Effect of Reynolds number on flow and heat transfer in incompressible forced convection over a 3D backward-facing step

A three-dimensional incompressible numerical model for the case of the 3D backward-facing step flow is established to investigate the characteristics of fluid flow and heat transfer in the low and middle Reynolds number ranges (200 ≤ Re≤1400). The governing equations, including continuous, unsteady Navier–Stokes and energy equations, are solved by the finite volume method in FLUENT. The simulation results show that the time averaged reattachment length reaches the peak value at Re = 1000, and subsequently decreases as the increase of Re. The formation of secondary peak Nu influenced by flow instability has a better contribution to the heat transfer at the center area. Taking away the hot fluid and carrying the cold fluid into the floor wall, which is caused by periodic instability, have positive effects on heat transfer enhancement.     

The flow near the edge of a disc at rest in a rotating fluid

Summary The Reynolds number Re being based on the angular velocity of the fluid and the radius of the disc, it is shown that within a distance O(Re ^-2/3) from the edge of the disc, the flow is determined by the Navier-Stokes equations. The boundary-value problem describing this flow is formulated. The asymptotic behaviour of its solution is investigated analytically and its complete numerical solution is evaluated. Results for various physical quantities, among them the additional torque due to the Navier-Stokes flow, are presented.     

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.     

A formulation for near-wall RANS/LES coupling

A near-wall eddy-viscosity formulation for LES is presented. A RANS-like eddy-viscosity corrected with the resolved turbulent stress is imposed in the near-wall region. The RANS eddy-viscosity is obtained from a resolved LES of channel flow at Re_τ = 395 and stored in a look-up table. When used with a wall stress model, this technique enables LES to be performed on coarse grids. Results are presented for channel flow at several Reynolds numbers up to Re_τ = 10,000. Various issues concerning the numerical behavior of the method are discussed.     

Global 2D stability analysis of the cross lid-driven cavity flow with a streamfunction-vorticity approach

We have recently analyzed the global two-dimensional (2D) stability of the staggered lid-driven cavity (LDC) flow with a higher order compact (HOC) approach. In the analysis, critical parameters are determined for both the parallel and anti-parallel motion of the lids and a detailed analysis has been carried out on either side of the critical values.

In this article, we carry out an investigation of flow stabilities inside a two-sided cross lid-driven cavity with a pair of opposite lids moving in both parallel and anti-parallel directions. On discretization, the governing 2D Navier–Stokes (N–S) equations describing the steady flow and flow perturbations results in a generalized eigenvalue problem which is solved for determining the critical parameters on four different grids. Elaborate computation is performed for a wide range of Reynolds numbers (Re) on either side of the critical values in the range 200 ? Re ? 10000. For flows below the critical Reynolds number Re_c, our numerical results are compared with established steady-state results and excellent agreement is obtained in all the cases. For Reynolds numbers above Re_c, phase plane and spectral density analysis confirmed the existence of periodic, quasi-periodic, and stable flow patterns.     

Buoyancy-driven convective heat transfer from a semi-circular cylinder for various confinements

Buoyancy-driven convective heat transfer from a semi-circular cylinder for various confinements has been studied using numerical simulations for wide ranges of parameters, Reynolds numbers (1?≤?Re?≤?50), Richardson numbers (0?≤?Ri?≤?2), Prandtl numbers (0.7?≤?Pr?≤?50) and confinement ratios (0.2?≤?β?≤?0.8). A hot semi-circular cylinder is symmetrically kept in a 2D rectangular confinement. The circular side of the cylinder faces the upstream flow and the fluid flows against gravity in the channel. The governing equations are numerically solved using FLUENT and the results obtained are presented in the form of isotherms, streamlines, pressure coefficients, drag coefficients, Nusselt numbers, etc. The highest value of pressure coefficient increases with blockage ratio for all cases. The drag coefficient decreases with Re and shows complex phenomena with change in Ri and blockage ratio of the channel. Pressure drag has contributed more as compared with viscous drag in all cases. The curved surface showed more heat transfer than the flat surface of the semi-circular cylinder. The value of β also has great influence at large value of Peclect numbers (=?2500). Overall average heat transfer in terms of average Nusselt number is a function of Ri, Re, Pr and β.     

Laminar flow in an annulus between two concentric rotating porous spheres

Summary An analysis is presented for the steady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric porous spheres with injection/suction at their boundaries. The inner sphere rotates with constant angular velocity about its own fixed axis, while the outer sphere is stationary. A solution of the Navier-Stokes equations is obtained by employing a regular perturbation technique. The solution obtained is in the form of a power series expansion in terms of the rotational Reynolds number Re, and an injection/suction Reynolds number Re _w , and is valid for small values of these parameters. Results for the velocity distributions, streamlines, and viscous torques for various values of the flow parameters Re, Re _w , and radius ratios are presented. Viscous torques at the inner and outer spheres are compared with those obtained from the numerical solution of the Navier-Stokes equations, in order to find the range of Re and Re _w for which this solution is accurate.     

Meshless local Petrov–Galerkin method-higher Reynolds numbers fluid flow applications

The meshless local Petrov–Galerkin (MLPG) primitive variable based method is extended to analyze the incompressible laminar fluid flow within or over some different two-dimensional geometries. Although still in laminar regions, the Reynolds numbers considered in this study are in the ranges for which, in the literature, the MLPG primitive variable based method has never produced stable solutions and comparable results with those of the conventional methods. The considered test problems include, a steady lid-driven cavity flow with Reynolds numbers up to and including 10,000, a flow over a backward-facing step at 800 Reynolds number, and a transient fluid flow past a circular cylinder with Reynolds numbers up to and including 200. The present method solves the incompressible Navier–Stokes (N–S) equations in terms of the primitive variables using the characteristic-based split (CBS) scheme for discretization. The weighting function in the weak formulation of the governing equations is taken as unity, and the field variables are approximated using the moving least square (MLS) interpolation. For validation purposes, the obtained results are compared with those of the conventional numerical methods. The agreements of the compared results reveal a step forward towards further applications of the MLPG primitive variable based approach.     

A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Hydrodynamic study of a rotating MHD flow in a cylindrical cavity by ultrasound Doppler shift method

An experimental study of a steady laminar magnetohydrodynamic (MHD) flow driven by a rotating disk at the top of a cylindrical cavity filled with water or mercury is presented. The velocity distributions were analysed using the ultrasound velocity (UVP) measuring technique. The uniform and constant applied magnetic field is directed along the axis of the cavity. The measurements were compared with results obtained from a numerical model based on a finite volume computational fluid dynamics (CFD) model. The effects of the magnetic field, the fluid and wall electrical conductivities, and the wall thickness are investigated through the conductance ratio k which characterises the influence of the wall on the closure of the electric current distribution. The other relevant parameters are the Hartmann number M, and the Reynolds number Re. The study was performed essentially for different values of Re ? 30,000 and M ? 260. There were close agreement between numerical results, the present ultrasonic measurements and other reported experimental and numerical works. The experiments have revealed something that has not been predicted numerically, the sidewall layer is unstable for special conditions of Hartmann and Reynolds numbers.     

Heat transfer asymptote in laminar flow of non-linear viscoelastic fluids in straight non-circular tubes

The fully developed thermal field in constant pressure gradient driven laminar flow of a class of non-linear viscoelastic fluids with instantaneous elasticity in straight pipes of arbitrary contour ∂D with constant wall flux is investigated. The non-linear fluids considered are constitutively represented by a class of single mode, non-affine constitutive equations. The driving forces can be large. Asymptotic series in terms of the Weissenberg number Wi are employed to expand the field variables. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour ∂D₀. The analytical method presented is capable of predicting the velocity and temperature fields in tubes with arbitrary cross-section. Heat transfer enhancement due to shear-thinning is identified together with the enhancement due to the inherent elasticity of the fluid. The latter is to a very large extent the result of secondary flows in the cross-section but there is a component due to first normal stress differences as well. Increasingly large enhancements are computed with increasing elasticity of the fluid as compared to its Newtonian counterpart. Order of magnitude larger enhancements are possible even with slightly viscoelastic fluids. The coupling between inertial and viscoelastic non-linearities is crucial to enhancement. Isotherms for the temperature field are discussed for non-circular contours such as the ellipse and the equilateral triangle together with the behavior of the average Nusselt number Nu, a function of the Reynolds Re, the Prandtl Pr and the Weissenberg Wi numbers. Analytical evidence for the existence of a heat transfer asymptote in laminar flow of viscoelastic fluids in non-circular contours is given for the first time. Nu becomes asymptotically independent from elasticity with increasing Wi, Nu = f(Pe, Wi) → Nu = f(Pe). This asymptote is the counterpart in laminar flows in non-circular tubes of the heat transfer asymptote in turbulent flows of viscoelastic fluids in round pipes. A different asymptote corresponds to different cross-sectional shapes in straight tubes. The change of type of the vorticity equation governs the trends in the behavior of Nu with increasing Wi and Pe. The implications on the heat transfer enhancement is discussed in particular for slight deviations from Newtonian behavior where a rapid rise in enhancement seems to occur as opposed to the behavior for larger values of the Weissenberg number where the rate of increase is much slower. The asymptotic independence of Nu from elasticity with increasing Wi is related to the extent of the supercritical region controlled by the interaction of the viscoelastic Mach number M and the Elasticity number E, which mitigates and ultimately cancels the effect of the increasingly strong secondary flows with increasing Wi to level off the enhancement. The physics of the interaction of the effects of the Elasticity E, viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generating the heat transfer enhancement is discussed.     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

基于特征线分离算法的大涡模拟

将基于特征线的分离算法与大涡模拟相结合,推导了不可压流大涡模拟有限元离散方程组,并将该方法应用于三维流场的层流及湍流非定常计算。将不同雷诺数下的三维顶盖驱动空腔流动计算结果与实验数据以及直接数据模拟结果进行对比,吻合较好,验证了方法的可靠性和准确性。     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part I: The variational formulation for a single evolution equation

The objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow, fluid flow and species transport processes with sharply defined phase-change interfaces. In this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non-linear time-dependent partial differential equations. In Part II of this paper¹ we extend the formulation given here for a single evolution equation to a system of coupled evolution equations. In Part III² we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method.     

Development of a miniature cyclone separator operating at low Reynolds numbers as a pre-separator for portable black carbon monitors

In this study, a cyclone separator that can be used as a sampling inlet for portable black carbon (BC) monitors operating at a flow rate of less than 200 mL/min was developed. A prototype was fabricated to evaluate its performance by experiments, and the cut-off size of the cyclone separator was predicted through numerical analysis by applying various turbulence models. The RNG k–ε model was found to be suitable for the analysis of the cyclone separator operating at Reynolds numbers of less than 1000. Cyclone separators were designed through simulation and fabricated for each operating flow rate (50, 100, 150, and 200 mL/min) of a BC monitor, and their performances were experimentally verified. Meanwhile, when the non-dimensional analysis method of the previous study conducted at Reynolds numbers of 1000 or higher was used, the cyclone separator operating at Reynolds numbers of less than 1000 also exhibited a similar linear tendency.     

Heat transfer characteristics of oscillating flow regenerator filled with circular tubes or parallel plates

Under assumption of small perturbation, linear thermoacoustic theory was applied to analyze heat transfer characteristics of compressible oscillating flow in two kinds of simple regenerators filled with circular tubes or parallel plates. Based on the cross-sectional oscillating velocity and temperature distributions, the exact expressions of Nusselt number were derived in complex notation. The Nusselt number is the function of Prandtl number, kinetic Reynolds number Re_ω and the third dimensionless variable, D. Here, the D is defined as the ratio of heat transfer capability aroused by mean temperature gradient and gas compressibility. Both the gas compressibility and mean temperature gradient effects were discussed and two corresponding Nusselt numbers were given. In particular, simpler expressions for these two Nusselt numbers were deduced for extreme values of Re_ω and D. Finally, combined effect of gas compressibility and non-zero mean temperature gradient on heat transfer characteristics were analyzed via D. The analysis shows that the mean temperature gradient gives predominant contribution to the heat transfer performance of oscillating flow regenerator.     

Boundary layer structure at the exit of a cascade of parallel thin plates

The laminar flow near the exit from a cascade of parallel thin plates is investigated. The transition from the Poiseuille flow developed between two parallel plates to a uniform flow in the wake is carried out in two stages using an analytical approach for large Reynolds numbers. The first stage involves the study of the regular disturbances in the wake, where the equations of motion are reduced to the Prandtl equations. The numerical solution of these equations and their asymptotic treatment in the near wake are carried out. Comparison of the results shows that the asymptotic approach enables an improved solution near the exit, where singular disturbances arise. The second stage requires study of this singular region, by introducing boundary layers surrounding the trailing edges. This study treats the interactions induced upstream of the trailing edges and acts to ensure the asymptotic matching of the Poiseuille flow upstream with the asymptotic solution of the near wake downstream.     

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.