Experimental investigations on two-phase flow past a sphere using digital Particle-Image-Velocimetry

Digital Particle-Image-Velocimetry was applied to investigate particle trajectories in a gas flow past a sphere. The particle displacement was determined by autocorrelation analysis of image sections. To enhance the signal/noise ratio a synthetic image with idealized particle pictures was generated from the real image. The autocorrelation function (ACF) was calculated using the Fast Hartley Transformation (FHT). The desired secondary maximum of this function was detected by an algorithm with subpixel resolution. A data validation step testing the plausibility of the velocity vectors completes the image analysis. Particle trajectories are traced with help of the particles' velocity vectors. The particle deposition on a sphere can be deduced from the course of these trajectories.List of symbols Cu Cunningham correction - e double distance between the limiting particle trajectory and the stagnation point axis - f focal length - H Hartley-Transform - M enlargement factor - S interference band spacing - x coordinate - y coordinate - x _p particle diameter - x _T droplet or sphere diameter - V image as grey value function - _.rel face velocity of droplet or sphere - particle image displacement - x particle image displacement in x-direction - y particle image displacement in y-direction - collection efficiency - wavelength of the laser light - _L fluid viscosity - _L fluid density - _p particle density     

Rarefied gas flow past a sphere with injection

Hypersonic rarefied gas flow over the windward face of a sphere is considered in the presence of distributed injection from the surface of the body. A similar problem was previously solved in [1–3] within the framework of continuum mechanics and in [4] on the basis of model kinetic equations. In the present study the calculations were carried out using the Monte Carlo method of direct statistical modeling [5, 6]. The injected gas was the same as the free-stream gas. A simple monatomic gas model with a rigid sphere interaction potential was employed. The reflection of the molecules from the surface of the body was assumed to be diffuse with total energy accommodation. The calculation procedure using weighting factors is described in [7]. The influence of injection on the mechanical and thermal effect of the gas flow on the body is investigated for various degrees of rarefaction of the medium and injection rates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 175–179, July–August, 1990.     

Isothermal flow past a blowing sphere

Steady, axisymmetric, isothermal, incompressible flow past a sphere with uniform blowing out of the surface is investigated for Reynolds numbers in the range 1 to 100 and surface velocities up to 10 times the free stream value. A stream-function-velocity formulation of the flow equations in spherical polar co-ordinates is used and the equations are solved by a Galerkin finite-element method. Reductions in the drag coefficients arising from blowing are computed and the effects on the viscous and pressure contributions to the drag considered. Changes in the surface pressure, surface vorticity and flow patterns for two values of the Reynolds number (1 and 40) are examined in greater detail. Particular attention is paid to the perturbation to the flow field far from the sphere.     

The flow past a freely rotating sphere

We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold \(Re^\mathrm{OS}=206\) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to \(Re\approx 270\) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold \(Re^\mathrm{SS}=212\). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.     

剥蚀引起的球锥体两相绕流效应

基于多学科理论建立了极端环境下球锥体烧蚀、剥蚀的数理模型。利用离散涡方法计算流场与球锥表面压力分布,采用三方程烧蚀模型计算热化学控制机制下的烧蚀速率;引入颗粒轨道模型求解剥蚀颗粒的运动,编程计算高温高压燃烧室内球锥形烧蚀试件的绕流场及剥蚀颗粒的运动轨迹。研究表明：烧蚀对球锥体的绕流影响显著,球锥体的涡云较快地转变为不对...     

Hypersonic flow past a supersonic two-phase source

The interaction of a uniform hypersonic gas flow with a supersonic two-phase gas-particle source is considered. In the symmetry-axis neighborhood between the bow and termination shock waves, an approximate analytical solution for the carrier-phase parameters is found. On the basis of parametric numerical calculations, the behavior of the particle trajectories and the concentration distribution in the shock layers are studied for both continuum and free-molecule flow regimes around the particles. The appearance of regions with multiple intersections of the particle trajectories and the formation of "layer structures" in the particle concentration distributions (particle accumulation regions near the envelopes of the particle trajectories) are indicated. The dependence of the number of the high concentration layers on the governing parameters is studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 134–147, May–June, 1998. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-00313) and the National Foundation for Natural Sciences of China (joint RFBR-NFNS grant No.96-01-00017c).     

Stokes flow past a porous sphere with an impermeable core

Stokes flow of a viscous, incompressible fluid past a porous sphere with an impermeable core using Darcy law for the flow in the porous region is discussed. The formulae for drag and torque are found by deriving the corresponding Faxen's laws. It is found that torque is always less than that on a solid sphere and it does not depend on the radius of the impermeable core. Some illustrative examples are discussed.     

Flow of a stream containing charged inertial particles past a conducting sphere

The problem of the flow of a hydrodynamic stream containing electrically charged particles past a conducting sphere is solved. The influence of the volume density of the electric charge and the potential of the sphere on the capture coefficient is determined for different values of the inertia parameter of the particles and different drag laws for them. It is shown that for an earthed sphere can appreciably exceed unity. In the formulation of the problem, it is assumed that the region of electrogasdynamic flow is bounded by two electrode grids, which simulate the exit of the source of charged particles and the surrounding electric conditions. The velocity field near the sphere is assumed irrotational. The problem is solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–26, July–August, 1982.I thank A. B. Vatazhin for assistance in the work.     

A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

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Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

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A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

An investigation of non-newtonian flow past a sphere

Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finite-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60.Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV_∞P/η₀, where η₀ is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Non-Newtonian flow past a sphere in a long cylindrical tube

Consideration is given to the problem of a sphere falling along the axis of a vertical cylindrical tube containing a viscoelastic fluid. Numerical predictions of the flow are obtained using a well established finite element Galerkin mixed formulation. The effect of elasticity on the streamline pattern, the drag and the stress field are discussed.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.