Steady Asymmetric Flow Research Articles

Rheological effects on purely-elastic flow asymmetries in the cross-slot geometry.

Viscoelastic flows in the cross-slot geometry can undergo a transition from a steady symmetric to a steady asymmetric flow state, ostensibly due to purely-elastic effects arising beyond a critical flow rate, or Weissenberg number Wi. However, some reports suggest that shear thinning of the fluid's viscosity may also play an important role in this transition. We employ a series of polymer solutions of varying rheological properties to investigate in detail how the interplay between fluid elasticity and shear thinning affects the onset and development of asymmetric flows in the cross-slot. Flow velocimetry is performed on each of the polymer solutions, and is used to assess the degree of flow asymmetry I in the cross-slot as a function of both Wi and a dimensionless parameter S quantifying the flow-rate-dependent extent of shear thinning. Typically, the flow field breaks symmetry as Wi is increased beyond a critical value, but the magnitude of I is found to also be dependent on S. For a few specific polymer solutions, the flow field recovers symmetry above a second, higher critical Wi as S becomes small. The experimental results are summarized in a flow state diagram in Wi-S space, showing the relationship between flow asymmetry and fluid rheology. Finally, to gain a deeper understanding of the effects of shear thinning, numerical simulations are performed using the linear simplified Phan-Thien-Tanner model. We demonstrate that the degree of both shear thinning and elasticity of the fluid, and their interplay, are important factors controlling elastic instabilities in the cross-slot geometry.

Investigation of the extensional properties of elasto-visco-plastic materials in cross-slot geometries

• Flow of an EVP material is studied in the CSER or the OSCER geometries. • The Saramito/HB model is used to account for elastic and plastic effects. • At higher ε ˙ , flow is destabilized either to time-dependent or steady asymmetric flows. • Values for the HB exponent > 0.5 must be avoided with EVP materials. The generation of a wide region of shear-free, strong extensional flow near a stagnation point has been proven to be a great challenge, especially when considering yield-stress fluids. Capillary break up and filament stretching rheometers cannot approximate such flow field when elasto-visco-plastic (EVP) materials are used, because solidification of the material near the solid disks induces strong shear contributions to the flow. To overcome these limitations, we propose the standard (CSER) and optimized (OSCER) cross-slot devices to investigate the extensional properties of yield-stress fluids. To examine and understand the nonlinear dynamics of elasto-visco-plastic materials in strong extension flows, we model the rheology of a typical Carbopol solution with the popular Saramito/Herschel-Bulkley (SRM/HB) model. Using our stabilized finite element method, PEGAFEM-V, we solve the momentum and mass conservation coupled with the constitutive equation in both cross-slot flow geometries for a wide range of flow parameters. At low extension rates, the flow is steady and symmetric, allowing for plastic effects to be evaluated. At higher extension rates, the interplay of material elasticity and plasticity destabilizes the flow leading either to time-dependent or steady asymmetric flows. Our analysis points out that adopting values for the HB exponent > 0.5 must be avoided when modeling typical EVP materials with the SRM/HB model. Finally, by performing a wide parametric analysis, we conclude that these instruments can produce ideal conditions for the characterization of various EVP materials in a wide spectrum of extension rates.

Symmetry-breaking bifurcations in two-dimensional square vortex flows

We present a theoretical study of spatial symmetries and bifurcations in a laterally bounded two-dimensional flow composed of approximately square vortices. The numerical setting simulates a laboratory experiment wherein a shallow electrolyte layer is driven by a plane-parallel force that is nearly sinusoidal in both extended directions. Choosing an integer or half-integer number of forcing wavelengths along each direction, we generate square vortex flows invariant under different spatial symmetries. We then map out the sequence of symmetry-breaking bifurcations leading to the formation of fully asymmetric flows. Our analysis reveals a gallery of pitchfork and Hopf bifurcations, both supercritical and subcritical in nature, resulting in either steady or time-dependent asymmetric flows. Furthermore, we demonstrate that different types of flows (steady, periodic, pre-periodic, or quasi-periodic), at times with twofold multiplicity, emerge as a result of symmetry-breaking bifurcations. Our results also provide new theoretical insights into previous experimental observations in quasi-two-dimensional square vortex flows.

Numerical investigation on the flow around an inclined prolate spheroid

Numerical simulations are performed for the flow around an inclined 5:2 prolate spheroid in a uniform freestream. The Reynolds number (Re = 300, 500, 700, and 1000) and incidence angle (α = 0°–90°) are considered as significant parameters affecting the wake transitions, where α = 0° indicates flow parallel to the major axis of the prolate spheroid, and the Re is based on the inflow velocity U0 and the volume-equivalent sphere diameter De of the spheroid. In the range considered of Re and α, eight flow regimes are identified: (i) steady axisymmetric (SA) flow regime; (ii) steady planar symmetric flow regime; (iii) steady asymmetric (SAS) flow regime; (iv) periodic planar symmetric flow regime with non-zero mean lift or “Zig-zig-like” (Zz-like) mode; (v) periodic asymmetric flow regime with double-sided vortex shedding; (vi) multi-periodic asymmetric flow regime with double-sided vortex shedding and low frequency modulation (MPADL); (vii) multi-periodic asymmetric flow regime with single-sided vortex shedding and low frequency modulation (MPASL); and (viii) weakly chaotic state. Three of them are new and first reported, i.e., SAS, MPADL, and MPASL modes. The wake structure of the Zz-like mode is different from that of the zig-zig mode in the sphere/disk wake with a pair of streamwise vortices extending to the near wake. It is found that the elongated body can delay the onset of unsteadiness at small incidence angles. A flow regime map in the considered (Re, α) space is then provided. Finally, the physical mechanisms of the low-frequency phenomena observed at different wake modes are explored.

An Effective Interface Tracking Method for Simulating the Extrudate Swell Phenomenon.

The extrudate swell, i.e., the geometrical modifications that take place when the flowing material leaves the confined flow inside a channel and moves freely without the restrictions that are promoted by the walls, is a relevant phenomenon in several polymer processing techniques. For instance, in profile extrusion, the extrudate cross-section is subjected to a number of distortions that are motivated by the swell, which are very difficult to anticipate, especially for complex geometries. As happens in many industrial processes, numerical modelling might provide useful information to support design tasks, i.e., to allow for identifying the best strategy to compensate the changes promoted by the extrudate swell. This study reports the development of an improved interface tracking algorithm that employs the least-squares volume-to-point interpolation method for the grid movement. The formulation is enriched further with the consistent second-order time-accurate non-iterative Pressure-Implicit with Splitting of Operators (PISO) algorithm, which allows for efficiently simulating free-surface flows. The accuracy and robustness of the proposed solver is illustrated through the simulation of the steady planar and asymmetric extrudate swell flows of Newtonian fluids. The role of inertia on the extrudate swell is studied, and the results that are obtained with the newly improved solver show good agreement with reference data that are found in the scientific literature.

Steady and Unsteady Asymmetric Flow Regions Past an Axisymmetric Body

The theoretical solution of the flowfield past an axisymmetric body flying at a high angle of attack at subsonic flow conditions is a challenging problem since it entails large areas of boundary-layer separation and a complex vortex sheet structure. At high angles of attack, the flow is asymmetric and shows a dependence on the orientation angle, provided the body surface has sufficient roughness. Regarding theoretical simulations based on the unsteady Reynolds-averaged Navier–Stokes equations, eddy-viscosity turbulence models fail to simulate the unsteady flow structure in the rear zone of the body, yielding sectional side forces far different from those measured in experiments. Solutions obtained in this work by using Reynolds stress turbulence models combined with scale-adaptive simulation past an ogive-cylinder configuration show their ability to reproduce the essential features of the unsteady flow in the rear body. The model appears to be a suitable tool to investigate the complexities of this type of flow.

Numerical simulations of flow past three circular cylinders in equilateral-triangular arrangements

Flow past three identical circular cylinders is numerically investigated using the immersed boundary method. The cylinders are arranged in an equilateral-triangle configuration with one cylinder placed upstream and the other two side-by-side downstream. The focus is on the effect of the spacing ratio , Reynolds number and three-dimensionality on the flow structures, hydrodynamic forces and Strouhal numbers, where is the cylinder centre-to-centre spacing and is the cylinder diameter. The fluid dynamics involved is highly sensitive to both and , leading to nine distinct flow structures, namely single bluff-body flow, deflected flow, flip-flopping flow, steady symmetric flow, steady asymmetric flow, hybrid flow, anti-phase flow, in-phase flow and fully developed in-phase co-shedding flow. The time-mean drag and lift of each cylinder are more sensitive to than while fluctuating forces are less sensitive to than . The three-dimensionality of the flow affects the development of the wake patterns, changing the ranges of different flow structures. A diagram of flow regimes, together with the contours of hydrodynamic forces, in the space, is given, providing physical insights into the complex interactions of the three cylinders.

Three-dimensional numerical modelling of flow field in shallow reservoirs

Shallow flows can play a significant role in sediment management for dam reservoirs (e.g. sedimentation in shallow reservoirs and free-flow flushing operation). When shallow flow emerges with symmetric or asymmetric patterns, the flow domain exhibits complex three-dimensional (3D) features (e.g. helical flows). This study focuses on the numerical modelling of the velocity field in shallow reservoirs with varying geometries and also varying bed conditions (i.e. flat and misshaped beds). A fully 3D numerical model using the finite-volume method was utilised to reproduce the 3D flow velocity field. The experimentally measured surface velocity in all cases and velocity profiles in one case were used to validate the model. The numerical results showed that a slight disturbance in the inflow boundary condition results in a steady asymmetric flow pattern in reservoirs with a higher defined shape factor, but does not affect the flow pattern in reservoirs with a lower defined shape factor. Nonetheless, the simulated and measured flow velocity fields are reasonably consistent in all cases. These results can be used to optimise the design of sand traps or water storage facilities, and also to optimise sediment management in existing reservoirs.

Oscillatory elastic instabilities in an extensional viscoelastic flow.

Dilute polymer solutions are known to exhibit purely elastic instabilities even when the fluid inertia is negligible. Here we report the quantitative evidence of two consecutive oscillatory elastic instabilities in an elongation flow of a dilute polymer solution as realized in a T-junction geometry with a long recirculating cavity. The main result reported here is the observation and characterization of the first transition as a forward Hopf bifurcation resulted in a uniformly oscillating state due to breaking of time translational invariance. This unexpected finding is in contrast with previous experiments and numerical simulations performed in similar ranges of the Wi and Re numbers, where the forward fork-bifurcation into a steady asymmetric flow due to the broken spatial inversion symmetry was reported. We discuss the plausible discrepancy between our findings and previous studies that could be attributed to the long recirculating cavity, where the length of the recirculating cavity plays a crucial role in the breaking of time translational invariance instead of the spatial inversion. The second transition is manifested via time aperiodic transverse fluctuations of the interface between the dyed and undyed fluid streams at the channel junction and advected downstream by the mean flow. Both instabilities are characterized by fluid discharge-rate and simultaneous imaging of the interface between the dyed and undyed fluid streams in the outflow channel.

Instability Mechanisms in Straight-Diverging-Straight Channels

The linear instability mechanisms of two-dimensional flows through straight-diverging-straight channels with variable angle of divergence, α and expansion ratio κ = D/d, have been studied here. The extreme limits of this problem are compared to those of the plane Poiseuille flow (α = 0o) and the sudden expansion flows (α = 90o). The base flow obtained by direct numerical simulations shows two picthfork type of bifurcations. At very low Reynolds numbers (Re) the flow is steady and symmetric with two symmetric separated regions at the walls. With an increase in Re the symmetric flow evolves into an asymmetric flow with separated regions of unequal lengths. The linear stability analysis on symmetric base flows confirms the primary instability to be a symmetry-breaking type thus leading to a picthfork bifurcation. With a further increase in Re the steady asymmetric flow develops into a base state with ‘tertiary’ separated region. The appearance of the tertiary separated region is a bifurcation of the flow, but is not related to an instability. The Re beyond which the tertiary flow develops is characterized for the various parameters and compared with literature for the sudden expansion flows (α = 90o). A fairly good comparison is observed for the higher expansion ratios (κ). While the sudden expansion flows have been studied extensively, very little is known about the straight-diverging-straight channels. Thus, the instability mechanisms including the primary and secondary bifurcations have been studied in detail for the intermediate α to derive a link between the plane Poiseuille flow and the sudden expansion flows.

Viscoelastic instabilities in micro-scale flows

Many artificial and natural fluids contain macromolecules, particles or droplets that impart complex flow behavior to the fluid. This complex behavior results in a non-linear relationship between stress and deformation standing in between Newton’s law of viscosity for an ideal viscous liquid and Hooke’s law for an ideal elastic material. Such non-linear viscoelastic behavior breaks down flow reversibility under creeping flow conditions, as encountered at the micro-scale, and can lead to flow instabilities. These instabilities offer an alternative to the development of systems requiring unstable flows under conditions where chaotic advection is unfeasible. Flows of viscoelastic fluids are characterized by the Weissenberg (Wi) and Reynolds (Re) numbers, and at the micro-scale flow instabilities occur in regions in the Wi–Re space typically unreachable at the macro-scale, namely high Wi and low Re. In this paper, we review recent experimental work by the authors on the topic of elastic instabilities in flows having a strong extensional component, including: flow through a hyperbolic contraction followed by a sudden expansion; flow in a microfluidic diode and in a flow focusing device; flow around a confined cylinder; flow through porous media and simplified porous media analogs. These flows exhibit different types of flow transitions depending on geometry, Wi and Re, including: transition from a steady symmetric to a steady asymmetric flow, often followed by a second transition to unsteady flow at high Wi; direct transition between steady symmetric and unsteady flows.

Symmetry-breaking Bifurcations in T-channel Flows: Effects of Fluid Viscoelasticity

It is well known that, beyond a critical aspect-ratio-dependent Reynolds number, the flow in a three-dimensional T-channel junction, i.e. two-opposing planar channel streams joining and turning through 90 degrees, can break symmetry. For the case of two square cross-section inlets and an outlet arm of equal area (i.e. aspect ratio of two) this bifurcation is to a steady asymmetric flow. This flow bifurcation has been proposed as a method of enhancing mixing in microfluidic channels where significant increases in mixing quality are observed beyond the bifurcation. In the current work we investigate numerically the effects of viscoelasticity on this supercritical pitchfork bifurcation using a numerical finite-volume method. Results from both the upper-convected Maxwell and Oldroyd-B models show that the instability occurs at lower Reynolds numbers for viscoelastic fluids in comparison to the Newtonian base case. At higher Deborah numbers the transition leads directly to an unsteady flow.

Instabilities in stagnation point flows of polymer solutions

A recently developed microfluidic device, the optimized shape cross-slot extensional rheometer or OSCER [S. J. Haward, M. S. N. Oliveira, M. A. Alves, and G. H. McKinley, “Optimized cross-slot flow geometry for microfluidic extensional rheometry,” Phys. Rev. Lett. 109, 128301 (2012)10.1103/PhysRevLett.109.128301], is used to investigate the stability of viscoelastic polymer solutions in an idealized planar stagnation point flow. Aqueous polymer solutions, consisting of poly(ethylene oxide) and of hyaluronic acid with various molecular weights and concentrations, are formulated in order to provide fluids with a wide range of rheological properties. Semi-dilute solutions of high molecular weight polymers provide highly viscoelastic fluids with long relaxation times, which achieve a high Weissenberg number (Wi) at flow rates for which the Reynolds number (Re) remains low; hence the elasticity number El = Wi/Re is high. Lower concentration solutions of moderate molecular weight polymers provide only weakly viscoelastic fluids in which inertia remains important and El is relatively low. Flow birefringence observations are used to visualize the nature of flow instabilities in the fluids as the volumetric flow rate through the OSCER device is steadily incremented. At low Wi and Re, all of the fluids display a steady, symmetric, and uniform “birefringent strand” of highly oriented polymer molecules aligned along the outflowing symmetry axis of the test geometry, indicating the stability of the flow field under such conditions. In fluids of El > 1, we observe steady elastic flow asymmetries beyond a critical Weissenberg number,Wicrit, that are similar in character to those already reported in standard cross-slot geometries [e.g., P. E. Arratia, C. C. Thomas, J. Diorio, and J. P. Gollub, “Elastic instabilities of polymer solutions in cross-channel flow,” Phys. Rev. Lett. 96, 144502 (2006)10.1103/PhysRevLett.96.144502]. However, in fluids with El < 1 we observe a sequence of time-dependent inertio-elastic instabilities beyond a critical Reynolds number, ${\mathop{\rm Re}\nolimits} _{crit} $ Re crit, characterized by high frequency spatiotemporal oscillations of the birefringent strand. By plotting the critical limits of stability for the various fluids in the Wi-Re operating space, we are able to construct a stability diagram delineating the distinct steady symmetric, steady asymmetric and inertio-elastic flow regimes in this idealized planar elongational flow device.

Effects of an inserted square cylinder on wall heat transfer in laminar channel flows

Effects of unsteady flows past a confined square cylinder on wall heat transfer are investigated numerically. Two Reynolds numbers and several the gap distances of the square cylinder are considered to analyze the geometric effects on the wall heat transfer. The results show the detailed description of flow structure and attendant heat transfer caused by the variation of the gap distance. As the gap distance increases, the flow changes from steady asymmetric flow to symmetric vortex shedding having the inversion of the Karman vortex street. Also, the square cylinder induces intensive flow instabilities and significant alteration of the thermal boundary layer. As a result, the temperature distributions on the channel walls become steeper and wall heat transfer is increased.

PREDICTION AND CONTROL OF FLOW ASYMMETRY OVER SLENDER CONICAL BODIES.

The supersonic locally-conical flow over a 5° semi-apex circular cone at a Mach number of 1.8 and angles of attack (AOA) ranging from 10° to 35° is studied computationally using the three-dimensional Navier-Stokes equations. An implicit upwind flux-difference splitting scheme was used to obtain the numerical solution of the governing equations. Three flow patterns were predicted depending on the AOA. These flow patterns are; steady symmetric flow at AOA = 10°, steady asymmetric flow for 15° AOA 25° and unsteady asymmetric flow with vortex shedding for AOA ?. 30° . A number of passive control methods including vertical, fixed or erected, fins, two side strakes and four strakes in two different arrangements, oblique and cross, are used to restore the flow symmetry over the cone. A comparison between the studied control methods is carried out to select the best method of control. The four cross-strake method was found to be the best of the tested control methods.

Two-dimensional interactions of non-isothermal counter-flowing streams in an adiabatic channel with aiding and opposing buoyancy

A numerical investigation of two-dimensional interaction of two non-isothermal opposing jets/streams of different fluids (same phase, miscible) in an adiabatic channel is carried out in the mixed convection regime. The thermal buoyancy as well as intrinsic buoyancy (inherent differences in the densities of the two fluid streams) has been considered through the Boussinesq approximation. The two buoyancy forces give rise to thermal and intrinsic Richardson numbers, RiT and RiC, respectively as the important dimensionless numbers governing the problem. Two configurations, namely; (i) the upper stream of heavier fluid at lower temperature and lower stream of lighter fluid at higher temperature (aiding buoyancy forces), and (ii) the upper stream of lighter fluid at lower temperature and lower stream of heavier fluid at higher temperature (opposing buoyancy forces), are considered. For both the scenarios, the simulations have been carried out for various combinations of RiT and RiC. The numerical experiments reveal the existence of three flow modes depending upon the flow configuration and the magnitude of the buoyancy forces. These are (i) steady asymmetric flow (ii) steady symmetric flow and, (iii) the unsteady, periodic, non-symmetric flow with formation of standing waves and vortex-shedding in the channel. At sufficiently large buoyancy levels, the unsteady periodic flow mode with standing waves is observed for the aiding configuration only. The mixing process is quantitatively monitored through a scalar mixing index that represents the mean square deviation of fluid temperature/mixture concentration from the bulk values at a given channel section. It is shown that in the steady symmetric flow mode, the buoyancy has a very slight effect on the mixing characteristics. However, the mixing is significantly enhanced for the unsteady, periodic standing wave mode.

An experimental investigation of moderate reynolds number flow in a T-Channel

An experimental investigation of water flow in a T-shaped channel with rectangular cross section (20 × 20 mm inlet ID and 20 × 40 mm outlet ID) has been conducted for a Reynolds number Re range of 56–422, based on inlet diameter. Dynamical conditions and the T-channel geometry of the current study are applicable to the microscale. 2-D planar particle imaging velocimetry (PIV) and laser-induced fluorescence (LIF) were used in multiple locations of the T-channel to investigate local dynamical behaviors. Steady symmetric and asymmetric flow regimes predicted in the literature, which is largely numerical, are experimentally verified. Unsteady flow regimes, which are numerically predicted to occur at higher Re but have not yet been experimentally characterized, are also examined, and real-time LIF results illuminate the evolution of unsteady structure. Experimental data of the present resolution and scope are not presently available for unsteady flow regimes. Time scales are presented for unsteady flow regimes, which are found to exhibit periodic behavior and to occur for Re ≥ 195. An unsteady symmetrical regime is identified for Re ≥ 350 that is detrimental to mixing. Momentum fields and dynamical behaviors of all flow regimes are characterized in detail, such that published mixing trends may be better understood. Results of all experimental trials were used to construct a regime map. A symmetric topology is found to be dominant for Re from 56 to 116, when flow is steady, and 350 to 422, when flow is characterized by unsteady stagnation-point oscillation in the T-channel junction. Asymmetric flow, which is positively indicated for mixing, is dominant for Re between 142 and 298, and the fluid interface exhibits both steady (two standing vortices) and unsteady (shear-layer type roll-up) behaviors. This result is based on multiple experiments and suggests a practical operating range of 142 ≤ Re ≤ 298 where asymmetric flow is highly likely to experimentally occur. The identification of an upper limit on Re, beyond which mixing appears negatively impacted by a more symmetrical momentum field, is practically important as pressure drops on the microscale are significant.

Investigation of steady laminar flows with an initial disturbance

Results of a parametric study of steady asymmetric flows are analyzed. Three-dimensional unsteady equations of hydromechanics for a compressible medium are solved by a time-dependent method. The range of the characteristic Reynolds number Re = 60–350 is considered. It is shown that a symmetric flow becomes asymmetric at Re = 90. This value can be considered as a threshold value for air. In the examples considered, the upper separation region is always smaller than the lower separation region owing to flow asymmetry in the vicinity of the left boundary of the domain of integration. The dependence of the separation region size on the Reynolds number is found.

Stability of flows past a pair of circular cylinders in a side-by-side arrangement

The stability and transition of flow past a pair of circular cylinders in a side-by-side arrangement are investigated by numerical simulations and linear stability analyses. Various flow patterns around the cylinders have been reported to appear due to an instability of the steady symmetric flow that is realized at small Reynolds numbers, among which deflected oscillatory flow is particularly noticeable. The physical origin of the flow is explored by bifurcation analyses of the numerical data. We found that the deflected oscillatory flow arises from the steady symmetric flow through sequential instabilities due to stationary and oscillatory unstable modes. Steady asymmetric flow with respect to the streamwise centreline between the two cylinders was also found to be induced by the instability due to a stationary mode in a very narrow range of the gap width between the two cylinders. We classify the instability modes of the steady symmetric flow into four groups in the parameter space of the gap width, and evaluate the critical Reynolds number for each mode of instability.

Transitions of axisymmetric flow between two corotating disks in an enclosure

Transitions of flow between two corotating disks in an enclosure are investigated numerically. The outer cylindrical boundary of the flow field is fixed, whereas the inner cylinder rotates together with the two disks. The flow is not only symmetric with respect to the inter-disk midplane but also axisymmetric around the axis of rotation at small Reynolds numbers although it becomes unstable to disturbances at large Reynolds numbers. Two kinds of instability modes are known, one of which breaks the axisymmetry of the flow to yield a polygonal flow pattern in the radial–tangential plane. Such instability occurs for small length ratios, the ratio of the length of cylinders (gap between two disks) to the width of the annulus. The other instability is a symmetry-breaking instability with respect to the inter-disk midplane retaining the axisymmetry, which occurs for intermediate length ratios. We investigate the latter symmetry-breaking instability assuming the axisymmetry of the flow field, in which we obtain steady axisymmetric flows numerically and analyze their linear stability. It is found that oscillatory flow as well as steady asymmetric flow appears resulting from the first instability even in the intermediate length ratios where the flow remains axisymmetric. Numerical simulations are also performed to obtain time-periodic flows, and bifurcation diagrams of the flow are depicted for various values of length ratio. The critical Reynolds numbers for the Hopf and symmetry-breaking pitchfork instabilities are evaluated and a transition diagram is obtained.