Rotational Flow in a Narrow Annular Gap Based on Lattice Boltzmann Method

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College of Astronautics，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract: For comprehensive characteristics of flow in a gas bearing，lattice Boltzmann method（LBM）is applied for study of the two-dimensional flow between two eccentric cylinders with the inner one rotating at a high speed. The flow pattern and circumferential pressure distribution are discussed based on critical issues such as eccentricity ranging from 0.2 to 0.9，clearance ratio varying from 0.005 to 0.01 and rotating speed in the range of 3×104—1.8×105 r/min.The analysis and discussion on the circumferential pressure distribution affirmed the quasilinear relation between the extremum pressure and rotating speed. Furthermore，a high eccentricity and small clearance ratio contributes most to the fluctuation of the circumferential pressure distribution. The flow pattern inside the channel exhibits separation vortex under a large eccentricity. The conclusions drawn in this work give rise to prediction of the flow pattern in the gas bearing which is beneficial for evaluating the performance of as well as instructing the design and development.

Key words：narrow annular；eccentric rotating cylinders；gas bearing；lattice Boltzmann method（LBM）

0 Introduction

In research，industrial，aeronautical fields etc.，bearings are widely used and is of great importance for the performance of critical mechanical compo?nents［1］. Compared with conventional bearings，gas bearings are preferred with the advantages of oil free，non-pollution and so on. Its extraordinary per?formance that could operate at high-speed of more than 104r/min brings itself many applications in the light-weights and high-speed machines［2-3］. The working mechanism of gas bearings，known as the hydrodynamic effect，is capacity formation for sup?porting of the rotor by pressure gradient generation with fluid passing through an unevenly channel. For this purpose，the magnitude of the clearance is mi?nuscule，usually even less than 100 μm. Up to now，flow mechanism in such a narrow clearance has still been a puzzle，which strongly deserves fur?ther consideration. In addition，the flow in the chan?nel significantly couples with heat transfer at high ro?tating speed and somehow affects the structure of bearing.

The study of flow in a narrow clearance be?tween eccentric cylinders dates back to 1886，when Reynold pioneered the famous differential equations describing the pressure distribution in journal bear?ings. Henceforth，researchers［3］revealed that the so?lution of Reynold equations is an approximation to the Navier-Stokes（N-S）equations when it comes to a small clearance ratio. As a result，application of the Reynold equations is significantly developed in the study of journal bearings. The theory based on Reynold equations had been improved continuously and formed a comprehensive hierarchy. For exam?ple，Reynold equations were utilized to individually analyze the hydrodynamic effect in simplified wedgeshaped channels［4-6］. Furthermore，conditions in the whole annular were also discussed extensively.Thermohydrodynamic models that coupled the ener?gy equation with Reynold equations were developed to predict the temperature rise and pressure distribu?tion of the air film［7-9］. Moreover，thermal structural effect［10-11］and cooling flow［12］were considered for further improvement of the results. Resulting in a lower capacity，the wall slip caused by gas rarefac?tion effect was proved to be nonnegligible at a high Knudsen number［13-14］. The aforementioned studies advance the gas bearing but focused little on the flow field of the air film，which are not accessible through Reynold equations.

The laminar flow between eccentric cylinders with one rotating wall has been widely discussed un?der large clearance ratio. Numerous methods have been applied to problems，such as finite element method［15］，mixed Galerkin method［16］，variational multiscale element free Galerkin method［17］，direct simulation Monte Carlo approach［18］，boundary ele?ment method［19］， multigrid finite difference scheme［20］and boundary integral equation meth?od［21］. Analytical solutions for Stokes flows between two infinitely long cylinder were given by Ballal et al.［22］with flow patterns investigated under different parameters. Separation vortex was generated when eccentricity of the channel exceeded a critical value.The positions where it started and ended，so called separation point and reattachment point，were inde?pendent of the speed of the inner cylinder and sym?metrically located along the whole annular. The ex?istence of the vortex was verified by experi?ments［23］，in which the torque and flow force exert?ed on the cylinders were examined. Considering the inertial effect，Andres［24］analyzed how the flow pat?tern transformed at varying Reynolds number and eccentricity. Different from Stokes flow，the sym?metry of the separation vortex no longer existed.Non-isothermal conditions were investigated with variable-viscosity by Dai et al［25］. The flow pattern was influenced by geometry as well as thermal boundary conditions. The stress patterns were stud?ied with varying boundary conditions and eccentrici?ties［26-27］. Taking rarefaction into account，gas flow between eccentric cylinders was investigated with outer cylinder rotating［18］. Under high Knudsen num?ber，velocity slip induced by rarefaction was capable of suppressing the separation vortex mentioned above.

All the studies were conducted with a clearance ratio dozens of times larger than that of gas bearing.In other words，situations with extreme low clear?ance ratio still require extra exploration. However，conventional continuum models are unable to give an accurate solution once it comes to the hydraulic diameter less than 1 mm. LBM based on kinetic the?ory had better performance in the application of mi?croscopic flow［28］. The movement of molecules is considered as cluster evolution，in which the mole?cules transfer and collide with each other，following energy and momentum conservation during colli?sion. Most importantly，LBM’s prominent compu?tational efficiency makes it competent for micro scale computational study.

To sum up，the design of gas bearings still lacks of theoretical support，especially for the highspeed flow in the narrow annular configuration. The small-scale rotating flow gives rise to sophisticated field distribution and coupling effect，which is sel?dom reported. This work aims to examine the flow mechanism in a narrow annulus which stands for the clearance in gas bearings. The flow pattern and pres?sure distribution are obtained and discussed in terms of eccentricity，rotating speed and clearance ratio.As Reynolds number grows，the flow becomes un?stable and a three-dimensional Taylor vortex ap?pears［29］. The fundamental work presented here fo?cuses on flow pattern at cross section and keeps the axial variation for further study and discussion.

1 Methods

For LBM，space is divided into a series of lat?tices that align with each other. The nodes on the lattices match the corresponding one on the adjacent lattice. Collision term in the equation is replaced by the Bhatnagar-Gross-Krook （BGK） operator.Thus，the Boltzmann equation can be described as

wherefis the density distribution function，cthe lat?tice velocity，andΩthe operator mentioned above.An approximation with a relaxation timeτand equi?librium distribution functionfeqis used to describe the procedure during collision. Meantime，velocities of the lattices are discretized into several directions.The discrete Boltzmann equation without external force is

whereiis the direction of particle velocity.The kine?matical viscosityνand relax timeτsatisfy

In the classic two-dimensional model（D2Q9），as shown in Fig.1，particle velocities are catego?rized into nine types.

Fig.1 D2Q9 model

Accordingly，different particle velocity vectors can be written as

where the lattice velocitycis defined byc=δx/δt.δxis the length of lattice andδtis the time step.Thus，the equilibrium distribution is

whereωiis the weighting factor in each direction：ω0=4/9，ωi=1/9（i=1，2，3，4），ωi=1/36（i=5，6，7，8）. Macroscopic parameters densityρ，velocityu，and pressurep，can be calculated by

The model of our calculation is shown in Fig.2.

Fig.2 Annular channel as in journal bearing

We consider a rigid system with the shapes of the boundary remaining a circle. The rotor，with ra?dius ofR1，rotates at a constant angular velocityωin the counterclockwise direction. The parameterθis circumferential degree that starts from the maxi?mum thickness，withθ=180° corresponding to the minimum thickness. The radius of the outer bound?ary，R2，equals to the radical clearanceCplus the rotor radiusR1.Therefore，the clearance ratiocis

Reynolds numberReis defined as

Eccentricityεis described byl，which is the distance between the centers of two circles

The circumferential angleθstarts from the maximum clearance and varies from 0 to 2π follow?ing the rotating direction. Therefore，the local clear?ance is

Non-slip velocity boundary condition is as?sumed at both sides of the channel. The carved boundary is treated by the method described in Ref.［30］. As shown in Fig.3，the open circle（rf）and grey solid circle（rb）represent the nodes in fluid and solid region，respectively. The solid black circle（rw）indicates intersection of the node links and the dashed line，which is the real curved boundary.

Fig.3 Method for curved boundary

Defineqas

The post-collision distribution functionf+for noderbcan be written as

whereeαˉ=-eα，rff=rf+eαˉδtanduf=u(rf，t).Hereuwstands for the wall velocity，which is set to zero at the outer circle and the velocity of the rotor at inner circle，respectively. Owing to the negligible radial pressure gradient，the pressure at the center line of the thickness is taken as the typical results for the whole circumferential distribution. The working fluid is set as isothermal gas. Ambient pressurePais set to 0.1 MPa. Dimensionless pressure is described by

The computational results are verified by com?paring pressure distribution with data published by Diprima et al.［3］as shown in Fig.4. For comparison，the definition of pressure (P-Pa)C2/(μωR21) is the same as the one employed in the above research withμstanding for the dynamic viscosity. The modi?fied Reynolds number is 0.8 with eccentricity and clearance ratio at 0.75 and 0.08，respectively. The distribution in low pressure zone match well with each other and diverge slightly in the high pressure zone with discrepancy less than 6%. To conclude，the model constructed by utilizing LBM is accurate enough for calculating the annular flow.

Fig.4 Verification of model by comparing with Diprima’s result

2 Results

For a comprehensive understanding of the an?nular flow characteristics，the pressure and flow dis?tribution are systematically studied in terms of the effect of eccentricity，rotor rotating speed and clear?ance ratio.

2.1 Effect of eccentricity

The distribution as well as the extremum of pressure are depicted in Figs.5，6 with various ec?centricities，while rotating speed and clearance ratio remain 3×104r/min and 0.01，respectively. The Reynolds number is estimated to be 200. The in?creasing eccentricity gives rise to the absolute value of the maximum pressure and the minimum pres?sure. The circumferential pressure distribution fluc?tuates in a quite low level with eccentricity less than 0.4. The values of extremum pressure change smoothly with an increasing eccentricity smaller than 0.6，while upsurge significantly once eccentrici?ty reaches 0.7. With a high eccentricity，the mini?mum thickness becomes smaller when the maxi?mum thickness becomes larger on the contrary. Un?der the circumstance，the hydrodynamic effect will be more efficient，which brings dramatic changes in pressure distribution.

Furthermore，with eccentricity increasing from 0.2 to 0.9，the ratio of the maximum pressure to the minimum pressure keeps rising，ranging from 1.007to 1.36. The maximum and minimum pressure ap?pears at upstream of the minimum thickness（θ=180°）and downstream of it，respectively. The posi?tions of extremum pressure accord well with the lu?brication theory and approach the minimum thick?ness where a large pressure gradient is generated. In Fig.6，the two extremum pressure positions locate nearly symmetrically along the line ofy=180° and vary evenly with increasing eccentricity.

Fig.6 Value and position of extremum pressure

With eccentricity approximately reaches 0.4，back flow is generated in annular flow. This phe?nomenon is commonly studied in large clearance ra?tio and low Reynolds number condition. Hence?forth，separation point and reattachment point are defined to describe the formation and distribution of the vortex. For example，in Ballal’s results［22］，the critical eccentricity was proved to be 0.324 24 with a clearance ratio of 0.5. For better understanding of back flow generation under extreme low clearance ratio，the streamline patterns are presented in Fig.7 at the eccentricity of 0.6. The typical images are shown with distribution at the maximum thickness，the separation and reattachment zones.

Upon generation，the vortex shifts along the channel and occupies the largest portion of the flow at the maximum thickness. The generation of sepa?ration vortex is caused by the existence of adverse pressure gradient along counterclockwise orienta?tion. In a Stokes flow，the separation and reattach?ment points distribute symmetrically along the cen?ter line of the cylinders. In our study with Reynolds number larger than 200，both points A and C shift to the direction towards the minimum thickness of the annulus which is closer to the reattachment point. Fig.8 presents the positions of the separation point and reattachment point against eccentricity. As eccentricity ascends，the contraction and expansion of the annular become more drastic and thus induces a larger adverse pressure gradient and a wider clear?ance around the maximum thickness. As a result，the separation vortex is more likely to generate and occupies larger region in the channel，which can ex?plain the approaching of points A and C.

The tangential velocity，non-dimensionalized byR1ω，is shown in Figs.9，10，with distribution along the minimum and maximum thickness of the channel. Based on the analytical solution of a Cou?ette flow，the variation of velocity is faster with a smaller favorable pressure gradient. Fig.5 demon?strates that minimum thickness suffers from favor?able pressure gradient. With the eccentricity increas?ing from 0.2 to 0.5，the drops of tangential velocity from the rotor to the stationary boundary become slower along the minimum thickness. Upon exceed?ing the eccentricity value of 0.5，the variation rate of velocity no longer follows the pattern due to fluid compressibility，geometry change and the drastic pressure gradient change. The minimum thickness under eccentricity of 0.2 could be eight times as much as the one at the eccentricity of 0.9. Addition?ally，the difference of pressure gradient is significant according to Fig.5. Consequently，the coupling ef?fects of pressure gradient and geometry contributes to the rate change of the velocity variation.

Fig.5 Non-dimensional circumferential pressure under dif?ferent eccentricities

Fig.9 Non-dimensionalized tangential velocity along mini?mum thickness

In Fig.10，the non-dimensionalized tangential velocity gradually decreases from the rotor to the stationary boundary along the maximum thickness.The variation rate of the velocity becomes larger as the eccentricity increases due to the adverse pres?sure gradient. Negative velocity appears only when the eccentricity equals to or be greater than 0.4，cor?responding to the back flow generation. As a result，the non-dimensionalized tangential velocity exhibits an increasing trend near to the stationary boundary with large eccentricity. In this region，the variation rate of the velocity also increases with eccentricity.

Fig.10 Non-dimensionalized tangential velocity along maxi?mum thickness

2.2 Effect of rotation speed

Fig.11 shows a comparison of pressure distribu?tions with rotation speed increasing from 3×104r/min to 1.8×105r/min with an interval of 3×104r/min. The calculation is implemented with clearance ratio and eccentricity fixed at 0.01 and 0.6，respec?tively. Both the absolute values of the maximum and minimum pressures increase almost linearly with the rotating speed，as shown in Fig.12. Posi?tions of both the maximum and the minimum pres?sures show no significant shift as rotating speed in?creases and mainly focuses within the area around 140°and 220°.

Fig.11 Non-dimensionalized pressure distribution under different rotating speeds

The non-dimensionalized tangential velocity along the minimum thickness and maximum thick?ness are shown in Figs.13，14. At the minimum thickness，the non-dimensionalized tangential veloc?ity is overall slightly higher with a larger rotating speed. At the region close to the rotor，the variation rate is larger at a lower rotating speed. As shown in Fig.14，the effect of rotor rotating speed seems neg?ligible at the maximum thickness as the curves al?most overlap with each other.

Fig.13 Non-dimensionalized tangential velocity along mini?mum thickness

Fig.14 Non-dimensionalized tangential velocity along maxi?mum thickness

The dependence of the positions of separation and reattachment point on the rotating speed is shown in Fig.15. The variation is confined in a range no more than 5 degree，indicating a negligible effect of the rotating speed on the location of the sep?aration vortex. The reattachment point is about 10°closer to the minimum thickness than the separation point. For flow in small-scale channel with Reyn?olds number no less than 200 in this paper，the posi?tions of separation and reattachment points almost remain stationary.

Fig.15 Positions of separation point and reattachment point against eccentricity

2.3 Effect of clearance ratio

Fig.16 depicts the pressure distributions when clearance ratios vary from 0.005 to 0.01. The rotat?ing speed and eccentricity are set to 3×104r/min and 0.6，respectively. Obviously，the value of extre?mum pressure decreases as clearance expands. Up?on reaching 0.008，there is no significant variation of the extremum pressure in terms of the clearance ratio，as shown in Fig.17. To sum up，a narrower channel leads to a larger extremum of pressures.The variation of the separation and reattachment point positions are confined in a relatively small range，indicating negligible effect of the clearance ratio.

Fig.16 Non-dimensionalized pressure distribution under dif?ferent clearance ratios

Fig.17 Value and position of extremum pressure

The velocity distribution along the minimum thickness is quite obscure，as shown in Fig.18.With the same rotating speed，the increase of clear?ance ratio could result in the expansion of the local clearances. Besides，the compressibility of the work?ing gas and nonlinear relation between pressure and clearance ratio could contribute to other sophisticat?ed factors.

Fig.18 Non-dimensionalized tangential velocity along the minimum thickness

The distribution of non-dimensionalized tangen?tial velocity along the maximum thickness of the channel is shown in Fig.19 at different clearance ra?tios. The extremum increases with enhancement of the clearance ratio. Overall，there is no tremendous variation of non-dimensionalized tangential velocity along the minimum and maximum thicknesses with clearance ratio varying. In Fig.20，the separation and reattachment points also remain almost station?ary，indicating that the effect of clearance ratio is negligible.

Fig.19 Non-dimensionalized tangential velocity along maxi?mum thickness

Fig.20 Positions of separation point and reattachment point against clearance ratio

3 Conclusions

The two-dimensional flow between eccentric cylinders with clearance ratio less than 0.01 is dis?cussed. The pressure distribution and flow pattern are analyzed in terms of clearance ratio，rotation speed and eccentricity. Large eccentricity，high ro?tating speed and low clearance ratio can generate a drastically fluctuating pressure distribution. Further?more，the extremum pressure is almost proportional to the rotating speed and upsurge significantly with eccentricity larger than 0.7 or clearance ratio smaller than 0.008. Meantime，the existence of an eccentric?ity-dependent separation vortex is affirmed with ec?centricity exceed a critical value lying between 0.3 and 0.4. Eccentricity contributes mostly to the posi?tions of extremum pressure and separation vortex.Considering the coupling effects of the pressure gra?dient and geometry，non-dimensionalized tangential velocity distributions are parametrically analyzed along the maximum and minimum thicknesses.