Numerical simulation of confined laminar jet flows

Two‐dimensional incompressible jet development inside a duct has been studied in the laminar flow regime, for cases with and without entrainment of ambient fluid. Results have been obtained for the flow structure and critical Reynolds number values for steady asymmetric jet development and for the onset of temporal oscillations, at various values of the duct‐to‐jet width ratio (aspect ratio). It is found that at low aspect ratios and Reynolds numbers, jet development inside the duct is symmetric. For larger aspect ratios and Reynolds numbers, the jet flow at steady state becomes asymmetric with respect to the midplane, and for still higher values, it becomes oscillatory with respect to time. When entrainment is present, the instabilities of asymmetric development and temporal oscillations occur at a much higher critical Reynolds number for a given aspect ratio, indicating that the stability of the jet flow is higher with entrainment. Copyright © 2000 John Wiley & Sons, Ltd.     

Unsteady laminar separated flow through constricted channel

In the present paper unsteady Navier-Stokes equations have been solved numerically by finite-difference technique in staggered grid distribution for a flow through a channel with locally symmetric and asymmetric constrictions. A coordinate stretching has been made to map the infinite irregular geometry into a finite regular computational domain. Pressure and pressure-velocity corrections scheme have been developed. Convergence criteria (in terms of continuity equation) has been achieved after few time iterations. The critical Reynolds number for asymmetric flow through a symmetric constriction has been found. Critical values depend on the area reduction and the length of the constriction. The increment of Reynolds number grows the asymmetry of the flow. The root mean square (r.m.s.) centreline vertical velocity for asymmetric flow through a symmetric constriction has been drawn at different Reynolds numbers. For flow through symmetric constriction the centreline vertical velocity shows finite oscillation behind the constriction at high Reynolds number.     

Linear stability analysis of flow in a periodically grooved channel

We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two‐dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady‐state, whereas a self‐sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty‐function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd.     

Viscosity effect on the flow patterns in T-type micromixers

Flow patterns and mixing of liquids with different viscosities in T-type micromixers are numerically investigated on the Reynolds number range from 1 to 250. The viscosity ratio of the mixing media varied from 1 to 2; its effect on the flow structure and the mixing is studied. The dependences of the mixing efficiency and the pressure difference in the channel on the viscosity ratio and the Reynolds number are obtained. It is shown that the viscosity ratio has a considerable effect on the flow structure before and after transition from the symmetric to the asymmetric flow pattern. The self-similar behavior of the asymmetric flow pattern is established.     

Multiplicity of Steady Two-Dimensional Flows in Two-Sided Lid-Driven Cavities

The two-dimensional steady incompressible flow in rectangular cavities is calculated numerically by a finite volume method. The flow is driven by two opposing cavity side walls which move with constant velocities tangentially to themselves. Depending on the cavity aspect ratio and the two side-wall Reynolds numbers different flow states exist. Their range of existence and the bifurcations between different states are investigated by a continuation method accurately locating the bifurcation points. When both side walls move in opposite directions up to seven solutions are found to exist for the same set of parameters. Three of these are point-symmetric and four are asymmetric with respect to the center of the cavity, if the side-wall Reynolds numbers have the same magnitude. When the walls move in the same direction, up to five different flow states are found. In this case only a single mirror symmetric solution exists for equal Reynolds numbers. Received 9 February 2000 and accepted 9 October 2000     

Effect of span length and temperature on the 3-D confined flow around a vortex promoter

This article presents a numerical study on the influence of span length and wall temperature on the 3-D flow pattern around a square section vortex promoter located inside a micro-channel in the low Reynolds number regime. The first objective of the work is to quantify the critical Reynolds number that defines the onset of vortex shedding and to identify the different regimes that appear as a function of the channel aspect ratio (span to height ratio). We found that the critical Reynolds number for the onset of the Karman street regime increases as the aspect ratio decreases. In particular, for the aspect ratio of 1/2 the critical Reynolds number is nearly six times the critical Reynolds number of the 2-D problem. An intermediate oscillating regime between the steady and the Karman street solutions was also found to exist within a rather wide range of Reynolds numbers for small channel aspect ratios. The second objective was to investigate the influence of the vortex promoter wall temperature on both vortex shedding and flow pattern. This has practical engineering implications because the working fluid considered in the article is water that has a viscosity that depends significantly on temperature and promotes a strong coupling between the momentum and energy equations that influences the system behaviour. Results indicate that high surface temperature on the prism promotes the onset of the Karman street, suggesting design guidelines for micro-channel based heat sinks that make use of vortex promoters.     

Effect of span length and temperature on the 3-D confined flow around a vortex promoter

This article presents a numerical study on the influence of span length and wall temperature on the 3-D flow pattern around a square section vortex promoter located inside a micro-channel in the low Reynolds number regime. The first objective of the work is to quantify the critical Reynolds number that defines the onset of vortex shedding and to identify the different regimes that appear as a function of the channel aspect ratio (span to height ratio). We found that the critical Reynolds number for the onset of the Karman street regime increases as the aspect ratio decreases. In particular, for the aspect ratio of 1/2 the critical Reynolds number is nearly six times the critical Reynolds number of the 2-D problem. An intermediate oscillating regime between the steady and the Karman street solutions was also found to exist within a rather wide range of Reynolds numbers for small channel aspect ratios. The second objective was to investigate the influence of the vortex promoter wall temperature on both vortex shedding and flow pattern. This has practical engineering implications because the working fluid considered in the article is water that has a viscosity that depends significantly on temperature and promotes a strong coupling between the momentum and energy equations that influences the system behaviour. Results indicate that high surface temperature on the prism promotes the onset of the Karman street, suggesting design guidelines for micro-channel based heat sinks that make use of vortex promoters.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios

This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.     

Flows through plane sudden‐expansions

A calculation method has been developed and used to represent flows downstream of plane symmetric expansions with dimensions and velocities encompassing laminar and turbulent flows. Except for very low Reynolds numbers, the flows are time‐dependent and asymmetric and the calculated results are appraised first in relation to published measurements of laminar flows and then to new measurements obtained at a Reynolds number of 26 500. The time‐dependent laminar simulations indicate that the critical Reynolds numbers are predicted with excellent accuracy for different expansion ratios and the details of the asymmetric velocity profiles are in good agreement with experimental measurements. The laminar flow calculations also show that increasing the thickness of the separating boundary layer leads to longer regions of separation and no dominant frequency for Reynolds numbers up to those at which the third separation region was observed. The turbulent flow simulations made use of the k–ε turbulence model and provided a satisfactory representation of measurements, except in regions close to the wall and within the recirculation regions. Also, the longer reattachment length was underestimated. Limitations are discussed in relation to these and higher‐order assumptions. Copyright © 2000 John Wiley & Sons, Ltd.     

On the validity of the perturbation approach for the flow inside weakly modulated channels

The equations governing the flow of a viscous fluid in a two‐dimensional channel with weakly modulated walls have been solved using a perturbation approach, coupled to a variable‐step finite‐difference scheme. The solution is assumed to be a superposition of a mean and perturbed field. The perturbation results were compared to similar results from a classical finite‐volume approach to quantify the error. The influence of the wall geometry and flow Reynolds number have extensively been investigated. It was found that an explicit relation exists between the critical Reynolds number, at which the wall flow separates, and the dimensionless amplitude and wavelength of the wall modulation. Comparison of the flow shows that the perturbation method requires much less computational effort without sacrificing accuracy. The differences in predicted flow is kept well around the order of the square of the dimensionless amplitude, the order to which the regular perturbation expansion of the flow variables is carried out. Copyright © 2002 John Wiley & Sons, Ltd.     

DNS of viscoelastic turbulent channel flow with rectangular orifice at low Reynolds number

Direct numerical simulations of turbulent viscoelastic-fluid flow in a channel with a rectangular orifice were performed to investigate the influence of viscoelasticity on turbulence statistics and turbulent structures downstream of the orifice. The geometry considered is periodic rectangular orifices with 1:2 expansion. The constitutive equation follows the Giesekus model, valid for polymer (or surfactant) solutions, which are generally capable of reducing the turbulent frictional drag in a smooth channel. The friction Reynolds number and the Weissenberg number were set to 100 and 20-30, respectively. A drag reduction of about 20% was achieved in the viscoelastic flows. The onset Reynolds number for the transition from a symmetric to an asymmetric state was found to be shifted to higher values than that for the Newtonian flow. In the viscoelastic flow, the turbulent kinetic energy was decreased and fewer turbulent eddies were observed, as the Kelvin-Helmholtz vortices were quickly damped. Away from the orifice, quasi-streamwise vortices in the viscoelastic flow were sustained for a longer period, accompanied by energy exchange from elastic energy of the viscoelastic fluid to kinetic energy.     

A numerical investigation of the influence of the aspect ratio in three‐dimensional separated flows

The influence of aspect ratio in three‐dimensional, numerical experiments of separated flows is studied in the case of the backward‐facing step at Reynolds numbers 600, 800, and 950. The computational domain is designed as an actual laboratory experiment. The governing equations are the steady state, isothermal, and incompressible Navier–Stokes equations. The expansion ratio of the computational domain is 1:2. The aspect ratio varies from 1:10 to 1:40. The results of the computations focus on the spanwise variations of the length and the strength of the two eddies along the lower and upper wall. It is concluded that both numerical and laboratory experiments should be designed with an aspect ratio of at least 1:20, if only the accuracy of the position of the detachment and the re‐attachment points matters. If the accuracy of the shear‐stress distributions is also taken into account, then an aspect ratio of at least 1:30 should be chosen. Finally, if the magnitudes of the vortex centers are also considered, then only the aspect ratio of 1:40 qualifies for a realization of two‐dimensional flow conditions in the plane of symmetry. This is contrary to the common practice in the field, at least from the side of laboratory experiments, where an aspect ratio of 1:10 is still considered adequate for this purpose. Copyright © 2011 John Wiley & Sons, Ltd.     

Three-dimensional numerical simulation of the wake flow of an afterbody at subsonic speeds

We numerically investigate the wake flow of an afterbody at low Reynolds number in the incompressible and compressible regimes. We found that, with increasing Reynolds number, the initially stable and axisymmetric base flow undergoes a first stationary bifurcation which breaks the axisymmetry and develops two parallel steady counter-rotating vortices. The critical Reynolds number (Re _cs) for the loss of the flow axisymmetry reported here is in excellent agreement with previous axisymmetric BiGlobal linear stability (BiGLS) results. As the Reynolds number increases above a second threshold, Re _co, we report a second instability defined as a three-dimensional peristaltic oscillation which modulates the vortices, similar to the sphere wake, sharing many points in common with long-wavelength symmetric Crow instability. Both the critical Reynolds number for the onset of oscillation, Re _co, and the Strouhal number of the time-periodic limit cycle, St_sat, are substantially shifted with respect to previous axisymmetric BiGLS predictions neglecting the first bifurcation. For slightly larger Reynolds numbers, the wake oscillations are stronger and vortices are shed close to the afterbody base. In the compressible regime, no fundamental changes are observed in the bifurcation process. It is shown that the steady state planar-symmetric solution is almost equal to the incompressible case and that the break of planar symmetry in the vortex shedding regime is retarded due to compressibility effects. Finally, we report the developments of a low frequency which depends on the afterbody aspect ratio, as well as on the Reynolds and on the Mach number, prior to the loss of the planar symmetry of the wake.     

Numerical study of flow and thermal behaviour of lid‐driven flows in cavities of small aspect ratios

Numerical study has been performed to investigate the effects of cavity shape on flow and heat transfer characteristics of the lid‐driven cavity flows. Dependence of flow and thermal behaviour on the aspect ratio of the cavities is also evaluated. Three types of the cross‐sectional shape, namely, circular, triangular, and rectangular, and four aspect ratios, 0.133, 0.207, 0.288, and 0.5, are taken into account to construct twelve possible combinations; however, attention is focused on the small‐aspect‐ratio situations. Value of the Reynolds number considered in this study is varied between 100 and 1800. For the cases considered in this study a major clockwise vortex driven by the moving lid prevailing in the cavity is always observed. When the Reynolds number is fixed, the rectangular cavity produces strongest lid‐driven flow, and the triangular cavity weakest. For the cases at small aspect ratio and low Reynolds number, the streamlines appear symmetric fore‐and‐aft with respect to the central line at x/L = 0.5. Data for the local and average Nusselt numbers are also provided. For rectangular cavities, it is observed that case 1/5R produces the highest average Nusselt number at any Reynolds number. Among the twelve possible geometric cases considered herein, the highest and lowest average Nusselt numbers are found with cases 1/6T and 1/2C, respectively. Copyright © 2006 John Wiley & Sons, Ltd.     

Visualization of three-dimensional flow in a four-roll mill

Flow visualization has been used to study the motion of a Newtonian fluid in a four-roll mill. Observations of the effect of increasing Reynolds number on the evolution of steady three-dimensional flow between the rollers are reported. The flow visualization experiments have been conducted for two values of the aspect ratio (defined as the ratio of roller length to gap width between rollers). The rollers have been constrained to rotate in a manner which generates an approximately two-dimensional pure extensional flow near the center of the apparatus at low speeds. Steady, symmetrically positioned vortices grow near the top and bottom walls of the container as the Reynolds number is increased up to a critcal value. An increase in Reynolds number beyond this critical value results in a loss of symmetry in the flow pattern. The nature of this transition is strongly dependent upon the aspect ratio.     

Effects of aspect ratio and orientation on the wake characteristics of low Reynolds number flow over a triangular prism

The wake characteristics of unconfined flows over triangular prisms of different aspect ratios have been numerically analysed in the present work. For this purpose, a fixed Cartesian-grid based numerical technique that involves the porous medium approach to mimic the effect of solid blockage has been utilised. Correspondingly, laminar flow simulations ranging from the sub-critical regime (before the onset of vortex shedding) to the super-critical regime have been considered here within the limits of two-dimensionality. In the sub-critical regime, correlations relating the wake bubble length with Reynolds number (Re) have been proposed for various aspect ratios. Also, the effects of aspect ratio and Reynolds Number on the drag force coefficient (C_D) have been characterised for two different geometrical orientations of the prism (base or apex facing the flow). Subsequently, the critical Reynolds number at the onset of vortex shedding has been predicted for each of the aspect ratio considered, by an extrapolation procedure. The unsteady flow characteristics of the super-critical regime are finally highlighted for different aspect ratios and triangular orientations considered in the study.     

Numerical study of magnetohydrodynamic laminar flow separation in a channel with smooth expansion

The flow of an electrically conducting incompressible viscous fluid in a plane channel with smooth expansion in the presence of a uniform transverse magnetic field has been analysed. A solution technique for the governing magnetohydrodynamic equations in primitive variable formulation has been developed. A co‐ordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. The governing equations are discretized using finite‐difference approximations in staggered grid. Pressure Poisson equation and pressure correction formulae are derived and solved numerically. It is found that with increase in the magnetic field, the size of the flow separation zone diminishes and for sufficiently large magnetic field, the separation zone disappears completely. The peak u‐velocity decreases with increase in the magnetic field. It is also found that the asymmetric flow in a symmetric geometry, which occurs at moderate Reynolds numbers, becomes symmetric with sufficient increase in the transverse magnetic field. Thus, a transverse magnetic field of suitable strength has a stabilizing effect in controlling flow separation, as also in delaying the transition to turbulence. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical investigation of wall effects in three‐dimensional,laminar flow over a backward facing step with a constant aspect and expansion ratio

The wall effects are investigated in the three‐dimensional laminar flow over a backward‐facing step. For this purpose, a numerical experiment is designed under actual laboratory conditions. The aspect ratio of the computational domain is 1:40 and the expansion ratio is 1:2. The Reynolds number ranges from 100 to 950. The governing equations are the steady state, isothermal and incompressible Navier–Stokes equations for Newtonian fluids. They are solved with a homemade Galerkin finite element code. The computations are validated with data from available laboratory and numerical experiments. The results focus on the variation of both velocity profiles and lengths of eddies along the lower and upper wall in the spanwise direction. Calculated streamlines in the streamwise and transverse direction show how the flow is distorted near the lateral wall and how it develops up to the plane of symmetry. The study of skin friction lines along the top and bottom wall of the domain reveals a flow that takes place in the spanwise direction. This spanwise component of the flow becomes more dominant with increasing Reynolds number and is impossible to be sustained at steady state for Reynolds numbers higher than 950 for this particular geometry. Copyright © 2012 John Wiley & Sons, Ltd.     

非对称槽道中涡旋波的特性研究

利用PIV流场显示技术，对振荡流体在非对称槽道中涡旋波的产生、发展和消失的规律进 行了实验研究和分析，测得了涡旋波流场的速度矢量图，阐明了涡旋波流场周期性变化的特 点. 结合涡动力学方程，深入分析并揭示了做周期性运动的流体能在槽道中产生波的特性这 一规律，从中发现：流体周期变化的非定常性和不对称的槽道结构是形成涡旋波流动的主要 因素. 本文对涡旋波流场中各个旋涡的速度分布和涡量进行了测量和计算，分析了涡旋波 强化传质的机理，并研究了Re数对涡旋波流动的影响