Three-dimensional hydrodynamic analysis of fluid flow between two eccentric rotating cylinders

Three-dimensional hydrodynamic analysis of fluid flow between two eccentric rotating cylinders S. A. GandjalikhanNassab & M. A. Mehrabian Department of Mechanical Engineering, Shahid Bahonar Universip, Kerman. Iran Abstract In this study, hydrodynamic characteristics of laminar fluid flow between two eccentric rotating cylinders with finite length is investigated. The analysis is based on the numerical solution of the full Navier-Stokes equations using CFD techniques. Considering the complexity of the physical geometry, conformal mapping is used to generate an orthogonal grid and the governing equations are transformed in the computational domain. Discretized forms of the transformed equations are obtained by control volume method and solved by SIMPLE algorithm. The numerical results of this analysis can be used to investigate the oil flow pattern in the journal bearings.to validate the computational results, comparison with the experimental data of other investigators is made, and reasonable agreement is found. 1 Introduction The accuracy of journal bearing performance prediction depends to a great extent on the ability to predict the oil flow pattern through the bearing. To reach this goal, a computational procedure for the numerical solution of three- dimensional incompressible viscous flow between two rotating eccentric cylinders with finite length is introduced m this work. This type of flow has been a subject of interest for many years. In 1968,the stability of viscous flow between eccentric rotating cylinders was studied by Ritchie [l].in that work, the stability of viscous flow between two eccentric cylinders has been analyzed for the case in which the inner cylinder rotates while the outer cylinder

184 Advmces irl Fluid Mechmks W remains stationary as in journal bearings, and where the difference in radii of cylinders is small in comparison with the mean radius. The linearized equations governing the marginal stability of axially periodic disturbances were derived for the case of infinitely long cylinders, and approximately solved to give an estimate for the critical Taylor number at which vortex flow occurs for a range of relative eccentricity of the cylinders. The linear stability of flow between two infinitely long eccentric rotating cylinders was studied by Di prima and Stuart [2]in 1972. Considering the fact that the basic flow depends on both radial and azimuthal directions, they argued that the entire flow field affects the stability characteristics and, therefore, a global stability analysis should be considered instead of the previously used local stability theory. In 197.5, the motion of viscous fluid flow contained between two rotating circular Cylinders whose axes were slightly apart was considered by Wood [3]. The result is an approximate analytical solution at large Reynolds numbers to the problem of viscous flow between two eccentric cylinders which are rotating in the same direction. No example has been calculated and, therefore, the results has not been compared with other solutions. A transient computational procedure for the numerical solution of the incompressible and laminar Navier-Stokes equations in doubly connected domains was described by Sood and Elrod [4] in 1974. For the purpose of illustration, the problem of flow between two eccentric cylinders, where the inner cylinder was rotating with unit speed, was studied.the differential equations were replaced by their finite-difference analogs and the resulting set of algebraic equations was solved iteratively. Solutions were obtained for both the Stokes flow (without the inertia) and the Navier-Stokes flow. It was found that the inclusion of inertia doesnot appreciably affect the accuracy of the solution. The hydrodynamic characteristics of the lubricant film in a journal bearing was studied by Medwell [5]in 1984. In that study, the steady, three-dimensional, laminar flow in a journal bearing with an axial groove at the lubricant inlet was considered. The governing equations only solved in the pressure zone of the flow by finite element method. Solutions were obtained for different values of the Reynolds number, eccentricity ratio and the length to diameter ratio. To study the effect of fluid inertia, an inertia correction factor was calculated for the load, power loss and the flow rate. In 1992, Dai, et al. [6] studied the effects of approximations which are usually considered in the analysis of hydrodynamic lubrication. In that study, they compared the results from three models for the isoviscous laminar flow in a long journal bearings. The first model was based on the full Navier-Stokes equations written in the bipolar coordinate system. The second model was lubrication theory in bipolar coordinate that neglected the fluid inertia. The third model was the classical lubrication theory of Reynolds that neglects both the fluid inertia and film curvature. The study demonstrated that on decreasing the clearance ratio, the results of both the Navier-Stokes equations and the bipolar lubrication theory converge monotonically to the results from the classical lubrication theory, one from below and the other from above.

Admcc.sill Fluid Mdxznics IV 1 8 5 Recently, computational fluid dynamic technique has been used for the thennohydrodynamic analysis of journal bearings. Tucker and Keog [7,8] in 1995 and 1996obtained solutions to a set of exact governing equations for both cases of stationary and orbiting center of the journal. In these studies, the continuity, Navier-stokes and energy equations were considered as governing equations in cylindrical coordinate system (r,o,z). The finite difference forms of the governing equations in cylindrical coordinate obtained by using finite volume method and were solved by SIMPLEC algorithm. By this method, the velocity, pressure and temperature distributions of lubricant flow in journal bearings were obtained. It is noted that using the cylindrical coordinate system for solving the governing equations, introduces an approximation in the analysis. Although, several studies have been done to analyze the fluid flow between two rotating eccentric cylinders, but there is a few studies in which the fluid flow pattern and the effect of different parameters on the flow were be under consideration in details. So, the present work focuses on this point by solving the exact governing equations. In this study, for prediction the hydrodynamic characteristics and the fluid flow pattern between two rotating eccentric cylinders, numerical solutions of the full three dimensional Navier-Stokes equations under laminar, isoviscous and steady conditions are obtained by CFD techniques. To avoid any approximation in the numerical solution, conformal mapping is used to generate an orthogonal grid and the descretized form of the governing equations are solved in the curvilinear coordinate system as a computational domain. 2 Governing equations The governing equations which are written for a three-dimensional, steady, incompressible laminar flow, consist of the continuity and Navier-Stokes equations. The non-dimensional forms of these equations in the Cartesian coordinate system, fig.( l), can be written as : -+-+-- au av aw-, ax ay az c - 1 du +-@v---)=-- d 1 au dp (2) dx Redx ay Fkdy az Redz ax ~,,w_'aw)+d(,w_lw)+d(w~~law,=-ap (4) Redx ax Reay dy dz

186 Advmces irl Fluid Mechmks W and the dimensionless variables are defmed as : * x x =- C ' y * =yz* 2, U* =!Lv* =A,$2 p* = -,Re=- p C' C V V v' p v 2 PVC P In these definitions, c is the radial clearance and v = r, 03 is the linear speed of inner cylinder. It should be noted that in fig.(1) and eqns.(1) to (4), asterisks have been dropped for convenience. oil feed hole Y Figure 1: Geometrical configuration of two eccentric cylinders 3 Transformation functions Because of the complex flow geometry in the (x,y) plane which is the region between two eccentric circles, the governing equations are transformed into a simple computational domain such that, the physical domain at each axial location z=cte, is conformally mapped into a rectangular computational domain. The transformation between physical and computational planes can be performed in two steps. Fig.(2) shows these transformations along with their transformation functions. Itis mentioned that the z-axis is the same for both computational and physical coordinates. From these transformation functions, the relations between physical and computational planes are obtained and the transformed forms of the governing equations for cp as a dependent variable in the computational plane can be written in the following common form : in which the values of A, B, C, rwand S, vary from one equation to other.

Admcc.sill Fluid Mdxznics IV 1 8 7 (a), Z - plane (b) > L- plane ari-z C,=a+iP=aZ-r. (c), 6- plane 6= 5+iq= -iln< Figure 2: Mapping of the flow field. 4 Boundary conditions The following conditions in the physical plane are considered : 1.Periodic boundary conditions in circumferential sense are imposed for all dependent variables. 2. No slip condition is used on the surfaces of inner and outer cylinders. 3.At the groove, where z I ALI2 and n - 0 I e rei+ n the following conditions are applied: Referring to figs.(1 and 2) and noticing that q=o,we can write v=w=o and U = vi,where vi is the dimensionless inlet fluid velocity at the groove which will be corrected at each iteration. 4. At the section z = 0 (mid-plane), the condition of symmetry, i e., zero axial gradient of dependent variables, is used. 5. At the section z =L/2,p = 0 (atmospheric gauge pressure) and zero axial gradient of velocity components will be considered. Also, to calculate vi at each iteration, the following continuity equation will be considered: in which,w is the axial velocity component at the section z = Ll2.

188 Advmces irl Fluid Mechmks W 5 Solution procedure Finite difference forms of the partial differential equations (5)were obtained by integrating over an elemental cell volume with staggered control volumes for the 5, q and z-velocity components. The discretized governing equations were numerically solved by SIMPLE algorithm of Patankar and Spalding [lo]. Numerical solutions were obtained iteratively by the line-by-line method with progressing in axial direction. The iterations were terminated when the sum of the absolute residuals was less than iw3for each equation. Numerical calculations were performed by writing a computer program in FORTRAN. As the result of grid tests for obtaining the grid-independent solutions, an optimum grid of 30x20~20,with clustering near the surface of inner cylinder, was used for the flow field calculations. 6 Results and discussion In order to validate the computational results,a test case was analyzed and the results are compared with the experimental data of Ref [9]. Fig(3) shows the pressure distribution around the inner cylinder of this test case. It can be seen that the pressure increases as the fluid passes through the converging region and decreases in the diverging zone. However, the agreement between the analysis and experiment is satisfactory. Figure 3: Pressure distribution around the surface of the inner cylinder at the midplane. Re=%, E= 0.5,~/ rs = 1 The system of rotating eccentric cylinders which are under study in this work, have an axial inlet groove located on the line of centers at the section of maximum gap [see tig(l)]. It is noted that in the bearings running under lubrication conditions, the clearance ratio is very small,but in order to show a typical flow field in physical domain in this study, numerical solution of the governing equations for the fluid flow between two eccentric cylinders with

Admcc.sill Fluid Mdxznics IV 1 8 9 large clearance ratio, c / rs = 1 (out of lubrication condition) is obtained. Figs.(4-a and b) show the velocity filed and streamlines of this flow at the midplane (z=o). These figures indicate a separated flow in the vicinity of maximum film thickness because the existence of an unfavorable pressure gradient in this region. This result agrees with numerical and experimental results of several other investigators. The pressure contours and also the pressure distribution on the surface of the inner cylinder are presented in figs.(5-a and b). As shown in these figures, the value of pressure increases in the converging zone of the flow and the maximum pressure occurs at a short distance downstream from the minimum film thickness. (a) velocity vectors (b) streamlines Figure 4: Velocity field and streamlines between two eccentric rotating cylinders. Re=35, E= 0.5,c/ rs = 1,LID=10 Also, these figures indicate a negative pressure in the diverging zone of the flow. If the value of pressure in this region falls below the vapor pressure, cavitation will occur. It is noted that the effect of cavitation is neglected in this study. 1, 2n 9 (a) pressure contours (b) pressure distribution Figure 5: Pressure contours for the flow and pressure distribution around the surface of the inner cylinder at the midplane. Re=35, &=0.5,c/rs =l,l/d=lo

190 Advmces irl Fluid Mechmks W In another case with c / rs = 0.1, the pressure distributions on the surface of inner cylinder for three different axial planes are shown in Fig.(6). It is seen that the pressure zone ( region with positive pressure ) is located almost in the converging region of the flow and in the diverging region, the value of fluid pressure decreases such that the negative pressure is developed. In reality, if the value of pressure falls below the vapor pressure in the diverging zone, cavitation occurs and there is a two phase flow of lubricant and vapor in that region. In this hydrodynamic analysis, it is assumed that there is no region under cavitation condition. Figure 6: Pressure distribution on the surface on inner cylinder at three different axial planes. Re=60, E= 0.5,c / rs = 0.1, L/D=7.5 Also, fig.(6) indicates that the absolute value of pressure in both converging and diverging domain decreases in the axial direction such that the maximum value of pressure occurs at the mid-plane and the value of pressure in the outlet-plane is equal to zero which is imposed as a boundary condition. 1... Figure 7 : Mean axial velocity distribution. Re=60,&=0.5,c/rs= O.l,L/D=7.5 The mean axial velocity distribution in the 5-z plane is shown in Fig(7). It can be seen that there is no axial velocity at the mid-plane. Then, for z >O, this

Admcc.sill Fluid Mdxznics IV 1 9 1 velocity component increases with increasing z in the pressure zone, where a positive squeeze effect ( favorable pressure gradient in axial direction ) is present. But in the diverging region, especially in the vicinity of the outlet plane (z = L/2 ), there is a negative squeeze effect ( unfavorable pressure gradient in axial direction, [see fig. (6) ] ) which generates a reverse flow such that there is an inflow into the flow field. This phenomena which behaves as a suction does not occur in the bearings under hydrodynamic lubrication conditions because of the cavitation effect which is neglected in the present analysis. Pressure distributions across the film thickness at the mid-plane of the bearing for four different circumferential sections (0 = O,d2,x,3x/2 ) are shown in Fig.@). It is seen that the pressure variation along the film thickness is small although the clearance ratio which is considered in this test case is very large in comparison to the hydrodynamic lubrication conditions. Thereby, the variation of pressure across the film thickness can be ignored for the lubricant flow in journal bearings, which can be considered as a reason for the Reynolds lubrication theory to predict accurately the hydrodynamic characteristics of journal bearings. 1 I I t -040 02 0.4 0.6 0.8 1 Figure 8: Pressure distribution across the film at the mid-plane Re=60, E = 0.5,c / rs = 0.1, L/D=7.5 7 Conclusion In this study, the analysis of hydrodynamic characteristics of fluid flow between two rotating eccentric cylinders is done using CFD techniquesto avoid any approximation, conformal mapping is used to transfer the governing equations to the computational plane. The transformed equations are solved for the velocity components and pressure by SIMPLE algorithm. The numerical results from this analysis can be used to investigate the journal bearing performance and the oil flow pattern through this system. The main conclusions can be summarized as follow: 1. In the fluid flow between two eccentric cylinders such as lubricant flow in journal bearings, pressure increases as the fluid passes through the

192 Advmces irl Fluid Mechunks W converging region of the flow and then falls in the diverging part, such that the maximum pressure occurs immediately downstream from the minimum film thickness. If the value of pressure in the diverging zone is reduced below the vapor pressure of the lubricant, cavitation will occur. 2. The unfavorable pressure gradient in the circumferential direction, which exists in the vicinity of the maximum film thickness, causes flow separation at this region. The extent of the recirculation region decreases in the axial direction. 3. A positive squeeze effect on the lubricant film is present in the converging region up to the minimum film thickness after which a negative squeeze effect is developed. In reality for the lubricant flow in journal bearings under lubrication conditions, the negative squeeze effect is disappeared due to the cavitation effects. References [l]ritchie, G. S., On the stability of viscous flow between eccentric rotating cylinders. Journal of Fluid Mechanics,Vol. 32, part 1,pp. 131-144,1968. [2] Di Prima, R. C. & Stuart, J. T. Non-Local effects in the stability of flow between eccentric rotating cylinders. Journal of Fluid Mechanics,Vol. 54, part 1,pp. 393-410,1972. [3] Wood, W. W. The asymptotic expansion at large Reynolds numbers for steady motion between non-coaxial rotating cylinders. Journal of Fluid Mechanics,Vol.3,pp. 1.59-175,1975. [4] Sood,R, C% Elrod Jr, H. G. Numerical solution of the incompressible Navier- Stokes equations In doubly-connected regions. AIAA Journal, Vol. 12, pp. 636-641,1974. [5] Medwell, J. 0.Finite element analysis of steadily loaded hydrodynamic journal bearings. Proc. of the dst Int. Con$ On Numerical Methods in Laminar and Turbulent Flow,part 1,pp,173-186,1985. [6] Dai, R. X., Dong, Q. & Szeri, A. Z. Approximations in hydrodynamic lubrication. Journal of Tribology,Trans. ASME, Vol. 114,pp. 14-25,1992. [7]Tucker, P. G. &L Keogh, P. S. A generalized CFD approach for journal bearing performance prediction. Proc, hstn Mechanical Engineers, Journal of Tribology,Vol. 209, Part J, pp. 99-108,1995. [S] Tucker, P. G. 8~ Keogh, P. S. On the dynamic thermal state in a hydrodynamic bearings with a whirling journal using CFD techniques. ASME Journal of Tribology,Vol. 118, pp. 356-363,1996. [9] Pan, T. C% Vohr, H. Supper laminar flow in bearings and seals. The symposium on lubrication in nuclear applications., pp. 216-245,1967. [lo]patankar,v.p. & Spalding, B. D. A calculation procedure for heat, mass and momentum transfer in three - dimensional parabolic flows. International Journal of Heat and Mass Transfer,Vol. 15,pp. 1787-1806,1972.