Design appropriate modeling for determining optimal friction reduction with surface textures

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Georgia Institute of Technology Marquette University Milwaukee School of Engineering North Carolina A&T State University Purdue University University of California, Merced University of Illinois,

1 Georgia Institute of Technology Marquette University Milwaukee School of Engineering North Carolina A&T State University Purdue University University of California, Merced University of Illinois, Urbana-Champaign University of Minnesota Vanderbilt University Design appropriate modeling for determining optimal friction reduction with surface textures Jonathon K. Schuh, Yong Hoon Lee, James T. Allison, Randy H. Ewoldt University of Illinois at Urbana-Champaign Fluid Power Innovation & Research Conference Minneapolis, MN October 10-12, 2016

2 Motivation Objective: decrease friction in lubricated sliding contact Decrease shear stress Increase separating normal force Method: use surface textures Want to determine optimal design of surface texture for decreasing friction Mathematical model needed 2

3 Building Intuition: Experimental Results [1] Schuh and Ewoldt, Tribology International, Model for normal force production? 3

4 Normal force Scaling Inertia: F N ~ rw 2 Viscous: F N ~ hw Cavitation: F N ~ f even ( W) Normal force due to viscous effects. Model using Stokes or Reynolds equation. [1] Schuh and Ewoldt, Tribology International,

5 Reynolds Equation 1 r r ru r ( ) + 1 r u q q + u z z 0 = -Ñp + hñ 2 u 1 st design choice: Stokes flow over Navier-Stokes. Stokes flow is linear, while Navier-Stokes is not. Linear problems are easier to solve for design Assume: One linear PDE that when solved gives h flow field: 0 ideal << for 1 design where it while 2pbe R 0 solved repeatedly. p z = 0 u q = 1 2hr 1 r u r = 1 2h p r = h 2 u r z 2 p q = h 2 u q z 2 p ( ( )) r z2 - zh r,q p q z2 - zh( r,q ) æ è ç ( ) + rw 1- Substitue u r and u q into continuity, integrate in z 1 r æ pö r è ç rh3 r ø + 1 æ r q è ç h 3 r pö q ø z h r,q ( ) = 6hW h q ö ø 5

6 Previous 1-D Theories Cartesian 1-D Unable to correctly predict sign from experiments. Need 2-D cylindrical coordinates 6

7 Solving Reynolds equation numerically 1 æ pö r r è ç rh3 r ø + 1 æ r q è ç Solve using pseudo-spectral method Langrange polynomial basis functions Zeros are Gauss-Lobatto- Legendre points [2,3] Accurate numerical evaluation of integrals [3] ε~e -N [4] Turn solving PDE into solving linear system wkp=wf [5,6] One matrix inversion gives all design variables: Ideal for design h 3 r pö q ø = 6hW h q [2] M.T. Heath, Scientific Computing: An Introductory Survey, 2002 [3] B. Fornberg, A practical Guide to Pseudospectral Methods, 1998 [4] M.Y. Hussaini and T.A. Zang, Annual Review of Fluid Mechanics, 1987 [5] D.A. Kopriva, Implementing spectral methods for partial differential equations, 2009 [6] G.E. Karniadakis and S. Sherwin, Spectral/hp element method for computational fluid dynamics,

8 Code Validation Analytical solution to Reynolds equation ( ) R o - R i 1 r Constant ( ) h = h q 1 d æ r dq è ç h 3 r dp ö dq ø = 6hW dh dq ( ) ( ) h = h 1 1-k q + h 1 1+k, k = h 0 j 2 h 1 F N = ò A æ æ 1- R ö i pda = 3 2 ç 2 hwr 2 æ Rj ö è ç R o ø è o è ç ø 1-k h 1 4 ö ø æ è ç ln ( ) ( k ) + 2( 1-k ) 2 1+k ö ø Expected exponential scaling seen. Code is validated. 8

9 Surface Textures R i p r r=r i = 0 p r = p q =-j /2 r q =j /2 q r R c R t Geometric Parameter R o Value 20 mm j = 2p N tex R o R c mm R t 3 mm φ π/5 rad 9

10 Determining R i Must use finite inner radius in discretized form 1/r will diverge if R i =0 Use R i =0.01 mm 0.05% of R o 10

11 Boundary condition at outer edge Either ( ) = 0 p r = R o p r r=r o = 0 Exp. Shapes Use p r r=r o = 0 No mass flux at boundary Other Shapes depend on BC 11

12 Regions of Applicability Reynolds equation is derived using assumptions Not valid for all input values Should only compare to experiments where equation is valid Neglect Inertia: Re h < 0.1 Re h 1 ( ) ( ) ( ) Negelct viscous heating: Na < 0.1 Na 1 Negelct gradients in flow: h 0 / R < 0.1 h 0 / R 1 12

13 Numerical Outputs 13

14 Shear Stress Comparison s = N å i=1 ( h a exp i -h a sim i ) 2 N Good agreement between experiments and simulations; σ=±3.1%. 14

15 Normal Force Comparison Correct sign of normal force given by simulations. s = N å i=1 ( F N exp i - F Nsim i ) 2 N Correct order of magnitude of force predicted. 15

16 Friction Coefficient Comparison m * = M / R F N Good agreement between experiments and simulations where Reynolds equation is valid 16

17 Future Work Use validated code for arbitrarily defined surfaces Ex: define surface using cubic splines Determine optimal design of texture with Newtonian fluid 17

18 Conclusions Developed code for solving the Reynolds equation in cylindrical coordinates Design appropriate model Validated the code against an analytical solution Expected scaling in error seen. Good agreement between code and experiments in region where Reynolds equation applies Correct sign of normal force Correct order of magnitude Optimal angle β exists for decreasing friction with asymmetric surface textures 18

19 Acknowledgements and Contact Information Ewoldt Research Group Funding Center for Compact and Efficient Fluid Power Allison Research Group Specific People Michael Johnston (MS UIUC, 2014) Nathan Bristow (BS UIUC, 2014) Nikita Dutta (REU summer 2014) Feargus MacFhionnlaoich (REU summer 2015) Jonathon Schuh 19

20 Appendix 20

21 Derivation of Reynolds equation Ñ iv = 0 -Ñp +hñ 2 v = 0 Non-Dimensionalization r * = r R, z* = z h 0 v * r = v r WR, v * q = v q WR, v * z = h 0 R 0 v z æ WR h 0 è ç R ö ø Ñ iv = 0 h 2 v r z = p 2 r h 2 v q z 2 = 1 r 0 = p z p q j R R c R t Integrate momentum in z and apply BC's v r = 1 p 2h r z2 - zh v q = 1 2hr ( ) p q z2 - zh h D L æ ( ) + rwç h - z Integrate continuity in z using Leibniz rule 1 r æ pö r è ç rh3 r ø + 1 æ r q è ç h 3 r è pö q ø h W ö ø = 6hW h q One equation for pressure, which when solved gives flow field 21

22 Solving Reynolds equation numerically 1 r æ pö r è ç rh3 r ø + 1 æ r q è ç h 3 r pö q ø ( ) = 6hW h ( ) = 6hW h q Ñ i h 3 Ñp q ò wñ i h 3 Ñp da = ò w6hw h q da A ò wh 3 Ñp iˆn ds - ò Ñw i( h 3 Ñp) da = ò w6hw h q da A A -ò Ñw i( h 3 Ñp) da = ò w6hw h q da A 0 A A A [2-4] [2] M.T. Heath, Scientific Computing: An Introductory Survey, 2002 [3] D.A. Kopriva, Implementing spectral methods for partial differential equations, 2009 [4] G.E. Karniadakis and S. Sherwin, Spectral/hp element method for computational fluid dynamics,

23 Pseudo-Spectral Discretization Solve equation in weak form [2,6,7] év æ pö r r è ç rh3 r ø + v æ h 3 pö ù ò ê r q è ç r q ø úda = ò v6hw h da A ë û q A Use integration by parts é v p - r rh3 r + v h 3 p ù ò ê ë q r q ú û dr dq = ò v6hwr h dr dq q A A [2] M.T. Heath, Scientific Computing: An Introductory Survey, 2002 [6] D.A. Kopriva, Implementing spectral methods for partial differential equations, [7] G.E. Karniadakis and S. Sherwin, Spectral/hp element method for computational fluid dynamics, 2005

24 Pseudo-Spectral Discretization Use change of variables to use Gauss-Lobatto-Legendre quadrature x = 2 R o - R i y = 2 j q æ è ç r - R o + R i 2 ö ø Write functions using basis functions v = p = h = N N åå i=1 N j=1 N åå l=1 N m=1 N åå q=1 r=1 v ij r ( x i )r ( y j ) p lm r ( x l )r ( y m ) h qr r ( x q )r ( y r ) Turns PDE into set of linear equations Kp = f [2] M.T. Heath, Scientific Computing: An Introductory Survey, 2002 [5] B. Fornberg, A practical Guide to Pseudospectral Methods, 1998 [6] D.A. Kopriva, Implementing spectral methods for partial differential equations, 2009 [7] G.E. Karniadakis and S. Sherwin, Spectral/hp element method for computational fluid dynamics,

25 Computed Fields 25

26 Future Work (Continued) Use code for restricted set of 2 nd order fluids b 11 =b 2 3-D flow theorem of Giesekus [7] p = p N + b 2 Dp N + b 2 b 1 Dt 4 g (1) :g (1) Determine optimal design of both fluid and texture ( ) [7] Bird et. al., Dynamics of Polymeric Liquids,

F O R SOCI AL WORK RESE ARCH

7 TH EUROPE AN CONFERENCE F O R SOCI AL WORK RESE ARCH C h a l l e n g e s i n s o c i a l w o r k r e s e a r c h c o n f l i c t s, b a r r i e r s a n d p o s s i b i l i t i e s i n r e l a t i o n