that the saturation time is a few nanoseconds, much less than the experimental time scale of 35 ns. ...
... In this section, we would like to investigate if this result would be the same for a wide distribution with the same statistics as the dominant single-mode initial conditions. Rikanati et al. [13] suggested a statistical model for the selfsimilar evolution of the KH instability in the incompressible regime. The building blocks of this model are the single-mode evolution and the merger rate of three neighboring vortices. ...
This paper brings new insights on an experiment, measuring the Kelvin-Helmholtz (KH) instability evolution, performed on the OMEGA-60 laser facility. Experimental radiographs show that the initial seed perturbations in the experiment are of multimode spectrum with a dominant single-mode of 16 μm wavelength. In single-mode-dominated KH instability flows, the mixing zone (MZ) width saturates to a constant value comparable to the wavelength. However, the experimental MZ width at late times has exceeded 100 μm, an order of magnitude larger. In this work, we use numerical simulations and a statistical model in order to investigate the vortex dynamics of the KH instability for the experimental initial spectrum. We conclude that the KH instability evolution in the experiment is dominated by multimode, vortex-merger dynamics, overcoming the dominant initial mode.
... The KH instability large-scale behavior has been thoroughly investigated through a wide range of experimental, numerical, and theoretical work resulting in an understanding of its growth rates and characteristics and of the main mechanisms dominating its evolution (see, for example, [6,7]). By implementing previously reported KH largescale instability growth rates [6,7], we try to model the spread angle of the SS instability as a function of the MR flow parameters. ...
... The KH instability large-scale behavior has been thoroughly investigated through a wide range of experimental, numerical, and theoretical work resulting in an understanding of its growth rates and characteristics and of the main mechanisms dominating its evolution (see, for example, [6,7]). By implementing previously reported KH largescale instability growth rates [6,7], we try to model the spread angle of the SS instability as a function of the MR flow parameters. In the following paragraphs, a brief description of the growth rates of the large-scale KH instability will be presented, followed by a detailed modeling of the SS instability evolution. ...
... As expected from dimensional considerations, the width of the KH large-scale turbulent mixing zone (TMZ) evolves with time according to ht cvt, where v is the shear velocity, t represents the evolution time, and c 0:19 0:01 is a dimensionless constant derived experimentally [6], numerically, and recently even theoretically [7]. It should be mentioned that, in most experiments, the instability growth is measured spatially, i.e., as a function of the advection distance from the mixing starting point. ...
Theoretical and experimental research, on the previously unresolved instability occurring along the slip stream of a shock-wave Mach reflection, is presented. Growth rates of the large-scale Kelvin-Helmholtz shear flow instability are used to model the evolution of the slip-stream instability in ideal gas, thus indicating secondary small-scale growth of the Kelvin-Helmholtz instability as the cause for the slip-stream thickening. The model is validated through experiments measuring the instability growth rates for a range of Mach numbers and reflection wedge angles. Good agreement is found for Reynolds numbers of Re 2 x 10(4). This work demonstrates, for the first time, the use of large-scale models of the Kelvin-Helmholtz instability in modeling secondary turbulent mixing in hydrodynamic flows, a methodology which could be further implemented in many important secondary mixing processes.
... At each merger, the wavelength will double and the mixing width will continue to grow. 28 The experiments and simulations below are too short lived for vortex merger to occur, so that this process is unimportant for the purposes of this work. ...
... Using this initial interface, assumptions about vortex-merger dynamics underlying statistical models of late time KH growth can be interrogated. 28 Future simulations including a new laser package in RAGE will better characterize the laser turn-off rarefaction and more accurately predict the impact of decompression and deceleration on the late-time growth of the interface. ...
We derive a model describing vorticity deposition on a high-Atwood number interface with a sinusoidal perturbation by an oblique shock propagating from a heavy into a light material. Limiting cases of the model result in vorticity distributions that lead to Richtmyer-Meshkov and Kelvin-Helmholtz instability growth. For certain combinations of perturbation amplitude, wavelength, and tilt of the shock, a regime is found in which discrete, co-aligned, vortices are deposited on the interface. The subsequent interface evolution is described by a discrete vortex model, which is found to agree well with both RAGE simulations and experiments at early times.
... While the appearance of the small-scale structures can be attributed to the increased surface roughness, it should be noted that the "hair-like" structure, in contrast to the more familiar "vortex" structure is a known feature of high Atwood number shear flow instability. This feature was observed in numerical simulations of multimode Kelvin-Helmholtz instability by Rikanati et al. 48 Images with 50 mg/cc foam have more "hair-like" structures than with 200 mg/cc foam, as expected, because the Atwood number is higher in the 50 mg/ cc foam. Figure 7 shows three measured density profiles as a function of distance in the vertical direction above the three vortices, as indicated by the arrows in Fig. 6(b). ...
... 23 Selfsimilarity of the later time KH growth was expected based on experimental KH data from shock tube experiments with liquids and gases. 23,24,48 The model predicts that the mixing width w(t) increases with time t in a very large Reynolds number self-similar regime and is proportional to the interface velocity V: w(t) ¼ d Vt, where d % 0.18 is a constant derived from classical fluid dynamics experiments 23,50 and V is the relative velocity between the two fluids at the interface. This expression follows from dimensional considerations, in which the only length-scale that can be formed from the available quantities V and t is proportional to Vt. ...
Kelvin-Helmholtz (KH) growth of pre-imposed 2D single-mode and 3D
broadband modulations was measured with side-on, x-ray radiography on
OMEGA. In experiments, a strong, laser-driven shock wave propagates
along the plane separating carbonized resorcinol foam (CRF) with a
density of 0.1 g/cc and Iodine-doped polystyrene (CH) with density of
1.4 g/cc. Modulations on the foam-CH interface grow due to KH
instability after the shock sets a flow of foam material along the
interface. The growth results of 2D and 3D modulations will be presented
along with comparisons with 2D simulations and 3D turbulent KH models.
... Any further growth is difficult to predict with linear theory. However, a simple non-linear model predicts that the maximum amplitude is 0.28 λ [64]. Since flow speeds in excess of 10 µm/ns are easily achieved in HED experiments, it appeared feasible to design an experiment that would show detectable KH growth in the 17 ns window available on Nike. ...
... In Fig. 4.5 a discrete vortex model with a constant circulation Γ = 4700 µm 2 /ns was used to calculate the time-dependent growth of the perturbations, starting from a compressed height of 44 µm [57,64]. In this vortex model the potential flow generated by an infinite string of line vortices was calculated. ...
The research discussed in this thesis represents work toward the demonstration of experimental designs for creating a Kelvin-Helmholtz (KH) unstable shear layer in a high-energy-density (HED) plasma. Such plasmas are formed by irradiating materials with several kilo-Joules of laser light over a few nanoseconds, and are defined as having an internal pressure greater than one-million atmospheres. Similar plasmas exist in laboratory fusion experiments and in the astrophysical environment. The KH instability is a fundamental fluid instability that arises when strong velocity gradients exist at the interface between two fluids. The KH instability is important because it drives the mixing of fluids and initiates the transition to turbulence in the flow. Until now, the evolution of the KH instability has remained relatively unexplored in the HED regime This thesis presents the observations and analysis of two novel experiments carried out using two separate laser facilities. The first experiment used 1.4 kJ from the Nike laser to generate a supersonic flow of Al plasma over a low-density, rippled foam surface. The Al flow interacted with the foam and created distinct features that resulted from compressible effects. In this experiment there is little evidence of the KH instability. Nevertheless, this experimental design has perhaps pioneered a new method for generating a supersonic shear flow that has the potential to produce the KH instability if more laser energy is applied. The second experiment was performed on the Omega laser. In this case 4.3 kJ of laser energy drove a blast wave along a rippled foam/plastic interface. In response to the vorticity deposited and the shear flow established by the blast wave, the interface rolls up into large vorticies characteristic of the KH instability. The Omega experiment was the first HED experiment to capture the evolution of the KH instability.
... They investigated the interaction between the most unstable short waves and long waves, observing rapid initial growth of short waves followed by the dominance of long waves in the instability. In 2003, Rikanati et al. 33 used a statistical mechanics approach to study the evolution of multi-mode incompressible KH instability at different density ratios. They derived linear growth coefficients for the mixing region. ...
A two-component discrete Boltzmann method (DBM) is employed to study the compressible Kelvin–Helmholtz (KH) instability with dual-mode interface perturbations, consisting of a fundamental wave and a second harmonic. The phase difference is analyzed in two distinct ranges, and the amplitude ratio is studied by varying the amplitude of either the first or second harmonic. The global average density gradient and the global mixing degree are analyzed from a hydrodynamic non-equilibrium perspective. The thermodynamic non-equilibrium (TNE) intensity is probed as a thermodynamic non-equilibrium variable. The system is also explored from a geometric perspective, with a focus on the rotation of two vortices, the mixing layer width, and the non-equilibrium area. Physically, under the influence of shear velocity, the fluid interface becomes distorted and progressively elongated, resulting in the formation of two small vortex structures and an enhancement of the physical gradient. The two vortices then begin to interact and merge into a single large vortex with complex fluid structures. Consequently, the physical gradient decreases, and the local TNE intensity weakens. Subsequently, the material interface elongates further, increasing the non-equilibrium region and enhancing the local TNE intensity. Finally, the physical gradient decreases due to dissipation and/or diffusion, weakening the local TNE intensity.
... The perturbation of the shear layer preceding the embedded shock led to CRVR formation. This process involved the slipstream originating from the shock-wave Mach reflection's triple point, its thickening due to small-scale Kelvin-Helmholtz vortices, aspects reported in prior studies [91,92]. ...
Vortex loops are compact toroidal structures wherein fluid rotation forms a closed loop around a fictitious axis, manifest in many natural occurrences. These phenomena result from brief impulses through vents or apertures in fluid systems, such as in caves, volcanic crusts, downbursts, or the descent of liquid droplets. The majority of naturally occurring and laboratory-generated vortex loops, studied for fundamental research on their formation, growth, instability, and dissolution, are classified as incompressible. This categorisation denotes negligible alterations in thermodynamic properties within the vortex loop. However, a distinct category of vortex loops emerges from processes involving artillery, shock tubes, explosions, chemical interactions, and combustion. This class primarily constitutes compressible vortex loops. Their presence in flow fields spans over a century, and they have been observed since the application of open-ended shock tubes to explore phenomena like diffracting shock waves, blast wave interactions with objects, and noise mitigation. The study and comprehension of compressible vortex loops and their interactions have historically relied heavily on optical techniques, lacking comprehensive insight into the intricate flow dynamics. Nevertheless, the advancements in flow visualisation tools and
computational capabilities in the 21st century have significantly aided scientists in scrutinising and characterising these vortex loops and their interactions in intricate detail. Unfortunately, a comprehensive review of the literature addressing compressible vortex loops originating from shock tubes, their evolution, and interactions
with shockwaves and various objects, including walls, appears lacking. This review article aims to address this gap.
... So far, the majority of the KH instability studies are limited to Newtonian fluids [12][13][14][15][16][17][18][19][20][21]. That occurring in the flow of non-Newtonian fluids has rarely been researched. ...
In this paper, the two-dimensional Kelvin–Helmholtz (KH) instability occurring in the shear flow of polymer fluids is modeled by the dissipative particle dynamics (DPD) method at the coarse-grained molecular level. A revised FENE model is proposed to properly describe the polymer chains. In this revised model, the elastic repulsion and tension are both considered between the adjacent beads, the bond length of which is set as one segment’s equilibrium length. The entanglements between polymer chains are described with a bead repulsive potential. The characteristics of such a KH instability in polymer fluid shear flow can be successfully captured in the simulations by the use of the modified FENE model. The numerical results show that the waves and vortexes grow more slowly in the shear flow of the polymer fluids than in the Newtonian fluid case, these vortexes become flat, and the polymer impedes the mixing of fluids and inhibits the generation of turbulence. The effects of the polymer concentration, chain length, and extensibility are also investigated regarding the evolution of KH instability. It is shown that the mixing of two polymer fluids reduces, and the KH instability becomes more suppressed as the polymer concentration increases. The vortexes become much longer with the evolution of the elongated interface as the chain length turns longer. As the extensibility increases, the vortexes become more flattened. Moreover, the roll-up process is significantly suppressed if the polymer has sufficiently high extensibility. These observations show that the polymer and its properties significantly influence the formation and evolution of the coherent structures such as the waves and vortexes in the KH instability progress.
... The vortices become large when they reach the end wall. The evolution of K-H vortices formed from the triple point is completely different from the K-H vortices formed in a steady [31][32][33] and transient jet [26,34]. ...
Over the last couple of decades, the shock-wave boundary-layer interaction has gained a lot of attention due to its practical importance in many engineering applications. It is a complex problem involving shock-wave bifurcation, boundary-layer separation, the interaction of contact discontinuity with the shock wave, the formation of shocklets, and the evolution of vortical structures having different length scales. It is difficult to capture experimentally the detailed flow field having the shocklets and vortices. Numerical solvers having negligible numerical dissipation are highly essential to predict these structures accurately. Over the years, many researchers have obtained a grid-converged solution for shock-wave boundary-layer interaction at Reynolds numbers of 200 and 1000. The shock-wave boundary-layer interaction at higher Reynolds numbers is not attempted due to the requirement of huge computational resources and challenges associated with convergence. In the investigation, the grid-converged solution is obtained for a Reynolds number of 2500 with a 13th order high-resolution hybrid scheme using 100 cores of a computational cluster equipped with 3.0 GHz Intel Xeon processors incorporating the MPI library for parallelization. The complex flow field is analyzed in detail using wall density, density gradient, vorticity, pressure, and enstrophy plots after validating the solver with the benchmark wall pressure,density, and v-velocity around the primary vortex provided by Zhou et al. (2018), Phys. Fluids, 30, 016102 for a Reynolds number 1000. It is observed that the triple point height and number of vortices in both separated zone and at the shear layer increase with an increase in Reynolds number. The grid-converged data obtained from the present simulation can be used as benchmark data to validate different numerical schemes/solvers at higher Reynolds numbers.
... Regularized and discretized versions of this equation have previously been used to model the Kelvin-Helmholtz instability [38]. Versions of this calculation with the vorticity assumed to be localized to points at the centers of vortices have also been used when modeling the late time behavior of both RM [65] and KH [64]. This localized vorticity assumption has also been used when modeling shock wave driven KH in the HED regime [32]. ...
A shock incident on a contact surface between two materials will deposit vorticity baroclinically given mis-alignments between pressure and density gradients. This vorticity will typically cause any perturbations on the pre-shock interface to grow. The process responsible for that growth will depend on the average tilt between interface and incoming shock, the details of the perturbation on the pre-shock interface, the material properties, and the strength of the incoming shock. If the shock front is parallel to the pre-shock interface the Richtmyer-Meshkov (RM) process will dominate perturbation growth. If the shock front is orthogonal to the pre-shock interface, the shear-driven Kelvin-Helmholtz (KH) instability will typically dominate perturbation growth. Previous theoretical work describing the shock-driven KH instability has assumed that the baroclinic vorticity deposition by the shock along the interface is independent of the post-shock shear flow across that same interface. However, this contradicts Stokes' theorem, which establishes a direct correspondence between the integral vorticity within some region and the shear flow across it through the definition of vorticity (). Here, I generalize a method for estimating integral baroclinic vorticity generation in the KH geometry, first presented in Hurricane, 2008, to both include an arbitrary tilt between the shock and interface, and to furthermore calculate the vorticity distribution along the interface. This vorticity deposition model was previously used in conjunction with a discrete vortex model (DVM), where the vorticity was assumed to be localized to a single point per wavelength. Here, I instead use it in conjunction with a vortex sheet model, which takes into account the full vorticity distribution along the interface. The vortex sheet model can capture both the early and late time behaviors for RM, KH, and oblique geometries so long as secondary processes remain unimportant. Interestingly, for an oblique interface with an appropriate perturbation, vortex sheet models predict that perturbation growth can be dominated by RM at early times while still asymptoting to a KH-like state at late times. I present OMEGA-EP experiments and simulations which confirm this result. I then use the vorticity deposition and vortex sheet models to predict what shock-interface interaction will evolve according to the DVM post-shock. I present simulations and preliminary experimental results which support this prediction.
... As observed in Section 4.3 (Fig. 23(c)), the increase in the vortex pairing time duration can be related to the decrease in the relative difference of circulation/energy of the trailing and the leading vortices [66]. The leap-frog motion is noticed in the current data between the leading and the trailing vortices which is similar to the results present in the literature [3,67]. ...
The interaction of trailing and leading vortex structures in low Reynolds number forced (varicose) jet with coflow is studied using Direct Numerical Simulation (DNS), where we report the dynamics of these vortices for a range of varying parameters. In a circular forced jet with Strouhal number 0.2, continuous formation of these vortices is observed which undergo pairing, tearing and disintegration. The forced jet is analyzed for varying frequency of forced perturbations, coflow temperature, turbulence intensity, momentum thickness, coflow intensity and Reynolds number at a constant forcing amplitude. The observation reveals that any reduction in momentum thickness increases the circulation as well as energy capacity of the leading and the trailing vortices, thereby enhancing the vortex pairing time. However, the hot coflow increases the strength of the leading vortex to such an extent that the generated trailing vortex is weak and dissipates even before it undergoes vortex pairing, while an incomplete disintegration of the leading vortex occurs in this case. Moreover, the strength of leading and trailing vortices is found to decrease with an increase in coflow power, thereby leading to a weaker vortex pairing. The decrease in Reynolds number also reduces the intensity of trailing vortices; whereas, at a lower Reynolds Number (Re) the trailing vortices dissipate soon after they form. The findings are achieved based on the analysis done by Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), fast Fourier transform (FFT), Q-criterion and vortex tracking method to understand the dynamic interaction and behavior of the vortices.
... For the non-equilibrium flow (figure 4b), we observed that the Mach stem is much shorter and the shock layers become thick and coarse because the thermochemical relaxation layer clings to the shock front. The K-H vortices along TD are not as evident as the former case because the density difference across TD declines in the non-equilibrium flow and hence suppresses the growth of the mixing zone (Rikanati, Alon & Shvarts 2003). ...
The deformation of the Mach stem in pseudo-steady shock wave reflections is investigated numerically and theoretically. The numerical simulation provides the typical flow patterns of Mach stem deformation and reveals the differences caused by high-temperature gas effects. The results also show that the wall jet, which causes Mach stem deformation, can be regarded as a branch of the mainstream from the first reflected shock. A new theoretical model for predicting the Mach stem deformation is developed by considering volume conservation. The theoretical predictions agree well with the numerical results in a wide range of test conditions. With this model, the wall-jet velocity and the inflow velocity from the Mach stem are identified as the two dominating factors that convey the influence of high-temperature thermodynamics. The mechanism of high-temperature gas effects on the Mach stem deformation phenomenon are then discussed.
... KH 涡的合并现象及其机理已经有一些比较深 入的研究[31,32] .在气泡结构的外缘, 高 χ 的区域, 二次的 KH 不稳定性开始发展, KH 涡结构逐渐变 得明显. ...
An important effect of the interfacial instability occurring at the interfaces of gases is to enhance the mixing of gases. In the present paper, the vortex/wall interactions at the late stage of the evolution of V shaped air/interface accelerated by weak shock wave in a duct is numerically simulated using high-resolution finite volume method with minimized dispersion and controllable dissipation (MDCD) scheme. The objective of the present paper is to study the mechanism of mixing enhancement due to the vortex/wall interactions. Because of the shock impingement, the Richtmyer-Meshkov instability is first developed. As a result, the baroclinic vorticity is deposited near the interface due to the misalignment of the density and pressure gradient right after the interaction of shock wave with V shaped interface, leading to the formation of vortical structures along the interface manifested by the Kelvin-Helmholtz instability. The vortices induce the rolling up and deformation of interface, and multi-scale vortical structures are generated because of the interaction and merging between vortices. This process eventually causes the turbulence mixing transition. The vortex induced velocity field drives the vortices to move to the lower/upper walls of the duct, leading to the complicated interaction between vortex and wall. It is observed in the numerical results that during the vortex/wall interaction, vortex is accelerated along the wall, leading to the stretching of material interface. Then the primary vortex will lift off from the wall and forms a second vortex. These two phenomena are the two main mechanisms of the mixing enhancement. Because of the inherent instability at the interface, the stretching of the interface will spread the area of instability. Furthermore, at the late stage of the interfacial instability, the flow near the interface is turbulent because of the rolling and pairing of the vortices. Therefore, the stretching of the interface will speed up the development of the interfacial turbulence and enhance the mixing. The vortex lifting off from the wall can directly speed up the mixing since it makes the heavy gas move directly into the light gas. To further determine which mechanism is dominant, we study the evolution of the mixing parameter derived from a fictitious fast chemical reaction model. It is shown that during the acceleration of the vortices along the wall and the stretching of the interface, the slope of the mixing parameter increases by a factor of 2, which indicates a significant mixing enhancement. And the vortices lifting off from the wall also shows considerable mixing enhancement but it is not so strong as the first mechanism.
... (增 长, 相 邻 的 涡 之 间 逐 渐 发 生 相 互 诱 导 和 并 对[16,17], 其中尖端位置的涡首先发生并对, 形成 最为明显的第一涡对, 如图 6 (a), t = 0.5 ms 时刻 的涡量分布图所示. 并对之后的涡依旧会与相邻 的涡发生相互诱导, 逐渐接近, 继续发生并对过 程, 流场逐渐发展成为 5 个明显的涡团, 如图 6 (b), t = 1.0 ms 时刻的结果所示. ...
Based on the mass fraction model of multicomponent mixture, the interactions between weak shock wave and "V" shaped air/SF6 interface with different vertex angles are numerical simulated. The numerical scheme used in the simulation is the high-resolution finite volume method with minimized dispersion and controllable dissipation scheme, in which the dissipation can be adjusted without affecting the already optimized dispersion property of the scheme. The grid sensitivity study is performed to guarantee that the resolution is sufficient in the numerical simulation. After the shock wave interacts with the interface, the baroclinic vorticity is deposited near the interface due to the misalignment of the density and pressure gradient, which is the manifestation of the Richtmyer-Meshkov instability, leading to the vortical structures forming along the interface. The interface perturbations lead to the bubbles and spikes appearing. The predicted leftmost interface displacement and interface width growth rate in the early stage of interface evolution agree well with the experimental results. The process of transition to turbulence at the material interface is studied in detail. The numerical results indicate that with the evolution of the interfacial vortical structure due to Kelvin- Helmholtz instability, the array of vortices begins to merge. As a result, the vortices accumulate in several distinct regions. It is in these regions that the multi-scale structures are generated because of the interaction between vortices. It is shown clearly that in the regions where vortices are accumulated, the fluctuation energy spectrum has many large and smallscale elements, which indicates there may be turbulent structures in these regions. To further examine if there is mixing transition in these regions, the characteristic length scales of the flow fields are calculated. The separation between the Lipemann-Taylor scale and inner viscous scale is observed based on the circulation-based Reynolds number, leading to the appearance of an uncoupled inertial range. The classical Kolmogorov -5/3 power law is also shown in the fluctuation energy spectrum, which means that the inertial range is developed. The appearing of this inertial range confirms that the mixing transition does occur, and the flow field near the material interface will develop into turbulence.
... The other simulated a nearly incompressible case, by using a higher pressure (and, hence, sound speed), so that M c ¼ 0.1. To interpret the results, it is helpful to recall that single vortices eventually reach a (full) saturated amplitude of 0.56 times their spacing, after which their amplitudes oscillate [28][29][30]. As a result, the impact of a smaller growth rate is to delay the growth but not ultimately to change the saturated amplitude. ...
We report the first observation, in a supersonic flow, of the evolution of the Kelvin-Helmholtz instability from a single-mode initial condition. To obtain these data, we used a novel experimental system to produce a steady shock wave of unprecedented duration in a laser-driven experiment. The shocked, flowing material creates a shear layer between two plasmas at high energy density. We measured the resulting interface structure using radiography. Hydrodynamic simulations reproduce the large-scale structures very well and the medium-scale structures fairly well, and imply that we observed the expected reduction in growth rate for supersonic shear flow.
Carbon nanomaterials are unique with excellent functionality and diverse structures. However, agglomerated structures are commonly formed because of small‐size effects and surface effects. Their hierarchical assembly into micro particles enables carbon nanomaterials to break the boundaries of classical Geldart particle classification before stable fluidization under gas‐solid interactions. Currently, there are few systematic reports regarding the structural evolution and fluidization mechanism of carbon nano agglomerations. Based on existing research on carbon nanomaterials, this article reviews the fluidized structure control and fluidization principles of prototypical carbon nanotubes (CNTs) as well as their nanocomposites. The controlled agglomerate fluidization technology leads to the successful mass production of agglomerated and aligned CNTs. In addition, the self‐similar agglomeration of individual ultralong CNTs and nanocomposites with silicon as model systems further exemplify the important role of surface structure and particle‐fluid interactions. These emerging nano agglomerations have endowed classical fluidization technology with more innovations in advanced applications like energy storage, biomedical, and electronics. This review aims to provide insights into the connections between fluidization and carbon nanomaterials by highlighting their hierarchical structural evolution and the principle of agglomerated fluidization, expecting to showcase the vitality and connotation of fluidization science and technology in the new era.
The interaction of trailing and leading vortex structures in low Reynolds number (R = 1000) forced (varicose) jet with coflow is studied using Direct Numerical Simulation (DNS), where we report the dynamics of these vortices for a range of varying parameters. In a circular forced jet with Strouhal number 0.2, continuous formation of these vortices is observed which undergo pairing, tearing and disintegration. The forced jet is analyzed for varying frequency of forced perturbations, coflow temperature, turbulence intensity, momentum thickness, coflow intensity and Reynolds number at a constant forcing amplitude (15%). The observation reveals that any reduction in momentum thickness increases the circulation as well as energy capacity of the leading and the trailing vortices, thereby enhancing the vortex pairing time. However, the hot coflow increases the strength of the leading vortex to such an extent that the generated trailing vortex is weak and dissipates even before it undergoes vortex pairing, while an incomplete disintegration of the leading vortex occurs in this case. Moreover, the strength of leading and trailing vortices is found to decrease with an increase in coflow power, thereby leading to a weaker vortex pairing. The decrease in Reynolds number also reduces the intensity of trailing vortices; whereas, at a lower Reynolds number (Re) the trailing vortices dissipate soon after they form. The findings are achieved based on the analysis done by Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), fast Fourier transform (FFT), Q-criterion and vortex tracking method to understand the dynamic interaction and behavior of the vortices.
The interaction of a planar shock wave with a triangle-shaped sulfur hexafluoride ((Formula presented.)) cylinder surrounded by air is numerically studied using a high resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The vortex dynamics of the Richtmyer–Meshkov instability and the turbulent mixing induced by the Kelvin–Helmholtz instability are discussed. A modified reconstruction model is proposed to predict the circulation for the shock triangular gas–cylinder interaction flow. Several typical stages leading the shock-driven inhomogeneity flow to turbulent mixing transition are demonstrated. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum in late time manifest the turbulent mixing transition for the present case. The analysis of variable-density energy transfer indicates that the flow structures with high wavenumbers inside the Kelvin–Helmholtz vortices can gain energy from the mean flow in total. Consequently, small scale flow structures are generated therein by means of nonlinear interactions. Furthermore, the occasional “pairing” between a vortex and its neighboring vortex will trigger the merging process of vortices and, finally, create a large turbulent mixing zone. © 2018 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature
We report the first observations of Kelvin-Helmholtz vortices evolving from well-characterized, dual-mode initial conditions in a steady, supersonic flow. The results provide the first measurements of the instability'svortex merger rate and supplement data on the inhibition of the instability's growth rate in a compressible flow. These experimental data were obtained by sustaining a shockwave over a foam-plastic interface with a precision-machined seed perturbation. This technique produced a strong shear layer between two plasmas at high-energy-density conditions. The system was diagnosed using x-ray radiography and was well-reproduced using hydrodynamic simulations. Experimental measurements imply that we observed the anticipated vortex merger rate and growth inhibition for supersonic shear flow.
A model of second harmonic generation by Kelvin-Helmholtz (KH) instability is presented. Weakly nonlinear theoretical results of the growth of the second harmonic agree well with the results of the LARED-S code. Nonlinear threshold of the KH instability for the single mode perturbation is determined by two-dimensional simulation.
A shock wave appearing in supersonic gas flow reflects in different ways depending on flow conditions. It can take the form of regular or irregular reflection. For the irregular reflection configuration of three shock waves and a slipstream arises. Mathematical investigations of the development of parameters across slipstream in triple shock configuration have been made with variation of the angle of incidence of the shock wave, the shock wave Mach number and the adiabatic index of the gas. It has been shown that the characteristic mixing parameters of the slipstream increase with the increase of Mach number of the flow and the decrease of the heat capacity ratio. This leads to an increase of vortex formation and an increase of the angular spread of the slipstream. It also has been shown that the angle between the reflected wave and the slipstream diminishes with the decrease in heat capacity ratio so that the value may become of the same order as the spread angle. This may lead to quantitative changes in the whole reflection pattern near the triple point. The evident dependence of slipstream instability magnitude on the physical and chemical transformation intensity in the fluid was previously experimentally observed. The results of an analytical investigation appeared to be in good agreement with the experimental data.
In this work, we examine the hydrodynamics of high-energy-density (HED) shear flows. Experiments, consisting of two materials of differing density, use the OMEGA-60 laser to drive a blast wave at a pressure of ∼50 Mbar into one of the media, creating a shear flow in the resulting shocked system. The interface between the two materials is Kelvin-Helmholtz unstable, and a mixing layer of growing width develops due to the shear. To theoretically analyze the instability's behavior, we rely on two sources of information. First, the interface spectrum is well-characterized, which allows us to identify how the shock front and the subsequent shear in the post-shock flow interact with the interface. These observations provide direct evidence that vortex merger dominates the evolution of the interface structure. Second, simulations calibrated to the experiment allow us to estimate the time-dependent evolution of the deposition of vorticity at the interface. The overall result is that we are able to choose a hydrodynamic model for the system, and consequently examine how well the flow in this HED system corresponds to a classical hydrodynamic description.
Counter-rotating vortex rings (CRVRs) are observed to form ahead of a primary
compressible vortex ring that is generated at the open end of a shock tube at
sufficiently high Mach numbers. In most of the earlier studies, the embedded shock
strength has been asserted as the cause for the formation of CRVRs. In the present
study, particle image velocimetry (PIV) measurements and high-order numerical
simulations show that CRVRs do not form in the absence of a Mach disk in the
sonic under-expanded jet behind the primary vortex ring. Kelvin–Helmholtz-type
shear flow instability of the slipstream originating from the triple point of the Mach
disk and subsequent eddy pairing, as observed by Rikanati et al. (Phys. Rev. Lett.,
vol. 96, 2006, art. 174503) in shock-wave Mach reflection, is found to be responsible
for CRVR formation. The growth rate of the slipstream in the present problem follows
the model proposed by them. The parameters influencing the formation of CRVRs as
well as their dynamics is investigated. It is found that the strength of the Mach disk
and its duration of persistence results in an exit impulse that determines the number
of CRVRs formed.
An experiment meant to investigate the evolution of single mode Kelvin–Helmholtz (KH) instability in the supersonic regime is presented and theoretically analyzed. This experiment is intended to provide a direct measurement of the two-dimensional vortex evolution so that the high-Mach-number effects can be measured. The proposed design takes advantage of the ability of OMEGA-EP to drive experiments for up to 30 ns to produce steady conditions for KH that endure long enough to observe substantial growth. KH growth for the proposed design has been analyzed using two-dimensional numerical simulations. The results were compared to synthetic temporal KH numerical simulations using non-dimensional scaling in the low and high Mach number regime. The comparisons show that the growth in the high Mach number regime is expected to be suppressed by up to a factor of two. The effects of two-dimensional rarefactions from the lateral boundaries of the experimental system were also investigated. It was found that they introduce no major uncertainties or hazards to the experiment. We produced simulated radiographs, which show that the proposed experimental system will enable observation of the KH structures. An experiment of this kind has not yet been performed, and therefore would serve to validate numerical results and analytical models presented here and in the literature.
Kelvin–Helmholtz (KH) turbulent mixing measurements were performed in experiments on the OMEGA Laser Facility [T.R. Boehly et al., Opt. Commun. 133 (1997) 495]. In these experiments, laser-driven shock waves propagated through low-density plastic foam placed on top of a higher-density plastic foil. Behind the shock front, lower-density foam plasma flowed over the higher-density plastic plasma. The interface between the foam and plastic was KH unstable. The experiments were performed with pre-imposed, sinusoidal 2D perturbations, and broadband 3D perturbations due to surface roughness at the interface between the plastic and foam. KH instability growth was measured using X-ray, point-projection radiography. The mixing layer caused by the KH instability with layer width up to ∼100 μm was observed at a location ∼1 mm behind the shock front. The measured mixing layer width was in good agreement with simulations using a K–L turbulent mixing model in the two-dimensional ARES hydrodynamics code. In the definition of the K–L model K stands for the specific turbulent kinetic (K) energy, and L for the scale length (L) of the turbulence.
The impact of harmonic perturbations on a turbulent mixing layer has been discussed. Any turbulent flow can be decomposed into a mean and fluctuating component by introducing an average used for deriving the Reynolds-averaged Navier-Stokes (RANS) equations. The phase-averaged parameters are divided into the time-averaged and the coherent parts, where the latter is represented as a sum of Fourier modes of the forcing frequency. The main part of the turbulent Reynolds stress (TRS) is generated by large coherent structures, which are an intrinsic feature of a turbulent mixing layer. A Kelvin-Helmholtz wave propagates downstream with phase velocity (V 1 + V 2)/2, and its growth or attenuation is completely determined by the time-averaged velocity and the eddy viscosity. The dynamics of the wave can be roughly estimated by the location of neutral stability curve.
The development of a new experimental system for generating a strong shear flow in a high-energy-density plasma is described in detail. The targets were designed with the goal of producing a diagnosable Kelvin–Helmholtz (KH) instability, which plays an important role in the transition turbulence but remains relatively unexplored in the high-energy-density regime. To generate the shear flow the Nike laser was used to drive a flow of Al plasma over a low-density foam surface with an initial perturbation. The interaction of the Al and foam was captured with a spherical crystal imager using 1.86keV X-rays. The selection of the individual targets components is discussed and results are presented.
A recent series of experiments on the OMEGA laser provided the first controlled demonstration of the Kelvin–Helmholtz (KH) instability in a high-energy-density physics context [E. C. Harding et al., Phys. Rev. Lett. 103, 045005, (2009); O. A. Hurricane et al., Phys. Plasmas 16, 056305, (2009)]. We present 3D simulations which resolve previously reported discrepancies between those experiments and the 2D simulation used to design them. Our new simulations reveal a three-dimensional mechanism behind the low density "bubble" structures which appeared in the experimental x-ray radiographs at late times but were completely absent in the 2D simulations. We also demonstrate that the three-dimensional expansion of the walls of the target is sufficient to explain the 20% overprediction by 2D simulation of the late-time growth of the KH rollups. The implications of these results for the design of future experiments are discussed. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4752018]
Shear-flow, Kelvin–Helmholtz (KH) turbulent mixing experiments were performed on the OMEGA Laser Facility [Boehly et al., Opt. Commun. 133, 495 (1997)] in which laser-driven shock waves propagated through a low-density plastic foam placed on top of a higher-density plastic foil. The plastic foil was comprised a thin iodine-doped plastic tracer layer bonded on each side to an undoped density-matched polyamide-imide plastic. Behind the shock front, lower-density foam plasma flowed over the higher-density plastic plasma, such that the interface between the foam and plastic was KH unstable. The initial perturbations consisted of pre-imposed, sinusoidal 2D perturbations, and broadband 3D perturbations due to surface roughness at the interface between the plastic and foam. KH instability growth was measured using side-on radiography with a point-projection 5-keV vanadium backlighter. Time-integrated images were captured on D-8 x-ray film. Spatial density profiles of iodine-doped plastic mixed with foam were inferred using x-ray radiographs. The mixing layer ensuing from the KH instability with layer width up to ~100 um was observed at a location ~1 mm behind the shock front. The measured mixing layer width was in good agreement with predictions based on a simple self-similar model of KH instability growth using an estimate of the shear velocity obtained from numerical simulations of the experiments. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4752015]
The first successful high energy density Kelvin-Helmholtz (KH) shear layer experiments (O.A. Hurricane et al. in Phys. Plasmas,
16:056305, 2009; E.C. Harding et al. in Phys. Rev. Lett., 103:045005, 2009) demonstrated the ability to design and field a target that produces, in a controlled fashion, an array of large diagnosable
KH vortices. Data from these experiments vividly showed the complete evolution of large (∼400μm) distinct eddies, from formation
to apparent turbulent break-up in the span of about 75ns. A second set of experiments, in which the density of a key carbon-foam
material was varied, was recently performed. The new series showed a great deal of fine-structure that was not as apparent
as in the original experiments. In this paper, the results of both experiments will be discussed along with supporting theory
and simulation. An attempt is made to connect these observations with some turbulent scale-lengths. Finally, we speculate
about the possible connection of these experiments to astrophysical contexts.
KeywordsKelvin-Helmholtz instability–Shear layer–Vortex
A laser initiated experiment is described in which an unstable plasma shear layer is produced by driving a blast wave along a plastic surface with sinusoidal perturbations. In response to the vorticity deposited and the shear flow established by the blast wave, the interface rolls up into large vortices characteristic of the Kelvin-Helmholtz instability. The experiment used x-ray radiography to capture the first well-resolved images of Kelvin-Helmholtz vortices in a high-energy-density plasma.
Rayleigh-Taylor (RT) and Richtmyer–Meshkov (RM) instabilities at the pusher–fuel interface in inertial confinement fusion (ICF) targets may significantly degrade thermonuclear burn. Present-day supercomputers may be used to understand the fundamental instability mechanisms and to model the effect of the ensuing mixing on the performance of the ICF target. Direct three-dimensional numerical simulation is used to investigate turbulent mixing due to RT and RM instability in simple situations. A two-dimensional turbulence model is used to assess the effect of small-scale turbulent mixing in the axisymmetric implosion of an idealized ICF target.
The statistics of the large scale vortex structure in turbulent mixing layers have been investigated theoretically. It is shown that similarity in the fully developed flow results in a common description of the Eulerian and Lagrangian statistics. In the Eulerian frame of reference, a conservation equation is derived and solved to show that the distribution of vortex circulation is lognormal. It is also shown that the standard deviation normalized by the mean value of the distribution depends only on the amalgamation mechanism. The value for pairing is in good agreement with experimental measurements. These results are used to calculate the life span and survival probabilities of the vortices in the Lagrangian frame of reference. These distributions are in good agreement with direct measurements of the life span probability and with space‐time correlation measurements, respectively. Some implications of these results on the dynamics of the large scale vortices in the fully developed turbulent flow are discussed. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/70591/2/PFLDAS-31-9-2533-1.pdf
The development of three-dimensional motions in a plane mixing layer was investigated experimentally. It is shown that superimposed on the primary, spanwise vortex structure there is a secondary, steamwise vortex structure. Three aspects of this secondary structure were studied. First, the spanwise vortex instability that generates the secondary structure was characterized by measurements of the critical Reynolds number and the spanwise wavelength at several flow conditions. While the critical Reynolds number was found to depend on the velocity ratio, density ratio and initial shear-layer-profile shape, the mean normalized wavelength is independent of these parameters. Secondly, flow visualization in water was used to obtain cross-sectional views of the secondary structure associated with the streamwise counter-rotating vortices. A model is proposed in which those vortices are part of a single vortex line winding back and forth between the high-speed side of a primary vortex and the low-speed side of the following one. Finally, the effect of the secondary structure on the spanwise concentration field was measured in a helium-nitrogen mixing layer. The spatial organization of the secondary structure produces a well-defined spanwise entrainment pattern in which fluid from each stream is preferentially entrained at different spanwise locations. These measurements show that the spanwise scale of the secondary structure increases with downstream distance.
The evolution of a randomly perturbed interface between unbounded incompressible fluids undergoing Rayleigh-Taylor instability is analyzed numerically and theoretically. Two-dimensional simulation results, obtained with an interface tracking code, are presented and compared with a theoretical model based on Young's two-phase flow description of the mixing process. The simplifications implied by self-similarity and by high drag enable simple analytic results to be obtained for the profiles of the average volume fractions and velocities of the two materials as a function of penetration depth. Agreement of the analytic results with the simulation data is demonstrated for a wide range of density ratios.
The phenomenon of vortex merging in a two-dimensional mixing layer is treated statistically by introducing the probability density of spacings between neighboring vortices. The rate of merging is the other statistical variable and is a function of the spacing. The mixing layer is assumed to be statistically uniform along the streamwise coordinate and not to change with time. The following further assumptions were made: no correlation between adjacent spacings, similarity of the probability density, etc., to all times, and that the rate of merging for a single pair of vortices is inversely proportional to the square of their distance. The probability density was obtained by solving the governing equation. The result agrees well with that data measured by Brown and Roshko.
The Richtmyer–Meshkov instability of a two-liquid system is investigated experimentally. These experiments utilize a novel technique that circumvents many of the experimental difficulties that have previously limited the study of Richtmyer–Meshkov instability. The instability is generated by vertically accelerating a tank containing two stratified liquids by bouncing it off of a fixed coil spring. A controlled two-dimensional sinusoidal initial shape is given to the interface by oscillating the container in the horizontal direction to produce standing waves. The motion of the interface is recorded during the experiments using standard video photography. Instability growth rates are measured and compared with existing linear theory. Disagreement between measured growth rates and the theory are accredited to the finite bounce length. When the linear stability theory is modified to account for an acceleration pulse of finite duration, much better agreement is attained. Late time growth curves of many different experiments seem to collapse to a single curve when correlated with the circulation deposited by the impulsive acceleration. A theory based on modeling the late time evolution of the instability using a row of vortices is developed. The growth curve given by this model has similar shape to those measured, but underestimates the late-time growth rate.
The nonlinear growth of the multimode Richtmyer-Meshkov instability in the limit of two fluids of similar densities (Atwood number A⃗0) is treated by the motion of point potential vortices. The dynamics of a periodic bubble array and the competition between bubbles of different sizes is analyzed. A statistical mechanics model for the multimode front mixing evolution, similar to the single-bubble growth and two-bubble interaction based model used by Alon et al. [Phys. Rev. Lett. 72, 2867 (1994)] for A=1, is presented. Using the statistical bubble merger model, a power law of t0.4 for the mixing zone growth is obtained, similar to that of the bubble front growth for the A=1 case and in good agreement with experiments and full numerical simulations.
It is observed experimentally that a spatially growing shear layer entrains an unequal amount of fluid from
each of the freestreams, resulting in a mixed fluid composition that favors the high-speed fluid. A simple argument is proposed, based on the geometrical properties of the large-scale now structures of the subsonic, fully
developed, two-dimensional mixing layer, which yields the entrainment ratio and growth of the turbulent mixing
layer. The predictions depend on the velocity and density ratio across the layer and are in good agreement with
measurements to date.
A theoretical model for the ablatively driven Rayleigh–Taylor (RT) instability single-mode and multimode mixing fronts is presented. The effect of ablation is approximately included in a Layzer-type potential flow model, yielding the description of both the single-mode evolution and the two-bubble nonlinear competition. The reduction factor of the linear growth rate due to ablative stabilization obtained by the model is similar to the Takabe formula. The single-bubble terminal velocity is found to be similarly reduced by ablation, in good agreement with numerical simulations. Two-bubble competition is calculated, and a statistical mechanics model for multimode fronts is presented. The asymptotic ablation correction to the classical RT αgt2 mixing zone growth law is derived. The effect of ablative stabilization on the allowed in-flight aspect ratio of inertial confinement fusion pellets is estimated using the results of the statistical mechanics model. © 1998 American Institute of Physics.
The development of free shear layers formed by the mixing of initially separated free streams is examined in a review of recent work. The mixing layer is viewed as a prototype for a class of inviscidly unstable free shear flows including jets and wakes, and the focus is on 2D homogeneous incompressible mixing layers. Major areas covered include dynamical processes in free shear layers, the influence of operational parameters, sensitivity to artificial excitations, and global feedback effects. Graphs, drawings, and photographs of characteristic phenomena are provided.
To evaluate the usefulness of digital image analysis in extracting quantitative information from flow pictures we have studied a 16 mm ciné film of a turbulent mixing layer. A sequence of 373 frames is digitized and analysed to isolate and measure the concentration eddies that constitute the large structure and to follow their individual evolution in time. As a result, statistics are given on the life history of the eddies, the structure of the amalgamation process and the amount of entrainment, as measured by area change, due to amalgamation as compared to the total. It is found that most of the entrainment occurs during the normal life of eddies and not during pairing. Mixing intermittency is computed from the observed shape of the eddies and seen to compare well with previous measurements. The significance of these results in modelling the mixing layer is discussed briefly and some comments are given on the general usefulness of the techniques presented.
The detailed dynamics of an unstable free shear layer are examined for a gravitationally stable or neutral fluid. This first article focuses on the part of the evolution that precedes the first subharmonic interaction. This consists of the transformation of selectively amplified sinusoidal waves into periodically spaced regions of vorticity concentration (the cores) joined by thin layers (the braids), in which vorticity is also concentrated. The thin layers are the channels along which vorticity is advected into the cores, and the cores provide the strain which creates the braids. For moderately long waves an analysis is given of the braid structure as a function of time. For gravitationally stable shear layers at high Reynolds numbers, the local vorticity reaches such large values as to cause secondary shear instability on a small (length) and short (time) scale. A physical account of the primary instability and its self-limiting mechanism is used as a basis for a computation, which yields growth rates and maximum amplitude as a function of initial layer parameters. The computation supplies the wavelength of waves that grow to achieve the largest (absolute) amplitude. Finally, the model makes it clear that, in the absence of secondary instability, this initial phase of the nonlinear development of the layer contributes only a modicum of additional mixing, especially at high Reynolds numbers.
The investigation of a separated shear layer was undertaken to clarify the non-linear mechanisms associated with instability and transition to turbulence. Such an investigation is of practical importance since profiles which resemble the separated shear layer are a common occurrence.
A two-dimensional free shear layer was formed by separation of a laminar boundary layer from a rearward-facing step. The free-stream speed was approximately 16 ft./sec. Hot-wire measurements were made in the region directly downstream of the plate trailing edge. The measurements included mean velocity profiles, frequency spectra of the longitudinal fluctuation, and root-mean-square amplitude and phase distributions of various spectral components of the longitudinal fluctuation. Several measurements were designed to detect the presence of periodic spanwise structure.
The most important findings were: Significant non-linear distortion of the initial unstable wave occurred without periodic spanwise structure.
Non-linear distortion was first manifest by the growth of a subharmonic oscillation, which was strongly intermittent. Numerous harmonics of the sub-harmonic oscillation were also present.
Strong evidence suggests that secondary instabilities were present, which created still higher frequencies.
A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel. The Reynolds number, based on the velocity difference and the thickness of the shear layer, varies from about 45, where the shear layer originates, to about 850 at a distance of 50 cm. Dye is injected between the two streams just before they are brought together, marking the vorticity-carrying fluid. Unstable waves grow, and fluid is observed to roll up into discrete two-dimensional vortical structures. These turbulent vortices interact by rolling around each other, and a single vortical structure, with approximately twice the spacing of the former vortices, is formed. This pairing process is observed to occur repeatedly, controlling the growth of the mixing layer. A simple model of the mixing layer contains, as the important elements controlling growth, the degree of non-uniformity in the vortex train and the ‘lumpiness’ of the vorticity field.
The initial growth of irregularities on an interface between two ; compressible fluids is studied for impulsive (i.e., shock) acceleration. It was ; found that the ultimate rate of growth is roughly the same as that given by the ; incompressible theory, if the initial comnpression of the irregularities and of ; the fluids is taken into account. (auth)
Results are presented of an experimental study of the stability of the interface of two gases traversed by ashockwave. It is found that the interface is unstable both in the case of shock wave passage from the lighter to the heavier gas and for passage in the opposite direction. The interface disturbance grows linearly with time in the first approximation.
Two-dimensional hydrodynamic codes are used to simulate the growth of perturbations at an interface between two fluids of different density due to Rayleigh-Taylor instability. Problems where the interface is subjected to a constant acceleration and where it is accelerated and decelerated by shock waves are considered. Emphasis is placed on the case when the initial perturbation consists of many different wavelength modes. Results are compared with the experimental data of Andronov et al. (1976) and Read (1983). The use of a simple empirical model of the mixing process based on the equations of two-phase flow is discussed.
Plane turbulent mixing between two streams of different gases (especially nitrogen and helium) was studied in a novel apparatus. Spark shadow pictures showed that, for all ratios of densities in the two streams, the mixing layer is dominated by large coherent structures. High-speed movies showed that these convect at nearly constant speed, and increase their size and spacing discontinuously by amalgamation with neighbouring ones. The pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainment rates. Large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle; it is concluded that the strong effects, which are observed when one stream is supersonic, are due to compressibility effects, not density effects, as has been generally supposed.
A statistical model of Rayleigh-Taylor bubble fronts in two dimensions is introduced. Float and merger of bubbles lead to a scale-invariant regime, with a stable distribution of scaled bubble radii and a constant front acceleration. The model is solved for a simple merger law, showing that a family of such stable distributions exists. The basins of attraction of each of these are mapped. The properties of the scale-invariant distributions for various merger laws, including a merger law derived from the Sharp-Wheeler model, are analyzed. The results are in good agreement with computer simulations. Finally, it is shown that for some merger laws, a runaway bubble regime develops. A criterion for the appearance of runaway growth is presented.
A renormalization-group fixed point is found, corresponding to chaotic mixing in the Rayleigh-Taylor instability problem. The outer envelope of the mixing region, adjacent to the heavy fluid, is dominated by a merger of unstable modes ( bubbles of light fluid) and dynamically changing length scales. A statistical model is introduced as an approximation to the full two-fluid Euler equation to describe the mixing envelope. Molecular-chaos and continuous-time approximations to this model define an approximate renormalization-group equation, which is shown to have a nontrivial fixed point.
The late time evolution and structure of 2D Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts is calculated, using a new statistical merger model based on the potential flow equations. The merger model dynamics are shown to reach a scale invariant reigme. It is found that the Rayleigh-Taylor front reaches a constant acceleration, growing as 0.05gt2, while the Richtmyer-Meshkov front grows as at0.4 where a depends on the initial perturbation. The model results are in good agreement with experiments and simulations.
The nonlinear evolution of large structure in Rayleigh-Taylor and Richtmyer-Meshkov bubble and spike fronts is studied numerically and explained theoretically on the basis of single-mode and two-bubble interaction physics at Atwood numbers A. Multimode Rayleigh-Taylor bubble (spike) fronts are found as hB = alphaBAgt2 [hs = alphasAgt2] with alphaB = 0.05, while Richtmyer-Meshkov bubble (spike) fronts are found as hB = aBtthetaB hs = astthetas\(A\) with thetaB = 0.4 at all A's. The dependence of these scaling laws and parameters on A and on initial conditions is explained.