Planar Shear Flow Research Articles

Laminar–turbulent coexistence in annular Couette flow

Annular Couette flow is the flow between two coaxial cylinders driven by the axial translation of the inner cylinder. It is investigated using direct numerical simulation in long domains, with an emphasis on the laminar–turbulent coexistence regime found for marginally low values of the Reynolds number. Three distinct flow regimes are demonstrated as the radius ratio $\unicode[STIX]{x1D702}$ is decreased from 0.8 to 0.5 and finally to 0.1. The high-$\unicode[STIX]{x1D702}$ regime features helically shaped turbulent patches coexisting with laminar flow, as in planar shear flows. The moderate-$\unicode[STIX]{x1D702}$ regime does not feature any marked laminar–turbulent coexistence. In an effort to discard confinement effects, proper patterning is, however, recovered by artificially extending the azimuthal span beyond $2\unicode[STIX]{x03C0}$. Eventually, the low-$\unicode[STIX]{x1D702}$ regime features localised turbulent structures different from the puffs commonly encountered in transitional pipe flow. In this new coexistence regime, turbulent fluctuations are surprisingly short-ranged. Implications are discussed in terms of phase transition and critical scaling.

Modification of three-dimensional instability in the planar shear flow around two circular cylinders in tandem

Direct stability analysis based on the Floquet theory has been employed to clarify the effects of planar shear on three-dimensional instabilities in the wake of two identical circular cylinders of diameter D in tandem arrangement. The center-to-center separation (Lx) in the range of 1.2 ≤ Lx/D ≤ 2.5 was considered. The onset of the three-dimensional instabilities was calculated, and the critical Reynolds number and corresponding spanwise wavenumber varying with the separation were discussed for different shear rates. Representative configurations were chosen to illustrate different transition scenarios, with the three-dimensional instabilities studied in detail for each case. It was found that three different effects of planar shear on the three-dimensional synchronous instability originally present in the otherwise uniform flow were identified depending on the separation. A subharmonic mode referred to as mode SS was observed to develop in the wake due to the flow asymmetry caused by the planar shear. This subharmonic mode differs from the C-type mode in terms of both the spatial structure and critical spanwise wavelength. Furthermore, the mode SS instability was found to be intensified as the shear becomes stronger, and it develops more rapidly than the synchronous modes.

Extension of classical stability theory to viscous planar wall-bounded shear flows

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternaleet al.,Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.

A linear system for pipe flow stability analysis allowing for boundary condition modifications

An accurate system to study the stability of pipe flow that ensures regularity is presented. The system produces a spectrum that is as accurate as Meseguer and Trefethen (2000), while providing flexibility to amend the boundary conditions without a need to modify the formulation. The accuracy is achieved by formulating the state variables to behave as analytic functions. We show that the resulting system retains the regular singularity at the pipe centre with a multiplicity of poles such that the wall boundary conditions are complemented with precisely the needed number of regularity conditions for obtaining unique solutions. In the case of axisymmetric and axially constant perturbations the computed eigenvalues match, to double precision accuracy, the values predicted by the analytical characteristic relations. The derived system is used to obtain the optimal inviscid disturbance pattern, which is found to hold similar structure as in plane shear flows.

Ring polymer dynamics and tumbling-stretch transitions in planar mixed flows.

The properties of dilute polymer solutions are governed by the conformational dynamics of individual polymers which can be perturbed in the presence of an applied flow. Much of our understanding of dilute solutions comes from studying how flows manipulate the molecular features of polymer chains out of equilibrium, primarily focusing on linear polymer chains. Recently there has been an emerging interest in the dynamics of nonlinear architectures, particularly ring polymers, which exhibit surprising out-of-equilibrium dynamics in dilute solutions. In particular, it has been observed that hydrodynamics can couple to topology in planar elongational and shear flows, driving molecular expansion in the nonflow direction that is not observed for linear chains. In this paper, we extend our understanding of dilute ring polymer dynamics to mixed flows, which represent flow profiles intermediate between simple shear or planar elongation. We map the conformational behaviors at a number of flow geometries and strengths, demonstrating transitions between coiled, tumbling, and stretched regimes. Indeed, these observations are consistent with how linear chains respond to mixed flows. For both linear and ring polymers, we observe a marked first-order-like transition between tumbling and stretched polymers that we attribute to a dynamic energy barrier between the two states. This manifests as bimodal extension distributions in a narrow range of flow strengths and geometries, with the primary difference between rings and linear chains being the presence of molecular expansion in the vorticity direction.

STUDY OF FLOW-INDUCED MOTION CHARACTERISTICS OF THREE TANDEM CIRCULAR CYLINDERS IN PLANAR SHEAR FLOW

Flow-induced motions of three elastically mounted circular cylinders subjected to the planar shear flow in tandem arrangement at $Re =100$ were numerically investigated by using semi-implicit characteristics-based split (CBS) finite element algorithm. Firstly, the results are compared with the existing data in the literature to verify the accurateness of the method. Then, the influence of some key parameters, such as shear ratio ($k$), natural frequency ratio ($r$) and reduced velocity ($U_{\rm r}$) on the characteristics of flow-induced dynamic responses and flow field of three circular cylinders with tandem arrangement was analyzed. The numerical results show that the shear ratio, natural frequency ratio and reduced velocity play an important role in the amplitude and trajectory of motion. The change of maximum amplitude of the upstream cylinder is similar to that of an isolated cylinder case with the increasing of $k$. The maximum amplitudes of the midstream and downstream cylinders increase, and the dual resonance is observed. Meanwhile, the increasing of $k$ causes the extension of the resonance region. With the increasing of $r$, the lock-in region of the upstream cylinder reduces in the in-line direction. However, for the midstream and downstream ones, the dual-lock-in region broadens. On the other hand, the prominent $X-Y$ trajectories of three circular cylinder resemble figure-eight shape and irregular one in the uniform flow. With the change of $k$, the motion orbit transforms from figure-eight to the raindrop shape in the lock-in region. In the larger shear ratio and higher natural frequency ratio case, the dual-raindrop is observed in the $X-Y$ vibrating trajectory of the midstream cylinder. Finally, the flow-induced motion mechanism of three tandem circular cylinders exposed to planar shear flow is revealed according to the analysis of the characteristic of the flow field.

On degenerate circular and shear flows: the point vortex and power law circular flows

We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. By using a Fourier multiplier approach, we construct energy functionals to deduce stability in a perturbative setting. For sufficiently high spherical harmonics, we can handle any circular flows with power law singularities or zeros as or , while for low frequencies we can treat circular flows close to the Taylor–Couette flow. Similar results apply in the planar shear flow case close to Couette.

Shear-induced diffusion in cohesive granular flows: effect of enduring clusters

We investigate the effect of intergranular cohesive forces on the properties of self-diffusion in dense granular flows. The study is based on a series of simulated plane shear flows at different inertial and cohesion numbers, in which transverse diffusivities are measured. Results evidence an increase in diffusivity by up to two orders of magnitude when introducing cohesion. This strong effect is analysed using the Green–Kubo framework, expressing the diffusivity in terms of instantaneous grain velocity fluctuations and their time correlation. This analysis shows that cohesion, by forming enduring clusters in the flow, enhances the velocity fluctuations and their time persistence, which both contribute to enhancing grain mixing and self-diffusion.

Shear-induced diffusion in non-local granular flows

We investigate the properties of self-diffusion in heterogeneous dense granular flows involving a gradient of stress and inertial number. The study is based on simulated plane shear flows with gravity and Poiseuille flows, in which non-local effects induce some creep flow in zones where stresses are below the yield criterion. Results show that shear-induced diffusion is qualitatively different in zones above and below the yield criterion. In sub-yield layers, diffusivity is no longer governed by instantaneous velocity fluctuations, and we evidence a direct scaling between diffusivity and local shear rate. This is interpreted by analysing the grain trajectories, which exhibit some caging in zones below the yield. Finally, we introduce an explicit scaling for the profile of local inertial number in these zones, which leads to a straightforward expression of the diffusivity as a function of the stress and position in non-local flows.

Effect of pressure on segregation in granular shear flows.

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

Thermodynamic integration for the determination of nonequilibrium entropy

Planar shear flow described by the SLLOD algorithm is a well-defined nonequilibrium steady state system with strong nonlinearity. Equipped with a numerical feedback thermostat it produces stable averages and fluctuations. We assume that moving infinitely slowly and isoenergetically along nonequilibrium states the excess rate at which heat is removed from the system can determine the work needed to move the system from equilibrium to steady states far from equilibrium. The excess heat rate is measured relative to the average heat rate of the corresponding steady state. As the best approximation, we use the configurational temperature to obtain entropy from the excess work in the typical shear rate range of simulations.

Transverse turbulent bands in rough plane Couette flow

In most subcritical planar shear flows, the transitional regime features oblique large-scale laminar-turbulent patterns. So far, such laminar-turbulent patterns have only been investigated in flows over perfectly smooth walls and little attention has been devoted to cases with rough surfaces as found in most practical engineering, urban applications, and in nature. In this study, we investigate laminar-turbulent patterns in plane Couette flow with one rough wall by means of direct numerical simulation, as a function of the Reynolds number and of the roughness height. The roughness is modeled using a force term in the Navier–Stokes equations. The focus of this study is on a new regime featuring non-oblique turbulent bands transverse to the motion of the walls, and separated by arbitrary long laminar gaps. This regime is found when the wall is sufficiently rough. This transverse turbulent band occurs at Reynolds numbers just below the onset of self-sustained turbulence found in the smooth wall case. The localized turbulence patches have a streamwise extent as large as 50–180 gap widths, decreasing with decreasing Reynolds number. The turbulent fraction as well as the band width show a linear relationship with the Reynolds number.

Vortices Enhance Diffusion in Dense Granular Flows.

This Letter introduces unexpected diffusion properties in dense granular flows and shows that they result from the development of partially jammed clusters of grains, or granular vortices. Transverse diffusion coefficients D and average vortex sizes ℓ are systematically measured in simulated plane shear flows at differing inertial numbers I revealing (i)a strong deviation from the expected scaling D∝d^{2}γ[over ˙] involving the grain size d and shear rate γ[over ˙] and (ii)an increase in average vortex size ℓ at low I, following ℓ∝dI^{-1/2} but limited by the system size. A general scaling D∝ℓdγ[over ˙] is introduced that captures all the measurements and highlights the key role of vortex size. This leads to establishing a scaling for the diffusivity in dense granular flow as D∝d^{2}sqrt[γ[over ˙]/t_{i}] involving the geometric average of shear time 1/γ[over ˙] and inertial time t_{i} as the relevant time scale. Analysis of grain trajectories is further evidence that this diffusion process arises from a vortex-driven random walk.

Understanding the destabilizing role for surface tension in planar shear flows in terms of wave interaction

The counter-intuitive destabilization of shear flow by surface tension is rationalized in terms of the interaction in a distance between interfacial counter-propagating vorticity waves on a Taylor-Caulfield setup. Interestingly the flow can be unstable even in the absence of stratification.

Vortex-induced vibrations of a cylinder in planar shear flow

The system composed of a circular cylinder, either fixed or elastically mounted, and immersed in a current linearly sheared in the cross-flow direction, is investigated via numerical simulations. The impact of the shear and associated symmetry breaking are explored over wide ranges of values of the shear parameter (non-dimensional inflow velocity gradient, $\unicode[STIX]{x1D6FD}\in [0,0.4]$) and reduced velocity (inverse of the non-dimensional natural frequency of the oscillator, $U^{\ast }\in [2,14]$), at Reynolds number $Re=100$; $\unicode[STIX]{x1D6FD}$, $U^{\ast }$ and $Re$ are based on the inflow velocity at the centre of the body and on its diameter. In the absence of large-amplitude vibrations and in the fixed body case, three successive regimes are identified. Two unsteady flow regimes develop for $\unicode[STIX]{x1D6FD}\in [0,0.2]$ (regime L) and $\unicode[STIX]{x1D6FD}\in [0.2,0.3]$ (regime H). They differ by the relative influence of the shear, which is found to be limited in regime L. In contrast, the shear leads to a major reconfiguration of the wake (e.g. asymmetric pattern, lower vortex shedding frequency, synchronized oscillation of the saddle point) and a substantial alteration of the fluid forcing in regime H. A steady flow regime (S), characterized by a triangular wake pattern, is uncovered for $\unicode[STIX]{x1D6FD}>0.3$. Free vibrations of large amplitudes arise in a region of the parameter space that encompasses the entire range of $\unicode[STIX]{x1D6FD}$ and a range of $U^{\ast }$ that widens as $\unicode[STIX]{x1D6FD}$ increases; therefore vibrations appear beyond the limit of steady flow in the fixed body case ($\unicode[STIX]{x1D6FD}=0.3$). Three distinct regimes of the flow–structure system are encountered in this region. In all regimes, body motion and flow unsteadiness are synchronized (lock-in condition). For $\unicode[STIX]{x1D6FD}\in [0,0.2]$, in regime VL, the system behaviour remains close to that observed in uniform current. The main impact of the shear concerns the amplification of the in-line response and the transition from figure-eight to ellipsoidal orbits. For $\unicode[STIX]{x1D6FD}\in [0.2,0.4]$, the system exhibits two well-defined regimes: VH1 and VH2 in the lower and higher ranges of $U^{\ast }$, respectively. Even if the wake patterns, close to the asymmetric pattern observed in regime H, are comparable in both regimes, the properties of the vibrations and fluid forces clearly depart. The responses differ by their spectral contents, i.e. sinusoidal versus multi-harmonic, and their amplitudes are much larger in regime VH1, where the in-line responses reach $2$ diameters ($0.03$ diameters in uniform flow) and the cross-flow responses $1.3$ diameters. Aperiodic, intermittent oscillations are found to occur in the transition region between regimes VH1 and VH2; it appears that wake–body synchronization persists in this case.

Laminar-turbulent patterns with rough walls

Oblique large-scale laminar-turbulent patterns are found near the onset of turbulence in subcritical planar shear flows. Their robustness to the introduction of wall roughness is investigated numerically in plane Couette flow as a function of the Reynolds number and the roughness height. The effect of roughness is considered on either one or two walls and is modeled numerically using a parametric model suggested recently [Busse and Sandham, J. Fluid Mech. 712, 169 (2012)]. In the case of two rough walls, the patterns are robust for a mean roughness height up to 10% of the wall gap, but the flow shows larger turbulent fractions with increasing roughness height. In the case of one rough wall only, the trend is similar, but the onset Reynolds number decreases faster with increasing roughness height. Roughness height levels above 15% of the wall gap give rise to new coherent structures, including turbulent bands with nonoblique interfaces. The energetic efficiency of the various regimes is investigated by monitoring the friction factor versus the friction Reynolds number. The mechanisms allowing for streamwise localization of the stripe patterns are discussed, with or without roughness, in the light of various low-order models.

Universal continuous transition to turbulence in a planar shear flow

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

Modified kinetic theory applied to the shear flows of granular materials

Granular materials are characterized by large collections of discrete particles, where the particle-particle interactions are significantly more important than the particle-fluid interactions. The current kinetic theory captures fairly accurately the granular flow behavior in the dilute case, when only binary interactions are significant, but is not accurate at all in the dense flow regime, where multi-particle interactions and contacts must be modeled. To improve the kinetic theory results for granular flows in the dense flow regime, we propose a Modified Kinetic Theory (MKT) model that utilizes the contact duration or cutoff time to account for the complex particle-particle interactions in the dense regime. The contact duration model, also called TC model, was originally proposed by Luding and McNamara [“How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model,” Granular Matter 1, 113 (1998)] to solve the inelastic collapse issue existing in the inelastic hard sphere model. This model defines a cutoff time tc such that dissipation is not counted if the time between two consecutive contacts is less than tc. As shown in their study, the use of a cutoff time tc can also reduce the dissipation during multi-particle contacts. In this paper we relate the TC model with the Discrete Element Method (DEM) by choosing the cutoff time tc to be the duration of contact calculated from the linear-spring-dashpot soft-sphere model of the DEM. We examine two types of granular flows, simple shear flow and the plane shear flow, and compare the results of the classical kinetic theory model, the present MKT model, and the DEM model. We show that the MKT model entails a significant improvement over the kinetic theory model for simple shear flows at inertial regimes. With the MKT model the calculations are close to the DEM results at solid fractions as high as 0.57. Even for the plane shear flows, where shear rate and solid fraction are inhomogeneous, the results of the MKT model agree very well with the DEM results.

Vortex-induced vibrations of a square cylinder under linear shear flow

This paper investigates the numerical vortex-induced vibration (VIV) of a square cylinder which is connected to a 2-DOF mass-spring system and is immersed in the planar shear flow by employing a characteristic-based split (CBS) finite element method (FEM). The reduced mass of the square cylinder is Mr = 2, while the reduced velocity, Ur, is changed from 3 to 12 with an increment of ΔUr = 1. The effects of some key parameters on the cylinder dynamic responses, vibrating frequencies, the flow patterns as well as the energy transferred between the fluid and cylinder are revealed. In this study, the key parameters are selected as follows: shear ratio (k = 0, 0.05 and 0.1) and Reynolds numbers (Re = 80 and 160). Numerical results demonstrate that the X–Y trajectories of the cylinder mainly appear as a symmetrical figure ‘8’ in uniform flow (k = 0) and an unsymmetrical figure ‘8’ and ‘O’ in shear flows (k = 0.05 and 0.1). The maximum oscillation amplitudes of the square cylinder in both the inline and transverse directions have distinct characteristics compared to that of a circular cylinder. Two kinds of flow patterns, ‘2S’ and ‘P + S’, are mainly observed under the shear flow. Also, the mean values of the energy of the cylinder system increase with the reduced velocity, while the root mean square (rms) of the energy reaches its peak value at reduced velocity Ur = 5.

The W-criterion for the onset of shear banding in complex fluids

A stability analysis of planar shear flow shear of a homogeneous, complex fluid predicts that shear banding instabilities can grow in fluids with a shear thinning strength above Wc = 1 and dampen out in fluids with W < 1. The shear thinning strength, \( W\left(\dot{\gamma}\right)=-\kern0.32em \partial \kern0.5em \log \eta /\partial \kern0.5em \log \dot{\gamma} \), arises naturally as the lead material function for the stability analysis of shear thinning fluids. The onset of shear banding is modeled as shear-thinning instability, which is attributed to anomalously strong shear thinning. Not considered here are inertial or elastic instabilities. In lack of suitable viscosity data from experiments, a Carreau powerlaw fluid and a Carreau-Yasuda powerlaw fluid serve as testbeds for the W-criterion. The analysis shows that the limiting high shear viscosity, η∞, plays an important role in shear banding and that the ratio of the limiting high shear viscosity and zero shear viscosity, η∞/η0, has to be sufficiently small for shear banding to occur. The main purpose of this brief communication is to share this new stability criterion. Extensive testing is still needed and is planned for future study.