The flow structure of dilute gas-particle suspensions behind a shock wave moving along a flat surface

The asymptotic and numerical investigations of shock-induced boundary layers in gas-particle mixtures are presented.The Saffman lift force acting on a particle in a shear flow istaken into account.It is shown that particle migration across the boundary layer leads tointersections of particle trajectories.The corresponding modification of dusty gas model isproposed in this paper.The equations of two-phase sidewall boundary layer behind a shock wave moving at aconstant speed are obtained by using the method of matched asymptotic expansions.Themethod of the calculation of particle phase parameters in Lagrangian coordinates isdescribed in detail.Some numerical results for the case of small particle concentration aregiven.     

The distribution of particles in a shock-induced boundary layer of a dusty gas over a solid surface

The laminar boundary layer behind a constant-speed shock wave moving through a dusty gas along a solid surface is studied. The Saffman lift force acting on a spherical particle in a gas boundary layer is taken into account. A method for calculating the density profile of dispersed phase near the wall is proposed and some numerical results are given. It is shown that behind the shock wave, there exists a curved thin layer where the density of particles is many times higher than the original one. This dust collection effect may be of essential importance to the problem of dust explosion in industry.     

激波诱导含灰气体边界层中的粒子分布

本文研究当激波沿着一个固体表面等速地穿越含灰气体运动时所诱导的层流边界层特性。考虑了作用在气体边界层中球形粒子的 Saffman 升力,建议了一种计算近壁区中弥散相密度剖面的方法,并给出了数值计算结果。本文结果表明:在激波后方存在着一个弯曲的薄层区域,其中的粒子密度可以比其波前原始值增加许多倍。这种粒子聚集效应对于工业中粉尘爆炸等实际问题具有重要意义。     

Single-Fluid Model of a Mixture for Laminar Flows of Highly Concentrated Suspensions

A model of laminar flow of a highly concentrated suspension is proposed. The model includes the equation of motion for the mixture as a whole and the transport equation for the particle concentration, taking into account a phase slip velocity. The suspension is treated as a Newtonian fluid with an effective viscosity depending on the local particle concentration. The pressure of the solid phase induced by particle-particle interactions and the hydrodynamic drag force with account of the hindering effect are described using empirical formulas. The partial-slip boundary condition for the mixture velocity on the wall models the formation of a slip layer near the wall. The model is validated against experimental data for rotational Couette flow, a plane-channel flow with neutrally buoyant particles, and a fully developed flow with heavy particles in a horizontal pipe. Based on the comparison with the experimental data, it is shown that the model predicts well the dependence of the pressure difference on the mixture velocity and satisfactorily describes the dependence of the delivered particle concentration on the flow velocity.     

Motion of a dusty gas at the entrance to a flat channel and a circular pipe

It is shown that at large Reynolds numbers, calculated from the entrance velocity and the half-width of the channel, four characteristic flow regions are formed in the entrance section. The equations describing the motion of the mixture in each of these regions are constructed by the method of matched expansions. An expression relating the particle concentration distribution at points remote from the entrance section to the particle concentration distribution in the boundary layer on a flat plate at points remote from the beginning of the plate is obtained. The dependence of the dispersed-phase concentration profile formed on the governing parameters is studied on the basis of a numerical solution. It is shown that as the contribution of the Saffman force to the interphase momentum transfer increases, the rise in particle concentration in the direction of the wall is replaced by a fall. A qualitative correspondence between the calculated particle concentration profiles and certain known experimental data is noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 80–87, November–December, 1988.     

可侵蚀地表上方含尘大气运动的数值模型

在稀相气固两相流的双流体模型框架下，导出含尘大气运动的基本方程，其中的源项考虑相间双向耦合作用；通过对基本方程进行无量纲化，求得控制气相和弥散相动力学行为的相似准则，包括弗洛德数、罗斯贝数、颗粒雷诺数、颗粒惯性参数、颗粒剪切横移参数、颗粒旋转横移参数和沙尘质量载荷率等. 作为一个模型问题，研究可侵蚀地表上方充分发展湍流大气边界层流动对土壤颗粒的气动卷扬作用，其中计及气动阻力、萨弗曼升力和重力等因素的影响. 为了克服由于沙尘跃移运动轨迹交叉导致流动参数多值性所造成的困难，引进拉格朗日坐标下弥散相连续方程. 对两种不同风速和4种不同粒径条件下的沙尘运动进行数值模拟，给出相应情况中沙尘运动特性和密度分布剖面并讨论风速和粒径等参数的影响，还细致分析了含尘大气边界层中沙尘与气流之间的能量传递过程. 所得的方程、准则和方法可以为土壤风蚀和沙尘暴等自然灾害的预报提供理论基础.     

Motion of a suspension in the laminar boundary layer on a flat plate

The boundary layer motion of a weak suspension is investigated with allowance for the effect on the particles not only of the Stokes force but also of the additional transverse force resulting from the transverse nonuniformity of the flow over the individual particle. As distinct from studies [1–3], in which the limiting values of the transverse force (Saffman force) were used [4], the velocity and density of the dispersed phase have been determined with allowance for the dependence of the Saffman force on the ratio of the Reynolds numbers calculated from the velocity of the flow over the individual particle and the transverse velocity gradient of the undisturbed flow, respectively [5, 6].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–73, January–February, 1992.In conclusion the authors wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions of the results.     

压力梯度对边界层两相流动稳定性的影响

分析了有压力梯度的边界层两相流动稳定性,推导出类似于Saffman理论的修正的稳定性方程,数值计算采用高精度的谱方法。结果说明,压力梯度对边界层两相流动稳定性有显著的影响,顺压梯度增强流动稳定性,而逆压梯度则促进流动失稳。在不同的压力梯度和浓度下,Stokes数对流动稳定性的影响是一致的,存在一个临界Stokes数,小Stokes数促进流动失稳,而大Stokes数则提高临界雷诺数,抑制流动失稳的最佳Stokes数为10的量级。     

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 34–42, November–December, 1993.In conclusion, the authors wishes to thank S. V. Manuilovich for assisting with the numerical calculations.     

Lagrangian model on the turbulent motion of small solid particle in turbulent boundary layer flows

.Intr0ductionSurfaceerosionofmaterialbysolid-particleimpactisanimportantprobleminmultiphaseflowindustriaIdevicesandthecharacteristicsoftheparticIe'smotioninaturbulentboundarylayerflowisthebaseofthestudyofthematerialsurfaceerosion.Manycalculationmodelshave…     

CFD-DEM simulation of the hole cleaning process in a deviated well drilling: The effects of particle shape

We investigate the effect of particle shape on the transportation mechanism in well-drilling using a three-dimensional model that couples computational fluid dynamics (CFD) with the discrete element method (DEM). This numerical method allows us to incorporate the fluid–particle interactions (drag force, contact force, Saffman lift force, Magnus lift force, buoyancy force) using momentum exchange and the non-Newtonian behavior of the fluid. The interactions of particle−particle, particle−wall, and particle−drill pipe are taken into account with the Hertz–Mindlin model. We compare the transport of spheres with non-spherical particles (non-smooth sphere, disc, and cubic) constructed via the multi-sphere method for a range of fluid inlet velocities and drill pipe inclination angles. The simulations are carried out for laboratory-scale drilling configurations. Our results demonstrate good agreement with published experimental data. We evaluate the fluid–particle flow patterns, the particle velocities, and the particle concentration profiles. The results reveal that particle sphericity plays a major role in the fluid–solid interaction. The traditional assumption of an ideal spherical particle may cause inaccurate results.     

湍流边界层中固体颗粒运动的数值模拟

根据Ｌａｇｒａｎｇｅ颗粒运动微分方程及不可压缩湍流边界层中流体的壁面速度分布规律，数值求解了颗粒在湍流边界层中的运动，考虑了Ｓａｆｆｍａｎ升为对颗粒运动的影响，壁面对运动阻力的影响，给出了固体颗粒沉积边壁，在边界层外缘上所需的最小速度和最小入射角，计算结果还表明边界层对固体颗粒撞击边壁的速度和入射角有较大影响，从数值结果可可以发现一个重要现象。     

HYDRODYNAMIC RESISTANCE EFFECT OF FLUID LAYER BETWEEN TWO IMMERSED APPROACHING PARTICLES

1. Introduction The mechanisms of impact and rebound of solid parti- cles in particulate flow systems are of interest over a wide range of application areas such as fluidized beds, pneu- matic transport, filtration processes, erosion and pollution control of suspended particles. In many cases, the colli- sions of particles against themselves and against walls may affect the properties of the mixture. Efforts have been made to describe the fundamental mechanics of particle collisions. The conta…     

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

Structure and stability of the interface between magnetic and conventional fluids. Model of a three-component medium

The structure of a flat interphase boundary between a magnetic suspension and a conventional immiscible fluid is investigated within the framework of the model of a three-component medium taking the dependence of the free energy of the system on the concentration gradients into account. It is shown that for certain values of the constitutive parameters the bulk magnetic particle concentration increases significantly inside the interfacial layer, i.e., the particles are significantly adsorbed on the interface. The dependence of the surface tension on the magnetic field strength is determined. It is shown that for certain problem parameters this dependence qualitatively corresponds to that obtained experimentally and described in the phenomenological theory developed by Golubyatnikov and Subkhankulov in 1986. In the case of strong particle adsorption the dependence of the surface tension on the magnetic particle concentration on the phase interface is significantly nonlinear. A refined model of the interface as a two-dimensional continuum with surface magnetization is constructed. Constitutive equations, conditions on the interface, and necessary stability conditions are obtained.     

The effect of non-linear interaction between gas and particle velocities on the hydrodynamic stability in the Blasius boundary layer

A weakly non-linear stability analysis of two phase flow in the Blasius boundary layer has been carried out. Two mathematical models have been established based on the perturbation shape preserved assumption and linear stability model of two phase flow proposed by Stuart [On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech. 4 (1958) 1-21] and Saffman [On the stability of laminar flow of dusty gas, J. Fluid Mech. 13 (1962) 120-128], respectively. The perturbation model and the perturbation energy balance equation are solved numerically with Chebyshev spectral method and artificial boundary condition. The numerical program adopted in the present study is verified by comparison with former works. The results show that the non-linear interaction between mean flow and perturbation reduces the growth rate of perturbation, while the non-linear interaction between particle phase and gas phase increases the growth rate of perturbation amplitude. The distortion of the mean flow caused by the Reynolds stress modifies the rate of transfer of energy from the mean flow to disturbance. The existence of particle alleviates the distortedness. The result also indicates that the weakly non-linear stability theory is consistent to linear stability theory, and the addition of fine and coarse particles reduces and increases the critical Reynolds number.     

Assessment of force models on finite-sized particles at finite Reynolds numbers

Finite-sized inertial spherical particles are fully-resolved with the immersed boundary projection method(IBPM) in the turbulent open-channel flow by direct numerical simulation(DNS). The accuracy of the particle surface force models is investigated in comparison with the total force obtained via the fully-resolved method. The results show that the steady-state resistance only performs well in the streamwise direction, while the fluid acceleration force, the added-mass force, and the shear-induced Saffman lift can effectively compensate for the large-amplitude and high-frequency characteristics of the particle surface forces, especially for the wall-normal and spanwise components. The modified steady-state resistance with the correction effects of the acceleration and the fluid shear can better represent the overall forces imposed on the particles, and it is a preferable choice of the surface force model in the Lagrangian point-particle method.     

Lift force exerted on a spherical particle in a laminar boundary layer

A spherical particle moving in an unbounded viscous shear flow is acted upon by a lift force [1, 2] which results from taking the inertial terms into account in the equations of motion. When the particle moves at the bottom of a laminar boundary layer the magnitude of the force differs from that obtained in [1, 2], The problem of determining the lift force exerted on the particle as a function of its distance from the wall has been solved by matched asymptotic expansions. The magnitude of the force is expressed in terms of a multiple integral which can be evaluated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–71, September–October, 1989.In conclusion, the author wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions.     

Agglomeration of inertial particles in a random rotating symmetric straining flow

This paper is concerned with the development and validation of a simple Lagrangian model for particle agglomeration in a turbulent flow involving the collision of particles in a sequence of correlated straining and vortical structures which simulate the Kolmogorov small scales of motion of the turbulence responsible for particle pair dispersion and collision. In this particular study we consider the collision rate of monodisperse spherical particles in a symmetric (pure) straining flow which is randomly rotated to create an isotropic flow. The model is similar to the classical model of Saffman and Turner (S&T) (1956) for the collision (agglomeration) of tracer particles suspended in a turbulent flow. However unlike S&T, the straining flow is not frozen in time persisting only for timescales ∼Kolmogorov timescale. Furthermore, we consider the collision of inertial particles as well as tracer particles, and study their behavior not only at the collision boundary but also in its vicinity. In the simulation, particles are injected continuously at the boundaries of the straining flow, the size of the straining region being typical of the Kolmogorov length scale η_K of the turbulence. For steady state conditions, we calculate the flux of particles colliding with a test particle at the centre of the straining flow and consider its dependence on the inertia of the colliding particles (characterized by the particle Stokes number, St). The model replicates the segregation and accumulation observed in DNS and in particular the maximum segregation for St ∼ 1 (where St is the ratio of the particle response time to the Kolmogorov timescale). We also calculate the contributions of the various turbulent forces in the momentum balance equation for satellite particles and show for instance that for small Stokes number, there is a balance between turbulent diffusion and turbophoresis (gradient of kinetic stresses) which in turn is responsible for the build-up of concentration at the collision boundary. As found in previous studies, for the case of inertialess tracer particles, the collision rate turns out to be significantly smaller than the S&T prediction due to a lowering of the concentration at the collision boundary compared to the fully mixed value. The increase in collision rate for St ∼ 0.5 is shown to be a combination of particle segregation (build-up of concentration near the collision boundary) and the decorrelation of the relative velocity between the local fluid and a colliding particle. The difference from the S&T value for the agglomeration kernel is shown to be a consequence of the choice of perfectly absorbing boundary conditions at collision and the influence of the time scale of the turbulence (eddy lifetime). We draw the analogy between turbulent agglomeration and particle deposition in a fully developed turbulent boundary layer.     

Influence of a counter rotating vortex pair on the stability of a jet in a cross flow: an experimental study by flow visualizations

 The loss of stability of a flow resulting from the interaction of a transverse jet and a crossflow has been studied experimentally by means of several nonintrusive optical methods (Laser-Induced Fluorescence, Particle Streak Velocimetry). In particular, flow visualizations have allowed measurement of the characteristic velocity of rotation of a counter-rotating vortex pair, which is the prevailing system of this complex flow. It is shown that the elliptical geometry of the cross sections of these vortex structures can cause the loss of stability of the jet according to the theory of Landman and Saffman (1987). The unstable jet state is characterized by the generation of transverse unsteady rotating structures on the jet boundaries. These transverse eddies are clearly identifiable and their regularity of occurrence is satisfactorily explained by the theory of Landman and Saffman. Therefore, this study shows for the first time that the longitudinal structures must be taken into account in the mechanism of stability arising from the meeting of a jet and a cross stream. Received: 7 May 1997/Accepted: 20 March 1998