A two-fluid model for pulsatile flow in catheterized blood vessels

The pulsatile flow of blood through a catheterized artery is analyzed, assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the peripheral region of plasma as a Newtonian fluid. The resulting non-linear implicit system of partial differential equations is solved using perturbation method. The expressions for shear stress, velocity, flow rate, wall shear stress and longitudinal impedance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio, amplitude, pulsatile Reynolds number ratio and peripheral layer thickness are discussed. It is observed that the velocity distribution and flow rate decrease, while, the wall shear, width of the plug flow region and longitudinal impedance increase when the yield stress increases. It is also found that the velocity increases, but, the longitudinal impedance decreases when the thickness of the peripheral layer increases. The wall shear stress decreases non-linearly, while, the longitudinal impedance increases non-linearly when the catheter radius ratio increases. The estimates of the increase in the longitudinal impedance are considerably lower for the present two-fluid model than those of the single-fluid model.     

Two-fluid non-linear model for flow in catheterized blood vessels

Blood flow through a catheterized artery is analyzed, assuming the flow is steady and blood is treated as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the plasma in the peripheral region as a Newtonian fluid. The expressions for velocity, flow rate, wall shear stress and frictional resistance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio and peripheral layer thickness are discussed. It is noticed that the velocity and flow rate decrease while the wall shear stress and resistance to flow increase when the yield stress or the catheter radius ratio increases while all the other parameters were held fixed. It is found that the velocity and flow rate increase while the wall shear stress and frictional resistance decrease with the increase of the peripheral layer thickness. The estimates of the increase in the frictional resistance are significantly very small for the present two-fluid model than those of the single-fluid Casson model.     

Laminar flow through channels with porous walls and with an applied transverse magnetic field

Summary The problem of two-dimensional steady laminar flow of a viscous incompressible and electrically conducting fluid through a channel with two equally porous walls in the presence of a transverse magnetic field has been extended to include all values of Hartmann number and small suction velocity at the walls. Expressions for the velocity components, the pressure and the wall friction in terms of the Hartmann number and the suction Reynolds number are given. It is found that the pressure drop in the major flow direction and the wall friction decrease with the increase in suction and increase with the increase in the strength of the magnetic field.     

磁场和Hall电流对狭窄动脉中血液流动的影响

A micropolar model for blood simulating magnetohydrodynamic flow through a horizontally nonsymmetric but vertically symmetric artery with a mild stenosis is presented. To estimate the effect of the stenosis shape, a suitable geometry has been considered such that the horizontal shape of the stenosis can easily be changed just by varying a parameter referred to as the shape parameter. Flow parameters, such as velocity, the resistance to flow （the resistance impedance）, the wall shear stress distribution in the stenotic region, and its magnitude at the maximum height of the stenosis （stenosis throat）, have been computed for different shape parameters, the Hartmann number and the Hall parameter. This shows that the resistance to flow decreases with the increasing values of the parameter determining the stenosis shape and the Hail parameter, while it increases with the increasing Hartmann number. The wall shear stress and the shearing stress on the wall at the maximum height of the stenosis possess an inverse characteristic to the resistance to flow with respect to any given value of the Hartmann number and the Hall parameter. Finally, the effect of the Hartmann number and the Hall parameter on the horizontal velocity is examined.     

倾斜非均匀磁场下导电方管磁流体管流的数值模拟研究

热核聚变反应堆液态金属包层应用中的一个重要问题是液态金属在导电管中流动和强磁场相互作用产生的额外的磁流体动力学压降.这种磁流体动力学压降远远大于普通水力学压降.美国阿贡国家实验室ALEX研究小组,对非均匀磁场下导电管中液态金属磁流体动力学效应进行了实验研究,其实验结果成为液态金属包层数值验证的标准模型之一.液态金属包层在应用中会受到不同方向的磁场作用,本文以ALEX的非均匀磁场下导电方管中液态金属管流实验中的一组参数为基础,保持哈特曼数、雷诺数和壁面电导率不变,采用三维直接数值模拟的方法,研究了外加磁场与侧壁之间的倾角对导电方管内液态金属流动的速度、电流和压降分布的影响.研究结果表明:沿流向相同横截面上的速度、电流以及压力分布均随磁场的倾斜而同向旋转.倾斜磁场均匀段,横截面上的高速区位于平行磁场方向的哈特曼层和平行层交叉位置,压力梯度随磁场倾角的增大先增大后减小.倾斜磁场递减段,在三维磁流体动力学效应作用下,横截面上的高速射流位置向垂直磁场方向偏移.磁场递减段的三维磁流体动力学压降随磁场倾角的增大而增大.随磁场倾斜,截面上的射流峰值逐渐减小,二次流增强,引发层流向湍流的转捩.      

A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels

In this paper, we discussed a mathematical model for two-layered non-Newtonian blood flow through porous constricted blood vessels. The core region of blood flow contains the suspension of erythrocytes as non-Newtonian Casson fluid and the peripheral region contains the plasma flow as Newtonian fluid. The wall of porous constricted blood vessel configured as thin transition Brinkman layer over layered by Darcy region. The boundary of fluid layer is defined as stress jump condition of Ocha-Tapiya and Beavers–Joseph. In this paper, we obtained an analytic expression for velocity, flow rate, wall shear stress. The effect of permeability, plasma layer thickness, yield stress and shape of the constriction on velocity in core & peripheral region, wall shear stress and flow rate is discussed graphically. This is found throughout the discussion that permeability and plasma layer thickness have accountable effect on various flow parameters which gives an important observation for diseased blood vessels.     

倾斜吹吸控制下湍流边界层减阻的直接数值模拟

雷诺切应力是壁湍流高摩擦阻力的重要来源, 有理论认为可以通过壁面生成负雷诺应力(数值上为正)的方式来削弱湍流流场中雷诺应力的分布, 以此获得流动减阻. 而通过对雷诺平均运动方程的法向二次积分, 可以发现壁面生成正雷诺应力(数值上为负)对壁面摩擦阻力系数才有负贡献. 文中在湍流边界层流动的控制区域下边界设置一系列倾斜狭缝, 利用该装置通过周期性吹吸的方法产生壁面生成正(负)雷诺应力, 并采用直接数值模拟方法考察和验证上文提到的减阻理论. 文中采用的湍流边界层流动模型, 其流动雷诺数(基于外流速度及动量损失厚度)从300 发展到860. 文中通过多组数值模拟算例, 考察了射流强度和频率对壁面摩擦阻力系数的影响, 并对比了壁面生成正或负雷诺应力对流动的影响. 研究表明, 壁面生成正雷诺应力控制的减阻率能达到3.26, 而壁面生成负雷诺应力控制的减阻效果较壁面生成正雷诺应力控制的要差; 壁面生成的正雷诺应力对壁面摩擦阻力有负贡献, 而壁面生成的负雷诺应力对壁面摩擦阻力有正贡献; 通过考察控制的收支比, 发现控制方案不能获得能量净收益.      

Wall functions for numerical modeling of laminar MHD flows

A general wall function treatment is presented for the numerical modeling of laminar magnetohydrodynamic (MHD) flows. The wall function expressions are derived analytically from the steady-state momentum and electric potential equations, making use only of local variables of the numerical solution. No assumptions are made regarding the orientation of the magnetic field relative to the wall, nor of the magnitude of the Hartmann number, or the wall conductivity. The wall functions are used for defining implicit boundary conditions for velocity and electric potential, and for computing mass flow and electrical currents in near wall-cells. The wall function treatment was validated in a finite volume formulation, and compared with an analytic solution for a fully developed channel flow in a transverse magnetic field. For the case with insulating walls, a uniform 20×20 grid, and Hartmann numbers Ha={10,30,100}, the accuracy of pressure drop and wall shear stress predictions was {1.1%,1.6%,0.5%}, respectively. Comparable results were obtained also with conducting Hartmann walls. The accuracy of predicted pressure drop and wall shear stress was essentially independent of the resolution of the Hartmann layers. When applied also to the parallel walls, the wall functions reduced the errors by a factor two to three. The wall functions can be implemented in any general flow solver, to allow accurate predictions at reasonable cost even for complex geometries and nonuniform magnetic fields.     

Structure of near wall turbulence downstream of a wall mounted protrusion: an interesting Reynolds stress suppression phenomena

A local suppression in the generation of near wall Reynolds stress is achieved by modifying the buffer region and sublayer (y ⁺ <30) of a turbulent pipe flow with a 16.4 wall unit high wall mounted protrusion. Multi-component, multi-point, time resolved laser Doppler velocimetry measurements are made in the undisturbed and modified ARL/PSU glycerin tunnel pipe flow at a Reynolds number of approximately 10000. A downstream converging flow field is produced by the divergence of the approaching mean flow around the protrusion. A pair of counter-rotating vortices, 15 wall units in diameter with common flow down, are generated by the protrusion and also contribute to the wall directed flow convergence. The convergence region is 15 wall units high and more than 100 wall units long and appears to decouple the near wall region from the outer turbulent wall layer. Locally, turbulent velocity fluctuations in the form of Reynolds stress producing events, sweeps and ejections, are retarded within this region. This results in a reduction in near wall uv Reynolds stress and local wall shear. Interestingly, the counter-rotating vortices act to increase turbulent diffusion in a manner which is uncorrelated with Reynolds stress generation.     

Magnetic field effect on heat transfer and fluid flow characteristics of blood flow in multi-stenosis arteries

Heat and fluid flow characteristics of blood flow in multi-stenosis arteries in the presence of magnetic field is considered. A mathematical model of the multi-stenosis inside the arteries is introduced. A finite difference scheme is used to solve the governing equations in terms of vorticity-stream function along with their boundary conditions. The effect of magnetic field and the degree of stenosis on wall shear stress and Nusselt number is investigated. It was found that magnetic field modifies the flow patterns and increases the heat transfer rate. The severity of the stenosis affects the wall shear stress characteristics significantly. The magnetic field torque will increase the thermal boundary layer thickness and the temperature gradient in the streaming blood, and hence increasing the local Nusselt number     

Study of laminar–turbulent flow transition under pulsatile conditions in a constricted channel

In this paper, direct numerical simulation is performed to investigate a pulsatile flow in a constricted channel to gain physical insights into laminar–turbulent–laminar flow transitions. An in-house computer code is used to conduct numerical simulations based on available high-performance shared memory parallel computing facilities. The Womersley number tested is fixed to 10.5 and the Reynolds number varies from 500 to 2000. The influences of the degree of stenosis and pulsatile conditions on flow transitions and structures are investigated. In the region upstream of the stenosis, the flow pattern is primarily laminar. Immediately after the stenosis, the flow recirculates under an adverse streamwise pressure gradient, and the flow pattern transitions from laminar to turbulent. In the region far downstream of the stenosis, the flow becomes re-laminarised. The physical characteristics of the flow field have been thoroughly analysed in terms of the mean streamwise velocity, turbulence kinetic energy, viscous wall shear stresses, wall pressure and turbulence kinetic energy spectra.     

Two-phase non-linear model for blood flow in asymmetric and axisymmetric stenosed arteries

The pulsatile flow of a two-phase model for blood flow through axisymmetric and asymmetric stenosed narrow arteries is analyzed, treating blood as a two-phase model with the suspension of all the erythrocytes in the core region as the Herschel-Bulkley material and plasma in the peripheral layer as the Newtonian fluid. The perturbation method is applied to solve the resulting non-linear implicit system of partial differential equations. The expressions for various flow quantities are obtained. It is found that the pressure drop, plug core radius, wall shear stress increase as the yield stress or stenosis height increases. It is noted that the velocity increases, longitudinal impedance decreases as the amplitude increases. For asymmetric stenosis, the wall shear stress increases non-linearly with the increase of the axial distance. The estimates of the increase in longitudinal impedance to flow of the two-phase Herschel-Bulkley material are significantly lower than those of the single-phase Herschel-Bulkley material. The results show the advantages of two-phase flow over single-phase flow in small diameter arteries with stenosis.     

An inclined wall jet: Mean flow characteristics and effects of acoustic excitation

 The mean velocity field of a 30° inclined wall jet has been investigated using both hot-wire and laser Doppler anemometry (LDA). Provided that the nozzle aspect ratio is greater than 30 and the inclined wall angle (β) is less than 50°, LDA measurements for various β show that the reattachment length is independent of the nozzle aspect ratio and the nozzle exit Reynolds number (in the range 6670–13,340). There is general agreement between the reattachment lengths determined by LDA and those determined using wall surface oil film visualisation technique. The role of coherent structures arising from initial instabilities of a 30° wall jet has been explored by hot-wire spectra measurements. Results indicate that the fundamental vortex roll-up frequency in both the inner and outer shear layer corresponds to a Strouhal number (based on nozzle exit momentum thickness and velocity) of 0.012. The spatial development of instabilities in the jet has been studied by introducing acoustic excitation at a frequency corresponding to the shear layer mode. The formation of the fundamental and its first subharmonic has been identified in the outer shear layer. However, the development of the first subharmonic in the inner shear layer has been severely suppressed. Distributions of mean velocities, turbulence intensities and Reynolds shear stress indicate that controlled acoustic excitation enhances the development of instabilities and promotes jet reattachment to the wall, resulting in a substantially reduced recirculation flow region. Received: 24 November 1998/Accepted: 24 August 1999     

Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall

An analysis has been carried out to study the effect of magnetic field on an electrically conducting fluid of second grade in a parallel channel. The coolant fluid is injected into the porous channel through one side of the channel wall into the other heated impermeable wall. The combined effect of inertia, viscous, viscoelastic and magnetic forces are studied. The basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations by using appropriate transformations for velocity and temperature. Numerical solutions of these equations are obtained with the help of Runge-Kutta fourth order method in association with quasi-linear shooting technique. Numerical results for velocity field, temperature field, skin friction and Nusselt number are presented in terms of elastic parameter, Hartmann number, Prandtl number and Reynolds number. Special case of our results is in good agreement with earlier published work.     

Turbulent separated reattached flow in a two-dimensional curved-wall diffuser

A turbulent separation-reattachment flow in a two-dimensional asymmetrical curved-wall diffuser is studied by a two-dimensional laser doppler velocimeter. The turbulent boundary layer separates on the lower curved wall under strong pressure gradient and then reattaches on a parallel channel. At the inlet of the diffuser, Reynolds number based on the diffuser height is 1.2×10⁵ and the velocity is 25.2m/s. The results of experiments are presented and analyzed in new defined streamline-aligned coordinates. The experiment shows that after Transitory Detachment Reynolds shear stress is negative in the near-wall backflow region. Their characteristics are approximately the same as in simple turbulent shear layers near the maximum Reynolds shear stress. A scale is formed using the maximum Reynolds shear stresses. It is found that a Reynolds shear stress similarity exists from separation to reattachment and the Schofield-Perry velocity law exists in the forward shear flow. Both profiles are used in the experimental work that leads to the design of a new eddy-viscosity model. The length scale is taken from that developed by Schofield and Perry. The composite velocity scale is formed by the maximum Reynolds shear stress and the Schofield-Perry velocity scale as well as the edge velocity of the boundary layer. The results of these experiments are presented in this paper.     

The steady/pulsatile flow and macromolecular transport in T-bifurcation blood vessels

IntroductionThefluiddynamicsofbloodcyclesystemplayanimportantroleinthepathogenesisofatherosclerosis.ThephysiologicalanatomyfoundthattheatherosclerosisappearsoftenatthebifurcationorcurvedflowareatoallkindsofRefs.[1 ,2 ] .Theshearstressvariesgreatlyinthoseareaandinfluencesthemacromoleculartransportacrossthebloodwall[3,4].Thus ,theinvestigationoftheflowandmacromoleculartransportinthesecomplexbloodvesselaandthecorrelationbetweenthemareinterestingtoresearchers.Intheseaspect,Liepschstudiedtheflowi…     

An experimental study of the dynamic characteristics in boundary layer in the flow past a two-dimensional circular cylinder at subcritical reynolds numbers

An experimental study of the dynamic characteristics of flow past a two-dimensional circular cylinder is described. The fluctuationsoof wall shear stress, surface-pressure and velocity of the flow are measured with hot-film, hot-wire and pressure transducer. The frequency feature of fluctuations of wall shear stress is given. The cross-correlation functions of these fluctuations at any two points are calculated. The experimental results reveal that there is an overall syncronous fluctuation, at the shedding frequency, in boundary layer in the flow past a two-dimensional circular cylinder at subcritical Reynolds number.     

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.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

Numerical prediction of turbulent boundary layer noise from a sharp-edged flat plate

An efficient hybrid uncorrelated wall plane waves–boundary element method (UWPW-BEM) technique is proposed to predict the flow-induced noise from a structure in low Mach number turbulent flow. Reynolds-averaged Navier-Stokes equations are used to estimate the turbulent boundary layer parameters such as convective velocity, boundary layer thickness, and wall shear stress over the surface of the structure. The spectrum of the wall pressure fluctuations is evaluated from the turbulent boundary layer parameters and by using semi-empirical models from literature. The wall pressure field underneath the turbulent boundary layer is synthesized by realizations of uncorrelated wall plane waves (UWPW). An acoustic BEM solver is then employed to compute the acoustic pressure scattered by the structure from the synthesized wall pressure field. Finally, the acoustic response of the structure in turbulent flow is obtained as an ensemble average of the acoustic pressures due to all realizations of uncorrelated plane waves. To demonstrate the hybrid UWPW-BEM approach, the self-noise generated by a flat plate in turbulent flow with Reynolds number based on chord Re_c = 4.9 × 10⁵ is predicted. The results are compared with those obtained from a large eddy simulation (LES)-BEM technique as well as with experimental data from literature.