Lip Vortex Research Articles

Entry flow and constitutive modelling of fluid S1

Experimental observations of the flow through an axisymmetric 4:1 contraction for the shear-thinning test fluid S1 are presented. In order to provide the basis for comparison with numerical simulations of the flow, the linear viscoelastic spectrum and the nonlinear parameters for three constitutive models are also obtained, based on the rheological characterisation of the fluid by Ooi and Sridhar [J. Non-Newtonian Fluid Mech. 52 (1994) 121–135]. The Phen-Thien-Tanner model provides the best fit to the data, but a good fit can also be obtained for the Giesekus model if the fit to the extensional stress growth coefficient is restricted to low strains. The results of flow visualisation experiments are presented for the flow of S1 through a 4:1 axisymmetric contraction with either an abrupt entry or a rounded re-entrant corner. The dimensionless vortex size increased with Weissenberg number for each geometry, with a larger vortex observed for the abrupt entry geometry for identical flow conditions. No elastic lip vortex or flow instability was observed for We ≤ 5.0.

急縮小流路における高分子流体の三次元可視化実験と流動解析

A three-dimensional flow visualization of a polyacrylamide/corn-syrup aqueous solution was carried out, using a 2.5:1 abrupt contraction with 3:1 aspect ratio. A numerical simulation of three-dimensional flow was also carried out with an Oldroyd-B model in the same geometry. The numerical results were compared with the visualization results.The experiment shows that a lip vortex takes place near the front wall at a low flow rate, but is not observed near the central part of the lip corner at the same flow condition. The lip vortex then develops with increasing flow rate into a larger corner vortex. Under the experimental conditions of the present paper, the lip vortex has an important influence on the development of vortex enhancement in the 3-D abrupt contraction. For some combinations of rheological parameters, the numerical results agreed qualitatively with the experimental results.

On vortex development in viscoelastic expansion and contraction flows

Vortex structure and formation are investigated for both expansion and contraction flows of viscoelastic fluids concentrating on a class of constitutive models due to Phan-Thien and Tanner. Expansion flows are treated in both two and three dimensions, and viscoelasticity is observed to suppress vortex activity. Two-dimensional contraction flows are analysed for highly elastic solutions. Different geometric aspect ratios, levels of inertia and elasticity are considered for these various flows and lip vortices are observed under certain circumstances.

Flows of polymer solutions through contractions Part 1: flows of polyacrylamide solutions through planar contractions

The present paper is devoted to the mechanism of vortex enhancement and the occurrence of lip vortices in contraction flows of viscoelastic fluids. The FENE-P model was used in order to simulate earlier experimental results of Evans and Walters for the flow of aqueous polyacrylamide solutions through planar contractions. The influence of the contraction ratio, the polyacrylamide concentration and the corner sharpness were investigated. In all cases, numerical simulation gave qualitative agreement with experiments.

Mechanism for entry flow instability through a forward-facing step channel

Flow observations through a forward-facing step channel were conducted to complete the mechanism for instability of planar entry flow of rather dilute polymer solutions, and the types of unstable flow patterns as well as the mechanism for the back-and-forth movement of the Goertler vortices have been studied in detail. Our observations have confirmed that four types of flow patterns exist in the unstable flow regime, where the Goertler vortices occur around the salient corner, and the manner of instability becomes more complex as the solution concentration increases. Furthermore, the appearance and coalescence of the Goertler vortex pairs cause their back-and-forth movement along the longitudinal direction of the channel, and three-dimensional lip vortices play an important role in the appearance of a pair of the Goertler vortices.

Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method

A combination of the finite element method and the finite volume method has been developed for time-dependent viscoelastic flow problems. The governing system of equations is decoupled provided that a reasonably small time-increment is used, which is obtained from a Courant-Friedrichs-Lewy (CFL) condition taking into account a local vorticity wave speed. The momentum equation is solved explicitly with respect to time by a finite element method. The constitutive equation for an upper-convected Maxwell model is solved implicitly by finite volume methods. Case studies have been conducted for start-up of planar Poiseuille flow and planar four-to-one contraction flow for upper-convected Maxwell and Oldroyd-B fluids. Numerical results agree with analytical solutions for shock propagation in start-up of Poiseuille flow with no upper limit on the Weissenberg number We. In planar 4:1 contraction flow, the size of the comer vortex compares well with experimental measurements and other numerical predictions. An attempt to seek a critical value of We was stopped at We = 2 because of transient behaviour of the viscoelatic flow. There are new findings: vorticity separation from a re-entrant corner and a temporary lip vortex in transient flow induced by an instantaneous increase of We.

A solver for time-dependent viscoelastic fluid flows

We present a solver for computing time-dependent viscoelastic flow in two space dimensions for UCM, Oldroyd-B and Giesekus fluids. The method is based on second-order accurate finite differences on composite overlapping grids, created with the grid generating program CMPGRD, which allows very complex geometries. Several numerical results are presented. We demonstrate transient viscoelastic instabilities in a channel flow with a cylinder. A study of a 2:1-contraction flow indicates why this geometry has been difficult to simulate. An example of a 4:1-contraction shows that the Giesekus model does give rise to vortices very similar to the so-called lip vortices observed in experiments. We also give an industrial application by demonstrating the elastic effects in a paper-coating flow.

A taylor-petrov-galerkin algorithm for viscoelastic flow

A viscoelastic flow is solved using a generalised Taylor-Galerkin/pressure correction scheme that incorporates consistent Petrov-Galerkin streamline upwinding within the discretisation of the constitutive equations. The numerical approach is indirect, in the sense that fractional equation solution stages are introduced within the framework of a time-stepping scheme. The Oldroyd-B and Phan-Thien-Tanner constitutive models are considered and the proposed Taylor-streamline upwind/Petrov-Galerkin algorithm is used to simulate flow through a 4:1 planar contraction. For the Oldroyd-B model, the algorithm indicates the onset of instability as elasticity is increased and a limiting Weissenberg number is observed, with no lip vortices apparent for values less than five. For a particular class of Phan-Thien-Tanner models, it is found that the algorithm is stable at high Weissenberg numbers. In this case the solutions exhibit a lip-vortex mechanism for the establishment of recirculating regions as has been observed in some experiments. Results presented for various combinations of fluid parameters suggest that the extensional behaviour of the viscoelastic model is the single most important factor governing the stability and convergence of the algorithms for highly elastic fluids.

Entry-flow calculations for the Oldroyd-B and FENE equations

Entry-flow calculations are presented for the Oldroyd-B and FENE equations in the limit of high contraction ratio in both planar and axisym- metric geometries. In addition to abrupt entry flow a cone-shaped entry flow of half-angle π/4 is also considered. A decoupled finite-difference scheme is used with time stepping to converge to the solution of the non-linear equations. For the solution of the stress equation a streamline integration technique is developed. For the FENE equation it is found that a reordering of the elliptic operator for the stream function enables larger time steps to be taken at high We when the polymers are near full extension. These calculations show that the FENE equation can predict much larger vortices than the Oldroyd-B equation when in the non-linear regime of the FENE spring. No vortices are observed for the Oldroyd-B equation in planar flow and only weak vortices when the flow is axisymmetric. For the cone entry flow, larger vortices are again predicted for the FENE equation but they remain localised near the entry, in a manner similar to lip vortices. Pressure drops are also considered and it is argued that at high We the non-dimensional decrease in pressure drop compared with the Newtonian pressure drop should be linear in We for the Oldroyd-B equation. An estimate of this pressure drop is made in the high We, but low concentration, limits.

Corner flow of a suspension of rigid rods

Stress singularities in the neighbourhood of sharp corners can be a source of severe problems in the numerical simulation of non-Newtonian flows leading to loss of convergence with grid refinement (G.G. Lipscombe, R. Kennings and M.M. Denn, J. Non-Newtonian Fluid Mech., 24 (1987) 85 [1]). For Newtonian flows the nature of this singularity is given by the analysis of Dean and Montagnon (W.R. Dean and P.E. Montagnon, Phil. Trans. R. Soc. London, Ser. A., 308 (1949) 199 [2]) in terms of similarity solutions. In this paper we extend this similarity analysis to a suspension of rigid rods. In the limit of nearly full extension the FENE constitutive model has the same behaviour as such a suspension. Our analysis predicts the possibility of lip vortices but their behaviour is somewhat inconsistent with those observed experimentally.

Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Anomalous flow patterns in viscoelastic entry flow through a planar contraction

This paper adds further experimental evidence to demonstrate that the flow characteristics in planar contraction flows of shear-thinning aqueous polyacrylamide solutions are far more complex than initially supposed, with anomalous flow phenomena appearing in the unstable flow regime. It is confirmed that a lip vortex occurs at the re-entrant corner and it appears at lower flow rates with an increase in contraction ratio.

Further Simulations of Viscoelastic Entry Flows Using a Heuristic Approach

The entry flow of viscoelastic fluids is simulated numerically for various tubular and planar abrupt geometries using the finite element method. Test fluids include polymer solutions (Boger fluids) and polymer melts (LDPE, PS) for which shear viscosity and normal stress data are available. Instead of using any established viscoelastic consti tutive equation, the Viscometric Approximation Theory (VAT) is used which simplifies the equations to their steady simple-shear flow counterpart thus overcoming problems of convergence. This heuristic approach, although not rigorous and consistent with consti tutive principles, is shown to capture intricate details of the flow field found experimen tally. These include the growth of the reservoir vortex with flowrate from a concave to a straight and then a convex shape in a 4:1 tubular contraction; the appearance of a second lip vortex in a 16:1 tubular contraction for low flow rates; the difference in vortex size and streamline patterns of LDPE and PS melts in a 6:1 planar contraction This heuristic ap proach is thus recommended for studying the effect of normal stresses in viscoelastic flows with a main direction for engineering calculations much as the lubrication approximation has been used in the past for purely viscous fluids.

On the inertial and extensional effects on the corner and lip vortices in a circular 4:1 abrupt contraction

An examination is presented of the effects of inertia on the circular abrupt contraction flow of generalized Newtonian and viscoelastic fluids, the extensional viscosity of which has been increased by means of a dependence on the fluid as well as the second invariants of the rate of deformation tensor. It is shown that the combination of inertia and elongational effects leads to the development of a small and strong lip vortex which may quickly envelop the stagnant corner vortex. On a macroscopic level, this phenomenon is also accompanied by an increase of the pressure drop.

Circular entry flows of fluid M1

The observations presented in circular entry flows for fluid M1 can be used to formulate conclusions about other viscoelastic fluids that fall within this general class of fluid behaviour. Evidence reported here shows that the lip and corner vortices observed in contraction flows of fluid M1 can exist either in isolation from each other, or in a figure-eight combination of both. A probable mechanism is proposed for the generation and growth of the lip vortex, based on the presence of an extenstionally formed, solid-like core of fluid near the contraction entry zone. The mechanism is tested using a purely Newtonian numerical simulation in a re-entrant tube entry flow geometry. End correction and extensional viscosity measurements, using orifice pressure drop techniques, reveal a maximum value of both entities and increasingly unstable values beyond those flow conditions.

Estimation of the elongational viscosity of polyethylene blends at high deformation rates

The Binding and Cogswell entry flow analyses were used to model the flow of low-density polyethylene (LDPE), linear low-density polyethylene (LLDPE), and a 50 50 blend of LLDPE with LDPE through a sudden contraction. The entrance angle of the converging stream is estimated by using the two previous analyses and compared to the flow visualization experiments. The entrance angle was well predicted, being more accurate in the case of the Binding analysis, when the entrance angle was small (large vortices). An approximate sink flow analysis is proposed which gave better predictions when the entrance angles were large. Since experimental results of the elongational viscosity at high deformation rates are not available, the elongational viscosity predicted from three previous entry flow analyses were compared to the predictions of the Larson (PEC) model. It was shown that both the strain hardening parameter of the PEC model and the distribution of the relaxation moduli and relaxation times determine the power-law indices t and n of the elongational viscosity and the viscosity, respectively, and therefore control the generation of lip vortices upstream of a contraction. The entrance pressure was shown to have the same dependence on deformation rate for both the Cogswell and Binding analyses. All three models predicted vortex growth if t is greater than n. Complex flow patterns were generated by the flow of the 50 50 blend at high deformation rates whereby a minimum in the entrance angle was observed.

Reentrant corner singularities in non-newtonian flow. Part I: theory

Biorthogonal series expansions are used to study the creeping flow of a corotational Maxwell fluid in plane reentrant and non-reentrant sectors. The governing equations are formulated in terms of stream function and Airy stress function only, the radial parts of which are determined by an infinite system of fourth-order nonlinear ordinary differential equations which are seen to be singular perturbations of the linear second-order equations for Stokes flow. Exact formal solutions in logarithmic series are derived, but do not readily enable the asymptotics of the corner singularities to be studied. Instead, approximate methods based on Picard linearization and dominant component analysis are used to decouple the system, whence the singularities are studied in terms of generalized hypergeometric equations and Meijer G-functions. Certain conditions on the roots of associated quartic equations are derived, the satisfaction of which would indicate the existence of lip vortices in a reentrant sector and the integrability of stress in a neighbourhood of the corner.

Further observations of elastic effects in tubular entry flows

The current understanding of tubular entry flows for Newtonian fluids and the influence of elasticity on these flows in the absence of shear thinning and significant fluid inertia is reviewed. While tubular entry flow for Newtonian fluids both with and without fluid inertia is now a solved problem, no single worker has been able to predict any of the significant elastic effects observed for constant viscosity elastic liquids in such flows. New elastic flow phenomena are reported which complicate the problem even more. Two distinctly different flow patterns are observed in a four to one contraction for two constant viscosity elastic fluids with essentially the same characteristics times. Tubular entry flows of elastic liquids develop (with increasing λγ) in the same way only if the contraction ratio is high enough. At sufficiently low contraction ratio, a lip vortex (at the tube entrance) and a corner vortex (in the upstream tube corner) can both be present and interacting with the lip vortex dominating the flow when vortex enhancement and growth occurs. The entry flow problem for elastic liquids is far more complex than originally imagined and is indeed a challenge for numerical simulation and selection of constutitive equations.