Simultaneous measurements of velocity and deformation in flows through compliant diaphragms

Flow through a circular orifice in a deformable diaphragm mounted in a pipe was studied experimentally as a simple yet suitable case for validating numerical fluid/structure interaction (FSI) codes including structures with significant deformation and strain. The flow was characterized using pressure taps, particle image velocimetry (PIV), and hot-film anemometry while deformation of the compliant diaphragm was determined directly from PIV images. The diaphragm material properties were measured independently by a uniaxial tensile testing machine. The diaphragm material modulus, orifice diameter, and pipe Reynolds number were varied over ranges appropriate for simulations of flows through heart valves. Pipe Reynolds numbers ranged from 600 (laminar upstream condition) to 8800 (turbulent upstream condition). The pressure drop across the diaphragm resulted in a concave deformation for all cases studied. For the range of Reynolds number tested, the Euler number decreased with increasing Reynolds number as a result of orifice expansion. The flow immediately downstream of compliant diaphragms was jet-like with strong inward radial velocity components and vena contracta. Laminar low Reynolds number flow (Re=600) through both rigid and compliant diaphragms yielded early and regular roll up of coherent vortex rings at a fixed frequency in contrast to turbulent higher Reynolds number flow (Re=3900), which yielded a broad range of vortex passage frequencies. Expansion of the compliant orifice for Re=3900 resulted in an initially broader slower jet with delayed shear layer development compared with the equivalent rigid case.     

柔性壁面湍流边界层相干结构控制的实验研究

本文利用热膜测速技术对刚性壁面和柔性壁面湍流边界层的流向速度分量进行了实验测量,首先研究了柔性壁面对平均速度分布和湍流度分布的影响,结果表明:柔性壁面的边界层速度分布在对数律层向上有所平移,缓冲层加厚,具有一般的壁面减阻特征;而柔性壁的湍流度比刚性壁的湍流度要低,分布也更为平坦。然后综合运用自相关法和条件采样技术研究了湍流近壁区的相干结构,结果表明:刚性壁自相关曲线的第二峰值出现的时间比柔性壁的短,柔性壁的猝发频率比刚性壁的低许多。实验结果表明柔性壁面具有一定的减阻作用。     

Flow-rate limitation in a uniform thin-walled collapsible tube,with comparison to a uniform thick-walled tube and a tube of tapering thickness

Previous experiments on a tapered-thickness tube showed qualitatively different behaviour from that exhibited by a uniform thick-walled tube. To understand whether the taper or the thinner wall was responsible, similar aqueous flow-limitation experiments were conducted on a uniform thin-walled tube of the same material, with all other experimental set-up the same. As in the thick tube, there was a dramatic reduction in flow-rate when collapse and flow limitation started, but during external pressure reduction, the limited flow-rate progressively increased, so that as in the tapered-thickness tube, there was little flow-rate increase when collapse ceased. Hysteresis was thus a prominent feature of the relationship between flow-rate and pressure drop along curves of constant upstream transmural pressure. Flow-rate limitation was mainly accompanied by large-amplitude self-excited oscillation for both increasing and decreasing external pressure, to an even greater extent than in the tapered-thickness tube. Clusters of points sharing the same pair of upstream transmural pressure and upstream driving pressure values were found, indirectly implying as in the tapered-thickness tube that the flow-limited flow-rate for a given pressure drop was not uniquely determined by upstream transmural pressure. Negative effort dependence was observed in all three tubes, but in the thin tube, as in the tapered-thickness tube, it was obscured for some values of upstream transmural pressure where low-frequency single-collapse-per-cycle oscillations occurred. Thus, the qualitatively unique properties of the tapered-thickness tube appear to be confined to the relative lack of hysteresis, and the oscillatory regime in which collapse ceased before the downstream end. The rest of the observed behaviours seem to be characteristic simply of more compliant tubing.     

Propagation of a Tollmien-Schlichting wave over the junction between rigid and compliant surfaces

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 31–48. Original Russian Text Copyright © 2004 by Manuilovich.     

Simulations of flow through open cell metal foams using an idealized periodic cell structure

A new approach to modeling the flow through a porous medium with a well defined structure is presented. This approach entailed modeling an idealized open cell metal foam based on a fundamental periodic unit of eight cells and solving the flow through the three-dimensional cellular unit. To model an infinitely large matrix, periodic boundary conditions were set on the walls parallel to the flow direction, while a pseudo-periodic boundary condition with a prescribed volumetric flow rate was set over the inlet–outlet pair of the unit cell. The pressure drop data of the flow through the cellular unit were then compared on a length-normalized basis against experimental data. The pressure drop values predicted by the simulations were consistently 25% lower than the values obtained in the experiments on a similar foam and under identical flow conditions. One explanation for the discrepancy between the two sets of data is the lack of pressure drop increasing wall effects in the simulations. The increase in the pressure drop from wall effects in the simulation was quantified.     

各向异性柔性壁上二维T-S波演化的数值研究

受自然界鸟类羽毛的柔性特征启发, 利用数值模拟的手段进行了各向异性柔性壁面对亚音速边界层中T-S(Tollmien-Schlichting)波空间演化的影响研究. 首先, 刚性壁面上的数值结果与线性理论预测的结果吻合得很好, 验证了所采用的高阶精度格心型有限差分方法的可靠性. 在此基础上, 将部分刚性壁面替换为柔性壁面, 结果表明柔性壁面能够减小甚至消除T-S波的不稳定增长区间, 即抑制T-S波的发展, 因而具有推迟边界层转捩的潜力. 柔性壁面的变形不仅有对应T-S波波形的成分, 还会因柔性段前缘引起波长更长, 与T-S波频率相同的壁面波动. 随后开展的参数研究表明, 增大壁面阻尼削弱了前缘引起的壁面波动; 增大壁面的刚度、张力以及弹性系数都会使得壁面的刚性增强, 整体变形幅度下降; 柔性壁面的支撑杠杆臂倾角越大, 壁面刚性越强. 以上参数的增大均会使得柔性壁面抑制T-S波的效果降低. 此外, 当流动反方向流过时, 抑制T-S波的效果也会明显下降. 这些研究结果旨在揭示鸟类高效飞行的部分奥秘, 为被动减阻提供新的思路.      

动脉瘤内流场以及瘤体尺寸的影响的数值研究

采用计算流体动力学(CFD)数值模拟的方法,在周期性脉动速度入流条件下,建立刚性动脉瘤模型并研究了动脉瘤模型中流场的特征(速度、压力、壁面剪切应力)。得到了脉动入流一个周期内流场特征的变化规律,发现动脉瘤的后端有相当高的压力和壁面剪切应力,而且高压力和壁面剪切应力分布的位置几乎是固定的。探讨了不同动脉瘤尺寸对内部流场的影响,动脉瘤的直径与瘤体长度之比越大,瘤壁承受的剪切应力就越大,动脉瘤破裂的危险性就越高。     

SIMPLE RESISTIVE MODELS FOR STEADY FLOW THROUGH UNIFORM AND TAPERED-STIFFNESS COLLAPSIBLE TUBES

An extremely basic model is postulated and examined numerically, to find out which aspects of observed steady-flow collapsible-tube behaviour are predicted and can be explained. The model in simplest form states that the tube has a fixed viscous resistance per unit length when not collapsed, and a higher one when collapsed. Collapse occurs where the falling internal pressure in the streamwise direction causes negative transmural pressure to pass a fixed threshold set by tube wall stiffness. This model suffices to explain (i) the sigmoidal dependence of pressure drop on flow rate when external pressure is fixed, (ii) the weak dependence of pressure-drop on flow rate when downstream transmural pressure is fixed and (iii) the weak dependence of flow rate on pressure drop when upstream transmural pressure is fixed. The effects of incorporating more realistic collapse behaviour (finite compliance once the tube buckles, varying compliance once opposite walls are in contact) on these dependencies are examined. The model is also used to explore the several qualitatively distinct configurations that may be taken up by a tube which varies in stiffness along its length.     

Boundary layer receptivity measurements on compliant surfaces

This paper describes receptivity measurements in a pre-transitional boundary layer flowing over either a rigid or a compliant surface. Fluctuating velocities and frequency spectra were determined on one rigid and nine compliant surfaces. The results showed that the near wall receptivity grows linearly with Re_θ. An empirical correlation of the gain frequency spectrum for a rigid wall was also established. For the compliant surfaces, the near wall gain is increased markedly near the leading edge of the plate due to the amplification of high and mid-frequencies. These frequencies are dissipated though as the flow progresses over the compliant surface such that the receptivity is lower on all the compliant surfaces than on the rigid surface at the trailing edge. An empirical correlation for the ratio of the gains on compliant and rigid surfaces in terms of the compliant surface coefficient ζ²/Eρ_CSL² and Re_θ was established. This correlation indicates that compliant surfaces can suppress receptivity by up to 25% for a Re_θ = 400.     

Exact solution for compressive flow of viscoplastic fluids under perfect slip wall boundary conditions

Based on the perfect slip condition between rigid walls and fluids, the compressive flow of Herschel-Bulkley fluids and biviscous fluids was studied. The explicit expressions of stresses and fluid velocity were given. To move the rigid walls for a Herschel-Bulkley fluid with the yield stress (τ₀), the mean pressure applied onto the rigid wall should be larger than 2τ₀/. No yield surface exists in the interior of the fluids when flow occurs. For a biviscous fluid, a critical load was given. The fluid behaves like the Bingham fluid when the external applied load onto the wall is larger than the critical load, otherwise the fluid is Newtonian. Received: 10 June 1997 Accepted: 22 September 1997     

Flow and heat/mass transfer in a wavy duct with various corrugation angles in two dimensional flow regimes

In this study, two dimensional heat/mass transfer characteristics and flow features were investigated in a rectangular wavy duct with various corrugation angles. The test duct had a width of 7.3 mm and a large aspect ratio of 7.3 to simulate two dimensional characteristics. The corrugation angles used were 100°, 115°, 130°, and 145°. Numerical analysis using the commercial code FLUENT, was used to analyze the flow features. In addition, the oil-lamp black method was used for flow visualization. Local heat/mass transfer coefficients on the corrugated walls were measured using a naphthalene sublimation technique. The Reynolds number, based on the duct hydraulic diameter, was varied from 700 to 5,000. The experimental results and numerical analysis showed interesting and detailed features in the wavy duct. Main flow impinged on upstream of a pressure wall, and the flow greatly enhanced heat/mass transfer. On a suction wall, however, flow separation and reattachment dominantly affected the heat/mass transfer characteristics on the wall. As the corrugation angle decreased (it means the duct has more sharp turn), the region of flow stagnation at the front part of the pressure wall became wider. Also, the position of flow reattachment on the suction wall moved upstream as the corrugation angle decreased. A high heat transfer rate appeared at the front part of the pressure wall due to main-flow impingement, and at the front part of the suction wall due to flow reattachment. The high heat/mass transfer region by the main-flow impingement and the circulation flow induced at a valley between the pressure and suction walls changed with the corrugation angle and the Reynolds number. As the corrugation angle decreased, the flow in the wavy duct changed to transition to turbulent flow earlier.     

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.     

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.     

Numerical study of effects of pulsatile amplitude for transitional turbulent pulsatile flow in pipes with ring‐type constrictions

The effects of pulsatile amplitude on sinusoidal transitional turbulent flows through a rigid pipe in the vicinity of a sharp‐edged mechanical ring‐type constriction have been studied numerically. Pulsatile flows were studied for transitional turbulent flow with Reynolds number (Re) of the order of 10⁴, Womersley number (Nw) of the order of 50 with a corresponding Strouhal number (St) of the order of 0.04. The pulsatile flow considered is a sinusoidal flow with dimensionless amplitudes varying from 0.0 to 1.0. Transitional laminar and turbulent flow characteristics in an alternative manner within the pulsatile flow fields were observed and studied numerically. The flow characteristics were studied through the pulsatile contours of streamlines, vorticity, shear stress and isobars. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity, turbulent shear stress are smaller during the acceleration phase when compared with those during deceleration period. Various parametric equations within a pulsatile cycle have also been formulated through numerical experimentations with different pulsatile amplitudes. In the vicinity of constrictions, the empirical relationships were obtained for the instantaneous flow rate (Q), the pressure gradient (dp/dz), the pressure loss (P_loss), the maximum velocity (V_max), the maximum vorticity (ζ_max), the maximum wall vorticity (ζ_w,max), the maximum shear stress (τ_max) and the maximum wall shear stress (τ_w,max). Elliptic relation was observed between flow rate and pressure gradient. Quadratic relations were observed between flow rate and the pressure loss, the maximum values of shear stress, wall shear stress, turbulent kinematic energy and the turbulent viscosity. Linear relationships exist between the instantaneous flow rate and the maximum values of vorticity, wall vorticity and velocity. The time‐average axial pressure gradient and the time average pressure loss across the constriction were observed to increase linearly with the pulsatile amplitude. Copyright © 1999 John Wiley & Sons, Ltd.     

Pulsatile magneto-biofluid flow and mass transfer in a non-Darcian porous medium channel

A numerical study of pulsatile flow and mass transfer of an electrically conducting Newtonian biofluid via a channel containing porous medium is considered. The conservation equations are transformed and solved under boundary conditions prescribed at both walls of the channel, using a finite element method with two-noded line elements. The influence of magnetic field on the flow is studied using the dimensionless hydromagnetic number, Nm, which defines the ratio of magnetic (Lorentz) retarding force to the viscous hydrodynamic force. A Darcian linear impedance for low Reynolds numbers is incorporated in the transformed momentum equation and a second order drag force term for inertial (Forchheimer) effects. Velocity and concentration profiles across the channel width are plotted for various values of the Reynolds number (Re), Darcy parameter (λ), Forchheimer parameter (Nf), hydro-magnetic number (Nm), Schmidt number (Sc) and also with dimensionless time (T). Profiles of velocity varying in space and time are also provided. The conduit considered is rigid with a pulsatile pressure applied via an appropriate pressure gradient term. Increasing the hydromagnetic number (Nm) from 1 to 15 considerably depresses biofluid velocity (U) indicating that a magnetic field can be used as a flow control mechanism in, for example, medical applications. A rise in Nf from 1 to 20 strongly retards the flow development and decreases the velocity, U, across the width of the channel. The effects of other parameters on the flowfield are also discussed at length. The flow model also has applications in the analysis of electrically conducting haemotological fluids flowing through filtration media, diffusion of drug species in pharmaceutical hydromechanics, and also in general fluid dynamics of pulsatile systems.     

Numerical Simulation of the Flow in the Carotid Bifurcation

Pulsatile flow through the three-dimensional carotid artery bifurcation has been studied using the artificial-compressibility method. The part of the flow with large inertia bifurcates and creates a very steep velocity gradient on the divider walls. The flow near the nondivider walls slows down because of dilation of the cross section and strong adverse pressure gradient. The secondary flow in the bifurcation region, which is similar to the Dean vortex in a curved pipe, is strong and very complex. The region of separation is not closed for the cases of steady and pulsatile flow. The extent of this region is small and the streamlines are smooth except in the decelerating phase of systole. The change of common-internal bifurcation angle (25°± 15°) for fixed internal–external bifurcation angle of 50° has more effect on the shear on the bifurcation-internal carotid wall and less effect on the shear on the common-internal carotid wall. The mean wall shears are not sensitive to the input flow-rate waveform for constant mean flow, but the maximum wall shears are. Received 3 January 1997 and accepted 11 April 1997     

Origin of flow asymmetry in planar nozzles with separation

An experimental investigation was conducted to study the mechanisms that lead to the origin of flow asymmetry in overexpanded planar nozzles, especially at low nozzle pressure ratios. Three Mach 2 planar nozzles with different divergent wall angles but same area-ratio were tested. For all three nozzles, a large portion of the dimensional pressure rise data across the separation shock shows the nature of boundary layer to be in the laminar/transitional state. Depending upon the local flow conditions, the flow can, therefore, experience either an early or a delayed separation on either wall. This can result in a free or a restricted shock separation condition on either wall which can initiate the beginning of flow asymmetry in nozzles at low nozzle pressure ratio. However, a higher nozzle wall angle was observed to prevent initiation of such a flow asymmetry. The present tests, therefore, indicate that in addition to the state of the boundary layer along the nozzle wall, the proximity of the separated shear layer to the nozzle walls also seems to play a dominant role in initiating conditions that favor the origin of flow asymmetry in nozzles. A significant drop in the shock unsteadiness levels was also indicated by increasing the wall angle.     

A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls

An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.The theory is applied to various cases including two- and three-dimensional disturbances, developing in boundary layers over isotropic and anisotropic compliant walls. The disturbances can be treated as either temporally or spatially growing. Eigenvalues are very accurately predicted by means of the theory, especially near points of neutral stability. The computational requirements are trivial compared with those required for full numerical solution of the Orr-Sommerfeld equation. For isotropic compliant walls the theory confirms the earlier result of Miles and Benjamin that the phase shift in the disturbance velocity across the critical layer plays a dominant role in destabilization of the Class B travelling-wave flutter through making irreversible energy transfer possible due to the work done by the fluctuating pressure at the wall. The theory elucidates the secondary role played by the phase shift occurring across the wall layer. Viscous effects are much more important for anisotropic compliant walls which admit substantial horizontal, as well as vertical, displacement. For these walls an important mechanism for irreversible energy transfer is the work done by fluctuating shear stress. This almost invariably has a stabilizing effect on the travelling-wave flutter. In addition there is a weaker effect arising from the effect of anisotropic wall compliance on the phase shift across the wall layer. This may be stabilizing or destabilizing.This work was carried out with the support of the Ministry of Defence (Procurement Executive) and the Office of Naval Research and was completed while P.W.C. and J.S.B.G. were on study leave at the Department of Aerospace Engineering, The Pennsylvania State University, and the Department of Mathematics, Iowa State University, Ames, respectively. They would like to express their gratitude to those institutions and the Office of Naval Research for financial support during their study leaves.     

动脉局部狭窄时脉动流的有限元分析

本文利用有限元方法研究动脉局部狭窄下的脉动流流场,重点考查在50％与80％面积狭窄下的速度分布、压力分布、壁面剪应力分布及流动分离情况。几何形状及边界条件均模拟相应的脉动流实验模型。采用测得的随时间变化的速度分布作为入口端条件,并利用罚函数和逆风格式等计算技巧得出了光滑的与实验基本相符的速度、压力波形。本文讨论了不同狭窄下速度、压力、壁面剪应力的分布形态,给出了脉动流中狭窄处局部流动分离的间歇性变化规律,并结合实验与临床应用进行了讨论。     

NUMERICAL SIMULATION OF STEADY FLOW IN A COMPLIANT TUBE OR CHANNEL WITH TAPERED WALL THICKNESS

Flow through compliant tubes with linear taper in wall thickness is numerically simulated by finite element analysis. Two models are examined: a compliant channel and an axisymmetric tube. For verification of the numerical method, flow through a compliant stenotic vessel is simulated and compared to existing experimental data. Steady two-dimensional flow in a collapsible channel with initial tension is also simulated and the results are compared with numerical solutions from the literature. Computational results for an axisymmetric tube show that as cross-sectional area falls with a reduction in downstream pressure, flow rate increases and reaches a maximum when the speed index (mean velocity divided by wave speed) is near unity at the point of minimum cross-sectional area, indicative of wave-speed flow limitation or “choking” (flow speed equals wave speed) in previous one-dimensional studies. For further reductions in downstream pressure, the flow rate decreases. Cross-sectional narrowing is significant but localized. For the particular wall and fluid properties used in these simulations, the area throat is located near the downstream end when the ratio of downstream-to-upstream wall thickness is 2; as wall taper is increased to 3, the constriction moves to the upstream end of the tube. In the planar two-dimensional channel, area reduction and flow limitation are also observed when outlet pressure is decreased. In contrast to the axisymmetric case, however, the elastic wall in the two-dimensional channel forms a smooth concave surface with the area throat located near the mid-point of the elastic wall. Though flow rate reaches a maximum and then falls, the flow does not appear to be choked.