Steady flow of power-law fluids across an unconfined elliptical cylinder

The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the power-law index (0.2?n?1.8), Reynolds number (0.01?Re?40) and the aspect ratio of the elliptic cylinder (0.2?E?5) on the local and global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning (n<1) behaviour and the aspect ratio E>1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand, for shear-thickening (n>1) fluid behaviour and for E<1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms of closure relations for easy use in a new application.     

Finite element simulation of deep bed filtration

In this paper, the numerical model for separation efficiency and transport in periodic porous media is studied. Finite element method was used to simulate the development of a predictive model of behavior of porous media during injection of particles. This paper describes the effects of injected particle size, Reynolds number and particle drag coefficient. The numerical results show that the separation efficiency increased with injected particle size increase. The separation efficiency is found to increase with increasing Reynolds number. For the effect of drag force, C_D, in porous media, numerical results show that for C_D<10 and C_D>100, the separation efficiency is not affected by drag coefficient in the range of drag coefficient from 10 to 100, and the separation efficiency significantly depends on the Reynolds number.     

Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium

A finite element analysis is performed to investigate the effects of uniform and non-uniform heating of bottom wall on natural convection flows within isosceles triangular enclosures filled with porous medium. The detailed analysis is carried out in two cases depending on various thermal boundary conditions:

(I)

two inclined walls are maintained at constant cold temperature while the bottom wall is uniformly heated;

(II)

two inclined walls are maintained at constant cold temperature while the bottom wall is non-uniformly heated.

The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Darcy number, Da (10^-5?Da?10^-3), Rayleigh number, Ra (10³?Ra?10⁶) and Prandtl number, Pr (0.026?Pr?1000) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It is observed that at small Darcy numbers, the heat transfer is primarily due to conduction irrespective of Pr. As the Darcy number increases, there is a change from conduction dominant regime to convection dominant regime. Flow circulations are also found to be strong functions of Pr at large Da (Da=10^-3) and multiple circulation cells occur at small Pr with Ra=10⁶. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case, but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. As average Nusselt number is same on both the inclined walls, the average Nusselt number for bottom wall is times that of the inclined wall which is well matched in two cases considered for verifying the thermal equilibrium of the system. The correlations are proposed for average Nusselt number as functions of Ra for various Darcy and Prandtl numbers.     

Effect of power-law index on critical parameters for power-law flow across an unconfined circular cylinder

The conditions for the formation of a wake and for the onset of wake instability for the flow of power-law fluids over an unconfined circular cylinder are investigated numerically by solving the continuity and momentum equations using FLUENT (version 6.2). The effect of power-law index on the critical Reynolds numbers, Strouhal number and drag coefficient has been presented over a wide range of power-law index (0.3?n?1.8) thereby establishing the limits of the flow without separation and the steady symmetric flow regimes, respectively. While both the shear-thinning (n<1) and the shear-thickening (n>1) seem to lower the value of the critical Reynolds number denoting the onset of wake instability as compared to that for Newtonian fluids, the effect is seen to be more prominent for shear-thickening fluids than that for shear-thinning fluids. The corresponding values of the critical Strouhal number (St_c) and drag coefficient have also been presented for the critical values of the Reynolds number. Included here are also a series of streamline plots showing the onset of asymmetry and of the time-dependent flow regime.     

Two-dimensional unsteady flow of power-law fluids over a cylinder

The unsteady flow of incompressible power-law fluids over an unconfined circular cylinder in cross-flow arrangement has been studied numerically. The two-dimensional (2-D) field equations have been solved using a finite volume method based solver (FLUENT 6.3). In particular, the effects of the power-law index (0.4?n?1.8) and Reynolds number (40?Re?140) on the detailed kinematics of the flow (streamline, surface pressure and vorticity patterns) and on the macroscopic parameters (drag and lift coefficients, Strouhal number) are presented in detail. The periodic vortex shedding and the evolution of detailed kinematics with time are also presented to provide insights into the nature of flow. The two-dimensional flow transits from steady to unsteady behaviour at a critical value of the Reynolds number Re∼(40-50) and the von-Karman vortex street is observed beyond the critical Reynolds number (Re). Obviously, both the lift coefficient and Strouhal number values are zero for the steady flow, but their values increase with the increasing Reynolds number (Re) in the unsteady flow regime. For highly shear-thickening fluids (n=1.8), the flow becomes unsteady at Re=40 and unsteadiness in the flow appears at Re=50 for all values of power-law index (n). As expected, the evolution of the kinematics and vortex shedding show a complex dependence on the flow parameters near the transition in the flow. For a fixed value of the Reynolds number (Re), the drag coefficient increases and lift coefficient decreases with increasing value of the power-law index (n). For a fixed value of the power-law index (n), the drag coefficient gradually increases with the Reynolds number (Re). Similar to the drag coefficient, lift coefficient also shows a complex dependence on the power-law index (n) near the transition zone. The value of the Strouhal number (St) decreases with the increasing value of the power-law index (n) at a fixed value of the Reynolds number (Re).     

黏性液体中单个上升气泡的流体动力学行为(英文)

The rising behavior of single bubbles has been investigated in six systems with different viscosity and Morton number(Mo) from 3.21×10-11 to 163. Bubbles with maximum equivalent diameter of up to 16 mm were investigated. The bubble Reynolds number(Re) ranged from 0.02 to 1200 covering 3 regimes in which two func-tions are obtained relating the drag coefficient,CD,with Re and Mo. It has been found that in the high Reynolds number regime the drag coefficient increases until the Reynolds number of about 1200. The classic expression of Jamialahmadi(1994) is improved and extended to high viscosity liquids. A new relationship for the aspect ratio of deformed bubbles in terms of Re,the Etvs number and Mo,applicable to a wide range of system properties,espe-cially in high viscosity liquids,is also suggested.     

Experimental investigation of pressure-drop characteristics across multi-layer porous metal structures

This study investigates the effect of airflow (in the range of 0–70 m s^?1) on the pressure-drop characteristics for a novel multi-layered, nickel-based porous metal, as a function of thickness (affected by sectioning) and density (affected by compression). In addition to generating unique data for these materials, the study highlights the need for precise pinpointing of the different flow regimes (Darcy, Forchheimer and Turbulent) in order to enable accurate determination of the permeability (K) and form drag coefficient (C) defined by the Forchheimer equation and to understand the complex dependence of length-normalised pressure drop on sample thickness.     

A drag correlation of fluid particles rising through stagnant liquids in vertical pipes at intermediate Reynolds numbers

A drag correlation for a fluid particle rising along the axis of a vertical pipe at low and intermediate Reynolds numbers, Re, is proposed by making use of available correlations and a numerical database accumulated by interface tracking simulations. The accuracy of the interface tracking method has been verified through comparisons between measured and predicted velocities of single drops in vertical pipes. Being similar to drag model for solid spheres proposed by Michaelides, the developed drag correlation takes into account inertial and wall effects as their linear combination. The correlation gives good estimation of the drag coefficient for fluid particles rising through stagnant liquids in vertical pipes under the conditions of 0.13?Eo?30, −10.0?log M?2.0, 0.083?Re<200, 0?κ?10.0 and λ?0.6, where Eo is the Eötvös number, M the Morton number, κ the viscosity ratio and λ the ratio of particle diameter to pipe diameter.     

Wall effects in flow past a circular cylinder in a plane channel: a numerical study

This paper describes a numerical study on the steady flow of an incompressible Newtonian fluid past a circular cylinder confined in a plane rectangular channel. Using FLUENT (version 6), two-dimensional steady state computations were carried out for an uniform inlet velocity and for different values of the Reynolds numbers in the range between 0.1 and 200 and blockage ratios (ratio of the channel width to the cylinder diameter) in the range between 1.54 and 20. The flow parameters such as drag coefficient, length of the recirculation zone, and the angle of separation are presented as functions of the Reynolds number and blockage ratio. The total drag coefficient (C_D) was found to decrease with an increase in the blockage ratio (λ) for a fixed value of the Reynolds number (Re) and to decrease with increasing Reynolds number for a fixed value of λ. Similarly, for a fixed value of λ, both the angle of separation and the length of the recirculation zone increase with the increasing Reynolds number.     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Drag of non-spherical solid particles of regular and irregular shape

The drag of a non-spherical particle was reviewed and investigated for a variety of shapes (regular and irregular) and particle Reynolds numbers (Re_p). Point-force models for the trajectory-averaged drag were discussed for both the Stokes regime (Re_p ? 1) and Newton regime (Re_p ? 1 and sub-critical with approximately constant drag coefficient) for a particular particle shape. While exact solutions were often available for the Stokes regime, the Newton regime depended on: aspect ratio for spheroidal particles, surface area ratio for other regularly-shaped particles, and min-med-max area for irregularly shaped particles. The combination of the Stokes and Newton regimes were well integrated using a general method by Ganser (developed for isometric shapes and disks). In particular, a modified Clift-Gauvin expression was developed for particles with approximately cylindrical cross-sections relative to the flow, e.g. rods, prolate spheroids, and oblate spheroids with near-unity aspect ratios. However, particles with non-circular cross-sections exhibited a weaker dependence on Reynolds number, which is attributed to the more rapid transition to flow separation and turbulent boundary layer conditions. Their drag coefficient behavior was better represented by a modified Dallavalle drag model, by again integrating the Stokes and Newton regimes. This paper first discusses spherical particle drag and classification of particle shapes, followed by the main body which discusses drag in Stokes and Newton regimes and then combines these results for the intermediate regimes.     

PRESSURE DROP AND HOMOGENIZATION EFFICIENCY OF A MOTIONLESS MIXER

Pressure drop and homogenization efficiency of a motionless mixer of helical type have been studied experimentally. For evaluation of the drag coefficient the equation has been proposed which is valid within the range of Reynolds number from 10^?2 to 10⁴ The efficiency of the motionless mixer for mixing of two Newtonian liquids has been investigated by using a decolourization method. For the operating conditions studied in this work it appears that there is the worse performance of the mixer about the Reynolds number of 50. No influence of volume flow rate ratios (from 1 to 10) upon the performance of the mixer has been observed. A higher number of mixing elements must be applied for homogemzation of liquids with viscosity ratios above 100 as compared with that for viscosity ratio 1.     

DEVELOPMENT OF NEW EMPIRICAL EQUATIONS FOR ESTIMATION OF DRAG COEFFICIENT,SHAPE DEFORMATION,AND RISING VELOCITY OF GAS BUBBLES OR LIQUID DROPS

Some new correlations are derived to estimate the drag coefficient, the shape deformation, and the rising velocity of particles moving in an infinite liquid medium. The correlations are derived in terms of the dimensionless groups such as Reynolds number (Re), Morton number (Mo), and Weber number (We). The derivations are based on the experimental data or some other correlations given in the literature. A single statement is proposed to estimate the drag coefficient for the spherical solid particles that may be applicable in the range of 0.5 < Re < 10⁵. Similarly, some other equations are also derived to estimate the drag coefficient, the shape deformation, or the rising velocity for gas bubbles or liquid drops. The drag equation is applicable in the range of 0.5 < Re < 100 and 9 × 10^?7 ≤ Mo ≤ 7; the shape deformation equation is applicable in the range of 0.5 < Re < 100 and 1.1 × 10^?5 ≤ Mo ≤ 7; and the rising velocity equation is applicable in the range of 0.1 < Re < 100 and 9 × 10^?7 ≤ Mo ≤ 80. The model predictions are compared with the experimental data and with the other correlations given in the literature. The results indicated that the model predictions are in a good agreement with the literature data.     

Stagnation-point flows in a porous medium

In this paper, we present non-linear exact and asymptotic solutions to a Navier-Stokes equation of Brinkman type proposed by Joseph et al. (Water Resour. Res. 18(4) (1982) 1049) for the flow in the stagnation-point laminar boundary layer on a cylinder or sphere if fibers of increasing concentration are uniformly added to a porous medium surrounding these blunt bodies. Although one cannot perform a rigorous averaging of the (u·∇)u term, one is able to gain useful insight into the transition in behavior that occurs between the classical solutions of Hiemenz (Dinglers Polytech. J. 326 (1911) 321) and Homann (Z. Angew. Math. Mech. 16 (1936) ; Forsch. Geb. Ingenieurwes. 7 (1936) 1) for the two-dimensional and axisymmetric stagnation-point boundary layers and the local expansion of the Brinkman solution for the flow past a cylinder or sphere in the stagnation regions as the Darcy permeability is decreased. In this analysis, a new fundamental dimensionless parameter emerges, β=υ/KA, where A is the characteristic velocity gradient 4U /D imposed by the external flow, υ is the kinematic viscosity and K, the Darcy permeability. β denotes the ratio of the square of two lengths, the classical boundary layer thickness for a high Reynolds number flow and the fiber-interaction layer thickness K^1/2. The exact solutions of the non-linear Brinkman equation for the stagnation-point flow presented herein show the structure of a new type of boundary layer that evolves as β varies from zero, the classical limit of the Hiemenz and Homann solutions, to β?1, the classical Brinkman limit where inertial effects are negligible. Using asymptotic analysis we shall show that when β?1 the classical boundary layer thickness decreases as β^-1/2. Because of the introduction of the Darcy term, the pressure field differs greatly from the classical stagnation-point flow. The pressure does not increase monotonically along the stagnation streamline, and for β?1 there is a pressure minimum that approaches the origin as β^-1/2.     

Natural Convection in Heat Generating Oval Porous Enclosures: A Non‐Darcian Model

This paper presents a series of numerical simulations dealing with the problem of natural convection flows and associated heat transfer in an enclosure filled with a fluid‐saturated porous medium. The analysis is based on the finite element technique and incorporates the Brinkman‐extended Darcy model for an oval enclosure. The numerical results obtained for a modified Rayleigh number, Ra, Darcy number, Da, offset, E, and eccentricity, e, are presented and discussed. The numerical predictions for a square enclosure compared well with published data. It is found that any increase in Da or Ra results in a higher fluid velocity that is responsible for shifting the core of the flow. Moreover, at higher ovality (E = 0.5), asymmetric flow is observed even at the lower range of Rayleigh number (Ra ? 20), which may be attributed to the effect of curved isothermal wall.     

Impaction of spherical particles on cylinders at moderate Reynolds numbers

Inertial and interceptive impaction of spherical particles on circular cylinders was investigated theoretically. The particles were considered to be suspended in a fluid moving steadily through a random array of parallel cylinders. Fluid flow fields around the cylinders were obtained by numerically solving the Navier-Stokes Equations subject to Kuwabara's zero vorticity boundary condition. These solutions were subsequently utilized in calculating particle trajectories and impaction efficiencies. The latter are presented as functions of Reynolds number (0·2 ? Re_c ? 40), particle inertial parameter (0 ? P ? 1000), particle to cylinder size ratio (0·001 ? K ? 1·0) and cylinder concentration (10^?4 ? c ? 0·111).The impaction efficiencies and critical inertial parameters differ from earlier theoretical predictions. The discrepancies are primarily attributable to the inaccurate flow field representations used by previous authors. The agreement between Subramanyam and Kuloor's experimental work and present theory is satisfactory.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

Numerical Simulation for Particle Penetration Depth Distribution in Deep Bed Filtration

A two‐dimensional model has been developed to simulate particle penetration through porous media. The particle penetration depends on many parameters including the Reynolds number, particle drag coefficient, the ratio of the diameter of injected to filtered particles, fluid velocity, and pore size, etc. The numerical model for separation efficiency in periodic porous media was studied. Previous work has described the effects of injected particle size, Reynolds number and particle drag coefficient. In this study, the porous media flow is modeled (solution of the Navier‐Stokes equations) by using the finite element method, and the analysis is restricted to the case of two‐dimensional periodic porous media. The effects of these factors and particle depth distribution in porous media are investigated. It is noted that the results for the three Reynolds numbers 1, 16.56, and 100, are qualitatively similar, and about 40 % of particles are trapped in the top part of the filter.     

Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers

A numerical investigation has been carried out to obtain the steady state drag coefficients and flow patterns of a single Newtonian fluid sphere sedimenting in power-law liquids. A finite difference method based simplified marker and cell (SMAC) algorithm has been implemented on a staggered grid arrangement to solve the continuity and momentum equations. For both phases, the convective terms have been discretized using the quadratic upstream interpolation for convective kinematics (QUICK) scheme, and diffusive and non-Newtonian terms with central differencing scheme. An exponential transformation has been applied in the radial direction for the continuous phase computational domain. In order to ensure the accuracy of the solver, extensive validation has been carried out by comparing the present results with the existing literature values for a few limiting cases. Further, in this study the effects of the Reynolds number (Re_o), internal to external fluid characteristic viscosity ratio (k) and power-law index (n_o) on the continuous phase flow field, pressure drag (C_dp), friction drag (C_df) and total drag (C_D) coefficients have been analyzed over the range of parameters: 5?Re_o?500, 0.1?k?50 and 0.6?n_o?1.6. Based on numerical results obtained in this work, a simple correlation has been proposed for the total drag coefficient, which can be used to predict the rate of sedimentation of a fluid sphere in power-law liquids.     

A new expression for spherical aerosol drag in slip flow regime

A 3D simulation study for an incompressible slip flow around a spherical aerosol particle was performed. The full Navier–Stokes equations were solved and the velocity jump at the gas–particle interface was treated numerically by imposition of the slip boundary condition. Analytical solution to the Stokesian slip flow past a spherical particle was used as a benchmark for code verification, and excellent agreement was achieved. The simulation results showed that in addition to the Knudsen number, the Reynolds number affects the slip correction factor. Thus, the Cunningham-based slip corrections must be augmented by the inclusion of the effect of Reynolds number for application to Lagrangian tracking of fine particles. A new expression for the slip correction factor as a function of both Knudsen number and Reynolds number was developed. The particle total drag coefficient was also correlated against Re and Kn over the range of gas–particle relative speeds yielding the incompressible slip flow from the Stokesian regime up to the threshold of compressibility. Inclusion of gas slip on the particle surface enhances the accuracy of particle drag force prediction up to 40.9% in the range of 0.01<Kn<0.1 and 0.125<Re<20 compared to the no-slip continuum drag values.