Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

用电化学方法测试动脉模型壁面剪应力

应用电化学方法,对动脉模型Ｔ型分叉部位流场壁面剪应力进行测试研究。测试了对于现有理论分析和数值计算都比较困难的高雷诺数（ＲＥ＝１０００－２０００）流动流场的壁面剪应力,并且对苦干不同雷诺数及不同支管分流情况进行了系列测试。通过实验发现,此部痊同时存在高剪应力区和低剪应力区,确定了它们的位置和剪应力的大上。系列测试还显示：随着雷诺数的变化,无量纲管应力有一定的变化;而当支管分流变化时,无量纲剪应力的     

Drag of a strongly heated sphere at small Reynolds numbers

Using an asymptotic small-perturbation method, the flow around a strongly heated sphere at small Reynolds numbers Re ≪ 1 is considered with account for thermal stresses in the gas in the higher-order approximations, beyond the Stokes one. It is assumed that the value of the Prandtl number Pr is arbitrary and the temperature dependence of the viscosity is described by a power law with an arbitrary exponent. In the O(Re²) and O(Re³ ln(Re)) approximations, the drag force of a heated sphere is found over a wide range of the ratios of sphere’s temperature to the gas free-stream temperature T _W /T _∞. The limits of applicability of the first (in Re) approximation are investigated, including the negative-drag effect, attributable to the action of the thermal stresses. The results are compared with numerical calculations of the flow around a hot sphere. The limits of applicability of the approximations found are examined. Similar results are obtained for the standard Navier-Stokes equations in which the thermal stresses are neglected.     

Some Reynolds number effects on two- and three-dimensional turbulent boundary layers

Experimental data for a two-dimensional (2-D) turbulent boundary layer (TBL) flow and a three-dimensional (3-D) pressure-driven TBL flow outside of a wing/body junction were obtained for an approach Reynolds number based on momentum thickness of Re _θ =23,200. The wing shape had a 3:2 elliptical nose, NACA 0020 profiled tail, and was mounted on a flat wall. Some Reynolds number effects are examined using fine spatial resolution (Δy ⁺=1.8) three-velocity-component laser-Doppler velocimeter measurements of mean velocities and Reynolds stresses at nine stations for Re _θ =23,200 and previously reported data for a much thinner boundary layer at Re _θ =5,940 for the same wing shape. In the 3-D boundary layers, while the stress profiles vary considerably along the flow due to deceleration, acceleration, and skewing, profiles of the parameter correlate well and over available Reynolds numbers. The measured static pressure variations on the flat wall are similar for the two Reynolds numbers, so the vorticity flux and the measured mean velocities scaled on wall variables agree closely near the wall. The stresses vary similarly for both cases, but with higher values in the outer region of the higher Re _θ case. The outer layer turbulence in the thicker high Reynolds number case behaves similarly to a rapid distortion of the flow, since stream-wise vortical effects from the wall have not diffused completely through the boundary layer at all measurement stations. Received: 9 June 2000/Accepted: 26 January 2001     

Leading edge film cooling heat transfer with high free stream turbulence using a transient liquid crystal image method

This paper studies film effectiveness and heat transfer coefficients on a large scale symmetric circular leading edge with three rows of film holes. The film hole configuration focuses on a smaller injection angle of 20° and a larger hole pitch with respect to the hole diameter (P/d=7.86). The study includes four blowing ratios (M=1.0,1.5,2.0 and 2.5), two Reynolds numbers (Re=30,000 and 60,000), and two free stream turbulence levels (nominally Tu=1% and 20% depending on the Reynolds number). A transient liquid crystal (LC) image technique is employed to obtain the film effectiveness and heat transfer coefficient distributions with high spatial resolutions of 0.6 mm in both streamwise and spanwise directions. Results are presented for detailed and spanwise averaged values of film effectiveness and Frössling number. Turbulence intensity has an attenuation on film effectiveness as well as on Frössling number for all blowing ratios at Re=30,000. Under high turbulence conditions the film effectiveness and Frössling number increase as blowing ratio increases from 1.0 to 2.0 for both Reynolds numbers. Further increasing the blowing ratio results in reverse effect. Increasing the Reynolds number from 30,000 to 60,000 results in increases in both the film effectiveness and Frössling number at high turbulence except for M=2.5. The blowing ratio of two shows a spatial coupling of the stagnation row of film holes with the second row (21.5°) of film holes which results in the highest film effectiveness and also the highest Frössling numbers.     

A three-dimensional simulation of a steady approach flow past a circular cylinder at low Reynolds number

The three-dimensional (3D) unsteady viscous wake of a circular cylinder exposed to a steady approach flow is calculated using a fractional-step finite-difference/spectral-element method. The calculated flow fields at Reynolds numbers of 100 (2D) and 200 (3D) are examined in detail. The flow field at Re = 100 is 2D as expected, while the flow field at Re = 200 has distinct 3D features, with spanwise wavelengths of about 3.75 cylinder diameters. The calculated results produce drag and lift coefficients and Strouhal numbers that agree extremely well with the experimental values. These 3D values at Re = 200 are in better agreement with experimental values than the results of a 2D calculation at Re = 200, which is expected. © 1998 John Wiley & Sons, Ltd.     

Simulations of Turbulent Flow in a Plane Asymmetric Diffuser

Large-eddy simulations (LES) of a planar, asymmetric diffuser flow have been performed. The diverging angle of the inclined wall of the diffuser is chosen as 8.5°, a case for which recent experimental data are available. Reasonable agreement between the LES and the experiments is obtained. The numerical method is further validated for diffuser flow with the diffuser wall inclined at a diverging angle of 10°, which has served as a test case for a number of experimental as well as numerical studies in the literature (LES, RANS). For the present results, the subgrid-scale stresses have been closed using the dynamic Smagorinsky model. A resolution study has been performed, highlighting the disparity of the relevant temporal and spatial scales and thus the sensitivity of the simulation results to the specific numerical grids used. The effect of different Reynolds numbers of the inflowing, fully turbulent channel flow has been studied, in particular, Re _b  = 4,500, Re _b  = 9,000 and Re _b  = 20,000 with Re _b being the Reynolds number based on the bulk velocity and channel half width. The results consistently show that by increasing the Reynolds number a clear trend towards a larger separated region is evident; at least for the studied, comparably low Reynolds-number regime. It is further shown that the small separated region occurring at the diffuser throat shows the opposite behaviour as the main separation region, i.e. the flow is separating less with higher Re _b . Moreover, the influence of the Reynolds number on the internal layer occurring at the non-inclined wall described in a recent study has also been assessed. It can be concluded that this region close to the upper, straight wall, is more distinct for larger Re _b . Additionally, the influence of temporal correlations arising from the commonly used periodic turbulent channel flow as inflow condition (similar to a precursor simulation) for the diffuser is assessed.     

Thermal-hydraulic performance of oval tubes in a cross-flow of air

The thermal-hydraulic performance of five oval tubes is experimentally investigated and compared with that for a circular tube in a cross-flow of air. The range of Reynolds numbers Re_D is approximately between 1,000 and 11,000. The nominal axis ratios R (major axis/minor axis) for three of the investigated oval tubes are 2, 3, and 4. Two other configurations of oval tubes are also tested, an oval tube R=3 with two wires soldered on its upper and lower top positions, and a cut-oval tube. The performance of the tubes is corrected for the effects of area blockage and turbulence intensity. The measurement results show that the mean Nusselt numbers Nu_D for the oval tubes are close to that for the circular tube for Re_D<4,000. For a higher Re_D, the Nu_D for the oval tubes is lower than that for the circular tube and it decreases with the increase in the axis ratio R. The drag coefficients C_d for the tubes are measured and the combined thermal-hydraulic performance is indicated by the ratio Nu_D/C_d, which shows a better combined performance for the oval tubes.     

A note on the numerical simulations of flow past a wavy square-section cylinder

The flow past a square-section cylinder with a geometric disturbance is investigated by numerical simulations. The extra terms, due to the introduction of mapping transformation simulating the effect of disturbance into the transformed Navier-Stokes equations, are correctly derived, and the incorrect ones in the previous literature are pointed out and analyzed. Furthermore, the relationship between the vorticity, especially on the cylinder surface, and the disturbance is derived and explained theoretically. The computations are performed at two Reynolds numbers of 100 and 180 and three amplitudes of waviness of 0.006, 0.025 and 0.167 with another aim to explore the effects of different Reynolds numbers and disturbance on the vortex dynamics in the wake and forces on the body. Numerical results have shown that, at the mild waviness of 0.025, the Kairmain vortex shedding is suppressed completely for Re = 100, while the forced vortex dislocation is appeared in the near wake at the Reynolds number of 180. The drag reduction is up to 21.6% at Re = 100 and 25.7% at Re = 180 for the high waviness of 0.167 compared with the non-wavy cylinder. The lift and the Strouhal number varied with different Reynolds numbers and the wave steepness are also obtained.     

Aerodynamic loading on a cylinder behind an airfoil

The interaction between the wake of a rotor blade and a downstream cylinder holds the key to the understanding and control of electronic cooling fan noise. In this paper, the aerodynamic characteristics of a circular cylinder are experimentally studied in the presence of an upstream NACA 4412 airfoil for the cylinder-diameter-based Reynolds numbers of Re_d=2,100–20,000, and the airfoil chord-length-based Reynolds numbers of Re_c=14,700–140,000. Lift and drag fluctuations on the cylinder, and the longitudinal velocity fluctuations of the flow behind the cylinder were measured simultaneously using a load cell and two hot wires, respectively. Data analysis shows that unsteady forces on the cylinder increase significantly in the presence of the airfoil wake. The dependence of the forces on two parameters is investigated, that is, the lateral distance (T) between the airfoil and the cylinder, and the Reynolds number. The forces decline quickly as T increases. For Re_c<60,000, the vortices shed from the upstream airfoil make a major contribution to the unsteady forces on the cylinder compared to the vortex shedding from the cylinder itself. For Re_c>60,000, no vortices are generated from the airfoil, and the fluctuating forces on the cylinder are caused by its own vortex shedding.     

Numerical study of the effects occurring near a circular cylinder in stratified fluid flows with short buoyancy periods

Flow around a two-dimensional circular cylinder of a stratified fluid with periods buoyancy T_b = 25.2 and 6.28 sec is studied numerically over a wide range of Reynolds and Froude numbers. It is found that in the presence of a perturbation ahead of a cylinder which moves downstream with increasing Reynolds number, the salinity isolines have the shape of a semi-circular comb with sharp teeth. The shape change of the attached waves and the occurrence of fluid layers of different densities in the cylinder wake are studied. In flows with a buoyancy period T_b = 6.28 sec at Reynolds numbers Re < 60, stagnant zones are found in the cylinder wake, and at Re > 60, these zones are absent.     

Vortex shedding in cylinder flow of shear-thinning fluids: I. Identification and demarcation of flow regimes

An experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers. The measurements of vortex shedding frequency and formation length (l_f) were carried out by laser-Doppler anemometry in Newtonian fluids and in aqueous polymer solutions of CMC and tylose. These were shear thinning and elastic at weight concentrations ranging from 0.1 to 0.6%. The 10 and 20 mm diameter cylinders (D) used in the experiments had aspect ratios of 12 and 6 and blockage ratios of 5 and 10%, respectively. The Reynolds number (Re^*) was based on a characteristic shear rate of U_∞/(2D) and ranged from 50 to 9×10³ thus encompassing the laminar shedding, the transition and shear-layer transition regimes. Increasing fluid elasticity reduced the various critical Reynolds numbers (Re_etr^*, Re_lf^*, Re_bbp^*) and narrowed the extent of the transition regime. For the 0.6% tylose solution the transition regime was even suppressed. On the other end, pseudoplasticity was found to be indirectly responsible for the observed reduction in Re_otr^*: it increases the Strouhal number which in turn increases the vortex filaments, precursors of the transition regime. Elasticity was better quantified by the elasticity number Re′/We than by the Weissenberg number. This elasticity number involves the calculation of the viscosity at a high characteristic shear rate, typical of the boundary layer, rather than at the average value (U_∞/(2D)) used for the Reynolds number, Re^*.     

Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity

A numerical study of the three-dimensional fluid flow has been carried out to determine the effects of the transverse aspect ratio, Ay, on the flow structure in two-sided non-facing lid-driven cavities. The flow is complex, unstable and can undergo bifurcation. The numerical method is based on the finite volume method and multigrid acceleration. Computations have been investigated for several Reynolds numbers and various aspect ratio values. At a fixed Reynolds number, Re=500, the three-dimensional flow characteristics are analyzed considering four transverse aspect ratios, Ay=1,0.75,0.5 and 0.25. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. An analysis of the flow evolution shows that, at Ay=0.75, the flow bifurcates to a periodic regime at (Re=600) with a frequency f=0.093 less than the predicted value in the cubical cavity. A correlation is established when Ay=0.5 and gives the critical Reynolds number value. At Ay=0.25, the periodic regime occurs at high Re value beyond 3500, after which the flow becomes chaotic. It is shown that, when increasing Ay over the unit, the flow in the cavity exhibits a complex behavior. The kinetic energy transmission from the driven walls to the cavity center is reduced at low Ay values.     

Numerical simulations of laminar flow over a 3D backward-facing step

A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.     

PIV–LES analysis of channel flow rotating about the streamwise axis

The flow field of a channel rotating about the streamwise axis is analyzed experimentally and numerically. The current investigations were carried out at a bulk velocity based Reynolds number of Re_m = 2850 and a friction velocity based Reynolds number of Re_τ = 180, respectively. Particle-image velocimetry (PIV) measurements are compared with large-eddy simulation data to show earlier direct numerical simulation findings to generate too large a reverse flow region in the center region of the spanwise flow. The development of the mean spanwise velocity distribution and the influence of the rotation on the turbulent properties, i.e., the Reynolds stresses and the two-point correlations of the flow, are confirmed in both investigations. The rotation primarily influences those components of the Reynolds shear stresses, which contain the spanwise velocity component. The size of the correlation areas and thus the length scales of the flow generally grow in all three coordinate directions leading to longer structures. Furthermore, experimental results of the same channel flow at a significantly lower bulk Reynolds number of Re_{m, l} = 665, i.e., a laminar flow in a non-rotating channel, are introduced. The experiments show the low Reynolds number flow to become turbulent under rotation and to develop the same characteristics as the high Reynolds number flow.     

Numerical investigation of turbulent free jet flows issuing from rectangular nozzles: the influence of small aspect ratio

In this research, the fluid and thermal characteristics of a rectangular turbulent jet flow is studied numerically. The results of three-dimensional jet issued from a rectangular nozzle are presented. A numerical method employing control volume approach with collocated grid arrangement was employed. Velocity and pressure fields are coupled with SIMPLEC algorithm. The turbulent stresses are approximated using k–e{\varepsilon} model with two different inlet conditions. The velocity and temperature fields are presented and the rates of their decay at the jet centerline are noted. The velocity vectors of the main flow and the secondary flow are illustrated. Also, effect of aspect ratio on mixing in rectangular cross-section jets is considered. The aspect ratios that were considered for this work were 1:1 to 1:4. The results showed that the jet entrains more with smaller AR. Special attention has been drawn to the influence of the Reynolds number (based on hydraulic diameter) as well as the inflow conditions on the evolution of the rectangular jet. An influence on the jet evolution is found for smaller Re, but the jet is close to a converged state for higher Reynolds numbers. The inflow conditions have considerable influence on the jet characteristics.     

Numerical simulation of two‐dimensional laminar slot‐jet impingement flows confined by a parallel wall

Two‐dimensional laminar incompressible impinging slot‐jet is simulated numerically to gain insight into flow characteristics.Computations are done for vertically downward‐directed slot‐jets impinging on a plate at the bottom and confined by a parallel surface on top. The behaviour of the jet with respect to aspect ratio (AR) and Reynolds number (Re) are described in detail. The computed flow patterns for various AR (2–5) and for a range of jet‐exit Reynolds numbers (100–500) are analysed to understand the flow characteristics. The transient development of the flow is also simulated for AR = 4 and Re = 300. It is found that the reattachment length is dependent on both AR and Reynolds number for the range considered. The correlation for reattachment length is suggested. The maximum resultant velocity V_rmax and its trajectory is reported. A detailed study of horizontal velocity profile at different downstream locations is reported. It is found that the effect of Reynolds number and AR is significant to the bottom wall vorticity in the impingement and wall jet regions. Copyright © 2007 John Wiley & Sons, Ltd.     

The effect of Reynolds number on a turbulent far-wake

The turbulent far-waked generated by a circular cylinder is investigated for two values (1350 and 4600) of the Reynolds number Re _θ (based on the free stream velocity and the momentum thickness). Two arrays of sixteen X-wires, eight in the (x,?y)-plane and eight in the (x,?z)-plane, are used to capture the main features of the large-scale motion in two orthogonal planes. Both the magnitude of the measured Reynolds stresses and the size of the two-point velocity and vorticity correlation contours increase with Reynolds number. The probability density function and spectra of the velocity signals also exhibit differences with Re _θ. A comparison of centerline turbulence intensities with those in the literature suggests that the Reynolds number dependence may disappear for Re _θ?5000.     

Dynamics of dual-particles settling under gravity

As a first step towards understanding particle–particle interaction in fluid flows, the motion of two spherical particles settling in close proximity under gravity in Newtonian fluids was investigated experimentally for particle Reynolds numbers ranging from 0.01 to 2000. It was observed that particles repel each other for Re>0.1 and that the separation distance of settling particles is Reynolds number dependent. At lower Reynolds numbers, i.e. for Re<0.1, particles settling under gravity do not separate.The orientation preference of two spherical particles was found to be Reynolds number dependent. At higher Reynolds numbers, the line connecting the centres of the two particles is always horizontal, regardless of the way the two particles are launched. At lower Reynolds numbers, however, the particle centreline tends to tilt to an arbitrary angle, even of the two particles are launched in the horizontal plane. Because of the tilt, a side migration of the two particles was found to exist. A linear theory was developed to estimate the side migration velocity. It was found that the maximum side migration velocity is approximately 6% of the vertical settling velocity, in good agreement with the experimental results.Counter-rotating spinning of the two particles was observed and measured in the range of Re=0–10. Using the linear model, it is possible to estimate the influence of the tilt angle on the rate of rotation at low Reynolds numbers. Dual particles settle faster than a single particle at small Reynolds numbers but not at higher Reynolds numbers, because of particle separation. The variation of particle settling velocity with Reynolds number is presented. An equation which can be used to estimate the influence of tilt angle on particle settling velocity at low Reynolds number is also derived.     

On the effect of flow direction on mixed convection from a horizontal cylinder

The influence of free stream direction on mixed (natural and forced) convective heat transfer from a circular cylinder is investigated. The cylinder, which has an isothermal surface, is placed with its axis horizontal and normal to the oncoming flow. The free stream direction varies between the vertically upward (parallel flow) and the vertically downward (contraflow) directions. The investigation is based on the time integration of the unsteady, two-dimensional equations of motion and energy until reaching steady conditions. The study is limited to Reynolds numbers up to Re = 40 and Grashoff numbers of Gr = Re². The results are compared with the available experimental data and the agreement is satisfactory.