Parallel Shear Flow Research Articles

Wavelet-based resolvent analysis of non-stationary flows

This work introduces a formulation of resolvent analysis that uses wavelet transforms rather than Fourier transforms in time. Under this formulation, resolvent analysis may extend to turbulent flows with non-stationary mean states. The optimal resolvent modes are augmented with a temporal dimension and are able to encode the time-transient trajectories that are most amplified by the linearised Navier–Stokes equations. We first show that the wavelet- and Fourier-based resolvent analyses give equivalent results for statistically stationary flow by applying them to turbulent channel flow. We then use wavelet-based resolvent analysis to study the transient growth mechanism in the near-wall region of a turbulent channel flow by windowing the resolvent operator in time and frequency. The computed principal resolvent response mode, i.e. the velocity field optimally amplified by the linearised dynamics of the flow, exhibits characteristics of the Orr mechanism, which supports the claim that this mechanism is key to linear transient energy growth. We also apply this method to non-stationary parallel shear flows such as an oscillating boundary layer, and three-dimensional channel flow in which a sudden spanwise pressure gradient perturbs a fully developed turbulent channel flow. In both cases, wavelet-based resolvent analysis yields modes that are sensitive to the changing mean profile of the flow. For the oscillating boundary layer, wavelet-based resolvent analysis produces oscillating principal forcing and response modes that peak at times and wall-normal locations associated with high turbulent activity. For the turbulent channel flow under a sudden spanwise pressure gradient, the resolvent modes gradually realign themselves with the mean flow as the latter deviates. Wavelet-based resolvent analysis thus captures the changes in the transient linear growth mechanisms caused by a time-varying turbulent mean profile.

Linear instability of viscous parallel shear flows: revisiting the perturbation no-slip condition

Linear stability analysis currently fails to predict turbulence transition in canonical viscous flows. We show that two alternative models of the boundary condition for incipient perturbations at solid walls produce linear instabilities that could be sufficient to explain turbulence transition. In many cases, the near-wall behaviour of the discovered instabilities is empirically indistinguishable from the classical no-slip condition. The ability of these alternative boundary conditions to predict linear instabilities that are consistent with turbulence transition suggests that the no-slip condition may be an overly simplified model of fluid–solid interface physics, particularly as a description of the flow perturbations that lead to turbulence transition in wall-bounded flows.

Brownian particle diffusion in generalized polynomial shear flows.

This study presents a mathematical framework for calculating the diffusion of Brownian particles in generalized shear flows. By solving the Langevin equationsusing stochastic instead of classical calculus, we propose a mathematical formulation that resolves the particle mean-square displacement (MSD) at all timescales for any two-dimensional parallel shear flow described by a polynomial velocity profile. We show that at long timescales, the polynomial order of time of the particle MSD is n+2, where n is the polynomial order of the transverse coordinate of the velocity profile. We generalize the method to resolve particle diffusion in any polynomial shear flow at all timescales, including the order of particle relaxation timescale, which is unresolved in current theories. Particle diffusion at all timescales is then studied for the cases of Couette and plane Poiseuille flows and a polynomial approximation of a hyperbolic tangent flow while neglecting the boundary effects. We observe three main stages of particle diffusion along the timeline for Couette and plane Poiseuille flows and four main stages for hyperbolic tangent flow. The particle MSD is distinctly different across these stages due to different dominant physical mechanisms. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles.

Single-cell mechanical assay unveils viscoelastic similarities in normal and neoplastic brain cells

Understanding cancer cell mechanics allows for the identification of novel disease mechanisms, diagnostic biomarkers, and targeted therapies. In this study, we utilized our previously established fluid shear stress assay to investigate and compare the viscoelastic properties of normal immortalized human astrocytes and invasive human glioblastoma (GBM) cells when subjected to physiological levels of shear stress that are present in the brain microenvironment. We used a parallel-flow microfluidic shear system and a camera-coupled optical microscope to expose single cells to fluid shear stress and monitor the resulting deformation in real time, respectively. From the video-rate imaging, we fed cell deformation information from digital image correlation into a three-parameter generalized Maxwell model to quantify the nuclear and cytoplasmic viscoelastic properties of single cells. We further quantified actin cytoskeleton density and alignment in immortalized human astrocytes and GBM cells via fluorescence microscopy and image analysis techniques. Results from our study show that contrary to the behavior of many extracranial cells, normal and cancerous brain cells do not exhibit significant differences in their viscoelastic properties. Moreover, we also found that the viscoelastic properties of the nucleus and cytoplasm as well as the actin cytoskeletal densities of both brain cell types are similar. Our work suggests that malignant GBM cells exhibit unique mechanical behaviors not seen in other cancer cell types. These results warrant future studies to elucidate the distinct biophysical characteristics of the brain and reveal novel mechanical attributes of GBM and other primary brain tumors.

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

Instability of mixing layers: Momentum and thermal transport in the continuum breakdown regime.

We examine the momentum and thermal transport in the continuum breakdown regime of a mixing layer flow, which exhibits Kelvin-Helmholtz instability under ideal continuum conditions. The Grad 13 moment model is used as it provides an adequate description of the flow physics (second-order accurate in Knudsen number) in the transition regime. Analytical solutions are developed under breakdown conditions for two-dimensional, compressible, parallel shear flows. It is shown that the deviation of viscous stress and heat flux from the Navier-Stokes-Fourier system follows two different scaling regimes depending upon the Mach number. At low Mach numbers, the departure of all stress and heat-flux components depends only upon the Knudsen number. At high Mach number, the scaling of shear stress and transverse heat flux depends on the product of the Knudsen and Mach numbers. The normal stresses depend individually on the Knudsen and Mach number. The scaling results are verified against numerical simulations of compressible mixing layers performed using the unified gas kinetic scheme for various degrees of rarefaction.

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Global Stability of a Conduction-Diffusion System with Superimposed Plane Parallel Shear Flows

Global Stability of a Conduction-Diffusion System with Superimposed Plane Parallel Shear Flows

Restratification in Late Winter Lakes Induced by Cabbeling

AbstractThe under‐ice fluid dynamics during late winter in many freshwater settings remain poorly understood. One example is how cabbeling, the generation of dense water by mixing water masses on different sides of the temperature of maximum density (Tmd), affects the vertical transport. Using high resolution numerical simulations of the development of a stratified parallel shear flow, we show that when the temperature stratification passes through Tmd, cabbeling modifies the three‐dimensional aspects of the instability with the net effect of forming an emergent, stable stratification after a brief period of strong mixing. This stratification effectively separates the quiescent upper layer from the turbulent one below, thereby limiting transport and mixing between them. We propose a simple model for the vertical fluid flux during the mixing regime, and discuss a potential mechanism responsible for the emergent stratification as well as its sensitivity to the bulk Richardson number.

Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

AbstractWe consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity , which is assumed to be small, and the wave number in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity , we obtain optimal decay estimates for large times, both in the enhanced dissipation regime and in the Taylor dispersion regime . Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart–Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.

Effects of the parallel flow shear on the ITG-driven turbulent transport in tokamak plasmas

The impact of the parallel flow shear on the tokamak plasma stability and turbulent transport driven by the ion temperature gradient (ITG) modes is analyzed by means of local gyrokinetic numerical analyses. It is shown that the parallel flow shear increases the ITG growth rate in the linear regime, and induces a broadening and shift of the radial spectrum. Then, the different effects of the finite parallel shear on the ITG turbulence characteristics are deeply analyzed in the nonlinear regime. These studies highlight that a reduction of the thermal-ion turbulent heat flux is induced by a complex mechanism involving the nonlinear generation of an enhanced zonal flow activity. Indeed, the turbulent sources of the zonal flows are increased by the introduction of the finite parallel flow shear in the system, beneficially acting on the saturation level of the ITG turbulence. The study has been carried out for the Waltz standard case below the critical threshold of the destabilization of the parallel velocity gradient instability, and then generalized to a selected pulse of a recent JET scenario with substantial toroidal rotation in the edge plasma region. It is, thus, suggested that the investigated complex mechanism triggered by the finite parallel flow shear reducing the ITG turbulent heat fluxes could be complementary to the well-established perpendicular flow shear in a region with sufficiently large plasma toroidal rotation.

Subharmonic eigenvalue orbits in the spectrum of pulsating Poiseuille flow

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr–Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

New direction and perspectives in elastic instability and turbulence in various viscoelastic flow geometries without inertia

We shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertialess flows with curved streamlines. Then we describe a theory of elastic turbulence and prediction of elastic waves Re ≪ 1 and Wi ≫ 1, which speed depends on the elastic stress similar to the Alfvén waves in magneto-hydrodynamics and in a contrast to all other, which speed depends on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvature, so parallel shear flows are proved linearly stable, similar to Newtonian parallel shear flows. However, the linear stability of parallel shear flows does not imply their global stability. Here we switch to the main subject, namely a recent development in inertialess parallel shear channel flow of polymer solutions. In such flow, we discover an elastically driven instability, elastic turbulence, elastic waves, and drag reduction down to relaminarization that contradict the linear stability prediction. In this regard, we discuss briefly normal versus non-normal bifurcations in such flows, flow resistance, velocity and pressure fluctuations, and coherent structures and spectral properties of a velocity field as a function of Wi at high elasticity number.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

A study of ions' shear flow instabilities and parallel electron current in pure pair‐ion and pair‐ion‐electron plasmas

AbstractThe pair‐ion plasma has attained special attention due to equal masses and equal but opposite charges. The electrostatic fluctuations with sheared effect are investigated in ideal low‐density and low‐temperature pure pair‐ion (PI) and pair‐ion electron (PIE) plasmas in different cases. These instabilities are also studied with homogeneous and inhomogeneous density effects. The growing D'Angelo mode can become the electrostatic instability with shear flow in uniform PIE plasma. It is observed that the shear flow also gives rise to an electrostatic instability in temperature anisotropic pure pair‐ion plasma. It is also seen that in pair‐ion plasma, the presence of small concentration of electrons results in significant changings in real frequency and growth rate of ion acoustic waves (IAWs). The effect of current driven field aligned flow of electrons on IAWs is also discussed in this article. Kinetic and fluid models are used to study the growth rate of IAWs when both the ionic species have similar shear flow and when both the ionic species have different shear flow.

A Marginal Stability Paradigm for Shear‐Induced Diapycnal Turbulent Mixing in the Ocean

AbstractTurbulent mixing induced by breaking internal waves is key to the ocean circulation and global tracer budgets. While the classic marginal shear instability of Richardson number ∼1/4 has been considered as potentially relevant to turbulent wave breaking, its relevance to flows that are not steady parallel shear flows has been suspect. We show that shear instability is indeed relevant in the ocean interior and propose a new marginal stability paradigm that relates the stability criterion based on Richardson number to one based on the ratio of Ozmidov and Thorpe turbulence scales. The new paradigm applies to both ocean interior and boundary layer flows. This allows for accurate quantification of the transition from downwelling to upwelling zones in a recently emerged paradigm of ocean circulation. Our results help climate models more accurately calculate the mixing‐driven deep ocean circulation and fluxes of tracers in the ocean interior.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

On the stability of parallel shear flows with embedded swirl

The temporal stability of inviscid parallel shear flows with embedded swirl is investigated. The base flow is divided into three regions: an inner irrotational jet flow, a thin swirl layer on the outer rim of the nozzle, and an external potential flow outside the nozzle. The swirl distribution in the thin shear layer is of solid-body rotation and of Rankine type outside the nozzle. The axial velocity profile in the jet base flow is assumed to be top-hat. The linear dispersion relation was derived and solved numerically. A parametric study on the effect of swirl parameter, S, and the swirl layer thickness fraction, d, on the amplification rates of the unstable modes is performed. A helical instability wave with azimuthal wave number, m = –3, is the most unstable mode with S = 0.8 and = 0.08. Our results indicate that the embedded thin layer swirl is capable of triggering the centrifugal instability waves to promote jet mixing. This powerful technique may be employed to mitigate jet noise without incurring significant thrust loss.

Inviscid instability of compressible exponential boundary layer flows

Based on the linearized Euler equations and the normal-mode approach, the Pridmore-Brown equation for acoustics and the stability of parallel shear flows is solved for a boundary layer flow with an exponential velocity profile. The general solution is derived in terms of the confluent Heun function, and in turn, using the boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity, the boundary value problem is converted to an algebraic eigenvalue problem. Solutions to the eigenvalue problem allow a comprehensive picture of the stability of compressible boundary layers. In particular, the complex eigenvalues ω are calculated as a function of the streamwise wavenumber α and the Mach number M. We observe that with growing wavenumbers, the number of eigenvalues increases discretely. Only for M > 1, unstable modes exist, and the boundary between neutrally stable and unstable modes is defined by the transonic line ω = α to be calculated from a degenerated eigenvalue equation. For large wall distances, the eigenfunctions converge exponentially toward zero. The spatial decay rate defines an “acoustic boundary layer thickness” δa, which indicates how far outside modes are still audible. For M > 1 and large α, δa diverges exponentially, i.e., in this parameter range, modes are perceptible even far from the boundary layer. A particularly steep rise of δa is observed when M > 2. Thus, there is a wavenumber range in which ωi and δa are both large and thus generates a particularly strong noise impact. From the eigenfunction for the unstable modes, a strong increase in the amplitude of acoustic waves can be identified in the vicinity of the wall, which indicates an accumulation and saturation of energy and thereafter leads to temporal instability and sound radiation in the free stream.