Breaking of waves of limiting amplitude over an obstacle

Homogeneous heavy fluid flows over an uneven bottom are studied in a long-wave approximation. A mathematical model is proposed that takes into account both the dispersion effects and the formation of a turbulent upper layer due to the breaking of surface gravity waves. The asymptotic behavior of nonlinear perturbations at the wave front is studied, and the conditions of transition from smooth flows to breaking waves are obtained for steady-state supercritical flow over a local obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 3–11, May–June, 2006.     

Theory of steady waves in horizontal flow with a linear velocity profile

The structure and characteristics of nonlinear steady waves on the surface of horizontal shear flow of an ideal homogeneous incompressible fluid of finite depth with a linear velocity profile are studied using two-dimensional theory and the Euler approach. The wave motion is considered irrotational. A modification of the first Stokes method is proposed that allows algebraic calculations of terms of perturbation series. Nonlinear dispersion relations are obtained and analyzed for both upstream and downstream traveling waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 43–48, May–June, 2006.     

Two-layer flow of magnetic fluids

The plane two-layer flow of viscous incompressible fluids with differentmagnetic properties between two horizontal rigid planes in a nonuniform traveling magnetic field is investigated. An arbitrary nonuniform periodic traveling magnetic field generates wavelike changes in the interface between the media and fluid flows with nonzero flow rates. From the given magnetic field the interface shape and the velocities, pressures and mean flow rates of the fluids are calculated. The cases of a cosine magnetic force and of the magnetic field created by ferromagnetic cylinders moving in a uniform magnetic field are considered. The effect of various parameters on the mean flow rates of the fluids is investigated.     

Single cavitation bubble generation and observation of the bubble collapse flow induced by a pressure wave

This study utilizes a U-shape platform device to generate a single cavitation bubble for a detailed analysis of the flow field characteristics and the cause of the counter jet during the process of bubble collapse caused by sending a pressure wave. A high speed camera is used to record the flow field of the bubble collapse at different distances from a solid boundary. It is found that a Kelvin–Helmholtz vortex is formed when a liquid jet penetrates the bubble surface after the bubble is compressed and deformed. If the bubble center to the solid boundary is within one to three times the bubble’s radius, a stagnation ring will form on the boundary when impinged by the liquid jet. The fluid inside the stagnation ring will be squeezed toward the center of the ring to form a counter jet after the bubble collapses. At the critical position, where the bubble center from the solid boundary is about three times the bubble’s radius, the bubble collapse flow will vary. Depending on the strengths of the pressure waves applied, the collapse can produce a Kelvin–Helmholtz vortex, the Richtmyer–Meshkov instability, or the generation of a counter jet flow. If the bubble surface is in contact with the solid boundary, the liquid jet can only move inside-out without producing the stagnation ring and the counter jet; thus, the bubble collapses along the radial direction. The complex phenomenon of cavitation bubble collapse flows is clearly manifested in this study.     

Stability of a fluid interface under tangential vibrations

The stability of the interface between two immiscible fluids of different density which occupy a plane horizontal layer performing harmonic horizontal oscillations is considered. Within the framework of the ideal fluid model a transformation reducing the problem of small plane perturbations to the Mathieu equation is found. Resonance instability domains associated with the formation of capillary-gravitational waves are investigated. A model which takes into account dissipation processes due to the presence of viscous friction is constructed. The role of the viscous dissipation in suppressing resonance instability is discussed. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–31, May–June, 1998. The work was carried out with partial support from the Russian Foundation for Basic Research (project No. 95-01-00386).     

Active Control of Wave Instabilities in Three-Dimensional Compressible Flows

In many fluid flows of practical importance transition is caused by the linear growth of wave instabilities, such as Tollmien–Schlichting waves, which eventually grow to a finite size at which stage secondary instabilities come into play. If transition is to be delayed or even avoided in such flows, then the linear growth of the disturbances must be prevented since control in the nonlinear regime would be a considerably more difficult task. Here a strategy for active control of two-dimensional incompressible and compressible Tollmien–Schlichting waves and its use in controlling the more practically relevant problem of crossflow instability which arises in swept-wing flows is discussed. The control is through an active suction/blowing distribution at the wall though the same result could be achieved by variable wall heating. In order to control the instability it is assumed that the wall shear stress and pressure are known from measurements. It is shown that, certainly at finite Reynolds numbers, it is sufficient to know the flow properties at a finite number of points along the wall. The cases of high and finite Reynolds numbers are discussed using asymptotic and numerical methods respectively. It is shown that a control strategy can be developed to stop the growth of all two-dimensional Tollmien–Schlichting waves at finite and large Reynolds numbers. Some discussion of nonlinear effects in the presence of active control is given and the possible control of other instability mechanisms investigated. Received 1 May 1998 and accepted 24 September 1998     

Effect of fluid compressibility on the linear stability of Poiseuille flow in a circular tube

The linear stability of a barotropic fluid flow in a circular absolutely rigid tube is considered. The behavior of perturbations in the form of monoharmonic waves resulting from the viscosity as well as the compressibility of the fluid is investigated. It is shown that the compressibility of the fluid affects the first type of perturbations only slightly and the second significantly and that the latter can be more dangerous from the standpoint of initiating instability, even for weakly compressible flows. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 5–10, March–April, 1994.     

Characteristics of unsteady type IV shock/shock interaction

Characteristics of the unsteady type IV shock/shock interaction of hypersonic blunt body flows are investigated by solving the Navier–Stokes equations with high-order numerical methods. The intrinsic relations of flow structures to shear, compression, and heating processes are studied and the physical mechanisms of the unsteady flow evolution are revealed. It is found that the instantaneous surface-heating peak is caused by the fluid in the “hot spot” generated by an oscillating and deforming jet bow shock (JBS) just ahead of the body surface. The features of local shock/boundary layer interaction and vortex/boundary layer interaction are clarified. Based on the analysis of flow evolution, it is identified that the upstream-propagating compression waves are associated with the interaction of the JBS and the shear layers formed by a supersonic impinging jet, and then the interaction of the freestream bow shocks and the compression waves results in entropy and vortical waves propagating to the body surface. Further, the feedback mechanism of the inherent unsteadiness of the flow field is revealed to be related to the impinging jet. A feedback model is proposed to reliably predict the dominant frequency of flow evolution. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.     

Influence of a homogeneous axial magnetic field on Taylor–Couette flow of ferrofluids with low particle–particle interaction

Experimental results concerning the stability of Couette flow of ferrofluids under magnetic field influence are presented. The fluid cell of the Taylor–Couette system is subject to a homogeneous axial magnetic field and the axial flow profiles are measured by ultrasound Doppler velocimetry. It has been found that an axial magnetic field stabilizes the Couette flow. This effect decreases with a rotating outer cylinder. Moreover, it could be observed that lower axial wave numbers are more stable at a higher axial magnetic field strength. Since the used ferrofluid shows a negligible particle–particle interaction, the observed effects are considered to be solely based on the hindrance of free particle rotation.     

Stationary simple waves in an electrically conducting magnetizable fluid

A study is made of the analog of Prandt1—Meyer flow in an incompressible electrically conducting ideal fluid that is magnetizable in accordance with an arbitrary isotropic law. It is shown that inhomogeneity of the magnetization in a conducting fluid makes possible the existence of stationary simple waves with varying magnetic permeability. For a paramagnetic fluid magnetized to saturation, the equations of these waves are integrated completely in the case of a magnetic field parallel to the velocity. Some regions of such flows of magnetizable fluids are discussed in the present paper for the example of the problem of flow around a perfectly conducting profile.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 136–143, September–October, 1980.I thank I.E. Tarapov for his interest in the work and valuable comments made in a discussion.     

Turbulent flow simulation of liquid jet emanating from pressure-swirl atomizer

The performances of three linear eddy viscosity models (LEVM) and one algebraic Reynolds stress model (ARSM) for the simulation of turbulent flow inside and outside pressure-swirl atomizer are evaluated by comparing the interface position with available experimental data and by comparing the turbulence intensity profiles at the atomizer exit. It is found that the turbulence models investigated exhibit zonal behaviors, i.e. none of the models investigated performs well throughout the entire flow field. The turbulence intensity has a significant influence on the global characteristics of the flow field. The turbulence models with better predictions of the turbulence intensity, such as Gatski-Speziale’s ARSM model, can yield better predictions of the global characteristics of the flow field, e.g. the reattachment lengths for the backward-facing step flow and the sudden expansion pipe flow, or the discharge coefficient, film thickness and the liquid sheet outer surface position for the atomizer flows. The standard k–ε model predicts stronger turbulence intensity as compared to the other models and therefore yields smaller film thickness and larger liquid sheet outer surface position. In average, the ARSM model gives both quantitatively and qualitatively better results as compared to the standard k–ε model and the low Reynolds number models.     

Special case of a traveling wave in one model of a multicomponent medium

The Kuropatenko model for a multicomponent medium whose components are polytropic gases is considered. It is assumed that, as x → ±∞, the multicomponent medium is in a homogeneous state with constant gas-dynamic parameters — velocity, pressure, and temperature. For the traveling wave flows, conditions similar to the Hugoniot conditions are obtained and used to uniquely determine the flow parameters for x → −∞ from the flow parameters x → +∞ and traveling wave velocity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 39–47, July–August, 2009.     

Two- and Four-Way Coupled Euler–Lagrangian Large-Eddy Simulation of Turbulent Particle-Laden Channel Flow

Large-eddy simulations (LES) of a vertical turbulent channel flow laden with a very large number of solid particles are performed. The motivation for this research is to get insight into fundamental aspects of co-current turbulent gas-particle flows, as encountered in riser reactors. The particle volume fraction equals about 1.3%, which is relatively high in the context of modern LES of two-phase flows. The channel flow simulations are based on large-eddy approximations of the compressible Navier–Stokes equations in a porous medium. The Euler–Lagrangian method is adopted, which means that for each individual particle an equation of motion is solved. The method incorporates four-way coupling, i.e., both the particle-fluid and particle–particle interactions are taken into account. The results are compared to single-phase channel flow in order to investigate the effect of the particles on turbulent statistics. The present results show that due to particle–fluid interactions the mean fluid profile is flattened and the boundary layer is thinner. Compared to single-phase turbulent flow, the streamwise turbulence intensity of the gas phase is increased, while the normal and spanwise turbulence intensities are reduced. This finding is generally consistent with existing experimental data. The four-way coupled simulations are also compared with two-way coupled simulations, in which the inelastic collisions between particles are neglected. The latter comparison clearly demonstrates that the collisions have a large influence on the main statistics of both phases. In addition, the four-way coupled simulations contain stronger coherent particle structures. It is thus essential to include the particle–particle interactions in numerical simulations of two-phase flow with volume fractions around one percent.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side

We study both experimentally and numerically the convective flow in a tall vertical slot with differently heated walls. The flow is investigated for the fluid with the Prandtl number Pr=26, which is large enough to ensure the traveling waves as primary instability and small enough to prevent boundary layer convection. The flow evolution is determined on the base of the visual observations, power spectra and amplitude analysis. In the numerical simulations of two- and three-dimensional flows, we accept an assumption of an infinite fluid layer. The satisfactory agreement with experiment is observed, and the sequence of convection states is discovered. It starts with a plane-parallel flow as primary solution, which becomes unstable to two counter-propagating waves. It is followed by a tertiary three-dimensional flow in the form of wavy traveling waves. As the Grashof number is increased even further, a chaotically oscillating cellular pattern consisting of the pieces of broken waves arises. The formation of a structure in the form of the vertical rolls chaotically modulated along axes concludes this complicated picture.     

Effect of Relaxation and Retardation Time on Peristaltic Transport of the Oldroydian Viscoelastic Fluid

The influence of relaxation and retardation time on peristaltic transport of an incompressible Oldroydian viscoelastic fluid by means of an infinite train of sinusoidal waves traveling along the walls of a two-dimensional flexible channel is investigated. A perturbation solution is obtained for the case in which the amplitude ratio (wave amplitude to channel half-width) is small. The results show that the values of the mean axial velocity of an Oldroydian viscoelastic fluid is smaller than that for a Newtonian fluid. The reflux phenomena are discussed. It is found that the critical reflux pressure gradient decreases with increasing retardation time and increases with increasing relaxation time. Numerical results are reported for different values of the physical parameters of interest. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 86–95, November–December, 2005.     

Gas-dynamic analogy for vortex free-boundary flows

The classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of long-wave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 8–15, May–June, 2007.     

Flow of electrolytes in a porous medium

A two-scale model of ion transfer in a porous medium is obtained for one-dimensional horizontal flows under the action of a pressure gradient and an external electric field by the method of homogenization. Steady equations of electroosmotic flows in flat horizontal nano-sized slits separated by thin dielectric partitions are averaged over a small-scale variable. The resultant macroequations include Poisson’s equation for the vertical component of the electric field and Onsager’s relations between flows and forces. The total horizontal flow rate of the fluid is found to depend linearly on the pressure gradient and external electric field, and the coefficients in this linear relation are calculated with the use of microequations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 162–173, July–August, 2008.     

Expanding the applications of holographic interferometry to the quantitative visualization of oscillatory thermofluid processes using temperature as tracer

 In this paper we discuss a novel application of holographic interferometry in the simultaneous quantitative visualization of high-speed, oscillatory flow and temperature fields in complex flow geometries. We consider cases of (i) self-sustained oscillatory flows with main flow imposed in grooved and communicating channels as well as (ii) oscillating thermofluid processes with zero mean velocity in a thermoacoustic refrigerator model. Examples showing unsteady temperature distributions obtained by real-time holographic interferometry combined with high-speed cinematography illustrate the possibilities of the approach introduced in the paper. Our study shows that temperature distributions accurately mirror flow structures in certain types of complex, unsteady flows, thus allowing, apart from the measurement of temperature profiles and heat transfer, also the measurement of oscillatory amplitudes, frequencies, wavelengths as well as the speed of propagation of traveling waves by applying digital image processing techniques. In the grooved and communicating channels it is possible to visualize the structure of the Tollmien–Schlichting waves through isotherms by using the infinite fringe field alignment of holographic interferometry. In the thermoacoustic refrigerator model, small amplitude temperature oscillations generated by the acoustic standing wave are visualized and measured. Image processing as well as data reduction procedures used in the analysis of these flow fields are discussed in the paper. Experimental data obtained by applying the techniques introduced in the paper show good agreement with theory and results of numerical simulations. Our study suggests that using temperature as tracer offers numerous advantages in the study of certain types of complex, unsteady flows. Received: 7 January 1997/Accepted: 18 September 1997     

Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.