Near-critical swirling flow of a viscoelastic fluid in a circular pipe

2017 ◽  
Vol 814 ◽  
pp. 325-360
Author(s):  
Zvi Rusak ◽  
Nguyen Ly ◽  
John A. Tichy ◽  
Shixiao Wang
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Vortex Breakdown ◽  
Shear Thinning ◽  
Weissenberg Number ◽  
Swirling Flows ◽  
Circular Pipe ◽  
Viscoelastic Characteristic ◽  
Mobility Parameter ◽  
Flow Inertia

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

Nonlinear Control of Axisymmetric Swirling Flows in a Long Finite-Length Pipe

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Lei Xu ◽  
Zvi Rusak ◽  
Shixiao Wang ◽  
Steve Taylor
Keyword(s):  
Reynolds Number ◽  
Feedback Control ◽  
High Reynolds Number ◽  
Vortex Breakdown ◽  
Basin Of Attraction ◽  
Swirling Flows ◽  
Active Feedback ◽  
Active Feedback Control ◽  
High Reynolds ◽  
Control Methodology

Feedback stabilization of inviscid and high Reynolds number, axisymmetric, swirling flows in a long finite-length circular pipe using active variations of pipe geometry as a function of the evolving inlet radial velocity is studied. The complicated dynamics of the natural flow requires that any theoretical model that attempts to control vortex stability must include the essential nonlinear dynamics of the perturbation modes. In addition, the control methodology must establish a stable desired state with a wide basin of attraction. The present approach is built on a weakly nonlinear model problem for the analysis of perturbation dynamics on near-critical swirling flows in a slightly area-varying, long, circular pipe with unsteady changes of wall geometry. In the natural case with no control, flows with incoming swirl ratio above a critical level are unstable and rapidly evolve to either vortex breakdown states or accelerated flow states. Following an integration of the model equation, a perturbation kinetic-energy identity is derived, and an active feedback control methodology to suppress perturbations from a desired columnar state is proposed. The stabilization of both inviscid and high-Re flows is demonstrated for a wide range of swirl ratios above the critical swirl for vortex breakdown and for large-amplitude initial perturbations. The control gain for the fastest decay of perturbations is found to be a function of the swirl level. Large gain values are required at near-critical swirl ratios while lower gains provide a successful control at swirl levels away from critical. This feedback control technique cuts the feed-forward mechanism between the inlet radial velocity and the growth of perturbation's kinetic energy in the bulk and thereby enforces the decay of perturbations and eliminates the natural explosive evolution of the vortex breakdown process. The application of this proposed robust active feedback control method establishes a branch of columnar states with a wide basin of attraction for swirl ratios up to at least 50% above the critical swirl. This study provides guidelines for future flow control simulations and experiments. However, the present methodology is limited to the control of high-Reynolds number (nearly inviscid), axisymmetric, weakly nonparallel flows in long pipes.

Dynamics of a macroscopic elastic fibre in a polymeric cellular flow

2017 ◽  
Vol 817 ◽  
pp. 388-405 ◽  
Author(s):  
Qiang Yang ◽  
Lisa Fauci
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Relaxation Times ◽  
Elastic Fibre ◽  
Weissenberg Number ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
Multiple Cells

We study the dynamics and transport of an elastic fibre in a polymeric cellular flow. The macroscopic fibre is much larger than the infinitesimal immersed polymer coils distributed in the surrounding viscoelastic fluid. Here we consider low-Reynolds-number flow using the Navier–Stokes/Fene-P equations in a two-dimensional, doubly periodic domain. The macroscopic fibre supports both tensile and bending forces, and is fully coupled to the viscoelastic fluid using an immersed boundary framework. We examine the effects of fibre flexibility and polymeric relaxation times on fibre buckling and transport as well as the evolution of polymer stress. Non-dimensional control parameters include the Reynolds number, the Weissenberg number, and the elasto-viscous number of the macroscopic fibre. We find that large polymer stresses occur in the fluid near the ends of the fibre when it is compressed. In addition, we find that viscoelasticity hinders a fibre’s ability to traverse multiple cells in the domain.

Vortex breakdown of compressible subsonic swirling flows in a finite-length straight circular pipe

2015 ◽  
Vol 781 ◽  
pp. 3-27 ◽  
Author(s):  
Zvi Rusak ◽  
Jung J. Choi ◽  
Nicholas Bourquard ◽  
Shixiao Wang
Keyword(s):  
Differential Equation ◽  
Mach Number ◽  
Finite Length ◽  
Compressible Flows ◽  
Vortex Breakdown ◽  
Nonlinear Partial Differential Equation ◽  
Swirling Flows ◽  
Circular Pipe ◽  
Flow Profile ◽  
Inlet Flow

A global analysis of steady states of inviscid compressible subsonic swirling flows in a finite-length straight circular pipe is developed. A nonlinear partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. The equation reflects the complicated thermo–physical interactions in the flows. Several types of solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a flow force condition describe the outlet state of the flow in the pipe. These solutions are used to form the bifurcation diagram of steady compressible flows with swirl as the inlet swirl level is increased at a fixed inlet Mach number. The approach is applied to two profiles of inlet flows, solid-body rotation and the Lamb–Oseen vortex, both with a uniform axial velocity and temperature. The computed results provide for each inlet flow profile theoretical predictions of the critical swirl levels for the appearance of vortex breakdown states as a function of the inlet Mach number, suggesting that the results are robust for a variety of inlet swirling flows. The analysis sheds light on the dynamics of compressible flows with swirl and vortex breakdown, and shows the delay in the appearance of breakdown with increase of the inlet axial flow Mach number in the subsonic range of operation. The present theory is limited to axisymmetric dynamics of swirling flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

A three-dimensional computation of the force and torque on an ellipsoid settling slowly through a viscoelastic fluid

1995 ◽  
Vol 283 ◽  
pp. 1-16 ◽  
Author(s):  
J. Feng Feng ◽  
D. D. Joseph ◽  
R. Glowinski ◽  
T. W. Pan
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Tilt Angle ◽  
Three Dimensional ◽  
Major Axis ◽  
Weissenberg Number ◽  
Second Order ◽  
The Body ◽  
Fall Velocity ◽  
Elasticity Number

The orientation of an ellipsoid falling in a viscoelastic fluid is studied by methods of perturbation theory. For small fall velocity, the fluid's rheology is described by a second-order fluid model. The solution of the problem can be expressed by a dual expansion in two small parameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stress. Then the original problem is split into three canonical problems: the zeroth-order Stokes problem for a translating ellipsoid and two first-order problems, one for inertia and one for second-order rheology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite element method with fictitious domains, and the force and torque on the body are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; normal stresses turn the major axis parallel to the fall. The competition of these two effects gives rise to an equilibrium tilt angle between 0° and 90° which the settling ellipsoid would eventually assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the perturbation results do not explain the sudden turning of a long body which occurs when a critical fall velocity is exceeded. This is not surprising because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with ‘shape tilting’ observed at low fall velocities.

Wall-separation and vortex-breakdown zones in a solid-body rotation flow in a rotating finite-length straight circular pipe

2014 ◽  
Vol 759 ◽  
pp. 321-359 ◽  
Author(s):  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Solid Body ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Swirling Flows ◽  
Flow Dynamics ◽  
Circular Pipe ◽  
Body Rotation ◽  
Solid Body Rotation

AbstractThe incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ${\it\omega}_{1}$. We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ${\it\omega}$. We prove that the base columnar flow is a unique steady-state solution when ${\it\omega}$ is below ${\it\omega}_{1}$. This state is asymptotically stable and a global attractor of the flow dynamics. However, when ${\it\omega}>{\it\omega}_{1}$, we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ${\it\omega}>{\it\omega}_{1}$, the base columnar flow is a min–max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min–max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007–1016), followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.

scholarly journals Comparison of Micro-Mixing in Time Pulsed Newtonian Fluid and Viscoelastic Fluid

Micromachines ◽  
2019 ◽  
Vol 10 (4) ◽  
pp. 262 ◽  
Author(s):  
Zhang ◽  
Zhang ◽  
Wu ◽  
Shen ◽  
Chen ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Newtonian Fluid ◽  
Viscoelastic Fluid ◽  
Flow Rate ◽  
Essential Role ◽  
Weissenberg Number ◽  
The Other ◽  
Fluid Mixing ◽  
Constant Flow ◽  
Constant Flow Rate

Fluid mixing plays an essential role in many microfluidic applications. Here, we compare the mixing in time pulsing flows for both a Newtonian fluid and a viscoelastic fluid at different pulsing frequencies. In general, the mixing degree in the viscoelastic fluid is higher than that in the Newtonian fluid. Particularly, the mixing in Newtonian fluid with time pulsing is decreased when the Reynolds number Re is between 0.002 and 0.01, while it is enhanced when Re is between 0.1 and 0.2 compared with that at a constant flow rate. In the viscoelastic fluid, on the other hand, the time pulsing does not change the mixing degree when the Weissenberg number Wi ≤ 20, while a larger mixing degree is realized at a higher pulsing frequency when Wi = 50.

Numerical Simulation of Confined Swirling Flows of Oldroyd Fluids

2009 ◽  
Author(s):  
Sahand Majidi ◽  
Ashkan Javadzadegan
Keyword(s):  
Numerical Simulation ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Maxwell Model ◽  
Reynolds Numbers ◽  
Weissenberg Number ◽  
Swirling Flows ◽  
Breakdown Phenomenon ◽  
Volume Method ◽  
Upper Convected Maxwell Model

The effect of a fluid’s elasticity has been investigated on the vortex breakdown phenomenon in confined swirling flow. Assuming that the fluid obeys upper-convected Maxwell model as its constitutive equation, the finite volume method together with a collocated mesh was used to calculate the velocity profiles and streamline pattern inside a typical lid-driven swirling flow at different Reynolds and Weissenberg numbers. The flow was to be steady and axisymmetric. Based on the results obtained in this work, it can be concluded that fluid’s elasticity has a strong effect on the secondary flow completely reversing its direction of rotation depending on the Weissenberg number. Even in swirling flows with low ratio of elasticity to inertia, vortex breakdown is postponed to higher Reynolds numbers. Also, the effect of retardation ratio on the flow structure of viscoelastic fluid with the Weissenberg number being constant was surveyed. Based on our results, by decreasing the retardation ratio the flow becomes Newtonian like.

Near-Critical Swirling Flow of Viscoelastic Fluids in a Pipe

2002 ◽  
Author(s):  
Zvi Rusak ◽  
John A. Tichy
Keyword(s):  
Relaxation Time ◽  
Viscoelastic Fluid ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Equations Of Motion ◽  
Fluid Viscosity ◽  
Retardation Time ◽  
Flow Perturbation ◽  
Boger Fluids ◽  
Flow Inertia

The interaction between flow inertia and elasticity in high Reynolds number, axisymmetric, and near-critical swirling flows of an incompressible and viscoelastic fluid in a straight circular pipe is studied. The stresses of the viscoelastic fluid are described by the Oldroyd-B constitutive model (representing the low constant-viscosity Boger fluids). A nonlinear small-disturbance analysis is developed from the governing equations of motion in order to understand the complicated interactions between flow inertia and fluid viscosity and elasticity. The effects of the fluid viscosity, relaxation time, and retardation time on the flow development in the pipe and on the critical swirl for vortex breakdown are explored. It is found that increasing the relaxation time with other parameters being fixed increases the critical swirl for vortex breakdown whereas increasing the retardation time with other parameters being fixed decreases the critical swirl for breakdown. Also, when the relaxation and retardation times are the same the critical swirl is the same as that of a Newtonian fluid. The viscoelastic characteristic times also effect the size of the flow perturbation. These results may explain the changes in the appearance of breakdown zones as function of Reynolds numbers (swirl level) that have been recently observed in the experiments by Stokes et al. (2001) where Boger fluids were used. This work extends for the first time the theory of vortex stability and breakdown to include effects of non-Newtonian fluids.

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