Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C _d≡ F/(ηU) = k ₀ + k ₂ Re²+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)^n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k ₀ is the drag coefficient for creeping flows. The proportionality constant k ₂ depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k ₂ on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.     

Analytical solutions of hydromagnetic boundary-layer flow of a non-Newtonian power-law fluid past a continuously moving surface

Summary The boundary-layer flow of a power-law non–Newtonian fluid over a continuously moving surface in the presence of a magnetic field B(x) applied perpendicular to the surface has been investigated. An analytical solution is obtained and compared with the numerical solution of the resulting non linear ordinary differential equation. The effects of the Stewart number (N) and the power law-index (n) on the velocity profiles and the skin-friction are studied.     

Non-isothermal flow of a power law fluid past a rectangular obstacle (of aspect ratio 1 × 2) in a channel: Drag and heat transfer

A two-dimensional numerical study has been carried out to investigate the drag and Nusselt number characteristics under forced convection conditions between a streaming power law liquid and a rectangle (with its longer side aligned with the direction of flow) placed symmetrically between two solid walls. In particular, the values of the individual and total drag coefficients, and of the Nusselt number are obtained as functions of the flow behaviour index (1.4 ⩾ n ⩾ 0.5), of Reynolds number (5 ⩽ Re ⩽ 40) and of the Peclet number (5 ⩽ Pe ⩽ 400) for a fixed value of the blockage ratio (1/8). Within these ranges of kinematic and rheological conditions, the drag and Nusselt number show only fair to moderate deviation from the corresponding Newtonian values at the same values of the Reynolds and Peclet numbers. Qualitatively speaking, the shear-thinning behaviour (n < 1) augments the drag and heat transfer while the shear-thickening behaviour (n > 1) causes both the drag and heat transfer to drop below the corresponding Newtonian values. The power-law fluid behaviour does not seem to alter the streamline, isovorticity and isotherm plots in a significant manner, except for the fact that the shear-thinning behaviour not only delays the formation of a visible wake but the resulting wake is also somewhat shorter than that in a Newtonian fluid. The shear thickening, on the other hand, has exactly the opposite influence on wake formation.     

Steady non–Newtonian flow past a circular cylinder: a numerical study

Summary. The steady and incompressible flow of non–Newtonian fluids past a circular cylinder is investigated for power law indices n between 0.2 and 1.4, blockage ratios of 0.037, 0.082 and 0.164, and the Reynolds numbers Re of 1, 20 and 40, using a stream function/vorticity formulation. The governing field equations have been solved by using a second-order accurate finite difference method to determine the drag coefficient, wake length, separation angle and flow patterns, and to investigate their dependence on power law index, blockage ratio and Reynolds number. The results reported here provide fundamental knowledge on the dependence of engineering flow parameters on blockage ratio and power law index, and further show that the effects on stream line and iso-vorticity patterns which result from an increase in the blockage ratio are similar to those which result from a decrease in the power law index.     

Power-law fluids flow and heat transfer over two tandem square cylinders: effects of Reynolds number and power-law index

The low-Reynolds numbers free-stream flow of power-law fluids and forced convection heat transfer around a square cylinder and two square cylinders in a tandem arrangement are numerically investigated. In the single cylinder case, the power-law index and Reynolds numbers range from n = 0.7 ? 1.4 and Re = 60 ? 160 at Pr = 0.7. In the tandem case, the spacing between the cylinders is four widths of each cylinder side and the power-law index ranges from 0.7 to 1.6 at Re = 40, 100, 160 and Pr = 0.7. All simulations are performed with a finite volume code based on the SIMPLEC algorithm and a non-staggered grid. The effect of spatial resolution on the results is also studied for a single cylinder and tandem cylinders. The mean and instantaneous streamlines, vorticity and temperature contours, the global quantities such as pressure and friction coefficients, the rms lift and drag coefficients, Strouhal and Nusselt numbers are determined and discussed for various power-law indexes at different Reynolds numbers. A comparison between the results of the single cylinder case and the two cylinders in tandem arrangement shows that there are relatively similar results for the single cylinder and the upstream cylinder of the tandem case.     

Laminar flow of a non-Newtonian fluid in channels with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing flow of a non-Newtonian fluid confined in two large plates separated by a small distance of h, with the upper plate stationary while the lower plate is uniformly porous and moving in the x-direction with constant velocity, is derived by accounting for the order of magnitude of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The one-dimensional governing equation for a power-law fluid flow confined between parallel plates, with the upper plate is stationary and the bottom plate subjected to sudden acceleration with a constant velocity in the x-direction and uniformly porous, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient velocity and thus the overall transient velocity distribution. The effects of mass suction/injection at the porous bottom plate on the flow of non-Newtonian fluids are examined for various values of time and power-law index. The results obtained from the present analysis are compared with the data available in the literature.     

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Summary The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.Nomenclature h Dimensionless microrotation - j Micro-inertia density - K Dimensionless parameter of vortex viscosity - m Falkner-Skan power-law parameter - Re Reynolds number - T Temperature - u, v Fluid velocities in thex andy directions, respectively - U Free stream velocity - x Streamwise coordinate along the body surface - y Coordinate normal to the body surface Greek symbols Thermal diffusivity - Wedge angle parameter - Spin gradient viscosity - Pseudo-similarity variable - Vortex viscosity - Absolute viscosity of the fluid - v Kinematic viscosity - Dimensionless temperature - Density of the micropolar fluid - Angular velocity of micropolar fluid - Stream function     

Modelling with viscous and viscoplastic materials under combining and separating flow configurations

Combining and separating incompressible flow of Newtonian and inelastic Herschel–Bulkley fluids is studied numerically employing a semi-implicit Taylor–Galerkin pressure-correction algorithm, where steady solutions are obtained through a transient finite element procedure. The influence of inertia and fluid rheology is analysed on flow patterns, velocity fields and pressure drops for various flow configurations, with fixed geometric gap width that stimulates the merging and splitting in the flow. For Newtonian fluids and at larger levels of inertia, the appearance of vortices was observed, with an increase in velocity differences and pressure drops across the channel. In this case, the numerical procedure was verified with good agreement against previous numerical and experimental observations. To extend the consideration to non-Newtonian inelastic materials, the material rheological characteristics were approximated with the use of the Herschel–Bulkley fluid model, incorporating the Ostwald–de Waele power-law model and viscoplastic yield stress. Findings for unyielded power-law fluids reveal slight increase in the size of the vortices as power index (m) was decreased. Variation of the consistency index (k) shows strong influence on the streamline patterns with a rapid increase in the vortex formation as k was decreased. For Bingham model solutions, devoid of shear-thinning and increasing yield stress, a higher value of Reynolds number is required for equivalent levels of vortex formation; also one observes the appearance of yielded and unyielded regions. Under Herschel–Bulkley modelling, there was little change noted in the kinematics, but some was apparent in rheological response. Once more, observations reveal the tendency to eliminate vortices at larger yield stress levels, with the appearance of unyielded regions.     

MHD Flow past a circular cylinder – a numerical study

 The steady, two-dimensional, incompressible MHD flow past a circular cylinder with an applied magnetic field parallel to the main flow is studied using the finite difference method. The magnetic Reynolds number is assumed to be small. Results are presented up to the Reynolds number R=500 and interaction parameter N=1.3. As N increases suppression of the separation is observed. Drag coefficient is decreasing for the small values of N and then increasing as N increases. It is found that a smaller value of far field distance is required as N increases. Received 6 March 2000     

Flow of power-law fluids in cone-plate viscometer

Summary The flow of a power-law fluid in a cone-plate viscometer has been considered. The first order solutions for velocity and pressure have been obtained. Effects of secondary flow on velocity components, pressure, rate of deformation tensors and apparent viscosity have been studied. Both the primary and secondary flows are influenced by flow behavior index n. In shear thinning fluids, the effects on both the primary and secondary flows are of similar nature, whereas in shear thickening fluids, they are of opposite nature. The net result is that for shear thinning fluids the effects of the primary and secondary flows are additive, whereas in shear thickening fluids they almost balance out each other and the resultant effect is insignificant. The other important observation is in Newtonian fluids the secondary flow makes a significant contribution at high value of Reynolds number, which depends on the radius, velocity, density, and viscosity. In power-law fluids, Reynolds number depends on an additional parameter n. With this change, the effect of secondary flows can be important even at low shear rates. A critical value of n is obtained at which flow is independent of shear rates.     

Peristaltic pumping of a micropolar fluid in a tube

Summary.  Peristaltic transport of a micropolar fluid in a circular tube is studied under low Reynolds number and long wavelength approximations. The closed form solutions are obtained for velocity, microrotation components, as well as the stream function and they contain new additional parameters namely, N the coupling number and m the micropolar parameter. In the case of free pumping (pressure difference Δp=0) the difference in pumping flux is observed to be very small for Newtonian and micropolar fluids but in the case of pumping (Δp>0) the characteristics are significantly altered for different N and m. It is observed that the peristalsis in micropolar fluids works as a pump against a greater pressure rise compared with a Newtonian fluid. Streamline patterns which depict trapping phenomena are presented for different parameter ranges. The limit on the trapping of the center streamline is obtained. The effects of N and m on friction force for different Δp are discussed. Received June 20, 2002; revised October 23, 2002 Published online: April 17, 2003 The authors thank the referees for pointing out some mistakes in the governing equations and for the suggestions to improve the presentation of the paper.     

Steady magnetohydrodynamic flow past a circular cylinder

The steady two-dimensional, viscous, electrically conducting flow around a circular cylinder is investigated. The flow and magnetic field are uniform and parallel at large distances from the cylinder. The equations and boundary conditions are derived for arbitrary values of R, R_mand β, where R is the Reynolds number, R_m the magnetic Reynolds number and β, the ratio of the square of the Alfvén speed to the square of the main stream speed. Because of the large number of parameters involved, the numerical solution is restricted to R = 40, R_m = 1 and infinity and 0 ? β ? 4. Also the cylinder is taken to be a perfect conductor, this avoids having to compute the magnetic field within the cylinder. The numerical computations for the non-magnetic case, i.e. β = 0, are presented and are found to be in good agreement with existing results. The effect of increasing the strength of the magnetic field (i.e. increasing β) on the drag coefficient, the size and position of the standing vortex and the increasing effect of the upstream propagation of disturbances are examined.     

Utilization of nonlinear diffusion solutions forn-diffusion and power-law materials

Summary The equation ofn-diffusion, as originally formulated by J. R. Philip for various problems involving unsteady turbulent flows, applies directly to the one-dimensional flow of a non-Newtonian power-law fluid as well as indirectly to the laminar boundary layer flow of such fluids over a flat plate. The purpose of this note is to utilize a simple relation between one-dimensionaln-diffusion and nonlinear diffusion with power-law diffusivity such that ifp(x, t) denotes ann-diffusion pressure thenc(x, t)=|p/x| satisfies the nonlinear diffusion equation with power law diffusivity. This means in particular that the large number of solutions presently known for nonlinear diffusion can be utilized in the context ofn-diffusion. Known solutions ofn-diffusion are obtained via this procedure as well as newn-diffusion solutions, including the source solution and a solution of the problem of fluid withdrawal at a constant flow rate for a non-Newtonian fluid in a porous medium of infinite extent. Solutions arising from other exact nonlinear diffusion profiles are also investigated as well as the limiting case ofn-diffusion forn tending to infinity and the results are displayed graphically.     

An Arbitrary Lagrangian-Eulerian finite element method

This paper describes an Arbitrary Lagrangian- Eulerian (ALE) finite element method for the simulation of fluid domains with moving structures. The fluid is viscous, incompressible and unsteady and the fluid motion is solved by a fractional step discretization of the Navier-Stokes equations. The emphasis is on convection dominated flows, and a three-step method is used for the convection term. The moving structure causes the mesh of the fluid domain to move, and a new algorithm is proposed to solve the important and crucial problem of the calculation of the mesh velocities. Numerical calculations of the added mass and added damping of a vibrating two-dimensional circular cylinder in the frequency Reynolds number range Re _w =20−2000 are performed to evaluate the proposed ALE finite element method. The numerically calculated added mass and added damping are compared to both analytical and numerical results. To further demonstrate the generality of the method, a numerical simulation of flow past an oscillating schematic sports car is presented.     

Low Reynolds number non-Newtonian flow in slowly varying axisymmetrical tubes

Summary Low Reynolds number flow of a non-Newtonian fluid through an axisymmetric tube whose radius varies slowly in the axial direction is analysed by asymptotic approximations. Expressions for the pressure drop along the tube and shear stress at the wall are obtained and compared with the Newtonian results. Some of the highlights of the investigations are (i) for a tube of exponentially increasing radius flow separation for R 3 is only possible at sufficiently long way down the tube while the low Reynolds number solutions do not break down when R 1 and (ii) for a locally constricted tube the Reynolds number for flow separation is less than its Newtonian value when 4K ₂>17K ₁. Here is a small parameter characterizing the radius variation,R the Reynolds number andK ₁ andK ₂ the visco-elastic and cross viscosity parameters respectively.With 1 Figure     

Numerical simulation of the motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow using DR-BEM

The motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow is studied using a boundary element numerical simulation. Non-linear effects from the dependency of the viscosity on the velocity field are treated in an implicit manner and the resultant domain integral is transformed into an equivalent series of boundary integrals using the Dual Reciprocity Method. The non-homogeneous (non-linear) system of algebraic equations resulting from the discretization of the boundary element formulation is solved using a modified Newton–Raphson method for drops with values of the power law index of $n=0.8$ and $0.6$ and compared to the corresponding Newtonian cases ($n=1$). The viscosity of the fluid inside the drop follows the truncated power law model. By using this model, the shear-thinning behaviour of the viscosity is correctly represented while avoiding the shear thickening which can be observed using the standard power law in small gradient flows. The simulations showed that the non-Newtonian drops had larger deformations than the corresponding Newtonian drops due to a general decrease in the viscosity. The value of the local viscosities was found to be dependant not only on the velocity field created by the motion of the internal cylinder, but strongly dependant on the surface tension forces. The rate of deformation of the drops was greater in the beginning of the simulation and decreased toward the end showing the drops found a more or less stable shape.     

Separation characteristics of fluid flow inside two parallel plates with wavy surface

Separation characteristics of fluid flow inside two parallel wavy plates for steady-laminar flow is investigated numerically in the present study. Governing equations are discretized using control volume based finite-volume method with collocated variable arrangement. SIMPLE algorithm is used and SIP solver is applied for solution of system of equations. Effect of surface waviness (defined by amplitude to average interwall spacing ratio, a/H) and aspect ratio (defined by wavelength to average interwall spacing ratio, w/H) on separation characteristics of fluid flow is presented. The present work has been carried out for surface waviness a/H=0-0.3, aspect ratio w/H=1.5-2.25. A critical Reynolds number (Re_c) is used to identify the appearance of first separation of fluid flow in the channel. Critical Reynolds (Re_c) number is calculated for wide range of surface waviness and aspect ratio. The structure of separation bubble depends strongly on waviness of the surface and aspect ratio for a particular Reynolds number and changes little with wave number (n). Finally pressure drop characteristics is presented in terms of average friction factor as a function of Reynolds number.     

Dependence of flow characteristics of rectangular cylinders near a wall on the incident velocity

Numerically simulated results are presented for a family of rectangular cylinders with aspect ratios r ₁ (=b/a with height a and width b) ranging from 0.1 to 1.0 (square cylinder) to gain a better insight into the dependency of the aerodynamic characteristics on the operational dimensionless parameters, namely Reynolds number Re and aspect ratio r ₁. This work describes the flow from a long cylinder of rectangular cross-section placed parallel to a wall and subjected to a uniform shear flow. The flow is investigated in the laminar Reynolds number range (based on the incident stream at the cylinder upstream face and the height of the cylinder) at cylinder to wall gap height 0.5 times the cylinder height. The governing unsteady Navier?CStokes equations are solved numerically through a finite volume method on a staggered grid system using QUICK scheme for convective terms. The resulting equations are then solved by an implicit, time-marching, pressure correction-based SIMPLE algorithm for Reynolds number up to 1,000. The critical Reynolds numbers at which vortex shedding from the cylinder is started are specified for both the cases: far from the wall and near to the wall. It is reported that the vortex shedding from the rectangular cylinder of lower aspect ratio r ₁ (???0.25) becomes regular and insensitive to the Reynolds number, while the aerodynamic characteristics of the rectangular cylinders with higher aspect ratio r ₁ (???0.5) are strongly dependent on the Reynolds number.     

Existence of extra vortex and twin vortex of anomalous mode in Taylor vortex flow with a small aspect ratio

Summary This experimental work on Taylor vortex flow in a gap with a small aspect ratio is concerned with two extra vortices and a twin vortex system, each of which depends on an anomalous cell of the anomalous mode. Extra vortices are smaller than other vortices such as defined cells. At any Reynolds number and aspect ratio extra vortices can be found at the corner of the end plate and inner rotating cylinder and at the corner of the end plate and outer stationary cylinder. For a one-cell flow (anomalous one-cell mode) in a symmetric system, an outer extra vortex develops and grows to the same size as the main cell, only in an aspect ratio of less than one. A twin vortex is observed to form when two vortices are aligned in the direction of the radius. There are three flow fields on the end plate; two are extra vortex flows and the other is the main cell flow. The flow direction of the anomalous cell is from the inner cylinder to the outer one, at the end plate opposite of the flow direction of the normal cell.Nomenclature R ₁ Radius of inner cylinder (2R ₁=40.19±0.006 mm) - R ₂ Radius of outer cylinder (2R ₂=60.11±0.024 mm) - R _r Radius ratio (R ₁/R ₂=0.669) - d Clearance between cylinders (R ₂–R ₁=9.96±0.025 mm) - L Height of working fluid - Aspect ratio=L/d - Rotational angular speed - Kinematic viscosity - Re Reynolds number=R ₁ d/ Other nomenclature is defined as it appears     

Vortex shedding suppression for laminar flow past a square cylinder near a plane wall: a two-dimensional analysis

Summary A numerical study on the uniform shear flow past a long cylinder of square cross-section placed parallel to a plane wall has been made. The cylinder is considered to be within the boundary layer of the wall. The maximum gap between the plane wall to the cylinder is taken to be 0.25 times the cylinder height. We investigated the flow when the regular vortex shedding from the cylinder is suppressed. The governing unsteady Navier-Stokes equations are discretized through the finite volume method on staggered grid system. A pressure correction based iterative algorithm, SIMPLER, has been used to compute the discretised equations iteratively. We found that the critical value of the gap height for which vortex shedding is suppressed depends on the Reynolds number, which is based on the height of the cylinder and the incident stream at the surface of the cylinder. At high Reynolds number (Re ≥ 500) however, a single row of negative vortices occurs for wall to cylinder gap height L ≥ 0.2. The shear layer that emerges from the bottom face of the cylinder reattaches to the cylinder itself at this gap hight.