a computational fluid dynamics analysis on air bearing...

faculty of mathematics and natural sciences

A computational fluid dynamics analysis on air bearing designs

Bachelor Project Mathematics

June 2015

Student: I.W. Tan

First supervisor: Prof.dr.ir. R.W.C.P. Verstappen

Second supervisor: Prof.dr. A.J. van der Schaft

Abstract

This report treats the results of a research project on simulating the air flow in aero-static thrust air bearings with the Open Source Computational Fluid Dynamics (CFD)software package OpenFOAM. The type of air bearing considered in this researchis a simple type circular bearing with eight orifices that are equally distributed overa concentric circle. In this report we first present some background information onair bearings, treat the fluid dynamics that is involved and consider the previouslyconducted research on computational models of air bearings. Next, we will discussthe simulations that we performed where we varied the choice of solvers, boundaryconditions, turbulence models and the viscosity parameter. Lastly, we will analyze theresults and give suggestions for future work on this subject. We find that manuallyaltering the viscosity via the Sutherland Coefficient As provides a stable initial flow,from which we can incrementally lower the viscosity back to values closer to the realis-tic value. Furthermore we find that the solvers sonicFoam and rhoCentralFoam showsimilar behaviour on our meshes and that sonicFoam and rhoSimpleFoam in Open-FOAM 1.7 perform differently from sonicFoam and rhoSimpleFoam in OpenFOAM2.3.

Keywords: aerostatic thrust air bearing, OpenFOAM, CFD

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Contents

1 Introduction 1

2 Background 32.1 Flat aerostatic thrust air bearings . . . . . . . . . . . . . . . . . . . . . . 32.2 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 General conservation principles . . . . . . . . . . . . . . . . . . . 42.2.2 Supersonic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Model 93.1 OpenFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 rhoSimpleFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 sonicFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 rhoCentralFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.1 The Realizable k-ε model . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 k-ω SST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Geometry and boundary conditions . . . . . . . . . . . . . . . . . . . . . 233.6 Schemes and Transport model . . . . . . . . . . . . . . . . . . . . . . . . 243.7 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results 274.1 Adapting the mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Changing several settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Initial pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Adjusting the time-step size . . . . . . . . . . . . . . . . . . . . . 35

4.3 Adjusting the viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Comparison to OpenFOAM 1.7 and a different geometry . . . . . . . . 42

5 Conclusion 47

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6 Future Work 51

A Additional results I

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Chapter 1

Introduction

This report treats the results of a research project on numerical simulations of aero-static thrust air bearings as commissioned by the company Schut Geometrical Metrol-ogy. Air bearings are bearings that use a thin film of air as a lubricant to reducethe friction between two moving surfaces. The company wishes to use the resultsfrom these simulations to improve the bearing design. Ultimately, the bearing has tosatisfy certain stability properties required for the implementation in high precisionmeasurement devices.

Figure 1.1: Three types of aerostatic circular thrust air bearings; 1.multiple orifices 2. single orifice 3. porous medium

In Figure 1.1, three types of flat aerostatic air bearings are illustrated. In thepast, the company has carried out several successful simulations on air bearings oftype 2, which contains a single inlet orifice in the center [6]. However, applying thesame modelling procedure to the geometry of the bearing of type 1 did not yield anyrealistic results. Therefore, the goal of this project was to find out how a bearingof type 1 can be simulated correctly. The simulations were carried out in the Open

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Source Computational Fluid Dynamics (CFD) software package OpenFOAM 2.3.0.This software provides features for applications ranging from complex turbulent flowinvolving chemical reactions to solid dynamics and electromagnetics [2].

When simulating fluid flow, it is important to tailor the numerical model to theflow dynamics involved. By making the right assumptions and simplifications, a lotof computation time can be saved without losing too much accuracy. After studyingsome of the available literature, we found that as the air flow exits the nozzle at thebottom of the bearing, it is subject to an extreme decrease in flow area. This canresult in locally supersonic flow, which requires the use of algorithms for compressiblefluids. This is presented in Chapter 2, along with a description of the bearing and thegoverning equations.

In reality, inevitable disturbances, such as disturbances in the load that is exertedon the bearing, will affect the height of the air gap. However, in this project weinvestigated a simplified situation in which the gap height is fixed. We have triedmultiple strategies to obtain a stable and realistic simulation, such as changing theboundary conditions, trying different solvers and manually altering the viscosity. Thespecific settings and procedures of the simulations are explained in Chapter 3.

The results from our simulations are presented in Chapter 4 together with ananalysis of the results in Chapter 5. We find that running a simulation with analtered viscosity is an effective strategy to obtain a stable initial flow for lowerand more realistic viscosities. Furthermore, we find that the solvers sonicFoamand rhoSimpleFoam in OpenFOAM 1.7 perform differently from sonicFoam andrhoSimpleFoam in OpenFOAM 2.3. Lastly, we find that the solvers rhoCentralFoamand sonicFoam show similar results in nearly all configurations, despite their differentnature.

Finally, we present some suggestions for future work in Chapter 6.

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Chapter 2

Background

2.1 Flat aerostatic thrust air bearings

Air bearings exist in different types, such as aerodynamical air bearings, which usetheir movement to create a thin high pressure air film between the surfaces, oraerostatic air bearings, in which the thin air film is externally pressurized. Moreover,there exist many other variable properties such as shape and material.

Figure 2.1: A schematic representation of the cross-section of an aero-static air bearing with a single orifice.Source: https://www.mech.kuleuven.be/apt/intro.en.html

In Figure 2.1, the cross-section of an aerostatic air bearing with a single orificeis shown. The shape and size of the air bearings considered in this research arecomparable to a hockey puck. An external compressor connected to the chamberinside the bearing, applies a supply pressure ps, which is higher than the atmosphericback pressure pa. The pressure difference induces an air flow which exits the chambervia the nozzle and flows into the air gap via the orifice. Finally, as it passes the edgeof the bearing, it can escape into the open air. The air bearing considered in this

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research has eight orifices distributed evenly over a concentric circle, comparable totype 1 in Figure 1.1.

2.2 Fluid dynamics

2.2.1 General conservation principles

Because the smallest length scale in this set up, the gap height under the bearing,is 4µm long, we can regard the gas as a continuum rather than as a collection ofdiscrete particles [15]. This means that we can use arbitrary control volumes toderive conservation equations for mass, momentum and scalar properties such astemperature and energy. By specifying the inflow, outflow, and sources for an arbitrarycontrol volume we obtain equations 2.1, 2.2 and 2.6 for mass, momentum and energyrespectively [8].

∂

∂ t

∫

Ω

ρ dΩ+

∫

S

ρv · n dS = 0, (2.1)

Equation 2.1 represents the conservation of mass, in which ρ represents thedensity, v the velocity vector and n the normal vector on the surface S of the controlvolume Ω. It states that the amount of mass in a control volume can only change bya mass flow through the boundary of that control volume.

∂

∂ t

∫

Ω

ρv dΩ +

∫

S

ρv v · n dS =

∫

S

T · n dS +

∫

Ω

ρb dΩ, (2.2)

Equation 2.2 represents the conservation of momentum, which is derived fromNewton’s second law of motion,

d(mv)dt

=∑

f . (2.3)

The tensor T in equation 2.2 represents the stress tensor and b represents the bodyforces. The equation states that the amount of momentum in a control volume canchange by advection, surface forces (for example pressure, shear stress or surfacetension) or body forces (for example gravity or the Coriolis force).In this research we will approximate air as a Newtonian fluid. Therefore, the stresstensor can be written as,

T= −

p+23µdiv v

I+ 2µD, (2.4)

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where p is the static pressure, µ is the dynamic viscosity, I is the unit tensor and D isthe deformation tensor [8],

D=12

grad v + (grad v)T

. (2.5)

Equation 2.1 and 2.2 together are known as the Navier-Stokes equations.

∂

∂ t

∫

Ω

ρh dΩ +

∫

S

ρhv · n dS =

∫

S

k grad T · n dS +

∫

Ω

(v · grad p + S : grad v) dΩ

+∂

∂ t

∫

Ω

p dΩ. (2.6)

Lastly, equation 2.6 represents the conservation of energy, in which h is the enthalpyper unit mass, k is the thermal conductivity defined as,

k = µcp

Pr, (2.7)

with cp the specific heat at constant pressure and Pr the Prandtl number, T is theabsolute temperature (K) and S is the viscous part of the stress tensor S = T+ pI [8].

2.2.2 Supersonic flow

In a lot of the previously conducted research on air bearings the presence of supersonicflow (Ma > 1) and an undesirable pressure depression attributed to a shock wave ismentioned [5, 6, 14]. Since our configuration is comparable to the configurations inthose articles, it is plausible that we will encounter supersonic flow too.

Shock waves

An important component of supersonic flow is the possible occurrence of shock waves.A shock wave is a thin region in the fluid across which the flow properties changedrastically. Because the highest velocity at which information on a fluid’s propertiestravels is the speed of sound, this information cannot travel upstream in supersonicflow. For the scope of this project it is sufficient to acknowledge the existence ofshock waves, in order to analyze the results of our model. More information on shockwaves can be found in [4] among many other text books.

Pressure depression

The undesirable pressure distribution mentioned earlier, is located close to the inletof the air bearing. This depression has a negative effect on the load capacity and has

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been studied extensively both theoretically and experimentally. Some researchers[20] attribute this depression to the transition from laminar to turbulent flow, but thiscontradicts the majority of articles on the subject, which show that the depressionexists due to shock waves in the bearing clearance [5, 7, 14]. In general, the latter isaccepted.

This explanation states that the air bearing can be considered as a converging-diverging duct, since the cross-sectional flow area at the entrance of the gap is smallerthan the flow area in the nozzle. From the area-velocity relation

dAA= (Ma2 − 1)

dvv

, (2.8)

we can see that the nature of the flow determines the effect of a change in cross-sectional flow area. For subsonic flow (Ma< 1), a decrease in area will accelerate theflow and an increase in area will decelerate the flow. For supersonic flow (Ma> 1) adecrease in area will decelerate the flow and an increase in area will accelerate theflow [4]. This explains how a small adjustment in supply pressure can induce a largechange in flow behaviour, as can be seen from figure Figure 2.2. If the differencebetween the supply pressure ps and the atmospheric pressure pa is large enough, theflow may become choked at the entrance of the gap and become supersonic.

Figure 2.2: A converging-diverging duct and the possible pressuredistributions and velocity profiles.Source: http://turbo.mech.iwate-u.ac.jp/Fel/turbomachines/stanford/images/LR_5.3.gif

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In Figure 2.2 the effect of the ratio of pa to ps on the flow in a converging divergingduct is illustrated. The pressure distributions that we encountered most often areof the type d, where the pressure difference is large enough to induce supersonicflow and a compression shock wave exists in the diverging duct that is the bearingclearance.

Because the transition from the inlet to the gap is at a right angle, the flowseparates from the boundary at high velocities. The resulting separation bubbleas illustrated in Figure 2.3 narrows the flow area even more. For a more detaileddescription of the fluid flow and the pressure depression often encountered in airbearings we refer to [7] and [14].

Figure 2.3: Qualitative representation of the shock/boundary layer in-teraction in the inlet region [7]

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Chapter 3

Model

3.1 OpenFoam

OpenFOAM is a free open source software package for applications ranging fromcomplex turbulent flow involving chemical reactions to solid dynamics and electro-magnetics [2]. In addition to the official software package that is managed by theOpenFOAM Foundation, there are numerous programmers that write extensions forthe existing OpenFOAM software themselves. The most of these unofficial extensionsare public too, but they have not been checked by the OpenFOAM Foundation. Often,these extensions contribute to new OpenFOAM versions after being processed bythe programmers of the OpenFOAM Foundation. The software is being used forcommercial as well as academic purposes and a lot of information can be found oninternet fora where users discuss their problems and experiences.

To perform a simulation in OpenFOAM, the user must create a case directory withat least the following sub-directories,

1. a time directory,

2. the "constant" directory,

3. the "system" directory.

The time directory, for example named 0 if t0 = 0, contains the boundary andinitial conditions for at least all variables that will be used in the calculations. Theconstant directory contains files that define the mesh as well as files in which thethermophysical and turbulence settings are specified. The system directory containsthe files in which the discretization schemes, linear system solvers, parallelisationsettings, solver settings and control settings are specified.

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3.2 Mesh

The mesh that we used was provided by the commissioning company and was createdin Mathematica [19]. It exploits the symmetry of the bearing by covering only 1

8 ofthe actual bearing geometry, which saves a lot of computation time. The mesh ismade of hexahedral cells exclusively because that is best suited for flow simulationsof very thin layers, such as our bearing clearance. Furthermore, by starting from athree dimensional structured hexahedral mesh it is less difficult to adopt the explicitmesh description format that OpenFOAM requires. The hexahedral mesh was fittedto the geometry of the bearing by transformations and removals of redundant cellswithout disturbing the structure and ordering of the mesh.

3.3 Solvers

We have used three solvers from the compressible solvers that are available in Open-FOAM 2.3.0, namely sonicFoam, rhoSimpleFoam and rhoCentralFoam. We willdescribe these three solvers below.

3.3.1 rhoSimpleFoam

The solver rhoSimpleFoam is a steady-state Semi-Implicit Method for PressureLinked Equations (SIMPLE) solver for laminar or turbulent Reynolds-averaged Navier-Stokes (RANS) flow of compressible fluids [2]. It uses the SIMPLE algorithm to solvethe coupled equations for velocity and pressure. The SIMPLE algorithm is an iterativealgorithm proven to be fairly efficient in solving steady state problems by the use of apredictor-corrector loop and under-relaxation factors [8]. This algorithm is also thebasis for many other iterative Navier-Stokes algorithms such as SIMPLEC, SIMPLER,PISO and PIMPLE. In Figure 3.1 a flow chart of the algorithm is illustrated.

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Start with un−1, ρn−1 and pn−1

k = 0 m = 0

Linearized momen-tum equation → um∗

Calculate mass fluxes mm∗l at

cell faces from ρm−1 and vm∗n

Pressure correction → p′

Correct mass fluxes at cell facesk++

Calculate um from pm

m++

Calculate other variables

Proceed to next time stepn++

Outerloopif k ≤

nNonOrthogonalCorrectors

if m <nOuter

Correctors

Figure 3.1: A flowchart of the SIMPLE algorithm for the nth time step

To calculate the values for the nth time step, the algorithm first enters a loopto calculate the pressure field and the velocity field, using the variable fields fromthe previous time step as initial values. The iterations of this loop are called outeriterations to distinguish them from inner iterations which are the iterations performedon linear systems with fixed coefficients [8]. These outer iterations continue as longas the pressure and velocity fields do not satisfy the specified residual tolerance andas long as the number of performed iterations m does not exceed the specified valueof nOuterCorrectors [11].

Lastly, all other variable fields, such as temperature and turbulent kinetic energyare calculated. After that, the algorithm proceeds to the next time step.

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We will now follow the derivations and steps as presented in [8] to explain howthe outer loop operates.The outer loop consists of three tasks. First, in case m= 0, the pressure and densityfields pn−1 and ρn−1 from the previous time step are used to solve the segregatedlinearized momentum equations. Else, in case m > 0, the pressure and densityfields pm−1 and ρm−1 from the previous outer iteration are used instead. Theselinearized momentum equations give a new estimate um∗ for the velocity field, wherethe asterisk is used to indicate that this estimate does not necessarily satisfy thecontinuity equation. The discretized segregated linearized momentum equation forthe ith component of the velocity estimate um∗ at the point P that is solved in thisfirst step of the outer iteration can be written as,

AuiP um∗

i,P +∑

l

Auil um∗

i,l =Qm−1ui−∆Ω

δpm−1

δx i

. (3.1)

Here m is the outer iteration counter and ∆Ω denotes the control volume centeredaround a cell face, which is not necessarily the same as the volume of the control vol-ume around point P. The coefficients AP and Al depend on the chosen discretizationschemes and represent the terms in the momentum equation that contain unknownvariable values, in this case um∗

i . The subscript l denotes the index of the neighbouringcells. The coefficient Qm−1

uicontains all terms of the momentum equation which do

not contain unknown variable values and thus are presumed known, except for thepressure gradient which is written explicitly in the last term in this equation [8].The pressure gradient is written explicitly here, because later on we want to relate acorrection of the velocity and density to a correction of the pressure.

Equation 3.1 can be rewritten as,

um∗i,P =

Qm−1ui−∑

l Auil um∗

i,l

AuiP

−∆Ω

AuiP

δpm−1

δx i

. (3.2)

Because the velocity field obtained from equation 3.2 together with the old densityvalues ρn−1 do not satisfy continuity, substituting the new interpolated cell face massfluxes m∗l into the continuity equation,

(ρm−1 −ρn−1)∆Ω∆t

+∑

l

m∗l =Q∗m, (3.3)

results in an imbalance Q∗m [8]. To enforce the conservation of mass it is necessary toimpose a correction step.

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This correction is the second task of the outer loop. Since we are dealing withcompressible flow, the mass flux varies not only with the velocity but with the densityas well. We must therefore calculate the density correction ρ′ and the normal velocitycorrection v′n such that the corrected mass flux for the mth outer iteration and lthcell face can be expressed as,

mml = ρ

ml vm

n,l Sl = (ρm−1 +ρ′)l(v

m∗n + v′n)l Sl , (3.4)

where Sl denotes the cell face area between the cell of point P and the lth neigbouringcell. The mass flux correction in terms of density and velocity can thus be expressedas,

m′l = (ρm−1 Sv′n)l + (v

m∗n Sρ′)l + (ρ

′ v′n S)l . (3.5)

Though methods exist to calculate the last term in equation 3.5 [8], we will neglectit here because it is of second order in terms of corrections. For a colocated variablearrangement, a mesh in which the velocity field and pressure field share the samegrid, the first term of equation 3.5 can be approximated by,

(ρm−1 Sv′n)l = − (ρm−1 S∆Ω)l

1Au

P

l

δp′

δn

l, (3.6)

where the overbar denotes an interpolated value and n is the coordinate of theoutward facing normal to the cell face [8]. If we assume the temperature to be fixedfor each outer iteration, we can approximate ρ′ in the second term of equation 3.5 by

ρ′ =

∂ ρ

∂ p

Tp′ = Cρ p′. (3.7)

Since we assumed that the fluid is a perfect gas, we can use the equation of state todetermine the coefficient Cρ by,

Cρ =1

RT. (3.8)

The second term of equation 3.5 can be written as,

(vm∗n Sρ′)l =

Cρm∗

ρm−1

l

p′l . (3.9)

If we now substitute equation 3.6 and 3.9 into equation 3.5 and neglect the last term,we get an expression for the mass flux correction in terms of known variable valuesand the pressure correction,

m′l = − (ρm−1S∆Ω)l

1Au

P

l

δp′

δn

l+

Cρm∗

ρm−1

l

p′l . (3.10)

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For the mass fluxes to satisfy the continuity equation, the corrected mass fluxes andthe corrected densities should satisfy,

ρ′P∆Ω

∆t+∑

l

m′l +Q∗m = 0. (3.11)

If we substitute equation 3.7 for ρ′P and equation 3.10 for m′l into equation 3.11, wecan write the pressure correction equation as,

AP p′P +∑

l

Al p′l = −Q∗m. (3.12)

After the resulting system of equations has been solved, only the last step of theouter loop remains. First, the corrected pressure is calculated. It has been foundthat adding only a portion of the pressure correction improves the convergence [8].This is done by choosing an under-relaxation factor αP ∈ [0, 1] ⊂ R which acts like apseudo-time step, such that,

pm = pm−1 +αP p′. (3.13)

This under relaxation factor is not used in the final outer iteration because the resultsof each time step must satisfy continuity. After this pressure correction, substitutingequation 3.13 and the expression um

i = um∗i + u′ into equation 3.2 gives,

u′i,P = −

∑

l Auil u′i,l

AuiP

−∆Ω

AuiP

δp′

δx i

P, (3.14)

from which the corrected velocity um is solved.

Via the file fvSolution the user can control several settings for the SIMPLEalgorithm. For example, the maximum number of outer iterations can be set throughthe variable nOuterCorrectors. Also, it is possible to solve the pressure correctionequation and calculate the mass fluxes at the boundary cell faces multiple times bysetting the variable nNonOrthogonalCorrectors larger than zero, to account forpossible grid non-orthogonality. The under-relaxation factors and residual tolerancescan also be specified there.

3.3.2 sonicFoam

The solver sonicFoam in OpenFOAM version 2.3 uses the PIMPLE algorithm, whichis a hybrid between the SIMPLE and PISO algorithms. The PISO algorithm is very sim-ilar to the SIMPLE algorithm, which we described above, except for two differences.

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It executes the outer loop only once for each time step and can therefore not useunder-relaxation factors, but on the other hand the pressure correction and resultingvelocity field are calculated more than once per time step, which we will call thePISO loop from now on. The PIMPLE algorithm combines SIMPLE and PISO suchthat both the outer loop and the PISO loop can be executed multiple times per timestep. For steady flows this allows larger time steps, as with the SIMPLE algorithm,since we can use the under-relaxation factor from the SIMPLE algorithm which actsas a pseudo time step.

Start with un−1, ρn−1 and pn−1

k = 0, l = 0, m = 0

Linearized momen-tum equation → um∗

Calculate mass fluxes mm∗l at

cell faces from ρm−1 and vm∗n

Pressure correction → p′

Correct mass fluxes at cell facesk++

Calculate um from pm

l++, m++

Calculate other variables

Proceed to next time stepn++

Outerloop

PISOloop

if k ≤nNonOrthogonal

Correctors

if l <nCorrectors

if m <nOuter

Correctors

Figure 3.2: A flowchart of the PIMPLE algorithm for the nth time step

In Figure 3.2 a flow chart of the PIMPLE algorithm is shown. The results from then−1th time step are used as initial values for the nth time step as the algorithm entersthe outer loop. First the discretized linearized segregated momentum equations are

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solved to obtain an estimate um∗, where again the asterisk is used to indicate that thisvalue does not necessarily satisfy the law of conservation of mass. Together with ρn−1,this estimate um∗ is then used to calculate an estimate of the mass fluxes throughthe cell faces mm∗

l . After that, the flux estimates are used to solve the imbalance ofthe continuity equation and subsequently to solve the pressure correction equation.This results in a correction on the pressure p′, a correction on the density ρ′ and acorrection on the cell face normal velocity v′n, by which the mass fluxes at the cell facescan be corrected. Depending on the variable nNonOrthogonalCorrectors, thesetwo steps can be executed multiple times to account for non-orthogonality. After that,the velocity cell center value estimates um∗

P are corrected and depending on the valueof nCorrectors the algorithm can perform another PISO iteration by returning tostep 2 of the outer loop. When the PISO loop is finished, the algorithm can performanother outer iteration, depending on the value of the variable nOuterCorrectors.We see that the PIMPLE algorithm is very similar to the SIMPLE algorithm. For amore detailed description of the PISO loop we refer to [8, 17].

3.3.3 rhoCentralFoam

The solver rhoCentralFoam is a density based solver for high speed compressibleflow [12]. Similarly to the SIMPLE based algorithms it is also suitable for implicittime-stepping. But whereas SIMPLE based algorithms use a pressure correction toallow this, the rhoCentralFoam solver uses an operator-splitting approach involvinga diffusion correction to allow this. This implies that in rhoCentralFoam for thevelocity and energy first an estimate based on the inviscid fluxes alone is calculated,which is then corrected by a diffusion correction. We will now present a more detaileddescription based on the article written by the programmers who have implementedrhoCentralFoam in OpenFOAM [9].

First, the values of u and ρ of the previous time step are used to compute thedensity-weighted field u= uρ. By using the Van Leer limiter and the value T fromthe previous time step, the cell face values u f ±, ρ f ± and Tf ± for the convective termsare then computed by interpolation for inward and outward flux separately. Here,the subscripts f + and f − denote the flow in the direction +S f and −S f respectively.The vector S f is defined as the face area vector normal to the boundary face surface,pointing out of the owner cell with a magnitude equal to the area of the boundaryface. After this, the remaining variables are calculated via the expressions as denotedin table 3.1.

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u f ± =u f ±

ρ f ±φ f ± = S f · u f ±

p f ± = ρ f ±RTf ± c f ± =Æ

γRTf ±

Texp = µ[(∇u)T − 23 tr(∇u)I]

Table 3.1: The variable expressions for the cell face values by inwardand outward flux [9].

Using these variables and updated µ and k, the continuity equation is solved for ρ.

Next, the inviscid convective momentum equation,

∂ u∂ t

I+∇ · [uu] +∇p = 0, (3.15)

is solved for u. This step is also known as the inviscid momentum prediction. Byusing this new ρ and u we then update u after which it is corrected by solving thediffusive velocity correction equation,

∂ (ρu)∂ t

V−∇ · (µ∇u)−∇ · (Texp) = 0, (3.16)

for u, where Texp, the part of the viscous stress tensor S containing inter-componentcoupling, is expressed explicitly in terms of the old value for u and the Laplacian term∇ · (µ∇u), the part of the viscous stress tensor without inter-component coupling istreated implicitly.Now, only the energy, temperature and pressure have to be calculated. This is doneby first solving the equation,

∂ E∂ t

I

+∇ · [u(E + p)] +∇ · (T · u) = 0, (3.17)

for E. We can then compute the temperature T from the equation,

T =1cv

Eρ−|u|2

2

, (3.18)

with our updated ρ, E and u. After that, the temperature T is corrected by solvingthe diffusive temperature correction,

∂ (ρcv T )∂ t

V−∇ · (k∇T ) = 0, (3.19)

for T . Lastly the pressure p is updated by p = ρRT .

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3.4 Turbulence models

To take into account that the flow is subject to turbulence, we used Reynolds-AveragedNavier-Stokes (RANS) turbulence models. For these models the Reynolds-AveragedNavier-Stokes equations are used which are slightly different from the Navier-Stokesequations we presented in chapter 2. For incompressible flow, the equations areobtained by decomposing the expressions for the velocity, pressure and viscous stresstensor into a mean component and a fluctuating component such that,

ui = ⟨ui⟩+ u′i, (3.20)

p = ⟨p⟩+ p′, (3.21)

Si j = ⟨Si j⟩+ S′i j. (3.22)

Here ⟨ui⟩, ⟨p⟩ and ⟨Si j⟩ denote the time averages defined as,

⟨ui⟩=1T

∫ t+T

t

ui(t′ )dt ′, (3.23)

for ⟨ui⟩ and defined in a similar way for ⟨p⟩ and ⟨Si j⟩. The variables u′i, p′ and S′i jdenote the fluctuating parts. This technique is called the Reynolds Decomposition[15]. When we substitute these expressions into the Navier-Stokes equations fromchapter 2 and then average the equations, an additional term arises in the momen-tum equations. This term contains the time average of the product of the velocityfluctuations, known as the Reynolds Stress,

Ri j = ⟨u′iu′j⟩. (3.24)

Because of the new unknown Reynolds Stress there are not enough equations todetermine all variables and therefore the problem is not closed anymore. Turbulencemodels provide closure to this problem by using additional equations based on exper-iments and assumptions to calculate the Reynolds Stress.

For compressible flow however, we have to take into account that the density isvariable as well. By applying the Reynolds Decomposition to the density and followingthe same procedure as before we would obtain a more difficult problem due to thearising correlation of ρ′ and u′i in the continuity equation and the triple correlationof ρ′, u′i and u′j in the momentum equation [18]. This problem is simplified byusing mean velocity components based on the mass-averaged values opposed to thetime-averaged values, also known as Favre-Averaging. In this case the velocity isexpressed as,

18

ui = ⟨ui⟩F + u′′i , (3.25)

where ⟨ ⟩F denotes the Favre-average given by,

⟨ui⟩F =1ρ

limT→∞

∫ t+T

t

ρ(x ,τ)ui(x ,τ)dτ, (3.26)

and where ρ denotes the time average. When we now substitute the Favre decom-positions of the velocity, specific internal energy, temperature and specific enthalpy,together with the Reynolds decompositions of the pressure, density and heat fluxvector into the conservation equations, we obtain the Favre-averaged equations [18].The Favre-averaged continuity and momentum equations differ from the laminarequations only by the Favre-Averaged Reynolds-Stress tensor,

τi j = ρu′′i u′′j , (3.27)

where u′′i and u′′j denote the fluctuating parts of the Favre decomposition.

3.4.1 The Realizable k-ε model

The realizable k-ε model uses the Turbulence Viscosity Hypothesis coined by Boussi-nesq in 1877, to provide closure to turbulence problem [15]. This hypothesis is anal-ogous to the assumption of the viscous stress relation for a Newtonian fluid, whichwe imposed earlier by employing equation 2.4. The Turbulence Viscosity Hypothesisstates that the Reynolds-stress-anisotropy tensor ai j defined as ai j ≡ ⟨uiu j⟩−

23 kδi j, is

linearly related to the mean rate-of-strain tensor S i j via the turbulent eddy viscosityνT as,

⟨uiu j⟩ −23

kδi j = −νT

∂ ⟨Ui⟩∂ x j

+∂ ⟨U j⟩∂ x i

. (3.28)

The realizable k-ε model then calculates the Reynolds stress by the relation,

νT = Cµk2

ε, (3.29)

where k and ε represent the turbulent kinetic energy and the turbulent dissipationrate respectively, which are calculated from two transport equations for each timestep. In contrary to the standard k-ε model where Cµ is a constant, in the Realizablek-εmodel Cµ is a variable and defined by the expressions as listed in Table 3.2, whereΩi j is the mean rate-of-rotation tensor viewed in a rotating reference frame with theangular velocity ωk [16].

19

Cµ = 1A0+As

kU∗ε

Ωi j = Ωi j − 2εi jkωk

U∗ ≡q

Si jSi j + Ωi jΩi j Ωi j = Ωi j − εi jkωk

Table 3.2: The expressions for Cµ.

Furthermore, the coefficients A0 and As are defined as in table 3.3.

A0 = 4.0 W =Si jS jkSki

S3

As =p

6cosφ S =Æ

Si jSi j

φ =13

cos−1(p

6W ) Si j =12

∂ u j

∂ x i+∂ ui

∂ x j

Table 3.3: The coefficients for A0 and As in Cµ [16].

The two additional equations of the k-ε model to close the problem are thetransport equations for the variables k and ε [3],

∂

∂ t(ρk) +

∂

∂ x j(ρku j) =

∂

∂ x j

µ+µt

σk

∂ k∂ x j

+ Pk + Pb −ρε− YM + Sk, (3.30)

∂

∂ t(ρε) +

∂

∂ x j(ρεu j) =

∂

∂ x j

µ+µt

σε

∂ ε

∂ x j

+ρ C1Sε−ρ C2ε2

k+pνε

+ C1εε

kC3εPb + Sε, (3.31)

where µt = ρνt , Pk represents the generation of turbulence kinetic energy due tothe mean velocity gradients, Pb is the generation of turbulence kinetic energy due tobuoyancy, Ym represents the contribution of the fluctuating dilatation in compressibleturbulence to the overall dissipation rate, Sk and Sε are user-defined source terms andC3e represents how much ε is affected by the buoyancy [10]. The other coefficientsfor the transport equations 3.30 and 3.31 can be found in table 3.4.

20

C1 = max

0.43,η

η+ 5

σε = 1.2

η =Æ

2Si jSi jkε

σk = 1.0

C1ε = 1.44 C2 = 1.9

Table 3.4: The coefficients for the transport equations of k and ε [16].

3.4.2 k-ω SST

The Shear Stress Transport (SST) k-ω turbulence model combines the k-ω modelwith the k-ε model such that k-ω is used in the inner region of the boundary layerand the k-ε in the free stream region. We will present the standard k-ω SST model asdescribed by NASA in [1], which is based on the article [13] by F.R. Menter in whichhe presented the model.

The turbulent viscosity is calculated as follows,

νt =a1k

max(a1ω,ΩF2), (3.32)

where,

Ω=Æ

2Wi jWi j,

Wi j =12

∂ ui

∂ x j−∂ u j

∂ x i

.

The transport equations for k and ω are,

∂ (ρk)∂ t

+∂ (ρu jk)

∂ x j= P − β∗ρωk+

∂

∂ x j

(µ+σkµt)∂ k∂ x j

, (3.33)

∂ (ρω)∂ t

+∂ (ρu jω)

∂ x j=γ

νtP − βρω2 +

∂

∂ x j

(µ+σωµt)∂ω

∂ x j

+

2(1− F1)ρσω2

ω

∂ k∂ x j

∂ω

∂ x j. (3.34)

21

With the following expression for P,

P = τi j∂ ui

∂ x j, (3.35)

τi j = µt

2Si j −23∂ uk

∂ xkδi j

−23ρkδi j, (3.36)

Si j =12

∂ u j

∂ x i+∂ ui

∂ x j

. (3.37)

The remaining variable expressions and constants can be found in table 3.5.

F1 = tanh ((arg1)4) F2 = tanh ((arg2)

2)

arg1 = min

max

p

kβ∗ωd

,500ν

d2ω

,4ρσω2 kC Dkωd2

arg2 = max

2

p

kβ∗ωd

,500ν

d2ω

C Dkω = max

2ρσω21ω∂ k∂ x j

∂ω∂ x j

, 10−20

α1 = 0.31

k = 0.41 β∗ = 0.09

γ1 =β1

β∗−σω1k2

p

β∗γ2 =

β2

β∗−σω2k2

p

β∗

σk1 = 0.85 σk2 = 1.0

σω1 = 0.5 σω2 = 0.856

β1 = 0.075 β2 = 0.0828

Table 3.5: Expressions and constants for the standard SST k-ω modelas presented by NASA [1].

The constants in table 3.5 that appear with subscript 1 and 2 are used in the equationswith the blending function F1. For example for β we have,

β = F1β1 + (1− F1)β2. (3.38)

22

3.5 Geometry and boundary conditions

y

x

Figure 3.3: The geometry we meshed and its position within the bearing(not on scale).

Number of orifices Bearing clearance (µm) Orifice radius (µm) Bearing radius (mm)

8 4 50 25

Table 3.6: The specifications and dimensions of the modeled air bearing

In the simulations we have carried out, we considered a vertically and horizontallystationary bearing floating at a constant height of 4µm. As illustrated in Figure 3.3,the mesh surrounds only one-eighth of the bearing with dimensions as listed in table3.6. Because of the symmetric shape of the bearing this is sufficient to determine theflow in the entire bearing clearance.

The mesh has four types of boundaries, namely the inlet at the top of the nozzle,the outlet at the edge of the bearing, the two symmetry planes on the clockwise andcounterclockwise sides and the solid walls consisting of the surface above which thebearing floats and the bearing itself.

As suggested in most of the literature on pressure driven flow simulations, wehave used an initial flow consisting of a fluid at rest with a uniform pressure. Duringthe first 5 × 10−5 s, we gradually increased the pressure difference by enhancingthe supply pressure to 6 bar and reducing the atmospheric back pressure to 1 bar.By gradually introducing this pressure difference, it is more likely that the flow willdevelop stably. At first we used an initial uniform pressure of 1.7 bar, but later wechanged it to 5 bar for two reasons. Firstly, based on results of simulations on thegrooved bearing in earlier research [6], we expected that a uniform pressure of 5 bar

23

is closer to the final converged situation than a uniform pressure of 1.7 bar. Secondly,since most of the instabilities we encountered originated near the inlet, we expectedit to be better to keep the inlet pressure more steady.

The majority of the simulations were executed with the boundary conditions asshown in table 3.7, but we have also tried others as described in Appendix A. Forexample for the inlet velocity, we have also used fixedValue and zeroGradientas well, and for the inlet pressure we have tried totalPressure too.

Variable Inlet Outlet Symmetry walls Solid walls

u pressureInlet-Velocity

zeroGradient slip fixedValue(0,0,0)

p fixedValue(Table)

fixedValue(Table)

zeroGradient zeroGradient

T fixedValue(293)

zeroGradient zeroGradient zeroGradient

k fixedValue(1× 10−6)

zeroGradient zeroGradient compressible::kqRWallFunctionvalue: 1 ×10−12

ε fixedValue(1)

zeroGradient zeroGradient compressible::epsilonWallFunctionvalue: 500000

ω fixedValue(1)

zeroGradient zeroGradient compressible::omegaWallFunctionvalue: 500000

αt zeroGradient zeroGradient zeroGradient compressible::alphatWallFunctionvalue: 0

µt zeroGradient zeroGradient zeroGradient compressible::mutkWallFunctionvalue: 0

Table 3.7: The boundary conditions for all patches.

3.6 Schemes and Transport model

In order to perform a simulation, the user first has to define the thermophysicalproperties in the file thermophysicalProperties. In listing 3.1 an overview ofthe settings that we have used for the thermophysical properties of the fluid can befound.

24

thermoType

20 type hePsiThermo;mixture pureMixture;transport sutherland;specie specie;thermo eConst;

25 equationOfState perfectGas;energy sensibleInternalEnergy;

30 mixture

specie

nMoles 1;35 molWeight 28.96;

thermodynamics

40 Cp 1004.5; // specific heat CpHf 2.544e+06; // heat of fusionCv 717; // specific heat Cv

45 transport

As 1.458e-06; // sutherland coefficientTs 110.4; // sutherland temperature

50

Listing 3.1: thermophysicalProperties line 18 to 50.

The settings we used, imply that we relate the pressure to the temperaturethrough the equation of state of a perfect gas and that we model the thermodynamicalbehaviour by a constant specific heat cp-model with evaluation of the internal energye and entropy s without considering the heat of formation. Furthermore, we haveused the Sutherland transport model, which relates the dynamic viscosity µ and thetemperature T via the expression,

µ=As

pT

1+Ts

T

, (3.39)

with the parameters As and Ts set as shown in listing 3.1.

25

3.7 Postprocessing

To process the results of our simulations, we have used the open source data analysisand visualisation application paraView combined with the OpenFOAM reader moduleas supplied by OpenFOAM. With paraView we could for example investigate thepressure distribution and velocity profiles throughout the entire mesh. With thesetools we inspected whether the results we received were realistic and whether theywere converging or not. We also used it extensively to determine our strategy toimprove future simulations.

26

Chapter 4

Results

4.1 Adapting the mesh

The first Mesh

(a) A close-up of the nozzle. (b) A close-up of the cross-section of thebearing with the y, z-plane.

Figure 4.1: The first mesh of the bearing with eight orifices.

For the first simulation, we used a mesh with a relatively uniform cell distribution,containing 814.464 cells of which 99.62% is hexahedral. Figure 4.1 contains twopictures of the mesh. The nozzle is octagonal instead of round, to make the meshingprocedure easier.

For this simulation we used the settings that led to good results in the past whenSchut performed simulations of a grooved air bearing. For those simulations theyused the solver sonicFoam together with the realizable k−ε turbulence model. Thegrooved bearing that they modeled is comparable to our bearing in dimensions, butthe design is different. It has only three orifices instead of eight, it contains recesses,also known as pockets, around the orifices and it has anchor shaped distribution

27

grooves from the orifices to about three-quarter of the bearing, as can be seen inFigure 4.2. Furthermore, in their simulations, only bearing clearances of 9µm andlarger were considered.

(a) A close-up of the air chamber above thenozzle and the recess around the orifice.

(b) An overview of the en-tire mesh.

Figure 4.2: The mesh of one-third of the grooved bearing, used by thecompany in earlier research.

The first simulation on the first mesh with the settings as described above crashedat t = 201.60×10−5 s with a final maximum velocity of 635 ms−1 and a final maximumpressure of 17.4 bar. In Figure 4.3 it can be seen that there is a pressure accumulationmoving downward through the nozzle, which then collapses back towards the inletafter which the simulation crashed. The pressure in the nozzle reaches values of over18 bar. Beyond the orifice however, very little happens. Only in the region near theoutlet a little activity can be seen.

#ID Solver Turbulence Model Progress Latest Time (×10−5 s) vmax (m s−1) pmax (bar)

5 sonicFoam Realizable k− ε Crash 201.60 635 17.4

Table 4.1: The results of the first simulation on the first mesh.

28

Figure 4.3: The pressure distribution over time in the y, z-plane. Simu-lation #5.

Second Mesh

The second mesh that we used is a finer mesh, see Figure 4.4. This mesh contains7.756.848 cells, of which 99.85% is hexahedral. This mesh has a lot more cells inthe nozzle than the previous mesh, which can be seen by comparing Figure 4.1a withFigure 4.4a.

(a) A close-up of the top of the nozzle. (b) A close-up of a cross-section of themesh with the y, z-plane.

Figure 4.4: The second mesh of the bearing with eight orifices.

The two simulations that we ran with the solver rhoSimpleFoam both show a

29

pressure distribution with multiple stationary circular pressure depressions aroundthe orifice. These depressions range from strong to invisible as you move away fromthe orifice, as can be seen in the top-view in Figure 4.5a and in a cross-section inFigure 4.5b.

(a) A top-view of the nozzle and the sur-rounding pressure depressions.

(b) The pressure in the y, z-plane.

(c) The normalised velocity vectors in thetransition from the orifice to the gap.

(d) The velocity magnitude in the inlet.

Figure 4.5: The results of the second mesh. Simulation #6.

Furthermore, the flow exits the orifice in multiple strong streams between whichthe flow is reversed, as indicated by the yellow arrows that represent the normalisedvelocity vectors in Figure 4.6.

The results of the simulation with sonicFoam barely show any movement norchange in pressure, except for in small regions near the inlet and outlet. In Fig-ure 4.7 and Figure 4.8 one can see that the fluid underneath the bearing is nearlystationary and that the pressure remains at the initial value of 1.7 bar almost every-where. Because this did not seem to change with time we pauzed simulation #8 att = 6.3× 10−5 s.

30

Figure 4.6: A top-view of the velocity profile. The yellow arrows repre-sent the normalised velocity vectors. Simulation #6.

#ID Solver Turbulence Model Progress Latest Time (×10−5 s) vmax (ms−1) pmax (bar)

6 rhoSimpleFoam Realizable k− ε Crash 1.6 702 2.14

7 rhoSimpleFoam Laminar Crash 1.5 646 2.06

8 sonicFoam Realizable k− ε Pauzed 6.3 365 6.02

Table 4.2: The results of the second mesh.

(a) The pressure distribution in a cross-section with the y, z-plane.

(b) A top view of the pressure distributionclose to the outlet.


31

(a) The velocity in a cross-section with they, z-plane.

(b) A top view of the velocity close to theoutlet.


Third Mesh

The final mesh that we have used for all simulations on this bearing from this pointon, contains 1.150.044 cells of which 99.74% is hexahedral. Because our geometrycontains a lot of thin regions, we could reduce the amount of cells without losing alot of accuracy. This is done by elongating the cells in the flow direction in regionswhere the flow is mostly laminar. To do so, we have used four gradients as visiblein Figure 4.9. Firstly, there is a vertical gradient in the nozzle, since the flow in thetop of the nozzle is irrelevant to the company compared to the flow in the bearingclearance.

(a) A close-up of the nozzle. (b) A cross-section of the nozzle with they, z-plane.

Figure 4.9: The third mesh of the bearing with eight orifices.

Secondly, there are gradients in the bearing clearance in the x and y direction,because we expect the flow to laminarize as it gets farther away from the orifice.Lastly, there is a vertical gradient in the z direction in the bearing clearance to make

32

(a) The top view of the final pressure dis-tribution.

(b) The pressure versus the y coordinate.

Figure 4.10: The pressure in simulation #10.

(a) The normalised velocity vectors in thetransition from the orifice to the gap.

(b) The velocity magnitude in the inlet.

Figure 4.11: The velocity in simulation #10.

the cells near the two solid walls more fine. Furthermore, this mesh has a squareorifice to make the meshing procedure easier.

We executed two simulations of which the behaviour is almost similar to the resultsof the simulations we performed with the second mesh. Similarly to simulation #8,the simulation with sonicFoam barely initiated a flow so we pauzed the simulation.The simulation with rhoSimpleFoam showed pressure depressions around the orificeagain. This time however, the pressure depressions were not static but they weremoving away from the orifice, similarly to a sound wave. Furthermore, the depressionswere divided into four directions, and there were more visible depressions than inthe previous simulations, as illustrated in Figure 4.10a. Lastly, the velocity profile inthe nozzle showed a higher velocity near the walls and corners, as can be seen inFigure 4.11b.

33

#ID Solver TurbulenceModel

Progress Latest Time(×10−5 s)

vmax (ms−1) pmax (bar)

9 sonicFoam Realizable k− ε Pauzed 22.00 335 6

10 rhoSimpleFoam Laminar Crash 20.10 1090 6.07

Table 4.3: The results of the third mesh.

4.2 Changing several settings

4.2.1 Initial pressure

After trying different meshes, we have experimented with some adjustments in thesettings such as changing the boundary conditions, the under-relaxation factors, theinitial uniform pressure value and the time-step size. Here we will only presentthe results of changing the initial pressure and the time-step size. The results ofchanging the boundary conditions and the under-relaxation factors are described inAppendix A. We must state however that from this point on, based on experiences ofOpenFOAM users online and based on the simulations we performed with differentboundary conditions as described in Appendix A, we continued with the inlet velocityboundary condition pressureInletVelocity instead of zeroGradient.

As mentioned in section 3.5, at some point we changed the initial uniform pressurefrom 1.7 bar to 5 bar. In the simulation with rhoSimpleFoam we still encounteredoscillations in the pressure, but this time they were originating from the outlet aswell, instead of only from the orifice. This can be seen in Figure 4.12. The behaviourof the sonicFoam solver was not noticeably different from before. There was stillvery little activity visible.

(a) The pressure. (b) The velocity.

Figure 4.12: The top view of a cross-section parallel to the x , y-plane.Simulation #19.

34

From this point on, we also considered the solver rhoCentralFoam which hasshown good performance in treating shock waves [12]. With an initial pressure of5 bar and no turbulence model, the flow behaviour was the same as what we haveseen from the sonicFoam solver so far. There was barely any activity except for insmall regions near the inlet and outlet.


Comment Progress Latest Time(×10−5 s)

vmax (m s−1) pmax (bar)

19 rhoSimpleFoam Laminar p0 = 5 barregular

Crash 1.64 357 5.08

23 sonicFoam Realizablek− ε

p0 = 5 barregular

Pauzed 34.10 367 6

31 rhoCentralFoam Laminar p0 = 5 barregular

Pauzed 10.00 121 6

Table 4.4: The results of changing the initial pressure to 5 bar.

4.2.2 Adjusting the time-step size

In order to reduce the oscillations we encountered with the solver rhoSimpleFoam,we have also carried out two simulations with smaller time-step sizes. By usinga smaller time-step, the simulation has more iterations to perform between eachincrease of pressure difference, which happens every 5× 10−7 s during the pressurebuild-up. First we executed simulation #15, which is the same as simulation #12but with a time-step size of 1× 10−8 s instead of 2× 10−7 s. When we pauzed thesimulation at t = 3.60× 10−5 s we noticed the oscillations again, which did not showany sign of damping.

Later on when we were using a mesh of a different bearing type (described inmore detail in section 4.4), we also performed a simulation with a smaller time-stepsize to compare. Instead of a time-step size of 5×10−8 s we used 1×10−8 s. Becauseboth simulations were started from the same initial condition at t = 5× 10−5 s, wecan see that this simulation crashed after exactly the same amount of time-stepswith exactly the same final maximum velocity and final maximum and minimumpressure. When analyzing the flow with ParaView we could again see the oscillationsoriginating from the orifice and from the outlet.

35

#ID Turbulence Model Time-step Progress Latest Time (×10−5 s) vmax (m s−1) pmax (bar)

15 Laminar 1× 10−8 Pauzed 3.60 1060 5.76

59 k−ω SST 5× 10−8 Crash 5.80 1630 6

60 k−ω SST 1× 10−8 Crash 5.16 1630 6

Table 4.5: The results of the rhoSimpleFoam solver with smaller time-step sizes.

4.3 Adjusting the viscosity

The last strategy we used to eliminate the instabilities that we encountered whenusing rhoSimpleFoam, was to adjust the dynamic viscosity. We manually altered thedynamic viscosity by changing the Sutherland coefficient As such that we obtained astable flow. Then, we gradually lowered it back to the realistic value while using eachresulting flow as the initial condition for the next simulation. For our first simulationwe changed the coefficient As from 1.458× 10−6 to 1.458× 10−3. As mentioned inSection 3.6 the Sutherland transport model relates the dynamic viscosity µ to thetemperature T by,

µ=As

pT

1+Ts

T

, (4.1)

where Ts denotes the Sutherland temperature and As denotes the Sutherland Coeffi-cient. This means that we increased the dynamic viscosity by a factor of 1000. Thissimulation converged very fast, even before the pressure build-up was finished. Foran initial pressure of 1.7 bar as well as for an initial pressure of 5 bar, there were nooscillations in neither the pressure nor the velocity, as demonstrated in Figure 4.13band Figure 4.13c.

36

(a) The y-axis. (b) A plot of the pressure versus the posi-tion on the y-axis.

(c) The top view of the pressure distributionwith 20 linear scaled contours.

(d) The normalised velocity vectors in thetransition from the nozzle to the gap.

Figure 4.13: The results of simulation #16.

Next, we did the same simulation but with a Sutherland coefficient of 1.458×10−4,which corresponds to a scaling factor of 100. After pausing the simulations at8.10× 10−5 s there were still oscillations which did not seem to subside. However,when we used the converged solution of the simulations with a scaling factor of 1000as the initial state, the simulation converged at t = 32.34× 10−5 s. In Figure 4.14the pressure, velocity and speed of sound versus the y coordinate are plotted. Tocalculate the speed of sound we used the approximate relation of the speed of soundcair to the temperature T in dry air given by,

cair = 331.3

√

√

1+T

273.15. (4.2)

We can see that in contrary to the pressure distribution in Figure 4.13b, this distribu-tion has a depression near the orifice.

After that, we used the converged solution of simulation #26 as an initial stateand lowered the scaling factor from 100 to 10. After 199.34× 10−5 s however, the

37

Figure 4.14: A plot of the pressure (black), velocity (blue) and speed ofsound (red) versus the y coordinate for simulation #26.

simulation still showed some oscillations which did not seem to subside. Becauselowering the Sutherland coefficient from 1.458×10−4 to 1.458×10−5 did not result ina stable flow, we performed a simulation with As = 7×10−5. Although this simulationdid not converge enough to satisfy the predefined tolerance values, the pressuredistribution seemed reasonably stable at t = 96.34 s.

For the solver rhoSimpleFoam we executed this viscosity simulation sequencewith the k−ω SST turbulence model as well. The simulation with rhoSimpleFoamand k −ω SST was able to converge with a viscosity scaling factor of 100 from astationary initial flow and from the final solution of simulation #22 as well. Both ofthese simulations converged towards a pressure distribution with a slightly differentpressure depression next to the orifice, as can be seen in Figure 4.15.

When we continued from the final state of simulation #32 with a scaling factorof 10 however, the simulation showed oscillations and did not show any sign ofconvergence at t = 140.04× 10−5 s. Also for a Sutherland coefficient of 7× 10−5,the simulation did not crash but the oscillations did not seem to be damping either.When we continued from #32 with a scaling factor of 1 the simulation crashed beforethe first write interval was completed.

38

(a) Simulation #28. (b) Simulation #32.

Figure 4.15: A close-up of the plots of the pressure versus the y position.

#ID Initial state As Progress Latest Time(×10−5 s)


16 1.7 bar 1.458× 10−3 Converged 4.42 287 6.02

17 1.7 bar 1.458× 10−4 Pauzed 8.10 1060 6.08

22 5 bar 1.458× 10−3 Converged 5.04 289 6.07

26 22 1.458× 10−4 Converged 32.34 703 6.11

37 26 7× 10−5 Pauze 96.34 772 6.09

33 26 1.458× 10−5 Pauze 199.34 1070 6.07

Table 4.6: The results of rhoSimpleFoam without a turbulence modeland with gradually lowering the Sutherland Coefficient to therealistic value.



28 5 bar 1.458× 10−4 Converged 25.26 714 6.12

32 22 1.458× 10−4 Pauzed 19.04 723 6.13

49 28 7× 10−5 Pauzed 70.26 822 6.09

45 32 1.458× 10−5 Pauzed 140.04 1030 6.08

44 32 1.458× 10−6 Crashed 19.32 - -

Table 4.7: The results of rhoSimpleFoam with the k − ω SST tur-bulence model and with gradually lowering the SutherlandCoefficient to the realistic value.

39

(a) The pressure in a point under-neath the nozzle.

(b) The velocity in a point underneaththe nozzle.

(c) The pressure in a point next tothe nozzle.

(d) The velocity in a point next to thenozzle.

Figure 4.16: The pressure and velocity values over time in two arbitrarypoints close to the orifice. Simulation #34.

Since the solver sonicFoam seemed to have a more stable behaviour, we used itto follow up on the final state of simulation #16 and #37 as listed in Table 4.8. Theresulting flow contained a lot of pressure waves originating from the nozzle. Althoughmost of these swells eventually faded out, at t = 990.32× 10−5 s the simulation stillwas not converged. In Figure 4.16 we can see that the oscillations seem to be damping.

Subsequently, we continued from the final state of simulation #34 with a Suther-land coefficient of 1.458× 10−4. This flow did not show any heavy oscillations andsteadily developed a pressure depression near the orifice, similarly to the previoussimulations with a viscosity scaling factor of 100. Although the flow was not con-verged yet at t = 1490.32 × 10−5 s, it seemed reasonably stable with a pressuredistribution as shown in Figure 4.17.

40

Figure 4.17: A plot of the pressure distribution along the y-axis ofsimulation #38.

Lastly, we used Sutherland coefficients of 1.458 × 10−5 and 1.458 × 10−6 incombination with the solvers rhoCentralFoam and sonicFoam, with the finalsolutions from simulations #26 and #37 as initial flows as listed in Table 4.8 andTable 4.9. These four simulations showed very similar results. At first the pressuredistribution contained some instabilities in the form of pressure oscillations originatingfrom the orifice with relatively high pressure values. Later on, these instabilitiesfaded out and all four pressure distributions converged towards a distribution asplotted in Figure 4.18. The pressure underneath the nozzle is at 9 bar significantlyhigher than the inlet pressure of 6 bar.

(a) The total plot. (b) A close-up of the area underneath thenozzle.

Figure 4.18: Two plots of the pressure along the y-axis of simulation#40.

41



34 16 1.458× 10−3 Converged 990.32 209 6.29

38 34 1.458× 10−4 Pauzed 1490.32 428 6.3

43 37 1.458× 10−5 Pauzed 140.34 599 8.58

41 37 1.458× 10−6 Pauzed 164.34 652 8.57

Table 4.8: The results of sonicFoam with the Realizable k−ε turbulencemodel and with gradually lowering the Sutherland Coefficientto the realistic value.

#ID Initial state TurbulenceModel

As Progress Latest Time(×10−5 s)


48 28 Laminar 7× 10−5 Pauzed 140.26 589 9.34

42 37 k−ω SST 1.458× 10−5 Pauzed 148.34 616 8.51

40 26 k−ω SST 1.458× 10−6 Pauzed 799.34 651 24

Table 4.9: The results from rhoCentralFoam and gradually loweringthe Sutherland Coefficient to the realistic value.

4.4 Comparison to OpenFOAM 1.7 and a different ge-ometry

Since the realistic results that were obtained in previous research [6] on modellingair bearings in OpenFOAM were obtained with OpenFOAM 1.7 and on a differentmesh, we decided to do a comparison test between the OpenFOAM versions 1.7 and2.3, and between the two meshes.

First, we investigated the performance of the solver rhoSimpleFoam in combi-nation with the k −ω SST turbulence model on the new mesh in OpenFOAM 2.3.This new mesh is a two dimensional mesh of a simple bearing with one orifice in thecenter, illustrated as type 2 in Figure 1.1. Since this bearing is radially symmetrical,it suffices to mesh only one radial slice of the bearing. Furthermore, this bearingis different from our bearing because it has a rounded transition from the nozzleto the bottom surface of the bearing, as illustrated in Figure 4.19, and the bearingclearance is 9µm. Again, we started out with an initial viscosity that is a factor1000 larger than the realistic value. This simulation converged in 5× 10−5 s. Whenwe adjusted the Sutherland coefficient such that the viscosity was only 100 times

42

Figure 4.19: A close-up of the transition from the nozzle to the gap inthe two dimensional mesh.

larger than the realistic value, the simulation crashed whether we used the previouslyconverged solution as the initial flow or not. The simulations performed with aSutherland coefficient of 1.458× 10−5 and 1.458× 10−6 crashed as well. After thatwe experimented with the option transonic and several limiters. The results ofthis can be found in Appendix A.

#ID Initialstate

Solver As Progress Latest Time(×10−5 s)


51 5 bar rhoSimpleFoam 1.458× 10−3 Converged 5.00 327 6

50 5 bar rhoSimpleFoam 1.458× 10−4 Crashed 6.00 1460 6

52 51 rhoSimpleFoam 1.458× 10−4 Crashed 8.00 1520 6

Table 4.10: Results from rhoSimpleFoam with the k −ω SST turbu-lence model on the 2D mesh of the simple type bearing inOF 2.3.

While the solver sonicFoam obtained good results in OpenFOAM 1.7 with thismesh, in OpenFOAM 2.3 the flow was not solved successfully even though exactlythe same settings were employed. Furthermore, using one of the converged so-lutions obtained from rhoSimpleFoam as the initial flow for the sonicFoam andrhoCentralFoam solvers, yielded interesting patterns of shock reflections prior to

43

crashing.

Mesh OpenFOAM 1.7 OpenFOAM 2.32D mesh of simple bearing Good BadMesh 3 of bearing with 8 orifices Bad Bad

Table 4.11: A comparison of the performance of sonicFoam in Open-FOAM 1.7 and OpenFOAM 2.3 and on different meshes.

Lastly, we performed two simulations in OpenFOAM 1.7 on the third mesh ofour bearing with multiple orifices and with exactly the same settings that gave suc-cessful results in the past on the grooved and simple bearing [6]. This simulationresulted in a flow that is comparable to the results of sonicFoam that we describedin previous sections, in the sense that there was barely a flow initiated. The solverrhoCentralFoam gave similar results. The simulations with rhoSimpleFoam allcrashed before the first write interval was reached.

#ID Initialstate

Solver As Progress Latest Time(×10−5 s)

vmax(m s−1)

pmax(bar)

61 5 bar sonicFoam Realizable k− ε Pauzed 300.00 421 6.38

64 5 bar rhoCentralFoam Laminar Crash 29.00 285 5.57

69 46 sonicFoam1 k−ω SST Crashed 4.00 671 5.75

71 46 rhoCentralFoam1 k−ω SST Crashed 3.20 593 5.55

72 46 sonicFoam k−ω SST Pauzed 50.00 528 9.37

68 46 rhoCentralFoam k−ω SST Pauzed 50.00 567 8.63

Table 4.12: Results from sonicFoam and rhoCentralFoam with therealistic Sutherland Coefficient on the 2D mesh of the simpletype bearing in OF 2.3.

1For these simulations we have changed nCorrectors from 1 to 2 and div(φ,ε) from boundedGauss upwind to Gauss upwind, which corresponds to the settings that gave successful results inon the other bearings in OpenFOAM 1.7.

44

#ID Initialstate

Solver TurbulenceModel

As Progress Latest Time(×10−5 s)

vmax(m s−1)

pmax(bar)

74 1.7 bar sonicFoam Realizablek− ε

1.458× 10−6 Pauzed 5 67 6

77 1.7 bar rhoCentralFoam k−ω SST 1.458× 10−6 Crashed 5.00 70 6.00

76 1.7 bar rhoSimpleFoam k−ω SST 1.458× 10−6 Crashed 0 - -



Table 4.13: The simulations performed on mesh #3 in OpenFOAM 1.7.

45

Chapter 5

Conclusion

Meshes

The first mesh that we used, as described in Section 4.1, results in a flow that isdifferent from all other flows that we obtained. We interpreted this difference as theconsequence of a shortage of cells in the nozzle. We think that the flow is not ableto develop properly because this mesh has only four columns of cells in the nozzlewhich are all adjacent to a boundary face with the boundary condition U = (0, 0, 0),as can be seen from Figure 4.1a.

Our next two meshes contain more cells in the nozzle and in combination withthe solver rhoSimpleFoam they result in flows that seem more realistic. Althoughboth meshes result in oscillations that grow stronger until the simulation crashes,the flows are developed in the entire mesh and the velocity profiles, as illustratedin Figure 4.5c and Figure 4.11a, seem plausible. The second mesh contains almostseven times as many cells as the third mesh but does not perform any better. If wecompare the results of simulation #7 and #10 as listed in Table 4.2 and in Table 4.3,we can even say that it performed worse, since it crashed earlier. Furthermore, eventhough the second mesh has more cells, the cells near the orifice are larger. Since thatis the region where we expect the flow to be the most turbulent, the second mesh isnot an efficient mesh. Despite the difference in cell size near the orifice we do notsee any difference in the representation of swirls or separation near the corner whenwe compare Figure 4.5c with Figure 4.11a. Because of these results we conclude thatthe additional cells in the second mesh are redundant and we continued the rest ofthe simulations with the third mesh.

rhoSimpleFoam

The results we obtained with the solver rhoSimpleFoam contain oscillations in thepressure and velocity. When we analyzed the results, we could see that the simulations

47

crashed whenever the oscillations grew to a certain intensity. Therefore we thinkthat the oscillations are closely related to the crashes.

After we switched to the new initial pressure, as described in section 4.2.1, wecould see that the oscillations now originated from the outlet as well. This suggeststhat the oscillations are an indication that the model has difficulties with settling theflux that the changing pressure difference between the inlet and outlet induces. Onthe other hand as mentioned in Section 4.2.1 as well, the results were not improvedby reducing the time-step size. Since this would allow the simulation to performmore iterations between each raise in pressure difference during the build-up phase,it does not seem like the oscillations occur because the pressure difference is built uptoo fast.

The best technique we found to prevent the simulations with rhoSimpleFoamfrom crashing, is to modify the dynamic viscosity through the Sutherland Coefficientas described in Section 4.3. Using a converged solution from a simulation with ahigh dynamic viscosity as the initial flow for a lower dynamic viscosity improved theconvergence. This can be seen by comparing the results of simulation #17 and #26,as listed in Table 4.6. If we compare Figure 2.2 with Figure 4.14, we can see thatour resulting flow corresponds reasonably well with the theory of supersonic flows inconverging and diverging ducts. The shape of our pressure and velocity plots conformroughly to the shape of the pressure and velocity plot in Figure 2.2 for situation d.Also, the point where the velocity exceeds the speed of sound and the fact that themaximum pressure and minimum velocity in our plot are at the same location are inaccordance with Figure 2.2. Furthermore, the occurrence of a pressure depressionin air bearings has been discussed extensively in literature [20, 5, 7, 14]. Thereforewe conclude that the results of these simulations are a decent approximation of thegeneral flow in an air bearing.

sonicFoam and rhoCentralFoam

For both sonicFoam and rhoCentralFoamwe saw very similar behaviour. Initiatingfrom a stationary flow resulted in a flow that barely moved, as described in Section4.1. When we used a viscosity scaling factor of 1 and 10 with a converged solutionas the initial flow, both simulations resulted in an increase in pressure which did notsubside as visible in Figure 4.18. Furthermore we found that sonicFoam performeddifferently in OpenFOAM 1.7 from how it performed in OpenFOAM 2.3 as listedin Table 4.11. When we compared the transport equations for the energy from thesource code, we found multiple differences. These differences might explain why theresults we obtained in OpenFOAM 1.7 are different from the results we obtained inOpenFOAM 2.3.

48

Turbulencemodel

From comparing simulations #6 and #7 we found that the performance of theRealizable k− ε and the Laminar dummy turbulence model is similar. This agreeswith the results obtained in previous research by Schut [6]. Furthermore, fromsimulation #17 and #28 we conclude that the k−ω SST performs better than theLaminar turbulence model, since simulation #28 converges whereas in simulation#17 the oscillations remained.

49

Chapter 6

Future Work

Based on our results we would suggest to start from less extreme conditions andgradually work towards the desired set-up. All successful simulations we have seenduring this research, which are the simulations with a higher viscosity and thesimulations with a larger bearing clearance, have in common that the conditionshave less impact on the velocity. The modified viscosity results in lower velocitiesbecause of a greater viscous drag whereas the larger bearing clearance reduces thelevel of constriction and thereby the intensity of the venturi effect. Therefore werecommend to first investigate whether the bearing can be modelled successfullywith larger bearing clearances or smaller pressure gradients. This may already resultin useful information and otherwise it may produce a good solution to use as theinitial condition for simulations with more challenging conditions.

Furthermore, we encountered a lot of things that we can not explain yet. Forexample why the velocity in the inlet in Figure 4.5d and Figure 4.11b has such anunexpected profile or why simulations #43, #41, #42 and #40 show such highpressure values near the orifice. Lastly and most importantly it might be rewarding toperform experiments with this type of bearing to determine whether the oscillationsthat we encountered are realistic or not. We assumed them to be a malfunction,because they seem to cause the simulations to crash. However because we have notfound results from experiments on air bearings like ours, we cannot exclude that theoscillations are a realistic part of the solution.

51

Bibliography

[1] The menter shear stress transport turbulence model, http://turbmodels.larc.nasa.gov/sst.html.

[2] Openfoam - the open source computational fluid dynamics toolbox, version 2.3.0.,www.openfoam.org.

[3] Realisable k-epsilon model – cfd-wiki, the free cfd reference, http://www.cfd-online.com/Wiki/Realisable_k-epsilon_model.

[4] John D. Anderson, Fundamentals of aerodynamics, second ed., McGraww Hill,1991.

[5] D. Dowson, Laboratory experiments and demonstrations in tribology: 7-externallypressurised air lubricated thrust bearings, Tribology International (1969).

[6] C. Draaijer and J. C. Nauta, Investigation of a cfd procedure for optimization ofthe air gap stability for aerostatic thrust bearings.

[7] Mohamed E. Eleshaky, Cfd investigation of pressure depressions in aerostaticcircular thrust bearings.

[8] Joel H. Ferziger and Milovan Periç, Computational methods for fluid dynamics,second ed., Springer-Verlag, 1999.

[9] Christopher J. Greenshields, Henry G. Weller, Luca Gasparini, and Jason M.Reese, Implementation of semi-discrete, non-staggered central schemes in a colo-cated, polyhedral, finite volume framework, for high-speed viscous flow, Interna-tional Journal for Numerical Methods in Fluids (2009).

[10] R. A. W. M. Henkes and C. J. Hoogendoorn, Scaling of the turbulent naturalconvection flow in a heated square cavity, Journal of Heat Transfer (1994).

[11] T. Holzmann, Openfoam guide - the pimple algorithm in openfoam,http://openfoamwiki.net/index.php/OpenFOAM_guide/The_PIMPLE_algorithm_in_OpenFOAM.

53

[12] Luis F. Gutiérrez Marcantoni, José P. Tamagno, and Sergio A. Elaskar, High speedflow simulation using openfoam, Mecánica Computacional (2012).

[13] F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering appli-cations, American Institute of Aeronautics and Astronautics Journal 32 (1994),no. 8, 1598 – 1605.

[14] H. Mori, A theoretical investigation of pressure depression in externally pressurizedgas-lubricated circular thrust bearings, Journal of Fluids Engineering (1961).

[15] Stephen B. Pope, Turbulent flows, Cambridge University Press, 2000.

[16] Tsan-Hsing Shih, William W. Liou, Aamir Shabbir, Zhigang Yang, and JiangZhu, A new k-ε eddy viscosity model for high reynolds number turbulent flows,Computers & Fluids 24 (1995), no. 3, 227 – 238.

[17] H. K. Versteeg and W. Malalasekera, An introduction to computational fluiddynamics, Longman Scientific & Technical, 1995.

[18] David C. Wilcox, Turbulence modeling for cfd, DCW Industries, Inc., 1993.

[19] Inc. Wolfram Research, Mathematica, Wolfram Research, Inc.

[20] S. Yoshimoto, M. Yamamoto, and K. Toda, Numerical calculations of pressuredistribution in the bearing clearance of circular aerostatic thrust bearings with asingle air supply inlet, Journal of Tribology (2006).

54

Appendix A

Additional results

As mentioned in Section 4.2.1 we performed some additional simulations in whichwe varied the boundary conditions, the under-relaxation factors, the limiters for theflux φ and the option transonic.

Different boundary conditions

Figure A.1: The velocity in a cross-section of the nozzle with the y, z-plane. Simulation #13.

First, we changed the inlet velocity boundary condition from zeroGradient tofixedValue (0, 0,−10) which corresponds to a downward flow of 10 m

s . Combinedwith the rhoSimpleFoam solver, this resulted in a flow which is less structured

I

than the flow that we obtained from rhoSimpleFoam with the zeroGradient inletvelocity boundary condition. This is visible in Figure A.1.

We proceeded by changing the inlet velocity to pressureInletVelocity andsubsequently to pressureInletOutletVelocity. With rhoSimpleFoam, thesetwo boundary conditions showed the same oscillations as we have seen before insimulation #10 in section 4.1. We could see the same outward moving pressuredepressions originating from the edges of the orifice. For the simulation with thesolver sonicFoam as well, changing the boundary condition from zeroGradientto pressureInletVelocity did not have any visible effect on the behaviour. Thefluid was still barely moving except for in small regions near the inlet and outlet.Nevertheless, based on experience from other OpenFOAM users online, we decided touse the pressureInletVelocity boundary condition for the inlet velocity bound-ary from this point on.Besides changes in the velocity boundary conditions, we have also changed theinlet pressure from the fixedValue boundary condition to totalPressure. Un-fortunately we could not analyze the resulting flow since this simulation crashed att = 4× 10−7 s, before the first write interval (the time interval after which the stateis stored in the computer) was reached.


BoundaryCondition

Progress Latest Time(×10−5 s)

vmax(ms−1)

pmax(bar)

11 rhoSimpleFoam Laminar pressureInletOutletVelocity

Pauzed 3.00 902 5

12 rhoSimpleFoam Laminar pressureInletVelocity

Pauzed 2.50 674 3.85

13 rhoSimpleFoam Laminar fixedValue Pauzed 2.6 25 4.36

17b rhoSimpleFoam Laminar totalPressure Crashed 0.04 - -

18 sonicFoam Realizablek− ε

pressureInletVelocity

Pauzed 11.00 208 6

Table A.1: The results of varying the boundary conditions for the inletvelocity and pressure.

Under-relaxation factors

In order to minimize the oscillations that we encountered in the simulations withrhoSimpleFoam, we modified the under-relaxation factors. The exact changes thatwe made are listed in Table A.2. Despite that the simulations lasted noticeably longer,the oscillations were still present. For the simulation with the initial pressure at 1.7bar, lowering the under-relaxation factors did not stabilize the flow sufficiently either.

II


Comment Progress Latest Time(×10−5 s)

vmax(m s−1)

pmax(bar)

14 rhoSimpleFoam Laminar p0 = 1.7 barURF:U : 0.7→ 0.5

Crash 4.50 991 6.05

20 rhoSimpleFoam Laminar p0 = 5 barURF:U : 0.7→ 0.5p : 0.3→ 0.2ρ : 0.05→ 0.025

Crash 2.90 618 5.51

21 rhoSimpleFoam Laminar p0 = 5 barURF:U : 0.7→ 0.3p : 0.3→ 0.1ρ : 0.05→ 0.015

Crash 5.40 1100 6.07

Table A.2: The results of changing the initial pressure to 5 bar andlowering the under-relaxation factors.

Limiters and transonic

We performed three simulations on the two dimensional mesh in which we exper-imented with the setting transonic and two different limiters. When we set theoption transonic to on, this resulted in a very wild flow with velocities of over 6000ms−1. The different limiters for the flux φ did not prevent the rhosimpleFoamsimulations from crashing either.

#ID Initial state Solver As Progress Latest Time(×10−5 s)

vmax m s−1 pmax (bar)

54 5 bar rhoSimpleFoamtransonic

1.458× 10−3 Crashed 2.00 3.55×1037

5.14

58 51 rhoSimpleFoamvanLeer limiter

1.458× 10−4 Crashed 6.64 1020 6.03

59 51 rhoSimpleFoamlinear limiter

1.458× 10−4 Crashed 5.80 1630 6.00

Table A.3: Results from rhoSimpleFoam with the k −ω turbulencemodel on the 2D mesh of the simple type bearing in OF 2.3.

III

a computational fluid dynamics analysis on air bearing...

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