Numerical experiments on the gas flow between eccentric rotating cylinders

A numerical experiment was carried out on the gas flow field between two eccentric cylinders, one of which is rotating. Attention was paid to the presence of separated recirculating regions from the continuum to the rarefied regimes. The direct simulations were performed by means of a Monte Carlo (DSMC) method and bi‐polar co‐ordinates were adopted. The calculations were relative to isothermal walls at the same temperature. Streamlines and velocity profiles were evaluated as functions of the Knudsen number, of the Mach number and of the geometric parameters. The gas considered was argon. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element solution of flow between eccentric cylinders with viscous dissipation

A computational study of viscous flow between two eccentrically rotating cylinders is presented in which the effect of viscous dissipation is taken into account. The space discretization is based on piecewise linear finite elements with velocity stabilization, while the method of characteristics is used for time integration. Numerical results illustrate the efficiency of the adopted approach.     

Flow of a viscoelastic fluid between eccentric cylinders

The velocity field in the annular region between two eccentric cylinders for a second-order fluid is determined. Second-order velocity terms appear only as a result of the interaction between imposed axial and planar motions. When only one of the two motions is imposed by the boundary conditions, the velocity field coincides with that of a Navier-Stokes fluid.The results of the present study are to be used in a future investigation of the stresses, forces, and torques acting on the walls of the cylinders to enable a rheometer based on the present boundary value problem to be suggested.     

A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

Stability of viscoelastic flow around periodic arrays of cylinders

Low Reynolds number flow of Newtonian and viscoelastic Boger fluids past periodic square arrays of cylinders with a porosity of 0.45 and 0.86 has been studied. Pressure drop measurements along the flow direction as a function of flow rate as well as flow visualization has been performed to investigate the effect of fluid elasticity on stability of this class of flows. It has been shown that below a critical Weissenberg number (We_c), the flow in both porosity cells is a two-dimensional steady flow, however, pressure fluctuations appear above We_c which is 2.95±0.25 for the 0.45 porosity cell and 0.95±0.08 for the higher porosity cell. Specifically, in the low porosity cell as the Weissenberg number is increased above We_c a transition between a steady two-dimensional to a transient three-dimensional flow occurs. However, in the high porosity cell a transition between a steady two-dimensional to a steady three-dimensional flow consisting of periodic cellular structures along the length of the cylinder in the space between the first and the second cylinder occurs while past the second cylinder another transition to a transient three-dimensional flow occurs giving rise to time- dependent cellular structures of various wavelengths along the length of the cylinder. Overall, the experiments indicate that viscoelastic flow past periodic arrays of cylinders of various porosities is susceptible to purely elastic instabilities. Moreover, the instability observed in lower porosity cells where a vortex is present between the cylinders in the base flow is amplifieds spatially, that is energy from the mean flow is continuously transferred to the disturbance flow along the flow direction. This instability gives rise to a rapid increase in flow resistance. In higher porosity cells where a vortex between the cylinders is not present in the base flow, the energy associated with the disturbance flow is not greatly changed along the flow direction past the second cylinder. In addition, it has been shown that in both flow cells the instability is a sensitive function of the relaxation time of the fluid. Hence, the instability in this class of flows is a strong function of the base flow kinematics (i.e., curvature of streamlines near solid surfaces), We and the relaxation time of the fluid.     

A mixed galerkin method for computing the flow between eccentric rotating cylinders

A mixed Galerkin technique with B-spline basis functions is presented to compute two-dimensional incompressible flow in terms of the primitive variable formulation. To circumvent the Babuska–Brezzi stability criterion, the artificial compressibility formulation of the equation of mass conservation is employed. As a result, the diagonal components of the matrix form in the governing equations are not singular. The B-spline basis is used because it is superior to other splines in providing computer solutions to fluid flow problems. One of the advantages of the B-spline basis is that it has excellent approximation properties. Numerical examples of applications of the mixed formulation are presented to demonstrate the convergence characteristics and accuracy of the present formulation. © 1998 John Wiley & Sons, Ltd.     

Fluid trapping by two partially shrouded rotating cylinders

In this paper we examine the prospect of using localized flow control for biomimetic fluid trapping. The problem is of interest for applications that call for guided transport of fluid volumes.The study shows that trapping can be achieved with the help of two partially shrouded rotating cylinders in a side-by-side arrangement. Secondary flows that manifest successful trapping resemble recirculation zones forming under the crests of peristaltic deformation waves, in particular with respect to their response to increasing incident flow velocity. Varying the rotation speed of the cylinders provides means to control the amount of trapped fluid.Numerical calculations to support these conclusions are presented in the paper for 0≤Re≤100 and h≈2, where Re and h are, respectively, the Reynolds number and the center-to-center distance between two cylinders divided by the cylinder diameter. Experimental validation of numerical results is performed for 0≤Re≤4.     

Laminar fluid convection between concentric and eccentric heated horizontal rotating cylinders for low-Prandtl-number fluids

Numerical experiments are performed to study rotational effects on the mixed convection of low-Prandtlnumber fluids enclosed between the annuli of concentric and eccentric horizontal cylinders. The inner cylinder is assumed to be heated and rotating. The rotational Reynolds number considered is in the range where the effect of Taylor vortices is negligible. The Prandtl number of the fluid considered is in the range 0·01–1·0. The Rayleigh number considered is up to 10⁶. A non-uniform mesh transformation technique coupled with the introduction of ‘false transient’ parameters to the vorticity and streamfunction-vorticity expressions was used to solve the governing set of equations. Results show that when the inner cylinder is made to rotate, the multicellular flow patterns observed in stationary cylindrical annuli subside in a manner depending on the Prandtl number of the fluids. Eventually the flow tends toward a uniform flow similar to that of a solid body rotation. For a fixed Rayleigh number and with a Prandtl number of the order of 1·0, when the inner cylinder is made to rotate, the mean Nusselt number is observed to decrease throughout the flow. For lower Prandtl number of the order 0·1–0·01 the mean Nusselt number remained fairly constant when the inner cylinder was made to rotate. The mean Nusselt numbers obtained were also compared with available data from other investigators.     

Numerical simulation of fluid flow of two rotating side-by-side circular cylinders by Lattice Boltzmann method

A numerical investigation of the two-dimensional laminar flow around side-by-side rotating circular cylinders using Lattice Boltzmann method is conducted. The effects of variation of rotational speed ratio β and different gap spacings g* at Reynolds number of 100 are studied. A various range of rotational speed ratio 0 ≤ β ≤ 2 for four different gap spacings of 3, 1.5, 0.7 and 0.2 are investigated. Flow conditions and its characteristics, such as lift and drag coefficients and Strouhal number, is studied. The results indicated that as β increases, the flow changes its condition from periodic to steady after a critical rotational speed. Results also indicated that variation of the gap spacing and rotational speed has significant effect on wake pattern. Wake pattern in turn has significant effect on the Strouhal number. Finally, the result is compared with experimental and other numerical data.     

The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders

Numerical solutions based on the method of fundamental solutions are discussed for Stokes flow inside a rectangular cavity in the presence of circular cylinders. The Stokeslets are used as the fundamental solutions to obtain the solution for the flow field by a linear combination of fundamental solutions. Flow results on the cellular structure of flow field resulting from the dynamics of cylinders and the horizontal walls of the cavity are reported for (i) one rotating cylinder in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions, (ii) two rotating cylinders kept apart in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions. The effect of aspect ratio of the rectangular cavity, direction of movement of the two parallel horizontal sides of the cavity and the diameter of the rotating cylinder on the flow structure are studied. The flow results obtained for the single cylinder case are in accordance with the results available in the literature. From the computational point of view, the present numerical procedure based on the method of fundamental solutions is efficient and simple to implement as compared to the mesh-dependent schemes, which needs complex mesh generation procedure for the multiply connected geometrical domains considered in this article.     

Measurements of pulsating and oscillating flows of non-Newtonian fluids through concentric and eccentric cylinders

The unsteady flow of non-Newtonian fluids through concentric and eccentric cylinders was investigated experimentally. Two experiments were carried out; one was pulsating flow and the other was flow under a constant pressure gradient with the inner cylinder oscillating longitudinally. The flow enhancement was examined and its dependence on the frequency of the oscillations and the eccentricity of the apparatus was determined.     

A multigrid finite difference approach to steady flow between eccentric rotating cylinders

A simple finite difference scheme over a non‐uniform grid is proposed to solve the two‐dimensional, steady Navier–Stokes equations. Instead of the Newton method, a more straightforward line search algorithm is used to solve the resultant system of non‐linear equations. By adopting the multigrid methodology, a fast convergence is achieved, at least for low‐Reynolds number flow. This scheme is applied, in particular, to flow between eccentric rotating cylinders. The computed results are shown in good agreement with some analytic findings. Copyright © 2000 John Wiley & Sons, Ltd.     

Flow between finite,steadily rotating eccentric cylinders

While concentric cylinder flows exhibit Ekman pumping at small values of the Reynolds number, once the container symmetry is broken, fluid particles migrate toward the horizontal midplane within the diverging part of the flow domain, and they move away from the midplane within the converging part of the domain. Under this condition, Ekman pumping is evident only at the widest and at the narrowest sections of the gap. When gradually increasing the Reynolds number, the flow acquires a cellular structure in a manner reminiscent of Benjamin's symmetric flow data. The cells spread from the stationary end-plates and link up at the center. The cells persist round the cylinder with varying strength and thus are truly toroidal. We show evidence of solution multiplicity at higher values of the Reynolds number by calculating two qualitatively different flows. One flow contains four cells, the other contains six, and both are supported by identical steady-state conditions.This material is based upon work supported by the National Science Foundation under Grant No. CTS-8920956.     

Emergence of taylor vortices between rotating eccentric cylinders

In a linear setting we examine the stability of the flow of a viscous incompressible liquid between eccentric cylinders, the inner cylinder rotating and the outer cylinder fixed. We consider the case of a narrow gap between the cylinders, which is characteristic for problems in lubrication theory. The main flow, perturbations, and the critical Reynolds number are found in the form of expansions in powers of the eccentricity to within O (³). The results obtained agree with the known experimental data for 0 0.5 and confirm the stabilizing influence of the eccentricity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 101–106, May–June, 1973.The author wish to thank N. P. Artemenko, A. D. Myshkis, and R. Di Prima for a useful discussion of the problem.     

Kinetic analysis of longitudinal Couette flow between coaxial cylinders

On the basis of a model kinetic equation, the rarefied gas flow between coaxial circular cylinders, of which the outer one is at rest while the inner one travels along its symmetry axis at a constant velocity, is studied. The problem is solved numerically in both the linear and nonlinear formulations by an implicit conservative method of second-order accuracy. The effect of the rarefaction, the cylinder radius ratio, and the inner cylinder velocity on the flow parameters is investigated. The limits of applicability of the linearized kinetic equation are established.     

Geometric shapes of the interface surface of bicomponent flows between two concentric rotating cylinders

In this paper, the shape problem of interface of bicomponent flows between two concentric rotating cylinders is investigated. With tensor analysis, the problem is reduced to an energy functional isoperimetric problem when neglecting the effects of the dissipative energy caused by viscosity. We derive the associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of the second order. Moreover, by considering the effects of the dissipative energy, we propose another total energy functional to characterize the geometric shape of the interface, and obtain the corresponding Euler-Lagrangian equation, which is also a nonlinear elliptic boundary value problem of the second order. Thus, the problem of the geometric shape is converted into a nonlinear boundary value problem of the second order in both cases.     

Three-dimensional spectral approximations to Stokes flow between eccentrically rotating cylinders

A three-dimensional spectral algorithm for the solution of Stokes flow between eccentrically rotating cylinders is described. Included in the model are pressure boundary conditions at the two ends of the finite length cylinders and the effect of a fluid line source on the inner cylinder. A comparison of results for the load and couple on the inner cylinder is made with those available from lubrication theory in the absence of a line source. Good agreement is shown for long, short and finite journal bearings when the various geometrical assumptions inherent in the lubrication analysis are satisfied.     

Flow of an electrorheological fluid between eccentric rotating cylinders

Electrorheological fluids have numerous potential applications in vibration dampers, brakes, valves, clutches, exercise equipment, etc. The flows in such applications are complex three-dimensional flows. Most models that have been developed to describe the flows of electrorheological fluids are one-dimensional models. Here, we discuss the behavior of two fully three-dimensional models for electrorheological fluids. The models are such that they reduce, in the case of simple shear flows with the intensity of the electric field perpendicular to the streamlines, to the same constitutive relation, but they would not be identical in more complicated three-dimensional settings. In order to show the difference between the two models, we study the flow of these fluids between eccentrically placed rotating cylinders kept at different potentials, in the setting that corresponds to technologically relevant problem of flow of electrorheological fluid in journal bearing. Even though the two models have quite a different constitutive structure, due to the assumed forms for the velocity and pressure fields, the models lead to the same velocity field but to different pressure fields. This finding illustrates the need for considering the flows of fluids described by three-dimensional constitutive models in complex geometries, and not restricting ourselves to flows of fluids described by one-dimensional models or simple shear flows of fluids characterized by three-dimensional models.     

Perturbation theory for viscoelastic fluids between eccentric rotating cylinders

The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.