Finite element calculation of viscoelastic flow in a journal bearing: II. Moderate eccentricity

Finite element calculations of two-dimensional flows of viscoelastic fluids in a journal bearing geometry reported in an earlier paper (J. Non-Newt. Fluid Mech. 16 (1984) 141-172) are extended to higher eccentricity (ρ = 0.4); at this higher eccentricity flow separation occurs in the wide part of the gap for a Newtonian fluid. Calculations for the second-order fluid (SOF), upper-convected Maxwell (UCM), and the Giesekus models are continued in increasing Deborah number for each model until either a limit point is reached or oscillations in the solution make the numerical accuracy too poor to warrant proceeding. No steady solutions to the UCM model were found beyond a limit point De_c, as was the case for results at low eccentricities. The value of De_c was moderately stabel to mesh refinement. A limit point also terminated the calculations with a SOF model, in contradiction to the theorems for uniqueness and existence for this model. The critical value of De increased drastically with increasing refinement of the mesh, as expected for solution pathology caused by approximation error. Calculations for the Giesekus fluid with the mobility parameter α ≠ O showed no limit points, but failed when irregular oscillations destroyed the quality of the solution. The behavior of the recirculation region of the flow and the load on the inner cylinder were very sensitive to the value of α used in the Giesekus model. The recirculation disappeared at low values of De except when the mobility parameter α was so small that the viscosity was almost constant over the range of shear rates in the calculations. The recirculation persisted over the entire range of accessible De for the UCM fluid, the limit of α = O of the Giesekus model. The behavior of the recirculation is coupled directly to the viscosity by calculations with an inelastic fluid with the same viscosity predicted by the Giesekus model.     

Injection and suction of an upper-convected maxwell fluid through a porous-walled tube

The steady-state, similarity solutions of the flow of an upper-convected Maxwell fluid through a tube with a porous wall are constructed by asymptotic and numerical analyses as functions of the direction of flow through the tube, the amount of elasticity in the fluid, as measured by the Deborah number De, and the degree of fluid slip along the tube wall. Fluid slip is assumed to be proportional to the local shear stress and is measured by a slip parameter β that ranges between no-slip (β = 1) and perfect slip (β = 0). The most interesting results are for fluid injection into the tube. For β = 1, the family of flows emanating from the Newtonian limit (De = 0) has a limit point where it turns back to lower values of De. These solutions become asymptotic to De = 0) and develop an O(De) boundary layer near the tube wall with singularly high stresses matched to homogeneous elongational flow in the core. This solution structure persists for all nonzero values of the slip parameter. For β ≠ 1, a family of exact solutions is found with extensional kinematics, but nonzero shear stress convected into the tube through the wall. These flows differ for low De from the Newtonian asymptote only by the absence of the boundary layer at the tube wall. Finite difference calculations evolve smoothly between the Newtonian-like and extensional solutions because of approximation error due to under-resolution of the boundary layer. The radial gradient of the axial normal stress of the extensional flow is infinite at the centerline of the tube for De > 1; this singularity causes failure of the finite difference approximations for these Deborah numbers unless the variables are rescaled to take the asymptotic behavior into account.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

Viscoelastic flow between eccentric rotating cylinders: unstructured control volume method

This paper reports a convergent numerical algorithm for the Upper-Convected Maxwell (UCM) fluid between two eccentric cylinders at various eccentricity ratios (?); the outer cylinder is stationary, and the inner one rotating. The problem is solved by an unstructured control volume method (UCV), which is designed for a general viscoelastic flow problem with an arbitrary computational domain. A self-consistent false diffusion technique and an iteration scheme are used in combination to solve the problem. The computations of the UCM fluid using the numerical algorithm are carried out to a higher value of the Deborah number (De) at each eccentricity tested than hitherto possible with previous numerical simulations. The solutions are compared with previous numerical results, confirming the effectiveness of the UCV method as a general technique for solving viscoelastic flow problems.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Effects of Permeable Cylinder on the Flow Structure in Deep Water

Flow behaviors around permeable cylinders were investigated using Particle Image Velocimetry technique in deep water. The height of deep water and free stream velocity were kept constant as h_w = 340 mm and U = 156 mm/s. To find out the effect of the permeable cylinders on the flow structure, eight different porosities (β = 0.4, 0.5, 0.6, 0.65, 0.7, 0.75, 0.8, and 0.85) were used. The results have indicated that the permeable cylinders are effective on the control of large-scale vortical structures downstream of the permeable cylinder. As the porosity increases, turbulent kinetic energy and Reynolds shear stress decrease. This means that the fluctuations in the wake region are significantly weakened by permeable cylinders. The permeable cylinders having the porosity higher than 0.6 do not pose an obstacle in the flow. Furthermore, for all diameter values of permeable cylinders, it can be concluded that the flow structures downstream of the permeable cylinder show similar trend with each other.     

Thermal study of an array of inline horizontal cylinders below a nearly adiabatic ceiling

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal cylinder in a horizontal array of cylinders consists of three isothermal cylinders, located underneath a nearly adiabatic ceiling is studied experimentally. A Mach–Zehnder interferometer is used to determine thermal field and smoke test is made to visualize flow field. Effects of the cylinders spacing to its diameter (S/D), and cylinder distance from ceiling to its diameter (L/D) on heat transfer from the centered cylinder are investigated for Rayleigh numbers from 1500 to 6000. Experiments are performed for an inline array configuration of horizontal cylinders of diameters D = 13 mm. Results indicate that due to the nearly adiabatic ceiling and neighboring cylinders, thermal plume resulted from the centered cylinder separates from cylinder surface even for high L/D values and forming recirculation regions. By decreasing the space ratio S/D, the recirculation flow strength increases. Also, by decreasing S/D, boundary layers of neighboring cylinders combine and form a developing flow between cylinders. The strength of developing flow depends on the cylinders Rayleigh number and S/D ratio. Due to the developing flow between cylinders, the vortex flow on the top of the centered cylinder appears for all L/D ratios and this vortex influences the value of local Nusselt number distribution around the cylinder.Variation of average Nusselt number of the centered cylinder depends highly on L/D and the trend with S/D depends on the value of Rayleigh number.     

A LOCAL MESH REFINEMENT ALGORITHM APPLIED TO TURBULENT FLOW

This paper presents a local mesh refinement procedure based on a discretization over internal interfaces where the averaging is performed on the coarse side. It is implemented in a multigrid environment but can optionally be used without it. The discretization for the convective terms in the velocity and the temperature equation is the QUICK scheme, while the HYBRID-UPWIND scheme is used in the turbulence equations. The turbulence model used is a two-layer k–ϵ model. We have applied this formulation on a backward-facing step at Re=800 and on a three-dimensional turbulent ventilated enclosure, where we have resolved a geometrically complex inlet consisting of 84 nozzles. In both cases the concept of local mesh refinements was found to be an efficient and accurate solution strategy. © 1997 by John Wiley & Sons, Ltd.     

CONTROL OF FLOW PAST BLUFF BODIES USING ROTATING CONTROL CYLINDERS

Computational results for control of flow past a circular cylinder using small rotating cylinders are presented. A well-proven stabilized finite-element method, that has been applied to various flow problems earlier, is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The formulation is first applied to study flow past an isolated rotating cylinder. Excellent match with experimental results, reported earlier, is observed. It is found that in purely two-dimensional flows, very high lift coefficients can be realized. However, it is observed, via three-dimensional Navier–Stokes simulations, that the end-effects and centrifugal instabilities along the cylinder span lead to a loss of lift and increase in drag. The aspect ratio of the cylinder plays an important role. The flow past a bluff body with two rotating control cylinders is studied using 2-D numerical simulations. The effect of the Reynolds number is studied by carrying out simulations for Re=10²and 10⁴. Finite element meshes with an adequate number of grid points are employed to resolve the flow in the gap between the main and control cylinders. Two values of the gap are considered: 0·01D and 0·075 D, where D is the diameter of the main cylinder. It is observed that when the control cylinders rotate at high speed, such that the tip speed is 5 times the free-stream speed, the flow at Re=100 achieves a steady state. For Re=10⁴, even though the flow remains unsteady, the wake is highly organized and narrower compared to the one without control. The results are in good agreement with the flow-visualization studies conducted by other researchers for bluff bodies using similar control concepts. In all the cases, a significant reduction in the overall drag coefficient and the unsteady aerodynamic forces acting on the main cylinder is observed. Results are also presented for the power requirements of the system for translation and rotation. It is found that the coefficient of power required for the rotation of control cylinders is significant for Re=100 but negligible for Re=10⁴flow. The size of the gap is found to be more critical for the Re=10⁴flows. This study brings out the relevance of the gap as a design parameter for such flow control devices.     

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L² = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L² = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0–4.5.     

TR-PIV measurement of the wake behind a grooved cylinder at low Reynolds number

A comparative study of the wakes behind cylinders with grooved and smooth surfaces was performed with a view to understand the wake characteristics associated with the adult Saguaro cacti. A low-speed recirculation water channel was established for the experiment; the Reynolds number, based on the free-stream velocity and cylinder diameter (D), was kept at Re_D=1500. State-of-the-art time-resolved particle image velocimetry (TR-PIV) was employed to measure a total of 20 480 realizations of the wake field at a frame rate of 250 Hz, enabling a comprehensive view of the time- and phase-averaged wake pattern. In comparison to the wake behind the smooth cylinder, the length of the recirculation zone behind the grooved cylinder was extended by nearly 18.2%, yet the longitudinal velocity fluctuation intensity was considerably weakened. A global view of the peaked spectrum of the longitudinal velocity component revealed that the intermediate region for the grooved cylinder, which approximately corresponds to the transition region where the shear layer vortices interact, merge and shed before the formation of the Karman-like vortex street, was much wider than that for the smooth one. The unsteady events near St=0.3-0.4 were detected in the intermediate region behind the grooved cylinder, but no such events were found in the smooth cylinder system. Although the formation of the Karman-like vortex street was delayed by about 0.6D downstream for the grooved cylinder, no prominent difference in the vortex street region was found in the far wake for both cylinders. The Proper Orthogonal Decomposition (POD) method was used extensively to decompose the vector and swirling strength fields, which gave a close-up view of the vortices in the near wake. The first two POD modes of the swirling strength clarified the spatio-temporal characteristics of the shear layer vortices behind the grooved cylinder. The small-scale vortices superimposed on the shear layers behind the grooved cylinder were found to be generated and convected downstream in the same phase, which would significantly reduce the fluctuating force on the cylinder surface.     

Multiple-time scale approach to stress growth in suspensions of rodlike macromolecules

The rheologial properties of a dilute suspension of rigid rodlike macromolecules at the inception of a steady homogeneous flow are examined within the framework of the kinetic theory of polymeric liquids. The cases of simple shear and elongational flows are considered under the assumption of small non-dimensional shear rates, ϵ = κ₀/6D, where D is the rotation diffusion of the particles. The method of multiple-time scales is used to solve the problem and the resulting transport coefficients are shown to reveal new features as compared with the results of the straightforward perturbation technique. In particular, the onset of stress oscillations leading to an undershoot following the overshoot of the shear viscosity and the overshoot of the normal stress difference are detected as ϵ approaches unity.     

Experimental analysis of thermal fields in horizontally eccentric cylindrical annuli

This paper presents new experimental results on thermal field and heat transfer in a two-dimensional annulus between horizontally eccentric cylinders. The study is conducted by means of optical techniques, for 1.07×10⁴≤Ra _L≤8.27×10⁴ and a wide eccentricity range. The horizontal eccentricity of the inner cylinder substantially alters the thermal field and the geometry of the plume, but, in analogy to the behaviour for vertical eccentricity, the average Nu is slightly affected in the investigated range of eccentricity. The concentric geometry is also considered mainly to validate the experimental technique and evaluate the accuracy of the adopted methodology by comparison with available results. Both shearing interferometer and reference beam interferometer are obtained by means of Wollaston prisms with appropriate splitting angles, so that the temperature and local Nu distributions may be evaluated quantitatively from the original pictures via digital image processing.     

On the multi-scale analysis of strongly non-linear forced oscillators

We consider the resonant response of strongly non-linear oscillators of the form ü + 2ϵηu + mu + ϵƒ(u) = 2ϵpcosΩt, where ƒ(u) is an odd non-linearity, ϵ need not be small, and m = −1, 0, or + 1. Approximate solutions are obtained using a multiple-scale approach with two procedural steps which differ from the usual ones: (1) the detuning is introduced in the square of the excitation frequency Ω and as a deviation from the so called backbone curve and (2) a new expansion parameter α = α(ϵ) is defined, enabling accurate low order solutions to be obtained for the strongly non-linear case.     

NUMERICAL CALCULATION OF 2D,UNSTEADY FLOW IN CENTRIFUGAL PUMPS: IMPELLER AND VOLUTE INTERACTION

A numerical model is developed for calculating the two-dimensional, unsteady, incompressible and turbulent flow within the rotating impeller and stationary volute of an industrial centrifugal pump. The objective is the investigation and comprehension of the instantaneous behaviour of centrifugal pumps, aiming at the reduction of vibrations, radial forces and hydraulic noise. The computation is performed within a blade-to-blade streamtube for the impeller and a tube normal to the axis of rotation for the volute. The equations to be solved are the unsteady Reynolds-averaged Navier–Stokes equations along with the continuity equation and the unsteady κ–ϵ equations for turbulence modelling. The finite volume method is applied for space discretization and an implicit scheme for time discretization. A multidomain overlapping grid technique is used for matching together the relative flow field calculated within the rotating impeller and the absolute one calculated within the stationary volute. In this way the impeller and volute interaction is directly taken into account. The numerical model is validated for a centrifugal pump of N _q=32 under design flow conditions. Comparisons between calculation and measurements show fairly good agreement.     

Analytical solutions for finite cylinders compressed between platens and the strain effect on the valence-band structure of Si1−xGex alloy

This paper presents an analytical solution for inhomogeneous strain and stress distributions within finite circular cylinders of Si_1−xGe_x alloy under compression test with end friction. The method follows Lekhnitskii’s stress function approach, but a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. Numerical results show that the axial, radial, circumferential and shear strains are all inhomogeneous within finite cylinders, and local strain concentrations near two end surfaces were usually developed as long as friction exists between end surfaces and loading platens. Moreover, by using envelope-function method, the effect of strain on the valence-band structure of Si_1−xGe_x alloy is also studied. It was found that strain can induce band splitting, alteration of the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1−xGe_x alloy.     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Experimental and theoretical observations of elastic instabilities in eccentric cylinder flows: local versus global instability

A theoretical and experimental investigation of the stability of the viscoelastic flow of a Boger fluid between eccentric cylinders is presented. In our theoretical study, a local linear stability analysis for the flow of an Oldroyd-B fluid suggests that the flow is elastically unstable for all eccentricities. A global solution to the stability problem is obtained by a perturbation eigenvalue analysis, incorporating the azimuthal variation of the base state flow at the same order as the streamwise variation of the stability function. A comparison between the local and global stability predictions is made. Flow visualization experiments with a solution of high molecular weight polyisobutylene dissolved in a viscous solvent clearly show the transition from a purely azimuthal flow to a secondary toroidal flow. Comparison of these experimental results with the local linear stability theory shows good agreement between the measured and predicted critical conditions for the onset of the non-inertial cellular instability at small δ, where δ is the eccentricity made dimensionless with the average gap thickness. At higher eccentricities, experiment and local linear stability theory cease to agree. Evidence will be given that this disagreement is due to a global affect, i.e. the convection of stress not included the local theory. Specifically, it is suggested that convection of polymeric stresses in the base flow as well as in the disturbance flow can stabilize the instabilities found in this geometry. Finally, the discovery of a new localized purely elastic instability associated with the recirculation flow in the co-rotating eccentric cylinder geometry is presented.     

A purely elastic transition in Taylor-Couette flow

Experimental evidence of a non-inertial, cellular instability in the Taylor-Couette flow of a viscoelastic fluid is presented. A linear stability analysis for an Oldroyd-B fluid, which is successful in describing many features of the experimental fluid, predicts the critical Deborah number,De _c , at which the instability is observed. The dependence ofDe _c on the value of the dimensionless gap between the cylinders is also determined.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.