Small Wave Numbers Research Articles

Effect of small heat release and viscosity on thermal-diffusive instability

The linear stability of a thermal reaction front has been investigated based on the thermal-diffusive model proposed by Zel’dovich and Frank-Kamenetskii, which is called ZFK model. In the framework of ZFK model, heat-conduction and mass-diffusion equations are treated without the effect of hydrodynamic flow. Then, two types of instability appear: cellular and oscillatory instabilities. The cellular instability has only positive real part of growth rate, while the oscillatory instability is accompanied with non-zero imaginary part. In this study, the effect of heat release and viscosity on both instabilities is investigated asymptotically and numerically. This is achieved by coupling mass-conservation and Navier–Stokes equations with the ZFK model for small heat release. Then, the stable range of Lewis number, where the real part of growth rate is negative, is widened by non-zero values of heat release for any wavenumber. The increase of Prandtl number also brings the stabilization effect on the oscillatory instability. However, as for the cellular instability, the viscosity leads to the destabilization effect for small wavenumbers, opposed to its stabilization effect for moderate values of wavenumber. Under the limit of small wavenumber, the property of viscosity becomes clear in view of cut-off wavenumber, which makes the real part of growth rate zero.

Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

Modulational instability of spin-orbit coupled Bose-Einstein condensates in discrete media

We address the impact of intra-site spin-orbit (SO) coupling and associated inter-component Rabi coupling on the modulational instability (MI) of plane-wave states in two-component discrete Bose-Einstein condensates (BECs). Conditions for the onset of the MI and the respective instability are found analytically. SO coupling allows us to produce the MI even for a small initial wavenumber (q<π/2) for miscible states. In particular, SO coupling introduces MI even in the absence of hopping coefficient, a concept which may have wider ramifications in heavy atomic BECs. We have also shown how our results of the linear stability analysis can be corroborated numerically. The fact that we have brought out the stability criteria in different domains of system parameters means that our model is tailor made experiments.

Effects of dust particles charged by inelastic collisions and by photoionization on Alfvén waves in a stellar wind

ABSTRACT Using a kinetic description of a homogeneous magnetized dusty plasma with Maxwellian distribution of electrons and protons and dust particles charged by inelastic collisions and by photoionization, we analyse the dispersion relation considering the case where waves and radiation propagate exactly parallel to the ambient magnetic field. The investigation emphasizes the changes that the photoionization process brings to the propagation and damping of the waves in a stellar wind environment, since Alfvén waves are believed to play a significant role in the heating and acceleration processes that take place in the wind. The results show that, in the presence of dust with negative equilibrium electrical charge, the Alfvén mode decouples into the whistler and ion cyclotron modes for all values of wavenumber, but when dust particles acquire neutral or positive values of electrical charge, these modes may couple for certain values of wavenumber. It is also seen that the whistler and ion cyclotron modes present null group velocity in an interval of small wavenumber, and that the maximum value of wavenumber for which the waves are non-propagating is reduced in the presence of the photoionization process. For very small values of wavenumber, the damping rates of the modes could change significantly from very small to very high values if the sign of the dust electrical charge is changed.

Instability of cumulation in converging cylindrical shock wave

The conditions of linear instability for a converging cylindrical shock wave in an arbitrary inviscid medium are obtained. The initial continuous cylindrical symmetry of the shock wave front is exchanged on a discrete symmetry that is determined by the most unstable small azimuthal dimensionless wave numbers 0&amp;lt;k&amp;lt;kth&amp;lt;1 of corrugation perturbations. Due to the long azimuthal wavelengths (λ=2πRs0/k, Rs0—the radius of the shock wave) of perturbations, the shape of the resulting shock wave front is not changed significantly, but the corresponding restriction of the internal energy cumulation can be caused by the intensification of the rotation of the medium behind the front. The instability and the restriction of cumulation are also possible in the case of the exponential rapid growth of the one-dimensional perturbations with k=0, when the shape of the shock front is not changed at all. The correspondence of present theory to the experimental and simulation data on underwater electrical explosion is considered.

Turbulent radiative diffusion and turbulent Newtonian cooling

Radiation transport plays an important role in stellar atmospheres, but the effects of turbulence are being obscured by other effects such as stratification. Using radiative hydrodynamic simulations of forced turbulence, we determine the decay rates of sinusoidal large-scale temperature perturbations of different wavenumbers in the optically thick and thin regimes. Increasing the wavenumber increases the rate of decay in both regimes, but this effect is much weaker than for the usual turbulent diffusion of passive scalars, where the increase is quadratic for small wavenumbers. The turbulent decay is well described by an enhanced Newtonian cooling process in the optically thin limit, which is found to show a weak increase proportional to the square root of the wavenumber. In the optically thick limit, the increase in turbulent decay is somewhat steeper for wavenumbers below the energy-carrying wavenumber of the turbulence, but levels off toward larger wavenumbers. In the presence of turbulence, the typical cooling time is comparable to the turbulent turnover time. We observe that the temperature takes a long time to reach equilibrium in both the optically thin and thick cases, but in the former, the temperature retains smaller scale structures for longer.

Modeling and analysis of cilia generated shear-rate dependant flow

The ciliated channel is a universal structure that is involved in organism-tube, and cilia beating makes a significant impact on bio-fluid transport. Inspired by the biological discovery of the cilia applications, in this novel study, we illustrate the characteristics of the mass transfer of non-Newtonian fluid in a ciliated channel. We assume that a metachronal wave of cilia generates a two-dimensional wavy channel. We consider Carreau materials, whose viscosities are dependent on both the infinite-shear-rate and zero-shear-rate. The transformation from fixed coordinates to moving coordinates is employed. We reduce the constitutive equations by hypotheses of small Reynolds number and wave number. The simplified system is solved via a perturbation method with a stream function. Finally, we sketch visualizations of velocity, pressure gradient, and pressure difference versus different parameters of physical interests. We analyze the effects of different boundary layers on fluid flow. Besides, graphical results for pressure rise, velocity field, and streamline are attained. This study is the provision of suggestive results in designing biomaterial nanotechnology.

Triglobal infinite-wing shock-buffet study

This article uses triglobal stability analysis to address the question of shock-buffet unsteadiness, and associated modal dominance, on infinite wings at high Reynolds number, expanding upon recent biglobal work, aspiring to elucidate the flow phenomenon's origin and characteristics. Infinite wings are modelled by extruding an aerofoil to finite aspect ratios and imposing a periodic boundary condition without assumptions on spanwise homogeneity. Two distinct steady base flows, spanwise uniform and non-uniform, are analysed herein on straight and swept wings. Stability analysis of straight-wing uniform flow identifies both the oscillatory aerofoil mode, linked to the chordwise shock motion synchronised with a pulsation of its downstream shear layer, and several monotone (non-oscillatory), spatially periodic shock-distortion modes. Those monotone modes become outboard travelling on the swept wing with their respective frequencies and phase speeds correlated with the sweep angle. In the limiting case of very small wavenumbers approaching zero, the effect of sweep creates branches of outboard and inboard travelling modes. Overall, triglobal results for such quasi-three-dimensional base flows agree with previous biglobal studies. On the contrary, cellular patterns form in proper three-dimensional base flow on straight wings, and we present the first triglobal study of such an equilibrium solution to the governing equations. Spanwise-irregular modes are found to be sensitive to the chosen aspect ratio. Nonlinear time-marching simulations reveal the flow evolution and distinct events to confirm the insights gained through dominant modes from routine triglobal stability analysis.

Variation and Episodes of Near-Inertial Internal Waves on the Continental Slope of the Southeastern East China Sea

Based on in situ observations, six episodes of near-inertial internal waves (NIWs) were detected on the East China Sea (ECS) continental slope, and the mechanisms and characteristics of them were examined. The generation mechanisms of the observed NIWs included typhoon, wind burst, lateral propagation, and energy transfer from low-frequency flow. The depth-integrated near-inertial kinetic energy (NIKE) showed no significant seasonal variation, and the annual mean NIKE and near-inertial currents were 400 J/m2 and 3.50 cm/s, respectively. Downward propagation of NIKE was evident in the small wavenumber band according to the rotary vertical wavenumber spectra. The NIKE was subsurface-intensified, and the near-inertial vertical shear reached 0.01 s−1. The vertical phase speeds of the NIWs ranged from 5 to 19 m/h. The frequencies of the NIWs were mostly red-shifted, however, blue-shift also existed. One episode had both blue- and red-shifted frequencies vertically, and had both upward and downward propagating vertical phase speeds. The e-folding times of the observed NIWs ranged from 4 to 11 days, which were influenced by successive wind bursts and background vorticity. On the left-hand side of Kuroshio, the background vorticity is usually positive; however, the NIWs were almost red-shifted, which resulted from the Doppler shift of the Kuroshio.

Analytic derivation of the inertial range of compressible turbulence

An analytic model for steady state turbulence is employed to obtain the inertial range power spectrum of compressible turbulence. We assume that for homogeneous turbulence, the timescales controlling the energy injected at a given wavenumber from all smaller wavenumbers are equal for each spatial component. However, the longitudinal component energy is diverted into compression, so the rate controlling the energy that is transferred to all larger wavenumbers by the turbulent viscosity is reduced. The resulting inertial range is a power law with an index of −2. Indeed, such power spectra were observed in various astrophysical settings and also in numerical simulations.

The scalar, vector, and tensor modes in gravitational wave turbulence simulations

We study the gravitational wave (GW) signal sourced by primordial turbulence that is assumed to be present at cosmological phase transitions like the electroweak and quantum chromodynamics phase transitions. We consider various models of primordial turbulence, such as those with and without helicity, purely hydrodynamical turbulence induced by fluid motions, and magnetohydrodynamic turbulence whose energy can be dominated either by kinetic or magnetic energy, depending on the nature of the turbulence. We also study circularly polarized GWs generated by parity violating sources such as helical turbulence. Our ultimate goal is to determine the efficiency of GW production through different classes of turbulence. We find that the GW energy and strain tend to be large for acoustic or irrotational turbulence, even though its tensor mode amplitude is relatively small at most wave numbers. Only at very small wave numbers is the spectral tensor mode significant, which might explain the efficient GW production in that case.

Linear stability analysis of two fluid columns of different densities and viscosities in a gravity field

The linear stability of a vertical interface separating two miscible fluid columns of different densities and viscosities under the influence of gravity is investigated. This flow possesses a time-dependent reference state (each column accelerates at different rates owing to their different densities) and the interface thickness grows as the square root of time (by diffusion). Numerical integration of the linear initial-value problem is carried out and discussed in detail as a function of vertical and spanwise wavenumbers and the flow parameters. Adjoint-based optimization is performed in order to determine initial conditions that lead to maximum growth of disturbances in finite time. Results indicate that the rate of growth of the perturbation energy at small wavenumbers (less affected by viscosity initially) is dominated by two-dimensional modes (no spanwise variation). Substantial transient growth is observed at higher wave modes initially, followed by asymptotic decay of the perturbations at large time. Sensitivity of perturbation growth with respect to initial time, density and viscosity ratios is investigated. This work is complementary to previous inviscid analysis of this configuration, which showed that the interface was unconditionally unstable at all wave modes, even in the presence of surface tension, and that instability grew as the exponential of time squared.

Effect of relaxation and retardation times on dusty Jeffrey fluid in a curved channel with peristalsis

In recent work, the Jeffrey liquid with uniform dust particles in a symmetric channel is studied. Moving sinusoidal wave is executed on the walls of the channel, which generates peristaltic transport in the fluid. The governing equations for fluid and dust particles have been formulated using stream function. Perturbation method is used to get analytical solution of the problem by using small wave number. Graphical analysis has been carried out for stream function and velocity of fluid and dust particles. Effects of different parameters such as curvature k, relaxation time [Formula: see text], wave number [Formula: see text] and retardation time [Formula: see text] are debated through graphs for both dust particles and fluid. The noteworthy outcomes are fluid velocity, pressure gradient in the region [Formula: see text] and bolus size increases by increasing [Formula: see text]

Dissipation in the magnetic turbulence of reversed field pinch plasmas

Reversed field pinch (RFP) plasmas are subject to tearing instability that creates a broad spectrum of magnetic fluctuations. The dominant fluctuations have poloidal and toroidal mode numbers (m,n)=(1,6−10) and can grow to 2–3% of the mean magnetic field. Through nonlinear coupling, this growth culminates in a strong reconnection event and broadening of the magnetic spectrum extending to the ion gyroradius scale. Multiple developments occur during the reconnection stage: ions and electrons are energized, magnetic fluctuation amplitudes increase, plasma flow is halted, and the toroidal magnetic flux increases in a sawtooth-like fashion as the RFP dynamo becomes stronger. Magnetic fluctuations are measured in the plasma edge at multiple radial locations from r/a = 0.75 to 0.96 to assess and characterize the magnetic turbulence. The measured spectrum perpendicular to the mean field, S(k⊥), can be fit to a model spectrum consisting of power-law and exponential component with one free parameter that characterizes dissipation. The measured dissipation is much larger than estimated from classical viscous or resistive dissipation, but it is consistent with a flow damping measurement of anomalous viscosity. The measurements show an evolution of the spectrum during which fluctuation power builds up in the smallest wavenumbers and cascades to the larger wavenumber due to the nonlinear coupling between the linear (m, n) = (1, &amp;gt; 6) and the nonlinear (m, n) = (0, 1) tearing modes.

Near-inertial dissipation due to stratified flow over abyssal topography

AbstractLinear theory for steady stratified flow over topography sets the range for topographic wavenumbers over which freely propagating internal waves are generated, and the radiation and breaking of these waves contribute to energy dissipation away from the ocean bottom. However, previous numerical work demonstrated that dissipation rates can be enhanced by flow over large scale topographies with wavenumbers outside of the lee wave radiative range. We conduct idealized 3D numerical simulations of steady stratified flow over 1D topography in a rotating domain and quantify vertical distribution of kinetic energy dissipation. We vary two parameters: the first determines whether the topographic obstacle is within the lee wave radiative range and the second, proportional to the topographic height, measures the degree of flow non-linearity. For certain combinations of topographic width and height, breaking occurs in pulses every inertial period, such that kinetic energy dissipation develops inertial periodicity. In these simulations, kinetic energy dissipation rates are also enhanced in the interior of the domain. In the radiative regime the inertial motions arise due to resonant wave-wave interactions. In the small wavenumber non-radiative regime, instabilities downstream of the obstacle can facilitate the generation and propagation of non-linearly forced inertial motions, especially as topographic height increase. In our simulations, dissipation rates for tall and wide non-radiative topography are comparable to those of radiative topography, even away from the bottom, which is relevant to the ocean where the topographic spectrum is such that wider abyssal hills also tend to be taller.

On the vertical shear instability in magnetized protoplanetary discs

ABSTRACT The vertical shear instability (VSI) is a robust phenomenon in irradiated protoplanetary discs (PPDs). While there is extensive literature on the VSI in the hydrodynamic limit, PPDs are expected to be magnetized and their extremely low ionization fractions imply that non-ideal magnetohydrodynamic effects should be properly considered. To this end, we present linear analyses of the VSI in magnetized discs with Ohmic resistivity. We primarily consider toroidal magnetic fields, which are likely to dominate the field geometry in PPDs. We perform vertically global and radially local analyses to capture characteristic VSI modes with extended vertical structures. To focus on the effect of magnetism, a locally isothermal equation of state is employed. We find that magnetism provides a stabilizing effect to dampen the VSI, with surface modes, rather than body modes, being the first to vanish with increasing magnetization. Subdued VSI modes can be revived by Ohmic resistivity, where sufficient magnetic diffusion overcomes magnetic stabilization, and hydrodynamic results are recovered. We also briefly consider poloidal magnetic fields to account for the magnetorotational instability (MRI), which may develop towards surface layers in the outer regions of PPDs. The MRI grows efficiently at small radial wavenumbers, in contrast to the VSI. When resistivity is considered, the VSI dominates over the MRI for Ohmic Elsässer numbers ≲0.09 at plasma beta parameter βZ ∼ 104.

Temporal instability of surfactant-laden compound jets with surface viscoelasticity

When compound jets are covered with surfactant on both interfaces, the property of those interfaces will be viscoelastic, unlike the Newtonian fluid interface. In this paper, the effects of surface viscoelasticity are principally studied through linear stability analysis and calculated using the Chebyshev spectral collocation method. Both surface elasticity and viscosity have properties to make the jet more stable, but the physical effects behind the two influencing mechanisms are different. When the inner and outer interfaces are separately covered with surfactant, the variations of surface viscoelastic parameters affect the growth rate and dominant wavenumber differently. In addition, the sensitivity of the maximum growth rate to the surface viscoelastic parameters depends on which interface the surfactant covers. A significant increase in the surface shear viscosity reduces the influence of surface elasticity on the maximum growth rate. The long-wave assumption was adopted to derive a one-dimensional dispersion equation in an algebraic form. The comparison between the simplified expression and the numerical results validated the applicability of the one-dimensional model in the vicinity of small wavenumber.

Diurnal continental shelf waves with a permeable coastal boundary: Application to the shelf northwest of Norway

Spatially damped continental shelf waves (CSWs) with diurnal tidal frequency outside Lofoten–Vesterålen in north-west Norway are studied theoretically for an idealized shelf topography. Wave damping is caused by the exchange of fluid on the shelf with an inner archipelago through a permeable coastline. This exchange is modelled by the application of a Robin condition at the coastal boundary. It is shown that CSWs with diurnal frequencies are possible in a small wave number range centred around zero group velocity. By calculating the nonlinear radiation stress components in the spatially damped CSWs, we find the time- and depth averaged Lagrangian mean drift current to second order along the coast. We show that the Lagrangian mean drift current is independent of the value of the damping coefficient, however small, as long as it is nonzero. This illustrates the singular behaviour of the Lagrangian wave drift problem for CSWs.

Subluminal electrostatic noise in isotropic space plasmas. General formulas and nonrelativistic thermal limit

The properties of the collective subluminal electrostatic fluctuations in isotropic plasmas are investigated using the covariant kinetic theory of linear fluctuations based on the correct momentum–velocity relation. The covariant theory correctly accounts for the differences in subluminal and superluminal fluctuations in contrast to the non-covariant theory. The general formalism developed here is valid in unmagnetized plasmas and in magnetized plasmas for wavevectors of electrostatic waves parallel to the direction of the uniform magnetic field. Of particular interest are potential differences between the covariant and the non-covariant approach and the consequences of these differences in modifying observational predictions. For thermal particle distributions of protons and electrons with nonrelativistic equal temperatures, the covariant and non-covariant theories yield exactly the same dispersion function and relation for weakly damped electrostatic waves. Also, the quasi-equilibrium wavenumber spectrum of collective thermal electrostatic noise agrees in both theories apart from the important wavenumber restriction |k|&amp;gt;kc=ωp,e/c. While the non-covariant analysis also yields eigenmode fluctuations at small wavenumbers with superluminal phase speeds, the correct covariant analysis indicates that subluminal electrostatic fluctuations are only generated at wavenumbers |k|&amp;gt;kc by spontaneous emission of the plasma particles. As a consequence, the nonrelativistic thermal electrostatic noise wavenumber spectrum is limited to the wavenumber range ωp,e≤|k|≤kmax. Within a linear fluctuation theory, superluminal electrostatic noise cannot be generated.

On the temporal linear stability of the asymptotic suction boundary layer

A temporal linear stability analysis of the asymptotic suction boundary layer is presented. For this, the Orr–Sommerfeld equation is solved in terms of generalized hypergeometric functions. Together with the corresponding boundary conditions, an algebraic eigenvalue problem is formulated. Thereof we derive the temporal continuous spectrum yielding a rather distinct spectrum if, for example, compared to the one from the Blasius solution. A second key result is that the discrete spectrum in the limits α→0,that is, small streamwise wave numbers, and Re→∞ is only present in the distinguished limit Re α=O(1). This results in a degenerated Orr–Sommerfeld equation and the expanded algebraic eigenvalue problem poses a lower limit of (Re α)min≈0.841 91. We show that this lower bound corresponds to a maximum extension of the viscous eigenfunction in the wall-normal direction. The full algebraic eigenvalue problem is numerically solved for the temporal case up to Re=6.0×106. Besides the further refined critical values αcr=0.155 46, ωcr=0.023 297, Recr=54 378.620 32, discrete spectra and eigenfunctions are examined and ω=ωr+iωi is the complex frequency. In particular, eigenvalue spectra are investigated with regard to their behavior due to a variation of the Reynolds number and the wave number, respectively, and only A-modes according to the definition of Mack [“A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer,” J. Fluid Mech. 73, 497–520 (1976)] were identified. From these, three different classes of eigenfunctions of the wall-normal disturbance velocity are presented. Finally, we find that the inviscid part of the eigenfunctions is dominant in wall-normal direction and only propagates in streamwise direction, while the viscous part is limited to the vicinity of the wall and propagates toward it in an almost perpendicular direction.