Initial Velocity Perturbations Research Articles

Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions

Abstract Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: − G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ . \left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1 − D 1-D to high dimension R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) .

Numerical Simulations of the Decaying Transverse Oscillations in the Cool Jet

In the present paper, we describe a 2.5D (two-and-a-half-dimensional) magnetohydrodynamic (MHD) simulation that provides a detailed picture of the evolution of cool jets triggered by initial vertical velocity perturbations in the solar chromosphere. We implement random multiple velocity, Vy, pulses of amplitude 20–50 km s−1 between 1 Mm and 1.5 Mm in the Sun’s atmosphere below its transition region (TR). These pulses also consist of different switch-off periods between 50 s and 300 s. The applied vertical velocity pulses create a series of magnetoacoustic shocks steepening above the TR. These shocks interact with each other in the inner corona, leading to complex localized velocity fields. The upward propagation of such perturbations creates low-pressure regions behind them, which propel a variety of cool jets and plasma flows in the localized corona. The localized complex velocity fields generate transverse oscillations in some of these jets during their evolution. We study the transverse oscillations of a representative cool jet J1, which moves up to the height of 6.2 Mm above the TR from its origin point. During its evolution, the plasma flows make the spine of jet J1 radially inhomogeneous, which is visible in the density and Alfvén speed smoothly varying across the jet. The highly dense J1, which is triggered along the significantly curved magnetic field lines, supports the propagating transverse wave of period of approximately 195 s with a phase speed of about 125 km s−1. In the distance–time map of density, it is manifested as a transverse kink wave. However, the careful investigation of the distance–time maps of the x- and z-components of velocity reveals that these transverse waves are actually of mixed Alfvénic modes. The transverse wave shows evidence of damping in the jet. We conclude that the cross-field structuring of the density and characteristic Alfvén speed within J1 causes the onset of the resonant conversion and leakage of the wave energy outward to dissipate these transverse oscillations via resonant absorption. The wave energy flux is estimated as approximately of 1.0 × 106 ergs cm−2 s−1. This energy, if it dissipates through the resonant absorption into the corona where the jet is propagated, is sufficient energy for the localized coronal heating.

Quasi-periodic spicule-like cool jets driven by Alfvén pulses

ABSTRACT We perform a 2.5D magnetohydrodynamic simulation to gain a comprehensive understanding of the formation of spicule-like cool jets caused by initial transverse velocity pulses akin to Alfvén pulses in the solar chromosphere. We invoke multiple velocity (Vz) pulses between 1.5 and 2.0 Mm in the solar atmosphere, which create the initial transverse velocity perturbations. These pulses transfer energy non-linearly to the field-aligned perturbations via the ponderomotive force. This physical process further creates magnetoacoustic shocks followed by quasi-periodic plasma motions in the solar atmosphere. The field-aligned magnetoacoustic shocks move upwards, which subsequently causes the quasi-periodic rise and fall of chromospheric plasma into the overlying corona as thin and cool spicule-like jets. The magnitude of the initial applied transverse velocity pulses is taken in the range of 50–90 km s−1. These pulses are found to be strong enough to generate spicule-like jets. We analyse the evolution, kinematics and energetics of these spicule-like jets. We find that the transported mass flux and kinetic energy density are substantial in the local solar corona. These mass motions generate in situ quasi-periodic oscillations on the scale of ≃ 4.0 min above the transition region.

Strong solutions of the equations for viscoelastic fluids in some classes of large data

We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. Our result mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preventing the formation of singularities in viscous flows. Moreover, for given initial velocity perturbation and zero initial deformation around the rest state, we find, as the elasticity coefficient or time goes to infinity, that(1)any straight line segment l0 consisted of fluid particles in the rest state, after being bent by a velocity perturbation, will turn into a straight line segment that is parallel to l0 and has the same length as l0.(2)the motion of the viscoelastic fluid can be approximated by a linear pressureless motion in Lagrangian coordinates, even when the initial velocity is large. Moreover, the above mentioned phenomena can also be found in the corresponding compressible fluid case.

Chaos in breaking waves

This study investigates the chaotic behavior of breaking waves by laboratory experiments and numerical modeling. Repeated laboratory runs with different initial velocity perturbations show that the wave profile before the wave breaks can be accurately reproduced, but the subsequent breaking process varies among runs, indicating the lack of repeatability of breaking waves in reality. Numerical simulations based on the Smoothed Particle Hydrodynamics method are further carried out to examine the repeatability of wave breaking process. Consistent with the laboratory observation, multiple numerical simulations with variations in initial conditions present highly repeatable velocity field and free surface profile in the potential flow region but considerable variation at the breaking and post-breaking processes. Comparison also shows that 3D vortex structures induced by breaking waves are different among cases. Analysis of particle trajectory reveals that there is a similar trajectory thus a minor trajectory divergence among particles that are initially located at the pre-breaking region and the flume bottom, which are not directly impacted by the breaking process. However, a much more significant particle trajectory divergence is observed among particles that are initially located at the wave-splash region and the bore propagation region. The rate of divergence of particle trajectory under breaking waves is further examined by computing the Lyapunov exponent, a widely used indicator of chaos. This study reveals that different initial velocity perturbations lead to variations of near-surface velocity at the onset of wave breaking, which eventually cause the development of drastically different breaking wave jets and splashes. Therefore, the process of wave breaking, like many other dynamic processes in nature, exhibits a chaotic behavior.

Coupling between velocity and interface perturbations in cylindrical Rayleigh–Taylor instability**Project supported by the National Natural Science Foundation of China (Grant Nos. 11275031, 11475034, 11575033, and 11274026) and the National Basic Research Program of China (Grant No. 2013CB834100).

Rayleigh–Taylor instability (RTI) in cylindrical geometry initiated by velocity and interface perturbations is investigated analytically through a third-order weakly nonlinear (WN) model. When the initial velocity perturbation is comparable to the interface perturbation, the coupling between them plays a significant role. The difference between the RTI growth initiated only by a velocity perturbation and that only by an interface perturbation in the WN stage is negligibly small. The effects of the mode number on the first three harmonics are discussed respectively. The low-mode number perturbation leads to large amplitudes of RTI growth. The Atwood number and initial perturbation dependencies of the nonlinear saturation amplitude of the fundamental mode are analyzed clearly. When the mode number of the perturbation is large enough, the WN results in planar geometry are recovered.

The effect of magnetic fields and ambipolar diffusion on the column density probability distribution function in molecular clouds

Simulations generally show that non-self-gravitating clouds have a lognormal column density ($\Sigma$) probability distribution function (PDF), while self-gravitating clouds with active star formation develop a distinct power-law tail at high column density. Although the growth of the power law can be attributed to gravitational contraction leading to the formation of condensed cores, it is often debated if an observed lognormal shape is a direct consequence of supersonic turbulence alone, or even if it is really observed in molecular clouds. In this paper we run three-dimensional magnetohydrodynamic simulations including ambipolar diffusion with different initial conditions to see the effect of strong magnetic fields and nonlinear initial velocity perturbations on the evolution of the column density PDFs. Our simulations show that column density PDFs of clouds with supercritical mass-to-flux ratio, with either linear perturbations or nonlinear turbulence, quickly develop a power-law tail such that $dN/d \log \Sigma \propto \Sigma^{-\alpha}$ with index $\alpha \simeq 2$. Interestingly, clouds with subcritical mass-to-flux ratio also proceed directly to a power-law PDF, but with a much steeper index $\alpha \simeq 4$. This is a result of gravitationally-driven ambipolar diffusion. However, for nonlinear perturbations with a turbulent spectrum ($v_{k}^{2} \propto k^{-4}$), the column density PDFs of subcritical clouds do retain a lognormal shape for a major part of the cloud evolution, and only develop a distinct power-law tail with index $\alpha \simeq 2$ at greater column density when supercritical pockets are formed.

Primordial black holes in linear and non-linear regimes

We revisit the formation of primordial black holes (PBHs) in the radiation-dominated era for both linear and non-linear regimes, elaborating on the concept of an apparent horizon. Contrary to the expectation from vacuum models, we argue that in a cosmological setting a density fluctuation with a high density does not always collapse to a black hole. To this end, we first elaborate on the perturbation theory for spherically symmetric space times in the linear regime. Thereby, we introduce two gauges. This allows to introduce a well defined gauge-invariant quantity for the expansion of null geodesics. Using this quantity, we argue that PBHs do not form in the linear regime irrespective of the density of the background. Finally, we consider the formation of PBHs in non-linear regimes, adopting the spherical collapse picture. In this picture, over-densities are modeled by closed FRW models in the radiation-dominated era. The difference of our approach is that we start by finding an exact solution for a closed radiation-dominated universe. This yields exact results for turn-around time and radius. It is important that we take the initial conditions from the linear perturbation theory. Additionally, instead of using uniform Hubble gauge condition, both density and velocity perturbations are admitted in this approach. Thereby, the matching condition will impose an important constraint on the initial velocity perturbations δh0 = −δ0/2. This can be extended to higher orders. Using this constraint, we find that the apparent horizon of a PBH forms when δ > 3 at turn-around time. The corrections also appear from the third order. Moreover, a PBH forms when its apparent horizon is outside the sound horizon at the re-entry time. Applying this condition, we infer that the threshold value of the density perturbations at horizon re-entry should be larger than δth > 0.7.

2D-DNS and 2D-RANS Simulations of Supersonic QUASI-2D Turbulent Reacting Shear Flow

<p>Numerical studies of quasi-2D supersonic turbulent hydrogen-air mixing and combustion in free shear layer configuration are performed using 2D-DNS [1] and RANS equations. In order to produce the roll-up and pairing of vortex rings, an unsteady boundary condition is applied at the inlet plane. Frequencies of<br /> initial velocity perturbations have been taken in accordance with linear stability theory. The influences of different inflow perturbations on mixing layer structure are presented. At the outflow, the non-reflecting boundary condition is adopted. In the case of RANS simulation two-parameter k-e turbulence model is used. Thermal conduction is described by Fourier’s law, while diffusion of species by Fick’s law. Equation of state for thermally perfect multispecies gas is used. Thermodynamic parameters, such as specific heat, enthalpy, entropy and internal energy are determined by fourth order degree polynomial formula, which has dependence on temperature. Temperature is determined using Newton-Raphson iteration procedure. The Wilke’s formula is used to determine the mixture viscosity coefficient. Approximation of convection terms are performed by the ENO-scheme of third-order accuracy and approximation of diffusion terms – by second-order central-difference operators. For the description of reaction pathways of hydrogen, a seven species chemical reaction model by Jachimowski is adopted. Chemical reaction source term implicitly includes in mass fraction transport equations, where linearization is applied using Taylor decomposition. The hydrogen flow parameters are M<sub>0</sub> = 2.0, T<sub>0</sub> = 2000 K, P<sub>0</sub> = 101325 Pa, and air flow parameters are M<sub>¥</sub> = 2.1, T<sub>¥</sub> = 2000 K, p<sub>¥</sub> = 101325 Pa. Convective Mach number is M<sub>c</sub> = 0.38, where effect of compressibility is significant.</p>

On the Second Harmonic Generation through Bell—Plesset Effects in Cylindrical Geometry

Generation of the second harmonic initiated by Bell—Plesset effects in a cylindrical geometry is studied analytically. For an initial single-mode velocity perturbation, the second-order mode-coupling formula is obtained by expanding the perturbation displacement and velocity potential up to the second-order accuracy. It is found that the initially symmetric interface evolves into a significant bubble-spike asymmetric pattern. The second-order solutions clearly show that the amplitude of the spike grows faster than that of the bubble. The temporal evolutions of the amplitudes of the bubble and spike are dependent on the interface velocity V0. The larger interface velocity leads to the smaller amplitude of the perturbation at an arbitrary interface position in a cylindrically convergent geometry.

Mechanism by which a frictionally confined rod loses stability under initial velocity and position perturbations

We present a theory to reveal for the first time the distinct mechanisms by which a compressed rod confined in a channel buckles in the presence of dry friction. Contrary to the case of a frictionless contact, with friction the system can bear substantially enhanced compressive load without buckling after its stiffness turns negative, and the onset of instability is strongly affected by the amount of perturbation set by the environment. Our theory, confirmed by simulations, shows that friction enhances stability by opening a wide stable zone in the perturbation space. Buckling is initiated when the applied compressive force is such that the boundary of the stable zone touches a point set by the environment, at a much higher critical load. Furthermore, our analysis shows that friction has a strong effect on the buckling mode; an increase in friction is found to lead to higher buckling modes.

Out-of-plane stability analysis of collinear spinning three-craft Coulomb formations

The results of a stability analysis focusing on the out-of-plane motion of collinear three-craft Coulomb formations with set charges are discussed. Such a formation is assumed to be spinning in deep space without relevant gravitational forces present. Assuming in-plane motion only with circular relative trajectories and initial position and velocity perturbations confined to the orbital plane, the previous work analytically proves marginal stability in the linear sense and numerically shows marginal stability in the short term. In this paper, the equations of motion are presented in the cylindrical coordinate frame in order to analyze the out-of-plane motion in more detail. The out-of-plane motion is shown to decouple to first order from the marginally stable in-plane motion. A simple control law is developed and applied, which directly controls the out-of-plane motion only within specified deadbands. For a wide variety of out-of-plane perturbations, the control law succeeds in preserving the in-plane variant shape despite some out-of-plane motion. A trend between the settling time and deadband, which defines the largest out-of-plane errors allowed before the controller is turned on, is determined, which illustrates how large the deadband may be before the in-plane motion is affected.

Coupling between interface and velocity perturbations in the weakly nonlinear Rayleigh-Taylor instability

Weakly nonlinear (WN) Rayleigh-Taylor instability (RTI) initiated by single-mode cosinusoidal interface and velocity perturbations is investigated analytically up to the third order. Expressions of the temporal evolutions of the amplitudes of the first three harmonics are derived. It is shown that there are coupling between interface and velocity perturbations, which plays a prominent role in the WN growth. When the “equivalent amplitude” of the initial velocity perturbation, which is normalized by its linear growth rate, is compared to the amplitude of the initial interface perturbation, the coupling between them dominates the WN growth of the RTI. Furthermore, the RTI would be mitigated by initiating a velocity perturbation with a relative phase shift against the interface perturbation. More specifically, when the phase shift between the interface perturbation and the velocity perturbation is π and their equivalent amplitudes are equal, the RTI could be completely quenched. If the equivalent amplitude of the initial velocity perturbation is equal to the initial interface perturbation, the difference between the WN growth of the RTI initiated by only an interface perturbation and by only a velocity perturbation is found to be asymptotically negligible. The dependence of the WN growth on the Atwood numbers and the initial perturbation amplitudes is discussed. In particular, we investigate the dependence of the saturation amplitude (time) of the fundamental mode on the Atwood numbers and the initial perturbation amplitudes. It is found that the Atwood numbers and the initial perturbation amplitudes play a crucial role in the WN growth of the RTI. Thus, it should be included in applications where the seeds of the RTI have velocity perturbations, such as inertial confinement fusion implosions and supernova explosions.

The dynamics of Rayleigh-Taylor stable and unstable contact discontinuities with anisotropic thermal conduction

We study the effects of anisotropic thermal conduction along magnetic field lines on an accelerated contact discontinuity in a weakly collisional plasma. We first perform a linear stability analysis similar to that used to derive the Rayleigh-Taylor instability (RTI) dispersion relation. We find that anisotropic conduction is only important for compressible modes, as incompressible modes are isothermal. Modes grow faster in the presence of anisotropic conduction, but growth rates do not change by more than a factor of order unity. We next run fully non-linear numerical simulations of a contact discontinuity with anisotropic conduction. The non-linear evolution can be thought of as a superposition of three physical effects: temperature diffusion due to vertical conduction, the RTI, and the heat flux driven buoyancy instability (HBI). In simulations with RTI-stable contact discontinuities, the temperature discontinuity spreads due to vertical heat conduction. This occurs even for initially horizontal magnetic fields due to the initial vertical velocity perturbation and numerical mixing across the interface. The HBI slows this temperature diffusion by reorienting initially vertical magnetic field lines to a more horizontal geometry. In simulations with RTI-unstable contact discontinuities, the dynamics are initially governed by temperature diffusion, but the RTI becomes increasingly important at late times. We discuss the possible application of these results to supernova remnants, solar prominences, and cold fronts in galaxy clusters.

A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell’s Demon which may separate fast and slow billiard particles.

Thermally induced van der Waals rupture of thin viscous fluid sheets

We consider the dynamics of a thin symmetric fluid sheet subject to an initial temperature profile, where inertia, viscous stresses, disjoining pressures, capillarity, and thermocapillarity are important. We apply a long-wave analysis in the limit where deviations from the mean sheet velocity are small, but thermocapillary stresses and heat transfer from the sheet to the environment are significant and find a coupled system of partial differential equations that describe the sheet thickness, the mean sheet velocity, and the mean sheet temperature. From a linear stability analysis, we find that a stable thermal mode couples the velocity to the interfacial dynamics. This coupling can be utilized to delay the onset of rupture or to promote an earlier rupture event. In particular, rupture can be induced thermally even in cases when the heat transfer to the surrounding environment is significant, provided that the initial phase shift between the initial velocity and temperature disturbances is close to ϕ = π/2. These effects suggest a strategy that uses phase modulation in the initial temperature perturbation related to the initial velocity perturbation that assigns priority of the rupture events at particular sites over several spatial periods.

RELAXATION OF PULSAR WIND NEBULA VIA CURRENT-DRIVEN KINK INSTABILITY

We have investigated the relaxation of a hydrostatic hot plasma column containing toroidal magnetic field by the Current-Driven (CD) kink instability as a model of pulsar wind nebulae. In our simulations the CD kink instability was excited by a small initial velocity perturbation and developed turbulent structure inside the hot plasma column. We demonstrated that, as envisioned by Begelman, the hoop stress declines and the initial gas pressure excess near the axis decreases. The magnetization parameter "σ", the ratio of the magnetic energy to the thermal energy for a hot plasma, declined from an initial value of 0.3 to about 0.01 when the CD kink instability saturated. Our simulations demonstrated that axisymmetric models strongly overestimate the elongation of the pulsar wind nebulae. Therefore, the previous requirement for an extremely low pulsar wind magnetization can be abandoned. The observed structure of the pulsar wind nebulae do not contradict the natural assumption that the magnetic energy flux still remains a good fraction of the total energy flux after dissipation of alternating fields.

THREE-DIMENSIONAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATIONS OF CURRENT-DRIVEN INSTABILITY. II. RELAXATION OF PULSAR WIND NEBULA

We have investigated the relaxation of a hydrostatic hot plasma column containing toroidal magnetic field by the current-driven (CD) kink instability as a model of pulsar wind nebulae. In our simulations, the CD kink instability is excited by a small initial velocity perturbation and develops a turbulent structure inside the hot plasma column. We demonstrate that, as envisioned by Begelman, the hoop stress declines and the initial gas pressure excess near the axis decreases. The magnetization parameter σ, the ratio of the Poynting to the kinetic energy flux, declines from an initial value of 0.3 to about 0.01 when the CD kink instability saturates. Our simulations demonstrate that axisymmetric models strongly overestimate the elongation of the pulsar wind nebulae. Therefore, the previous requirement for an extremely low pulsar wind magnetization can be abandoned. The observed structure of the pulsar wind nebulae does not contradict the natural assumption that the magnetic energy flux still remains a good fraction of the total energy flux after dissipation of alternating fields.

Macroscopic parameters of three-dimensional flows in free shear turbulence

The initial stage of the onset of turbulence in a three-dimensional compressible inviscid shear flow is studied. An initial deterministic velocity perturbation in the form of one or several Fourier modes leads to the development of a cascade of instabilities, which is numerically simulated. The influence exerted on the formation of the cascade of instabilities and the transition to turbulence by the size of the computational domain, the shear layer width, and the initial conditions is analyzed. It is shown that the mechanism of turbulence onset is essentially three-dimensional. The influence of various flow parameters and initial conditions on the formation of the turbulence cascade is studied numerically.

Modelling Mass Transfer Boundary Layer Instability in the CO-Based VAPEX Process

Abstract VAPEX (vapour extraction) is a promising technique for the recovery of heavy oil and bitumen reservoirs, especially for cases where steam-assisted gravity drainage and other thermal recovery methods are not economical. In the VAPEX process, a solvent is injected into the reservoir to reduce the oil viscosity and mobilize it towards the production well. CO2-based VAPEX is an attractive option from both economic and environmental perspectives. In CO2-based VAPEX, unlike other hydrocarbon solvents, dissolution of CO2 into the oil can result in a density increase of the diluted oil. As a consequence, the diluted oil has a higher density than the immobile oil beneath and a gravitationally unstable diffusive boundary layer is induced, which may lead to natural convection. In this study, a mathematical model for the diffusive boundary layer in the CO2-oil contact region in the VAPEX process is developed and the possibility of convective mixing occurrence is examined using linear stability analysis, based on the amplification of the initial velocity perturbations. It is found that in field-scale cases in the VAPEX process, the Rayleigh number is much smaller than the critical Rayleigh number, (Rac), and natural convection cannot happen in this process. Introduction The world's total reserve of heavy oil and bitumen is about six trillion barrels, which is about six times the amount of conventional resources(1). A major part of these resources are in Canada, Venezuela and the United States. Most of these reserves are at such depths that open-pit mining cannot be used economically, and in situ methods have to be used to reduce the viscosity of the oil-in-place and mobilize it. Either thermal methods or non-thermal methods can be used to recover these reserves. The viscosity of oil is a strong function of temperature and decreases sharply with increasing temperature. Currently steam-assisted gravity drainage (SAGD), a thermal method, is a popular scheme for the recovery of heavy oil and bitumen and has been successfully applied in several fields. Despite the success of this process for some reservoirs, there are many reservoirs that SAGD cannot be applied due to excess heat loss, which makes it uneconomical to operate. In thin reservoirs, the need for steam increases and the steam-to-oil ratio (SOR) is prohibitively high. Many oil and bitumen reservoirs have a bottom aquifer, and heat loss to the water can make the process unfeasible(2). There are also reservoir conditions where SAGD may not be applied, such as when water saturation is high or porosity is low. In cases where SAGD cannot be applied, VAPEX is an option for the recovery of these resources. The initial development of the VAPEX process was first introduced by Butler and Mokrys(3) as a solvent analogue to steam-assisted gravity drainage. The steam chamber in SAGD is replaced by a solvent chamber in VAPEX. In the VAPEX process, a solvent is injected near its dew point (where both solubility and diffusivity of the vapour solvent into oil are at their maximums) and forms a solvent chamber within the reservoir(4).