A computational fluid dynamics analysis on air bearing designs

Transcription

1 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

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3 Abstract This report treats the results of a research project on simulating the air flow in aerostatic thrust air bearings with the Open Source Computational Fluid Dynamics (CFD) software package OpenFOAM. The type of air bearing considered in this research is a simple type circular bearing with eight orifices that are equally distributed over a concentric circle. In this report we first present some background information on air bearings, treat the fluid dynamics that is involved and consider the previously conducted research on computational models of air bearings. Next, we will discuss the simulations that we performed where we varied the choice of solvers, boundary conditions, turbulence models and the viscosity parameter. Lastly, we will analyze the results and give suggestions for future work on this subject. We find that manually altering the viscosity via the Sutherland Coefficient A s provides a stable initial flow, from which we can incrementally lower the viscosity back to values closer to the realistic value. Furthermore we find that the solvers sonicfoam and rhocentralfoam show similar behaviour on our meshes and that sonicfoam and rhosimplefoam in Open- FOAM 1.7 perform differently from sonicfoam and rhosimplefoam in OpenFOAM 2.3. Keywords: aerostatic thrust air bearing, OpenFOAM, CFD ii

4 Contents 1 Introduction 1 2 Background Flat aerostatic thrust air bearings Fluid dynamics General conservation principles Supersonic flow Model OpenFoam Mesh Solvers rhosimplefoam sonicfoam rhocentralfoam Turbulence models The Realizable k-ε model k-ω SST Geometry and boundary conditions Schemes and Transport model Postprocessing Results Adapting the mesh Changing several settings Initial pressure Adjusting the time-step size Adjusting the viscosity Comparison to OpenFOAM 1.7 and a different geometry Conclusion 47 iii

5 6 Future Work 51 A Additional results I iv

6 Chapter 1 Introduction This report treats the results of a research project on numerical simulations of aerostatic thrust air bearings as commissioned by the company Schut Geometrical Metrology. Air bearings are bearings that use a thin film of air as a lubricant to reduce the friction between two moving surfaces. The company wishes to use the results from these simulations to improve the bearing design. Ultimately, the bearing has to satisfy certain stability properties required for the implementation in high precision measurement 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 the past, the company has carried out several successful simulations on air bearings of type 2, which contains a single inlet orifice in the center [6]. However, applying the same modelling procedure to the geometry of the bearing of type 1 did not yield any realistic results. Therefore, the goal of this project was to find out how a bearing of type 1 can be simulated correctly. The simulations were carried out in the Open 1

7 Source Computational Fluid Dynamics (CFD) software package OpenFOAM This software provides features for applications ranging from complex turbulent flow involving chemical reactions to solid dynamics and electromagnetics [2]. When simulating fluid flow, it is important to tailor the numerical model to the flow dynamics involved. By making the right assumptions and simplifications, a lot of computation time can be saved without losing too much accuracy. After studying some of the available literature, we found that as the air flow exits the nozzle at the bottom of the bearing, it is subject to an extreme decrease in flow area. This can result in locally supersonic flow, which requires the use of algorithms for compressible fluids. This is presented in Chapter 2, along with a description of the bearing and the governing equations. In reality, inevitable disturbances, such as disturbances in the load that is exerted on the bearing, will affect the height of the air gap. However, in this project we investigated a simplified situation in which the gap height is fixed. We have tried multiple strategies to obtain a stable and realistic simulation, such as changing the boundary conditions, trying different solvers and manually altering the viscosity. The specific settings and procedures of the simulations are explained in Chapter 3. The results from our simulations are presented in Chapter 4 together with an analysis of the results in Chapter 5. We find that running a simulation with an altered viscosity is an effective strategy to obtain a stable initial flow for lower and more realistic viscosities. Furthermore, we find that the solvers sonicfoam and rhosimplefoam in OpenFOAM 1.7 perform differently from sonicfoam and rhosimplefoam in OpenFOAM 2.3. Lastly, we find that the solvers rhocentralfoam and sonicfoam show similar results in nearly all configurations, despite their different nature. Finally, we present some suggestions for future work in Chapter 6. 2

8 Chapter 2 Background 2.1 Flat aerostatic thrust air bearings Air bearings exist in different types, such as aerodynamical air bearings, which use their movement to create a thin high pressure air film between the surfaces, or aerostatic 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 aerostatic air bearing with a single orifice. Source: In Figure 2.1, the cross-section of an aerostatic air bearing with a single orifice is shown. The shape and size of the air bearings considered in this research are comparable to a hockey puck. An external compressor connected to the chamber inside the bearing, applies a supply pressure p s, which is higher than the atmospheric back pressure p a. The pressure difference induces an air flow which exits the chamber via the nozzle and flows into the air gap via the orifice. Finally, as it passes the edge of the bearing, it can escape into the open air. The air bearing considered in this 3

9 research has eight orifices distributed evenly over a concentric circle, comparable to type 1 in Figure Fluid dynamics General conservation principles Because the smallest length scale in this set up, the gap height under the bearing, is 4 µm long, we can regard the gas as a continuum rather than as a collection of discrete particles [15]. This means that we can use arbitrary control volumes to derive conservation equations for mass, momentum and scalar properties such as temperature and energy. By specifying the inflow, outflow, and sources for an arbitrary control volume we obtain equations 2.1, 2.2 and 2.6 for mass, momentum and energy respectively [8]. t Ω ρ dω + S ρv n ds = 0, (2.1) Equation 2.1 represents the conservation of mass, in which ρ represents the density, v the velocity vector and n the normal vector on the surface S of the control volume Ω. It states that the amount of mass in a control volume can only change by a 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 from Newton s second law of motion, d(mv) = f. (2.3) dt The tensor T in equation 2.2 represents the stress tensor and b represents the body forces. The equation states that the amount of momentum in a control volume can change by advection, surface forces (for example pressure, shear stress or surface tension) or body forces (for example gravity or the Coriolis force). In this research we will approximate air as a Newtonian fluid. Therefore, the stress tensor can be written as, T = p + 23 µ div v I + 2µD, (2.4) 4

10 where p is the static pressure, µ is the dynamic viscosity, I is the unit tensor and D is the deformation tensor [8], D = 1 2 grad v + (grad v) T. (2.5) Equation 2.1 and 2.2 together are known as the Navier-Stokes equations. ρh dω + ρhv n ds = k grad T n ds + (v grad p + S : grad v) dω t Ω S S Ω + t Ω p dω. (2.6) Lastly, equation 2.6 represents the conservation of energy, in which h is the enthalpy per unit mass, k is the thermal conductivity defined as, k = µ c p Pr, (2.7) with c p the specific heat at constant pressure and Pr the Prandtl number, T is the absolute temperature (K) and S is the viscous part of the stress tensor S = T + pi [8] Supersonic flow In a lot of the previously conducted research on air bearings the presence of supersonic flow (Ma > 1) and an undesirable pressure depression attributed to a shock wave is mentioned [5, 6, 14]. Since our configuration is comparable to the configurations in those 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 change drastically. Because the highest velocity at which information on a fluid s properties travels is the speed of sound, this information cannot travel upstream in supersonic flow. For the scope of this project it is sufficient to acknowledge the existence of shock waves, in order to analyze the results of our model. More information on shock waves can be found in [4] among many other text books. Pressure depression The undesirable pressure distribution mentioned earlier, is located close to the inlet of the air bearing. This depression has a negative effect on the load capacity and has 5

11 been studied extensively both theoretically and experimentally. Some researchers [20] attribute this depression to the transition from laminar to turbulent flow, but this contradicts the majority of articles on the subject, which show that the depression exists due to shock waves in the bearing clearance [5, 7, 14]. In general, the latter is accepted. This explanation states that the air bearing can be considered as a convergingdiverging duct, since the cross-sectional flow area at the entrance of the gap is smaller than the flow area in the nozzle. From the area-velocity relation da A = (Ma2 1) dv v, (2.8) we can see that the nature of the flow determines the effect of a change in crosssectional flow area. For subsonic flow (Ma < 1), a decrease in area will accelerate the flow and an increase in area will decelerate the flow. For supersonic flow (Ma > 1) a decrease in area will decelerate the flow and an increase in area will accelerate the flow [4]. This explains how a small adjustment in supply pressure can induce a large change in flow behaviour, as can be seen from figure Figure 2.2. If the difference between the supply pressure p s and the atmospheric pressure p a is large enough, the flow may become choked at the entrance of the gap and become supersonic. Figure 2.2: A converging-diverging duct and the possible pressure distributions and velocity profiles. Source: stanford/images/lr_5.3.gif 6

12 In Figure 2.2 the effect of the ratio of p a to p s on the flow in a converging diverging duct is illustrated. The pressure distributions that we encountered most often are of the type d, where the pressure difference is large enough to induce supersonic flow and a compression shock wave exists in the diverging duct that is the bearing clearance. Because the transition from the inlet to the gap is at a right angle, the flow separates from the boundary at high velocities. The resulting separation bubble as illustrated in Figure 2.3 narrows the flow area even more. For a more detailed description of the fluid flow and the pressure depression often encountered in air bearings we refer to [7] and [14]. Figure 2.3: Qualitative representation of the shock/boundary layer interaction in the inlet region [7] 7

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14 Chapter 3 Model 3.1 OpenFoam OpenFOAM is a free open source software package for applications ranging from complex turbulent flow involving chemical reactions to solid dynamics and electromagnetics [2]. In addition to the official software package that is managed by the OpenFOAM Foundation, there are numerous programmers that write extensions for the existing OpenFOAM software themselves. The most of these unofficial extensions are public too, but they have not been checked by the OpenFOAM Foundation. Often, these extensions contribute to new OpenFOAM versions after being processed by the programmers of the OpenFOAM Foundation. The software is being used for commercial as well as academic purposes and a lot of information can be found on internet fora where users discuss their problems and experiences. To perform a simulation in OpenFOAM, the user must create a case directory with at 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 t 0 = 0, contains the boundary and initial conditions for at least all variables that will be used in the calculations. The constant directory contains files that define the mesh as well as files in which the thermophysical and turbulence settings are specified. The system directory contains the files in which the discretization schemes, linear system solvers, parallelisation settings, solver settings and control settings are specified. 9

15 3.2 Mesh The mesh that we used was provided by the commissioning company and was created in Mathematica [19]. It exploits the symmetry of the bearing by covering only 1 8 of the actual bearing geometry, which saves a lot of computation time. The mesh is made of hexahedral cells exclusively because that is best suited for flow simulations of very thin layers, such as our bearing clearance. Furthermore, by starting from a three dimensional structured hexahedral mesh it is less difficult to adopt the explicit mesh description format that OpenFOAM requires. The hexahedral mesh was fitted to the geometry of the bearing by transformations and removals of redundant cells without 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 will describe these three solvers below rhosimplefoam The solver rhosimplefoam is a steady-state Semi-Implicit Method for Pressure Linked Equations (SIMPLE) solver for laminar or turbulent Reynolds-averaged Navier- Stokes (RANS) flow of compressible fluids [2]. It uses the SIMPLE algorithm to solve the coupled equations for velocity and pressure. The SIMPLE algorithm is an iterative algorithm proven to be fairly efficient in solving steady state problems by the use of a predictor-corrector loop and under-relaxation factors [8]. This algorithm is also the basis 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. 10

16 Start with u n 1, ρ n 1 and p n 1 k = 0 m = 0 Linearized momentum equation u m Calculate mass fluxes ṁ m at l cell faces from ρ m 1 and v m n if m < nouter Correctors if k nnonorthogonal Correctors Pressure correction p Correct mass fluxes at cell faces k++ Outer loop Calculate u m from p m m++ Calculate other variables Proceed to next time step n++ 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 loop to calculate the pressure field and the velocity field, using the variable fields from the previous time step as initial values. The iterations of this loop are called outer iterations to distinguish them from inner iterations which are the iterations performed on linear systems with fixed coefficients [8]. These outer iterations continue as long as the pressure and velocity fields do not satisfy the specified residual tolerance and as long as the number of performed iterations m does not exceed the specified value of noutercorrectors [11]. Lastly, all other variable fields, such as temperature and turbulent kinetic energy are calculated. After that, the algorithm proceeds to the next time step. 11

17 We will now follow the derivations and steps as presented in [8] to explain how the outer loop operates. The outer loop consists of three tasks. First, in case m = 0, the pressure and density fields p n 1 and ρ n 1 from the previous time step are used to solve the segregated linearized momentum equations. Else, in case m > 0, the pressure and density fields p m 1 and ρ m 1 from the previous outer iteration are used instead. These linearized momentum equations give a new estimate u m for the velocity field, where the asterisk is used to indicate that this estimate does not necessarily satisfy the continuity equation. The discretized segregated linearized momentum equation for the ith component of the velocity estimate u m at the point P that is solved in this first step of the outer iteration can be written as, A u i P um + i,p l A u iu m = Q m 1 l i,l u i Ω δp m 1. (3.1) δx i Here m is the outer iteration counter and Ω denotes the control volume centered around a cell face, which is not necessarily the same as the volume of the control volume around point P. The coefficients A P and A l depend on the chosen discretization schemes and represent the terms in the momentum equation that contain unknown variable values, in this case u m. The subscript l denotes the index of the neighbouring i cells. The coefficient Q m 1 u i contains all terms of the momentum equation which do not contain unknown variable values and thus are presumed known, except for the pressure 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 a correction of the velocity and density to a correction of the pressure. Equation 3.1 can be rewritten as, Q m 1 u m = u i l A u iu m l i,l i,p A u i P Ω A u i P δp m 1. (3.2) δx i Because the velocity field obtained from equation 3.2 together with the old density values ρ n 1 do not satisfy continuity, substituting the new interpolated cell face mass fluxes ṁ l into the continuity equation, (ρ m 1 ρ n 1 ) Ω t + l ṁ l = Q m, (3.3) results in an imbalance Q [8]. To enforce the conservation of mass it is necessary to m impose a correction step. 12

18 This correction is the second task of the outer loop. Since we are dealing with compressible flow, the mass flux varies not only with the velocity but with the density as well. We must therefore calculate the density correction ρ and the normal velocity correction v such that the corrected mass flux for the mth outer iteration and lth n cell face can be expressed as, ṁ m l = ρ m l v m n,l S l = (ρ m 1 + ρ ) l (v m n + v n ) l S l, (3.4) where S l denotes the cell face area between the cell of point P and the lth neigbouring cell. The mass flux correction in terms of density and velocity can thus be expressed as, ṁ 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 neglect it here because it is of second order in terms of corrections. For a colocated variable arrangement, a mesh in which the velocity field and pressure field share the same grid, the first term of equation 3.5 can be approximated by, 1 δp (ρ m 1 Sv ) n l = (ρ m 1 S Ω) l, (3.6) A u P δn l where the overbar denotes an interpolated value and n is the coordinate of the outward facing normal to the cell face [8]. If we assume the temperature to be fixed for each outer iteration, we can approximate ρ in the second term of equation 3.5 by ρ ρ = p = C ρ p. (3.7) p T Since we assumed that the fluid is a perfect gas, we can use the equation of state to determine the coefficient C ρ by, C ρ = 1 RT. (3.8) The second term of equation 3.5 can be written as, Cρ ṁ (v m Sρ ) n l = p. (3.9) l ρ m 1 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 values and the pressure correction, 1 δp Cρ ṁ = ṁ l (ρm 1 S Ω) l + p A u P δn l ρ. (3.10) m 1 l 13 l l l l

19 For the mass fluxes to satisfy the continuity equation, the corrected mass fluxes and the corrected densities should satisfy, ρ P Ω t + l ṁ + l Q = 0. (3.11) m If we substitute equation 3.7 for ρ and equation 3.10 for P ṁ into equation 3.11, we l can write the pressure correction equation as, A P p + A P l p = l Q. (3.12) m l After the resulting system of equations has been solved, only the last step of the outer loop remains. First, the corrected pressure is calculated. It has been found that adding only a portion of the pressure correction improves the convergence [8]. This is done by choosing an under-relaxation factor α P [0, 1] which acts like a pseudo-time step, such that, p m = p m 1 + α P p. (3.13) This under relaxation factor is not used in the final outer iteration because the results of each time step must satisfy continuity. After this pressure correction, substituting equation 3.13 and the expression u m = u m + u into equation 3.2 gives, i i u = l A u iu l i,l i,p A u Ω δp i P A u, (3.14) i δx P i P from which the corrected velocity u m is solved. Via the file fvsolution the user can control several settings for the SIMPLE algorithm. For example, the maximum number of outer iterations can be set through the variable noutercorrectors. Also, it is possible to solve the pressure correction equation and calculate the mass fluxes at the boundary cell faces multiple times by setting the variable nnonorthogonalcorrectors larger than zero, to account for possible grid non-orthogonality. The under-relaxation factors and residual tolerances can also be specified there sonicfoam The solver sonicfoam in OpenFOAM version 2.3 uses the PIMPLE algorithm, which is a hybrid between the SIMPLE and PISO algorithms. The PISO algorithm is very similar to the SIMPLE algorithm, which we described above, except for two differences. 14

20 It executes the outer loop only once for each time step and can therefore not use under-relaxation factors, but on the other hand the pressure correction and resulting velocity field are calculated more than once per time step, which we will call the PISO loop from now on. The PIMPLE algorithm combines SIMPLE and PISO such that both the outer loop and the PISO loop can be executed multiple times per time step. 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 acts as a pseudo time step. Start with u n 1, ρ n 1 and p n 1 k = 0, l = 0, m = 0 Linearized momentum equation u m Calculate mass fluxes ṁ m at l cell faces from ρ m 1 and v m n if m < nouter Correctors if k nnonorthogonal Correctors if l < ncorrectors Pressure correction p Correct mass fluxes at cell faces k++ Calculate u m from p m l++, m++ PISO loop Outer loop Calculate other variables Proceed to next time step n++ 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 the n 1th time step are used as initial values for the nth time step as the algorithm enters the outer loop. First the discretized linearized segregated momentum equations are 15

21 solved to obtain an estimate u m, where again the asterisk is used to indicate that this value does not necessarily satisfy the law of conservation of mass. Together with ρ n 1, this estimate u m is then used to calculate an estimate of the mass fluxes through the cell faces ṁ m. After that, the flux estimates are used to solve the imbalance of l the 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 a correction on the cell face normal velocity v, by which the mass fluxes at the cell faces n can be corrected. Depending on the variable nnonorthogonalcorrectors, these two steps can be executed multiple times to account for non-orthogonality. After that, the velocity cell center value estimates u m are corrected and depending on the value P of ncorrectors the algorithm can perform another PISO iteration by returning to step 2 of the outer loop. When the PISO loop is finished, the algorithm can perform another outer iteration, depending on the value of the variable noutercorrectors. We see that the PIMPLE algorithm is very similar to the SIMPLE algorithm. For a more detailed description of the PISO loop we refer to [8, 17] rhocentralfoam The solver rhocentralfoam is a density based solver for high speed compressible flow [12]. Similarly to the SIMPLE based algorithms it is also suitable for implicit time-stepping. But whereas SIMPLE based algorithms use a pressure correction to allow this, the rhocentralfoam solver uses an operator-splitting approach involving a diffusion correction to allow this. This implies that in rhocentralfoam for the velocity 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 detailed description based on the article written by the programmers who have implemented rhocentralfoam in OpenFOAM [9]. First, the values of u and ρ of the previous time step are used to compute the density-weighted field û = uρ. By using the Van Leer limiter and the value T from the previous time step, the cell face values û f ±, ρ f ± and T f ± for the convective terms are 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 boundary face. After this, the remaining variables are calculated via the expressions as denoted in table

22 u f ± = û f ± ρ f ± φ f ± = S f u f ± p f ± = ρ f ± RT f ± c f ± = γrt f ± T exp = µ[( u) T 2 3 tr( u)i] Table 3.1: The variable expressions for the cell face values by inward and outward flux [9]. Using these variables and updated µ and k, the continuity equation is solved for ρ. Next, the inviscid convective momentum equation, û + [uû] + p = 0, (3.15) t I is solved for û. This step is also known as the inviscid momentum prediction. By using this new ρ and û we then update u after which it is corrected by solving the diffusive velocity correction equation, (ρu) (µ u) (T exp ) = 0, (3.16) t V for u, where T exp, the part of the viscous stress tensor S containing inter-component coupling, 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 is treated implicitly. Now, only the energy, temperature and pressure have to be calculated. This is done by first solving the equation, Ê + [u(ê + p)] + (T u) = 0, (3.17) t I for Ê. We can then compute the temperature T from the equation, T = 1 Ê c v ρ u 2, (3.18) 2 with our updated ρ, Ê and u. After that, the temperature T is corrected by solving the diffusive temperature correction, (ρcv T) (k T) = 0, (3.19) t for T. Lastly the pressure p is updated by p = ρrt. V 17

23 3.4 Turbulence models To take into account that the flow is subject to turbulence, we used Reynolds-Averaged Navier-Stokes (RANS) turbulence models. For these models the Reynolds-Averaged Navier-Stokes equations are used which are slightly different from the Navier-Stokes equations we presented in chapter 2. For incompressible flow, the equations are obtained by decomposing the expressions for the velocity, pressure and viscous stress tensor into a mean component and a fluctuating component such that, u i = u i + u i, (3.20) p = p + p, (3.21) S i j = S i j + S. i j (3.22) Here u i, p and S i j denote the time averages defined as, u i = 1 T t+t t u i (t ) dt, (3.23) for u i and defined in a similar way for p and S i j. The variables u i, p and S i j denote the fluctuating parts. This technique is called the Reynolds Decomposition [15]. When we substitute these expressions into the Navier-Stokes equations from chapter 2 and then average the equations, an additional term arises in the momentum equations. This term contains the time average of the product of the velocity fluctuations, known as the Reynolds Stress, R i j = u i u. (3.24) j Because of the new unknown Reynolds Stress there are not enough equations to determine all variables and therefore the problem is not closed anymore. Turbulence models provide closure to this problem by using additional equations based on experiments and assumptions to calculate the Reynolds Stress. For compressible flow however, we have to take into account that the density is variable as well. By applying the Reynolds Decomposition to the density and following the same procedure as before we would obtain a more difficult problem due to the arising correlation of ρ and u in the continuity equation and the triple correlation i of ρ, u and u in the momentum equation [18]. This problem is simplified by i j using mean velocity components based on the mass-averaged values opposed to the time-averaged values, also known as Favre-Averaging. In this case the velocity is expressed as, 18

24 where F denotes the Favre-average given by, u i F = 1 ρ lim T u i = u i F + u i, (3.25) t+t t ρ(x, τ)u i (x, τ)dτ, (3.26) and where ρ denotes the time average. When we now substitute the Favre decompositions of the velocity, specific internal energy, temperature and specific enthalpy, together with the Reynolds decompositions of the pressure, density and heat flux vector into the conservation equations, we obtain the Favre-averaged equations [18]. The Favre-averaged continuity and momentum equations differ from the laminar equations only by the Favre-Averaged Reynolds-Stress tensor, τ i j = ρu i u j, (3.27) where u and u denote the fluctuating parts of the Favre decomposition. i j The Realizable k-ε model The realizable k-ε model uses the Turbulence Viscosity Hypothesis coined by Boussinesq in 1877, to provide closure to turbulence problem [15]. This hypothesis is analogous to the assumption of the viscous stress relation for a Newtonian fluid, which we imposed earlier by employing equation 2.4. The Turbulence Viscosity Hypothesis states that the Reynolds-stress-anisotropy tensor a i j defined as a i j u i u j 2 3 kδ i j, is linearly related to the mean rate-of-strain tensor S i j via the turbulent eddy viscosity ν T as, u i u j 2 3 kδ Ui i j = ν T + U j. (3.28) x j x i The realizable k-ε model then calculates the Reynolds stress by the relation, ν T = C µ k 2 ε, (3.29) where k and ε represent the turbulent kinetic energy and the turbulent dissipation rate respectively, which are calculated from two transport equations for each time step. In contrary to the standard k-ε model where C µ is a constant, in the Realizable k-ε model C µ is a variable and defined by the expressions as listed in Table 3.2, where Ω i j is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity ω k [16]. 19

25 C µ = U 1 A 0 +A s ku ε Ω i j = Ω i j 2ε i jk ω k S i j S i j + Ω i j Ω i j Ω i j = Ω i j ε i jk ω k Table 3.2: The expressions for C µ. Furthermore, the coefficients A 0 and A s are defined as in table 3.3. A 0 = 4.0 W = S i js jk S ki S 3 A s = 6 cos φ S = S i j S i j φ = 1 3 cos 1 ( 6W ) S i j = 1 2 uj x i + u i x j Table 3.3: The coefficients for A 0 and A s in C µ [16]. The two additional equations of the k-ε model to close the problem are the transport equations for the variables k and ε [3], t (ρk) + t (ρε) + (ρku j ) = µ + µ t x j x j σ k (ρεu j ) = x j x j k + P k + P b ρε Y M + S k, (3.30) x j µ + µ t ε ε 2 + ρ C 1 Sε ρ C 2 σ ε x j k + νε ε + C 1ε k C 3εP b + S ε, (3.31) where µ t = ρν t, P k represents the generation of turbulence kinetic energy due to the mean velocity gradients, P b is the generation of turbulence kinetic energy due to buoyancy, Y m represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, S k and S ε are user-defined source terms and C 3e represents how much ε is affected by the buoyancy [10]. The other coefficients for the transport equations 3.30 and 3.31 can be found in table

26 C 1 η = max 0.43, = 2S i j S i j k ε η η + 5 σ ε = 1.2 σ k = 1.0 C 1ε = 1.44 C 2 = 1.9 Table 3.4: The coefficients for the transport equations of k and ε [16] k-ω SST The Shear Stress Transport (SST) k-ω turbulence model combines the k-ω model with the k-ε model such that k-ω is used in the inner region of the boundary layer and the k-ε in the free stream region. We will present the standard k-ω SST model as described by NASA in [1], which is based on the article [13] by F.R. Menter in which he presented the model. The turbulent viscosity is calculated as follows, where, ν t = a 1 k max(a 1 ω, ΩF 2 ), (3.32) Ω = 2W i j W i j, W i j = 1 ui u j. 2 x j x i The transport equations for k and ω are, (ρk) t (ρω) t + (ρu jk) = P β ρωk + (µ + σ k µ t ) k x j x j x j + (ρu jω) x j = γ ν t P βρω 2 + x j 2(1 F 1 ) ρσ ω2 ω (µ + σ ω µ t ) ω x j, (3.33) k ω. (3.34) x j x j + 21

27 With the following expression for P, u i P = τ i j, (3.35) x j τ i j = µ t 2S i j 2 u k δ i j 2 3 x k 3 ρkδ i j, (3.36) S i j = 1 uj + u i. (3.37) 2 x i x j The remaining variable expressions and constants can be found in table 3.5. F 1 = tanh ((arg 1 ) 4 ) F 2 = tanh ((arg 2 ) 2 ) k arg 1 = min max β ωd, 500ν, 4ρσ ω2 k k d 2 ω C D kω d 2 arg 2 = max 2 β ωd, 500ν d 2 ω C D kω = max 2ρσ 1 k ω ω2 ω, 10 x j x 20 α 1 = 0.31 j k = 0.41 β = 0.09 γ 1 = β 1 β σ ω1k 2 β γ 2 = β 2 β σ ω2k 2 β σ k1 = 0.85 σ k2 = 1.0 σ ω1 = 0.5 σ ω2 = β 1 = β 2 = Table 3.5: Expressions and constants for the standard SST k-ω model as presented by NASA [1]. The constants in table 3.5 that appear with subscript 1 and 2 are used in the equations with the blending function F 1. For example for β we have, β = F 1 β 1 + (1 F 1 )β 2. (3.38) 22

28 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) Table 3.6: The specifications and dimensions of the modeled air bearing In the simulations we have carried out, we considered a vertically and horizontally stationary bearing floating at a constant height of 4 µm. As illustrated in Figure 3.3, the mesh surrounds only one-eighth of the bearing with dimensions as listed in table 3.6. Because of the symmetric shape of the bearing this is sufficient to determine the flow 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 and counterclockwise sides and the solid walls consisting of the surface above which the bearing floats and the bearing itself. As suggested in most of the literature on pressure driven flow simulations, we have used an initial flow consisting of a fluid at rest with a uniform pressure. During the first s, we gradually increased the pressure difference by enhancing the 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 will develop stably. At first we used an initial uniform pressure of 1.7 bar, but later we changed it to 5 bar for two reasons. Firstly, based on results of simulations on the grooved bearing in earlier research [6], we expected that a uniform pressure of 5 bar 23

29 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 expected it to be better to keep the inlet pressure more steady. The majority of the simulations were executed with the boundary conditions as shown in table 3.7, but we have also tried others as described in Appendix A. For example for the inlet velocity, we have also used fixedvalue and zerogradient as well, and for the inlet pressure we have tried totalpressure too. Variable Inlet Outlet Symmetry walls Solid walls u p T k ε ω pressureinlet- Velocity fixedvalue (Table) fixedvalue (293) fixedvalue ( ) fixedvalue (1) fixedvalue (1) zerogradient slip fixedvalue (0,0,0) fixedvalue (Table) zerogradient zerogradient zerogradient zerogradient zerogradient zerogradient zerogradient compressible:: kqrwallfunction value: zerogradient zerogradient compressible:: epsilonwallfunction value: zerogradient zerogradient compressible:: omegawallfunction value: α t zerogradient zerogradient zerogradient compressible:: alphatwallfunction value: 0 µ t zerogradient zerogradient zerogradient compressible:: mutkwallfunction value: 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 thermophysical properties in the file thermophysicalproperties. In listing 3.1 an overview of the settings that we have used for the thermophysical properties of the fluid can be found. 24

30 thermotype { 20 type hepsithermo ; mixture puremixture ; transport sutherland ; specie specie ; thermo econst ; 25 equationofstate perfectgas ; energy sensibleinternalenergy ; } 30 mixture { specie { nmoles 1; 35 molweight ; } thermodynamics { 40 Cp ; // specific heat Cp Hf e +06; // heat of fusion Cv 717; // specific heat Cv } 45 transport { As e - 06; // sutherland coefficient Ts ; // sutherland temperature } 50 } Listing 3.1: thermophysicalproperties line 18 to 50. The settings we used, imply that we relate the pressure to the temperature through the equation of state of a perfect gas and that we model the thermodynamical behaviour by a constant specific heat c p -model with evaluation of the internal energy e and entropy s without considering the heat of formation. Furthermore, we have used the Sutherland transport model, which relates the dynamic viscosity µ and the temperature T via the expression, µ = A s T 1 + T, (3.39) s T with the parameters A s and T s set as shown in listing

31 3.7 Postprocessing To process the results of our simulations, we have used the open source data analysis and visualisation application paraview combined with the OpenFOAM reader module as supplied by OpenFOAM. With paraview we could for example investigate the pressure distribution and velocity profiles throughout the entire mesh. With these tools we inspected whether the results we received were realistic and whether they were converging or not. We also used it extensively to determine our strategy to improve future simulations. 26

32 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 the bearing 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 cells of which 99.62% is hexahedral. Figure 4.1 contains two pictures of the mesh. The nozzle is octagonal instead of round, to make the meshing procedure easier. For this simulation we used the settings that led to good results in the past when Schut performed simulations of a grooved air bearing. For those simulations they used the solver sonicfoam together with the realizable k ε turbulence model. The grooved bearing that they modeled is comparable to our bearing in dimensions, but the 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

33 grooves from the orifices to about three-quarter of the bearing, as can be seen in Figure 4.2. Furthermore, in their simulations, only bearing clearances of 9 µm and larger were considered. (a) A close-up of the air chamber above the nozzle and the recess around the orifice. (b) An overview of the entire mesh. Figure 4.2: The mesh of one-third of the grooved bearing, used by the company in earlier research. The first simulation on the first mesh with the settings as described above crashed at t = s with a final maximum velocity of 635 m s 1 and a final maximum pressure of 17.4 bar. In Figure 4.3 it can be seen that there is a pressure accumulation moving downward through the nozzle, which then collapses back towards the inlet after which the simulation crashed. The pressure in the nozzle reaches values of over 18 bar. Beyond the orifice however, very little happens. Only in the region near the outlet a little activity can be seen. #ID Solver Turbulence Model Progress Latest Time ( 10 5 s) v max (m s 1 ) p max (bar) 5 sonicfoam Realizable k ε Crash Table 4.1: The results of the first simulation on the first mesh. 28

34 Figure 4.3: The pressure distribution over time in the y, z-plane. Simulation #5. Second Mesh The second mesh that we used is a finer mesh, see Figure 4.4. This mesh contains cells, of which 99.85% is hexahedral. This mesh has a lot more cells in the nozzle than the previous mesh, which can be seen by comparing Figure 4.1a with Figure 4.4a. (a) A close-up of the top of the nozzle. (b) A close-up of a cross-section of the mesh 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

35 pressure distribution with multiple stationary circular pressure depressions around the orifice. These depressions range from strong to invisible as you move away from the orifice, as can be seen in the top-view in Figure 4.5a and in a cross-section in Figure 4.5b. (a) A top-view of the nozzle and the surrounding pressure depressions. (b) The pressure in the y, z-plane. (c) The normalised velocity vectors in the transition 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 which the flow is reversed, as indicated by the yellow arrows that represent the normalised velocity vectors in Figure 4.6. The results of the simulation with sonicfoam barely show any movement nor change in pressure, except for in small regions near the inlet and outlet. In Figure 4.7 and Figure 4.8 one can see that the fluid underneath the bearing is nearly stationary and that the pressure remains at the initial value of 1.7 bar almost everywhere. Because this did not seem to change with time we pauzed simulation #8 at t = s. 30

36 Figure 4.6: A top-view of the velocity profile. The yellow arrows represent the normalised velocity vectors. Simulation #6. #ID Solver Turbulence Model Progress Latest Time ( 10 5 s) v max (m s 1 ) p max (bar) 6 rhosimplefoam Realizable k ε Crash rhosimplefoam Laminar Crash sonicfoam Realizable k ε Pauzed Table 4.2: The results of the second mesh. (a) The pressure distribution in a crosssection with the y, z-plane. (b) A top view of the pressure distribution close to the outlet. Figure 4.7: The results of the second mesh. Simulation #8. 31

37 (a) The velocity in a cross-section with the y, z-plane. (b) A top view of the velocity close to the outlet. Figure 4.8: The results of the second mesh. Simulation #8. Third Mesh The final mesh that we have used for all simulations on this bearing from this point on, contains cells of which 99.74% is hexahedral. Because our geometry contains a lot of thin regions, we could reduce the amount of cells without losing a lot of accuracy. This is done by elongating the cells in the flow direction in regions where the flow is mostly laminar. To do so, we have used four gradients as visible in Figure 4.9. Firstly, there is a vertical gradient in the nozzle, since the flow in the top of the nozzle is irrelevant to the company compared to the flow in the bearing clearance. (a) A close-up of the nozzle. (b) A cross-section of the nozzle with the y, 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

38 (a) The top view of the final pressure distribution. (b) The pressure versus the y coordinate. Figure 4.10: The pressure in simulation #10. (a) The normalised velocity vectors in the transition 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 square orifice to make the meshing procedure easier. We executed two simulations of which the behaviour is almost similar to the results of 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 orifice again. This time however, the pressure depressions were not static but they were moving away from the orifice, similarly to a sound wave. Furthermore, the depressions were divided into four directions, and there were more visible depressions than in the previous simulations, as illustrated in Figure 4.10a. Lastly, the velocity profile in the nozzle showed a higher velocity near the walls and corners, as can be seen in Figure 4.11b. 33

39 #ID Solver Turbulence Model Progress Latest Time ( 10 5 s) v max (m s 1 ) 9 sonicfoam Realizable k ε Pauzed rhosimplefoam Laminar Crash p max (bar) Table 4.3: The results of the third mesh. 4.2 Changing several settings Initial pressure After trying different meshes, we have experimented with some adjustments in the settings such as changing the boundary conditions, the under-relaxation factors, the initial uniform pressure value and the time-step size. Here we will only present the results of changing the initial pressure and the time-step size. The results of changing the boundary conditions and the under-relaxation factors are described in Appendix A. We must state however that from this point on, based on experiences of OpenFOAM users online and based on the simulations we performed with different boundary conditions as described in Appendix A, we continued with the inlet velocity boundary condition pressureinletvelocity instead of zerogradient. As mentioned in section 3.5, at some point we changed the initial uniform pressure from 1.7 bar to 5 bar. In the simulation with rhosimplefoam we still encountered oscillations in the pressure, but this time they were originating from the outlet as well, instead of only from the orifice. This can be seen in Figure The behaviour of the sonicfoam solver was not noticeably different from before. There was still very 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

40 From this point on, we also considered the solver rhocentralfoam which has shown good performance in treating shock waves [12]. With an initial pressure of 5 bar and no turbulence model, the flow behaviour was the same as what we have seen from the sonicfoam solver so far. There was barely any activity except for in small regions near the inlet and outlet. #ID Solver Turbulence Model 19 rhosimplefoam Laminar p 0 = 5 bar regular 23 sonicfoam Realizable k ε Comment Progress Latest Time ( 10 5 s) p 0 = 5 bar regular 31 rhocentralfoam Laminar p 0 = 5 bar regular v max (m s 1 ) Crash Pauzed Pauzed p max (bar) Table 4.4: The results of changing the initial pressure to 5 bar 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 using a smaller time-step, the simulation has more iterations to perform between each increase of pressure difference, which happens every s during the pressure build-up. First we executed simulation #15, which is the same as simulation #12 but with a time-step size of s instead of s. When we pauzed the simulation at t = s we noticed the oscillations again, which did not show any sign of damping. Later on when we were using a mesh of a different bearing type (described in more detail in section 4.4), we also performed a simulation with a smaller time-step size to compare. Instead of a time-step size of s we used s. Because both simulations were started from the same initial condition at t = s, we can see that this simulation crashed after exactly the same amount of time-steps with exactly the same final maximum velocity and final maximum and minimum pressure. When analyzing the flow with ParaView we could again see the oscillations originating from the orifice and from the outlet. 35

41 #ID Turbulence Model Time-step Progress Latest Time ( 10 5 s) v max (m s 1 ) p max (bar) 15 Laminar Pauzed k ω SST Crash k ω SST Crash Table 4.5: The results of the rhosimplefoam solver with smaller timestep sizes. 4.3 Adjusting the viscosity The last strategy we used to eliminate the instabilities that we encountered when using rhosimplefoam, was to adjust the dynamic viscosity. We manually altered the dynamic viscosity by changing the Sutherland coefficient A s such that we obtained a stable flow. Then, we gradually lowered it back to the realistic value while using each resulting flow as the initial condition for the next simulation. For our first simulation we changed the coefficient A s from to As mentioned in Section 3.6 the Sutherland transport model relates the dynamic viscosity µ to the temperature T by, µ = A s T 1 + T, (4.1) s T where T s denotes the Sutherland temperature and A s denotes the Sutherland Coefficient. This means that we increased the dynamic viscosity by a factor of This simulation converged very fast, even before the pressure build-up was finished. For an initial pressure of 1.7 bar as well as for an initial pressure of 5 bar, there were no oscillations in neither the pressure nor the velocity, as demonstrated in Figure 4.13b and Figure 4.13c. 36

42 (a) The y-axis. (b) A plot of the pressure versus the position on the y-axis. (c) The top view of the pressure distribution with 20 linear scaled contours. (d) The normalised velocity vectors in the transition 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 , which corresponds to a scaling factor of 100. After pausing the simulations at 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 1000 as the initial state, the simulation converged at t = s. In Figure 4.14 the pressure, velocity and speed of sound versus the y coordinate are plotted. To calculate the speed of sound we used the approximate relation of the speed of sound c air to the temperature T in dry air given by, c air = T (4.2) We can see that in contrary to the pressure distribution in Figure 4.13b, this distribution has a depression near the orifice. After that, we used the converged solution of simulation #26 as an initial state and lowered the scaling factor from 100 to 10. After s however, the 37

43 Figure 4.14: A plot of the pressure (black), velocity (blue) and speed of sound (red) versus the y coordinate for simulation #26. simulation still showed some oscillations which did not seem to subside. Because lowering the Sutherland coefficient from to did not result in a stable flow, we performed a simulation with A s = Although this simulation did not converge enough to satisfy the predefined tolerance values, the pressure distribution seemed reasonably stable at t = s. For the solver rhosimplefoam we executed this viscosity simulation sequence with the k ω SST turbulence model as well. The simulation with rhosimplefoam and k ω SST was able to converge with a viscosity scaling factor of 100 from a stationary initial flow and from the final solution of simulation #22 as well. Both of these simulations converged towards a pressure distribution with a slightly different pressure depression next to the orifice, as can be seen in Figure When we continued from the final state of simulation #32 with a scaling factor of 10 however, the simulation showed oscillations and did not show any sign of convergence at t = s. Also for a Sutherland coefficient of , 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 before the first write interval was completed. 38

44 (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 A s Progress Latest Time ( 10 5 s) v max (m s 1 ) bar Converged bar Pauzed bar Converged Converged Pauze Pauze p max (bar) Table 4.6: The results of rhosimplefoam without a turbulence model and with gradually lowering the Sutherland Coefficient to the realistic value. #ID Initial state A s Progress Latest Time ( 10 5 s) v max (m s 1 ) 28 5 bar Converged Pauzed Pauzed Pauzed Crashed p max (bar) Table 4.7: The results of rhosimplefoam with the k ω SST turbulence model and with gradually lowering the Sutherland Coefficient to the realistic value. 39

45 (a) The pressure in a point underneath the nozzle. (b) The velocity in a point underneath the nozzle. (c) The pressure in a point next to the nozzle. (d) The velocity in a point next to the nozzle. Figure 4.16: The pressure and velocity values over time in two arbitrary points close to the orifice. Simulation #34. Since the solver sonicfoam seemed to have a more stable behaviour, we used it to follow up on the final state of simulation #16 and #37 as listed in Table 4.8. The resulting flow contained a lot of pressure waves originating from the nozzle. Although most of these swells eventually faded out, at t = s the simulation still was 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 Sutherland coefficient of This flow did not show any heavy oscillations and steadily developed a pressure depression near the orifice, similarly to the previous simulations with a viscosity scaling factor of 100. Although the flow was not converged yet at t = s, it seemed reasonably stable with a pressure distribution as shown in Figure

46 Figure 4.17: A plot of the pressure distribution along the y-axis of simulation #38. Lastly, we used Sutherland coefficients of and in combination with the solvers rhocentralfoam and sonicfoam, with the final solutions from simulations #26 and #37 as initial flows as listed in Table 4.8 and Table 4.9. These four simulations showed very similar results. At first the pressure distribution contained some instabilities in the form of pressure oscillations originating from the orifice with relatively high pressure values. Later on, these instabilities faded out and all four pressure distributions converged towards a distribution as plotted in Figure The pressure underneath the nozzle is at 9 bar significantly higher than the inlet pressure of 6 bar. (a) The total plot. (b) A close-up of the area underneath the nozzle. Figure 4.18: Two plots of the pressure along the y-axis of simulation #40. 41

47 #ID Initial state A s Progress Latest Time ( 10 5 s) v max (m s 1 ) Converged Pauzed Pauzed Pauzed p max (bar) Table 4.8: The results of sonicfoam with the Realizable k ε turbulence model and with gradually lowering the Sutherland Coefficient to the realistic value. #ID Initial state Turbulence Model A s Progress Latest Time ( 10 5 s) v max (m s 1 ) Laminar Pauzed k ω SST Pauzed k ω SST Pauzed p max (bar) Table 4.9: The results from rhocentralfoam and gradually lowering the Sutherland Coefficient to the realistic value. 4.4 Comparison to OpenFOAM 1.7 and a different geometry Since the realistic results that were obtained in previous research [6] on modelling air bearings in OpenFOAM were obtained with OpenFOAM 1.7 and on a different mesh, we decided to do a comparison test between the OpenFOAM versions 1.7 and 2.3, and between the two meshes. First, we investigated the performance of the solver rhosimplefoam in combination 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 the center, 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 bearing is different from our bearing because it has a rounded transition from the nozzle to the bottom surface of the bearing, as illustrated in Figure 4.19, and the bearing clearance is 9 µm. Again, we started out with an initial viscosity that is a factor 1000 larger than the realistic value. This simulation converged in s. When we adjusted the Sutherland coefficient such that the viscosity was only 100 times 42

48 Figure 4.19: A close-up of the transition from the nozzle to the gap in the two dimensional mesh. larger than the realistic value, the simulation crashed whether we used the previously converged solution as the initial flow or not. The simulations performed with a Sutherland coefficient of and crashed as well. After that we experimented with the option transonic and several limiters. The results of this can be found in Appendix A. #ID Initial state Solver A s Progress Latest Time ( 10 5 s) v max (m s 1 ) 51 5 bar rhosimplefoam Converged bar rhosimplefoam Crashed rhosimplefoam Crashed p max (bar) Table 4.10: Results from rhosimplefoam with the k ω SST turbulence model on the 2D mesh of the simple type bearing in OF 2.3. While the solver sonicfoam obtained good results in OpenFOAM 1.7 with this mesh, in OpenFOAM 2.3 the flow was not solved successfully even though exactly the same settings were employed. Furthermore, using one of the converged solutions obtained from rhosimplefoam as the initial flow for the sonicfoam and rhocentralfoam solvers, yielded interesting patterns of shock reflections prior to 43

49 crashing. Mesh OpenFOAM 1.7 OpenFOAM 2.3 2D mesh of simple bearing Good Bad Mesh 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 of our bearing with multiple orifices and with exactly the same settings that gave successful results in the past on the grooved and simple bearing [6]. This simulation resulted in a flow that is comparable to the results of sonicfoam that we described in previous sections, in the sense that there was barely a flow initiated. The solver rhocentralfoam gave similar results. The simulations with rhosimplefoam all crashed before the first write interval was reached. #ID Initial state Solver A s Progress Latest Time ( 10 5 s) v max (m s 1 ) p max (bar) 61 5 bar sonicfoam Realizable k ε Pauzed bar rhocentralfoam Laminar Crash sonicfoam 1 k ω SST Crashed rhocentralfoam 1 k ω SST Crashed sonicfoam k ω SST Pauzed rhocentralfoam k ω SST Pauzed Table 4.12: Results from sonicfoam and rhocentralfoam with the realistic Sutherland Coefficient on the 2D mesh of the simple type bearing in OF For these simulations we have changed ncorrectors from 1 to 2 and div(φ, ε) from bounded Gauss upwind to Gauss upwind, which corresponds to the settings that gave successful results in on the other bearings in OpenFOAM

50 #ID Initial state Solver Turbulence Model bar sonicfoam Realizable k ε A s Progress Latest Time ( 10 5 s) v max (m s 1 ) Pauzed bar rhocentralfoam k ω SST Crashed bar rhosimplefoam k ω SST Crashed bar rhosimplefoam k ω SST Crashed bar rhosimplefoam k ω SST Crashed p max (bar) Table 4.13: The simulations performed on mesh #3 in OpenFOAM

51 46

52 Chapter 5 Conclusion Meshes The first mesh that we used, as described in Section 4.1, results in a flow that is different from all other flows that we obtained. We interpreted this difference as the consequence of a shortage of cells in the nozzle. We think that the flow is not able to develop properly because this mesh has only four columns of cells in the nozzle which 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 with the solver rhosimplefoam they result in flows that seem more realistic. Although both meshes result in oscillations that grow stronger until the simulation crashes, the flows are developed in the entire mesh and the velocity profiles, as illustrated in Figure 4.5c and Figure 4.11a, seem plausible. The second mesh contains almost seven times as many cells as the third mesh but does not perform any better. If we compare 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, even though the second mesh has more cells, the cells near the orifice are larger. Since that is the region where we expect the flow to be the most turbulent, the second mesh is not an efficient mesh. Despite the difference in cell size near the orifice we do not see any difference in the representation of swirls or separation near the corner when we compare Figure 4.5c with Figure 4.11a. Because of these results we conclude that the additional cells in the second mesh are redundant and we continued the rest of the simulations with the third mesh. rhosimplefoam The results we obtained with the solver rhosimplefoam contain oscillations in the pressure and velocity. When we analyzed the results, we could see that the simulations 47

53 crashed whenever the oscillations grew to a certain intensity. Therefore we think that the oscillations are closely related to the crashes. After we switched to the new initial pressure, as described in section 4.2.1, we could see that the oscillations now originated from the outlet as well. This suggests that the oscillations are an indication that the model has difficulties with settling the flux that the changing pressure difference between the inlet and outlet induces. On the other hand as mentioned in Section as well, the results were not improved by reducing the time-step size. Since this would allow the simulation to perform more 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 up too fast. The best technique we found to prevent the simulations with rhosimplefoam from crashing, is to modify the dynamic viscosity through the Sutherland Coefficient as described in Section 4.3. Using a converged solution from a simulation with a high dynamic viscosity as the initial flow for a lower dynamic viscosity improved the convergence. 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 that our resulting flow corresponds reasonably well with the theory of supersonic flows in converging and diverging ducts. The shape of our pressure and velocity plots conform roughly 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 the maximum pressure and minimum velocity in our plot are at the same location are in accordance with Figure 2.2. Furthermore, the occurrence of a pressure depression in air bearings has been discussed extensively in literature [20, 5, 7, 14]. Therefore we conclude that the results of these simulations are a decent approximation of the general flow in an air bearing. sonicfoam and rhocentralfoam For both sonicfoam and rhocentralfoam we saw very similar behaviour. Initiating from a stationary flow resulted in a flow that barely moved, as described in Section 4.1. When we used a viscosity scaling factor of 1 and 10 with a converged solution as the initial flow, both simulations resulted in an increase in pressure which did not subside as visible in Figure Furthermore we found that sonicfoam performed differently in OpenFOAM 1.7 from how it performed in OpenFOAM 2.3 as listed in Table When we compared the transport equations for the energy from the source code, we found multiple differences. These differences might explain why the results we obtained in OpenFOAM 1.7 are different from the results we obtained in OpenFOAM

54 Turbulencemodel From comparing simulations #6 and #7 we found that the performance of the Realizable k ε and the Laminar dummy turbulence model is similar. This agrees with the results obtained in previous research by Schut [6]. Furthermore, from simulation #17 and #28 we conclude that the k ω SST performs better than the Laminar turbulence model, since simulation #28 converges whereas in simulation #17 the oscillations remained. 49

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56 Chapter 6 Future Work Based on our results we would suggest to start from less extreme conditions and gradually work towards the desired set-up. All successful simulations we have seen during this research, which are the simulations with a higher viscosity and the simulations with a larger bearing clearance, have in common that the conditions have less impact on the velocity. The modified viscosity results in lower velocities because of a greater viscous drag whereas the larger bearing clearance reduces the level of constriction and thereby the intensity of the venturi effect. Therefore we recommend to first investigate whether the bearing can be modelled successfully with larger bearing clearances or smaller pressure gradients. This may already result in useful information and otherwise it may produce a good solution to use as the initial condition for simulations with more challenging conditions. Furthermore, we encountered a lot of things that we can not explain yet. For example why the velocity in the inlet in Figure 4.5d and Figure 4.11b has such an unexpected profile or why simulations #43, #41, #42 and #40 show such high pressure values near the orifice. Lastly and most importantly it might be rewarding to perform experiments with this type of bearing to determine whether the oscillations that 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 not found results from experiments on air bearings like ours, we cannot exclude that the oscillations are a realistic part of the solution. 51

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58 Bibliography [1] The menter shear stress transport turbulence model, larc.nasa.gov/sst.html. [2] Openfoam - the open source computational fluid dynamics toolbox, version , [3] Realisable k-epsilon model cfd-wiki, the free cfd reference, cfd-online.com/wiki/realisable_k-epsilon_model. [4] John D. Anderson, Fundamentals of aerodynamics, second ed., McGraww Hill, [5] D. Dowson, Laboratory experiments and demonstrations in tribology: 7-externally pressurised air lubricated thrust bearings, Tribology International (1969). [6] C. Draaijer and J. C. Nauta, Investigation of a cfd procedure for optimization of the air gap stability for aerostatic thrust bearings. [7] Mohamed E. Eleshaky, Cfd investigation of pressure depressions in aerostatic circular thrust bearings. [8] Joel H. Ferziger and Milovan Periç, Computational methods for fluid dynamics, second ed., Springer-Verlag, [9] Christopher J. Greenshields, Henry G. Weller, Luca Gasparini, and Jason M. Reese, Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flow, International Journal for Numerical Methods in Fluids (2009). [10] R. A. W. M. Henkes and C. J. Hoogendoorn, Scaling of the turbulent natural convection flow in a heated square cavity, Journal of Heat Transfer (1994). [11] T. Holzmann, Openfoam guide - the pimple algorithm in openfoam, PIMPLE_algorithm_in_OpenFOAM. 53

59 [12] Luis F. Gutiérrez Marcantoni, José P. Tamagno, and Sergio A. Elaskar, High speed flow simulation using openfoam, Mecánica Computacional (2012). [13] F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, American Institute of Aeronautics and Astronautics Journal 32 (1994), no. 8, [14] H. Mori, A theoretical investigation of pressure depression in externally pressurized gas-lubricated circular thrust bearings, Journal of Fluids Engineering (1961). [15] Stephen B. Pope, Turbulent flows, Cambridge University Press, [16] Tsan-Hsing Shih, William W. Liou, Aamir Shabbir, Zhigang Yang, and Jiang Zhu, A new k-ε eddy viscosity model for high reynolds number turbulent flows, Computers & Fluids 24 (1995), no. 3, [17] H. K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics, Longman Scientific & Technical, [18] David C. Wilcox, Turbulence modeling for cfd, DCW Industries, Inc., [19] Inc. Wolfram Research, Mathematica, Wolfram Research, Inc. [20] S. Yoshimoto, M. Yamamoto, and K. Toda, Numerical calculations of pressure distribution in the bearing clearance of circular aerostatic thrust bearings with a single air supply inlet, Journal of Tribology (2006). 54

60 Appendix A Additional results As mentioned in Section we performed some additional simulations in which we varied the boundary conditions, the under-relaxation factors, the limiters for the flux 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 to fixedvalue (0, 0, 10) which corresponds to a downward flow of 10 m s. Combined with the rhosimplefoam solver, this resulted in a flow which is less structured I