Stability of dispersed-particle shape at small Reynolds numbers

The results of experimental studies of the conditions of loss of stability of the shape of a single dispersed-phase inclusion (droplet and bubble) during its motion in a viscous fluid at low Reynolds numbers are presented. It is shown that in the conditions considered the deformation of an initially spherical inclusion occurs due to the development of the Rayleigh-Taylor instability, as a critical value of the Bond number is attained. It is found that the onset of deformation of the phase interface and the instability mechanism depend strongly on the particle motion regime. A range of critical Reynolds numbers, corresponding to the boundaries of the regions of the Rayleigh-Taylor and Kelvin-Helmholtz instabilities, is determined.     

Hydrodynamic drag of an ellipsoidal droplet at small Reynolds numbers

At present, there is an absence of the accurate data on the influence of the shape of a droplet on its hydrodynamic drag and mass transfer without which the design of mass transfer apparatus is impossible [1–3]. Most often it is assumed that the drag of an ellipsoidal liquid droplet as it moves along the axis of symmetry is determined by the product of the drag of a spherical liquid droplet and a coefficient which takes into account the shape and is determined from the drag of a solid ellipsoid for which the exact solutions are known. It is shown below that this assumption is not always valid.Translated from Tzvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–8, May–June, 1987.     

Drag of a strongly heated sphere at small Reynolds numbers

Using an asymptotic small-perturbation method, the flow around a strongly heated sphere at small Reynolds numbers Re ≪ 1 is considered with account for thermal stresses in the gas in the higher-order approximations, beyond the Stokes one. It is assumed that the value of the Prandtl number Pr is arbitrary and the temperature dependence of the viscosity is described by a power law with an arbitrary exponent. In the O(Re²) and O(Re³ ln(Re)) approximations, the drag force of a heated sphere is found over a wide range of the ratios of sphere’s temperature to the gas free-stream temperature T _W /T _∞. The limits of applicability of the first (in Re) approximation are investigated, including the negative-drag effect, attributable to the action of the thermal stresses. The results are compared with numerical calculations of the flow around a hot sphere. The limits of applicability of the approximations found are examined. Similar results are obtained for the standard Navier-Stokes equations in which the thermal stresses are neglected.     

Spatio-temporal instability of two-layer liquid film at small Reynolds numbers

The onset of instability with respect to the spatio-temporally growing disturbance in a viscosity-stratified two-layer liquid film flow is analyzed. The known results obtained from the temporal theory of instability show that the flow is unstable in the limit of zero Reynolds numbers. The present theory predicts the neutral stability in the same limit. The discrepancy is explained. Based on the mechanical energy equation, a new mechanism of instability is found. The new mechanism is associated with the convective nature of the disturbance that is not Galilei invariant.     

Diffusion to a particle at large péclet numbers and small Reynolds and Hartmann numbers

A study is made of the influence of a homogeneous magnetic field on the mass transfer for a spherical solid particle and a liquid drop in a flow of a viscous electrically conducting fluid. The previously obtained [1] velocity field of the fluid is used to calculate the concentration distribution in the diffusion boundary layer, the density of the diffusion flux, and the Nusselt number, which characterizes the mass transfer between the particle and the surrounding medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 189–192, January–February, 1980.     

Peristaltic flow at finite Reynolds numbers

In this article the author discusses the results of a numerical investigtion of peristaltic flow at finite Reynolds numbers and finite wave numbers and amplitudes of the traveling wave at the channel walls. The limits of applicability of the data of the asymptotic analysis carried out [6] by means of separate expansions in powers of the Reynolds number and the wave number are determined. It is shown that with increase in the Reynolds number the possibility of transition, under certain conditions, to the flow structure corresponding to nonaxial trapping is preserved.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–15, May–June, 1985.The author wishes to thank E. M. Zhukhovitskii for his interest in the work.     

Transverse galloping at low Reynolds numbers

The possibility of transverse galloping of a square cylinder at low Reynolds numbers (Re≤200, so that the flow is presumably laminar) is analysed. Transverse galloping is here considered as a one-degree-of-freedom oscillator subjected to fluid forces, which are described by using the quasi-steady hypothesis (time-averaged data are extracted from previous numerical simulations). Approximate solutions are obtained by means of the method of Krylov-Bogoliubov, with two major conclusions: (i) a square cylinder cannot gallop below a Reynolds number of 159 and (ii) in the range 159≤Re≤200 the response exhibits no hysteresis.     

Drag measurements on long thin cylinders at small angles and high Reynolds numbers

Measurements of the drag caused by turbulent boundary layer mean wall shear stress on cylinders at small angles of attack and high length Reynolds numbers (8×10⁶<Re_L<6×10⁷) are presented. The use of a full-scale, high-speed towing tank enabled the development of turbulent boundary layers on cylinders made of stainless steel, aluminum, titanium, and polyvinyl chloride. The diameter of all cylinders in this experiment was 12.7 mm; two cylinder lengths, 3.05 m and 6.10 m, were used, corresponding to aspect ratio values L/a=480 and 960, respectively. Materials of various densities were towed at critical angles, resulting in linear cylinder geometry for tow speeds ranging from 2.6 m/s to 20.7 m/s and angles between 0° and 12°. Towing angles were measured with digital photography, and streamwise drag was measured with a strut-mounted load cell at the tow point. The measured tangential drag was very sensitive to small increases in angle at all tow speeds. A momentum thickness length scale is proposed to scale the tangential drag coefficient. The effects of the cross-flow resulting from the small angles of tow have a significant effect on the tangential drag coefficient values. A scaling for the orthogonal force on the cylinders was determined and provides a correction to published normal drag coefficient values for pure cross-flow. The presence of the axial turbulent boundary layer has a significant effect on these orthogonal forces.     

Annular liquid jets at high Reynolds numbers

The flow of annular liquid jets at high Reynolds numbers is analysed by means of the finite element method and the full‐Newton iteration scheme. Results have been obtained for various values of the inner to the outer diameter ratio and for non‐zero surface tension, using extremely long meshes. The annular film moves far from the symmetry axis at low values of the Reynolds number. At higher Reynolds numbers, the film moves towards the axis of symmetry and appears close to very far downstream, forming a round jet. Asymptotic results for the radius of the resulting round jet are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

EHD flows at large electric Reynolds numbers

The motion of a medium consisting of neutral particles and charged particles of single sign is studied under the assumption that the electric Reynolds number (R_q=u/bE) is large. We calculate the freezingin integral and the Bernoulli and Cauchy-Lagrange integrals, study the fluid motion in a stream tube, and formulate the boundary layer problem.     

Interaction of a swarm of particles with a pulsating liquid at small Reynolds numbers

Equations are obtained for the effective viscosity of a liquid that is being filtered in a polydisperse cloud of particles, and for the force of interaction between the liquid and the particles in a nonstationary stream. These equations are solved for the limiting cases of small and large frequencies of the liquid pulsations. When a liquid flows over a concentrated system of particles, the latter influence the structure of the flow in the vicinity of each particle, leading, in particular, to an appreciable change in the relations between the shear velocities and the stresses produced in the system, and between the relative velocity and the force of interaction of the liquid with the particles. The problem of determining these relations, which is of considerable practical interest, has stimulated a diligent investigation of such a constricted flow of a liquid (see, e.g., the review in [1, 2]). However, in view of the difficulties of the analysis of the flow even in a regular lattice made up of regularly disposed particles, it is customary to introduce in theoretical papers dealing with the subject definite model assumptions that give rise to a considerable element of empiricism. An example is the known cell model of constricted flow [1], according to which hydrodynamic screening of each particle by the neighboring particles is taken into account semiempirically, by introducing a certain sphere concentric with the particle, on the surface of which the perturbation introduced into the stream by the given particle should vanish to some degree. This model was recently used again to calculate both the viscous-interaction force [3] and the effective viscosity [4] in stationary flow. Analogous results for nonstationary flow, insofar as the author knows, do not exist at all. In the present paper, these questions are considered more rigorously on the basis of the point-force approximation proposed in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–113, September–October, 1971.The author is grateful to V. G. Markov for useful discussion of the model and of the results of the present work.     

Free and confined jets at low Reynolds numbers

Free jets, and jets with tubular confinements, are investigated in the jet Reynolds number regime 80 Re_j 1000 being of interest for micro-jet pumps, among other applications. For issuing the jets, conventional (single-hole) nozzles as well as dual-hole nozzles of a particular design are used. Both flow visualization and LDA measurement indicate that, in agreement with previous findings, the jets issuing from conventional nozzles remain laminar up to large distances from the orifice. Thus there is but little entrainment of ambient fluid, and the performance of conventional nozzles in micro-jet pumps is rather poor. The dual-hole nozzles, on the other hand, are found to enforce transition to turbulent flow near the orifices. As a result, the entrainment rate is considerably increased, and the performance of jet pumps is improved when the dual-hole nozzles are applied. The experimental data are found to be in fair agreement with predictions based on mass and momentum balances.     

Instability of the interface in two-layer flows with large viscosity contrast at small Reynolds numbers

The Kelvin–Helmholtz instability is believed to be the dominant instability mechanism for free shear flows at large Reynolds numbers. At small Reynolds numbers, a new instability mode is identified when the temporal instability of parallel viscous two fluid mixing layers is extended to current-fluid mud systems by considering a composite error function velocity profile. The new mode is caused by the large viscosity difference between the two fluids. This interfacial mode exists when the fluid mud boundary layer is sufficiently thin. Its performance is different from that of the Kelvin–Helmholtz mode. This mode has not yet been reported for interface instability problems with large viscosity contrasts.These results are essential for further stability analysis of flows relevant to the breaking up of this type of interface.     

Fluid oscillations in pipes at moderate Reynolds numbers

Little is known of the transition of the laminar motion of an oscillating fluid into turbulent motion and the resistance to motion in this region. The theoretical calculation of the critical value of the Reynolds number is very complex in this case and has not yet been successfully accomplished [1, 2]. Some experimental data on this subject are presented in [3]. Below are presented results of measurements of the critical values of the Reynolds number and the resistance forces for laminar and turbulent regimes of fluid oscillations in pipes.     

Packet of gravity surface waves at high Reynolds numbers

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the third power of the steepness. The conditions, under which the packet envelope evolution is described by the nonlinear Schrödinger equation with a dissipative term linear in the amplitude, are determined. The rule, in accordance with which the term of this type can be correctly added in the evolutionary equation of an arbitrary order is formulated.     

The wake of falling disks at low Reynolds numbers

We visualized the wake structure of circular disks falling vertically in quiescent water.The evolution of the wake was shown to be similar to the flow patterns behind a fixed disk.The Reynolds number,Re = Ud/ν,is in the range of 40 200.With the ascension of Reynolds numbers,a regular bifurcation occurred at the first critical Reynolds number Re c 1,leading to a transition from an axisymmetric wake structure to a plane symmetric one;A Hopf bifurcation took place at the second critical Reynolds number Re c 2,as the wake structure became unsteady.Plane symmetry of the wake structure was first lost as periodic vortex shedding appeared,but recovered at higher Reynolds number.The difference between the two critical Reynolds numbers was found to be shape-dependent,as we compared our results for thin discs with those for other falling bodies,such as spheres and cones.This observation could be understood in terms of the instability mechanism of the vortical structure.     

Model-based control of vortex shedding at low Reynolds numbers

Model-based feedback control of vortex shedding at low Reynolds numbers is considered. The feedback signal is provided by velocity measurements in the wake, and actuation is achieved using blowing and suction on the cylinder’s surface. Using two-dimensional direct numerical simulations and reduced-order modelling techniques, linear models of the wake are formed at Reynolds numbers between 45 and 110. These models are used to design feedback controllers using \(\mathcal {H}_\infty \) loop-shaping. Complete suppression of shedding is demonstrated up to Re \(=\) 110—both for a single-sensor arrangement and for a three-sensor arrangement. The robustness of the feedback controllers is also investigated by applying them over a range of off-design Reynolds numbers, and good robustness properties are seen. It is also observed that it becomes increasingly difficult to achieve acceptable control performance—measured in a suitable way—as Reynolds number increases.