Numerical simulation of turbulent flow through series stenoses

The flow fields in the neighbourhoods of series vascular stenoses are studied numerically for the Reynolds numbers from 100 to 4000, diameter constriction ratios of 0.2–0.6 and spacing ratios of 1, 2, 3, 4 and ∞. In this study, it has been further verified that in the laminar flow region, the numerical predictions by k–ω turbulence model matched those by the laminar‐flow modelling very well. This suggests that the k–ω turbulence model is capable of the prediction of the laminar flow as well as the prediction of the turbulent stenotic flow with good accuracy. The extent of the spreading of the recirculation region from the first stenosis and its effects on the flow field downstream of the second stenosis depend on the stenosis spacing ratio, constriction ratio and the Reynolds number. For c₁ = 0.5 with c₂ ≤ c₁, the peak value of wall vorticity generated by the second stenosis is always less than that generated by the first stenosis. However, the maximum centreline velocity and turbulence intensity at the second stenosis are higher than those at the first stenosis. In contrast, for c₁ = 0.5 with c₂ = 0.6, the maximum values at the second stenosis are much higher than those at the first stenosis whether for centreline velocity and turbulence intensity or for wall vorticity. The peak values of the wall vorticity and the centreline disturbance intensity both grow up with the Reynolds number increasing. The present study shows that the more stenoses can result in a lower critical Reynolds number that means an earlier occurrence of turbulence for the stenotic flows. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite difference and finite element methods for mhd channel flows

A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive-variable and streamfunction–vorticity finite difference formulations. However, the primitive-variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet-like characteristics.     

Side‐wall effects on the bifurcation of the flow through a sudden expansion

Three‐dimensional computations have been performed to study the flow through a symmetric sudden expansion with an expansion ratio of 3 at low Reynolds numbers. The aspect ratio of the flow channel is allowed to vary within a wide range to examine its influence on the flow which bifurcates from a symmetric state to an asymmetric state. The results reveal that the critical Reynolds number of the symmetry‐breaking bifurcation increases while the aspect ratio is reduced. The flow behaviour near the side walls is illustrated by using limiting streamlines. The origin of the singular points identifiable on the side wall can be traced back to the recirculating flows and the relevant reattachment/separation points in the core of the channel. It is seen that the determination of the exact critical Reynolds number is not trivial because it depends on how to define asymmetric flow. Computations have also been conducted to show that a slight asymmetry in the channel geometry causes a smooth transition from symmetric to non‐symmetric states. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Investigation on time scales in a low Reynolds number jet flow using particle-tracking velocimetry

The near flow field of an axisymmetric water jet at Reynolds numbers between 2000 and 5000 is investigated using Particle-Tracking Velocimetry. Measurements are taken in the longitudinal section (along the mean flow) and in cross-sections (orthogonal to the mean flow). From the former, correlation coefficients of the two in-plane velocity components in a Lagrangian framework are obtained: thus Lagrangian integral scales can be computed. Those of the streamwise velocity (axial) component increase on moving away from the centreline, whereas the opposite happens for the vertical velocity (radial) component: integral time scales of the two components are almost equal at the interface between jet and ambient fluids. On the other hand, integral scales are almost constant or increase slightly with the axial direction. In cross-sections, fluid ejection and injection from the jet centreline are observed to be connected to counter-rotating vortices (mushroom): their number and size change with Reynolds number in agreement with results from other authors. The maximum ejection velocity (orthogonal to the mean jet flow), at 3 nozzle diameters downstream of the outlet, is found to be one half of the mean outlet velocity.     

Flow fields in a simulated axisymmetric terminal aneurysm with symmetric and asymmetric outflows

Laser-Doppler velocimetric measurements and flow visualization were performed in a glass axisymmetric aneurysm model with symmetric and asymmetric outflows through the branches. The bifurcation angle was fixed at 140°, and the Reynolds number based on the steady bulk average velocity and diameter of the affarent conduit was 500. The flow characteristics such as flow separation in the afferent conduit and flow activity inside the aneurysm for the symmetric and asymmetric outflow cases were compared in detail, and the case that is susceptible to thrombosis was identified. In addition, the onset of transition from laminar to turbulent flow inside the aneurysm was evidenced by the presence of vortex breakdown and the steep increase in the fluctuation level. Finally, the effect of pulsation on the flow pattern in the aneurysm was examined.     

Numerical studies of fluid flow through tubes with double constrictions

The flow fields in the neighbourhood of double constrictions in a circular cylindrical tube were studied numerically. The effects on the streamline, velocity and vorticity distributions as the flow passes through the constrictions in the tube were studied in the Reynolds number range 5–200. Double constrictions with dimensionless spacing ratios of 1, 2, 3 and ∞ were studied for a 50% constriction. It is noted that when the Reynolds number is below 10, no recirculation region is formed in the above constricted flow. For Reynolds numbers greater than 10, a recirculation region forms downstream of each of the constrictions. For constriction spacing ratios of 1, 2, and 3, when the Reynolds number is high, a recirculation region spreads between the valley of the constrictions. The recirculation region formed between the two constrictions has a diminishing effect on the generation of wall vorticity near the second constriction area. In general, the peak value of wall vorticity is found slightly upstream of each of the constrictions. When the Reynolds number is increased, the peak wall vorticity value increases and its location is moved upstream. Maximum wall vorticity generated by the first constriction is found to be always greater than the maximum wall vorticity generated by the second constriction. The extent of this spreading of the recirculation region from the first constriction and its effects on the second constriction depend on the constriction spacing ratio and the flow Reynolds number.     

Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion

In this work we present an investigation of viscoelastic flow in a planar sudden expansion with expansion ratio D/d = 4. We apply the modified FENE–CR constitutive model based on the non-linear finite extensibility dumbbells (FENE) model. The governing equations were solved using a finite volume method with the high-resolution CUBISTA scheme utilised for the discretisation of the convective terms in the stress and momentum equations. Our interest here is to investigate two-dimensional steady-state solutions where, above a critical Reynolds number, stable asymmetric flow states are known to occur. We report a systematic parametric investigation, clarifying the roles of Reynolds number (0.01 < Re < 100), Weissenberg number (0 < We < 100) and the solvent viscosity ratio (0.3 < β < 1). For most simulations the extensibility parameter of the FENE model was kept constant, at a value L² = 100, but some exploration of its effect in the range 100–500 shows a rather minor influence. The results given comprise flow patterns, streamlines and vortex sizes and intensities, and pressure and velocity distributions along the centreline (i.e. y = 0). For the Newtonian case, in agreement with previous studies, a bifurcation to asymmetric flow was observed for Reynolds numbers greater than about 36. In contrast viscoelasticity was found to stabilise the flow; setting β = 0.5 and We = 2 as typical values, resulted in symmetric flow up to a Reynolds number of about 46. We analyse these two cases in particular detail.     

How to estimate the low wavenumber turbulence frequency spectrum

 An estimate of the low wavenumber component of surface turbulence shear stress as a function of frequency has been obtained through measurements of the correlations of the longitudinal component of turbulence velocity made close to the surface at y ⁺=7. The data were acquired in a fully-developed turbulent pipe flow at a Reynolds number (based on centreline velocity and pipe diameter) of 268000, using two single hot-wire anemometer probes. A novel data analysis procedure has been introduced to establish the accuracy limits of the low wavenumber turbulence energy estimate for frequencies in the similarity regime of wall turbulence and the results are compared with other measurement techniques. Received: 18 November 1993/Accepted: 21 April 1997     

Large eddy simulation of boundary layer flow under cnoidal waves

Water waves in coastal areas are generally nonlinear, exhibiting asymmetric velocity profiles with different amplitudes of crest and trough. The behaviors of the boundary layer under asymmetric waves are of great significance for sediment transport in natural circumstances. While previous studies have mainly focused on linear or symmetric waves, asymmetric wave-induced flows remain unclear, particularly in the flow regime with high Reynolds numbers.Taking cnoidal wave as a typical example of asymmetric waves, we propose to use an infinite immersed plate oscillating cnoidally in its own plane in quiescent water to simulate asymmetric wave boundary layer. A large eddy simulation approach with Smagorinsky subgrid model is adopted to investigate the flow characteristics of the boundary layer. It is verified that the model well reproduces experimental and theoretical results. Then a series of numerical experiments are carried out to study the boundary layer beneath cnoidal waves from laminar to fully developed turbulent regimes at high Reynolds numbers, larger than ever studied before.Results of velocity profile, wall shear stress, friction coefficient, phase lead between velocity and wall shear stress, and the boundary layer thickness are obtained. The dependencies of these boundary layer properties on the asymmetric degree and Reynolds number are discussed in detail.     

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 least-squares finite element method for incompressible Navier-Stokes problems

A least-squares finite element method based on the velocity–pressure–vorticity formulation was proposed for solving steady incompressible Navier-Stokes problems. This method leads to a minimization problem rather than to the saddle point problem of the classic mixed method and can thus accommodate equal-order interpolations. The method has no parameter to tune. The associated algebraic system is symmetric and positive definite. In order to show the validity of the method for high-Reynolds-number problems, this paper provides numerical results for cavity flow at Reynolds number up to 10 000 and backward-facing step flow at Reynolds number up to 900.     

Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer

In the present paper we discuss the magnetohydrodynamic (MHD) peristaltic flow of a hyperbolic tangent fluid model in a vertical asymmetric channel under a zero Reynolds number and long wavelength approximation. Exact solution of the temperature equation in the absence of dissipation term has been computed and the analytical ex- pression for stream function and axial pressure gradient are established. The flow is analyzed in a wave frame of reference moving with the velocity of wave. The expression for pressure rise has been computed numerically. The physical features of pertinent parameters are analyzed by plotting graphs and discussed in detail.     

Numerical investigation of particle lateral migration in straight channel flows using a direct-forcing immersed boundary method

Inertia-induced cross-stream migration has been recently exploited for precise position of particles in confined channel flows. In this work, a three-dimensional finite volume based immersed boundary method has been developed to study the lateral migration and hydrodynamic self-assembly of neutrally-buoyant particles in pressure-driven flows. Simulation results show that, in 2D channel flows, the equilibrium position for a circular particle is closer to the centreline for larger particle Reynolds number due to the increasing flow rate, while in 3D square duct flow, the equilibrium position for a spherical particle is near a face centre and is closer to the wall for larger particle Reynolds number. Self-assembly of a pair of particles is observed in 3D square duct flows but not in 2D channel flows. Mechanisms for the self-assembly are discussed.     

The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length

Polymer solutions and melts can exhibit large upstream corner and lip vortices, unstable and diverging flow and an enhanced pressure drop when flowing through a geometry containing a constriction. In the present work, we use a planar microfluidic device to show that the length of the downstream constriction plays an important role in the upstream kinematics and the extra pressure drop. That is, the elastic flow phenomena observed upstream of a constriction during entry flows of polymer solutions are not exclusively a result of the stretching dynamics induced by the converging flow—the downstream relaxation events are, at least, equally important. Flow visualization experiments with semi-dilute solutions of a high molecular weight polymer showed that large stable symmetric vortices could be reduced to highly chaotic asymmetric flow, merely by increasing the length of the constriction—the Reynolds number and elasticity number were both held constant. This was accompanied by a higher extra pressure. These results support the hypothesis that elastic flow instabilities originate downstream of the constriction (at the expansion) and move progressively upstream with time and/or flowrate. These findings may also partly explain the discrepancies commonly observed between the results of entry flow experiments and numerical simulations, in which the downstream geometry is very rarely considered. Lastly, we illustrate how to minimize the occurrence of unstable flow upstream of a constriction, which is a necessary condition for closed microrheometry devices used to characterize low viscosity elastic fluids.     

Effect of symmetry and asymmetry of turbulent structures on the interaction region of a plane jet

Mean and rms velocity and temperature measurements are made in the near field of a plane jet for two different sets of conditions at the beginning of the interaction region. In the first, the jet has a nearly top-hat velocity profile and laminar boundary layers at the nozzle exit. Schlieren photography and spectral coherence measurements show that the mixing layer structures are strongly organised and symmetric about the centreline. In the second, the jet issues from a two-dimensional duct with a nearly fully developed velocity profile so that, upstream of the interaction region, turbulent structures should be three-dimensional and asymmetric about the jet centreline. The resulting weak interaction obtained in this case contrasts with the strong interaction measured in the first case.     

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.     

Simulations of fibre orientation in dilute suspensions with front moving in the filling process of a rectangular channel using level-set method

The simulation of fibre orientation in dilute suspension with front moving is carried out using the projection and level-set methods. The motion of fibres is described using the Jeffery equation, and the contribution of fibres to the flow is accounted for by the configuration-field method. The dilute suspension of short fibres in Newtonian fluids is considered. The governing Navier–Stokes equation for the fluid flow is solved using the projection method with finite difference scheme, while the fibre-related equations are directly solved with the Runge–Kutta method. In the present study for fibres in dilute suspension flow for injection molding, the effects of various flow and material parameters on the fibre orientation, the velocity distributions and the shapes of the leading flow front are found and discussed. Our findings indicate that the presence of fibre motion has little influence on the front shape in the ranges of fibre parameters studied at the fixed Reynolds number. Influence of changing fibre parameters only causes variation of front shape in the region near the wall, and the front shape in the central core area does not vary much with the fibre parameters. On the other hand, the fibre motion has strong influence on the distributions of the streamwise and transverse velocities in the fountain flow. Fibre motion produces strong normal stress near the wall which leads to the reduction of transversal velocity as compared to the Newtonian flow without fibres, which in turn, leads to the increased streamwise velocity near the wall. Thus, the fibre addition to the flow weakens the strength of the fountain flow. The Reynolds number has also displayed significant influence on the distribution of the streamwise velocity behind the flow front for a given fibre concentration. It is also found that the fibre orientation is not always along the direction of the velocity vector in the process of mold filling. In the region of the fountain flow, the fibre near the centreline is more oriented across the streamwise direction compared to that in the region far behind the flow front. This leads to the fact that the fibre near the centreline in the region of fountain flow is more extended along the transverse direction. As the fibre orientation in the suspension flow and the shape of the flow front have great bearing on the quality of the product made from injection molding, this study has much implications for engineering applications. These results can also be useful in other fields dealing with fibre suspensions.     

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Finite difference solutions have been obtained by the perturbation method to investigate the influence of shear thinning and elasticity on the flow around an inclined circular cylinder of finite length in a uniform flow. In this numerical analysis a generalized upper-convected Maxwell model, in which the viscosity changes according to the Cross model, has been used.The local flow over the cylinder is only slightly deflected. However, in the wake flow behind the cylinder the particle path is remarkably influenced by the axial flow and rapidly flows up parallel to the cylinder's axis. Then it gradually rejoins direction of the incoming flow. It is found that viscoelastic fluids are prone to flow axially in the vicinity of the cylinder. The numerical predictions generally agree with the flow visualization results.The numerical solutions also demonstrate that elasticity has a strong effect on the velocity profile especially around both ends of the cylinder; elasticity increases the asymmetric profiles of both circumferential velocity and axial velocity with respect to equal to 90° and decreases a difference in the circumferential velocity between the windward end and the leeward end.For non-Newtonian fluids, the length of the wake flow is influenced by not only the Reynolds number but also the cylinder diameter and it is larger for the cylinder with the smaller diameter at the same Reynolds number.Partly presented at the 9th Australasian Fluid Mechanics Conference, University of Auckland, New Zealand, 8–12 December, 1986     

Wake transition for surface mounted rectangular cylinder due to incoming shear flow

Three-dimensional unsteady wake characteristics have been investigated numerically in flow past surface mounted finite-height rectangular cylinder using Open Source Field Operation and Manipulation. Effect of impinging shear (shear intensity, K) on transitional characteristics of wake flow has been studied using iso-Q surfaces for Reynolds number (Re) in the range from 150 to 250. Various flow regimes, such as steady flow, symmetric and asymmetric modes of vortex shedding have been identified based on the values of Re and K for different side ratios (SR) of the cylinder. Unsteady wake oscillations have been analyzed using time signal of transverse velocity component in the wake. These signals have been decomposed into different component signals using Hilbert-Huang transformation (HHT). Variation of frequency and energy density with time of the decomposed signals has been presented in the form of Hilbert spectra. Effects of Re, SR and K on wake oscillation frequency have been illustrated in the form of marginal spectra. Time-delay reconstructions and Poincare sections have been examined to study periodic and aperiodic nature of the wake flow. Non-stationarity associated with the wake fluctuation is quantified in terms of degree of stationarity. Symmetric and asymmetric modes have been confirmed using singular value decomposition of the vorticity field and presented using dynamic modes. Growth rate and frequency of the modes corresponding to symmetric shedding are found to be lower than those for asymmetric shedding. In addition, variation in mean drag coefficient has been reported with change in Re and K for each value of SR.