STEADY FLOW GENERATED BY AN INTERFACIAL WAVE IN AN OSCILLATING CELL

Heat or mass transfer from spherical particles in oscillatory flow has important applications in combustion and spray drying. This work provides a parametric investigation of drag forces experienced by, and transport of a passive scalar from, an isolated rigid fixed sphere in steady and oscillatory axisymmetric flows. At Schmidt (Prandtl) number of 1, oscillatory flows with Reynolds numbers in the range 1–100 and oscillation amplitudes in the range 0.05–5 sphere diameters are investigated using numerical simulation. Scalar concentration is uniform on the surface of the sphere and zero in the far field. Coefficients of peak drag for steady and oscillatory flows are presented and compared to values obtained from Basset’s analytical solution for Stokes flow, and the relative contributions of the added mass, Stokes drag, and Basset history terms are examined. At the higher Reynolds numbers and amplitudes, it is found that the time-average mass transfer rate can be more than double that for diffusion in quiescent fluid, or in Stokes flow. Time-average Sherwood (Nusselt) numbers for oscillatory flows asymptote to the Stokes limit at low oscillation amplitude, regardless of Reynolds number. An unexpected result is that at intermediate Reynolds numbers and oscillation amplitudes, it is possible to depress the time-average mass-transfer coefficient slightly below that for Stokes flow. Within the Reynolds number range considered, Sherwood–Nusselt numbers in steady flow are found to be always higher than for an oscillatory flow of the same root-mean-square (rms) velocity.

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.

The dynamics of the interface of liquids with a high viscosity contrast, performing harmonic oscillations with zero mean in a straight slot channel, is experimentally investigated. The boundary is located across the channel and oscillates along the channel with a harmonic change in the flow rate of the fluid pumped through the channel. Owing to the high contrast of viscosities, the motion of the more viscous liquid obeys Darcy's law, while the low-viscosity liquid performs “inviscid” oscillations. The oscillations of the interface occur in the form of an oscillating flat tongue of low-viscosity liquid that periodically penetrates into the more viscous one. The interface oscillations lead to the manifestation of two effects. One of these consists of changes in the averaged shape of the interface and the liquid contact line. The interface in the cell plane takes the form of a “hill,” the dynamical equilibrium of which is maintained by oscillations, while the deformation of the boundary is proportional to the amplitude of the oscillations and vanishes in their absence. The second effect consists of the development of finger instability of the oscillating boundary, which manifests itself in the periodic development of fingers of low-viscosity liquid at part of a period. The instability develops in a threshold manner when the relative amplitude of the interface oscillations reaches a critical value. It is found that the instability has a local character and manifests itself in those regions of the interface where the amplitude of the oscillations reaches a critical value. The stability threshold decreases with the dimensionless frequency.

This paper reports on an extensive parameter space study of two-dimensional simulations of a circular cylinder forced to oscillate transverse to the free-stream. In particular, the extent of the primary synchronization region, and the wake modes and energy transfer between the body and the fluid are analyzed in some detail. The frequency range of the primary synchronization region is observed to be dependent on Reynolds number, as are the wake modes obtained. Energy transfer is primarily dependent on frequency at low amplitudes of oscillation, but primarily dependent on amplitude at high amplitudes of oscillation. However, the oscillation amplitude corresponding to zero energy transfer is found to be relatively insensitive to Reynolds number. It is also found that there is no discernible change to the wake structure when the energy transfer changes from positive to negative.

The problem of the onset of instability in a liquid layer flowing down a vibrating inclined plane is formulated. For the solution of the problem, the Fourier components of the disturbance are expanded in Chebychev polynomials with time-dependent coefficients. The reduced system of ordinary differential equations is analysed with the aid of Floquet theory. The interaction of the long gravity waves, the relatively short shear waves and the parametrically resonated Faraday waves occurring in the film flow is studied. Numerical results show that the long gravity waves can be significantly suppressed, but cannot be completely eliminated by use of the externally imposed oscillation on the incline. At small angles of inclination, the short shear waves may be exploited to enhance the Faraday waves. For a given set of relevant flow parameters, there exists a critical amplitude of the plane vibration below which the Faraday wave cannot be generated. At a given amplitude above this critical one, there also exists a cutoff wavenumber above which the Faraday wave cannot be excited. In general the critical amplitude increases, but the cutoff wavenumber decreases, with increasing viscosity. The cutoff wavenumber also decreases with increasing surface tension. The application of the theory to a novel method of film atomization is discussed.

The formation of sprays from a liquid film on a vibrating surface is used by ultrasonic atomizers for applications ranging from humidification to metal–powder manufacturing. The received opinion in the literature is that droplets are formed periodically from the apexes of an orderly pattern of standing capillary waves, with a wavelength that can be related to vibration frequency by stability analysis. It is described how this assumption may be incorrect in that, after droplet formation commences, the orderliness of the standing–wave pattern is lost due to one or more secondary instability phenomena. These phenomena, which lead to disorderliness, are investigated by using high–speed imaging techniques and a low–frequency vibrating film to model the high–frequency case, because of the difficulty of penetrating clouds of small droplets in the latter case. Different modes of droplet formation are identified and the flow patterns responsible for these modes are discussed. Physical mechanisms are proposed from which it is deduced that only a proportion of standing–wave crests can eject droplets, for a given wall–vibration period, and the identity of these ejecting waves should vary from period to period. The model thus developed demonstrates an apparently random ejection of droplets from one wave cell, even though the model itself is deterministic. The disorder of the capillary waves and the occurrence of several droplet–formation routes are sufficient to explain the range of droplet sizes that is produced by ultrasonic atomization.

In this study, the two-dimensional steady flow of power-law fluids past a semicircular cylinder (flat face oriented upstream) has been investigated numerically. The governing equations (continuity, momentum, and energy) have been solved in the steady symmetric flow regime over the range of the Reynolds number (0.01 ≤ Re ≤ 25), power-law index (0.2 ≤ n ≤ 1.8), and Prandtl number (0.72 ≤ Pr ≤ 100). Extensive new results reported here endeavor to elucidate the role of power-law index (0.2 ≤ n ≤ 1.8) on the critical Reynolds number denoting the onset of flow separation (Re c ) and of vortex shedding (Re c ). In shear-thinning fluids, both of these transitions are seen to be delayed than that in Newtonian and shear-thickening fluids. Furthermore, the influence of the Reynolds and Prandtl numbers, power-law index on drag phenomenon, and heat characteristics of semicircular cylinder have been studied in the steady flow regime. Finally, the present numerical values of the critical Reynolds numbers and the average Nusselt number have been correlated by simple forms which are convenient for interpolating these results for the intermediate values of the governing parameters in a new application.

In this paper, the stability of supersonic contact discontinuities in the three-dimensional compressible isentropic steady Euler flows is investigated by using the nonlinear geometric optics. We construct the asymptotic expansions of highly oscillatory contact discontinuities when a planar contact discontinuity is perturbed by a small amplitude high frequency oscillatory incident wave, and deduce there exists a large amplification of amplitudes in the reflected and refracted oscillatory waves when the high frequency oscillatory wave strikes the contact discontinuity front at three critical angles. Moreover, we obtain that the leading profiles of highly oscillatory waves are described by an initial boundary value problem of Burgers-transport equations, and the leading profile of contact discontinuity front satisfies an initial value problem of a Hamilton-Jacobi equation, respectively. The amplification phenomenon shows that this supersonic contact discontinuity is only weakly stable in the sense of Wang and Yu [“Stability of contact discontinuities in three-dimensional compressible steady flows,” J. Differ. Equ. 255, 1278–1356 (2013)].

Faraday wave instability characteristics of a single droplet in ultrasonic atomization and the sub-droplet generation mechanism

Vortex structure of steady flow in a rectangular cavity

The formation of Faraday waves (FWs) at the surfactant-covered free surface of a vertically vibrated liquid layer is considered. The layer is subjected to a vertical temperature gradient. The surfactant is insoluble. Linear stability analysis and the Floquet method are used for disturbances with arbitrary wave numbers to find the regions of critical vibration amplitude where FWs are generated. The problem is considered for the semi-infinite liquid layer, as well as for the layer of a finite depth. It is shown numerically, that in the semi-infinite case the critical tongue of a neutral stability curve corresponding to the lowest value of the forcing amplitude is related to the subharmonic instability mode. It changes to the harmonic one in the case of finite depth. The influence of thermocapillary Marangoni number on the critical amplitude of FWs is studied. The growth of that number stabilizes the system, however, this effect is very weak.

Through the method of visible observation, experimental study on transient-state characteristics of temperature on the interface of stratified fluids is made in three different experimental tubes. On the basis of experimental study, mechanism of temperature oscillation and heat transfer behaviors on the interface of stratified fluids are analyzed under transient-state conditions in the density lock. The results show that the temperature of work fluid near the interface has two kinds of instabilities—high amplitude oscillation and low amplitude oscillation. High amplitude oscillation results from the consequent breaking of the interface and will bring vigorous heat transfer between stratified fluids. After the high amplitude oscillation takes place, temperature jump of work fluid will arise on the interface. However, low amplitude oscillation is only dynamic wave of interface around its balanced position, and it will not cause vigorous heat transfer. Besides, comparative experiment on transient-state characteristics of temperature is done in experimental pipes with different diameters, the results show that high amplitude oscillation is more likely to take place in experimental pipes with larger diameters. Therefore, by virtue of its small diameters, honey-comb channel in density lock can curb the occurrence of high amplitude oscillation, efficiently decreasing heat transfer between stratified fluids.

Steady and pulsatile flow distribution in a multiple branching network with physiological applications

This study investigates the triggering of vortex shedding by transverse and in-line forced oscillation of a circular cylinder placed in a uniform stream below the critical Reynolds number 47. The effect of Reynolds number, frequency and amplitude of cylinder oscillation is investigated numerically. The effect of Reynolds number, amplitude of oscillation, and frequency of oscillation is investigated. Time-mean and rms values of lift, drag, base pressure, and torque coefficients, as well as mechanical energy transfer, are plotted against Reynolds number and amplitude of oscillation. Periodic vortex shedding is more easily triggered for a cylinder oscillated transversely than for in-line oscillation. Lower Reynolds number, amplitude of oscillation, and time is needed to reach periodic vortex shedding for the transversely oscillated cylinder.

5R6. Theory and Applications of Viscous Fluid Flows. - RK Zeytounian (12 Rue Saint-Fiacre, Paris, 75002, France). Springer-Verlag, Berlin. 2004. 488 pp. ISBN 3-540-44013-5. $109.00.Reviewed by MF Platzer (Dept of Aeronaut and Astronaut, Naval Postgraduate Sch, Code AA/PL, Monterey CA 93943-5000).Starting with the derivation of the Navier-Stokes equations for viscous heat-conducting fluids the author proceeds to discuss various forms of these equations, including the special cases of compressible isentropic viscous flow of polytropic gases and viscous incompressible fluid flow. He then discusses the Orr-Sommerfeld theory for the plane Poiseuille flow as well as other basic flow cases, such as steady flow through an arbitrary cylinder, annular flow between concentric cylinders, Benard thermal convection flow, Benard-Marangoni flow induced by tangential gradients of variable surface tension, flow due to a rotating disc, and Rayleigh flow caused by an impulsively started flat plate. The next three chapters are devoted to the very large and very low Reynolds number limits and to the low Mach number incompressible limit. In the chapter on very large Reynolds number flow the author discusses the application of the method of matched asymptotic expansions to the two-dimensional steady flat-plate flow problem and delineates the relationship of the unsteady Navier-Stokes equations to the inviscid Euler, the Prandtl boundary layer, the one-dimensional gas dynamics and the Rayleigh compressible flow equations. He also discusses the triple deck concept, laminar flow separation on a circular cylinder, and the three-dimensional boundary layer equations. In the chapter on very low Reynolds numbers, the unsteady-state matched Stokes-Oseen solution for the flow past a sphere and the flow over an impulsively started circular cylinder are discussed, followed by a consideration of the Stokes and Oseen steady-state compressible flow equations and the asymptotic analysis for small Reynolds number flows on a rotating disc. In the next chapter on low Mach number incompressible limit, the author discusses subtleties involved in analyzing unsteady weakly compressible flows; flow in a bounded cavity and through large aspect ratio channels. He then provides further examples by analyzing the acoustic streaming effect caused by an oscillating circular cylinder, the incompressible flow past a rotating and translating cylinder, the Ekman and Stewartson layers on rotating cylinders, and the Benard-Marangoni thermo-capillary instability problem due to heating of a horizontal viscous liquid from below. Also presented are some aspects of non-adiabatic viscous atmospheric flows and a few other topics, such as the entrainment of a viscous fluid in a two-dimensional cavity and the laminar boundary layer separation phenomenon near the leading-edge region of an airfoil and on an impulsively started cylinder. In this regard, he emphasizes the need for the simultaneous solution of the boundary layer and inviscid flow equations in order to remove the singularity at the separation point, as implemented in the viscous-inviscid interaction procedures. The next two chapters are devoted to a discussion of the existence, regularity and uniqueness of solutions for the viscous incompressible and compressible flow equations and the stability theory of fluid motion. In particular, the Guiraud-Zeytounian asymptotic approach to nonlinear hydrodynamic stability is elucidated and applied to the Rayleigh-Benard convection problem, followed by an analysis of the Benard-Marangoni thermo-capillary instability problem and the Couette-Taylor viscous flow between two rotating cylinders. The final chapter of Theory and Applications of Viscous Fluid Flows presents the finite-dimensional dynamical systems approach to turbulence by reviewing the Landau-Hopf, Ruelle-Takens-Newhouse, Feigenbaum and Pomeau-Manneville transition scenarios to turbulence. The book is ended by giving examples of strange attractors occurring in various fluid flows, such as in viscous isobaric wave motions or in the flow of an incompressible but thermally conducting liquid down a vertical plane (the Benard-Marangoni problem for a free-falling vertical film). It is evident from this brief summary that the author’s emphasis is on the mathematical aspects of the viscous flow equations and their various asymptotic limit cases and analytical solution methods. His choice of topics and flow problems is meant to provide young researchers in fluid mechanics, applied mathematics and theoretical physics with an up-to-date presentation of some key problems in the analysis of viscous fluid flows. Although the author intentionally limited himself to a select few topics, teachers of advanced viscous flow courses and researchers in this field will welcome this book for its thorough review of current work and the listing of 1156 relevant papers. In my judgment, it meets the stated objective of bridging the gap between standard undergraduate texts in fluid mechanics and specialized monographs.