Int. Jnl. of Multiphysics Volume 3 Number

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1 Int. Jnl. of Multihysics Volume 3 Number Simulation of filtration rocesses in deformable media Part 3.: Basic concets and article-fluid force imlementation of a non-sherical dirt article solver Gernot Boiger, Marianne Mataln and Wilhelm Brandstätter ICE Strömungsforschung GmbH, Hautlatz 3, 8700 Leoben, Austria Montanuniversität Leoben, ranz-josefstrasse 8, 8700 Leoben, Austria ABSTRACT A Lagrangian solver to realistically model large, non-sherical dirt articles and their behaviour in the vicinity of deformable filtration fibres has been rogrammed. While this aer focuses on basic solver concets as well as drag force imlementations, a related article, concerning the realisation of interaction effects and result verification, is forthcoming, [3]. Within the framework of a digitally reconstructed, deformable filter fibre geometry, the solver traces the governing multi hysics effects down to the occurrence of single force- and torque vectors. In order to go from an initial, sherical article model [], to a more sohisticated, non-sherical model, the caabilities of a Six Degrees of reedom Solver have been included in the rogramming. A anel model and the concet of satellite hel oints are used to handle articles that encomass several fluid calculation cells. An innovative drag force imlementation allows the consideration of rotational- and shear flow effects on article motion. Results are evaluated and comared to an analytical formulation.. INTRODUCTION The Oen Source Comutational luid Dynamics (CD) toolbox OenOAM has served as a rogramming environment for the develoment of a novel, deterministic, micro scale, fluid-article-fibre filtration solver [], []. This C++ based simulator is suosed to become an imortant tool for the CD design of filter media for automotive lubrication. It was created to consider all hysically relevant effects that go along with, or lead to a micro scale, dirt article deosition in a realistically reconstructed filtration fibre geometry. Concerning this subect, two aers, have been reviously ublished. While [] focuses on fibre deformation and fluid structure interaction (SI) effects, [] describes the develoment of a large, sherical article model for filtration alications. A second article [3] on the non-sherical article model is forthcoming. It is closely related to this work and is to be seen as a sequel. It mainly concerns itself with the handling of article interaction with its surroundings. Moreover it deals with the creation of a simlified, semi-analytical aroach to verify solver functionality and result quality. The current work resents a significant extension of the sherical dirt article model formulated in []. It describes the basic concets as well as the essential drag force imlementation method behind our novel, realistic, Lagrangian, non-sherical article model.

2 408 Simulation of filtration rocesses in deformable media Some of the main asects of the develoment effort behind this aer are: Imlementation of a Six Degrees of reedom (DO) solver for the Lagrangian article momentum equations (PME) in OenOAM. Design of an exlicit, force- and torque vector model, that reduces the modelling to the mere formulation of single force effects. Introduction of an adative time steing scheme for exlicit Euler discretization of the PME, [4]. Device of a surface hel oint scheme to account for large article effects in terms of fluid article, article fibre and article article interaction. Creation and verification of a drag force imlementation that uses a combination of non-sherical, semi-emirical drag force formulas [3], a anel method to consider free flow swirling effects, a lugging method to include inter article and article fibre hydrodynamics and a simle adotion of basic concets known from the immersed boundary method [6]. An efficient article fluid, two-way couling method []. Certain, imortant limitations.. ILTRATION SOLVER The original filtration solver, resented in [] and [] is caable of solving comlex fluid article fibre interaction roblems for microscoic, digitally reconstructed fibre samles ranging u to a few hundred microns in diameter and thickness. To reconstruct the microscoic geometry as accurately as ossible, inut data from comuter tomograhic scans is used. Rudimentary stacks of grey scale images are being digitalized by Matlab utilities. An exemlary result is shown in igure. Both the fibres and their surrounding sace are being transformed into structured grid meshes that reresent the solid and fluid framework for Comutational luid Dynamics (CD) analysis. igure Digitalization by matlab utilities. Transferring stacks of grey scale images (left) to fully digitalized data (right).

3 Int. Jnl. of Multihysics Volume 3 Number Because of the geometry range of ~00µm, the highly viscous, Newtonian oil current featuring a kinematic fluid viscosity of η f ~0-4 m /s, and relatively slow flow velocities of <0.4 m/s, the local Reynolds numbers in the fibre vicinity are exected to be mostly <0.5, but surely <.0. urthermore Knudsen numbers range well below Thus continuum equations are valid and the consideration of diffusion effects on article motion becomes unnecessary. After all a simle, incomressible, laminar and isothermal fluid solver can handle the situation. or quite some time, deformation effects have been susected to have significant imact on the filter characteristics of a fibre. Therefore the original solver was designed to handle fluid-structure interaction effects (SI). The SI utility features a stiff, exlicit couling and uses only one, single finite volume solver to handle the governing fluid dynamics as well as the structural mechanics on the solid side, []. The original article model was a sherical, Lagrangian, fully deterministic (nonstochastic) aroach with the caability to interact with the surrounding, Eulerian fluid fibre framework. Each article can extend well beyond the borders of a single calculation cell and can sense and affect fluid conditions within a multile cell region of the fluid mesh. In extension of the sherical article model, a highly detailed, more sohisticated and more accurate, non-sherical article model was develoed and can hereby be resented.. DRAG ORCES AND PARTICLE RELAXATION TIMES An obvious reason to go from a mere sherical dirt article descrition to a more realistic, non-sherical aroach lies within a significant difference in drag-force-to-mass-ratio. A good way to demonstrate the difference is to take a look at sherical and non-sherical article relaxation times τ of mass equivalent articles. With m being the article mass and ρ being the article density, the diameter of a mass equivalent shere D sh can be written as: D sh m = 6 3 πρ () Since the article Reynolds numbers under consideration, range significantly below, Stokes drag conditions can be assumed. Thus the exression for the article relaxation time for sherical articles τ,sh in the flow domain is given by: τ = sh, D sh ρ 8µ f () Here µ f is the dynamic fluid viscosity. or non-sherical articles a drag force correlation, roosed by Hölzer & Sommerfeld, [3] shall be chosen. In this case the definition of the non-sherical article relaxation time τ,nonsh is much more comlex and reads: C0 =, C + C u + C u τ nonsh rel 3 rel (3)

4 40 Simulation of filtration rocesses in deformable media Here u rel is the relative fluid article velocity and the constants C 0, C, C and C 3 are: C 0 = A m ρ f, ell f (4) Where ρ f is the fluid density and A f,ell is the frontal area of an ellisoid article, roected onto a lane, erendicular to the relative fluid-article velocity vector. C µ f = + D Φ Φ 8 sh length (5) C µ f = 3 34 Dsh Φ / (6) 04. ( log Φ ) Φ C 3 = 0. 4 cross (7) In Equ.5 to Equ.7 Φ, Φ cross and Φ length are the shae deendent, overall shericity, lengthwise shericity and cross-wise shericity, resectively, [3]. The comarison of Equ. and Equ.3 shows that non-sherical article relaxation times are generally lower than those of mass equivalent sheres. In [4] the arameter α ax is introduced to measure deviation from sherical shae. It reresents the medium, relative half axis deviation around D sh and is defined as: ( a D ) + ( b D ) + ( c D ) sh sh sh α ax = 3D sh (8) Here a, b and c are the lengths of the three article half axis, whereby a b c. Using α ax as a arameter, it becomes aarent that, the further the article shae deviates from being a shere (higher α ax ), the smaller τ,nonsh will be, [4]. A corresonding lot of the situation, as seen in igure, reveals that non-sherical article relaxation times show a deendency on local fluid conditions, while sherical article relaxation times do not. urthermore the results in igure show that non-sherical article relaxation times for highly non-sherical articles (α ax ) amount to less than /5 th of sherical relaxation times. Therefore a mere sherical article model would significantly underestimate fluid skin friction- and form drag forces on suosedly arbitrarily shaed dirt articles. One obvious consequence of disregarding article shae effects for filtration simulation would be a certain overestimation of filter fibre efficiencies MODEL BASICS This chater lists some basic concets and innovative imlementation schemes that had to be chosen and/or develoed in order to create a suitable framework for the non-sherical article model.

5 Int. Jnl. of Multihysics Volume 3 Number T,sh T,nonsh (α ax =0) 0.8 T /T,sh [-] T,nonsh (α ax =0.) T,nonsh (α ax =0.6) 0. T,nonsh (α ax =) Re [-] igure Sherical (blue) and non sherical (red, orange, yellow, turquoise) article relaxation time behaviour against article Reynolds number. Increasing α ax (0-) leads to lower τ,nonsh. All values are scaled by τ,sh (Re = 0).. PARTICLE GEOMETRY: ELLIPSOID SHAPE The non-sherical article shae reresentation is chosen to be an ellisoid with three indeendent, geometrical degrees of freedom. This choice offers the versatility to aroximate many shaes from sticks to lates (see igure 3) on the one hand and can be mathematically described retty easily, as seen in Equ.9, on the other hand. x y z a + b + c = (9) Here x, y and z are coordinates of a Lagrangian, co-rotational coordinate system, with base vectors being aligned along the article s main axis (see chater 3.). igure 3 The ellisoid shae can aroximate a wide variety of geometries, e.g. lates and sticks.

6 4 Simulation of filtration rocesses in deformable media The ellisoid article volume V is given by: V = 4 abc (0) 3 π An essential quantity for calculating skin friction forces on the article, is the article surface area A. In this work A for ellisoids shall be aroximated by a comaratively simle aroximation formula, roosed by Thomsen, [5]: ( ) A 4 a b a c b c () With.6075, this formula is reorted to yield a maximum of +/.06 % deviation about the correct result.. EULER AND LAGRANGE COORDINATE SYSTEM The fluid and SI calculations are based uon well known, Eulerian rinciles and require only one Cartesian coordinate system, with base vectors e x, e y and e z and coordinates, x, y and z. In the course of the SI calculation however, the fluid mesh actually works as Lagrangian mesh that adusts itself to dislaced fibres. Whereas the fibre structure mesh itself remains in its original osition, after the fashion of a tyical Eulerian mesh, []. or the article calculation the artly Lagrangian character of the fluid mesh is comletely irrelevant. The article solver does not require searate meshing. To account for article osition X and orientation, an additional co-rotational coordinate system is introduced. The article coordinate system, with base vectors e x, e y and e z being aligned along the main article axis, as seen in igure 4, originates from the article mass centre. Its coordinates are written as x, y and z. igure 4 Exemlary ellisoid article with co-rotational coordinate system. The relationshi of any single oint P within the Eulerian system, to the corresonding oint P within the co-rotational Lagrangian system, is given by the following formula: P = ( P X e e () P), n n n= x, y, z

7 Int. Jnl. of Multihysics Volume 3 Number Here the index n denotes the axe directions x, y and z resectively and the base vectors of the article coordinate system are given by: e e, e, e = ( ) n, n, n, n, 3 According to Equ. the transformation oeration T can be defined as: ( ) = ( ) T, X A X P P (3) (4) With A being the transformation matrix: e e e e e e A = e e z, z, x, x, x, 3 y, y, y, 3 e z,3 (5) Accordingly the retransformation from P to P is comuted as: P= ( P e e X n ) + n, n= x, y, z P (6) Here the retransformation oeration T is formulated by using the transosed transformation matrix A T : ( ) = T T, X A X P ( ) + P A similar multile coordinate system aroach is used by Rosdahl, [8]. His solver uses a third, additional, co-moving coordinate system, which also originates from the mass centre of the article and is aligned along the basis of the outer, inertial, Eulerian coordinate system..3 SIX DEGREES O REEDOM SOLVER Non sherical article motion in general consists of three degrees of translational as well as three degrees of rotational freedom. This is why the original, sherical article solver had to be transformed into a more general, six DO solver. It can now solve the translational and rotational, Lagrangian equations of motion for N external forces, which act on the article: (7) du dt 3 = πρ abc N 4 = (8) Here u is the article velocity and ρ the article density. d dt I n ω = n, ( ) ( r ) (9) Hereby I n, is the article moment of inertia around the rotational axe n of any acting torque vector. The direction of the lever is given by r. I n, is calculated by use of the article

8 44 Simulation of filtration rocesses in deformable media moment of inertia tensor I and by use of Equ.. I in its most general form can be described by using a continuous, satial density ρ s (x, y, z), [8]: ρ,, I = ( x y z) r E r r dxdydz V s ( ) s k k k Here V s describes the local sace which comletely encomasses the obect, r k is the distance vector to the axe of rotation, E stands for the identity matrix and denotes the outer roduct. (0) I = I n n n, () Inserting the exressions for standard, ellisoid, rincile moments of inertia, this finally amounts to: I m = b c n a ( + ) + ( + c ) n + ( a + b ) n 5 n, x, y,, z ().4 SINGLE ORCE AND TORQUE VECTORS A number of individual, translational and rotational force- and torque effects with influence on article traectory and deosition behaviour have to be accounted for. Those effects can be arted into three basic categories: article-fluid (see chater 4), article-fibre [3] and article-article [3] interactions. It is inaccurate to traditionally model one individual article-deosition effect, without taking into account the interaction with other articles, or a changing fluid field [], []. Thus in this work all modelling is broken down to the level of individual force effects and their resulting torques. The following interaction forces are relevant: article wall imact force, wall article fibre interaction force, fibre article article imact force, collision article fluid interaction (drag) force, fluid force due to ressure gradient (form drag), ressure force due to shear flow (shear drag), shear gravity, g A simultaneous calculation of fluid, ressure and shear would yield an overestimation of fluid article interaction forces. Deending on the secific mode of oeration, either fluid or ressure / shear are calculated. In order to model resulting torques T, the exact ositions r of acting forces need to be known. This is why a surface hel oint method (see chater.5) was devised. igure 5 illustrates an assembly of small, non-sherical articles and the corresonding system of acting forces and torques, that cause translation and rotation..5 PARTICLE SHAPE CONCEPTS In order to consider rotational effects, collision imact scenarios or any shae-related henomenon, the moving obect has to extend beyond a simle, oint-like reresentation. Thus the surface hel oint method, as well as a simle anel method to discretize the article surface are introduced.

9 Int. Jnl. of Multihysics Volume 3 Number igure 5 articles. Illustration of acting forces and torques on an assembly of non-sherical.5. Surface and ressure/velocity hel oints A cloud of u to M= 68 hel oints er article is used (see igure 6). 8 surface hel oints are ositioned directly at the surface of the article to serve as collision detectors and ressure/velocity robes. An additional 48 ressure/velocity hel oints are located at crucial ositions of the article anel model (see chater.5.) and detect local fluid field conditions. The hel oints surround the ellisoid at constant ositions HP m within the framework of the co-rotational article coordinate system. Thus each hel oint conserves its relative osition to the article centre, while the article moves arbitrarily within the Eulerian fluid domain, dragging along the Eulerian hel oint ositions HP m. igure 6 Non-sherical article with 8 surface hel oints and 48 ressure/ velocity hel oints.

10 46 Simulation of filtration rocesses in deformable media The hel oints essentially serve two uroses:. In their function as ressure/velocity hel oints they detect local fluid conditions.. They redefine the current article movement by tracking the individual, roected traectory. Collision that might occur along this course are detected. Using a simle, temoral Euler discretization [4], the hel oint osition HP i m at time i can be roected to its new osition HP i+ m, at time i+, after article time ste t. The new osition can be calculated as: i+ i HPm = HPm + ( ω r + u ) t h (3) Here r h is the hel oint distance vector to the article mass centre. This article rogression scheme is only used if collision events are to be exected. The linear traectory i HP is robed for obstacles. If a collision occurs at osition X coll before HP i+ m is reached, the fraction f m is set to: + m HP i m f m = X HP coll i+ m i HP m i HP m (4) Then the new article time ste t f * is calculated as: * t = min( f ) t f m f (5) Now the actual article movement is conducted. If no collision events are to be exected, the solver conducts translational and rotational oerations on the article mass centre X and on the article base vectors e,n. To find the new hel oint ositions at time i+, a simle coordinate transformation suffices: i+ i HPm = HPm e e X n + n, n= x, y, z + i+ (6) Note the fact that the co-rotational hel oint ositions HP m remain unchanged at all times..5. Panel method While the surface hel oint scheme has been designed to aid in the modelling of collisions and in the detection of local flow field conditions, a simle anel method is introduced to get a hold of hydrodynamic drag- and lift forces. Within the co-rotational coordinate system, the fixed hel oint ositions are used as a framework to encase the ellisoid with a system of edges and anels (see igure 7)..6 ADAPTIVE TIME STEPPING Because of the simle, exlicit, temoral Euler discretization that is used for article rogression, certain numerical instabilities have to be oosed [8]-[]. Studies on the subect have led to the develoment of an adative time steing scheme, that comensates

11 Int. Jnl. of Multihysics Volume 3 Number igure 7 Non-sherical article surrounded by hel oints and anels. individual article needs for higher time resolution. With σ rel,ud being the user-definable, relative standard deviation between a numerical and an analytical article seed u curve, the number of article sub time stes J can be calculated for two levels of accuracy. or σ rel,ud.% u f : t f J = σ τ D u rel UD (, ρ,, µ, sh f f ) (7) or.% u f <σ rel,ud 4.0% u f : t f J = 0. σ τ D u rel UD (, ρ,, µ, sh f f ) (8) Hereby u f stands for the uniform velocity of the surrounding fluid field. A full reort on the results has been ublished in [4]. 3 PARTICLE MOMENTUM EQUATION All changes in translational and rotational motion of a article stem from the sum of acting forces. Alying Newton s second law, the translational article momentum equation (PME) for arbitrarily shaed articles and arbitrary flow conditions, is given. The PME resents the framework for all article-motion modelling behind the non-sherical article solver. It can be written as: m du dt = d + h + Magnus + Saffman + axen + g + b + VM + Basset + N i= e, i (9) Steady State orces Unsteady orces Event orces

12 48 Simulation of filtration rocesses in deformable media Here d is the drag force, h the hydrodynamic lift force, Magnus is the Magnus force, Saffman is the Saffman force, axen is the axen force, g is the gravity force, b is the buoyancy force, VM is the added (virtual) mass effect, Basset is the Basset (history) force and e,i stands for the i-th of N event forces [6], [4], [5]. The individual force contributions, summarized in Equ.9 can be divided into three main categories: steady state forces, unsteady forces and event forces. The secialization of Equ.9 for small, sherical articles, that are not two-way-couled to a uniform, surrounding fluid, gives the Basset Bousinesque Oseen (BBO) equation [3], [7] without axen terms or interaction with solids or other articles. In this case the individual force contributions can be formulated as seen in igure 9. Steady state forces Drag force Hydrodynamic lift force Magnus force Saffman force ormulation π f d = c D u u u u d( Re ) sh 4 h = 0 π Magnus = D 3 ρ u u u ω 8 Saffman =, 65D sh f f f sh ρ ρµ f f (( u ω u u f ) ( f ) ) u f ( ) f f ( ) Gravity force g = π 3 D ρ g sh 6 Unsteady forces Event forces Buoyancy force Added mass Basset force wall, article, fibre interaction b = π 3 D ρ g sh f 6 π 3 Vm = D ρ u& u& sh f f ( u& u& u f u 3 f ) ( ) Basset = D πρ µ 0 dt + sh f f t t t N ei =, i= 0 ( ) igure 9 ormulation of force contributions as they would look like in order to turn the PME in Equ.9 into the classic BBO equation for small, sherical, non-couled articles.

13 Int. Jnl. of Multihysics Volume 3 Number In igure 9, c d denotes the drag coefficient and Re stands for the article Reynolds number. Usually a PME formulation like the one in Equ.9 is used for small articles and the classical Euler-Lagrange aroach [4]. This work however, treats large articles that san multile fluid cells, and still retains the tyical Euler-Lagrange methodology. Thus a secifically adusted, numerical scheme to model article-fluid interaction becomes necessary. Comarable rograms, like that of Schütz [7], are using article-related re-meshing of the fluid grid. 4 PARTICLE LUID INTERACTION To maximize calculation efficiency, a detailed drag imlementation, secifically adated to the case of dirt article filtration in lubricants has been created. The article fluid interaction model, consists of two alternative modules: ree low article fluid module ibre vicinity article fluid module Dirt articles are inected into the free flow regime ustream of the filter fibre geometry, where they occur in very low volume fractions. Particle-article interaction and hydrodynamic article imact on the fluid can be neglected here. As soon as the articles reach the fibre vicinity, the two-way couling takes effect and inter-article- as well as full article-fluid interaction becomes relevant. Those fundamentally different situations require searate drag modelling schemes in order to guarantee a good balance between accuracy and efficiency. 4. REE LOW PARTICLE-LUID INTERACTION MODULE Within the free flow regime, all article interactions with their surroundings are handled by the free flow module. In this zone, the most imortant asects of the revailing hydrodynamic situation are:.) The ratio between article diameter D sh and minimal distance to the nearest fibre (wall) boundary atch h w can be considered as small. Wall roximity has no effect on article drag..) Due to very low article volume fractions, the ratio between D sh and the minimal distance between neighbouring articles h can be considered as small. No hysical nor hydrodynamic article interaction takes lace. 3.) Due to very low article Reynolds number Re and very low article volume fraction, the hydrodynamic article effect on the fluid can be neglected. No twoway-couling is necessary. In the free flow regime it is rimarily imortant to gras torques, acting on the article. Rotational effects due to non-uniform flow fields can lead to a re-alignment of the articles, so that average enetration deth and filter fibre efficiencies are being influenced. In that context the anel descrition (see chater.5.) of the ellisoid shae is of secial imortance. The article is enclosed by M anels and each anel is subect to drag forces d, (which consist of ressure- and shear flow contribution,, and t, resectively) and hydrodynamic lift forces h,. orces that are better calculated by considering the entire article are: gravity g, buoyancy b and N event forces Σ e,i. Thus the adated PME within the free flow regime looks like Equ.30. m du dt M N (, τ, h, ) g b e, i = i= = + + (30)

14 40 Simulation of filtration rocesses in deformable media In analogy, torque effects on non-sherical rotation are described by: I dω dt M N ( τ h ) +,,, i = i= = r + r + r r ei, (3) Here r stands for the distance vector of each surface anel centre HP to the article mass centre X and r i denotes the distance vector from X to any article hel oint HP i that senses an imact event. The Basset history force and the added, virtual mass are being neglected because of the lack of strongly in stationary, relative article-fluid flow. Comaring Equ.30 with Equ.9, the following arallels can be drawn: = + d Saffman τ ( ) τ ( ) ( τ ) Magnus τ h axen h h (3) igure 0 shows a sketch of how the individual force contributions act on each anel and affect the article. HP r X u rel, r r + n h Panel d in HP d τ in out d HP + out h igure 0 Sketch of local force balance and force effect on anel centre. In igure 0, in stands for the force contribution of the incoming current, while out is the force contribution of the outgoing current as it would look like if it were deviated by the anel surface. The total force acting on each anel anel is given by:

15 Int. Jnl. of Multihysics Volume 3 Number = + anel d h (33) The following sub chaters describe the rocedural calculation of free flow drag- and lift force as well as torque effects within the module. 4.. ORCE CALCULATION irst the drag force contribution d, on each anel has to be calculated. The drag force term consists of a form drag- and a shear drag contribution,, and T,. While T, acts erendicular to the anel surface normal n,,, acts arallel to n,. However, since d,i is suosed to act in the direction of u rel,, the following ratio has to hold: τ,, = n e, urel, n e, urel, (34) Here e urel, is the base vector of the relative fluid-article-anel velocity encountered at the anel centre. The total anel drag coefficient c d,anel deends on the form drag coefficient c d, and on the shear drag coefficient c d,shear and is given by: c = c + c d, anel d, d, shear (35) In Equ.36 to Equ.40, form drag- and shear drag vectors are listed, as well as the total anel drag vector and its deendence on form and shear contribution,, * and T, * resectively. igure shows a sketch of the situation. = c c A u n e, ( + d, d, shear ) ρ f rel, (, urel, ) = c + c A ρ u e τ ( ), d, d, shear f rel, urel, * = c A u u, ρ d, f rel, rel, * = c A ρ u u τ, d, shear f rel, rel, = c A u u ρ d, d, anel f rel, rel, n, ( ) n, eurel, n, (36) (37) (38) (39) (40) In Equ.36 through Equ.40, A is the anel surface area and u rel, is the relative fluidarticle-anel velocity u f -u,. The article-anel velocity is given by the velocity of the article mass centre u and the rotational velocity contribution: u = u + r ω, (4)

16 4 Simulation of filtration rocesses in deformable media The anel Reynolds number is written as Re and is defined by using the hydraulic diameter d h, of the anel and the kinematic fluid viscosity η f : Re u = d rel, h, η f (4) u rel, * τ, *, d,, τ, igure Sketch of form- and shear force contribution to anel drag force. Secondly the hydrodynamic lift force h,, which stems from the deviation of the fluid at the anel, is calculated. The hydrodynamic lift is connected to in,, out, and d, via a simle, local force balance (see igure 0): = in, d, h, out, (43) Note that the suerscrit denotes the fact that force values are scaled by the acting surface area A. In addition to Equ.43, h, is defined to act erendicular to d,i, so that: h, =0 d, (44) The drag d, is given by Equ.40, while in can be easily derived out of the local fluid field information, obtained by the ressure/velocity hel oints. = ρ u u in, f rel, rel, (45) While the value of out is not known in advance, its base vector e out, is given because of anel orientation n, and relative anel-fluid velocity u rel,. e out, = ( u n u n rel, (, rel, ), ) ( u n rel, ( u, rel, ) n, ) Equ.43 and Equ.44 constitute a system of four equations and four unknowns: The three comonents of hydrodynamic lift h,,x, h,,y, h,,z and the absolute value of deviated flow momentum out,. The solution yields the following exressions for the local, hydrodynamic lift force vector h, and the vector of deviated fluid momentum out, : (46)

17 Int. Jnl. of Multihysics Volume 3 Number = c u u, ρ h danel, f rel, rel, urel, u n u ( ) rel,, rel, u rel, ( n u n, rel, ), (47) = c u, ρ out d, anel f rel, u urel, ( n u ) rel,, rel, u n u n rel, (, rel, ), (48) Due to non-couling, the wake of the article is not simulated in the free flow module. Therefore a anel has to face the current in order to yield accetable d, and h, results. The condition for calculating the individual force balance and for considering the anel is: u n rel,,! 0 (49) Consequentially the overall, unscaled drag force d unsc and hydrodynamic lift force h unsc are given by the contributions of all N considered anels: unsc d unsc h N = A, = N = A, = d h (50) (5) 4.. Weighing method and torque effect calculation The d, and h, calculation serves a useful urose: to obtain an idea of the force distribution over the article surface. This is necessary to gras rotational fluid field effects on the aligning article. In order to imrove the quantitative estimate on overall drag and lift force contributions, the anel results are being scaled to fit a newly found, emirical drag law for non-sherical articles that has been resented by Hölzer & Sommerfeld, [3]: c d somm = / Re Φ Re Φ Re Φ length ( log Φ). Φ cross (5) Here Φ, Φ length and Φ cross, are the standard-, the lengthwise- and crosswise-shericity, resectively. Using the Hölzer-Sommerfeld aroach, the over all scaled drag force d sc on the article can be calculated. In order to get a hold of realistic, rotational torque effects, the originally calculated, unscaled force contributions are scaled by the ratio d sc / d unsc : sc sc d = d, un d, d sc sc d = h, un h, d (53) (54)

18 44 Simulation of filtration rocesses in deformable media Then the more accurate, scaled, rotational torque effects are comuted using Equ.3: I dω M N = sc ( r r r sc d + h ) + i dt,, e, i = i= (55) igure shows a test case, where a longish, non-sherical article aroaches a valve of higher flow velocities and lower ressure. As hysically lausible and exected, the given drag imlementation models the occurring shear flow and ressure gradients over the article surface in such a way that the article aligns itself along the fluid stream lines. The translational and angular velocity vectors adat to the local fluid field conditions which leads to a article-sli effect (see igure ). igure Ellisoid article accelerating towards valve. Alignment along the current stream lines. Particle takes u its most stable osition of least drag- and lift forces. This behaviour causes the non-sherical sli effect with relevance for filtration efficiency and article enetration deth [3] Particle in fibre vicinity As soon as the article enters into the vicinity of the fibre geometry, the hydrodynamic situation changes as a whole and the fibre vicinity drag module takes over. The situation features the following characteristics:.) Particle wall flow effects can no longer be neglected since the ratio between article diameter and minimal article-wall (fibre) distance h is er definition no longer small..) Particles accumulate at the fibre in considerable volume fractions and the ratio between article diameter and medium, minimal article-article distance h is no longer small either. Particles interact hydrodynamically and hysically by lugging each others flow ath. 3.) The high article volume fractions, lead to a lugging of the fluid flow ath, diverting the flow and causing increased ressure dro over the filter. Hydrodynamic article imact on the fluid (two-way couling) becomes essential.

19 Int. Jnl. of Multihysics Volume 3 Number Emirical exressions, describing every one of those effects individually, can be found in literature [7], [3]. The fibre vicinity drag module incororates all those effects in a dynamic combination to a highly comlex, multi-arameter interaction situation. Local fluid cells, encomassed by articles are lugged by orosity adustment. The rocedure is analogous to the one resented in [] but has been extended to non-sherical article shaes, refined and quantified as shall be seen in the following. The lugging causes the fluid to be diverted around the fluid cell which leads to a local ressure build u i, that can be sensed by any of the N ressure hel oints HP i at the article surface (see igure 3). Since each ressure hel oint reresents /N th of the entire article surface area A and since ressure always acts erendicular to the local surface normal n i, the total ressure force on the article can be written as: = N An i i i i= (56) or infinitesimally fine grid sacing and an infinitely large number of ressure hel oints this exression amounts to: = nda= dv A V (57) The second, decisive force contribution results from viscosity effects (see igure 4). Because of a lack of wall boundary conditions at the border between lugged and unlugged cells, no zero velocity condition can be introduced at the article surface. What haens is that an effective zero velocity condition is imosed along a virtual surface including all cell centres ust within the article borders. Therefore local shear forces Ti at the hel oint ositions can be aroximated by using the velocity value of the nearest, unlugged fluid cell u f,i, at distance h u,i erendicular to the article surface. Thus the overall shear force T on the article can be calculated as: N = µ τ f i= urel, i h A e e n i urel, i urel, i i ui, ( ( )) (58) or infinitesimally fine grid sacing and an infinitely large number of ressure hel oints this exression amounts to: = u n da µ τ f A f (59) Where u f is the Jacobean of u f. Under the condition of incomressibility, div u f =0, this exression can be exanded and generalized. With τ being the viscous shear stress tensor, this amounts to: = τ n da= τ dv τ A V (60)

20 46 Simulation of filtration rocesses in deformable media Pressure and shear stress contributions to the overall drag force on an ellisoid article are reresented in igures 3 and 4. igure 3 Pressure force contribution to over all fluid-article force. Exemlary, two-way couled ellisoid. Pressure build u in frontal article area. ormation of ressure gradient across article surface. igure 4 Shear stress contribution to over all fluid-article force. Exemlary, twoway couled ellisoid. Plugging equals a zero flow velocity boundary condition at engulfed cell centres. Boundary layer is aroximated. Shear stresses can be derived. As a consequence of Equ.57 and Equ.60 the entire PME for the fibre vicinity module can be written as: m du dt m = ( + τ ) ρ (6)

21 Int. Jnl. of Multihysics Volume 3 Number or a limited number of discretizing surface elements N this exression yields: m du dt N N urel, i = An+ h A e i i i η e n f i urel,i urel, i i= i= ui, ( ( )) (6) It has to be stated that this drag- and lift force imlementation is grid deendent. The alied meshes however, are structured grids with never changing resolution. An exact knowledge about article shae and surface structure of actual dirt articles is not given. Yet lausibility commands the following statements to hold: Arbitrarily shaed dirt articles rather behave non-sherically than sherically. Dirt articles have rough surface structures than smooth surfaces. igure 5 qualitatively shows how some two-way couled, non-sherical articles can affect the surrounding fluid flow. To quantify the results, a fluid-article force evaluation within the fibre vicinity module igure 5 low field deviation by ellisoid articles getting stuck in simlified fibre structure. low field before inection of multile non-sherical articles (left). Deviated flow field after article inection and imact on fibres (right). has been conducted. Results have been comared to the corresonding values yielded by the free flow module, which is based on semi-emirical correlations (Hölzer/Sommerfeld, [3]), and to anlytical formulations (Stokes drag). Here the outcome shall be brievely discussed for the secial case of equiaxed (=sherical) ellisoids. The fibre vicinity drag module gives a binary, coarse, grid sacing s deendent reresentation of the article surface. Consequentially the module yields significantly higher fluid-article forces than a corresonding reresentation of smoothly surfaced obects. However, the qualitative drag behaviour against Re is fine. igure 9 summarizes the drag force behaviour of simle Stokes-flow-sheres and fibre-vicinity-module- sheres. The ratio S = s/d sh is used as a arameter. or suosedly, arbitrarily surfaced articles the CD results can be exected to be more aroriate, than any smooth surface reresentation. Still, correction functions have been introduced to comensate for surface roughness, and numerical resolution effects on a user

22 48 Simulation of filtration rocesses in deformable media defined basis. Because of the good qualitative behaviour of the solution, the finding of a suitable correction function is comaratively simle. Possible arameters of deendence are the article Reynolds number Re and the grid sacing ratio S. Exemlary cases within the arameter ranges 0.05 Re.0 and 0.05 S 0.5 have been evaluated. Note that the article model is, as of now, declared valid only for creeing flow conditions Re < 0.5. The correction function ζ is defined via the c d values of the analytical Stokes results c d,stokes and the model results c d,model : log( c ) d,mod el ς( S,Re ) = log( c ) dstokes, (63) An evaluation of the Re -influence shows, that for Re <0.5 the correction ζ does hardly vary with Re, if comared to the local average ζ, as seen in Equ.64 and in igure 6. ς Re ς 05. Re = 0 0 (64) An evaluation of the S-deendence shows, that the formulation of a simle correction igure 6 Plot of ζ against Re. or Re < 0.5 there is no relevant result deendence on Re. equation is ossible, as seen in igure 7. A linear fit to the ζ(s) results gives: ζ Re. S. S. < ( ) = (65) A third order, olynomial fit gives:

23 Int. Jnl. of Multihysics Volume 3 Number igure 7 Plot of ζ against S. Results (red) are fitted linearly (blue) and by a third order olynomial (green). 3 ζ Re. S 80. S S. 7860S < ( ) = + + (66) Here S stands for S-S 0, with S 0 =0.05. The smoothness correction ζ(s) is valid for Re < 0.5 and 0.05 S 0.5. Within that region, the corrected c d -values show an overall, relative, medium deviation from analytical results of ~8.% (linear fit) and ~5.% (olynomial fit). Similar results can be obtained for arbitrarily shaed ellisoids. A consideration of igure 7, Equ.65 and Equ.66 yields the surrising result that ζ will have to be smaller for larger S-values than for fine grid sacing, even though the shae reresentation gets worse. The exlanation for this is given by the fact that, with larger S, the closed fluid cell volume V block decreases as comared to the analytic volume of the obect V a, until S~0.5. or S-values larger than 0.5, the large article model forfeits its validity anyway. igure 9 shows analytical, un-corrected and smoothness-corrected model results in terms of c d -values. 5 CONCLUSION A non-sherical dirt article model for filtration alications has been rogrammed under the Oen Source CD tool box OenOAM. This work has resented the most essential, underlying concets behind the new solver. Moreover the two drag-/lift force modules for the free flow regime and the filter fibre vicinity have been thoroughly exlained and investigated. Thereby the efficient, orosity based, two-way couling strategy was discussed. A related, sequel article [3] is to be ublished as well. It focuses on event force modelling in terms of article-article, article-wall and article-fibre interaction effects. In addition to that, [3] deals with data conditioning and the develoment of a semi-analytical result

24 430 Simulation of filtration rocesses in deformable media igure 9 Plot of log(c d ) against log(re ). Averaged (over S range), original model results (urle) are fitted by olynomial smoothness correction, using Equ.66 (blue) to analytical Stokes drag results (yellow). Model is valid within the Stokes drag regime, log(re ) verification scheme. Based on the geometrically versatile shae of ellisoids, dirt articles, that feature six degrees of motional freedom, can now be modelled. A wide variety of article shaes, - anything from lates to sticks to simle sheres -, can be aroximated by this concet. The develoment effort behind this aer has entailed several rominent sub stes, such as: Design of an exlicit, force- and torque vector model, which reduces the modelling to the mere formulation of single force effects. Introduction of an adative time steing scheme for exlicit Euler discretization of the PME, [4]. Device of a surface hel oint scheme to account for large article effects in terms of fluid article, article fibre and article article interaction. Alication of a drag force imlementation that uses a combination of non-sherical, semi-emirical drag force formulas [3], a anel method to consider free flow swirling effects, a lugging method to include inter article and article fibre hydrodynamics and a simle adatation of basic concets known from the immersed boundary method [6]. Imlementation and descrition of an efficient article fluid, two way couling method []. This aer resents the framework for large, non-sherical, versatile articles with multile abilities. The concet of additional event forces, which can be added to the PME, enables the modular addition of individual interaction effects like those, resented in [3]:

25 Int. Jnl. of Multihysics Volume 3 Number Particle-article interaction effects Particle interaction with wall boundary atches Particle-fibre interaction effects The nature of the solver is such, that other, additional features could be imlemented retty easily in the future. Some ossible examles are: Electro static interaction Diffusion contributions to article motion Particle-article skin friction forces In a further ste, the fully assembled article module can be easily added into any laminar, continuum based, Eulerian fluid solver. Since validation is crucial as well, extensive measures to verify solver functionality and result quality had to be taken. Article [3] reorts of the develoment of a semi-analytical verification scheme to back u CD results. In the future this CD tool will enable the virtual, urely digital re-design of filter fibre materials. It will then be ossible to uload artificially created fibre structures, conduct the CD analysis and to use the results as a good estimate on the exected erformance of the newly designed fibre material. 6 REERENCES. M. Mataln, G.Boiger, W. Brandstätter, B. Gschaider, (008). Simulation of Particle iltration Processes in Deformable Media, Part : luid-structure Interaction, ICE Stroemungsforschung GmbH., Montanuniversitaet Leoben. Int.Journal of Multihysics, Vol.,( No.), July 008, ();. G. Boiger, M. Mataln, W. Brandstätter, B. Gschaider, (008). Simulation of Particle iltration Processes in Deformable Media, Part : Large Particle Modelling, ICE Stroemungsforschung GmbH., Montanuniversitaet Leoben. Int.Journal of Multihysics, Vol.,(No.), July 008,. 9 06(6)8; 3. G. Boiger, M. Mataln, W. Brandstätter, (009). Simulation of Particle iltration Processes in Deformable Media, Part 3.: Interaction modelling and solver verification of a non-sherical dirt article solver, ICE Stroemungsforschung GmbH., Montanuniversitaet Leoben. Article in review since Jan 009. Int.Journal of Multihysics; 4. G. Boiger, M. Mataln, W. Brandstätter, (009). Adative time steing for exlicit Euler imlementation of sherical and non-sherical article seed u. ICE Stroemungsforschung GmbH., Montanuniversitaet Leoben. Article in review since Jan 009. Int.Journal of Multihysics; 5. K. Thomsen, (004). Simlified Surface area of Ellisoid. Numericana.com, inal Answers. 6. S.Schütz, M.Piesche, G.Gorbach, M.Schilling, C.Seyfert, P.Kof, T.Deuschle, N.Sautter, E.Po, T.Warth. (007). CD in der mechanischen Trenntechnik. Chemie Ingeneur Technik 007, 79, No.II. DOI: 0.00/cite ; 7. S.Schütz, M.Schilling, M.Piesche. (007). Bestimmung der Widerstandskraft und des Transortverhaltens kugelförmiger Partikel in Strömungen mit Hilfe moderner CD-Werkzeuge. Chemie Ingeneur Technik 007, 79, No.. DOI: 0.00/cite ; 8. L.Rosendahl. (999). Using a multi-arameter article shae descrition to redict the motion of Non-Sherical article shaes in swirling flow. Institute of Energy Technology, Aalborg University. Alied Mathematical Modelling 4 (000), 5.

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