Large eddy simulation of vortex shedding and pressure
fluctuation in aerostatic bearings
Jincheng Zhu a,Han Chen a,b,Xuedong Chen a,n
a State Key Laboratory of Digital Manufacturing Equipment and Technology,Huazhong University of Science and Technology,
Wuhan430074,China
b Department of Mechanics,Huazhong University of Science and Technology,Wuhan430074,China
最薄手机a r t i c l e i n f o
Article history:
Received21May2012
Accepted5March2013
薛家兄弟的玩具Available online28April2013
Keywords:
Aerostatic bearing
Large eddy simulation
Vortex shedding前列腺肥大吃什么药
Pressure fluctuation
Vibration
a b s t r a c t
In aerostatic bearings,high speed air flow may induce small vibration,which has been
harmful to the improvement of moving and positioning accuracy of aerostatically
supported devices in ultra-precision applications.In this paper,the transient flow field
in the aerostatic bearing is numerically investigated using the large eddy simulation
接济的近义词
method.Turbulent structures are studied and vortex shedding phenomenon is discovered
in the bearing recess.Our computational results demonstrate that vortex shedding caus
pressure fluctuation in the bearing clearance.Relationship between pressure fluctuation
and bearing vibration is established bad on our simulation results and experimentally
measured vibration strength.
&2013Elvier Ltd.All rights rerved.
1.Introduction
Aerostatic bearings have been widely ud 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)verely 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 rearch efforts have been made on the air flow field in aerostatic bearings.舞台设计
Recently,the relationship between the high speed air flow and the small vibration in aerostatic bearings has been realized by many rearchers.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 the air vortices are responsible for the instability of the aerostatic bearing.Aoyama et al.(2006)also obrved this air vortex flow by CFD simulation and reached a similar conclusion,and accordingly propod 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 flow
Contents lists available at SciVer ScienceDirect
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Journal of Fluids and Structures
0889-9746/$-e front matter&2013Elvier Ltd.All rights rerved.
dx.doi/10.1016/j.jfluidstructs.2013.03.012
n Corresponding author.Tel./fax:+862787557325.
E-mail address:chenxd@mail.hust.edu(X.Chen).
Journal of Fluids and Structures40(2013)42–51
field or averaged the flow field in a Reynolds Averaged Navier –Stokes (RANS)n.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 ud 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 increas with increasing pressure fluctuation induced by vortex shedding in the bearing recess.2.Numerical modeling 2.1.LES
In 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 i
ðρ~u
i Þ¼0;ð1Þ∂ðρ~u i Þþ∂j ðρ~u i ~u j Þ¼−∂p i þ∂~s ij j −∂
j
ð~τij Þ;ð2Þ
the Favre filter is the density-weighted filter,where density and pressure are spatial filtered (denoted by “–”)while velocity
is 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
放生范逸臣~s ij ¼μ∂~u i ∂x j þ∂~u j ∂x i −23δij ∂~u k ∂x k
;ð3Þ~τij ¼ρðu i u j $
−~u
i ~u j Þ;ð4
Þ
J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5143
where τij needs to be modeled using a SGS model.Large 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 volu
me 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 cond order upwind interpolation is chon for the density,turbulent kinetic energy and turbulent dissipation rate,while the bounded central differencing is chon for momentum interpolation in LES.As the transient formulation,the cond order implicit scheme is adopted.The Non-Iterative Time-Advancement (NITA)scheme (Issa,1986)is ud to improve the computational efficiency,and the time step size Δt ¼1Â10−8s is chon according to the CFL condition u Δt =Δx o 1,where Δx is the size of control volume.2.2.Computational domain
For 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 12ctions along the circumferential direction,and only one ction (Fig.2)is ud 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 rece
ss,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 ud in the orifice region (RANS region);the flow in the air film is suppod to be laminar and LES is adopted in the recess region.An RANS –LES interface is ud to connect the orifice region and the recess region.2.3.Computational mesh
Fig.3shows the computational mesh ud in the LES,where non-conformal mesh is ud.It is known that the accuracy of LES is nsitive to mesh resolution,so more refined mesh is generated in the recess region.Mesh independence tests (e Table 1)are performed until further refinement of the mesh results in insignificant changes in the
computational
东莞旅游景点
Fig.1.Schematic of the aerostatic bearing.
Fig.2.Computational domain and ELES model.
J.Zhu et al./Journal of Fluids and Structures 40(2013)42–51
44
results.The parameters in Table 1are described as follows.The total number and the volume of the mesh in various regions
are listed.The non-dimensional distance y +can reflect wall-adjacent mesh resolution,which is defined as y þ¼ffiffiffiffiffiffiffiffi
ρτw 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 coar mesh ca and the fine mesh ca,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 conditions
As 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 ud 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 ud as the initial field of LES,which can help LES to converge quickly.
2.5.Validation of numerical model
In 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 en 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 aerostatic
bearings.
X
Z
Fig.3.Computational mesh.
Table 1
Mesh 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.Mesh
Recess Orifice Gas film Max y +
Total number
Mean E ðp A Þ
Fluctuation s ðp A Þ
Δmin
段奕宏主演的电影Δmax Δmin Δmax Δmin Δmax Coar 1.11129020.60
76.27
1755
71610
1.526807535062862Fine
0.43
437
0.6
519644
350463
67
J.Zhu et al./Journal of Fluids and Structures 40(2013)42–5145
3.Transient flow characteristics 3.1.Flow structures and vortex shedding
Fig.5displays the streamlines and the pressure contours computed from steady RANS simulation wh
en P s ¼4atm,in which flow paration and vortex formation can been en 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 ries of vortices with varying sizes and shapes.The vortex shedding phenomenon can be obrved.Specifically,the toroidal spanwi 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 fluctuation
As shown in Fig.5,pressure depression (sudden descent and ascent)can be obrved 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 ar
e induced,as shown in Fig.6(b).Similarly,the positions and the magnitudes of the pressure minima are transiently changing.Fig.7shows an instantaneous pressure distribution on the
02
4
6
p / P 0
r / r 2
Fig.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)42–51
46
