Three-dimensional flow of Newtonian and Boger fluids in square–square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streak-line photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re₂ ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De₂) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De₂ ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.     

Computing non‐Newtonian fluid flow with radial basis function networks

This paper is concerned with the application of radial basis function networks (RBFNs) for solving non‐Newtonian fluid flow problems. Indirect RBFNs, which are based on an integration process, are employed to represent the solution variables; the governing differential equations are discretized by means of point collocation. To enhance numerical stability, stress‐splitting techniques are utilized. The proposed method is verified through the computation of the rectilinear and non‐rectilinear flows in a straight duct and the axisymmetric flow in an undulating tube using Newtonian, power‐law, Criminale–Ericksen–Filbey (CEF) and Oldroyd‐B models. The obtained results are in good agreement with the analytic and benchmark solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical technique for laminar swirling flow at the interface between porous and homogenous fluid domains

There have been a few recent numerical implementations of the stress‐jump condition at the interface of conjugate flows, which couple the governing equations for flows in the porous and homogenous fluid domains. These previous demonstration cases were for two‐dimensional, planar flows with simple geometries, for example, flow over a porous layer or flow through a porous plug. The present study implements the interfacial stress‐jump condition for a non‐planar flow with three velocity components, which is more realistic in terms of practical flow applications. The steady, laminar, Newtonian flow in a stirred micro‐bioreactor with a porous scaffold inside was investigated. It is shown how to implement the interfacial jump condition on the radial, axial, and swirling velocity components. To avoid a full three‐dimensional simulation, the flow is assumed to be independent of the azimuthal direction, which makes it an axisymmetric flow with a swirling velocity. The present interface treatment is suitable for non‐flat surfaces, which is achieved by applying the finite volume method based on body‐fitted and multi‐block grids. The numerical simulations show that a vortex breakdown bubble, attached to the free surface, occurs above a certain Reynolds number. The presence of the porous scaffold delays the onset of vortex breakdown and confines it to a region above the scaffold. Copyright © 2008 John Wiley & Sons, Ltd.     

Multiphase non‐Newtonian effects on pulsatile hemodynamics in a coronary artery

Hemodynamic stresses are involved in the development and progression of vascular diseases. This study investigates the influence of mechanical factors on the hemodynamics of the curved coronary artery in an attempt to identify critical factors of non‐Newtonian models. Multiphase non‐Newtonian fluid simulations of pulsatile flow were performed and compared with the standard Newtonian fluid models. Different inlet hematocrit levels were used with the simulations to analyze the relationship that hematocrit levels have with red blood cell (RBC) viscosity, shear stress, velocity, and secondary flow. Our results demonstrated that high hematocrit levels induce secondary flow on the inside curvature of the vessel. In addition, RBC viscosity and wall shear stress (WSS) vary as a function of hematocrit level. Low WSS was found to be associated with areas of high hematocrit. These results describe how RBCs interact with the curvature of artery walls. It is concluded that although all models have a good approximation in blood behavior, the multiphase non‐Newtonian viscosity model is optimal to demonstrate effects of changes in hematocrit. They provide a better stimulation of realistic blood flow analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

Non‐Newtonian blood flow dynamics in a right internal carotid artery with a saccular aneurysm

Flow dynamics plays an important role in the pathogenesis and treatment of cerebral aneurysms. The temporal and spatial variations of wall shear stress in the aneurysm are hypothesized to be correlated with its growth and rupture. In addition, the assessment of the velocity field in the aneurysm dome and neck is important for the correct placement of endovascular coils. This work describes the flow dynamics in a patient‐specific model of carotid artery with a saccular aneurysm under Newtonian and non‐Newtonian fluid assumptions. The model was obtained from three‐dimensional rotational angiography image data and blood flow dynamics was studied under physiologically representative waveform of inflow. The three‐dimensional continuity and momentum equations for incompressible and unsteady laminar flow were solved with a commercial software using non‐structured fine grid with 283 115 tetrahedral elements. The intra‐aneurysmal flow shows complex vortex structure that change during one pulsatile cycle. The effect of the non‐Newtonian properties of blood on the wall shear stress was important only in the arterial regions with high velocity gradients, on the aneurysmal wall the predictions with the Newtonian and non‐Newtonian blood models were similar. Copyright © 2005 John Wiley & Sons, Ltd.     

The combined effects of non‐planarity and asymmetry on primary and secondary flows in the small bronchial tubes

The laminar flow in the small bronchial tubes is quite complex due to the presence of vortex‐dominated, secondary flows. In this paper, we report the results of a numerical investigation of the simultaneous effects of asymmetric and non‐planar branching on the primary and secondary flows in the small bronchial tubes, i.e. generations 6–12. We simulate steady‐state inspiratory flow at a Reynolds number of 1000 in three‐generation, asymmetric planar and non‐planar bronchial tube models. The non‐planar model was defined by applying a 90° out‐of‐plane rotation to the third‐generation branches. A detailed mesh refinement study was performed in order to demonstrate mesh independence. Significant differences were observed between flows in the planar and non‐planar models. An uneven mass flow distribution was observed in the non‐planar model in contrast to the evenly distributed mass flow in the planar model. The secondary flows created symmetric vortex patterns in the planar model, whereas vortex symmetry was lost in the non‐planar model. These results illustrate the importance of incorporating asymmetry in addition to non‐planarity in the geometric models. Copyright © 2008 John Wiley & Sons, Ltd.     

Effects of the side walls on the unsteady flow of a second-grade fluid in a duct of uniform cross-section

In this paper, the effects of the side walls on the unsteady flow of a second-grade fluid in a duct of rectangular cross-section are considered. Two types of unsteady flows are investigated. One of them is the unsteady flow in a duct of rectangular cross-section moving parallel to its length and the other is the unsteady flow due to an applied pressure gradient in a duct of rectangular cross-section whose sides are at rest. It is shown that a Newtonian fluid reaches steady-state earlier than a second-grade fluid and the effect of the side walls on a second-grade fluid is more effective than that on a Newtonian fluid.     

Turbulent flow of Newtonian and shear-thinning liquids through a sudden axisymmetric expansion

 Measurements are reported for the turbulent flow through a sudden expansion of a moderately elastic shear-thinning liquid and also for two Newtonian liquids. The differences in the mean velocity fields for the two fluid types are relatively small, including the length of the recirculation region which is essentially unaffected by the fluid rheology. Although turbulent kinetic energy levels for the non-Newtonian fluids are always lower than for the Newtonian fluids, no significant difference is found in the relative contributions to the turbulent kinetic energy of the axial, radial and tangential normal stresses. Since the vorticity thicknesses are much the same for all flows, viscoelasticity appears to be responsible for the reduced levels of turbulent kinetic energy for the non-Newtonian fluids. Received: 6 November 1998/Accepted: 27 January 1999     

Numerical solutions of pulsating flow and heat transfer characteristics in a channel with a backward-facing step

The incompressible laminar flow of air and heat transfer in a channel with a backward-facing step is studied for steady cases and for pulsatile inlet conditions. For steady flows the influence of the inlet velocity profile, the height of the step and the Reynolds number on the reattachment length is investigated. A parabolic entrance profile was used for pulsatile flow. It was found with amplitude of oscillation of one by Re=100 that the primary vortex breakdown through one pulsatile cycle. The wall shear rate in the separation zone varied markedly with pulsatile flows and the wall heat transfer remained relatively constant. The time-average pulsatile heat transfer at the walls was greater as with steady flow with the same mean Reynolds number.     

Effect of magnetic Reynolds number on the two‐dimensional hydromagnetic flow around a cylinder

Numerical experiments have been conducted to study the effect of magnetic Reynolds number on the steady, two‐dimensional, viscous, incompressible and electrically conducting flow around a circular cylinder. Besides usual Reynolds number Re, the flow is governed by the magnetic Reynolds number R_m and Alfvén number β. The flow and magnetic field are uniform and parallel at large distances from the cylinder. The pressure Poisson equation is solved to find the pressure fields in the entire flow region. The effects of the magnetic field and electrical conductivity on the recirculation bubble, drag coefficient, standing vortex and pressure are presented and discussed. For low interaction parameter (N<1), the suppression of the flow‐separation is nearly independent of the conductivity of the fluid, whereas for large interaction parameters, the conductivity of the fluid strongly influences the control of flow‐separation. Copyright © 2008 John Wiley & Sons, Ltd.     

Flow of a shear-thinning liquid in a cylindrical container with a rotating end wall

The flow patterns produced by rotating one end wall of a circular cylinder completely filled with a strongly shear-thinning viscoelastic liquid have been investigated using the laser-induced fluorescence flow visualization technique. An intense toroidal vortex is produced in the vicinity of the rotating end wall with outward spiraling flow over the end wall itself. This vortex drives a second countercirculating vortex of low intensity in the region of the stationary end wall. Under some circumstances an axial jet of fluid is observed moving away from the rotating end wall. This jet showed evidence of instability, whereas all flows were otherwise completely steady. The double-vortex structure is different from those recently observed in either a Newtonian or slightly shear-thinning liquid or in the low Reynolds number flow of an elastic liquid. There are, however, similarities with older work for a viscoelastic liquid at relatively high Reynolds numbers. The observations highlight the suitability of the cylinder/rotating end wall configuration as a sensitive test case for computational work.     

Non‐Newtonian blood flow study in a model cavopulmonary vascular system

A transient haemodynamic study in a model cavopulmonary vascular system has been carried out for a typical range of parameters using a finite element‐based Navier–Stokes solver. The focus of this study is to investigate the influence of non‐Newtonian behaviour of the blood on the haemodynamic quantities, such as wall shear stress (WSS) and flow pattern. The computational fluid dynamics (CFD) model is based on an artificial compressibility characteristic‐based split (AC‐CBS) scheme, which has been adopted to solve the Navier–Stokes equations in space–time domain. A power law model has been implemented to characterize the shear thinning nature of the blood depending on the local strain rate. Using the computational model, numerical investigations have been performed for Newtonian and non‐Newtonian flows for different frequencies and input pulse forms. The haemodynamic quantities observed in total cavopulmonary connection (TCPC) for the above conditions suggest that there are considerable differences in average (about 25–40%) and peak (about 50%) WSS distributions, when the non‐Newtonian behaviour of the blood is taken into account. The lower WSS levels observed for non‐Newtonian cases point to the higher risk of lesion formation, especially at higher pulsation frequencies. A realistic pulse form is relatively safer than a sinusoidal pulse as it has more energy distributed in the higher harmonics, which results in higher average WSS values. The present study highlights the importance of including non‐Newtonian shear thinning behaviour for modelling blood flow in the vicinity of repaired arterial connections. Copyright © 2010 John Wiley & Sons, Ltd.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

A 2D boundary element method for simulating the deformation of axisymmetric compound non‐Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non‐Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non‐Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non‐Newtonian material. By transforming the integral representation for the velocity to cylindrical co‐ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two‐dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non‐Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd‐B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break‐up mechanism of compound drops in relation to the specific non‐Newtonian character of the membrane. Copyright © 1999 John Wiley & Sons, Ltd.     

Global linear stability analysis of time‐averaged flows

The global linear stability analysis (LSA) of stationary/steady flows has been applied to various flows in the past and is fairly well understood. The LSA of time‐averaged flows is explored in this paper. It is shown that the LSA of time‐averaged flows can result in useful information regarding its stability. The method is applied to study flow past a cylinder at Reynolds number (Re) beyond the onset of vortex shedding. Compared with the direct numerical simulation, LSA of the Re=100 steady flow severely underpredicts the vortex shedding frequency. However, the LSA of the time‐averaged flow results in the correct value of the non‐dimensional frequency, St, of the associated instability. Copyright © 2007 John Wiley & Sons, Ltd.     

Thin film flow of non‐Newtonian MHD fluid on a vertically moving belt

Thin film flow of an Oldroyd 6‐constant fluid on a vertical moving belt is investigated analytically and numerically. The governing equations for the flow field are derived for a steady one‐dimensional flow. The effect of constant applied magnetic field is included and its influence on the flow field is studied. The nonlinear governing equations are solved analytically and the exact solution is obtained in an elegant way. Numerical solutions are also obtained using higher‐order Chebyshev spectral methods. The influence of various non‐Newtonian parameters, gravitational force and applied magnetic field is investigated. The results showing the effect of gravity, magnetic field and material constants α₁ and α₂ are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Steady and pulsating flow characteristics in straight tubes with and without a lateral circular protrusion

 Flow characteristics in straight tubes with and without a lateral circular protrusion had been investigated using Particle Image Velocimetry over a range of Reynolds numbers from 400 to 1400, and at Womersley number of 65. The practical interest of the flows considered lies mainly in blood flows through arteries with saccular aneurysm. Both steady and pulsating flow experiments had been conducted. It was found that under the steady flow conditions, a recirculating vortex would be formed inside the circular protrusion. The maximum strength of the vortex would be as low as 10% of the bulk mean velocity in the main tube at the highest Reynolds number tested (i.e. at 1400). Under the pulsating flow conditions, the vortex appeared and disappeared at different phase of a cycle. The sequence was only punctuated by quasi-inviscid flow behavior. The steady flow results only resembled those of the pulsating ones for about 1 10 of the time at each cycle. Received: 13 August 1997/Accepted: 30 June 1998