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Bulletin of the Transilvania University of Braşov Vol. 13(48) - 2006
Numerical study of the flow in an eccentric cylindrical gap with a feed hole
N. Scurtu[* ]P. Stücke** C. Egbers*
Abstract: A numerical investigation on the flow of eccentric Taylor-Couette system with rotating inner cylinder and fixed outer cylinder is presented. The flow between eccentric rotating cylinders is of interest in the fluid dynamic community, for instance, in the lubrication technology due to the strong effect of both the Reynolds number (Re) and eccentricity. Here, Re is defined as Re= UcLc/ν where Uc is the rotational speed of the inner cylinder, Lc is the mean value of the gap width and ν the kinematic viscosity of the fluid. The main flow field has been examined, the topology of its streamlines and the transition region from the laminar Couette-flow to the Taylor-vortex-flow in different eccentric arrangements of the cylinders. The effect of the eccentricity on flow patterns was studied for three different values of the eccentricity ε relative to the medium gap width 0.25, 0.5 and 0.75, respectively.
For the classical Taylor-Couette system with rotating inner cylinder and fixed outer cylinder, the fluid flow passes through stable circular Couette flow for Re < Recr and steady axisymmetric Taylor vortex flow for ReRecr. The value of Recr was found to increase by increasing the eccentricity. However, if one of the cylinders is displaced to an eccentric position, the rotational symmetry is broken and a fully three-dimensional flow can be observed.
Typically, the Reynolds equations, which are an idealized form of the Navier-Stokes equations, are used to compute the flow in journal bearings yielding a pressure distribution in relation to mean clearance and eccentricity. This pressure distribution, which can be calculated analytically, is compared with computed data by solving the Navier-Stokes equations directly. For small clearances an outstanding agreement of the pressure distribution is obtained for low Re however, for a larger gaps between inner and outer cylinder a substantial deviation can be found.
Furthermore, a feed hole is applied to the external cylinder to model the flow in journal bearings. The fluid flow, which is fed through the hole in the outer cylinder, generates a cross flow mixing with the circumferential main flow of the system and exits at both ends of the system, equally. In this case the pressure distribution differs essentially from the idealized distribution given by the Reynolds equations.
Keywords: Taylor-Couette flow, Numerical simulation.
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Bulletin of the Transilvania University of Braşov Vol. 13(48) - 2006
1. Introduction
The flow between eccentric rotating cylinders is of interest in fluid dynamics due to the strong effect of Reynolds number and eccentricity, both. Also it is of considerable technical importance, for example in the lubrication technology.
For the classical Taylor-Couette system with rotating inner cylinder and fixed outer cylinder, the characteristic of the flow will change with increasing Reynolds number from a stable circular Couette flow towards a steady axisymmetric Taylor vortex flow. The value of the critical Reynolds number is Recr. However, when displacing one of the cylinders to an eccentric position, the rotational symmetry is broken and a fully three-dimensional flow will be obtained.
The dependency of Recr on the eccentricity ε has been studied experimentally and analytically by DiPrima and Stuart [1]. It is well known that at an eccentricity larger than 0.3, an eddy occurs in the region of widest gap resulting in the separation of the main circumferential flow from the outer wall and its re-attachment downstream. For ε < 0.6 a non-axisymmetric Taylor vortex flow was observed by e.g. Karasudani [2], whereas no steady flow could be observed for ε > 0.6. Recent numerical results concerning the effect of eccentricity in the Couette-Taylor system with wide gap η = 0.5 were given by Shu et al. [4].
2. Simulation results
In this paper an approach is presented, which combines the finite-element spatial discretization and the θ-scheme time approximation and is used to solve the three-dimensional unsteady incompressible Navier-Stokes equations. Very good agreement of the numerical results obtained by this method and the experimental data for the concentric Couette-Taylor system with asymmetric boundary conditions and small gap, are described in [3] and [5], respectively.
Fig. 1 Three dimensional computational domain, (left) azimuthal section
and (right) axial section
Figure 1 shows a schematic configuration of the three dimensional eccentric system. The eccentricity between the rotating inner cylinder (radius R1, angular velocity ω1 and the fixed outer cylinder (radius R2) is be given by the distance e.
The geometric parameters are the mean gap width H0 = R2-R1, the normalized clearance ψ = H0/R1, the normalized eccentricity ε = e/H0 and the aspect ratio Γ = B/H0. The conventional Reynolds number is defined by Re = R1ω1H0/υ. For the numerical computations we consider a quadratic system, i.e. B = 2R1. Numerical simulations have been carried out for three values of the normalized clearance ψ = 0.2, 0.1 and 0.077 and for three values of the eccentricity ε = 0.25, 0.5 and 0.75, respectively.
Fig. 2 Recirculation vortex in axial section at ψ = 0.2, ε = 0.75 and (left) Re = 110, (right) Re = 200.
The velocity field and the recirculation vortex is represented in Fig. 2 in an axial section at ψ = 0.2, ε = 0.75 and Re = 110 in the left figure and Re = 200 in the right one. One can observe that for high Reynolds numbers vortices occur also at the interface between the main circumferential flow and the recirculation flow. In Fig. 3 the velocity field is represented for ψ = 0.077 and ε = 0.25 at Re = 180 in different azimuthal sections. The Taylor vortices are not developed in the whole gap. The vortices occur first in the region downstream of the wide gap and develop by increasing Re also in the small gap zone.
Fig. 3 Development of the Taylor vortices at ψ = 0.077, ε = 0.25 and
Re = 180 in the cylinder gap.
This phenomenon can be observed better in Fig. 4, where the iso-surface of the velocity magnitude magU = 0.5 is represented in the gap between the cylinders. In Fig. 4, one can see that downstream from the maximum gap the Taylor vortices were fully developed, but upstream from the widest gap the radial velocity component is very small.
Fig. 4 Iso-surface of the velocity magnitude in eccentric gap at ψ = 0.2,
ε = 0.5 and e = 150, (left) frontal view and (right) back view.
At ε = 0.5 and Re = 280 one can observe in Fig. 5 the recirculation flow, corresponding to the red vectors, which occurs in the wide gap area and in particular between the Taylor vortices.
For ε = 0.75 the flow passes from Couette flow directly into a turbulent form without formation of organized Taylor vortices. Moreover, the numerical results presented in this paper confirm the increase of Recr with increasing ε studied experimentally in [1,2].
Fig. 5 Taylor vortices and recirculation flow at ψ = 0.077, ε = 0.75
and Re = 280
The circumferential pressure distribution is compared in Fig. 6 with the analytical results obtained by solving the idealized Reynolds equation. At small gap values, ψ = 10, very good agreement of the pressure distribution is found for low Re. But for larger Reynolds numbers, a substantial deviation can be found. Additionally, a feed hole is applied to the outer cylinder to approximate the flow in journal bearings. In this case a quantity of fluid flows through the feed hole into the system and leaves the system equally at both ends. If the feed hole is positioned at φ = 180° the pressure distribution of the eccentric system differs essentially from the idealized distribution given by the Reynolds equations.
Fig. 6 Circumferential pressure distribution at ψ = 0.1, ε = 0.5 and Re = 12,
(left) standard Couette system and (right) Couette system with
feed hole in the outer cylinder.
In the future numerical simulations of the eccentric cylindrical system with smaller gaps i.e., with a normalized clearance ψ < 0.01 are planed, which are to be accompanied by experiments. Moreover, variations of the feed hole position in relation to the maximum gap will be incorporated into the program yielding a brought analysis of the velocity and pressure fields.
References
1. DiPrima, R.C., Stuart J.T.: Flow between eccentric rotating cylinders. In J. Lub. Tech. Trans. ASME, vol. F 94, A. 1972, p. 266-274.
2. Karasudani, T.: Non-axis-symmetric Taylor vortex flow in eccentric rotating cylinders. In: J. Phys. Soc. Japan, vol. 56, A. 1987, p. 855-858.
3. Meincke ,O., Scurtu, N., Egbers ,C., Bänsch, E.: On the influence of boundary conditions in Taylor-Couette flows. In Lecture Notes in Physics, vol. 549, Springer, A. 2000.
4. Shu, C., Wang, L., Chew, Y.T., Zhao N.: Numerical study of eccentric Couette-Taylor flows and effect of eccentricity on flow patterns. In Theoret. Comp. Fluid Dynamics, vol. 18, A. 2004, p. 43-59.
5. Stücke, P., Scurtu, N., Egbers, C.: Numerische Untersuchung der Strömung in Gleitlagern. In PAMM, vol. 5, A. 2005, p. 555 - 556.
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Bulletin of the Transilvania University of Braşov Vol. 13(48) - 2006
Studiul numeric al curgerii intr-o fanta cilindrica cu un orificiu de alimentare
Rezumat: Se prezinta un studiu numeric al curgerii intr-un sistem Taylor-Couette cu clindru interior rotativ si cu cilindru exterior fix. Curgerea intre cilindrii rotativi excentrici este de interes in Mecanica Fluidelor, de exemplu in tehnologia lubrificatiei, datorita importantului efect al numrului Reynolds si al excentricitatii. Sunt examinate campul curgerii medii, topologia liniilor de curent, si tranzitia de la curgerea laminara de tip Couette la curgerea cu vartejuri de tip Taylor, in diverse configuratii excentrice ale cilindrilor. Este studiat efectul excentricitatii asupra comportamentului curgerii pentru trei valori ale parametrului de excentricitate e, si anume 0.25, 0.5 si 0.75, relativ la latimea medie a fantei.
Cuvinte cheie: curgerea Taylor-Couette, simulare numerica.
[* ]*Lehrstuhl für Aerodynamik und Strömungslehre, Brandenburgische Technische Universität Cottbus.
** Institut für Kraftfahrzeugtechnik, Westsächsische Hochschule, Zwickau.