Three‐dimensional numerical simulation of red blood cell motion in Poiseuille flows

An immersed boundary method based on an FEM has been successfully combined with an elastic spring network model for simulating the dynamical behavior of a red blood cell (RBC) in Poiseuille flows. This elastic spring network preserves the biconcave shape of the RBC in the sense that after the removal of the body force for driving the Poiseuille flow, the RBC with its typical parachute shape in a tube does restore its biconcave resting shape. As a benchmark test, the relationship between the deformation index and the capillary number of the RBCs flowing through a narrow cylindrical tube has been validated. For the migration properties of a single cell in a slit Poiseuille flow, a slipper shape accompanied by a cell membrane tank‐treading motion is obtained for Re , and the cell mass center is away from the center line of the channel due to its asymmetric slipper shape. For the lower Re ?0.0137, an RBC with almost undeformed biconcave shape has a tumbling motion. A transition from tumbling to tank‐treading happens at the Reynolds number between 0.0137 and 0.03. In slit Poiseuille flow, the RBC can also exhibit a rolling motion like a wheel during the migration when the cell is released in the fluid flow with φ = π/2 and θ = π/2 (see Figure 12 for the definition of φ and θ). The lower the Reynolds number, the longer the rolling motion lasts; but the equilibrium shape and position are independent from the cell initial position in the channel. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparison of the traffic performance of a two-axle four wheel drive (4WD), rear wheel drive (RWD), and front wheel drive (FWD) vehicle on loose sandy sloped terrain

The traffic performances during driving and braking of a 5.88 kN weight wheeled vehicle with two-axle four wheel drive, rear wheel drive, and front wheel drive running up and down a loose sandy sloped terrain were compared by means of a simulation. For the given dimensions of the vehicle and the given terrain-wheel system constants, the relationship between the effective tractive and braking effort of the vehicle, the amount of sinkage of the front and rear wheels, the total amount of sinkage of the vehicle, and the slip ratio were calculated to estimate the optimum height of force of application and the optimum eccentricity of the center of gravity of the vehicle. It was observed that, during driving action, the maximum effective tractive effort of the four wheel drive vehicle (4WD) was larger than that of the rear wheel drive vehicle (RWD), which in turn was greater than that of the front wheel drive vehicle (FWD). During the braking action, the effective braking effort at skid -20% of the four wheel vehicle (4WB) was larger than that of the front wheel brake vehicle (FWB), in turn greater than that of the rear wheel brake vehicle (RWB), when the two-axle four wheel vehicle is moving up or down the loose sandy sloped terrain. The maximum terrain slope angle up which the two-axle wheeled vehicle is able to move during driving action was found to be about 0.067π rad for the 4WD vehicle, about 0.031π rad for the RWD vehicle, and about 0.017π rad for the FWD vehicle. The effective braking effort at skid-20% of 4WB, FWB and RWB was found to decrease with slope angle.     

Direct numerical simulation of a turbulent boundary layer up to Reθ = 2500

Direct numerical simulations (DNSs) of a turbulent boundary layer (TBL) with Re_θ = 570-2560 were performed to investigate the spatial development of its turbulence characteristics. The inflow simulation was conducted in the range Re_θ = 570-1600 by using Lund’s method. To resolve the numerical periodicity induced by the recycling method, we adopted a sufficiently long streamwise domain of x/θ_in,i = 1000 (=125δ_0,i), where θ_in,i is the inlet momentum thickness and δ_0,i is the inlet boundary layer thickness in the inflow simulation. Furthermore, the main simulation with a length greater than 50δ₀ was carried out independently by using the inflow data, where δ₀ is the inlet boundary layer thickness of the main simulation. The integral quantities and the first-, second- and higher-order turbulence statistics were compared with those of previous data, and good agreement was found. The present study provides a useful database for the turbulence statistics of TBLs. In addition, instantaneous field and two-point correlation of the streamwise velocity fluctuations displayed the existence of the very large-scale motions (VLSMs) with the characteristic widths of 0.1-0.2δ and that the flow structure for a length of approximately ∼6δ fully occupies the streamwise domain statistically.     

Adhesive contact on power-law graded elastic solids: The JKR–DMT transition using a double-Hertz model

A cohesive zone model of axisymmetric adhesive contact between a rigid sphere and a power-law graded elastic half-space is established by extending the double-Hertz model of Greenwood and Johnson (1998). Closed-form solutions are obtained analytically for the surface stress, deformation fields and equilibrium relations among applied load, indentation depth, inner and outer radii of the cohesive zone, which include the corresponding solutions for homogeneous isotropic materials and the Gibson solid as special cases. These solutions provide a continuous transition between JKR and DMT type contact models through a generalized Tabor parameter μ$\mu$. Our analysis reveals that the magnitude of the pull-off force ranges from (3+k)πRΔγ/2$(3+k)\pi R\mathrm{\Delta }\gamma /2$ to 2πRΔγ$2\pi R\mathrm{\Delta }\gamma$, where k$k$, R$R$ and Δγ$\mathrm{\Delta }\gamma$ denote the gradient exponent of the elastic modulus for the half-space, the radius of the sphere and the work of adhesion, respectively. Interestingly, the pull-off force for the Gibson solid is found to be identically equal to 2πRΔγ,$2\pi R\mathrm{\Delta }\gamma ,$ independent of the corresponding Tabor parameter. The obtained analytical solutions are validated with finite element simulations.     

Saint-Venants principle and the plane elastic wedge

The stress field due to self-equilibrating loading on the inner or outer arc of a plane strain elastic wedge sector is affected by two agencies: a geometric effect of increasing or decreasing area, and decay as anticipated by Saint-Venants principle (SVP) . When the load is applied to the inner arc the two effects act in concert ; however, when the load is applied to the outer arc the two effects act in opposition and for a wedge angle in excess of the half-space, 2α > π, for the symmetric case, and for 2α > 1.43π for the asymmetric case, the geometric effect is dominant over Saint-Venant decay and stress level increases as one moves away from the outer arc, confirming the inapplicability of SVP. This is additional to previously reported difficulties at these angle when a self-equilibrated load on the inner arc decays at the same rate as does a concentrated moment, and is explained in terms of the interaction of a near-field geometric effect and a far-field stress interference effect at a traction-free edge. For wedge angle 2α = 2π the unique Modes I and II inverse square root stress singularities at the crack tip, which are at the heart of Linear Elastic Fracture Mechanics (LEFM) , can be attributed to this inapplicability for just one symmetric and one asymmetric eigenmode.     

Low- and high-speed structures in the outer region of an adverse-pressure-gradient turbulent boundary layer

Large- and very large-scale structures in the form of elongated regions of low and high streamwise momentum have been studied in the outer region of a turbulent boundary layer subjected to a strong adverse pressure gradient. Large sets of streamwise–spanwise instantaneous velocity fields are acquired by particle image velocimetry at three wall-normal positions (0.2δ, 0.5δ, 0.8δ) at three different streamwise locations and at 0.1δ at the last streamwise location which allows us to study the wall-normal and streamwise variations of the structures. Subsequently, a pattern-recognition method and a classification scheme are employed in order to detect, classify and characterize the structures in an efficient and rigorous manner. Like in the case of zero-pressure-gradient turbulent boundary layers, long meandering streaky regions of low and high momentum are observed in the outer region of the present flow but they appear less frequently; especially in the lower part (at 0.1δ and 0.2δ) of the large-velocity-defect zone, i.e. near detachment. The dimensions of these large structures scale on boundary-layer thickness (δ) and are generally comparable to those previously reported for such structures in the overlap region of zero-pressure-gradient turbulent boundary layers. Interestingly, the adverse pressure gradient does not significantly affect the dimensions and arrangement of the large-scale structures in the upper part (at 0.5δ and 0.8δ) a segment of the outer region where the scaled Reynolds stresses also remain fairly self-similar.     

Control of shock unsteadiness in shock boundary-layer interaction on a compression corner using mechanical vortex generators

An experimental study was conducted to control the unsteadiness of separation shock in a Mach 2 24° compression ramp-induced interaction using mechanical vortex generators (VG). Control devices in the form of an array of single-row delta-ramps were placed upstream of the interaction region and tested for two streamwise locations with respect to the boundary layer thickness (δ) at the interaction location and height ‘h’ of the delta-ramps, i.e., at 27.5δ or h/δ?=?0.65 and at 12.5δ or h/δ?=?0.26, respectively. Surface oil study revealed traces of streamwise counter-rotating vortex pairs generated downstream of these devices. Measurements using pressure-sensitive paint also showed a spanwise sinusoidal pattern of wall pressure variation indicating generation of streamwise vortices from these control devices. These vortices, on interaction with the reverse flow in the separation bubble, replaced a well-defined separation line (for no control) by a highly corrugated separation line. In the region of separation, the mean pressure distribution gets modified while the peak rms value in the intermittent region of separation showed significant changes. Additionally, the spanwise spacing ‘s’ of the vertex of the delta ramps seemed to be an important parameter in controlling the peak rms value. A decrease in this spacing, i.e., VG1 with s =? 0, significantly reduced the peak rms value (by 50 and 35 %) while an increase in the spacing, i.e., VG2 with s =? 1?mm, consistently showed an increase (by 12 and 30 %) in the separation shock unsteadiness relative to no control, irrespective of their placement location (of h/δ?=?0.65 and 0.26, respectively).     

Interaction of a point sink with a transmitted shock wave in water

The structure of a wave of rarefaction (relief wave) created by the interaction of a shock wave with a point sink is considered. A singular region occurs in the relief wave in the angular range π/2≤θ≤3/2π; in this region the pressure exceeds that in the transmitted wave. Qualitative comparison is made with experimental results.     

Effects of the roughness height in turbulent boundary layers over rod- and cuboid-roughened walls

Direct numerical simulations (DNSs) of spatially developing turbulent boundary layers (TBLs) over two-dimensional (2D) rod-roughened walls and three-dimensional (3D) cuboid-roughened walls are conducted to investigate the effects of the roughness height on the flow characteristics in the outer layer. The rod elements are periodically aligned along the downstream direction with a pitch of p_x/θ_in = 12, and the cuboid elements are periodically staggered with a pitch of p_x/θ_in = 12 and p_z/θ_in = 3, where p_x and p_z are correspondingly the streamwise and spanwise pitches of the roughness and θ_in is the momentum thickness at the inlet. The first surface roughness is placed 80θ_in downstream from the inlet, leading to a step change from a smooth to rough surface. The rod and cuboid roughness height (k) is varied in the range of 0.1 ≤ k/θ_in ≤ 1.8 (13 ≤ δ/k ≤ 285), respectively (δ is the boundary layer thickness), and the Reynolds number based on the momentum thickness (θ) is varied in the range of Re_θ = 300 ~ 1400. For each case, the self-preservation form of the velocity-defect and the turbulent Reynolds stresses is achieved along the downstream direction. As the roughness height increases, the roughness function (ΔU⁺) extracted from the mean velocity profiles increases, although the velocity-defect profiles for the rough-wall cases show good agreement with the profile from the smooth-wall case. The magnitude of the Reynolds stresses in the outer layer increases with an increase of k/δ. The outer layer similarity between the flows over the rough and smooth-walls is found when δ/k ≥ 250 and 100 for the 2D rod and 3D cuboid, respectively. The continuous increase of the Reynolds stresses in the outer layer with an increase of k/δ is explained by a large population of very long structures over the rough-wall flows. Because the characteristic width of the structures increases continuously with an increase of k/δ for the rod and cuboid roughness, a wide width of the structures leads to frequent spanwise merging between adjacent structures. The active spanwise merging events with an increase of k/δ increase the streamwise coherence of the structures with the appearance of significant meandering.     

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) _λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).     

p-version least squares finite element formulation for two-dimensional,incompressible, non-Newtonian isothermal and non-isothermal fluid flow

This paper presents a p- version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, C⁰-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.     

An analysis of steady/unsteady electroosmotic flows through charged cylindrical nano-channels

The steady/unsteady electroosmotic flow in an infinitely extended cylindrical channel with diameters ranging from 10 to 100 nm has been investigated. A mixture of (NaCl + H₂O) is considered for the numerical calculation of the mass, potential, velocity, and mixing efficiency. Results are obtained for the channel diameters are small, equal, or greater than the electric double layer (EDL) both for steady and unsteady cases. In the present discussion, a symmetrical distribution of mole fractions is considered at the wall interface. Hence, the velocity and potential are symmetrical in nature toward the centerline of the channel, and also identical in nature at maximum and minimum time levels (i.e., at π/2 and 3π/2 for a periodic function) in the transient case. In case of steady flows, the velocity and potential satisfy the chemical equilibrium condition at the centerline. It is observed that the electric double layer reaches a local equilibrium in the presence of electroosmosis when the channel length is long compared to the characteristic hydraulic diameter and the flow is essentially one-dimensional, which depends only on channel diameter. Comparisons of NP (Nernst Plank) model with PB (Poisson–Boltzmann) model are achieved out for different published results at larger channel diameters.     

Junction Between a Plate and a Rod of Comparable Thickness in Nonlinear Elasticity

We analyze the asymptotic behavior of a junction problem between a plate and a perpendicular rod made of a nonlinear elastic material. The two parts of this multi-structure have small thicknesses of the same order δ. We use the decomposition techniques obtained for the large deformations and the displacements in order to derive the limit energy as δ tends to 0.     

基于低雷诺数串置翼型气动特性研究

基于低雷诺数,应用二维CFD方法,对串置翼型的气动特性进行了数值模拟。比较了串置翼型布局与常规单独翼型的升阻特性。分析了翼差角度对串置翼升阻特性的影响,并且在保持鸭翼±5°偏角不变的同时,研究了两翼之间的距离以及安装的相对高度对整个串置翼型升阻特性的影响。发现上鸭翼与主翼相结合具有相对较高的最大升力系数和临界迎角, 可以显著改善串置翼型的气动特性。     

Characterization of the flow for a single fluid in an excavation damaged zoneCaractérisation de l'écoulement d'un fluide monophasique en zone endommagée

The aim of this Note is to quantify the change of characteristics of the media of an Excavated Damaged Zone (EDZ) affected by several fractures. For this, we consider Darcy flow through matrix blocks and fractures with permeability of order ε²δ^θ and 1 respectively. ε is the size of a typical porous block, δ representing the relative size of the fracture and θ is a parameter characterising the permeability ratio. We derive the global behavior from the limit as ε and δ tend to zero. The resulting homogenized equation is of dual-porosity type for θ=2, but it is a simple-porosity model with effective coefficients for θ>2, and there is no flow at the macroscopic level when 0<θ<2. To cite this article: B. Amaziane et al., C. R. Mecanique 332 (2004).     

Crack paths in three-dimensional elastic solids. i: two-term expansion of the stress intensity factors—application to crack path stability in hydraulic fracturing

The aim of this work is to lay theoretical foundations for the prediction of crack paths within the theory of quasistatic LEFM under the most general hypotheses: arbitrary three-dimensional geometry, arbitrary loading. This objective requires to derive the expression of the stress intensity factors along the crack front after an arbitrary infinitesimal propagation. Only the first two terms of their expansion in powers of the crack extension length δ, proportional to δ⁰ = 1 and δfn1fn2, are considered in this paper. Fully general formulae for these terms are obtained by combining arguments of dimensional analysis (scale changes) and regularity properties (continuity, differentiability) of the stresses at a fixed, given point with respect to δ for δ = 0 derived from the Bueckner–Rice weight function theory. This notably allows to extend the Cotterell–Rice criterion for stable rectilinear propagation of a mode I crack under plane strain conditions to the three-dimensional case. As an application, a penny-shaped crack induced by hydraulic fracturing is considered. Conclusions concerning the influence of the orientation and depth of such a crack upon the stability of its coplanar propagation seem to be compatible with experimental evidence.     

An experimental study of choked foam flows in a convergent-divergent' nozzle

Choked flow of a foam in a convergent-divergent nozzle has been investigated. The foam consisted of air and a solution of a surface active agent in water. The upstream gas-liquid volume ratio $\text{δ}{}_0$ was in the range 0.053–1.57. The experimental results are in very good agreement with a homogeneous frictionless nozzle flow theory, assuming isothermal behaviour of the gas and no relative motion between the phases, for throat gas-liquid volume ratios $\text{δ}{}_1$ as high as 0.8; for ratios in the range $\text{0.8 < δ}{}_{\mathrm{t}}\text{< 2.98}$ the agreement, while only approximate, is still quite close. Departures from the homogeneous theory are explained in terms of (a) the failure of the assumption of the isothermal behaviour and (b) the existence of relative velocity between the phases. The latter effect predominates at low values of $\text{δ}{}_1$ but at large values, it appears that both contribute to errors in the predictions.     

Interaction between an incompressible flow and elastic cantilevers of circular cross-section

The purpose of this work is to study the deformation of elastic cantilevers due to hydrodynamic forces by coupled fluid–structure interaction simulations. The cantilever is placed in a rectangular duct and the Reynolds number based on bulk velocity and cantilever diameter is 400. Reduced velocities in the range π/4 to 2π are studied, which covers both un-synchronised motion and the initial branch of synchronisation. The cantilever surface is represented by a virtual boundary method which replaces a solid object in flow by additional force distribution to satisfy local boundary condition. The flow field is solved using a Cartesian finite difference code and the deformation of the cylinder a finite element approach using one-dimensional beam elements is used.     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.