Stochastically driven instability in rotating shear flows

Origin of hydrodynamic turbulence in rotating shear flows is investigated. The particular emphasis is on flows whose angular velocities decrease but specific angular momenta increase with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Such a mismatch between the linear theory and observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and the corresponding turbulence therein is ruled out. The present work explores the effect of stochastic noise on such hydrodynamic flows. We focus on a small section of such a flow which is essentially a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disk. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities, and hence large energy dissipations, that presumably generate instability. A range of angular velocity profiles (for the steady flow), starting with the constant angular momentum to that of the constant circular velocity are explored. It is shown that the growth and roughness exponents calculated from the contour (envelope) of the perturbed flows are all identical, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand origin of instability and turbulence in the three-dimensional Rayleigh stable rotating shear flows by introducing additive stochastic noise to the underlying linearized governing equations. This has important implications in resolving the turbulence problem in astrophysical hydrodynamic flows such as accretion disks.

Stochastically driven instability in rotating shear flows

Origin of hydrodynamic turbulence in rotating shear flows is investigated. The particular emphasis is on flows whose angular velocities decrease but specific angular momenta increase with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Such a mismatch between the linear theory and observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and the corresponding turbulence therein is ruled out. The present work explores the effect of stochastic noise on such hydrodynamic flows. We focus on a small section of such a flow which is essentially a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disk. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities, and hence large energy dissipations, that presumably generate instability. A range of angular velocity profiles (for the steady flow), starting with the constant angular momentum to that of the constant circular velocity are explored. It is shown that the growth and roughness exponents calculated from the contour (envelope) of the perturbed flows are all identical, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand origin of instability and turbulence in the three-dimensional Rayleigh stable rotating shear flows by introducing additive stochastic noise to the underlying linearized governing equations. This has important implications in resolving the turbulence problem in astrophysical hydrodynamic flows such as accretion disks.

Magnetohydrodynamic stability of stochastically driven accretion flows

We investigate the evolution of magnetohydrodynamic (or hydromagnetic as coined by Chandrasekhar) perturbations in the presence of stochastic noise in rotating shear flows. The particular emphasis is the flows whose angular velocity decreases but specific angular momentum increases with increasing radial coordinate. Such flows, however, are Rayleigh stable but must be turbulent in order to explain astrophysical observed data and, hence, reveal a mismatch between the linear theory and observations and experiments. The mismatch seems to have been resolved, at least in certain regimes, in the presence of a weak magnetic field, revealing magnetorotational instability. The present work explores the effects of stochastic noise on such magnetohydrodynamic flows, in order to resolve the above mismatch generically for the hot flows. We essentially concentrate on a small section of such a flow which is nothing but a plane shear flow supplemented by the Coriolis effect, mimicking a small section of an astrophysical accretion disk around a compact object. It is found that such stochastically driven flows exhibit large temporal and spatial autocorrelations and cross-correlations of perturbation and, hence, large energy dissipations of perturbation, which generate instability. Interestingly, autocorrelations and cross-correlations appear independent of background angular velocity profiles, which are Rayleigh stable, indicating their universality. This work initiates our attempt to understand the evolution of three-dimensional hydromagnetic perturbations in rotating shear flows in the presence of stochastic noise. © 2013 American Physical Society.

Representing spray zone with cross flow as a well-mixed compartment in a high shear granulator

The spray zone is an important region to control nucleation of granules in a high shear granulator. In this study, a spray zone with cross flow is quantified as a well-mixed compartment in a high shear granulator. Granulation kinetics is quantitatively derived at both particle-scale and spray zone-scale. Two spatial decay rates, DGSDR (droplet-granule spatial decay rate) ζDG and DPSDR (droplet-primary particle spatial decay rate) ζDP, which are functions of volume fraction and diameter of particulate species within the powder bed, are defined to simplify the deduction. It is concluded that in cross flow, explicit analytical results show that the droplet concentration is subject to exponential decay with depth which produces a numerically infinite depth of spray zone in a real penetration process. In a well-mixed spray zone, the depth of the spray zone is 4/(ζDG + ζDP) and π2/3(ζDG + ζDP) in cuboid and cylinder shape, respectively. The first-order droplet-based collision rates of, nucleation rate B0 and rewetting rate RW0 are uncorrelated with the flow pattern and shape of the spray zone. The second-order droplet-based collision rate, nucleated granule-granule collision rate RGG, is correlated with the mixing pattern. Finally, a real formulation case of a high shear granulation process is used to estimate the size of the spray zone. The results show that the spray zone is a thin layer at the powder bed surface. We present, for the first time, the spray zone as a well-mixed compartment. The granulation kinetics of a well-mixed spray zone could be integrated into a Population Balance Model (PBM), particularly to aid development of a distributed model for product quality prediction.

Transition in homogeneously heated inclined plane parallel shear flows

The transition of internally heated inclined plane parallel shear flows is examined numerically for the case of finite values of the Prandtl number Pr. We show that as the strength of the homogeneously distributed heat source is increased the basic flow loses stability to two-dimensional perturbations of the transverse roll type in a Hopf bifurcation for the vertical orientation of the fluid layer, whereas perturbations of the longitudinal roll type are most dangerous for a wide range of the value of the angle of inclination. In the case of the horizontal inclination transverse roll and longitudinal roll perturbations share the responsibility for the prime instability. Following the linear stability analysis for the general inclination of the fluid layer our attention is focused on a numerical study of the finite amplitude secondary travelling-wave solutions (TW) that develop from the perturbations of the transverse roll type for the vertical inclination of the fluid layer. The stability of the secondary TW against three-dimensional perturbations is also examined and our study shows that for Pr=0.71 the secondary instability sets in as a quasi-periodic mode, while for Pr=7 it is phase-locked to the secondary TW. The present study complements and extends the recent study by Nagata and Generalis (2002) in the case of vertical inclination for Pr=0.

Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices

A numerical continuation method has been carried out seeking solutions for two distinct flow configurations, planar Couette flow (PCF) and laterally heated flow in a vertical slot (LHF). We found that the spanwise vortex solution in LHF identifies a new solution in PCF. The vortical structure of our new solution has the shape of a hairpin observed ubiquitously in high-Reynolds-number turbulent flow, and we believe this discovery may provide the paradigm for a hierarchical organization of coherent structures in turbulent shear layers.

Symmetry of coherent vortices in plane couette flow

A numerical continuation method is carried out in a homotopy space connecting two different flows, the Plane Couette Flow (PCF) and the Laterally Heated Flow in a vertical slot (LHF). This numerical continuation method enables us to obtain an exact steady solution in PCF. The new solution has the shape of hairpin vortices (HVS: hairpin vortex solution), which is observed ubiquitously in turbulent shear flows.

Flow field analysis of some mixing and conveying screw element regions, within a closely intermeshing, co-rotating twin-screw extruder.

Grafting of antioxidants and other modifiers onto polymers by reactive extrusion, has been performed successfully by the Polymer Processing and Performance Group at Aston University. Traditionally the optimum conditions for the grafting process have been established within a Brabender internal mixer. Transfer of this batch process to a continuous processor, such as an extruder, has, typically, been empirical. To have more confidence in the success of direct transfer of the process requires knowledge of, and comparison between, residence times, mixing intensities, shear rates and flow regimes in the internal mixer and in the continuous processor.The continuous processor chosen for the current work in the closely intermeshing, co-rotating twin-screw extruder (CICo-TSE). CICo-TSEs contain screw elements that convey material with a self-wiping action and are widely used for polymer compounding and blending. Of the different mixing modules contained within the CICo-TSE, the trilobal elements, which impose intensive mixing, and the mixing discs, which impose extensive mixing, are of importance when establishing the intensity of mixing. In this thesis, the flow patterns within the various regions of the single-flighted conveying screw elements and within both the trilobal element and mixing disc zones of a Betol BTS40 CICo-TSE, have been modelled using the computational fluid dynamics package Polyflow. A major obstacle encountered when solving the flow problem within all of these sets of elements, arises from both the complex geometry and the time-dependent flow boundaries as the elements rotate about their fixed axes. Simulation of the time dependent boundaries was overcome by selecting a number of sequential 2D and 3D geometries, used to represent partial mixing cycles. The flow fields were simulated using the ideal rheological properties of polypropylene and characterised in terms of velocity vectors, shear stresses generated and a parameter known as the mixing efficiency. The majority of the large 3D simulations were performed on the Cray J90 supercomputer situated at the Rutherford-Appleton laboratories, with pre- and postprocessing operations achieved via a Silicon Graphics Indy workstation. A mechanical model was constructed consisting of various CICo-TSE elements rotating within a transparent outer barrel. A technique has been developed using coloured viscous clays whereby the flow patterns and mixing characteristics within the CICo-TSE may be visualised. In order to test and verify the simulated predictions, the patterns observed within the mechanical model were compared with the flow patterns predicted by the computational model. The flow patterns within the single-flighted conveying screw elements in particular, showed good agreement between the experimental and simulated results.

The flow behaviour of shallow fluidised beds

The flow behaviour of shallow gas-fluidised beds was studied. experimentally using a rotational viscometer, and an inclined open channel. Initially, tests were carried out with the viscometer in order to establish qualitative trends in the flow properties of a variety of materials over a wide range of fluidising conditions. Also, a technique was developed which enabled quantitative viscosity data to be extracted from the experimental results. The flow properties were found to be sensitive to the size, size-range and density of the fluidised material, the type of distributor used, and the moisture content of the fluidising gas. Tests in beds up to 120 mm deep showed that the fluidity of the bed improves with reduction in depth; and indicated a range of flow behaviour from shear-thinning to Newtonian, depending chiefly on fluidising velocity .. Later, an apparatus was built which provided for a steady, continuous flow of fluidised material down an inclined open channel of 3m length x 0.15m square, up to a mass flowrate of 10 kg/s (35 ton/hr). This facility has enabled data to be obtained that is of practical value in industrial applications; which is otherwise difficult in view of the present limited understanding of the true mechanism of fluidised flow. A correlation has been devised, based on analogy with laminar liquid flow, which describes the channel flow behaviour with reasonable accuracy over the whole range of shear-rates used. 1he channeI results indicated that at low fluidiising velocities the flow was adversely affected by settlement of a stagnant layer of particles on to the distributor, which gave rise to increased flow resistance. Conversely, at higher fluidising velocities the resistance at the distributor appeared to be less than at the walls. In view of this, and also because of the disparity in shear-rates between the two types of apparatus, it is not possible as yet to predict exactly the flow behaviour in an open channel from small-scale viscometer tests.

Transition in plane parallel shear flows heated internally

The stability of internally heated inclined plane parallel shear flows is examined numerically for the case of finite value of the Prandtl number, Pr. The transition in a vertical channel has already been studied for 0≤Pr≤100 with or without the application of an external pressure gradient, where the secondary flow takes the form of travelling waves (TWs) that are spanwise-independent (see works of Nagata and Generalis). In this work, in contrast to work already reported (J. Heat Trans. T. ASME 124 (2002) 635-642), we examine transition where the secondary flow takes the form of longitudinal rolls (LRs), which are independent of the steamwise direction, for Pr=7 and for a specific value of the angle of inclination of the fluid layer without the application of an external pressure gradient. We find possible bifurcation points of the secondary flow by performing a linear stability analysis that determines the neutral curve, where the basic flow, which can have two inflection points, loses stability. The linear stability of the secondary flow against three-dimensional perturbations is also examined numerically for the same value of the angle of inclination by employing Floquet theory. We identify possible bifurcation points for the tertiary flow and show that the bifurcation can be either monotone or oscillatory. © 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

The behaviour of granular material in pure shear, direct shear and simple shear

In biaxial compression tests, the stress calculations based on boundary information underestimate the principal stresses leading to a significant overestimation of the shear strength. In direct shear tests, the shear strain becomes highly concentrated in the mid-plane of the sample during the test. Although the stress distribution within the specimen is heterogeneous, the evolution of the stress ratio inside the shear band is similar to that inferred from the boundary force calculations. It is also demonstrated that the dilatancy in the shear band significantly exceeds that implied from the boundary displacements. In simple shear tests, the stresses acting on the wall boundaries do not reflect the internal state of stress but merely provide information about the average mobilised wall friction. It is demonstrated that the results are sensitive to the initial stress state defined by K0 = sh/sv. For all cases, non-coaxiality of the principal stress and strain-rate directions is examined and the corresponding flow rule is identified. Periodic cell simulations have been used to examine biaxial compression for a wide range of initial packing densities. Both constant volume and constant mean stress tests have been simulated. The characteristic behaviour at both the macroscopic and microscopic scales is determined by whether or not the system percolates (enduring connectivity is established in all directions). The transition from non-percolating to percolating systems is characterised by transitional behaviour of internal variables and corresponds to an elastic percolation threshold, which correlates well with the establishment of a mechanical coordination number of ca. 3.0. Strong correlations are found between macroscopic and internal variables at the critical state.

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow.

Cell exclusion in couette flow:evaluation through flow visualisation and mechanical forces

Cell exclusion is the phenomenon whereby the hematocrit and viscosity of blood decrease in areas of high stress. While this is well known in naturally occurring Poiseuille flow in the human body, it has never previously been shown in Couette flow, which occurs in implantable devices including blood pumps. The high-shear stresses that occur in the gap between the boundaries in Couette flow are known to cause hemolysis in erythrocytes. We propose to mitigate this damage by initiating cell exclusion through the use of a spiral-groove bearing (SGB) that will provide escape routes by which the cells may separate themselves from the plasma and the high stresses in the gap. The force between two bearings (one being the SGB) in Couette flow was measured. Stained erythrocytes, along with silver spheres of similar diameter to erythrocytes, were visualized across a transparent SGB at various gap heights. A reduction in the force across the bearing for human blood, compared with fluids of comparable viscosity, was found. This indicates a reduction in the viscosity of the fluid across the bearing due to a lowered hematocrit because of cell exclusion. The corresponding images clearly show both cells and spheres being excluded from the gap by entering the grooves. This is the first time the phenomenon of cell exclusion has been shown in Couette flow. It not only furthers our understanding of how blood responds to different flows but could also lead to improvements in the future design of medical devices.

Effect of oscillatory flow on nucleation kinetics of butyl paraben

More than 165 induction times of butyl paraben-ethanol solution in a batch moving fluid oscillation baffled crystallizer with various amplitudes (1-9 mm) and frequencies (1.0-9.0 Hz) have been determined to study the effect of COBR operating conditions on nucleation. The induction time decreases with increasing amplitude and frequency at power density below about 500 W/m3; however, a further increase of the frequency and amplitude leads to an increase of the induction time. The interfacial energies and pre-exponential factors in both homogeneous and heterogeneous nucleation are determined by classical nucleation theory at oscillatory frequency 2.0 Hz and amplitudes of 3 or 5 mm both with and without net flow. To capture the shear rate conditions in oscillatory flow crystallizers, a large eddy simulation approach in a computational fluid dynamics framework is applied. Under ideal conditions the shear rate distribution shows spatial and temporal periodicity and radial symmetry. The spatial distributions of the shear rate indicate an increase of average and maximum values of the shear rate with increasing amplitude and frequency. In continuous operation, net flow enhances the shear rate at most time points, promoting nucleation. The mechanism of the shear rate influence on nucleation is discussed.

A compartmental CFD-PBM model of high shear wet granulation

The conventional, geometrically lumped description of the physical processes inside a high shear granulator is not reliable for process design and scale-up. In this study, a compartmental Population Balance Model (PBM) with spatial dependence is developed and validated in two lab-scale high shear granulation processes using a 1.9L MiPro granulator and 4L DIOSNA granulator. The compartmental structure is built using a heuristic approach based on computational fluid dynamics (CFD) analysis, which includes the overall flow pattern, velocity and solids concentration. The constant volume Monte Carlo approach is implemented to solve the multi-compartment population balance equations. Different spatial dependent mechanisms are included in the compartmental PBM to describe granule growth. It is concluded that for both cases (low and high liquid content), the adjustment of parameters (e.g. layering, coalescence and breakage rate) can provide a quantitative prediction of the granulation process.