Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Forces acting on particles in separated wall‐bounded shear flow

Trajectories of solid particles in laminar and turbulent flow over a backward‐facing step (BFS) were numerically computed by integrating the equation of motion for particles. The various forces acting on the particles [5],[6] were calculated for a variety of flow Reynolds numbers and for different particle characteristics such as the Stokes number and the particle‐to‐fluid density ratio. The investigation was conducted for the distinct flow regimes of the BFS flow separately. Generally, the drag and gravitation were found to be the most significant forces. The lift and history force were the next most important, mostly two orders of magnitude smaller, but in some cases closing up to the other two in importance. The pressure and virtual mass effects were very small for the majority of cases. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of mass loading on particle-laden turbulent channel flow

Results are presented for the behavior of particle-laden gases in a moderate Reynolds number vertical channel down flow. The effects of mass loading on the gas-phase turbulence and particle concentration are studied. A direct numerical simulation including models of wall-particle interaction is conducted. It is confirmed that particle feedback causes the turbulence intensities to become more non-isotropic as the particle loading is increased. The particle concentration exhibits a maximum close to the wall and a slight increase in the middle of the channel. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

A vortex method for fluid-particle systems

We consider a two-dimensional, dilute fluid-particle system with low Reynolds number for the flow around the particles and high Reynolds number for the bulk flow. We use a vortex method to calculate numerically the incompressible fluid phase. For the compressible particle phase we use a particle method and Voronoi diagrams to calculate the particle density. We use the Stokes-Oseen formula to represent approximately the force of the fluid on the particles. We give the results of a numerical experiment that show the effect of fluid particle interaction on the bulk flow.     

两相流中柱状固粒对流体湍动特性影响的研究

对含柱状固粒的两相流场,建立了包含柱状固粒对流场影响的流体脉动速度方程,在求解脉动速度方程的基础上,经平均得到流体的湍流强度和雷诺应力.将该方法用于槽流湍流场的求解,并与单相流实验结果进行了比较.计算中变化柱状固粒的参数,给出了固粒的体积分数、长径比、松驰时间对流场湍动特性的影响,说明粒子对流场的湍动特性起着抑制作用,其抑制的程度与粒子的体积分数、长径比成正比,与粒子的松弛时间成反比.     

A Lagrangian simulation of particle deposition in a turbulent boundary layer in the presence of thermophoresis

Knowledge of particle deposition in turbulent flows is often required in engineering situations. Examples include fouling of turbine blades, plate-out in nuclear reactors and soot deposition. Thus it is important for numerical simulations to be able to predict particle deposition. Particle deposition is often principally determined by the forces acting on the particles in the boundary layer. The particle tracking facility in the CFD code uses the eddy lifetime model to simulate turbulent particle dispersion, no specific boundary layer being modelled. The particle tracking code has been modified to include a boundary layer. The non-dimensional yplus, y⁺, distance of the particle from the wall is determined and then values for the fluid velocity, fluctuating fluid velocity and eddy lifetime appropriate for a turbulent boundary layer used. Predictions including the boundary layer have been compared against experimental data for particle deposition in turbulent pipe flow. The results giving much better agreement. Many engineering problems also involve heat transfer and hence temperature gradients. Thermophoresis is a phenomena by which small particles experience a force in the opposite direction to the temperature gradient. Thus particles will tend to deposit on cold walls and be repulsed by hot walls. The effect of thermophoresis on the deposition of particles can be significant. The modifications of the particle tracking facility have been extended to include the effect of thermophoresis. A preliminary test case involving the deposition of particles in a heated pipe has been simulated. Comparison with experimental data from an extensive experimental programme undertaken at ISPRA, known as STORM (Simplified Tests on Resuspension Mechanisms), has been made.     

Numerical simulation of three-dimensional lee waves behind an isolated hill

Stably stratified fluid flow over smooth hills is an important process in environmental fluid mechanics. The aim of this article is to point out some of the key features of this phenomenon from both the physical and the mathematical modeling point of view. Three-dimensional laminar flow over a smooth, axisymmetric hill is considered for a range of values of the Froude number and the Reynolds number. The numerical simulations are performed on the basis of the Boussinesq approximation model, solved with a high-resolution compact finite-difference numerical scheme. The numerical results obtained are discussed with respect to the observed physical phenomena, taking into account the separate role of varying the velocity (Reynolds number) and stratification (Froude number). The mathematical modeling and numerical solvability issues encountered are discussed in detail, pointing out the key role of the computational setup and boundary conditions.     

Nonstationary mass transfer of a reacting spherical particle in laminar stream of a viscous fluid : PMM vol. 38, n 6, 1974, pp. 1041–1047

The process of the formation of a stationary mass transfer mode for a moving reacting particle is examined. An analytic expression valid for a nonstationary distribution of the concentration of matter in a steady stream of viscous fluid, flowing past a spherical particle, was obtained for the case when at a certain instant a chemical reaction of the first order begins at the surface of the sphere. The problem is solved for small finite Reynolds and Péclet numbers. The solution of the corresponding stationary problem has been obtained in [1]. Paper [2] examined a nonstationary heat transfer of a fluid spherical drop in an inviscid flow with spasmodic change of initial temperature at high Péclet numbers. Paper [3] contains an analysis of the problem of a nonstationary heat transfer of a rigid spherical particle for small Reynolds and Péclet numbers at spasmodic change of temperature of the particle surface. The results obtained in [3] can be used to describe the mass transfer for a moving reacting particle only in the case of a diffusion mode of the chemical reaction.     

微极液体在两个多孔圆盘间的MHD流动及其热传导

两个平行的无限大多孔圆盘,圆盘表面有均匀注入时,数值地研究圆盘间不可压缩导电微极流体,在横向外加磁场作用下的轴对称稳定层流.运用von Krmn的相似变换,将非线性运动的控制方程转化为无量纲形式.使用基于有限差分格式的算法,在相应的边界条件下,求解简化后耦合的常微分方程组.讨论Reynolds数、磁场参数、微极参数和Prandtl数,对流动速度和温度分布的影响.在特殊情况下,所得结果与已有文献的工作有着很好的一致性.研究表明,圆盘表面的传热率随着Rynolds数、磁场参数和Prandtl数的增加而增加;剪切应力随着注入的增加而减少,但它随着外部磁场的加强而增加.和Newton流体相比较,微极流体的剪切应力因素较弱,有利于聚合体加工过程中流动和温度的控制.     

Comparison of turbulence models in simulating swirling pipe flows

Swirling flow is a common phenomenon in engineering applications. A numerical study of the swirling flow inside a straight pipe was carried out in the present work with the aid of the commercial CFD code fluent. Two-dimensional simulations were performed, and two turbulence models were used, namely, the RNG k–ε model and the Reynolds stress model. Results at various swirl numbers were obtained and compared with available experimental data to determine if the numerical method is valid when modeling swirling flows. It has been shown that the RNG k–ε model is in better agreement with experimental velocity profiles for low swirl, while the Reynolds stress model becomes more appropriate as the swirl is increased. However, both turbulence models predict an unrealistic decay of the turbulence quantities for the flows considered here, indicating the inadequacy of such models in simulating developing pipe flows with swirl.     

Particle-Laden Turbulent Channel Flow and Particle-Wall Interactions

The behavior of particle-laden gases in a turbulent channel flow is studied at moderate Reynolds number. Effects of the wallparticle interaction models on the velocity, statistics, and dispersion of the particles are presented. The results were obtained from a direct numerical simulation with particle feedback on the gas phase. The models were found to considerably influence the studied particle properties. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

NSM solution for unsteady MHD Couette flow of a dusty conducting fluid with variable viscosity and electric conductivity

The effects of dependence on temperature of the viscosity and electric conductivity, Reynolds number and particle concentration on the unsteady MHD flow and heat transfer of a dusty, electrically conducting fluid between parallel plates in the presence of an external uniform magnetic field have been investigated using the network simulation method (NSM) and the electric circuit simulation program Pspice. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field perpendicular is applied to the plates. We solved the steady-state and transient problems of flow and heat transfer for both the fluid and dust particles. With this method, only discretization of the spatial co-ordinates is necessary, while time remains as a real continuous variable. Velocity and temperature are studied for different values of the viscosity and magnetic field parameters and for different particle concentration and upper wall velocity.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

Magnetohydrodynamics flow of a Burgers’ fluid in an orthogonal rheometer

The magnetohydrodynamics flow of an electrically conducting, incompressible Burgers’ fluid in an orthogonal rheometer is investigated. An exact solution is obtained. The effects of various dimensionless parameters existing in the model on the velocity field, vorticity and traction are studied graphically. It is noted that boundary layers form for a variety of reasons. It form as the Reynolds number increases. Also, as the Weissenberg number increases a distinct boundary layer formation is observed. It can develop at low Reynolds number provided the Weissenberg number is sufficiently high, however, it is not possible in the case of a Newtonian fluid. It is shown that no torque is exerted by the fluid on one of the disks. Results are compared with Oldroyd-B fluid.     

Prandtl's Mixing Length Model - Revisited

The paper is concerned with a modification of Prandtl's mixing length model of Reynolds stresses in fully developed turbulent channel flows. Here it is a well established fact that Prandtl's model falls short to describe the Reynolds stresses correctly very close to the wall. Furthermore, the resulting solution of the time averaged velocity fails to describe the channel flow correctly from the wall to its center. To overcome these shortcomings, the only characteristic mixing length in Prandtl's model is replaced by separate mixing lengths for velocity fluctuations parallel to the wall and normal to the wall, respectively. The modified model describes the mean velocity, all Reynolds stresses, and the functional dependence between Reynolds number based on the mean velocity and the one based on the friction velocity. For all Reynolds numbers these results - and corresponding results for the production terms of Reynolds stresses and the energy balance of the mean flow - agree quantitatively with experimental data and with data obtained by numerical simulations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

计及Hall和离子滑移电流的旋转流体在与时间相关初值条件下的MHDCouette流动

考虑Hall和离子滑移电流的影响,在旋转系统中研究导电流体非稳定的MHD Couette流动．在小数值磁场Reynolds数假定下,推导出基本的控制方程,使用著名的Laplace变换技术,数值地求解该基本方程．分两种情况:磁场相对于流体或者移动平板固定时,得到速度和表面摩擦力统一的闭式表达式．用图形讨论了问题的不同参数,对速度和表面摩擦力的影响．所得结果显示,主流速度u和次生速度v随着Hall电流而增大．离子滑移电流的增大,也会导致主流速度u的增大,但会使次生速度v减小．还给出了旋转、Hall和离子滑移参数的综合影响,确定了次生运动对流体流动的贡献．     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Fully-developed free-convective flow of micropolar and viscous fluids in a vertical channel

The problem of fully-developed laminar free-convection flow in a vertical channel is studied analytically with one region filled with micropolar fluid and the other region with a viscous fluid. Using the boundary and interface conditions proposed by previous investigators, analytical expressions for linear velocity, micro-rotation velocity and temperature have been obtained. Numerical results are presented graphically for the distribution of velocity, micro-rotation velocity and temperature fields for varying physical parameters such as the ratio of Grashof number to Reynolds number, viscosity ratio, width ratio, conductivity ratio and micropolar fluid material parameter. It is found that the effect of the micropolar fluid material parameter suppress the velocity whereas it enhances the micro-rotation velocity. The effect of the ratio of Grashof number to Reynolds number is found to enhance both the linear velocity and the micro-rotation velocity. The effects of the width ratio and the conductivity ratio are found to enhance the temperature distribution.     

矩形微通道热沉内单相稳态层流流体的流动与传热分析

采用解析方法分析了矩形微通道热沉内单相稳态层流流体的流动与传热．基于y方向流速和导热不变的假设,建立流体在矩形微通道内流动的流速方程和传热的温度方程,进而推导出Nusselt数和Poiseuille数的理论表达式．通过计算结果可以看出,推导的Nusselt数和Poiseuille数的解析解与其他文献的结果吻合较好,而且当宽高比趋于无穷大时,Nusselt数和Poiseuille数分别趋近于8.235和96,这与其他文献结果完全相同．在Reynolds数相同时,摩擦因数随着宽高比的增加而增加,而在相同宽高比时,摩擦因数随Reynolds数的增加而减小．