Parallel Shear Flow Research Articles

Bursty magnetic reconnection under slow shock‐generated whistler waves

AbstractNonlinear dynamics of magnetic reconnection with an external sub‐Alfvenic parallel shear flow is investigated by using two‐dimensional compressible Hall MHD simulation. Two pairs of slow shocks in the inflow region are generated by the sub‐Alfvenic shear flow in the MHD simulation. With inclusion of Hall effects, it is found that whistler waves are generated in the downstream region of slow shocks. The whistler waves propagating toward the reconnection region drive a large bursty enhancement in magnetic reconnection during its decaying phase.

Stability of natural convection in a vertical couple stress fluid layer

The stability of buoyancy-driven parallel shear flow of a couple stress fluid confined between vertical plates is investigated by performing a classical linear stability analysis. The plates are maintained at constant but different temperatures. A modified Orr–Sommerfeld equation is derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number Gc, critical wave number ac and critical wave speed cc are computed for wide ranges of couple stress parameter Λc and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. The value of Prandtl number, at which the transition from stationary to travelling-wave mode takes place, increases with increasing Λc. The couple stress parameter shows destabilising effect on the convective flow against stationary mode, while it exhibits a dual behaviour if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.

Ion-temperature-gradient sensitivity of the hydrodynamic instability caused by shear in the magnetic-field-aligned plasma flow

The cross-magnetic-field (i.e., perpendicular) profile of ion temperature and the perpendicular profile of the magnetic-field-aligned (parallel) plasma flow are sometimes inhomogeneous for space and laboratory plasma. Instability caused either by a gradient in the ion-temperature profile or by shear in the parallel flow has been discussed extensively in the literature. In this paper, (1) hydrodynamic plasma stability is investigated, (2) real and imaginary frequency are quantified over a range of the shear parameter, the normalized wavenumber, and the ratio of density-gradient and ion-temperature-gradient scale lengths, and (3) the role of inverse Landau damping is illustrated for the case of combined ion-temperature gradient and parallel-flow shear. We find that increasing the ion-temperature gradient reduces the instability threshold for the hydrodynamic parallel-flow shear instability, also known as the parallel Kelvin-Helmholtz instability or the D'Angelo instability. We also find that a kinetic instability arises from the coupled, reinforcing action of both free-energy sources. For the case of comparable electron and ion temperature, we illustrate analytically the transition of the D'Angelo instability to the kinetic instability as (a) the shear parameter, (b) the normalized wavenumber, and (c) the ratio of density-gradient and ion-temperature-gradient scale lengths are varied and we attribute the changes in stability to changes in the amount of inverse ion Landau damping. We show that near a normalized wavenumber k⊥ρi of order unity (i) the real and imaginary values of frequency become comparable and (ii) the imaginary frequency, i.e., the growth rate, peaks.

Comoving frames and symmetry-related motions in parallel shear flows

AbstractParallel shear flows have continuous symmetries of translation in the downstream and spanwise directions. As a consequence, flow states that differ in their spanwise or downstream location but are otherwise identical are dynamically equivalent. In the case of travelling waves, this trivial degree of freedom can be removed by going to a frame of reference that moves with the state, thereby turning the travelling wave in the laboratory frame into a fixed point in the comoving frame of reference. We here discuss a general approach, the method of comoving frames, by which the symmetry related motions can also be removed for more complicated and dynamically active states and demonstrate its application for several examples. For flow states in the asymptotic suction boundary layer (ASBL) we show that in the case of the long-period oscillatory edge state we can find local phase speeds which remove the fast oscillations and reveal the slow vortex dynamics underlying the burst phenomenon. For spanwise translating states we show that the method removes the drift but not the dynamical events that cause the big spanwise displacement. For a turbulent case we apply the method to the spanwise shifts and find slow components that are correlated over very long times. Calculations for plane Poiseuille flow show that the long correlations in the transverse motions are not specific to the ASBL.

The deviation from parallel shear flow as an indicator of linear eddy-viscosity model inaccuracy

A marker function designed to indicate in which regions of a generic flow field the results from linear eddy-viscosity turbulence models are plausibly inaccurate is introduced. The marker is defined to identify regions that deviate from parallel shear flow. For two different flow fields it is shown that these regions largely coincide with regions where the prediction of the Reynolds stress divergence is inaccurate. The marker therefore offers a guideline for interpreting results obtained from Reynolds-averaged Navier-Stokes simulations and provides a basis for the further development of turbulence model-form uncertainty quantification methods.

Kelvin–Helmholtz instability in the presence of variable viscosity for mudflow resuspension in estuaries

The temporal stability of a parallel shear flow of miscible fluid layers of different density and viscosity is investigated through a linear stability analysis and direct numerical simulations. The geometry and rheology of this Newtonian fluid mixing can be viewed as a simplified model of the behavior of mudflow at the bottom of estuaries for suspension studies. In this study, focus is on the stability and transition to turbulence of an initially laminar configuration. A parametric analysis is performed by varying the values of three control parameters, namely the viscosity ratio, the Richardson and Reynolds numbers, in the case of initially identical thickness of the velocity, density and viscosity profiles. The range of parameters has been chosen so as to mimic a wide variety of real configurations. This study shows that the Kelvin–Helmholtz instability is controlled by the local Reynolds and Richardson numbers of the inflection point. In addition, at moderate Reynolds number, viscosity stratification has a strong influence on the onset of instability, the latter being enhanced at high viscosity ratio, while at high Reynolds number, the influence is less pronounced. In all cases, we show that the thickness of the mixing layer (and thus resuspension) is increased by high viscosity stratification, in particular during the non-linear development of the instability and especially pairing processes. This study suggests that mud viscosity has to be taken into account for resuspension parameterizations because of its impact on the inflection point Reynolds number and the viscosity ratio, which are key parameters for shear instabilities.

Evaluation of vortex criteria by virtue of the quadruple decomposition of velocity gradient tensor

Vortices play a crucial role in fluid dynamics, which is closely related to fluid diffusion mixing, force, heat, and noise. Five widely-used vortex identification criteria, i.e. the -criterion, Q-criterion, -criterion, ci-criterion, and 2-criterion are analyzed, and four of them are compared with each other based on the velocity-gradient-tensor decomposition method. A new quadruple decomposition method (QDM) is introduced for the first time, so far as we know, to decompose fluid motions into four fundamental components: dilatation, axial deformation along the principal axes of the strain-range sensor, planar motion, and pure shearing. This method helps make the kinematic implications of the four vortex identification criteria more clear. It is found that the mean rotation of fluid elements always contains the pure shearing motion. Non-zero mean rotation does not guarantee the existence of the spiraling streamlines, e.g. in a typically parallel shear flow. A positive Q value indicates the strength of the pure rotation of a fluid element in the 2D complex eigenvalue plane on top of the axial deformation, which however is a sufficient but not a necessary condition for the existence of pure rotation. The -criterion can correctly tell the existence of pure rotation, but cannot accurately determine its strength. The ci-value represents the absolute strength of the pure rotation, which is the combined effect of the canonical rotation in the complex eigenvector plane and the pure shearing. The proposed QDM enables us to achieve a deeper understanding of vortices and motions in fluid dynamics.

Generation of surface waves by shear-flow instability

AbstractWe consider the linear stability of an inviscid parallel shear flow of air over water with gravity and capillarity. The velocity profile in the air is monotonically increasing upwards from the sea surface and is convex, while the velocity in the water is monotonically decreasing from the surface and is concave. An archetypical example, the ‘double-exponential’ profile, is solved analytically and studied in detail. We show that there are two types of unstable mode which can, in some cases, co-exist. The first type is the ‘Miles mode’ resulting from a resonant interaction between a surface gravity wave and a critical level in the air. The second unstable mode is an interaction between surface gravity waves and a critical level in the water, resulting in the growth of ripples. The gravity–capillary waves participating in this second resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. In both cases, the Reynolds stresses of an exponentially growing wave transfer momentum from the vicinity of the critical level to the zone between the crests and troughs of a surface wave.

Electromagnetic effects in the stabilization of turbulence by sheared flow

We have extended our study of the competition between the drive and stabilization of plasma microinstabilities by sheared flow to include electromagnetic effects at low plasma β (the ratio of plasma to magnetic pressure). The extended system of characteristic equations is formulated, for a dissipative fluid model developed from the gyrokinetic equation, using a twisting mode representation in sheared slab geometry and focusing on the ion temperature gradient mode. Perpendicular flow shear convects perturbations along the field at the speed we denote as Mcs (where cs is the sound speed). is required to make the system characteristics unidirectional and inhibit eigenmode formation, leaving only transitory perturbations in the system. This typically represents a much larger flow shear than in the electrostatic case, which only needs M > 1. Numerical investigation of the region shows the driving terms can conflict, as in the electrostatic case, giving low growth rates over a range of parameters. Also, at modest drive strengths and low β values typical of experiments, including electromagnetic effects does not significantly alter the growth rates. For stronger flow shear and higher β, geometry characteristic of the spherical tokamak mitigates the effect of an instability of the shear Alfvén wave, driven by the parallel flow shear.

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

On the nonlinear stability of parallel shear flow in the presence of a coplanar magnetic field

The nonlinear stability of laminar flow, in the x-direction, between two parallel planes in the presence of a coplanar magnetic field has been studied using Lyapunov direct method with either rigid or stress-free boundary planes. By defining a Lyapunov function it is proved that the laminar solutions of the system are nonlinearly unconditionally and asymptotically stable for all Reynolds numbers and magnetic Reynolds numbers if the perturbations are two-dimensional and depend only on y, z and t.

On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances

AbstractSmall-amplitude perturbations are governed by the linearized Navier–Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the Orr–Sommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius and asymptotic suction boundary layers, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. For the Blasius boundary layer, we highlight two particular properties of the CS: (i) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers $\pm {k}_{2} $, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (ii) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when ${R}^{- 1} \ll \omega \ll 1$, where $R$ is the Reynolds number based on the local boundary-layer thickness. For the asymptotic suction boundary layer, which is an exactly parallel flow, both temporal and spatial CS may be defined mathematically. However, at a finite $R$ neither of them represents the physical process of free-stream vortical disturbances penetrating into the boundary layer. The latter must instead be characterized by a peculiar type of continuous modes whose eigenfunctions increase exponentially with the distance from the wall. In the limit $R\gg 1$, all three types of CS are identical at leading order, and hence can be used to represent free-stream vortical disturbances and their entrainment. Low-frequency disturbances are found to generate a large-amplitude streamwise velocity in the boundary layer, which is reminiscent of longitudinal streaks.

On the nonlinear destabilization of stably stratified shear flow

AbstractA weakly nonlinear analysis of the bifurcation of the stratified Ekman boundary-layer flow near a critical bulk Richardson number is conducted and compared to a similar analysis of a continuously stratified parallel shear flow subject to Kelvin–Helmholtz instability. Previous work based on asymptotic expansions and predicting supercritical bifurcation at Prandtl number $Pr\lt 1$ and subcritical bifurcation at $Pr\gt 1$ for the parallel base flow is confirmed numerically and through fully nonlinear temporal simulations. When applied to the non-parallel Ekman flow, weakly nonlinear analysis and fully nonlinear calculations confirm that the nature of the bifurcation is dominantly controlled by $Pr$, although a sharp threshold at $Pr= 1$ is not found. In both flows the underlying physical mechanism is that the mean flow adjusts so as to induce a viscous (respectively diffusive) flux of momentum (respectively buoyancy) that balances the vertical flux induced by the developing instability, leading to a weakening of the mean shear and mean stratification. The competition between the former nonlinear feedback, which tends to be stabilizing, and the latter, which is destabilizing and strongly amplified as $Pr$ increases, determines the supercritical or subcritical character of the bifurcation. That essentially the same competition is at play in both the parallel shear flow and the Ekman flow suggests that the underlying mechanism is valid for complex, non-parallel stratified shear flows.

The effects of a shear flow on nonlinear evolutions of double tearing mode in Hall magnetohydrodynamics

Abstract The effects of a shear flow on nonlinear evolution of double tearing mode in Hall magnetohydrodynamics are investigated. The parallel shear flow displays a suppressing effect on the double tearing mode. Due to the effect of the shear flow, the relative displacements between the magnetic islands are changed, and therefore their mutual interactions become weak. Furthermore, the nonlinear impulsive growth phase of the double tearing mode is delayed with increasing the shear flow velocity. When the magnetic islands are pushing against each other, the width of current sheets decreases drastically, and therefore the Hall effects are dominant and the magnetic reconnection enhances greatly.

Irreversible compressible work and available potential energy dissipation in turbulent stratified fluids

Although it plays a key role in the theory of stratified turbulence, the concept of available potential energy (APE) dissipation has remained until now a rather mysterious quantity, owing to the lack of rigorous result about its irreversible character or energy conversion type. Here, we show by using rigorous energetics considerations rooted in the analysis of the Navier–Stokes for a fully compressible fluid with a nonlinear equation of state that the APE dissipation is an irreversible energy conversion that dissipates kinetic energy into internal energy, exactly as viscous dissipation. These results are established by showing that APE dissipation contributes to the irreversible production of entropy, and by showing that it is a part of the work of expansion/contraction. Our results provide a new interpretation of the entropy budget, that leads to a new exact definition of turbulent effective diffusivity, which generalizes the Osborn–Cox model, as well as a rigorous decomposition of the work of expansion/contraction into reversible and irreversible components. In the context of turbulent mixing associated with parallel shear flow instability, our results suggests that there is no irreversible transfer of horizontal momentum into vertical momentum, as seems to be required when compressible effects are neglected, with potential consequences for the parameterizations of momentum dissipation in the coarse-grained Navier–Stokes equations.

T cells utilize lamellipodia and filopodia for optimal intraluminal path finding (P5097)

Abstract Immune cells circulating in inflamed blood vessels undergo series of adhesion cascades to leave out of blood vessels to infiltrate into inflamed tissues. It has been shown that leukocytes crawl a considerable distance before they undergo transendothelial migration (TEM), but how they steer their crawling direction to find TEM sites is not well understood. To determine whether flow direction or EC orientation has predominant roles on the direction of T cell crawling, we fabricated well-aligned EC layers by culturing ECs on nanogrooved surfaces and applied shear flow either parallel or perpendicular to the direction of EC orientation. Regardless of flow direction, T cells crawled along the EC orientation. Then, we further identified guiding cues on ECs and found out that T cells tend to crawl along the valleys of topographical landscapes of EC layers while avoiding nuclei of ECs. By using pharmacological inhibitors for Arp 2/3, which nucleate actin filament branching to form lamellipodia, and cdc42, a small GTPase regulating filopodia formation, we found out that T cells utilize lamellipodia to sense topography of EC layers and filopodia to sense nuclei of ECs. Importantly, inhibitor-treated T cells crawled much longer distances than untreated T cells before they underwent TEM, suggesting that lamellipodia and filopodia of T cells are critical for optimal intraluminal path finding.

An explicit algebraic Reynolds-stress and scalar-flux model for stably stratified flows

AbstractThis work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth-order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a$K\text{{\ndash}} \omega $model (where$K$is turbulent kinetic energy and$\omega $an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability.

Small amplitude oscillatory shear flow superposed on parallel or perpendicular (orthogonal) steady shear of polydisperse linear polymers: The MLD model

The “naïve” polydisperse MLD model is used to analyze small amplitude oscillatory shear flow superposed on steady parallel shear flow [1]. Analytical results are derived for the dynamic moduli, G‖′(γ˙,ω) and G‖″(γ˙,ω), as a function of the molecular weight distribution, applied steady shear rate and the spectrum of longest orientational relaxation times of the system. The analysis reveals that there are two distinct factors contributing to deviations of the dynamic moduli from their equilibrium, no steady flow values, G′(0,ω) and G″(0,ω). The first is an orientational effect that reduces the moduli for all ω, but not uniformly, by reducing the projection of the chain segments on the applied incremental oscillatory deformation. The second effect is a relaxation spectrum cut-off phenomenon that truncates the low ω dynamic moduli selectively as the orientational relaxation time is systematically reduced by CCR. The dynamic moduli for superposed small oscillations perpendicular a steady flow, G⊥′(γ˙,ω) and G⊥″(γ˙,ω), are also calculated analytically for the naïve polydisperse MLD model again assuming there is no stretch. Under this assumption the results are identical to those for small strain oscillations parallel to a steady flow which is contrary to experiment. The relaxation spectrum cut-off phenomenon in steady shear flow is well established both experimentally and theoretically dating back to the seminal work of Bersted [2,3]. The analytic results are compared with direct numerical simulations of the diluted stretch tube polydisperse MLD model [4] in small amplitude oscillations superposed on parallel steady shear flow. Quantitative differences betwen the analytical results generated from the naïve MLD model and the diluted stretch tube MLD model are identified and interpreted using the analytic solution structure for G‖∗(γ˙,ω) and G⊥∗(γ˙,ω) which applies to both models. The combination of generally relevant analytical results and direct numerical simulation provide a direct means to interpret the dynamic moduli G‖∗(γ˙,ω) and G⊥∗(γ˙,ω) in terms of molecular parameters characterizing the system using the well-established MLD model.

Transient dynamics by continuous-spectrum perturbations in stratified shear flows

AbstractThe transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.

Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system

The linear stability of parallel shear flows for an inviscid generalized two-dimensional (2D) fluid system, the so-called α turbulence system, is studied. This system is characterized by the relation q = −( − Δ)α/2ψ between the advected scalar q and the stream function ψ. Here, α is a real number not exceeding 3 and q is referred to as the generalized vorticity. In this study, a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity. A stability analysis is then performed for a sheet vortex that violates the stability condition. The instability of a sheet vortex in the 2D Euler system (α = 2) is referred to as a Kelvin–Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < α < 3. The sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time. The growth rate is finite and depends on the wavenumber of the perturbation as k3 − α for 1 < α < 3, where k is the wavenumber of the perturbation. In contrast, for 0 < α ⩽ 1, the growth rate is infinite. In other words, a transition of the growth rate of the perturbation occurs at α = 1. A physical model for KH instability in the generalized 2D fluid system, which can explain the transition of the growth rate of the perturbation at α = 1, is proposed.