气浮轴承的fluent 三维仿真

Large eddy simulation of vortex shedding and pressurefluctuation in aerostatic bearingsJincheng Zhu a,Han Chen a,b,Xuedong Chen a,na State Key Laboratory of Digital Manufacturing Equipment and Technology,Huazhong University of Science and Technology,Wuhan430074,Chinab Department of Mechanics,Huazhong University of Science and Technology,Wuhan430074,Chinaa r t i c l e i n f oArticle history:Received21May2012Accepted5March2013Available online28April2013Keywords:Aerostatic bearingLarge eddy simulationVortex sheddingPressure fluctuationVibrationa b s t r a c tIn aerostatic bearings,high speed air flow may induce small vibration,which has beenharmful to the improvement of moving and positioning accuracy of aerostaticallysupported devices in ultra-precision applications.In this paper,the transient flow fieldin the aerostatic bearing is numerically investigated using the large eddy simulationmethod.Turbulent structures are studied and vortex shedding phenomenon is discoveredin the bearing recess.Our computational results demonstrate that vortex shedding causespressure fluctuation in the bearing clearance.Relationship between pressure fluctuationand bearing vibration is established based on our simulation results and experimentallymeasured vibration strength.&2013Elsevier Ltd.All rights reserved.1.IntroductionAerostatic bearings have been widely used in ultra-precision moving and positioning equipments.Due to the merit of near-zero friction and low heat generation,applications of aerostatic bearings make it possible for supported devices to realize nanometer positioning accuracy.However,with the increasing demand of positioning accuracy,the inherent small vibration on the order of nanometers(Kawai et al.,2005)severely damages stability and precision of the bearing,especially in sub-nanometer positioning equipments.To understand and eventually suppress this harmful vibration,traditional design and analysis methods for mass flow rate and load carrying capacity do not suffice anymore,and lots of research efforts have been made on the air flow field in aerostatic bearings.Recently,the relationship between the high speed air flow and the small vibration in aerostatic bearings has been realized by many researchers.Kawai et al.(2005)studied the nano-vibration in ultra-precision machine tools and attributed it to air turbulence due to bearing surface roughness.Chen and He(2006)found vortex flow structures in the bearing recess by computational fluid dynamics(CFD)simulation of the steady air flow field,and suggested that these air vortices are responsible for the instability of the aerostatic bearing.Aoyama et al.(2006)also observed this air vortex flow by CFD simulation and reached a similar conclusion,and accordingly proposed a new restrictor design to weaken the vibration. Zhang et al.(2007)analyzed the high Reynolds number(Re)flow in the bearing clearance,and reduced the vibration of aerostatic bearings by flow laminarization.In a recent work,Yoshimura et al.(2012)attributed nano-vibration of aerostatic bearings with surface restriction to pressure fluctuation at the bearing outlet due to atmospheric turbulence.Although the flow-induced nature of bearing vibration has generally been recognized,the previous works only assumed a steady flowContents lists available at SciVerse ScienceDirectjournal homepage:/locate/jfsJournal of Fluids and Structures0889-9746/$-see front matter&2013Elsevier Ltd.All rights reserved./10.1016/j.jfluidstructs.2013.03.012n Corresponding author.Tel./fax:+862787557325.E-mail address:chenxd@(X.Chen).Journal of Fluids and Structures40(2013)42–51field or averaged the flow field in a Reynolds Averaged Navier –Stokes (RANS)sense.Since this flow-induced vibration is apparently a time dependent process,time dependent is necessary to investigate the transient air flow field in the bearing clearance in order to further understand this harmful small vibration.To numerically analyze the detailed flow characteristics in aerostatic bearings,the full Navier –Stokes equations for compressible fluids have to be solved.Since the high speed air flow in the bearing gap near the orifice outlet is turbulent,RANS simulation is usually employed,and numerical results demonstrate adequate accuracy in predicting mean flow characteristics (Chen and He,2006;Li and Ding,2007).Pressure depressions (Eleshaky,2009;Yoshimoto et al.,2007)and vortex flow structures (Chen et al.,2011)near the orifice outlet have also been reported using RANS simulation.However,RANS simulation adopts a statistical turbulent model and details of turbulent structures remain unresolved.Ideally,direct numerical simulation (DNS)can resolve the whole spectrum of turbulent scales as no turbulent model is assumed,but its computational cost is prohibitively huge.In large eddy simulation (LES),large scale turbulent eddies are solved directly and small scale turbulence is modeled by sub-grid scale models.Thus,coherent turbulent structures can be obtained with acceptable computational cost in LES,which has been validated in various applications (Cheng et al.,2012;Lam et al.,2010;Tucker,2011).In a previous study (Chen et al.,2011),steady RANS simulation method was used to study air flow fields in various aerostatic bearings with different parameters and recess shapes,and the relationship between vortex strength and vibration energy of the bearing was established.However,no transient flow characteristics in the aerostatic bearing could be resolved.In this paper,the transient air flow field is investigated numerically using the LES method.Our simulation results reveal vortex shedding and pressure fluctuation in the bearing recess.Vibration of the bearing is also measured experimentally,and it is demonstrated that vibration strength of the bearing increases with increasing pressure fluctuation induced by vortex shedding in the bearing recess.2.Numerical modeling 2.1.LESIn LES,large eddies of turbulence are directly resolved and eddies with scales smaller than grid spacing are modeled.The governing equations employed in LES are the time-dependent Favre Filtered Navier –Stokes equations,including continuity and momentum equations:∂ρ∂t þ∂∂x iðρ~ui Þ¼0;ð1Þ∂ðρ~u i Þþ∂j ðρ~u i ~u j Þ¼−∂p i þ∂~s ij j −∂jð~τij Þ;ð2Þthe Favre filter is the density-weighted filter,where density and pressure are spatial filtered (denoted by “–”)while velocityis density-weighted (e üρÃ=ρ,n denotes a general variable).In Eq.(2),s ij is the viscous stress tensor and τij is the subgrid-scale (SGS)stress,which are defined as~s ij ¼μ∂~u i ∂x j þ∂~u j ∂x i −23δij ∂~u k ∂x k;ð3Þ~τij ¼ρðu i u j $−~ui ~u j Þ;ð4ÞJ.Zhu et al./Journal of Fluids and Structures 40(2013)42–5143where τij needs to be modeled using a SGS rge turbulent eddies can be resolved directly by Eqs.(1)and (2),and turbulent eddies with scales smaller than grid size are modeled.As a SGS model,the Wall-Adapting Local Eddy-Viscosity (WALE)model (Nicoud and Ducros,1999)is adopted in this paper.In this work,LES simulations were performed in the CFD software ANSYS Fluent using the finite volume method.In Fluent,the Pressure-Implicit with Splitting of Operators (PISO)algorithm (Issa,1986)is adopted as the pressure –velocity coupling scheme.In order to minimize numerical dissipation,the second order upwind interpolation is chosen for the density,turbulent kinetic energy and turbulent dissipation rate,while the bounded central differencing is chosen for momentum interpolation in LES.As the transient formulation,the second order implicit scheme is adopted.The Non-Iterative Time-Advancement (NITA)scheme (Issa,1986)is used to improve the computational efficiency,and the time step size Δt ¼1Â10−8s is chosen according to the CFL condition u Δt =Δx o 1,where Δx is the size of control putational domainFor generality and simplicity,a circular pad aerostatic bearing with a single central orifice restrictor is considered as shown in Fig.1.The outer diameter of the bearing is d 2¼20mm and the orifice diameter is d 0¼0.15mm.The cylindrical recess has a diameter d 1¼3mm and depth H ¼0.1mm.The air film thickness is h ¼10μm.In our LES calculations,the air flow domain is divided into 12sections along the circumferential direction,and only one section (Fig.2)is used as the computational domain to reduce the computational cost.This simplification is reasonable for qualitative study on turbulent structures in the aerostatic bearing.In order to allow for a fine resolution of turbulent structures in the bearing recess,the Embedded LES (ELES)modeling technique in Fluent is adopted.Specifically,the computational domain is divided into three regions:orifice,recess and gas film (Fig.2).The realizable k −εmodel is used in the orifice region (RANS region);the flow in the air film is supposed to be laminar and LES is adopted in the recess region.An RANS –LES interface is used to connect the orifice region and the recess putational meshFig.3shows the computational mesh used in the LES,where non-conformal mesh is used.It is known that the accuracy of LES is sensitive to mesh resolution,so more refined mesh is generated in the recess region.Mesh independence tests (see Table 1)are performed until further refinement of the mesh results in insignificant changes in thecomputationalFig.1.Schematic of the aerostatic bearing.putational domain and ELES model.J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5144results.The parameters in Table 1are described as follows.The total number and the volume of the mesh in various regionsare listed.The non-dimensional distance y +can reflect wall-adjacent mesh resolution,which is defined as y þ¼ffiffiffiffiffiffiffiffiρτw p y =μ,where y is the distance from the wall to the center of the first neighboring mesh,and τw is the wall shear stress.To resolve accurately turbulent eddies in the near-wall regions,y +is always guaranteed to be less than 1with local mesh refinement.As the calculation results,the mean values and the standard deviations of p A are compared between the coarse mesh case and the fine mesh case,where p A is the time variation (as described in Section 3.2)of area-weighted averaged pressure on the wall 2.2.4.Boundary and initial conditionsAs shown in Fig.2,pressure inlet boundary condition is specified at the orifice inlet,in which turbulent intensities of 1%,5%and 10%are considered;atmospheric pressure is specified at the bearing outlet;two symmetric boundaries are adopted on the two surfaces in the circumferential direction.On the solid walls,no-slip and no heat transfer conditions are specified.In addition,all the walls are assumed to be perfectly smooth.The air used in the simulations is assumed to obey the ideal gas law,hence the density varies according to the state equation.Other physical constants such as viscosity,molecular weight,specific heat and thermal conductivity are 1.7894Â10−5kg/(m s),28.966Â10−3kg/mol,1006.43J/(kg K)and 0.0242W/(m K),respectively.A steady RANS simulation result is used as the initial field of LES,which can help LES to converge quickly.2.5.Validation of numerical modelIn order to justify our numerical model,the existing experiment data (Yoshimoto et al.,2007)of pressure distribution of the aerostatic bearing are utilized as a comparison.Fig.4shows the comparison with our numerical result,where both the realizable k −εRANS result and the LES result are plotted.The LES result is the statistical mean pressure distribution in the bearing clearance.As can be seen in the figure,there is almost no discrepancy between our CFD results and the experimental data except for the region where r /r 2is between 0.034and 0.2.In this region near the orifice outlet,the LES result shows better agreement with experimental data than the RANS one.Therefore,the LES method can be employed in the calculation of the flow field of aerostaticbearings.XZputational mesh.Table 1Mesh refinement study,where Δdenotes mesh volume (μm 3),p A is the time variation of area-weighted averaged pressure (Pa)on the wall 2,E and s denote the mean value and standard deviation,respectively.MeshRecess Orifice Gas film Max y +Total numberMean E ðp A ÞFluctuation s ðp A ÞΔminΔmax Δmin Δmax Δmin Δmax Coarse 1.11129020.6076.271755716101.526807535062862Fine0.434370.651964435046367J.Zhu et al./Journal of Fluids and Structures 40(2013)42–51453.Transient flow characteristics 3.1.Flow structures and vortex sheddingFig.5displays the streamlines and the pressure contours computed from steady RANS simulation when P s ¼4atm,in which flow separation and vortex formation can been seen in the recess near the orifice outlet.It is noted that steady RANS simulation results in axisymmetric flow structures.Fig.6displays the corresponding instantaneous flow field obtained by LES at different times.In Fig.6(a),the iso-surfaces of instantaneous vorticity are depicted.In contrast to the single axisymmetric vortex in Fig.5,the coherent turbulent structure in the recess contains a series of vortices with varying sizes and shapes.The vortex shedding phenomenon can be observed.Specifically,the toroidal spanwise vortices develop after impinging of the orifice outflow on the bottom wall of the bearing,and then stretch in the radial direction along the wall surface with growing size through rolling-up process,and the convected wall vortices quickly break into more sophisticated small eddies downstream and finally are dissipated due to air viscosity.This vortex shedding phenomenon can also be explained as a typical flow pattern of the impinging jet (Lee and Lee,2000),since the high speed orifice outflow impinges perpendicularly on the solid wall.3.2.Pressure depression and fluctuationAs shown in Fig.5,pressure depression (sudden descent and ascent)can be observed near the orifice outlet where the minimum pressure occurs in the vortex core.However,with the vortex shedding displayed in the LES result,more local pressure minima corresponding to vortex centers are induced,as shown in Fig.6(b).Similarly,the positions and the magnitudes of these pressure minima are transiently changing.Fig.7shows an instantaneous pressure distribution on the0246p / P 0r / r 2Fig.4.Pressure distribution along radial direction in the aerostatic bearing:comparison of results among LES,RANS and the existing experiment.Fig.5.Streamlines and pressure contours obtained from steady RANS simulation.J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5146bottom wall (the recess region)of the bearing.It can be seen that the repeated pressure up-and-downs are present not only in the radial direction,but also in the circumferential direction.Fig.8plots pressure variations with time at three different locations (y ¼0.055mm,r ¼0.2,0.5,1.0mm),which are distributed along the radial direction in the bearing recess.Pressure fluctuation can be clearly seen in this figure,which also weakens along the radial direction.Table 1lists some statistical results of the area-weighted averaged pressure variation with time,where s ðp A Þcan represent the intensity of pressure fluctuation.In our simulations,when the intensities of inflow turbulence are respectively set as 1%,5%and 10%,the corresponding s ðp A Þare 62,65and 73,hence the influence of inflow turbulent level on pressure fluctuation is very small.In summary,associated with repeated shedding and downstream advection of vortices is repeated pressure depression (in space)and fluctuation (in time).It should be noted that this repeated pressure depression and fluctuation is not resolvable in RANS due to its essence of statistical averaging.It is manifest that LES results indicate unsteady flow characteristic in the aerostatic bearing even if initial and boundary conditions are all constant.4.DiscussionsAs is known,air supply pressure has a significant influence on the flow field in the aerostatic bearing.Therefore,air supply pressure values are varied in our numerical simulations.Fig.9shows the contours of instantaneous vorticity in the bearing recess when P s ¼2,3and 4atm.From this figure,the air flow is steady and remains in the laminar regime when the supply pressure is 2atm,while becomes unsteady and so vortex shedding is observed in the other two cases.Furthermore,associated with increasing air supply pressure,vorticity magnitude of the flow increases correspondingly.It suggests that the flow in the bearing recess has a transition from lamina to turbulence with increasing supply pressure.The correspondinglocal Reynolds number Re ¼_m=πr μat the recess entrance (r ¼r 0)in each case is calculated,as shown in Table 2.It can be inferred that the flow transition will appear when the local Reynolds number is beyond a certain critical valuebetweenFig.6.Iso-surfaces of instantaneous vorticity (a)and pressure (b).J.Zhu et al./Journal of Fluids and Structures 40(2013)42–51471000and 1500.Once beyond this critical Reynolds number,the air flow is turbulent and so results in pressure fluctuation.It can be observed from Fig.10that this pressure fluctuation becomes more severe as supply pressure increases.For comparison,the transient flow field of the non-recessed bearing (other dimensions being the same)is also calculated.Fig.11displays the instantaneous flow field obtained from LES,where the streamlines show that only one tiny vortex is present near the orifice outlet,and the flow is steady and no air vortex shedding appears.The absence of vortex shedding in the non-recessed bearing can also be explained by the smaller Reynolds number which confines the local flow in a laminarregime.Fig.7.Instantaneous pressure distribution on the bottom wall (the recess region)of the bearing.3.403.443.483.523.56p (P a )5Fig.8.Time 1.0mm).Fig.9.Contours of instantaneous vorticity under different supply pressure values.J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5148-300-1500150300t (ms)f o n o i t a u t c u l F P A (P a )Fig.10.Pressure fluctuations with different supply pressurevalues.Fig.11.Instantaneous flow field of non-recessed bearing.Fig.12.Photo of experimental set-up.Table 2Local Reynolds number computed in different cases.Case Recess P s (atm)_m(kg/s)Re 1Yes 2 2.66Â10−66312Yes 3 6.39Â10−615163Yes 4 1.06Â10−525154No44.52Â10−61070J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5149Vibration of the recessed bearing and the non-recessed bearing is also experimentally measured.Fig.12shows our experimental set-up.The two bearing pads have the same surface roughness.The tested bearing is placed on a marble base (with roughness less than 1μm),and air supply pressure can be regulated with a proportional throttle valve.External loads are applied on top of the aerostatic bearing.The parallelization of the bearing is assured by the displacement sensor (LVDT)which measures the bearing clearance height at three locations.Signals of the vibration acceleration of the aerostatic bearing are obtained by an accelerometer (PCB CA-YD-106),a data acquisition device (LMS SCADAS III),and a data analysis software platform (LMS Test Lab).The spectrum characteristics of bearing vibration acceleration are plotted in Fig.13.It can be seen that the amplitude of vibration acceleration increases with increasing air supply pressure,and the vibration of the non-recessed bearing (Fig.13(b))is much weaker than that of the recessed bearing (Fig.13(a)).Comparing our LES and experimental results,the qualitative behavior of pressure fluctuation is consistent with that of the bearing vibration when subjected to the different air supply pressure values.Although the vibration is also measured in the recessed bearing when P s ¼2atm,it may attribute to bearing surface roughness effect.Specially,compared with the results of the recessed bearing,the non-recessed bearing study clearly indicates that the vibration is relatively small when no pressure fluctuation is induced.Therefore,it can be concluded that vortex shedding results in remarkable pressure fluctuation and can induce small vibration of aerostatic bearings.5.ConclusionsThis paper is focused on the transient flow characteristics in ultra-precision aerostatic bearings.In order to capture turbulent structures and fluctuations,LES method is employed to numerically calculate the transient flow field in the bearing clearance.Vortex structures and pressure fluctuation in the bearing clearance are analyzed.Qualitative behavior of pressure fluctuation,as well as vibration of aerostatic bearing,is also discussed.If ever possible,however,the quantity analysis of the relationship between pressure fluctuation and small vibration of aerostatic bearings is needed in the future work.From the results in this paper,the following conclusions can be drawn:1.Vortex shedding is present in the aerostatic bearing recess for sufficiently large air supply pressure,i.e.,air vortices are repeatedly generated,shed,advected downstream and dissipated.2.Repeated pressure depression (in space)and fluctuation (in time)can be observed in the bearing clearance when vortex shedding occurs.3.Once the flow Reynolds number is beyond a certain value at the recess inlet,vortex shedding in the bearing clearance results in remarkable increase of pressure fluctuation and vibration of the aerostatic bearing.AcknowledgmentsThis study was supported by the National Basic Research and Development Program of China (No.2009CB724205)and the National Natural Science Foundation of China (Nos.51121002and51175196).Fig.13.Influence of air supply pressure on vibration acceleration of the aerostatic bearings in frequency domain.(a)Recessed bearing and (b)non-recessed bearing.J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5150J.Zhu et al./Journal of Fluids and Structures40(2013)42–5151 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