Abrupt Axisymmetric Contractions Research Articles

Numerical Simulation of Turbulent Pipe Flow Through an Abrupt Axisymmetric Constriction

Turbulent flow through an abrupt axisymmetric contraction has been investigated in a computational study. The full three-dimensional Navier-Stokes equations have been solved in cylindrical coordinates. The ring-like obstruction was accounted for by an immersed boundary method implemented in an open source code. The results were compared with data from an experimental study and a surprisingly good agreement was found in the separated-flow region downstream of the obstruction. A distinctly higher turbulence level was predicted upstream of the obstruction as compared with the measured data. Due to the imposed streamwise periodicity the simulated configuration mimicked a sequence of equally spaced obstructions and this periodicity gives rise to substantially higher turbulence levels than in a single-ring configuration. Particular attention was paid to the possible occurrence of mean-flow asymmetries, as frequently reported in planar-symmetric configuration. No sign of the Coanda effect was observed in the present study, for which a physical explanation was provided.

Classical Nucleation Theory Applied to Homogeneous Bubble Nucleation in the Continuous Microcellular Foaming of the Polystyrene−CO2 System

In the continuous production of microcellular thermoplastic foam, a polymer−physical foaming agent (PFA) solution is subjected to a rapid pressure drop through an extrusion foaming die. Simulations were run for the flow of a polymer−PFA solution through an extrusion foaming die with an abrupt axisymmetric contraction. The pressure drops across the die obtained through the simulations showed good qualitative agreement with experimental pressure drop measurements on the foaming extrusion die obtained in our laboratory. Field values of pressure, temperature, and velocity were obtained at each point in the foaming die. Once the values of pressure and temperature were obtained along each point in the foaming die, classical nucleation theory for bubble nucleation was invoked to predict the local bubble nucleation rate downstream of the saturation surface. The hydrodynamic constraints to the nucleation rate were calculated by using a modified form of the classical nucleation theory that accounted for the diffusional and viscosity constraints to the rate of homogeneous nucleation. The capillarity approximation was found not to be valid for bubble nucleation of CO2 in polymers; a correction accounting for the curvature dependence of surface tension was applied to get nonzero nucleation rates for the system to reconcile theoretically predicted rates with experimental observations.

Steady flow simulation of a polymer‐diluent solution through an abrupt axisymmetric contraction using internally consistent rheological scaling

AbstractIn this work, a new methodology is developed that describes the viscoelastic scaling of a polymer‐physical foaming agent (PFA) solution in a detailed and internally consistent manner. The approach is new in that while previous researchers have largely focused on scaling down experimentally obtained high pressure polymer‐PFA solution viscosity data onto a master curve for the viscosity of the undiluted polymer melt at a reference temperature and atmospheric pressure, we have generated the shear viscosity data required for our simulations by systematically scaling up the viscosity values obtained from measurements on a pure polymer melt to the desired temperature, pressure, and concentration values characterizing the flow. Simulations have been run for the flow of a polymer‐PFA solution through an extrusion foaming die with an abrupt axisymmetric contraction and good qualitative agreement is obtained with experimental pressure drop measurements obtained previously in our laboratory. The pressure drop rates and temperature rise rates have been estimated at the surface of incipient nucleation. Because of the short residence times in the die for the microcellular foaming process, approximating the flow through the die as a single phase flow in our simulations still gives useful insights into the dynamics of the flow. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007

An approximate solution to flow through a contraction for high Trouton ratio fluids

An approximate analytical solution is presented to predict the flow, vortex and pressure drop behaviour in abrupt axisymmetric contractions. This solution is developed to identify the effect of fluids which exhibit large Trouton ratios at large extensional rates despite having a constant shear viscosity. This behaviour approximates that shown in dilute polymeric solutions, although any transient effects are neglected by removing any strain dependence of the extensional viscosity and by assuming that the steady-state extensional viscosity undergoes a step change from a Trouton ratio of 3 to a large Trouton ratio at a critical rate. The predictions compare well to available experimental results for dilute to semi-dilute solutions of a flexible polymer, polystyrene, prior to the onset of elastic instabilities and for dilute solutions of a semi-rigid polymer, xanthan. The results show that vortex behaviour is determined by the fluid’s characteristic time (through the magnitude of the Weissenberg number of the flow), while the steady-state extensional viscosity is of far more importance than transient effects in determining the steady-state pressure drop.

Flow of Some Carboxymethylcellulose Solutions Through Abrupt Axisymmetric Contractions. Experimental Study and Modelling of Shear Thinning and Elongational Effects

Abstract The focus of this paper will be on the modelling and simulation of contraction flow, with marked aspect ratio β = 6, 9, 12. Two fluid families are considered: a glycerol Newtonian solution and carboxy-methyl-cellulose (CMC) solutions which present particular rheological properties. Their shear thinning character are modelled by a Cross formula over a large scale of shear rates. The elongational properties are taken via a simplified Ericksen model into account. Experimental velocity profiles are determined using the Laser Doppler Anemometry (L.D.A) technique. They are found to be in good agreement with numerical velocity profiles obtained using a finite volume method with extra source terms traducing the particular rheological behaviour proposed here. The simulations allow to determine the different values of an elongational parameter μ3. Then, some numerical results concerning the total energy losses are presented using the usual concept of the equivalent length.

Computations of non‐isothermal viscous and viscoelastic flows in abrupt contractions using a finite volume method

A finite volume method is applied to numerical simulations of steady isothermal and non‐isothermal flows of fluids obeying different constitutive equations: Newtonian, purely viscous with shear‐thinning properties (Carreau law) and viscoelastic Upper Convected Maxwell differential model whose temperature dependence is described by a William‐Landel‐Ferry equation. The flow situations concern various abrupt axisymmetric contractions from 2:1 to 16:1. Such flow geometries are involved in polymer processing operations. The governing equations are discretized on a staggered grid with an upwind scheme for the convective‐type terms and are solved by a decoupled algorithm, stabilized by a pseudo‐transient stress term and an elastic viscous stress splitting technique. The numerical results highlight the influence of temperature on the flow situations, and also the complex behaviour of the materials under non‐isothermal conditions.

Quantitative NMR velocity imaging of a main-chain liquid crystalline polymer flowing through an abrupt contraction

The flow of isotropic and liquid crystalline (LC) hydroxypropylcellulose (HPC) aqueous solutions into an abrupt axisymmetric contraction has been quantitatively measured by pulsed-field-gradient NMR techniques. Steady-state axial velocity profiles, acquired upstream of the contraction, reveal a large contraction entry length for the LC solution. This entry flow field exists over an order of magnitude change in flow rate and is attributed to elasticity that is associated with polydomain liquid crystallinity. Pronounced, off-centerline velocity maxima (in an axisymmetric flow field) were present upstream of the contraction, in the entry flow region. Apparently, a more viscous and elastic core of fluid was present along the centerline; this fluid resisted elongational strain more than the fluid closer to the walls. Quantitative velocity profiles were extracted from displacement distributions and corrected for elongational dispersion. The isotropic solution velocity profiles matched those obtained from viscoelastic simulations using an approximate Doi-Edwards model, parameterized with independent rheological data.

Numerical simulation of complex flows of non‐Newtonian fluids using the stream tube method and memory integral constitutive equations

AbstractIn this paper a memory integral viscoelastic equation is considered for simulating complex flows of non‐Newtonian fluids by stream tube analysis. A formalism is developed to take into account co‐deformational memory equations in a mapped computational domain where the transformed streamlines are parallel and straight. The particle‐tracking problem is avoided. Evolution in time and related kinematic quantities involved with a K‐BKZ integral constitutive model are easily taken into account in evaluating the stresses. Successive subdomains, the stream tubes, may be considered for computing the main flow in abrupt axisymmetric contractions from the wall to the central flow region. The ‘peripheral stream tube’ close to the duct wall is determined by developing a non‐conventional modified Hermite element. A mixed formulation is adopted and the relevant non‐linear equations are solved numerically by the Levenberg‐Marquardt algorithm. Although the singularity at the section of contraction is not involved explicitly, the results obtained for the peripheral stream tube clearly show the singularity effects and the extent of the recirculating zone near the salient corner. The algorithm is stable even at high flow rates and provides satisfactory solutions when compared with similar calculations in the literature.

A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction

This paper reports the first convergent numerical algorithm for the steady inertialess flow of an Upper-Convected Maxwell (UCM) fluid through a four-to-one abrupt axisymmetric contraction. The Finite Volume Method (FVM) is adopted along with a stream function-vorticity approach in the Elastic Viscous Split Stress (EVSS) form with a first-order upwind approximation applied to the convective terms of the stress constitutive equation. A staggered volume discretization of the flow variables eliminates the stress singularity from the computational domain without loosing any flow physics. The volume integrals of the governing equations over the flow domain result in a system of nonlinear algebraic equations that are solved iteratively by a semi-implicit line-to-line method with a pseudo-trasient term added to the stress constitutive equation. Computations of an UCM fluid using this method are carried out to a much higher value of the Deborah number ( De) than previous numerical simulations using the Finite Element Method (FEM). The solutions are found to be smooth, stable, and convergent with the finger stress tensor remaining posiitive-definite throughout the flow domain. Calculations are not performed above De = 6.25 because of the decreasing pseudo-time step constraint at higher elasticity. The finite volume algorithm approximates better solutions upon mesh refinement, demonstrates the smoothness and the mathematical well-posedness of this problem, and predicts a Newtonian-like flow structure near the singularity for an UCM fluid. Using this new method, the inertialess flow of an UCM fluid through 4:1 abrupt axisymmetric contraction, for the first time, produces larger corner vortices at higher values of De.

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.

LDV measurements of viscoelastic flow transitions in abrupt axisymmetric contractions: Interaction of inertia and elasticity

Laser Doppler velocimetry (LDV) measurements of the axial and radial velocity components are reported for the axisymmetric contraction flow of a viscoelastic solution of polyisobutylene dissolved in tetradecane. This fluid exhibits shear-thinning of both the viscosity and first normal stress difference; the viscosity is well described by the Carreau-Yasuda model. Shear-rate-dependent Reynolds and Deborah numbers are used to described the flow conditions. Measurements at low flow rates for contraction ratios of 2:1, 4:1, and 8:1 indicate that the flow near the contraction plane scales with the small tube radius. A viscoelastic corner vortex is detected at moderate flow rates, yet vortex growth is not observed with increased flow rate, because inertial forces press the vortex into the outer corner. At high flow rates, where both the Reynolds number and Deborah number are large, the flow near the contraction plane is divided into an accelerating core and an outer region where the flow retains its upstream profile. The magnitude of the local rate-of-strain ▪ is calculated from the axial and radial velocity profiles. In the core, the magnitude of ▪ >50s −1 and its value is dominated by the axial gradients ∂ v/∂ z and ∂ v r /∂ z. Large extension rates along the centerline suggest that extensional thinning occurs in this core region. Finite element calculations for a Carreau-Yasuda fluid with the same viscosity as the PIB/C14 solution and under the same flow conditions do not predict the two-regions of the flow.