Parallel Shear Flow Research Articles

Quasilinear transport due to the magnetic drift resonance with the ion temperature gradient instability in a rotating plasma

The quasilinear transport fluxes due to the ion temperature gradient instability are calculated in a toroidal plasma, in which the magnetic drift resonance is treated rigorously. The effects of the equilibrium parallel flow and flow shear on the radial particle and heat fluxes are studied numerically in detail. In the radial component of parallel viscosity, there exist the pinches driven by the density gradient, the temperature gradient, and the curvature of the background magnetic field. The direction of these pinches is discussed. It is found that each pinch can be inward or outward, which depends crucially on the resonance condition.

Linear sound amplification and absorption in a corrugated pipe

Linear sound propagation in an axisymmetric corrugated pipe with shear flow is studied numerically and experimentally. The acoustic and hydrodynamic perturbations are described by the linearized Euler equations (LEEs) in a parallel shear flow. Wave propagation and scattering are computed by means of a multimodal method where the disturbances are expressed as a linear combination of acoustic modes and hydrodynamic modes. The Floquet-Bloch approach is used to calculate the wavenumber in the periodic system. Both sound amplification and absorption, depending on the Strouhal number, are well predicted compared to experiments, which means that the flow-acoustic coupling in the system is effectively described by the present model. It is also shown that the corrugated pipe can amplify the sound even if the shear layer over the cavities is stable everywhere.

A singular vortex Rossby wave packet within a rapidly rotating vortex

This paper describes the quasi-steady régime attained by a rapidly rotating vortex after a wave packet has interacted with it. We consider singular, nonlinear, helical, and shear asymmetric modes within a linearly stable, columnar, axisymmetric, and dry vortex in the f-plane. The normal modes enter resonance with the vortex at a certain radius rc, where the phase angular speed is equal to the rotation frequency. The related singularity in the modal equation at rc strongly modifies the flow in the 3D helical critical layer, the region where the wave/vortex interaction occurs. This interaction induces a secondary mean flow of higher amplitude than the wave packet and that diffuses at either side of the critical layer inside two spiral diffusion boundary layers. We derive the leading-order equations of the system of nonlinear coupled partial differential equations that govern the slowly evolving amplitudes of the wave packet and induced mean flow a long time after this interaction started. We show that the critical layer imposes its proper scalings and evolution equations; in particular, two slow times are involved, the faster being secular. This system leads to a more complex dynamics with respect to the previous studies on wave packets where this coupling was omitted and where, for instance, a nonlinear Schrödinger equation was derived [D. J. Benney and S. A. Maslowe, “The evolution in space and time of nonlinear waves in parallel shear flows,” Stud. Appl. Math. 54, 181 (1975)]. Matched asymptotic expansion method lets appear that the neutral modes are distorted. The main outcome is that a stronger wave/vortex interaction takes place when a wave packet is considered with respect to the case of a single mode. Numerical simulations of the leading-order inviscid Burgers-like equations of the derived system show that the wave packet rapidly breaks and that the vortex, after intensifying in the transition stage, is substantially weakened before the breaking onset. This breaking could give a dynamical explanation of the formation of an inner spiral band through the prism of the critical layer theory.

Zonal flow generation in parallel flow shear driven turbulence

Generation of zonal flow in parallel flow shear driven turbulence is discussed. Nonlinear dynamics is formulated by calculating energy transfer in the wave number space. It is shown that zonal flows can be generated (gain energy) from the primary mode which is driven by parallel flow shear. As a result, helical flow pattern can develop in turbulent plasmas. Our results imply that zonal flow can be generated in 3D parallel flow shear driven turbulence, which indicates that zonal flows are ubiquitous in turbulent plasmas, either 2D or 3D. Implications for turbulent momentum transport in laboratory and astrophysical plasmas are discussed.

Negative viscosity from negative compressibility and axial flow shear stiffness in a straight magnetic field

Negative compressibility ion temperature gradient (ITG) turbulence in a linear plasma device controlled shear de-correlation experiment can induce a negative viscosity increment. However, even with this negative increment, we show that the total axial viscosity remains positive definite, i.e., no intrinsic axial flow can be generated by pure ITG turbulence in a straight magnetic field. This differs from the case of electron drift wave turbulence, where the total viscosity can turn negative, at least transiently. When the flow gradient is steepened by any drive mechanism, so that the parallel shear flow instability (PSFI) exceeds the ITG drive, the flow profile saturates at a level close to the value above which PSFI becomes dominant. This saturated flow gradient exceeds the PSFI linear threshold, and grows with ∇Ti0 as |∇V∥|/|k∥cs|∼|∇Ti0|2/3/(k∥Ti0)2/3. This scaling trend characterizes the effective stiffness of the parallel flow gradient.

The characteristics of billows generated by internal solitary waves

The spatial and temporal development of shear-induced overturning billows associated with breaking internal solitary waves is studied by means of a combined laboratory and numerical investigation. The waves are generated in the laboratory by a lock exchange mechanism and they are simulated numerically via a contour-advective semi-Lagrangian method. The properties of individual billows (maximum height attained, time of collapse, growth rate, speed, wavelength, Thorpe scale) are determined in each case, and the billow interaction processes are studied and classified. For broad flat waves, similar characteristics are seen to those in parallel shear flow, but, for waves not at the conjugate flow limit, billow characteristics are affected by the spatially varying wave-induced shear flow. Wave steepness and wave amplitude are shown to have a crucial influence on determining the type of interaction that occurs between billows and whether billow overturning can be arrested. Examples are given in which billows (i) evolve independently of one another, (ii) pair with one another, (iii) engulf/entrain one another and (iv) fail to completely overturn. It is shown that the vertical extent a billow can attain (and the associated Thorpe scale of the billow) is dependent on wave amplitude but that its value saturates once a given amplitude is reached. It is interesting to note that this amplitude is less than the conjugate flow limit amplitude. The number of billows that form on a wave is shown to be dependent on wavelength; shorter waves support fewer but larger billows than their long-wave counterparts for a given stratification.

Three-dimensional study of parallel shear flow around an obstacle in water channel and air tunnel

The purpose of this study is to show the effect of some physical parameters on a parallel shear flow. The variation of the velocity inlet and the influence of an obstacle placed at the bottom of a channel were analyzed. The governing equations based on K –e model in a line source downstream around a three-dimensional obstacle are determined by the finite volume method with SIMPLEC algorithm. The obtained results have allowed to establish the dynamic characteristics of this kind of flow. Horizontal and vertical velocity fields, Reynolds shear stress field, turbulence intensity and scalar concentration in the water channel and a wind tunnel were exposed. These results permit us to localize the maximum and minimum zones of turbulence, which help us to exploit those fields for different purposes such as energy extraction and port infrastructure.

A dynamic-solver–consistent minimum action method: With an application to 2D Navier–Stokes equations

This paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method (MAM) for a spatially extended system when employing the large deviation principle (LDP) to study the effects of small random perturbations. A dynamic solver is used to approximate the unperturbed system, and a minimum action method is used to approximate the LDP, which corresponds to solving an Euler–Lagrange equation related to but more complicated than the unperturbed system. We will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the LDP, based on which we propose to define both the dynamic solver and the MAM on the same approximation space for spatial discretization. The semi-discrete LDP can then be regarded as the exact LDP of the semi-discrete unperturbed system, which is a finite-dimensional ODE system. We achieve this methodology for the two-dimensional Navier–Stokes equations using a divergence-free approximation space. The method developed can be used to study the nonlinear instability of wall-bounded parallel shear flows, and be generalized straightforwardly to three-dimensional cases. Numerical experiments are presented.

Effect of Horizontal AC Electric Field on the Stability of Natural Convection in a Vertical Dielectric Fluid Layer

The stability of buoyancy-driven parallel shear flow of a dielectric fluid confined between differentially heated vertical plates is investigated under the influence of a uniform horizontal AC electric field. The resulting generalized eigenvalue problem is solved numerically using Chebyshev collocation method with wave speed as the eigenvalue. The critical Grashof number Gc, the critical wave number αc and the critical wave speed cc are computed for wide ranges of AC electric Rayleigh number Rea and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. It is found that the AC electric Rayleigh number is to instill instability on convective flow against both stationary and travelling-wave mode disturbances. Nonetheless, the value of Prandtl number at which the transition from stationary to travelling-wave mode takes place is found to be independent of AC electric Rayleigh number. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.

On the Nonlinear Stability of Plane Parallel Shear Flow in a Coplanar Magnetic Field

Lyapunov direct method has been used to study the nonlinear stability of laminar flow between two parallel planes in the presence of a coplanar magnetic field for streamwise perturbations with stress-free boundary planes. Two Lyapunov functions are defined. By means of the first, it is proved that the transverse components of the perturbations decay unconditionally and asymptotically to zero for all Reynolds numbers and magnetic Reynolds numbers. By means of the second, it is showed that the other components of the perturbations decay conditionally and exponentially to zero for all Reynolds numbers and the magnetic Reynolds numbers below \(\frac{\pi ^2}{2M}\), where M is the maximum of the absolute value of the velocity field of the laminar flow.

Structure formation in parallel ion flow and density profiles by cross-ferroic turbulent transport in linear magnetized plasma

In this paper, we show the direct observation of the parallel flow structure and the parallel Reynolds stress in a linear magnetized plasma, in which a cross-ferroic turbulence system is formed [Inagaki et al., Sci. Rep. 6, 22189 (2016)]. It is shown that the parallel Reynolds stress induced by the density gradient driven drift wave is the source of the parallel flow structure. Moreover, the generated parallel flow shear by the parallel Reynolds stress is found to drive the parallel flow shear driven instability D'Angelo mode, which coexists with the original drift wave. The excited D'Angelo mode induces the inward particle flux, which seems to help in maintaining the peaked density profile.

Drift–Alfven turbulence of a parallel shearing flow of the finite beta plasma with warm ions

It was predicted [Mikhailenko et al., Phys. Plasmas 23, 020701 (2016)] that two distinct drift–Alfven instabilities may be developed in the parallel shearing flow of finite beta plasmas (1≫β≫me/mi) with comparable ion and electron temperatures. The first one is the shear-flow-modified drift–Alfven instability, which develops due to the inverse electron Landau damping and exists in the shearless plasma as well. The second one is the shear-flow-driven drift–Alfven instability, which develops due to the combined effect of the velocity shear and ion Landau damping and is absent in the shearless plasma flows. In the present paper, these drift–Alfven instabilities are examined numerically and analytically by including the electromagnetic response of the ions. The levels of the drift–Alfven turbulence, resulted from the development of both instabilities, are determined from the renormalized nonlinear dispersion equation, which accounts for the nonlinear effect of ion scattering by the electromagnetic turbulence. The renormalized quasilinear equation for the ion distribution function, which accounts for the same nonlinear effect of ion scattering, is derived and employed for the analysis of the ion viscosity and ions heating resulting from the interactions of ions with drift–Alfven turbulence.

Elastic instabilities in parallel shear flows of a viscoelastic shear-thinning liquid

Experimental results from two parallel shear flows of a highly shear-thinning viscoelastic polymer solution show that at low flow rates, the mean velocity profiles are steady and in agreement with analytical expectations. At higher flow rate, however, instability arises and the flow becomes weakly time dependent.

Transition in the asymptotic suction boundary layer over a heated plate

The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analysed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and highlights the intricate interactions between thermal and shear forces.

Stability of natural convection in a vertical layer of Brinkman porous medium

A classical linear stability theory is applied to emphasize the effect of inertia on the stability of buoyancy-driven parallel shear flow in a vertical layer of porous medium. The Lapwood–Brinkman model with fluid viscosity different from effective viscosity is used to describe the flow in a porous medium. The resulting eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Darcy–Rayleigh number \(R_\mathrm{Dc} \), the critical wave number \(a_\mathrm{c}\) and the critical wave speed \(c_\mathrm{c}\) are computed over a wide range of values of the Darcy–Prandtl number \(Pr_\mathrm{D}\) and the Darcy number \({\tilde{D}}a\). Depending on the choice of physical parameters, instability occurs due to the presence of inertia. The value of \(Pr_\mathrm{D}\) at which the transition from stationary to traveling-wave mode instability takes place increases with decreasing \({\tilde{D}}a\). Besides, the effect of decreasing \({\tilde{D}}a\) shows destabilizing effect if the instability is via stationary mode, and on the contrary, it exhibits a dual behavior if the instability is through traveling-wave mode. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state. In the energy spectrum, transition of instability from one type to another is found to take place as a function of \(Pr_\mathrm{D}\). The disturbance kinetic energy due to surface drag and viscous force plays no significant role in the stability of flow throughout the domain of \(Pr_\mathrm{D}\) considered.

Turbulence Dynamics with the Coupling of Density Gradient and Parallel Velocity Gradient in the Edge Plasmas

AbstractTheory of parallel shear flow driven instability (PSFI) and its impact on turbulence dynamics and transport are presented. The mode is linearly unstable when the parallel flow shear exceeds a critical value. The quasilinear particle flux contains both outward and inward components. Nonlinear dynamics is formulated in terms of hydrodynamic helicity balance. The result implies that once excited, PSFI with helicity may spread from the excited region to stable regions. Implication for turbulence in scrape off layer plasmas is discussed. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Inviscid instability of two-fluid free surface flow down an incline

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The ...

Minimal flow perturbations that trigger kinematic dynamo in shear flows

Parallel shear flows cannot be kinematic dynamos on their own (Zel’dovich, Sov. Phys. JETP, vol. 4, 1957, pp. 460–462), but the addition of small flow perturbations can trigger dynamo action. Using an optimization algorithm inspired by Willis (Phys. Rev. Lett., vol. 109 (25), 2012, 251101) and Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we identify the smallest perturbation that when added to Kolmogorov flow can trigger dynamo action at some fixed value of the magnetic Reynolds number. In this way we numerically measure the fragility of the Zel’dovich anti-dynamo theorem. The minimal perturbations have surprisingly simple spatial structures. Their magnitudes vary inversely proportional to the magnetic Reynolds number and are always much larger than theoretical lower bounds calculated here using the methods of Proctor (Geophys. Astrophys. Fluid Dyn., vol. 98 (3), 2004, pp. 235–240; J. Fluid Mech., vol. 697, 2012, pp. 504–510).

Influence of equilibrium shear flow in the parallel magnetic direction on edge localized mode crash

The influence of the parallel shear flow on the evolution of peeling-ballooning (P-B) modes is studied with the BOUT++ four-field code in this paper. The parallel shear flow has different effects in linear simulation and nonlinear simulation. In the linear simulations, the growth rate of edge localized mode (ELM) can be increased by Kelvin-Helmholtz term, which can be caused by the parallel shear flow. In the nonlinear simulations, the results accord with the linear simulations in the linear phase. However, the ELM size is reduced by the parallel shear flow in the beginning of the turbulence phase, which is recognized as the P-B filaments' structure. Then during the turbulence phase, the ELM size is decreased by the shear flow.

Laminar–Turbulent Transition Control Using Passivity Analysis of the Orr–Sommerfeld Equation

The control problem for linearized two-dimensional wall-bounded parallel shear flows is considered as a means of preventing the laminar-to-turbulent transition. The linearized Navier–Stokes equations are reduced to the Orr–Sommerfeld equation with wall-normal velocity actuation entering through the boundary conditions on the wall. An analysis of the work-energy balance is used to identify appropriate sensor outputs that could lead to a passive closed-loop system using simple feedback laws. These sensor outputs correspond to the second spatial derivative (normal direction) of the streamwise velocity at the wall, and it is demonstrated that they can be realized by pressure measurements made at appropriate locations along the wall. Spatial discretization of the Orr–Sommerfeld equation in the wall-normal direction is accomplished using Hermite cubic finite elements, and the resulting spectrum is shown to agree with literature values for both plane Poiseuille flow and the Blasius boundary layer. Both cases are shown to be stabilized by simple constant-gain output feedback using the special choice of sensor measurement. In each case, this holds for a single gain and a wide range of Reynolds numbers and wave numbers. However, the closed-loop system produced in the Poiseuille case is shown to be passive (i.e., it has a positive-real transfer function), whereas in the Blasius case, it is nonminimum-phase and hence is not passive.