The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

The generalizations of the Onsager model for the radial boundary layer and the Carrier-Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier-Maslen models, because they are not restricted to the limit A >> 1, though they are restricted to the limit R e >> 1 and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter A = root m Omega(2)R(2)/2k(B)T). This parameter A is the ratio of the peripheral speed, Omega R, to the most probable molecular speed, root 2k(B)T/m, the Reynolds number Re = rho w Omega R(2)/mu, where m is the molecular mass, Omega and R are the rotational speed and radius of the cylinder, k(B) is the Boltzmann constant, T is the gas temperature, rho(w) is the gas density at wall, and mu is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier-Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used `diffuse reflection' boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a `temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier-Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.

Removal of unwanted fluid

This work is concerned with the removal of unwanted fluid through the source-sink pair. The source consists of fluid issuing out of a nozzle in the form of a jet and the sink is a pipe that is kept some distance from the source pipe. Of concern is the percentage of source fluid sucked through the sink. The experiments have been carried in a large glass water tank. The source nozzle diameter is 6Â mm and the sink pipe diameter is either 10 or 20Â mm. The horizontal and vertical separations and angles between these source and sink pipes are adjustable. The flow was visualized using KMnO 4 dye, planer laser induced fluorescence and particle streak photographs. To obtain the effectiveness (that is percentage of source fluid entering the sink pipe), titration method is used. The velocity profiles with and without the sink were obtained using particle image velocimetry. The sink flow rate to obtain a certain effectiveness increase dramatically with lateral separation. The sink diameter and the angle between source and the sink axes don't influence effectiveness as much as the lateral separation.

Unsteady Rotating Flow Over an Impulsively Rotating Infinite Disk with Axial Magnetic Field and Suction

The unsteady rotating flow of an incompressible laminar viscous electrically conducting fluid over an impulsively rotated infinite disk in the presence of magnetic field and suction is investigated. We have considered the situation where there is a steady state initially (i.e., at t = 0, the fluid is rotating with constant angular velocity over a stationary disk). Then at t > 0, the disk is suddenly rotated with a constant angular velocity either in the same direction or in opposite direction to that of the fluid rotation which causes unsteadiness in the flow field. The effect of the impulsive motion is found to be more pronounced on the tangential shear stress than on the radial shear stress. When the disk and the fluid rotate in the same direction, the tangential shear stress at the surface changes sign in a small time interval immediately after the start of the impulsive motion.

Removal of unwanted fluid

This work is concerned with the removal of unwanted fluid through the source-sink pair. The source consists of fluid issuing out of a nozzle in the form of a jet and the sink is a pipe that is kept some distance from the source pipe. Of concern is the percentage of source fluid sucked through the sink. The experiments have been carried in a large glass water tank. The source nozzle diameter is 6 mm and the sink pipe diameter is either 10 or 20 mm. The horizontal and vertical separations and angles between these source and sink pipes are adjustable. The flow was visualized using KMnO4 dye, planer laser induced fluorescence and particle streak photographs. To obtain the effectiveness (that is percentage of source fluid entering the sink pipe), titration method is used. The velocity profiles with and without the sink were obtained using particle image velocimetry. The sink flow rate to obtain a certain effectiveness increase dramatically with lateral separation. The sink diameter and the angle between source and the sink axes don't influence effectiveness as much as the lateral separation.

Effect of hinged leaflets on vortex pair generation

We experimentally study the effect of having hinged leaflets at the jet exit on the formation of a two-dimensional counter-rotating vortex pair. A piston-cylinder mechanism is used to generate a starting jet from a high-aspect-ratio channel into a quiescent medium. For a rigid exit, with no leaflets at the channel exit, the measurements at a central plane show that the trailing jet in the present case is never detached from the vortex pair, and keeps feeding into the latter, unlike in the axisymmetric case. Passive flexibility is introduced in the form of rigid leaflets or flaps that are hinged at the exit of the channel, with the flaps initially parallel to the channel walls. The experimental arrangement closely approximates the limiting case of a free-to-rotate rigid flap with negligible structural stiffness, damping and flap inertia, as these limiting structural properties permit the largest flap openings. Using this arrangement, we start the flow and measure the flap kinematics and the vorticity fields for different flap lengths and piston velocity programs. The typical motion of the flaps involves a rapid opening and a subsequent more gradual return to its initial position, both of which occur when the piston is still moving. The initial opening of the flaps can be attributed to an excess pressure that develops in the channel when the flow starts, due to the acceleration that has to be imparted to the fluid slug between the flaps. In the case with flaps, two additional pairs of vortices are formed because of the motion of the flaps, leading to the ejection of a total of up to three vortex pairs from the hinged exit. The flaps' length (L-f) is found to significantly affect flap motions when plotted using the conventional time scale L/d, where L is the piston stroke and d is the channel width. However, with a newly defined time scale based on the flap length (L/L-f), we find a good collapse of all the measured flap motions irrespective of flap length and piston velocity for an impulsively started piston motion. The maximum opening angle in all these impulsive velocity program cases, irrespective of the flap length, is found to be close to 15 degrees. Even though the flap kinematics collapses well with L/L-f, there are differences in the distribution of the ejected vorticity even for the same L/L-f. Such a redistribution of vorticity can lead to important changes in the overall properties of the flow, and it gives us a better understanding of the importance of exit flexibility in such flows.

The generalized Onsager model for a binary gas mixture

The Onsager model for the secondary flow field in a high-speed rotating cylinder is extended to incorporate the difference in mass of the two species in a binary gas mixture. The base flow is an isothermal solid-body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. Explicit expressions for the radial variation of the pressure, mass/mole fractions, and from these the radial variation of the viscosity, thermal conductivity and diffusion coefficient, are derived, and these are used in the computation of the secondary flow. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter epsilon = (Delta m/m(av)), where Delta m is the difference in the molecular masses of the two species, and the average molecular mass m(av) is defined as m(av) = (rho(w1)m(1) + rho(w2)m(2))/rho(w), where rho(w1) and rho(w2) are the mass densities of the two species at the wall, and rho(w) = rho(w1) + rho(w2). The equation for the master potential and the boundary conditions are derived correct to O(epsilon(2)). The leading-order equation for the master potential contains a self-adjoint sixth-order operator in the radial direction, which is different from the generalized Onsager model (Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the leading approximation for the secondary flow, followed by a solution of the diffusion equation for the leading correction to the species mole fractions. The O(epsilon) and O(epsilon(2)) equations contain inhomogeneous terms that depend on the lower-order solutions, and these are solved in a hierarchical manner to obtain the O(epsilon) and O(epsilon(2)) corrections to the master potential. A similar hierarchical procedure is used for the Carrier-Maslen model for the end-cap secondary flow. The results of the Onsager hierarchy, up to O(epsilon(2)), are compared with the results of direct simulation Monte Carlo simulations for a binary hard-sphere gas mixture for secondary flow due to a wall temperature gradient, inflow/outflow of gas along the axis, as well as mass and momentum sources in the flow. There is excellent agreement between the solutions for the secondary flow correct to O(epsilon(2)) and the simulations, to within 15 %, even at a Reynolds number as low as 100, and length/diameter ratio as low as 2, for a low stratification parameter A of 0.707, and when the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio 2 Delta m/(m(1) + m(2)) is as high as 0.5. Here, the Reynolds number Re = rho(w)Omega R-2/mu, the stratification parameter A = root m Omega R-2(2)/(2k(B)T), R and Omega are the cylinder radius and angular velocity, m is the molecular mass, rho(w) is the wall density, mu is the viscosity and T is the temperature. The leading-order solutions do capture the qualitative trends, but are not in quantitative agreement.

Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid

The influence of surfactant on the breakup of a prestretched bubble in a quiescent viscous surrounding is studied by a combination of direct numerical simulation and the solution of a long-wave asymptotic model. The direct numerical simulations describe the evolution toward breakup of an inviscid bubble, while the effects of small but non-zero interior viscosity are readily included in the long-wave model for a fluid thread in the Stokes flow limit. The direct numerical simulations use a specific but realizable and representative initial bubble shape to compare the evolution toward breakup of a clean or surfactant-free bubble and a bubble that is coated with insoluble surfactant. A distinguishing feature of the evolution in the presence of surfactant is the interruption of bubble breakup by formation of a slender quasi-steady thread of the interior fluid. This forms because the decrease in surface area causes a decrease in the surface tension and capillary pressure, until at a small but non-zero radius, equilibrium occurs between the capillary pressure and interior fluid pressure. The long-wave asymptotic model, for a thread with periodic boundary conditions, explains the principal mechanism of the slender thread's formation and confirms, for example, the relatively minor role played by the Marangoni stress. The large-time evolution of the slender thread and the precise location of its breakup are, however, influenced by effects such as the Marangoni stress and surface diffusion of surfactant. © 2008 Cambridge University Press.

Characteristics of flow fields induced by interfacial waves in two-layer fluid

When designing deep ocean structures, it is necessary to estimate the effects of internal waves on the platform and auxiliary parts such as tension leg, riser and mooring lines. Up to now, only a few studies are concerned with the internal wave velocity fields. By using the most representative two-layer model, we have analyzed the behavior of velocity field induced by interfacial wave in the present paper. We find that there may exist velocity shear of fluid particles in the upper and lower layers so that any structures in the ocean are subjected to shear force nearby the interface. In the meantime, the magnitude of velocity for long internal wave appears spatially uniform in the respective layer although they still decay exponentially. Finally, the temporal variation for Stokes and solitary waves are shown to be of periodical and pulse type.

Modelling the entrainment and motion of particles in a gap: Application to abrasive wear

The microscale abrasion or ball-cratering test is being increasingly applied to a wide range of bulk materials and coatings. The response of materials to this test depends critically on the nature of the motion of the abrasive particles in the contact zone: whether they roll and produce multiple indentations in the coating, or slide causing grooving abrasion. Similar phenomena also occur when hard contaminant particles enter a lubricated contact. This paper presents simple quantitative two-dimensional models which describe two aspects of the interaction between a hard abrasive particle and two sliding surfaces. The first model treats the conditions under which a spherical abrasive particle of size d can be entrained into the gap between a rotating sphere of radius R and a plane surface. These conditions are determined by the coefficients of friction between the particle and the sphere, and the particle and the plane, denoted by μs and μp respectively. This model predicts that the values of (μs + μp) and 2μs should both exceed √2d/R for the particles to be entrained into the contact. If either is less than this value, the particle will slide against the sphere and never enter the contact. The second model describes the mechanisms of abrasive wear in a contact when an idealized rhombus-sectioned prismatic particle is located between two parallel plane surfaces separated by a certain distance, which can represent either the thickness of a fluid film or the spacing due to the presence of other particles. It is shown that both the ratio of particle size to the separation of the surfaces and the ratio of the hardnesses of the two surfaces have important influences on the particle motion and hence on the mechanism of the resulting abrasive wear. Results from this model are compared with experimental observations, and the model is shown to lead to realistic predictions. © IMechE 2003.

Statistics and dynamics of turbulence-flame alignment in premixed combustion

The geometric alignment of turbulent strain-rate structures with premixed flames greatly influences the results of the turbulence-flame interaction. Here, the statistics and dynamics of this alignment are experimentally investigated in turbulent premixed Bunsen flames using high-repetition-rate stereoscopic particle image velocimetry. In all cases, the statistics showed that the most extensive principal strain-rate associated with the turbulence preferentially aligned such that it was more perpendicular than parallel to the flame surface normal direction. The mean turbulence-flame alignment differed between the flames, with the stronger flames (higher laminar flame speed) exhibiting stronger preferential alignment. Furthermore, the preferential alignment was greatest on the reactant side of the mean flame brush. To understand these differences, individual structures of fluid-dynamic strain-rate were tracked through time in a Lagrangian manner (i.e., by following the fluid elements). It was found that the flame surface affected the orientation of the turbulence structures, with the majority of structures rotating as they approached the flame such that their most extensive principal strain-rate was perpendicular to the flame normal. The maximum change in turbulent structure orientation was found to decrease with the strength of the structure, increase with the strength of the flame, and exhibit similar trends when the structure strength and flame strength were represented by a Karlovitz number. The mean change in orientation decreased from the unburnt to burnt side of the flame brush and appears to be influenced by the overall flame shape. © 2011 The Combustion Institute.