Translational oscillations of a microsphere in superfluid helium

The translational motion of a microsphere (radius 100 μm) in liquid helium is investigated. The sphere is levitating inside a superconducting capacitor and oscillates about its equilibrium position. The velocity amplitude and the resonance frequency are measured as a function of driving force and temperature (0.35 K up to 2.2 K). By increasing the driving force we first find a linear regime (laminar flow) which changes abruptly into a nonlinear one (turbulent flow). For temperatures below 0.7 K the linear drag is given by ballistic roton and phonon scattering whereas for temperatures above 1.1 K the hydrodynamic force on the sphere is described by Stoke's solution. In the turbulent regime, above a temperature independent threshold velocity, we find the drag force to be given by turbulence in the superfluid component plus an essentially laminar drag by the normal component.     

Mean force on a small sphere in a sound field in a viscous fluid

A mean force exerted on a small rigid sphere by a sound wave in a viscous fluid is calculated. The force is expressed as a sum of drag force coming from the external steady flow existing in the absence of the sphere and contributions that are cross products of velocity and velocity derivatives of the incident field. Because of the drag force and an acoustic streaming generated near the sphere, the mean force does not coincide with the acoustic radiation pressure, i.e., the mean momentum flux carried by the sound field through any surface enclosing the sphere. If the sphere radius R is considerably smaller than the viscous wave penetration depth delta, the drag force can give the leading-order contribution (in powers of delta/R) to the mean force and the latter can then be directed against the radiation pressure. In another limit, delta< or =R, the drag force and acoustic streaming play a minor role, and the mean force reduces to the radiation pressure, which can be expressed through source strengths of the scattered sound field. The effect of viscosity can then be significant only if the incident wave is locally plane traveling.     

Calculation of the force characteristics of a multi-coil suspension of a superconducting sphere

The theory presented in the accompanying paper [Ref. 1: Zh. Tekh. Fiz. 67(1), 3 (1997); Tech. Phys. 42, 1 (1997)] for the calculation of the magnetic field and the force characteristics of an electromagnetic suspension for a superconducting body is used to carry out specific calculations of the force characteristics of a multi-coil suspension of a superconducting sphere. Formulas are obtained for the dependence of the force and the stiffness, from which it is possible to approach the problem of the stable suspension of a superconducting sphere in the field of a system of circular currents. The problem of determining the magnetic field at the sphere is analyzed and the conditions are found for which it is less than the critical field. Zh. Tekh. Fiz. 67, 10–16 (January 1997)     

介观尺度流体绕流球体的耗散粒子动力学模拟

采用耗散粒子动力学(dissipative particle dynamics, DPD)方法, 对两平行平板间流体绕流三维球体进行了计算. 球体和平行平板由达到平衡状态的冻结DPD粒子组成, 流体在不同无量纲外力驱动下流动, 球体受力由组成球体的所有冻结DPD粒子求和得到. 流动达到充分发展后, 输出球体在流动方向的受力, 并计算球体的阻力系数, 与文献中的关联式进行了对比. 结果表明, 在Re≤qslant 100的范围内, DPD方法能较准确地计算出阻力系数, 在较大雷诺数时, 由于流     

Calculations of the flow resistance and heat emission of a sphere in the laminar and high-turbulent gas flows

An early drag crisis can occur at high turbulence of incoming gas flow to a sphere. To study the influence of a crisis on heat transfer from a sphere to gas, a numerical experiment was carried out in which the free gas flow around a sphere with a temperature lower than the sphere temperature was simulated for two cases. The flow was laminar in the first case and highly turbulent in the second case. To take into account turbulence, the kinematic coefficient of turbulent viscosity with a value, which is much higher (up to 2000 times) than that for physical viscosity, was introduced. The results of calculations show that the early drag crisis occurs at Reynolds numbers of about 100 and results in considerable (by four to seven times) decrease in the hydrodynamic force and sphere drag coefficient C _d . The early drag crisis is also accompanied by the crisis of heat transfer from a sphere to gas with a decrease in Nusselt numbers Nu by three to six times.     

Experimental verification of the early crisis of drag using a single sphere

The recently discovered phenomenon of drag anomaly (early crisis) observed for droplets in a two-phase turbulent flow at transition Reynolds numbers (Re = 10–100) is experimentally supported in the case of a gas flow about a single hard sphere as well. Using a specially designed torsion balance, the drag force of a foam plastic sphere 3 mm in diameter placed in a blower-produced turbulent jet is measured. It is found that the turbulence of the flow about the sphere must be high for the drag anomaly to arise.     

Motion of a spherical body in a viscous suspension

The problem of drag of an arbitrary-size solid sphere with its motion in a uniform monodispersed viscous suspension is considered in the Stokes approximation. The expression for the effective suspension viscosity is derived in the first approximation over the volume concentration of the viscous suspension. The coefficient before the concentration is found in the form of an explicit analytical function, which depends on the ratio of sizes of the dispersed particles and the body. The found coefficient coincides with Einstein’s result at the limit of “point” dispersed particles, the size of which is negligibly small compared with the size of the moving sphere, but can substantially differ from it in the case of finite-size particles.     

介观尺度流体绕流球体的耗散粒子动力学模拟

采用耗散粒子动力学(dissipative particle dynamics, DPD)方法, 对两平行平板间流体绕流三维球体进行了计算. 球体和平行平板由达到平衡状态的冻结DPD粒子组成, 流体在不同无量纲外力驱动下流动, 球体受力由组成球体的所有冻结DPD粒子求和得到. 流动达到充分发展后, 输出球体在流动方向的受力, 并计算球体的阻力系数, 与文献中的关联式进行了对比. 结果表明, 在Re≤qslant 100的范围内, DPD方法能较准确地计算出阻力系数, 在较大雷诺数时, 由于流体的压缩性导致计算结果出现差异.     

圆形线圈对超导球体的悬浮力特性分析

针对超导球体和圆形线圈之间的力的作用,介绍了一种简化而有效的计算悬浮力的方法。在此基础上,对线圈和超导球体的几何形状尺寸以及两者之间的相对位置关系对悬浮力的影响进行了综合的分析和讨论;对悬浮力在随着线圈和超导球体的具体参数的变化规律做了定量的分析。总结出的结论为超导悬浮系统的分析和设计提供重要的参照。     

The Oseen drag on a sphere and the method of induced forces

The method of induced forces, which was used previously to derive a scheme for the evaluation of the Oseen drag exerted on a circular cylinder, is applied now to the problem of the Oseen drag experienced by a sphere. By taking into account the zeroth and first force multipole alone, very satisfactory values for the drag coefficient are obtained for Reynolds numbers up to order 10².     

煤粉颗粒所受Magnus力的数值模拟

本文采用数值模拟方法计算了煤粉颗粒所受的Ｍａｇｎｕｓ力,考察了颗粒的旋转速度、流动Ｒｅ数对Ｍａｇｎｕｓ力的影响,并且给出了相应的关系公式．通过比较煤粉颗粒在运动过程中所受到的Ｍａｇｎｕｓ力与气动力的大小,得出：当煤粉颗粒的旋转速度为１８００转／分时,其大小约为气动力的１％,因此在实际的煤粉颗粒受力分析中是完全可以忽略该力的。     

Effect of the gas flow geometry and turbulence on the hydrodynamic drag of a body in the flow

The effect of the incoming flow geometry on the hydrodynamic drag of a body is investigated in a numerical experiment simulating a free gas flow past a sphere as well as flows in cylindrical tubes of various radii, in a confuser, and a diffuser. The results of calculations lead to the conclusion that the confinement of the flow by the tube walls, its contraction and expansion may change the hydrodynamic force and the drag acting on the body insignificantly (not more than by 30%). This cannot explain the early drag crisis, in which the values of these quantities decrease by 4–7 times for Reynolds numbers on the order of 100. This phenomenon is explained theoretically by the effect of strong turbulence of the incoming flow to the body.     

Stability analysis of a spherical electrostatic suspension in a liquid in the induction approximation

In [1] the image force was shown to impose additional conditions for the electrostatic suspension of a sphere without dynamic control of the electrode potential, and the dependence of the critical voltage between the electrodes on the sphere radius was derived experimentally. In this work, this dependence is found analytically by calculating electrical forces in the third-order approximation in shift of the sphere from equilibrium.     

Drag and Lift Forces Acting on a Sphere in Shear Flow of Power-Law Fluid

Laminar flow of a power-law fluid over a sphere is considered for unbounded shear flow. The Navier–Stokes equations with power-law viscosity are solved numerically using an in-house developed CFD package. Vorticities structures downstream of particle are suppressed for powerlaw fluid. The shear rate influence on drag force is negligible for power index close to unit, and the drag force appreciably decreases with falling power index. For small Reynolds numbers, the lift force coefficient monotonically decreases against the power index and exhibits an opposite behavior for moderate values of Reynolds numbers. The results of the parametric studies are used to derive correlations for the drag force and to detect the hydrodynamic differences from uniform flow. The investigation parameters varied within the following ranges: power-law index 0.3 ≤ n ≤ 1, Reynolds number 0 < Re ≤ 150, and dimensionless shear rate 0.05 ≤ s ≤ 0.4.     

Calculation of the flow about a sphere and the drag of the sphere under laminar and strongly turbulent conditions

To investigate the influence of a strongly turbulent incoming flow on the hydrodynamic drag of a body and occurrence of the early crisis of drag, a numerical experiment is conducted in which a free gas flow about a sphere is simulated for two cases, namely, for a laminar flow and for a strongly turbulent flow. Turbulence is simulated by assuming a high kinematic coefficient of turbulent viscosity. Calculation data lead us to conclude that the early crisis of drag at Reynolds numbers near 100, which shows up as a considerable (four-to sevenfold) decrease in the hydrodynamic force and the drag coefficient of the body, can be explained by the strong turbulence of the incoming flow.     

蒸汽爆炸粗混合过程中的蒸发阻力模型

在蒸汽爆炸的粗混合过程中,由于液体的快速蒸发,高温金属液滴的周围会产生一层很薄的蒸汽膜,此时液滴周围的边界层流动与没有液体蒸发时有很大的不同,因此,常温情况下的小球在连续液体中运动时的通用阻力模型在这种情况下是不适用的．本文通过受力分析,考虑了高温小球受力的分布和表面蒸发对小球周围力的影响,从阻力的基本机理上分析了蒸发状态下小球的运动阻力,分别提出了高温颗粒穿过自由表面时与其在液体中运动时的蒸发阻力模型．分析表明,当小球温度高于2500 K,特别是在靠近自由表面的区域,由于小球表面液体蒸发而产生的蒸发阻力作用非常明显．分析指出,小球的入水初速、小球表面的液体蒸发速率以及汽膜厚度都是影响小球运动阻力大小的重要因素．     

Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

We consider the flow of a dilute gas around a macroscopic heavy object. The state of the gas is described by an extended Boltzmann equation where the interactions between the gas molecules and the object are taken into account in computing the rate of change of the distribution function of the gas. We then show that the extended Boltzmann is equivalent to the usual Boltzmann equation, supplemented by boundary conditions imposed on the distribution function at the surface of the object. The remainder of the paper is devoted to a study of the solution of the extended Boltzmann equation in the case that the mean free path of a gas molecule is small compared to some characteristic dimension of the macroscopic object. We show that the Chapman-Enskog normal solution of the ordinary Boltzmann equation is not in general a solution of the extended equation near the surface of the object and must be supplemented by a boundary layer term. We then introduce a projection operator method which allows us to decompose the solution of the extended equation into a normal solution part and a boundary layer part when the gas flow is sufficiently slow. As a specific example of the method we consider the flow around a sphere, and derive the Stokes-Boussinesq form for the frequency-dependent force on the sphere for arbitrary slip coefficient. This derivation is the first one that starts from the Boltzmann equation for a general dilute gas and incorporates the effect of the boundary layer on the drag force.Work supported by the National Science Foundation.     

Numerical simulation of the sedimentation of a sphere in a sheared granular fluid: a granular Stokes experiment

We study, computationally, the sedimentation of a sphere of higher mass in a steady, gravity-driven granular flow of otherwise identical spheres, on a rough inclined plane. Taking a hydrodynamic approach at the scale of the particle, we find the drag force to be given by a modified Stokes law and the buoyancy force by the Archimedes principle, with excluded volume effects taken into account. We also find significant differences between the hydrodynamic case and the granular case, which are highlighted.     

Calculation of the drag and heat transfer from a sphere in the gas flow in a cylindrical channel

A numerical experiment on the simulation of heat transfer from a sphere to a gas flow in a cylindrical channel in the Stokes and transient flow regimes has been described. Radial and axial profiles of the gas temperature and the dependences of drag coefficient C_d of the body and Nusselt number Nu on Reynolds number Re have been calculated and analyzed. The problem of the influence of the early drag crisis for a sphere on its heat transfer to the gas flow has been considered. The estimation of this phenomenon has shown that the early drag crisis of the sphere in a strongly turbulent flow causes a reduction in heat transfer from the sphere to the gas by three to six times (in approximately the same proportion as for its drag coefficient).     

Sound radiation from point-driven shell structures

A numerical study is presented showing the structural response and sound radiation from a range of thin shell structures excited by a point force: a baffled flat plate, a sphere, a family of spheroids and a family of closed circular cylinders. All the structures have the same material properties, thickness and total surface area so the asymptotic modal density is the same. Dramatic differences are shown in the total radiated sound power for the different shells. It was already known that the flat plate and the sphere behave very differently. These results show that the cylinders and, particularly, the spheroids show patterns that are not intermediate between the two but instead display new features: in certain frequency ranges the radiated sound power can be at least an order of magnitude greater than either the plate or the sphere.