Numerical-Analytical Investigation of Multimodal Solutions of the Jeffery-Hamel Problem for a Convergent Channel

In the Jeffery-Hamel problem for a convergent channel, for small, moderately large and asymptotically large Reynolds numbers Re, multimodal, symmetrical and asymmetrical solutions not available in the literature are obtained and studied for the entire range of admissible convergence angles. For asymptotically small Re, these solutions depend singularly on Re. A number of novel hydrodynamic effects is established and commented on.     

Linear stability analysis of flow in a periodically grooved channel

We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two‐dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady‐state, whereas a self‐sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty‐function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd.     

Effect of the form of the electrodes on the current distribution in a magnetogasdynamic channel

A solution is given to the plane problem of the flow of a conducting gas across a homogeneous magnetic field in a magnetogasdynamic channel taking account of the Hall effect at small magnetic Reynolds numbers. The channel is formed by two long electrodes, and the cross section of the channel varies slightly and periodically along the gas flow. It is assumed that the electromagnetic forces are small. It is shown that the current distribution in the channel is nonuniform to a consider able degree and that inverse currents can form at the electrodes, with both subsonic and supersonic flows of the conducting gas. Transverse motion of the gas, due to a change in the cross section of the channel, leads to an increase of Joule energy losses. In [1] the current distribution was obtained in a flat channel formed by infinite dielectric walls, with the flow of a steady-state stream of plasma through the channel across a homogeneous magnetic field. With interaction between the flow and the magnetic field, closed current loops develop in the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–33, November–December, 1970.     

Turbulent bore in a supercritical flow over an irregular bed

Breaking waves in a free-surface homogeneous fluid flow in the neighborhood of a local variation in the channel depth are studied experimentally and theoretically. The structure of both a steady-state hydraulic jump generated by a local obstacle in the channel and an unsteady wave configuration consisting of two turbulent bores in the problem of lock failure is studied. Using the turbulent bore model [1], analytic profiles of breaking waves are obtained and the time-dependent problem is numerically investigated and compared with experimental data. It is shown that the model [1] with a hydrostatic pressure distribution over the depth adequately describes both the location and the structure of the steady-state and unsteady wave fronts.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 62–70. Original Russian Text Copyright © 2005 by Gusev and Lyapidevskii.     

Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method

In this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different α, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.     

Side‐wall effects on the bifurcation of the flow through a sudden expansion

Three‐dimensional computations have been performed to study the flow through a symmetric sudden expansion with an expansion ratio of 3 at low Reynolds numbers. The aspect ratio of the flow channel is allowed to vary within a wide range to examine its influence on the flow which bifurcates from a symmetric state to an asymmetric state. The results reveal that the critical Reynolds number of the symmetry‐breaking bifurcation increases while the aspect ratio is reduced. The flow behaviour near the side walls is illustrated by using limiting streamlines. The origin of the singular points identifiable on the side wall can be traced back to the recirculating flows and the relevant reattachment/separation points in the core of the channel. It is seen that the determination of the exact critical Reynolds number is not trivial because it depends on how to define asymmetric flow. Computations have also been conducted to show that a slight asymmetry in the channel geometry causes a smooth transition from symmetric to non‐symmetric states. Copyright © 2007 John Wiley & Sons, Ltd.     

On the topological bifurcation of flows around a rotating circular cylinder

Flow fields around a rotating circular cylinder in a uniform stream are computed using a low dimensional Galerkin method. Reslts show that the formation of a Fopple vortex pair behind a stationary circular cylinder is caused by the structural instability in the vicinity of the saddle located at the rear of the cylinder. For rotating cylinder a bifurcation diagram with the consideration of two parameters, Reynolds numberRe and rotation parameter α, is built by a kinematic analysis of the steady flow fields. The project supported by the National Natural Science Foundation of China     

Subsonic Radiation Gas Dynamics in a Laser Plasma Generator Channel

The radiation gasdynamic processes in the channel of an air laser plasma generator operating at atmospheric pressure are analyzed. In the multigroup approximation a numerical radiation gasdynamic model is formulated on the basis of the equations of motion of a viscous heat-conducting gas and the selective thermal radiation transport equation. Laminar and turbulent subsonic generator operation regimes are considered.For the purpose of approximately describing the turbulent gas and plasma mixing the Navier-Stokes equations averaged after Reynolds and the k-ε turbulence model are used. The problem is solved in the time-dependent two-dimensional axisymmetric formulation.Strong radiation-gasdynamic interaction regimes are investigated. In these regimes the energy losses due to radiation from the high-temperature region of the laser plasma and the absorption of its thermal self-radiation by the surrounding plasma and gas layers (radiation reabsorption) appreciably affect the gasdynamic flow structure. Two methods of integrating the selective thermal radiation transport equation in the generator channel are discussed. In one of these the thermal radiation transport is calculated inside the heated volume and in the other the radiation heat fluxes are calculated on the surfaces bounding the volume. The results of calculating the spectral and integral radiation heat fluxes on the inner surface of the generator are given.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 126–143.Original Russian Text Copyright © 2005 by Surzhikov.     

Use of diverging or converging arrangement of plates for the control of chaotic mixing in symmetric sinusoidal plate channels

This paper presents an experimental flow visualization study of the effect that the variation of converging or diverging angles plays in the flow field of a symmetric sinusoidal channel. The experiments were performed in a water tunnel and the visualization technique was laser illumination of seeded particles whose traces were captured using long time exposure photography. Geometrical parameters such as wave amplitude, wavelength and distance between plates were kept constant, while the Reynolds number and divergence or convergence angles were varied. It was found from the experiments that the divergence of the plates is a good way to promote chaotic mixing in channel flows, as the flow becomes more unstable for diverging channels. For the case of converging channels, the flow becomes very stable even for large values of the Reynolds number. These results were compared with those of a channel formed by a pair of sinusoidal parallel plates.     

Computation of internal turbulent flow with a large separated flow region

An implicit two-equation turbulence solver, KEM. in generalized co-ordinates, is used in conjunction with the three-dimensional incompressible Navier–Stokes solver, INS3D, to calculate the internal flow in a channel and a channel with a sudden 2:3 expansion. A new and consistent boundary procedure for a low Reynolds number form of the κ-ε turbulence model is chosen to integrate the equations up to the wall. The high Reynolds number form of the equations is integrated using wall functions. The latter approach yields a faster convergence to the steady-state solution than the former. For the case of channel flow, both the wall-function and wall-boundary-condition approaches yield results in good agreement with the experimental data. The back-step (sudden expansion) flow is calculated using the wall-function approach. The predictions are in reasonable agreement with the experimental data.     

An investigation of bimodal jet trajectory in flow through scaled models of the human vocal tract

Pulsatile two-dimensional flow through static divergent models of the human vocal folds is investigated. Although the motivation for this study is speech production, the results are generally applicable to a variety of engineering flows involving pulsatile flow through diffusers. Model glottal divergence angles of 10, 20, and 40° represent various geometries encountered in one phonation cycle. Frequency and amplitude of the flow oscillations are scaled with physiological Reynolds and Strouhal numbers typical of human phonation. Glottal velocity trajectories are measured along the anterior–posterior midline by using phase-averaged particle image velocimetry to acquire 1,000 realizations at ten discrete instances in the phonation cycle. The angular deflection of the glottal jet from the streamwise direction (symmetric configuration) is quantified for each realization. A bimodal flow configuration is observed for divergence angles of 10 and 20°, with the flow eventually skewing and attaching to the vocal fold walls. The deflection of the flow toward the vocal fold walls occurs when the forcing function reaches maximum velocity and zero acceleration. For a divergence angle of 40°, the flow never attaches to the vocal fold walls; however, there is increased variability in the glottal jet after the forcing function reaches maximum velocity and zero acceleration. The variation in the jet trajectory as a function of divergence angle is explained by performance maps of diffuser flow regimes. The smaller angle cases are in the unstable transitory stall regime while the 40° divergent case is in the fully developed two-dimensional stall regime. Very small geometric variations in model size and surface finish significantly affect the flow behavior. The bimodal, or flip-flopping, glottal jet behavior is expected to influence the dipole contribution to sound production.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Dynamic NMR microscopy measurement of the dynamics and flow partitioning of colloidal particles in a bifurcation

The flow and distribution of Newtonian, polymeric and colloid suspension fluids at low Reynolds numbers in bifurcations has importance in a wide range of disciplines, including microvascular physiology and microfluidic devices. A bifurcation consisting of circular capillaries laser etched into a hard polymer with inlet diameter 2.50 ± 0.01 mm, bifurcating to a small diameter outlet of 0.76 ± 0.01 mm and a large diameter outlet of 1.25 ± 0.01 mm is examined. Four distinct fluids (water, 0.25%wt xanthan gum, 8 and 22%vol hard-sphere colloidal suspensions) are flowed at flow rates from 10 to 30 ml/h corresponding to Reynolds numbers based on the entry flow from 0.001 to 8. PGSE NMR techniques are applied to obtain dynamic images of the fluids inside the bifurcation with spatial resolution of 59 × 59 μm/pixel in plane over a 200-μm-thick slice. Velocity in all three spatial directions is examined to determine the impact of secondary flows and characterize the transport in the bifurcation. The velocity data provide direct measurement of the volumetric distribution of the flow between the two channels as a function of flow rate. Water and the 8% colloidal suspension show a constant distribution with increasing flow rate, the xanthan gum shows an increase in fluid going into the larger outlet with higher flow rate, and the 22% colloidal suspension shows a decrease in fluid entering the larger channel with higher flow rate. For the colloidal particle flow, the distribution of colloid particles down the capillary is determined by examining the spectrally resolved propagator for the oil inside the core–shell particles in a direction perpendicular to the axial flow. Using dynamic magnetic resonance microscopy, the potential for using magnetic resonance for “particle counting” in a microscale bifurcation is thus demonstrated.     

Stabilization and evolution of parameters of a symmetric laminar flow in a plane channel with sudden expansion

A problem is formulated for computing the fields of parameters of a stationary laminar symmetric flow. A two-dimensional flow in a channel with a sudden change in the cross-sectional area is computed. The evolution of a three-dimensional perturbation inserted into the channel at the initial stage of computations is analyzed. It is demonstrated that the parameters of a two-dimensional flow in the channel at a Reynolds number Re = 50 become stabilized at a dimensionless time t > 20, whereas the steady state is reached under the same conditions at t ≈ 100. At a distance of approximately 10h (h is the channel width at the entrance), the flow becomes one-dimensional, but the streamwise component of the velocity vector remains a function of the streamwise coordinate owing to flow compressibility. __________ Translated from PrikladnayaMekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 35–42, January–February, 2007.     

Experimental investigation of instabilities in flow through a long square channel rotating about the transverse axis

The results of an experimental investigation of bifurcation phenomena in a laminar flow through a rotating square channel approximately 50 channel widths long are presented. A comparison with known results of the numerical modeling of bifurcations of developed steady-state flow is carried out. A map of the steady and unsteady flow regimes is plotted. The effect of artificially generated input perturbations on the conditions of onset of longitudinally oriented vortex structures in the neighborhood of the elevated-pressure side of channels of lesser length is investigated.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–93, March–April, 1996.     

Instabilities of a gas-liquid flow between two inclined plates analyzed using the Navier–Stokes equations

The paper is devoted to a theoretical analysis of a counter-current gas-liquid flow between two inclined plates. We linearized the Navier–Stokes equations and carried out a stability analysis of the basic steady-state solution over a wide variation of the liquid Reynolds number and the gas superficial velocity. As a result, we found two modes of the unstable disturbances and computed the wavelength and phase velocity of their neutral disturbances varying the liquid and gas Reynolds number. The first mode is a “surface mode” that corresponds to the Kapitza's waves at small values of the gas superficial velocity. We found that the dependence of the neutral disturbance wavelength on the liquid Reynolds number strongly depends on the gas superficial velocity, the distance between the plates and the channel inclination angle for this mode. The second mode of the unstable disturbances corresponds to the transition to a turbulent flow in the gas phase and there is a critical value of the gas Reynolds number for this mode. We obtained that this critical Reynolds number weakly depends on both the channel inclination angle, the distance between the plates and the liquid flow parameters for the conditions considered in the paper. Despite a thorough search, we did not find the unstable modes that may correspond to the instability in frame of the viscous (or inviscid) Kelvin–Helmholtz heuristic analysis.     

Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion

In this work we present an investigation of viscoelastic flow in a planar sudden expansion with expansion ratio D/d = 4. We apply the modified FENE–CR constitutive model based on the non-linear finite extensibility dumbbells (FENE) model. The governing equations were solved using a finite volume method with the high-resolution CUBISTA scheme utilised for the discretisation of the convective terms in the stress and momentum equations. Our interest here is to investigate two-dimensional steady-state solutions where, above a critical Reynolds number, stable asymmetric flow states are known to occur. We report a systematic parametric investigation, clarifying the roles of Reynolds number (0.01 < Re < 100), Weissenberg number (0 < We < 100) and the solvent viscosity ratio (0.3 < β < 1). For most simulations the extensibility parameter of the FENE model was kept constant, at a value L² = 100, but some exploration of its effect in the range 100–500 shows a rather minor influence. The results given comprise flow patterns, streamlines and vortex sizes and intensities, and pressure and velocity distributions along the centreline (i.e. y = 0). For the Newtonian case, in agreement with previous studies, a bifurcation to asymmetric flow was observed for Reynolds numbers greater than about 36. In contrast viscoelasticity was found to stabilise the flow; setting β = 0.5 and We = 2 as typical values, resulted in symmetric flow up to a Reynolds number of about 46. We analyse these two cases in particular detail.     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Solutions of the Jeffery-Hamel Problem Regularly Extendable in the Reynolds Number

The problem of steady viscous flow in a convergent channel is analyzed analytically and numerically for small, moderately large and asymptotically large Reynolds numbers over the entire range of allowed convergence angles. Attention is focused on regularly extendable problem solutions, for which purpose a high-accuracy hybrid numerical-analytical method of accelerated convergence and extension in a parameter is developed. For sufficiently large angles, the existence of tri-modal regimes symmetrical about the bisectrix and containing in- and outflow regions is established. The evolution of the velocity profiles with unbounded increase in the Reynolds number is investigated. Flow regimes for the critical convergence angle, which cannot be regularly extended in the parameter, are also studied. Several novel hydromechanical effects are noted.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.