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Using a combination of laser–plasma interactions and magnetic confinement configurations, a conceptual fusion reactor is proposed in this paper. Our reactor consists of the following: (1) A background plasma of boron11 and hydrogen ions, plus electrons, is generated and kept for a certain time, with densities of the order of a mg/cm3 and temperatures of tens of eV. Both the radiation level and the plasma thermal pressure are thus very low. (2) A plasma channel is induced in a solid target by irradiation with a high power laser that creates a very intense shock wave. This mechanism conveys the acceleration of protons in the laser direction. The mechanisms must be tuned for the protons to reach a kinetic energy of 300–1200 keV where the pB11 fusion cross section is significantly large (note that this value is not a temperature). (3) Those ultra-fast protons enter the background plasma and collide with boron11 to produce three alphas. Fusion born alphas collide with protons of the plasma and accelerate them causing a chain reaction. (4) A combination of an induction current and a magnetic bottle keeps the chain reaction process going on, for a pulse long enough to get a high energy gain. (5) Materials for the background plasma and the laser target must be replaced for starting a new chain reaction cycle.
Two-dimensional (2-D) numerical simulations based on the Eulerian–Lagrangian method that take droplet break-up into account are conducted to clarify the mean structure of gaseous detonation laden with a dilute water spray. The premixed mixture is a slightly diluted stoichiometric hydrogen–oxygen mixture at low pressure. The simulated results are analysed via 2-D flow fields and statistical Favre spatiotemporal averaging techniques. Gaseous detonation with water droplets (WD) propagates stably with a velocity decrease compared with the dry Chapman–Jouguet speed. The mean structure of gaseous detonation with dilute water spray shares a similar structure as the one without water spray. However, the hydrodynamic thickness is changed due to the interaction with water spray. Overall interphase exchanges (mass, momentum and energy) that take place within the hydrodynamic thickness induce a decrease of the detonation velocity and lower the level of fluctuations downstream of the mean leading shock wave. Droplet break-up occurs downstream of the induction zone and in our case, the water vapour from the evaporation of water spray does not affect the reactivity of gaseous detonation. The laminar master equation for gaseous detonation laden with inert WD shows that the hydrodynamic thickness should rely on the gaseous sound speed, and works well as the working mixture is weakly unstable and its cellular structure is regular. The droplet flow regimes and break-up modes have also been determined. The characteristic lengths of detonation and interphase exchanges have been ordered under the present simulation conditions and have been shown to be intimately intertwined.
A systematic analysis was carried out to study the effect of shock waves on copper sulfate crystal in such a way that its optical properties and surface morphological properties were examined for different number of shock pulses (0, 1, 3, 5, and 7) with the constant Mach number 1.7. The test crystal of copper sulfate was grown by slow evaporation technique. The surface morphological and optical properties were scrutinized by optical microscope and ultraviolet–visible spectrometer, respectively. On exposing to shock waves, the optical transmission of the test crystal started increasing from the range of 35–45% with the increase of shock pulses and thereafter started decreasing to 25% for higher number of applied shocks. The optical band transition modes and optical band gap energies were calculated for pre- and post-shock wave loaded conditions. The experimentally obtained data prove that the optical constants such as absorption coefficient, extinction coefficient, skin depth, optical density, and optical conductivity are strongly altered, so also the optical transmission due to the impact of shock waves. Hence, shock wave induced high transmission test crystal can be used as an appropriate candidate for ultraviolet light filter applications.
We present a theory for the evolution of a one-dimensional, steady-state detonation reaction zone to a two-dimensional reaction zone, when the explosive experiences a sudden loss of side-on confinement as a boundary of the explosive is impulsively withdrawn. Our focus is on condensed-phase explosives, which we describe as having a constant adiabatic gamma equation of state and an irreversible, state-independent reaction rate. We consider two detonation models: (i) the instantaneous reaction heat release Chapman–Jouguet (CJ)-limit and (ii) the spatially resolved reaction heat-release Zel’dovich–von Neumann–Döring (ZND) model, in the limit where only a small fraction of the energy release is resolved (the SRHR-limit). Two competing rarefaction waves are generated by this loss of confinement: (i) a smooth wave coming off the full length of the withdrawn boundary and (ii) a singular fan spreading out from the point where the detonation shock and the withdrawn boundary meet. For the CJ-limit, in all cases the singular rarefaction fan eventually dominates the competition to control the steady-state behaviour. For the SRHR-limit, the spatially resolved heat release moderates this competition. When the withdrawal speed is fast, the rarefaction fan dominates; when the withdrawal speed is slower, the smooth rarefaction eventually dominates, although the flow features a fan at early times. By examining the mathematical properties of the steady two-dimensional fan-based solution, we set down a mechanism for this transition in behaviours.
An intensive overview of the fundamentals and physical principles on which seismic methods are based. It provides the necessary related geophysical background to understand seismic data and, hence, the reader will obtain a more clear understanding of how to properly process the data in order to ultimately obtain better seismic images that are used for accurate interpretation. With various examples, this includes the theory of elasticity, the wave equation, the types of seismic waves, single-layer reflector models, seismic events, etc.
In turbulent flows subject to strong background rotation, the advective mechanisms of turbulence are superseded by the propagation of inertial waves, as the effects of rotation become dominant. While this mechanism has been identified experimentally (Dickinson & Long, J. Fluid Mech., vol. 126, 1983, pp. 315–333; Davidson, Staplehurst & Dalziel, J. Fluid Mech., vol. 557, 2006, pp. 135–144; Staplehurst, Davidson & Dalziel, J. Fluid Mech., vol. 598, 2008, pp. 81–105; Kolvin et al.Phys. Rev. Lett., vol. 102, 2009, 014503), the conditions of the transition between the two mechanisms are less clear. We tackle this question experimentally by tracking the turbulent front away from a solid wall where jets enter an otherwise quiescent fluid. Without background rotation, this apparatus generates a turbulent front whose displacement recovers the
law classically obtained with an oscillating grid (Dickinson & Long, Phys. Fluids, vol. 21 (10), 1978, pp. 1698–1701) and we further establish the scale independence of the associated transport mechanism. When the apparatus is rotating at a constant velocity perpendicular to the wall where fluid is injected, not only does the turbulent front become mainly transported by inertial waves, but advection itself is suppressed because of the local deficit of momentum incurred by the propagation of these waves. Scale-by-scale analysis of the displacement of the turbulent front reveals that the transition between advection and propagation is local both in space and spectrally, and takes place when the Rossby number based on the considered scale is of order unity, or equivalently, when the scale-dependent group velocity of inertial waves matched the local advection velocity.
When open-cut mines are eventually abandoned, they leave a large hole with sloping sides. The hole fills with rain water, and there is also contaminated run-off from surrounding land, that moves through the rock and eventually through the sloping sides of the abandoned mine. This paper considers a two-dimensional unsteady model motivated by this leaching flow through the rock and into the rain-water reservoir. The stability of the interface between the two fluids is analysed in the inviscid limit. A viscous Boussinesq model is also presented, and a closed-form solution is presented to this problem, after it has been linearized in a manner consistent with Boussinesq theory. That solution suggests that the interfacial zone is effectively neutrally stable as it evolves in time. However, an asymptotic theory in the interfacial region shows the interface to be unstable. In addition, the nonlinear Boussinesq model is solved using a spectral method. Interfacial travelling waves and roll-up are observed and discussed, and compared against the predictions of asymptotic Boussinesq theory.
We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional non-linear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behaviour of minimizers once the mass tends to a critical value.
Fluid flow through pipe-like conduits embedded in viscously deformable material occurs in many natural systems, including magma transport in the Earth’s mantle and channelized water flow beneath glaciers. Here, we present and explore a model of fluid flow in viscously deformable conduits that unifies previously published models of magmatic and glacial systems. Previous results for magmatic systems have demonstrated the existence of solitary wave solutions for the case of laminar flow in Newtonian conduits. Here we extend these models to allow turbulent fluid flow in power-law materials consistent with models used in subglacial hydrology. The generalized model encompasses both laminar and turbulent fluid flow, and the solid matrix may deform according to any power-law rheology. A quasilinear approximation of the governing equations is introduced, along with an initial condition that develops into a perfect step shock. This initial condition is used in numerical solution of the full nonlinear system where a dispersive wave train forms at shock time. We show that solitary wave solutions exist for all parameters. Rheology-dependent flattening of the wave peaks is investigated. In the limit of a perfectly plastic matrix, the solitary waves approach square waves asymptotically. Motivated by subglacial hydrology models, we study the effect of discharge-dependent melting on evolution of the solitary waves. We find that melting focuses at the wave peaks, causing the waves to grow and accelerate over time.
To date, the influence of nonlinear stratifications and two layer stratifications on internal wave propagation has been studied for two-dimensional wave fields in a Cartesian geometry. Here, we use a novel wave generator configuration to investigate transmission in nonlinear stratifications of an axisymmetric internal wave. We demonstrate that, despite the additional geometric complexity, with associated features such as an inhomogeneous spatial distribution of the energy flux, results for plane waves can be generalised to axisymmetric wave fields. Two configurations are studied, both theoretically and experimentally. In the case of a free incident wave, a transmission maximum is found in the vicinity of evanescent frequencies. In the case of a confined incident wave, resonant effects, in the sense of constructive interference, lead to enhanced transmission rates from an upper layer to a layer below. We consider the oceanographic relevance of these results by applying them to an example oceanic stratification, finding that there can be real-world implications.
Richtmyer–Meshkov instability of the SF6 gas layer surrounded by air is experimentally investigated. Using the soap film technique, five kinds of gas layer with two sharp interfaces are generated such that the development of each individual interface is highlighted. The flow patterns are determined by the amplitudes and phases of two corrugated interfaces. For a layer with both interfaces planar, the interface velocity shows that the reflected rarefaction waves from the second interface accelerate the first interface motion. For a layer with the second interface corrugated but the first interface planar, the reflected rarefaction waves make the first interface develop with the same phase as the second interface. For a layer with the first interface corrugated but the second interface planar, the rippled shock seeded from the first interface makes the second interface develop with the same phase as the first interface and the layer evolves into an ‘upstream mushroom’ shape. For two interfaces corrugated with opposite (the same) phase but a larger amplitude for the first interface, the layer evolves into ‘sinuous’ shape (‘bow and arrow’ shape, which has never been observed previously). For the interface amplitude growth in the linear stage, the waves’ effects are considered in the model to give a better prediction. In the nonlinear stage, the effect of the rarefaction waves on the first interface evolution is quantitatively evaluated, and the nonlinear growth is well predicted. It is the first time in experiments to quantify the interfacial instability induced by the rarefaction waves inside the heavy gas layer.
Motivated by planetary-driven applications and experiments in non-spherical geometries, we study compressible fluid modes in rotating rigid ellipsoids. Such modes are also required for modal acoustic velocimetry (MAV), a promising non-invasive method to track the velocity field components in laboratory experiments. To calculate them, we develop a general spectral method in rigid triaxial ellipsoids. The description is based on an expansion onto global polynomial vector elements, satisfying the non-penetration condition on the boundary. Then, we investigate the diffusionless compressible modes in rotating (and magnetised) rigid ellipsoids. The spectral description is successfully benchmarked against three-dimensional finite-element computations and analytical predictions. A spectral convergence is obtained. Our results have direct implications for MAV in experiments, for instance in the ZoRo experiment (gas-filled rigid spheroid). So far, deformation and rotational effects have been theoretically considered separately, as small perturbations of the solutions in non-rotating spheres. We carefully compare the perturbation approach, in this illustrative geometry, to the polynomial solutions. We show that second-order ellipticity effects are often present, even in weakly deformed ellipsoids. Moreover, high-order effects due to rotation and/or ellipticity should be observed for some acoustic modes in experimental conditions. Thus, perturbation theory should be used with care in MAV. Instead, the spectral polynomial method paves the way for future MAV applications in fluid experiments with rigid ellipsoids.
The aim of this study is to understand the dynamics of Rossby waves induced by a localised and periodic ‘plunger’ forcing – imposed on a background flow – which is intended as an elementary representation of transient mesoscale eddy forcing in the ocean. We consider linearised dynamics and its quasi-nonlinear extension, and focus on the rotating shallow-water model. The plunger induces a spectrum of Rossby waves that drive zonal momentum flux convergence at the forced latitudes. This behaviour has a robust and significant dependence on the background flow, which we treat as zonal and uniform. We systematically analyse this dependence using two methods. First, we use the eddy geometry formulation, in which Reynolds stresses are expressed in terms of eddy elongation and eddy tilt parameters, and consider the relationship between eddy geometry and zonal momentum redistribution. Second, we implement decompositions of flow responses into linear dynamical eigenmodes and compare with expectations from linear Rossby wave theory. Both methods compliment each other and aid the understanding of zonal momentum redistribution and its dependence on uniform background flow. We find that this dependence is determined by two factors: (i) dispersion-constrained resonance with the plunger forcing and (ii) efficiency of nonlinear eddy self-interactions. These results significantly improve our understanding of shallow-water Rossby waves, and may also be applied towards the development of parameterisations of oceanic mesoscale eddies.
This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.
The interaction of a shock wave and a water droplet embedded with a vapour cavity is experimentally investigated in a shock tube for the first time. The vapour cavity inside the droplet is generated by decreasing the surrounding pressure to the saturation pressure, and an equilibrium between the liquid phase and the gas phase is obtained inside the droplet. Direct high-speed photography is adopted to capture the evolution of both the droplet and the vapour cavity. The formation of a transverse jet inside the droplet during the cavity-collapse stage is clearly observed. Soon afterwards, at the downstream pole of the droplet, a water jet penetrating into the surrounding air is observed during the cavity-expansion stage. The evolution of the droplet is strongly influenced by the evolution of the vapour cavity. The phase change process plays an important role in vapour cavity evolution. The effects of the relative size and eccentricity of the cavity on the movement and deformation of the droplet are presented quantitatively.
Internal waves shoaling on the continental slope can break and form materially coherent vortices called boluses. These boluses are able to trap and transport material up the continental slope, yet the global extent of bolus transport is unknown. Previous studies of bolus formation primarily focused on systems consisting of two layers of uniform density, which do not account for the presence of ocean pycnoclines of finite thickness. We use hyperbolic tangent profiles to model the density stratification in our simulations and demonstrate the impact of the pycnocline on the bolus. A spectral clustering method is used to objectively identify the bolus as a Lagrangian coherent structure that contains the material advected upslope. The bolus size and displacement upslope are examined as a function of the pycnocline thickness, incoming wave energy, density change across the pycnocline and topographic slope. The dependence of bolus transport on the pycnocline thickness demonstrates that boluses in continuous stratifications tend to be larger and transport material further than in corresponding two-layer stratifications.
We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modelling the turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (non-Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the fourth-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the ‘normal’ closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourth-order moment, Landau fluid closures are required.
Cavitating flow over a circular cylinder is investigated over a range of cavitation numbers (
) for both laminar (at Reynolds number
) and turbulent (at
) regimes. We observe non-cavitating, cyclic and transitional cavitation regimes with reduction in free-stream
. The cavitation inside the Kármán vortices in the cyclic regime, is significantly altered by the onset of ‘condensation front’ propagation in the transitional regime. At the transition, an order of magnitude jump in shedding Strouhal number (
) is observed as the dominant frequency shifts from periodic vortex shedding in the cyclic regime, to irregular–regular vortex shedding in the transitional regime. In addition, a peak in pressure fluctuations, and a maximum in
based on cavity length are observed at the transition. Shedding characteristics in each regime are discussed using dynamic mode decomposition. A numerical method based on the homogeneous mixture model, fully compressible formulation and finite rate mass transfer developed by Gnanaskandan & Mahesh (Intl J. Multiphase Flow, vol. 70, 2015, pp. 22–34) is extended to include the effects of non-condensable gas (NCG). It is demonstrated that the condensation fronts observed in the transitional regime are supersonic (referred to as ‘condensation shocks’). In the presence of NCG, multiple condensation shocks in a given cycle are required for complete cavity condensation and detachment, as compared to a single condensation shock when only vapour is present. This is explained by the reduction in pressure ratio across the shock in the presence of NCG, effectively reducing its strength. In addition, at
(near transition from the cyclic to the transitional regime), the presence of NCG suppresses the low frequency irregular–regular vortex shedding. Vorticity transport at
, in the transitional regime, indicates that the region of attached cavity is nearly two-dimensional, with very low vorticity, affecting Kármán shedding in the near wake. Majority of vortex stretching/tilting and vorticity production is observed following the cavity trailing edge. In addition, the boundary-layer separation point is found to be strongly dependent on the amounts of vapour and gas in the free stream for both laminar and turbulent regimes.
In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function
, and the associated plasma response function
. We then consider a one-dimensional (1-D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourth-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1-D closures at higher-order moments than the fourth order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3-D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the fourth-order moment level. In appendix A, we provide comprehensive tables with Padé approximants of
up to the eighth-pole order, with many given in an analytic form.
Hysteresis phenomena in forced gravity–capillary waves on deep water where the minimum phase speed
are experimentally investigated. Four kinds of forcings are considered: two-dimensional/three-dimensional air-blowing/air-suction forcings. For a still-water initial condition, as the forcing speed increases from zero towards a certain target speed (
), there exists a certain critical speed (
) at which the transition from linear to nonlinear states occurs. When
, steady linear localized waves are observed (state I). When
, steady nonlinear localized waves, including steep gravity–capillary solitary waves, are observed (state II). When
, periodic shedding phenomena of nonlinear localized depressions are observed (state III). When
, steady linear non-local waves are observed (state IV). Next, with these state-II, III and IV waves as new initial conditions, as the forcing speed is decreased towards a certain target speed (
), a certain critical speed (
) is identified at which the transition from nonlinear to linear states occurs. When
, relatively steeper steady nonlinear localized waves, including steeper gravity–capillary solitary waves, are observed. When
, linear state-I waves are observed. These are hysteresis phenomena, which show jump transitions from linear to nonlinear states and from nonlinear to linear states at two different critical speeds. For air-blowing cases, experimental results are compared with simulation results based on a theoretical model equation. They agree with each other very well except that the experimentally identified critical speed (
) is different from the theoretically predicted one.