A SMOOTHED DISSIPATIVE PARTICLE DYNAMICS METHODOLOGY FOR WALL-BOUNDED DOMAINS

Transcription

1 A SMOOTHED DISSIPATIVE PARTICLE DYNAMICS METHODOLOGY FOR WALL-BOUNDED DOMAINS APPROVED: by Jun Yang A Dssertaton Submtted to the Faculty of WORCESTER POLYTECHNIC INSTITUTE n partal fulfllment of the requrement for the degree of Doctor of Phlosophy n Mechancal Engneerng Aprl 5, 013 Dr. Nkolaos A. Gatsons, Advsor Professor, Mechancal Engneerng Department, WPI Dr. Davd J. Olnger, Commttee Member Assocate Professor, Mechancal Engneerng Department, WPI Dr. Smon W. Evans, Commttee Member Assstant Professor, Mechancal Engneerng Department, WPI Dr. Marcus Sarks, Commttee Member Professor, Department of Mathematcs, WPI Dr. George E. Karnadaks, Commttee Member Professor, Dvson of Appled Mathematcs, Brown Unversty Dr. Mark W. Rchman, Graduate Commttee Representatve Assocate Professor, Mechancal Engneerng Department, WPI

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3 ABSTRACT Ths work presents the mathematcal and computatonal aspects of a smooth dsspatve partcle dynamcs wth dynamc vrtual partcle allocaton method (SDPD-DV) for modelng and smulaton of mesoscopc fluds n wall-bounded domans. The SDPD-DV method s realzed wth flud partcles, boundary partcles and dynamcally allocated vrtual partcles near sold boundares. The physcal doman n SDPD-DV contans external and nternal sold boundares, perodc nlets and outlets, and the flud regon. The sold boundares of the doman are represented wth boundary partcles whch have an assgned poston, wall velocty, and temperature upon ntalzaton. The flud doman s dscretzed wth flud partcles placed n a global ndex. The algorthm for nearest neghbor partcle search s based on a combnaton of the lnked-cell and Verlet-lst approaches and utlzes large rectangular cells for computatonal effcency. The densty model of a flud partcle n the proxmty of a sold boundary ncludes the contrbuton from the vrtual partcles n ts truncated support doman. The thermodynamc propertes of a vrtual partcle are dentcal to those of the correspondng flud partcle. A perodc boundary partcle allocaton method s used at perodc nlets and outlets. Models for the conservatve and dsspatve forces on a flud partcle n the proxmty of a sold boundary are presented and nclude the contrbutons of the vrtual partcles n ts truncated support doman. The ntegraton of the flud partcle poston and momentum equatons s accomplshed wth an mplementaton of the velocty-verlet algorthm. The ntegraton s supplemented by a bounceforward algorthm n cases where the vrtual partcle force model s not able to prevent partcle penetraton. The ntegraton of the entropy equaton s based on the Runge-Kutta scheme. In sothermal smulatons, the pressure of a flud partcle s obtaned by an artfcal compressblty formulaton for lquds and the deal gas law for compressble fluds. Samplng methods used for 3

4 partcle propertes and transport coeffcents n SDPD-DV are presented. The self-dffuson coeffcent s obtaned by an mplementaton of the generalzed Ensten and the Green-Kubo relatons. Feld propertes are obtaned by samplng SDPD-DV outputs on a post-processng grd that allows harnessng the partcle nformaton on desred spato-temporal scales. The sothermal (wthout the entropy equaton) SDPD-DV method s verfed and valdated wth smulatons n bounded and perodc domans that cover the hydrodynamc and mesoscopc regmes. Verfcaton s acheved wth SDPD-DV smulatons of transent, Poseulle, body-force drven flow of lqud water between plates separated by 10-3 m. The velocty profles from the SDPD-DV smulatons are n very good agreement wth analytcal estmates and the feld densty fluctuaton near sold boundares s shown to be below 5%. Addtonal verfcaton nvolves SDPD-DV smulatons of transent, planar, Couette lqud water flow. The top plate s movng at and separated by 10-3 m from the bottom statonary plate. The numercal results are n very good agreement wth the analytcal solutons. Addtonal SDPD-DV verfcaton s accomplshed wth the smulaton of a bodyforce drven, low-reynolds number flow of water over a cylnder of radus. The SDPD-DV feld velocty and pressure are compared wth those obtaned by FLUENT. An extensve set of SDPD-DV smulatons of lqud water and gaseous ntrogen n mesoscopc perodc domans s presented. For the SDPD-DV smulatons of lqud water the mass of the flud partcles s vared between 1.4 and real molecular masses and ther correspondng sze s between 1.08 and 33 physcal length scales. For SDPD-DV smulatons of gaseous ntrogen the mass of the flud partcles s vared between and real molecular masses and ther correspondng sze s between. 10 and physcal length scales. The equlbrum states are obtaned and show that the partcle speeds scale nversely wth 4

5 partcle mass (or sze) and that the translatonal temperature s scale-free. The self-dffuson coeffcent for lqud water s obtaned through the mean-square dsplacement and the velocty auto-correlaton methods for the range of flud partcle masses (or szes) consdered. Varous analytcal expressons for the self-dffusvty of the SDPD flud are developed n analogy to the real flud. The numercal results are n very good agreement wth the SDPD-flud analytcal expressons. The numercal self-dffusvty s shown to be scale dependent. For flud partcles approachng asymptotcally the mass of the real partcle the self-dffusvty s shown to approach the expermental value. The Schmdt numbers obtaned from the SDPD-DV smulatons are wthn the range expected for lqud water. The SDPD-DV method (wth entropy) s verfed and valdated wth smulatons wth an extensve set of smulatons of gaseous ntrogen n mesoscopc, perodc domans n equlbrum. The smulatons of N (g) are performed n rectangular domans wth n the range ~ m, wth flud mass n the range ~ kg. The mass of the flud partcles s vared between 118 and real molecular masses. The self-dffuson coeffcent for N (g) at equlbrum states s obtaned through the mean-square dsplacement for the range of flud partcle masses (or szes) consdered. The numercal self-dffuson s shown to be scale dependent. The smulatons show that self-dffuson decreases wth ncreasng mass rato. For a gven mass rato, ncreasng the smoothng length, ncreases the self-dffuson coeffcent. The shear vscosty obtaned from SDPD-DV s shown to be scale free and n good agreement wth the real value. We examne also the effects of tmestep n SDPD-DV smulatons by examnng thermodynamc parameters at equlbrum. These results show that the tme step can lead to a sgnfcant error dependng on the flud partcle mass and smoothng 5

6 length. Fluctuatons n thermodynamc varables obtaned from SDPD-DV are compared wth analytcal estmates. Addtonal verfcaton nvolves SDPD-DV smulatons of steady planar thermal Couette flow of N (g). The top plate at temperature T 1 =330K s movng at V xw =30m/s and s separated by 10-4 m from the bottom statonary plate at T =300K. The SDPD-DV velocty and temperature felds are n excellent agreement wth those obtaned by FLUENT. 6

7 ACKNOWLEDGEMENTS I would lke to express the deepest apprecaton to my advsor, Prof. Gatsons, who contnually and convncngly conveyed a sprt of adventure n regard to research and scholarshp, and an exctement n regard to teachng. You have shaped the professonal that I have become and further defned my sense of scentfc rgor and engneerng creatvty. Thank you for the years of support and nspraton. Great thanks are gven to Dr. Raffaele Potam whose ntal work on SDPD-DV has been the bass for the current mplementaton of SDPD-DV. Hs assstance was essental and nvaluable to the advancements of SDPD-DV pursued by me. I am very apprecatve for the nterest that Professor Karnadaks expressed n ths work. Hs nputs strengthened ths work and deepened my understandng. I would lke to thank also all the members of my commttee for ther tme, patence, and expertse. Several of the WPI faculty provded nvaluable expertse on many aspects of ths research. However, I would lke to specfcally acknowledge the efforts of Sa Najaf not only for hs expertse, contnued support and endless patence wth the Lnux unntated, but also regardng lfe overall. I would lke to thank all of the teachers, coaches and professors who have collectvely molded the person that I am today. I would lke to specfcally thank Prof. Dapeng Hu of the Dalan Unv. of Technology n Chna, t s your dscplne and scrutny of detal that s nstlled n all the work that I do, and at tmes of need, t s your teachngs that resound n my mnd and enforce the purty of scence and the rgor that must defne all that t touches. 7

8 I would lke to extend the sncerest thanks and apprecaton to all of the frends that I have made here at WPI throughout the years. You are too numerous to name yet too dear to forget. Grattude s extended to the Mechancal Engneerng Department, and especally to Glora Boudreau, Stata Cannng, Barbara Edlbert, Barbara Furhman and Patrca Rheaume. Most of mportant, I would lke to thank my parents. Ther contnued love, support, and admraton are empowerng. From my parents, I learned the value of educaton both from ther nstructon and nfluence. My success s a reflecton of ther hard work and sacrfce. I thank them for all that I am and all that they have done for me. Specal thanks go to my frends I met n Chrstan Gospel Church of Worcester for ther contnuous help and support throughout the years that I had spent at WPI. Ths work was partally supported by AFOSR s Computatonal Mathematcs Program under Grant FA I would also lke to acknowledge the support obtaned through Teachng Assstantshps obtaned from the Mechancal Engneerng Department at WPI. Ths work s dedcated to my mother and father. 8

9 TABLE OF CONTENTS ABSTRACT... 3 ACKNOWLEDGEMENTS... 7 TABLE OF CONTENTS... 9 LIST OF FIGURES LIST OF TABLES NOMENCLATURE INTRODUCTION Fundamental Spatotemporal Scales n Fluds DPD Overvew SPH Overvew SDPD Method: Lterature Revew and Outstandng Issues Boundary Condton for Wall-Bounded Domans SDPD Self-Densty Applcatons of the Full Non-sothermal SDPD Model Objectves and Approach SDPD-DV METHODOLOGY AND IMPLEMENTATION Overvew of the SDPD Method Mathematcal and Computatonal Aspects of the SDPD-DV method Boundary Partcle Loadng and Global Indexng Flud Partcle Loadng and Global Indexng Nearest Neghbor Partcle Search (NNPS) Flud Partcle Densty Evaluaton Flud Partcle Pressure and Temperature Evaluaton Flud Partcle Force Evaluaton

10 ..7 Integraton of Flud Partcle Poston, Momentum and Entropy Equatons Partcle Propertes and Transport Coeffcents Instantaneous and Sample-Averaged Flud Feld Propertes VERIFICATION, VALIDATION, AND ERROR OF THE ISOTHERMAL SDPD-DV Transent Body-Force Drven Planar Poseulle Flow Input Condtons and Computatonal Parameters Results and Dscusson Transent Planar Couette Flow Input Condtons and Computatonal Parameters Results and Dscusson Steady Low-Re Incompressble Flow over a Cylnder Input Condtons and Computatonal Parameters Results and Dscusson Equlbrum State and Self-Dffuson Coeffcent Input Condtons and Computatonal Parameters Results and Dscusson VERIFICATION, VALIDATION AND ERROR ANALYSIS OF SDPD-DV Equlbrum State and Transport Coeffcents Steady Planar Thermal Couette FLow SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary and Conclusons Recommendatons for Future Work REFERENCES APPENDIX A. SDPD Analytc Self-dffuson Coeffcent APPENDIX B. Boundary Method n 3D wth Lucy s Smoothng Functon

11 LIST OF FIGURES Fgure 1. The DPD cutoff radus used n calculatng nterpartcle forces Fgure. Dscrete SPH partcles representng the flud and the support doman of the nterpolatng functon around an SPH partcle Fgure 3. Contrbuton of neghborng partcles n partcle support doman to the number densty evaluaton Fgure 4. General flow chart of SDPD-DV and post-processng... 5 Fgure 5. Physcal wall-bounded doman wth an exteror wall and an nteror sold body showng the flud partcles (FPs), boundary partcles (BPs), and vrtual partcles (VPs) used n the SDPD-DV. Inlets and outlets are consdered perodc. The large rectangular cells are used for nearest neghbor partcle search (NNPS) Fgure 6. Normal vector for a BP on a sold boundary Fgure 7. Rectangular cells used for NNPS n the nteror, near a sold and near a perodc boundary Fgure 8. Support doman for a flud partcle (FP) for three cases consdered n SDPD-DV Fgure 9. Dynamc vrtual partcle allocaton (DVPA) method used n SDPD-DV for densty and force evaluaton. A vrtual partcle (VP) s generated for a flud partcle (FP) near a sold boundary represented by boundary partcles (BP) Fgure 10. Reflectve bounce-forward method n SDPD-DV appled n a case where a flud partcle (FP) penetrates a sold boundary represented by boundary partcles (BP) Fgure 11. Samplng used nn SDPD-DV for evaluaton of the self-dffuson coeffcent. The delay tme s τ. The total number of samples s M, and K s the number of samples n delay tme. The tme orgns are ndexed from m=1 ~ (M-K+1)

12 Fgure 1. Post-processng grd used for evaluaton of feld (Euleran) nstantaneous and tmeaveraged propertes from SDPD-DV partcle samples Fgure 13. SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. Physcal doman showng the BPs and FPs Fgure 14. Sample-averaged flud densty ρ(z d ) and velocty Vx(Z d,t) from SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1,000 kg/m 3, T =300 K, η =10-3 kg m -1 s -1, and f x =10-4 m/s. The analytcal profles are plotted for verfcaton (Morrs et al. 1997) Fgure 15. Forces due to vrtual partcles from SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1,000 kg/m 3, η =10-3 kg m -1 s -1, and f x =10-4 m/s Fgure 16. SDPD-DV smulatons of transent planar Couette flow. Physcal doman showng the BPs and FPs Fgure 17. Sample-averaged flud densty ρ(z d )and velocty V x (Z d, t) from SDPD-DV smulatons of transent, planar, Couette flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1000 kg/m 3, T =300 K, η =10-3 kg m -1 s -1, and V xw =1.5x10-5 m/s. The analytcal profles are plotted for verfcaton (Morrs, Fox and Zhu, 1997) Fgure 18. Computatonal domans used n SDPD-DV and FLUENT smulatons of steady, low- Reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s. (a) SDPD- DV doman wth L X =0.1 m, L Y =0.015 m, L Z =0.1 m showng the BPs on the cylnder and FPs n the doman. (b) FLUENT doman L X =0.3 m, L Y =0.015 m, L Z =0.3 m ncludes an array of 1

13 cylnders to smulate the perodc flow. The nsert shows the extend of the SDPD-DV physcal doman. (c) Planes P1,P,P3 and contours C1,C,C3,C4 used for comparson of SDPD-DV and FLUENT results Fgure 19. Sample-averaged V x (r) and P(r) on y = 0 m, y = m, y = m planes from SDPD-DV smulatons. V x (x,z) and P(r) from FLUENT smulaton. Steady, low- Reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s Fgure 0. Sample-averaged V x (r) and P(r) on y = m plane and along C1,C,C3,C4 from SDPD-DV smulatons. V x (x,z) and P(r) from FLUENT smulaton. Steady, low- Reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s Fgure 1. Phase plot v x -v y from SDPD-DV smulatons of H O(l) wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1 and N (g) wth ρ a =1,184 kg/m 3, T = 300 K, η =10-5 kg m -1 s -1. Results show the scale effects of flud partcle sze on velocty Fgure. Average translatonal temperature from SDPD-DV smulatons of H O(l) wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1 and N (g) wth ρ a =1,184 kg/m 3, T = 300 K, η =10-5 kg m -1 s -1. (a) Average translatonal temperature as a functon of tme (Case 1, H O(l)). (b) Average translatonal temperature and standard devaton as a functon of m and D for H O(l). (c) Average translatonal temperature and standard devaton as a functon of m and D for N (g) Fgure 3. Self-dffuson coeffcent and Schmdt number from SDPD-DV smulatons of H O(l), ρ a =1,000 kg/m 3, η=10-3 kg m -1 s -1, T=300K for varous szes of the SDPD-DV flud 13

14 partcles. Expermental and analytcal estmates, as well as analytcal estmates usng SDPD-DV parameters are shown for valdaton and verfcaton Fgure 4: Self-dffuson coeffcent from SDPD-DV smulatons of N (g) for values of h and m /m N(g) used n Cases 1-0. Analytcal estmates are from Eq.(4.4) and data by Holz and Sacco (000) Fgure 5 : Shear vscosty from SDPD-DV smulatons of N (g) for value of m /m N(g) used n Cases 1-0. Intal nput vscosty η(t=0) s plotted for verfcaton Fgure 6: Effects of tme step on equlbrum temperature, translatonal temperature and pressure from SDPD-DV smulatons of N (g). Results are for Case 9, 10, and Fgure 7: Effects of tme step on equlbrum temperature, translatonal temperature and pressure from SDPD-DV smulatons of N (g). Results are for Case 17, 18, and Fgure 8: Computatonal domans used n SDPD-DV and FLUENT smulatons of steady N (g) Couette flow wth ρ a =1.184 kg/m 3, η 0 = kg m -1 s -1, κ 0 =0.06 W m -1 K -1. The upper wall T 1 = 300 K, lower wall T = 300 K and V xw =30m/s Fgure 9: Sample-averaged V x (r) and T(r) on y = 0 m, y = m, y = m planes from SDPD-DV and FLUENT smulatons of steady state Couette flow. The doman has L X =10-4 m, L Y = m, L Z =10-4 m, upper wall T 1 = 300 K, lower wall T = 300 K, and V xw =30 m/s. The flud s N (g) wth ρ a =1.184 kg/m 3, η 0 = kg m -1 s -1 and, κ 0 =0.06 W m -1 K Fgure 30: Sample-averaged flud temperature T(r d ) and velocty V x (r d ) of steady-state Couette flow from SDPD-DV and FLUENT smulatons. The doman has L X =10-4 m, L Y = m, L Z =10-4 m, upper wall T 1 = 300 K, lower wall T = 300 K and V xw =30 m/s. The flud s N (g) wth ρ a =1.184 kg/m 3, η 0 = kg m -1 s -1 and, κ 0 =0.06 W m -1 K

15 Fgure 31: Functon h for Lucy kernel n -D and 3-D Fgure 3: Functon h for Lucy kernel n -D and 3-D Fgure 33: Functon h for Lucy kernel n -D and 3-D Fgure 34: Functon h for Lucy kernel n -D (Vazquez-Quesada formulaton) and n 3-D Fgure 35: Functon h for Lucy kernel n -D (corrected) and 3-D

16 LIST OF TABLES Table 1. Propertes of gaseous ntrogen and lqud water at STP Table. Nearest Neghborng Partcle Search Algorthm used n SDPD-DV Table 3. Dynamc Vrtual Partcle Allocaton and Densty Evaluaton Algorthm used n SDPD- DV Table 4. Tme ntegraton n SDPD-DV. Velocty-Verlet s used for the partcle poston and momentum equaton, and Runge-Kutta s used for the entropy equaton Table 5. Bounce-forward Method used n SDPD-DV n case of partcle penetraton Table 6: Input parameters used n SDPD-DV smulatons of transent planar Poseulle flow, transent planar Couette flow, and flow over a cylnder Table 7. Input and derved parameters n SDPD-DV smulatons of mesoscale flows of lqud water at equlbrum states n rectangular domans wth perodc boundares Table 8. Input and derved parameters n SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares Table 9: Input parameters n full-set SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares Table 10: Derved parameters n SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares Table 11: Fluctuatons n temperature, densty, pressure and velocty, from SDPD-DV and analytcal expressons Table 1: Input parameters used n SDPD-DV non-equlbrum smulatons of Couette flow

17 NOMENCLATURE Boldface denotes a vector. The magntude of a vector s denoted usng the same symbol as the vector, but wthout boldface. Duplcate use of a symbol, or usage not defned below, wll be clarfed wthn the text. c Sound speed Kn Knudsen Number cv Specfc heat capacty at constant m Mass C V Heat capacty at constant volume Mean free path D E Indvdual partcle dameter Densty Energy N P Number of partcles Vscosty Pressure f Body Force r poston C F D F R F h Total conservatve force Total dsspatve force Total random force Smoothng length Thermal conductvty S t T v V Entropy Tme step Temperature velocty Volume Bulk vscosty k B Bolztmann constant 17

18 1. INTRODUCTION The numercal modelng and smulaton of gases, lquds and mult-phase systems above the atomc spatotemporal scales and below macroscopc scales has become ncreasngly mportant due to numerous applcatons n physcal, bologcal and engneerng systems. Ths regme can be characterzed as mesoscopc and requres new mathematcal and computatonal methods because the tradtonal atomstc (mcroscopc) and hydrodynamc (macroscopc) descrptons are not vald for ether computatonal or theoretcal reasons. One of the key characterstcs of the mesoscopc flow regme s the presence of thermal fluctuatons. The Smooth Dsspatve Partcle Dynamcs (SDPD) developed by Espanol and Revenga (003) has been proposed as a method approprate for mescoscopc flows wth fluctuatons. The SDPD nvokes the Smoothed Partcle Hydrodynamcs (SPH) whch s a well-developed method for the Naver-Stokes equatons (Lu and Lu, 003). Usng the GENERIC framework developed by Ottnger (005) to descrbe hydrodynamc fluctuaton, Espanol and Revenga arrved to SDPD dscrete equatons, whch nclude thermal fluctuatons. A three-dmensonal mplementaton of the SDPD method n unbounded domans wth dynamc vrtual partcle allocaton (SDPD-DV) has been pursued n the Computatonal Gas&Plasma Dynamcs Lab (CGPL) at WPI (Yang et al., 011; Yang et al., 01; Yang et al., 013; Gatsons et al., 013). Ths work on SDPD-DV s part of the effort at the CGPL to develop smulaton methods that apply to the nvestgaton of mcro- and nanoscale flows and nvolves the Drect Smulaton Monte Carlo (DSMC) method for gases (Gatsons et al., 010; 013) and partcle-n-cell for plasmas (Gatsons et al., 009). There are three goals n ths work: Revse and further mplement exstng algorthms n SDPD-DV, as well as develop and mplement new algorthms. 18

19 Valdate and verfy the algorthms of the SDPD-DV code by comparsons wth expermental, analytcal and numercal solutons. Investgate thermodynamc propertes and transport coeffcents wth applcatons of the SDPD-DV code to mesoscopc systems n equlbrum. We revew below characterstc length, tme scales and non-dmensonal parameters used to characterze lquds and gases n order to establsh notons of mportance to mesoscale, those of contnuum, local thermodynamc equlbrum, and fluctuatons Fundamental Spatotemporal Scales n Fluds The molecular nteracton featured n all states of matter (solds, lquds, and gases) can be represented by a force due to the Lennard-Jones potental (Allen and Tldesley, 1987) gven by 1 6 r r Vj r 4 cj dj, (1.1) where, r s the dstance separatng the molecules and j, cj and dj are parameters partcular to the par of nteractng molecules, s a characterstc energy scale, and s a characterstc length scale. The force between two molecules can be acheved by dervatve of Eq.(1.1) 13 7 V j r 48 r dj r Fj r cj r. (1.) The characterstc atomc tme scale assocated wth ths Lennard-Jones molecular nteracton s m, (1.3) where m s the mass of the ndvdual molecule. The mean molecular spacng assumng that molecules are n contact wth one another s defned as 1/3, (1.4) 19

20 where the macroscopc densty s. For H O(l) typcal scales are shown n Table 1. For gases addtonal physcal scales can be defned (Vncent and Kruger, 1975; Gombos, 1994). The number of molecules n one mole of gas s the Avogadro number N A mol. The volume occuped by 1 mole of gas at a gven temperature and pressure s constant Vm.414 l/mol at STP. Gases obey the perfect gas law P nk T (1.5) B where, n s the number densty, 3 k B m kg (s K) s the Bolztmann constant, T (K) s the temperature. An mportant length scale n a gas s the mean molecular spacng defned as 1/3 n. (1.6) The molecular dameter of a gas molecule D s another physcal scale whch cannot be precsely defned but can be derved from vscosty measurements. Table 1 provdes estmates based on the hard-sphere model. For gases undergong bnary collsons (dlute gases), t s requred that D. (1.7) Transport propertes (such as vscosty and thermal conductvty) n a gas relate to collsons. The characterstc collson tme n a dlute gas derved from the hard-sphere collson frequency s 1 1 c D nv (1.8) where v s an average relatve speed. The average dstance between collsons, called the mean free path, s gven by 1 (1.9) Dn 0

21 The mean-square thermal molecular speed of a molecule for a gas n equlbrum s gven c 3kT B (1.10) m The propertes of a typcal gas and lqud at standard condtons are shown n Table 1. Table 1. Propertes of gaseous ntrogen and lqud water at STP. Property N (g) H O (l) Molecular dameter, D m m Number densty, n m m -3 Intermolecular spacng, δ m m Molecular speed, c 500 m/s 1,000 m/s Mean free path, λ m m Collson tme, c s s Whle all flud systems consst of molecules, macroscopc varables are obtaned from averages of propertes that depend on molecular veloctes. These macroscopc averages can be functons of tme and space. The contnuum hypothess allows us to descrbe the state of the flud usng a number of thermodynamc varables (or felds) that depend on poston r and tme t, for example, vr,t, r,t, Tr, t, Pr, t. One must be able to defne samplng volumes, nfntesmal subsystems, wth enough molecules so that statstcal averages can be performed and provde a homogenous thermodynamc feld. The local thermodynamc equlbrum hypothess mples that the thermodynamc varables for these nfntesmal subsystems vary wth tme and space but satsfy the same relatons as the equlbrum thermodynamcs propertes. Once these felds (or pont varables) are defned conservaton 1

22 equatons can be obtaned whch contan dsspatve fluxes (the stress tensor, the heat flux, and dffuson flux). Phenomenologcal expressons are used to relate the dsspatve fluxes J, t wth correspondng conjugate thermodynamc forces X, t Zarate and Sengers, 006] r r n the form gven [Ortz De J r, t M r, t X r, t, (1.11) The functonal dervatves of the dsspatve fluxes M, t r are phenomenologcal coeffcents, commonly referred to as the Onsager coeffcents, and must be symmetrc whch mples,, M r t M r t. The dependence of the Onsager coeffcents, n prncple, on space and tme through the local state varables, n most practcal applcatons can be neglected. For example, the heat flux Qr,t s gven by Fourer s law wth thermal conductvty Q( r, t) T ( r, t). (1.1) Wth the phenomenologcal models defned, conservaton laws, such as the Naver-Stokes equatons are derved. It s well known that fluctuatons exst n fluds n thermodynamc equlbrum as ntroduced by Landau and Lfshtz (Statstcal Physcs, Part 1, Ch XII, 1980) as well as those n non-equlbrum (Ortz De Zarate and Sengers, 006). These fluctuatons are spontaneous varatons of thermodynamc varables about ther mean values and are characterzed by correlaton functons of the fluctuatng propertes. Fluctuatons n systems n thermal equlbrum can be derved from statstcal physcs (Landau and Lfshtz, Statstcal Physcs, Part 1, Ch XII, 1980) and knetc theory n case of gases (Groot and Mazur, Ch IX, 196). The fluctuaton for a thermodynamc quantty f about ts mean f measured n a volume V s gven as

23 where The root-mean-square fluctuaton s gven by the varance f f f, (1.13) 1/ 1/ ( f) f f. (1.14) and the relatve fluctuaton s nversely proportonal to N 1/ ( f ) 1 f. (1.15) N where the denote the mean values of lengthy expressons. For example, n an nfnte flud n equlbrum the varance n number of partcles n a volume V wth temperature T s gven by (Landau and Lfshtz, 1980, Ch XII; Ortz de Zarate and Sengers, 006, Ch.3) N k TN V k TN B B T V P T V, (1.16) where the sothermal compressblty s T 1 V V P. (1.17) T In a gven volume element V, the varance n macroscopc densty s ndependent of tme t and equals to (Ortz de Zarate and Sengers, 006, Ch.3) S V E m k T where mse TkBT. Smlarly, the varance n pressure s V T B, (1.18) P k T V B ( P) kbt V S T. (1.19) The varance n volume s gven by V V kbt kbttv. (1.0) P T 3

24 For dlute gas 1/ P and the above expressons become T N N, (1.1), (1.) n P k T, (1.3) V ( P) B V Meanwhle n general, the varance n temperature s V. (1.4) n kt B T. (1.5) C The varance n entropy s S CP. (1.6) V where C V and C P are the heat capacty of the body as a whole at constant volume and constant pressure respectvely. In general fluctuatons ncrease wth decreasng volume V, where the property f s measured. It should be noted that the fluctuatons of velocty are statstcally ndependent of those of the other thermodynamc quanttes. The varance of each Cartesan component of the velocty s equal to 3kT V B (1.7) m Fluctuatng hydrodynamcs for fluds n thermodynamc equlbrum can be descrbed by the usual hydrodynamc equatons (e.g. Naver-Stokes) wth ntroducton of random nose terms. One approach ntroduced by Landau and Lfshtz (Statstcal Physcs, Part, Ch IX, 1980) s 4

25 referred to as stochastc forcng, and treats the dsspatve fluxes as the sum of an average plus a fluctuaton term, whch s defned as the fluctuatng dsspatve flux. The fluctuaton term J, t where J (, ) (, ) (, ) (, ) a r t M r t X r t J r t (1.8) r should satsfy that J ( r, t) 0, (1.9) J (, t) J (, t) C, t t r r r r, (1.30) C k TM r, r r r. (1.31) B The expressons n Eq. (1.31) s the so called fluctuaton-dsspaton theorem (FDT). Varous extensons of fluctuatng hydrodynamcs to systems of thermal non-equlbrum under the local equlbrum assumpton have been proposed. (de Groot and Mazur, 196; Ottnger, 005). A non-dmensonal parameter used to descrbe the applcablty of the contnuum approach n a gaseous flow s the local Knudsen Number gven by Kn. (1.3) L In ths defnton L s the local length scale of the gradent of a macroscopc quantty f r, t f L, (1.33) f (Brd, 007; Gombos, 1994). When the Knudsen number s used to defne the valdty of the Naver-Stokes equatons, t s often requred that Kn In order to smulate mesoscopc flows, new mathematcal and computatonal methods are requred because the tradtonal atomstc (mcroscopc) and hydrodynamc (macroscopc) 5

26 descrptons are not vald for ether computatonal or theoretcal reasons. The fundamental method approprate for molecular (atomstc) scales s Molecular Dynamcs (MD) (Allen and Tldesley, 1987; Hale, 1997). For example, a smulaton of 1 mcron volume at STP of N (g) and H O(l) contans and molecules respectvely. A smulaton based on MD s certanly not feasble. Varous partcle and hybrd (partcle-flud) smulaton methods coverng from atomstc to hydrodynamc scales are revewed n Koumoutsakos (005). From the contnuum approach, the Naver-Stokes cannot represent the thermal fluctuaton n mesoscale. Several models and numercal methods have been proposed for mesoscopc flud dynamcs, derved from ether bottom-up (molecular) and from top-down (hydrodynamc) scale. We revew below the Dsspatve Partcle Dynamcs (DPD) and Smoothed Partcle Hydrodynamcs (SPH). 1.. DPD Overvew A fundamental method for mesoscopc domans s the Dsspatve Partcle Dynamcs (DPD) method ntroduced by Hoogerbrugge and Koelman (199). The method can be nterpreted as a coarsenng approach, where the DPD partcles represent clusters of molecules nteractng by means of repulsve, dsspatve and random forces. Ths clusterng permts use of larger ntegraton tme steps and allows smulaton of spatal scales much larger than those covered by MD (Keaveny et al., 005). The statstcal mechancs context behnd DPD s presented by Espanol and Warren (1995). In DPD the flud regon s dscretzed by a number of flud partcles each havng mass, and gorvened by Newton s equaton of moton (Hoogerbrugge and Koelman, 199) d r dt v, (1.34) 6

27 dv m dt F. (1.35) where v s ts velocty, r s ts poston. F s the total force exerted on partcle by partcle j, gven as the sum of nterpartcle forces, consstng of a conservatve component C F j, a dsspatve component D F j, and a random component C j C rj R F j (Hoogerbrugge and Koelman, 199) F F e (1.36) rj rj rj j / r j j j D D F / v r, (1.37) j rj rj rj R R F /. (1.38) j R j All partcles j n a sphere of radus r c, whch s called cutoff radus, are nteractng wth partcle as shown n Fgure 1. F The conservatve force s usually a soft repulson gven by C D R rj a max 1 rj / r j c,0. The strength r j and r j D R random force, are coupled by rj rj 1 rj / rc and R are coupled by varable wth zero mean and unt varance. R max,0 n the dsspatve force and. The coeffcents kt. The j n Eq. (1.38) s a symmetrc Gaussan random B r c Fgure 1. The DPD cutoff radus used n calculatng nterpartcle forces. 7

28 The total force exerted on partcle s gven as (Espanol and Warren, 1995) F F F F. (1.39) C D R j j j j j j The tme evoluton of the poston and momentum equaton of a DPD partcle usng a tmestep dt s gven by dr v dt, (1.40) 1 C D R dv F. dt F dt F dt (1.41) m There have been many mprovements to DPD snce ts ntroducton and the method has been appled to a varety of mesoscale systems, ncludng bnary mmscble fluds (Novk and Coveney, 1997), collodal behavor (Dzwnel et al., 006), DNA n mcrochannels (Symeonds et al., 006), two phase flows (Twar and Abraham, 006a), flow over rotatng cylnder (Haber et al., 006), nanojet breakup (Twar and Abraham, 006b), polymers (Symenods et al., 006; Symeonds and Karnadaks, 006; Pan et al., 008), and water n mcrochannels (Kumar et al., 009), evaluaton of transport propertes (Rpoll et al., 001), and fluds out of equlbrum (Rpoll and Ernst, 004). A consderable amount of effort has been devoted to boundary condtons n DPD. These studes nclude the non-slp boundary condton (Hoogerbrugge and Koelman, 199; Espanol and Warren, 1995; Wllemsen et al., 000; Xu and Meakn, 009; Wang et al., 006; Duong-Hong et al., 004; Pvkn and Karnadaks, 005), slp boundary condtons (Smatek et al., 008), perodc boundary condtons (Chatterjee, 007), and wall reflecton laws (Revenga et al., 1999). Attempts to arrve at an energy-conservng DPD have appeared by addng a random heat term (Avalos and Macke, 1997; Macke et al., 1999) or mechancal energy (Chaudhr and 8

29 Lukes, 009). A generalzed form of DPD ncorporatng an nternal energy and a temperature varable for each partcle was presented by Espanol, (1997) and Rpoll et al. (1998). The energyconservng DPD model was used to nvestgate the heat conducton n nanofluds by He and Qao (008) SPH Overvew The SDPD method has ts orgns on Smoothed Partcle Hydrodynamcs (SPH), whch was orgnally developed for modelng of astrophyscal phenomena (Lucy, 1977; Benz et al., 1989). SPH was extended to smulate problems of contnuum sold and flud mechancs. Several forms of SPH equatons can be derved based on the form of equatons and the partcle approxmaton nvolved. We revew the SPH dervaton for the compressble, vscous Naver-Stokes equatons wrtten n terms of feld varables densty r,t, velocty vr,t and nternal energy. er, t., n the case of free external force (Lu and Lu, 003) D r, t, t r, t v r, Dt x Dv r, t 1 P Dt r, t x x De r, t P, t v r Dt r, t x r, t, (1.4) (1.43) (1.44) The superscrpts and denote the coordnate drectons. The vscous stress s proportonal to shear stress wth dynamc vscosty by v v ( ) v x x 3 (1.45) where the shear stress 9

30 v v v x x 3 (1.46) The nternal energy per unt mass e( r, t) for an deal gas s gven n terms of specfc heat capacty c V by and the pressure can be expressed as e r t c T (1.47), V P( r, t) 1 ( r, t) e( r, t) (1.48) The energy Eq. (1.44) assumes that heat flux and the body forces are neglected. The dervaton of SPH equatons encompasses two steps: (a) the ntegral approxmaton of flud felds such as r,t, vr,t, er, t and ther dervatves, and (b) the partcle approxmaton of these felds. In SPH the flud s represented by a fnte number of partcles each wth mass as shown n Fgure (a). The partcle approxmaton of any feld f x s gven by where W x x j, h j1 j m and volume V, N m j f x f x j W x x j, h (1.49) s the smoothng kernel functon or smoothng functon, h s the smoothng length defnng the nfluence volume of the smoothng functon W x x j, h Fgure (b). as shown n The smoothng functon W x x, h n SPH must be normalzed over ts support doman as follows and be symmetrc W x x, h dx1. (1.50) 30

31 x x W x x, h dx 0. (1.51) W x x, h j h j (a) (b) Fgure. Dscrete SPH partcles representng the flud and the support doman of the nterpolatng functon around an SPH partcle. The dscrete counterparts of the constant and lnear consstency condtons as expressed n Eq. (1.50) and Eq. (1.51) are N j1 N j1 W x x j, h x j 1, (1.5) x x j W x x j, h x j 0. (1.53) As dscussed by Lu and Lu (003), the dscretzed consstency condtons are not always satsfed due to the unbalanced partcle dstrbuton n cases where the support doman ntersects wth the boundary or support doman s rregularly dstrbuted. For the Naver-Stokes equatons, Eq. (1.4)-(1.44) can be expressed n form of SPH equatons (Lu and Lu, 003) gven as 31

32 N mw, (1.54) j j j1 dv p W W dt x x N N p j j j j j mj m j j 1 j j1 j (1.55) de dt 1 p p W (1.56) N j j mj vj. j1 j x SPH s a well-developed method and a plethora of revew papers and books have appeared ncludng, Monaghan (1994), Randles and Lbersky (1996), Lu and Lu (003), Monaghan (005), L and Lu (007), Monaghan (009). There has benn a wde range of SPH applcatons and a comprehensve revew s outsde of the scope of ths work. Studes nclude, ncompressble fluds (Morrs et al., 1997; Ellero et al., 007), free surface flow (Monaghan, 1994; Fang et al., 009), vscoelastc flow (Vazquez-Quesada and Ellero, 01; Ellero et al., 006), error estmaton (Amcarell et al., 011; Fateh and Manzar, 011), mmersed boundares (Heber and Koumoutsakos, 008), mult-phase multscale flow (Hu and Adams, 006), thermal fluctuatons n vscoelastc flud (Vazquez-Quesada et al., 009 c), and phase separatng flud mxture (Theulot et al., 005) SDPD Method: Lterature Revew and Outstandng Issues The SDPD was developed by Espanol and Revenga (003) from a top-bottom approach from SPH as the thermodynamcally consstent alternatve to DPD. The SDPD nvokes the Smoothed Partcle Hydrodynamcs (SPH), appled to the dscretzaton of the compressble, vscous Naver- Stokes equatons wrtten n terms of feld varables densty r,t, velocty vr,t and entropy S r, t whch s gven by Batchelor (1967) D r, t Dt 3 r, t v, (1.57)

33 Dvr, t P 1 v F, (1.58) Dt, t, t, t 3 v r r r T DS r, t T Dt t t. (1.59) r, r, The materal dervatve, followng a flud partcle, s D Dt t v. In the above system P s pressure, F s the external force exerted per unt mass, s the shear vscosty, s the bulk vscosty, and s the thermal conductvty. The vscous heatng feld ( r, t) s defned by v : v v (1.60) where the traceless symmetrc part of the velocty gradent tensor s 1 1 T v v v v (1.61) 3 Through the use of the GENERIC framework developed by Ottnger (005) to descrbe hydrodynamc fluctuatons, Espanol and Revenga arrved to the SDPD dscrete equatons that nclude thermal fluctuatons. In SDPD the ndependent varables are poston r t, velocty v t and entropy St of each flud partcle, a devaton from most mesoscopc and hydrodynamc models that nvolve the energy of the partcle. dr v dt, (1.6) m d v = F dt F dt F, (1.63) C D R TdS EV dt ECdt ER. (1.64) where F C, FD and F R are the conservatve term, dsspatve term and random terms; and ER 33 EV, EC are vscous, conductve and random terms. Detals of the formulaton are presented n Ch.. SDPD requres a state equaton to be expressed n the form E N, V, S. Then, the pressure Pr, t and temperature Tr, t are obtaned through the Maxwell relatons

34 E N, V, S Pr, t, (1.65) V T r, t E N, V, S S r, t (1.66) The work done by Espanol and Revenga (003) follows the dea ntroduced by Espanol and Serrano (1999) about treatng an SPH or a DPD partcle as movng thermodynamc subsystems. In both of ther works, the Sackur-Tetrode equaton for nternal energy E S, N, V s employed to close the system. However, the testng case by Espanol and Revenga (003) only nvolved the determnstc part of system equaton. Espanol and Serrano (1999) perform test wth a full model to show the fluctuaton n entropy wthn equlbrum system are n the order of Boltzmann constant and around a constant value. All of the testng cases are under reduced unt. The SDPD nvestgatons so far addressed fundamental ssues of the method. Serrano (006) compared the SDPD and the Vorono flud partcle model n a shear statonary flow. He found the effcency of the two methods comparable. The accuracy of the Vorono approach was found superor for regular ordnate systems whle SDPD produced more accurate results for arbtrary, dsordered confguratons. An ncompressble, sothermal SDPD model was used by Ltvnov et al. (008) to model polymer molecules n suspenson. The entropy equaton was decoupled from the governng equatons. A quantc splne kernel functon was appled to ther model. Perodc boundary condtons are consdered to model the bulk flud. Vrtual partcles mrrored at the surface of sold boundary are mplemented when the support doman of a flud partcle overlaps wth the sold wall surface. They concluded that the vrtual partcles would ntroduce errors on curved surfaces as Morrs et al. (1997) ponted out. In ther model, the SDPD nteract hydrodynamcally as well as by an addtonally fntely extendable nonlnear elastc sprngs. In the test cases, the 34

35 radus of gyraton and the end-to-end radus are compared wth analytcal solutons at dfferent chan lengths. A statc structure factor s employed to extract the statc factor exponent. A further test nvolved the Rouse mode. A form of dffuson coeffcent evaluated from the mean square dsplacement of the center of mass s ntroduced and compared to an analytcal formula for the D case. All parameters n these cases are n reduced unts. Ltvnov et al. (008) concluded that the confnement of sold boundary affects the polymer confguraton statstcs and nduces ansotropc effects. Ther model s a gudelne for further handlng realstc mcrofludc applcatons. Vazquez-Quesada et al. (009) nvestgated the consstent scalng of thermal fluctuatons n SDPD by takng the pont of vew that the SDPD partcles are real portons of flud materal nstead of only just smple Lagrangan movng nodes. They used the sothermal SDPD model and normalzed unts n the smulatons. They ponted out that the determnstc part of SDPD governng equaton s scale free whle the velocty varance from the stochastc part shows dependence on the physcal length of SDPD partcle. The tests are performed on collodal partcles (radus R 0.1 n reduced unt) suspended n a Newtonan flud (vscosty 0.06, mass densty 1, temperature T 1) wth three dfferent partcle resolutons (number of partcle N 484,1600,6400 ) n a box sze L 1). Ther results show that the velocty varance vary wth the resoluton of solvent, however t does not affect the velocty varance of the collod partcle. As a further proof of ther result, they examned a polymer molecule n a suspenson wth dfferent resoluton of solvent ( N 900,3600,14400 ), and added fntely extensble nonlnear elastc sprng forces between the polymer molecules. The model utlzed by Vazquez-Quesada et al. (009) for vscoelastc fluds s dentfed as SPH wth thermal fluctuatons, whch s actually the SDPD form added by a dmensonless 35

36 conformaton tensor to characterze the elongaton of the polymer molecules wthn the flud partcles. They provded a full scheme of SDPD wth closed equatons such as total energy, entropy, conformaton tensor. The entropy functon s gven by the logarthm of the number of mcrostates and coupled wth total energy and conformaton tensor. Ther smulatons are performed n a D perodc square box of length L 1 and N 400 flud partcles. They consdered unform shear flow wth densty 1, knematc vscosty 1, and Kolmogorov flow at Reynolds number Re 1. The dynamcs of the conformaton tensor are studed n terms of the dynamcs of ts egenvalues and egenvectors and have a non-elgble contrbuton to thermal fluctuatons. They state that the ntroducton of thermal fluctuatons n the vscoelastc flud partcle model s analogous to the stochastc contrbuton ntroduced by Landau and Lfshtz (Statstcal Physcs, part, Ch. IX, 1980) n fluctuatng hydrodynamcs. Ltvnov et al. (009) provded an analytcal expresson of the self-dffuson coeffcent for sothermal ncompressble SDPD model. The coeffcent n the self-dffuson coeffcent formula depends on the quantc splne smoothng functon. Ther expresson was tested by several smulatons performed n a 3D perodc box wth N 3,375 SDPD partcles. The smulatons assumed 1, kt 1, B m 3 L N, 1.5 L, h L/15 wth resultng selfdffuson D 1. The smulatons vared the dynamc vscosty. The actual self-dffuson coeffcent and Schmdt number defned as Sc D was calculated from the mean-square dsplacement and was compared wth the analytcal value under dfferent dynamc vscosty. Schmdt number that depends on flud partcle sze does not always agree wth the theoretcal predctons, and extensve prelmnary computatons are needed to characterze the dffuson propertes of the solvent. They pont out that the dffuson propertes depend on the choce of the smoothng length. They also state that ther smulatons do not show the sold-lke structures 36

37 found n DPD smulatons at hgh coarse-granng levels as dscussed by Pvkn and Karnadaks (006). Ban et al. (01) utlzed the SDPD method to model sold partcles n suspenson. They start from a reduced set of Naver-Stokes equaton for ncompressble flow to obtan an SDPD model. Closure s obtaned wth a state equaton n the form of artfcal compressblty gven by Batchelor (1967). The non-slp velocty boundary condton s embedded on all sold-lqud nterface by applyng frozen partcle as descrbed by Morrs et al. (1997). The velocty dynamcally assgned to a frozen partcle s calculated based on the dstance to the tangent plane of the closest pont to flud partcle. Therefore, t requres that the cutoff radus of support doman should be smaller than the smallest surface (both of convex and concave) curvature radus. The total force exerted on sold partcle would be the summaton of all forces exerted on each frozen partcles n sold partcle. They use the Velocty Verlet algorthm as the tme ntegrator. Ther numercal smulatons are carred out wth reduced unts, ncludng cases under non-brownan and Brownan condtons. Several tests under non-brownan condton nvolved flow through a fxed crcular or sphercal object n a perodc array, a partcle movng n a Newtonan flud under unsteady stuatons, a partcle rotatng under shear flow, and hydrodynamc nteractons between two approachng spheres. The Brownan dsk (n D) and Brownan sphere (3D) are nvestgated wth thermal fluctuatons producng ts ultmate Brownan dffusve dynamcs. The mean square dsplacement n a D doman s analyzed, and the dffusonal behavors are studed for both cases. A corrected form of dffuson coeffcent related to drag force s provded and s related to the Ensten-Stokes equaton for the actual dffuson coeffcent from mean square dsplacement. A more complcated smulaton nvolves a 37

38 collodal partcle n the vcnty of an external boundary. The splt form of the dffuson coeffcent s ntroduced whch depends on the dstance to the boundary Boundary Condton for Wall-Bounded Domans Among the outstandng theoretcal and computatonal ssues n SDPD s ts mplementaton n domans wth sold boundares of arbtrary geometry. We address ths ssue n ths work by developng an SDPD method for wall-bounded domans. We revew boundary condton approaches n DPD and SPH n order to provde the necessary background for our method. In mesh-free methods the wall and ts effects on the flud are modeled usng varous types and layers of ghost partcles or frozen partcles. Ghost partcles can be loaded ntally wth statc propertes (Morrs et al., 1997) or can be dynamcally generated wth propertes updated durng the smulaton (Randles and Lbersky, 1996). Statc ghost or vrtual partcles are preloaded as unformly dstrbuted layers (Duong-Hon et al., 004) or as nteractng partcles n the flow and sold boundary regons loaded wth the same densty as the flud partcles. (Hoogerbrugge and Koelman, 199; Boek et al., 1997; Revenga et al., 1999; Lu and Lu, 003). Dynamcally allocated ghost or vrtual partcles are generated by reflectng neghborng flud partcles whch may lead to an mperfect representaton of a curved boundary at low resoluton (Morrs et al., 1997). The dffculty arses n assgnng the physcal propertes of such ghost partcles and n defnng the nteracton forces between them and the flud partcles n the proxmty of the wall. Such forces must ensure that a flud partcle does not penetrate the wall and that the no-slp condton s enforced. In general, the flud partcle force s composed of a repulsve and a dsspatve term. The soft repulsve force actng on a flud partcle near a wall from the neghborng flud partcles may not be suffcently strong to prevent wall penetraton. To 38

39 overcome ths problem a stronger repulsve force between the flud and the wall ghost partcles, n the Lennard-Jones form, has been used n SPH (Monaghan, 1994). Dfferent solutons have been proposed n DPD for mposng wall condtons, such as ncreasng the repulsve force coeffcent for wall/flud nteracton (Pvkn and Karnadaks, 006), ncreasng the wall partcle densty (Fedosov et al., 008) or mposng bounce-back and bounce-forward boundary condtons (Revenga et al., 1999; Pvkn and Karnadaks, 005). Some mplementatons of SDPD used dynamcal ghost or vrtual partcle, ncludng Hu and Adams (006), Ltvnov et al. (008). A Lees Edwards boundary condton s appled for sold boundary by Ltvnov et al. (010). Quesada (009c) used the dstance between a flud partcle and the sold wall n order to evaluate the truncated area of the support doman. Ths area s then used to evaluate the densty and force on the flud partcle. Ths method may lead to an overestmaton of the flud partcle densty compared to the nteror regon. The effectveness of wall reflectons was dscussed by Revenga et al. (1999), ncludng specular reflecton, bounce-forward reflecton, Maxwellan reflecton, and bounce-back reflecton. Pvkn and Karnadaks (005) placed an extra thn layer of DPD partcles nsde the doman and adjacent to the sold boundary wth an adjusted wall-flud conservatve force parameter, whch s estmated accordng to the flud densty, to hold the no-slp boundary condtons. A smlar model was developed by Pvkn and Karnadaks (006) to control the densty fluctuatons near a sold boundary by applyng an adaptve force drected perpendcular to the wall. The force s also adjusted accordng to the densty and bns adjacent to the sold boundary. These adapted models are verfed and compared by Fedosov et al. (008). Another algorthm that modfes the boundary force s mplemented by Altenhoff et al. (007), n whch the boundary force s estmated by the probablty densty functon of the force contrbutons n 39

40 the bn adjacent to sold wall. A phase-feld nterface representaton to DPD for mposng the no-slp boundary condton was proposed by Xu and Meakn (009), whch consders the sold boundary as a phase ndcated by a varable SDPD Self-Densty In ths work, we use the summaton form n Eq. (1.54) for evaluaton of densty whch s a drect way of the approxmaton of SPH to the densty tself. Ths form nvolves the contrbuton from neghborng partcles n support doman by smoothng functon. Flebbe et al. (1994) suggested that the contrbuton of the self-densty should be dscounted snce overestmaton arses when self-densty n nvolved. Whtworth et al. (1995) suggested that although an overestmate n densty arses from the thermal fluctuaton ntally wth randomly dstrbuted partcles, t s elmnated after the system equlbrates Applcatons of the Full Non-sothermal SDPD Model The SDPD nvestgatons so far addressed fundamental ssues of the method but consdered only the sothermal formulaton, whch nvolves contnuty and momentum equatons alone (Ltvnov et al., 008; Ltvnov et al. 010, Vazquez-Quesada, 009 b; Ban et al., 01). One of the dffcultes assocated wth the full SDPD s that t requres a formulaton of the state equaton ndcated n Eq. (1.64)-(1.66). Espanol and Revenga (003) performed SDPD smulatons for an deal gas and used the Sackur-Tetrode equaton for E S, N, V wth only the determnstc part n Eq.(1.63) and (1.64). The full thermodynamcally consstent SDPD model has been valdated by Vazquez-Quesada (009 a) based on a Fourer problem that nvolves a flud between two wall at dfferent temperature. For lquds, Vazquez-Quesda (009) derved a 40

41 state equaton through a second-order Taylor expanson around the equlbrum state and appled t to vscoelastc flud and collodal suspenson flow Objectves and Approach Ths work s part of a research effort at the CGPL at WPI to develop a SDPD-DV methodology and apply to the nvestgaton of mesoscopc wall-bounded flows. There are three goals: Frst, to revse and further mplement exstng algorthms n SDPD-DV, as well as develop and mplement new algorthms. Second, to valdate and verfy the algorthms of the SDPD-DV code by comparsons wth expermental, analytcal and numercal solutons. Thrd, to nvestgate thermodynamc propertes and transport coeffcents wth applcatons of the SDPD-DV code to mesoscopc systems n equlbrum. The objectves and approaches are dvded n three man categores. 1. Revse and further mplement exstng algorthms n SDPD-DV, as well as develop and mplement new algorthms n order to acheve a fully functonal SDPD-DV methodology: a. Revse the perodc boundary condtons algorthm and the mplementaton of the perodc boundary cells searchng lst. b. Modfy evaluaton of smoothng functon n order to nclude the contrbuton of the self-densty for each flud partcle. c. Modfy and further mplement the dynamc vrtual partcle allocaton algorthm for the modelng of sold boundary n order to mnmze the truncaton error of densty. d. Develop and mplement the boundary normal vector algorthm used n the reflecton of the dynamcally allocated vrtual partcles. 41

42 e. Implement the algorthm for the contrbuton to the boundary force from the vrtual partcles. f. Implement a temperature boundary condton n the dynamc vrtual partcle allocaton model. g. Implement a bounce-forward algorthm for a sold boundary wth arbtrary shape and orentaton. h. Revse and rewrte portons of the Velocty Verlet ntegraton method for the poston and momentum equatons.. Develop and mplement a Runge Kutta ntegraton algorthm for the entropy equaton. j. Implement the artfcal ncompressblty method for modelng lqud flows. k. Implement a temperature power law for the shear vscosty, bulk vscosty and heat conductvty that appear n the momentum and energy SDPD equatons. l. Develop and mplement algorthms for the evaluaton of transport propertes such as dffuson coeffcent, shear vscosty and heat conductvty based on mean square dsplacement (MSD) and velocty autocorrelaton functon (VACF). m. Develop analytcal formulas of the self-dffuson coeffcent based on the SDPDflud followng Ltvnov et al. (009).. Valdate and verfy the SDPD-DV mplementaton: a. Valdate the mplementaton of boundary partcle and flud partcle loadng, dynamc vrtual partcle allocaton, perodc boundary partcle allocaton, flud feld propertes samplng, and bounce forward reflecton. Compare wth analytcal solutons for transent body-drven Poseulle flow of water between 4

43 statonary nfnte parallel plates of 10-3 m heght. Calculate the components of the forces attrbuted to the dynamcally allocated vrtual partcles and compared wth prevous DPD nvestgatons. b. Verfy movng and no-slp boundary condtons acheved by the dynamc vrtual partcle method by comparsons of SDPD-DV smulatons wth transent Couette flow of water between statonary nfnte parallel plates of 10-3 m heght. c. Verfy the ablty of the SDPD-DV to smulate curved 3D sold boundares, and the evaluaton of pressure by comparsons of SDPD-DV results of steady, low-re ncompressble flow over a cylnder of radus 0.0 m wth results from FLUENT. d. Perform smulatons of Poseulle flow between two nfnte parallel plates at dfferent temperature for verfcaton of the non-sothermal SDPD-DV. e. Perform smulatons of Couette flow between two nfnte parallel plates at dfferent temperature for verfcaton of the movng wall boundares wth constant temperature. 3. Investgate mesoscopc flows usng the sothermal and non-sothermal SDPD-DV mplementatons. a. Perform smulatons of H O(l) and N (g) at equlbrum states. Evaluate the selfdffuson coeffcent, shear vscosty, translatonal temperature, thermal speed, and thermal fluctuatons. Valdate and verfy by comparsons wth analytcal and expermental values. b. Examne the scale dependence n SDPD-DV and characterze the effects of partcle mass, partcle volume, smoothng length, and tme step. 43

44 The presentaton of ths work s organzed n the followng manner. In Chapter, the overvew of the SDPD-DV methodology s presented, as well as the mathematcal and numercal aspect, as pertanng to ts mplementaton wth dynamc vrtual partcle and non-sothermal model, s presented n detal for each aforementoned code modfcaton or addton. In Chapter 3, an extensve set of benchmark tests that cover the hydrodynamc and mesoscopc regmes s presented and used for verfcaton, valdaton and error analyss of the SDPD-DV method. The frst verfcaton of SDPD-DV nvolves comparsons wth analytcal solutons for a body-force drven, transent, Poseulle flow of water between parallel plates of 10-3 m heght. The second verfcaton test nvolves the transent, Couette flow of water between parallel plates of 10-3 m heght. The thrd test used for verfcaton nvolves the low-reynolds number, ncompressble flow over a cylnder of radus 0.0 m. An extensve set of SDPD-DV smulatons of lqud water and gaseous ntrogen n mesoscopc perodc domans s also presented. In Chapter 4 the SDPD- DV code wth entropy equaton s appled to mcroscale non-sothermal case studes. The valdaton and verfcaton ncludes smulaton of equlbrum gaseous ntrogen n perodc domans and the evaluaton of transport coeffcents. Fluctuatons n thermodynamc varables are evaluated and compared wth analytcal estmates. Results from SDPD-DV smulaton of non-sothermal ntrogen Couette flow are verfed wth results from FLUENT. Conclusons and recommendatons for future work are presented n Chapter 5. 44

45 . SDPD-DV METHODOLOGY AND IMPLEMENTATION We revew n ths chapter the basc elements of the SDPD model and present the dscrete SDPD equatons. We then present the major algorthmc features of the SDPD-DV mplementaton developed for smulaton of mesoscopc fluds n wall-bounded domans. Algorthms presented nclude the partcle loadng ndexng; the neghborng partcle search; densty and force evaluaton on nteror domans, sold boundary and perodc boundares; pressure evaluaton; ntegraton of SDPD-DV equatons; evaluaton of partcle transport propertes; evaluaton of sample-averaged partcle propertes. Materal n ths chapter appear n Gatsons et al. (013)..1 Overvew of the SDPD Method The SDPD dervaton of Espanol and Revenga (003) starts wth the Naver-Stokes equatons wrtten n the materal dervatve form (.e. followng a flud partcle) wth ndependent feld (Euleran) varables the densty ( r, t), velocty vr (, t), and entropy S( r, t) nstead of the E( r, t), D r, t Dt v r r r r, t v, (1.57) Dvr, t P 1 v, (1.58) Dt, t, t, t 3 T DS r, t T Dt t t. (1.59) r, r, The dervaton arrves frst at a dscrete SPH-type set of the determnstc Naver-Stokes equatons for r, v and S. The doman contanng flud of total mass M F and volume V T, followng SPH, s dscretzed wth a number of N F ponts each one representng a 45

46 thermodynamc closed system (equvalent to a materal volume or a flud partcle). Each partcle has constant mass m M N F, (.1) F and s descrbed by ndependent varables poston, velocty and entropy at tme t r( t) x ( t), y ( t), z ( t), v ( t) v ( t), v ( t), v ( t), S ( t). (.) x y z The SDPD dscretzaton s consstent wth a set of N F thermodynamcs systems (or Lagrangan flud partcles). Alternatvely, the SDPD dscretzaton can be consdered as a set of N F grd nodes whch are movng wth the materal (or partcle) velocty. These two vews are dentcal and can provde preferable vewponts durng analyss. The number densty d of the SDPD partcle follows SPH and s defned as a summaton of neghborng partcles as shown n Fgure 3 j Fgure 3. Contrbuton of neghborng partcles n partcle support doman to the number densty evaluaton. d W r rj, h, (.3) j 46

47 where, W( r, h ) s the nterpolant (smoothng) functon wth a fnte support h satsfyng normalzaton condton W r, h dr 1. (.4) Several optons are avalable for the nterpolant functon and a detaled lst of rules to construct smoothng functons can be found on Lu and Lu (003). The sum n Eq. (.3) s extended to all the j partcles that are wthn the fnte support h of partcle, ncludng the contrbuton from the partcles tself. The dstance between partcles and j s r rj r j xj, yj, zj, (.5) A dmensonally consstent partcle volume V s defned as the nverse of the partcle number densty V 1. (.6) d The feld densty ( r, t) at the partcle poston r () t of partcle s obtaned as [ r ( t)] ( t) m d ( t). (.7) Smlarly to the SPH approach, the dscrete value at a pont r (or poston of a partcle) for any feld varable ( r, t) can be calculated by nterpolaton are: W ( r rj ) j j ( r ) W ( r r ). (.8) j The determnstc SDPD equatons consstent wth the Naver-Stokes Eq. (1.57)-(1.59) dr dt 47 j v, (.9)

48 dv P P j 5 Fj Fj m F 5 j j j j j j dt j d d r v e e v, (.10) j 3 j dd j 3 j dd j ds T dt F j Tj. (.11) j dd j Ths determnstc dynamcs s ntroduced nto the GENERIC framework of Ortega (005). Ths s done n order to ntroduce the stochastc part for the system dynamcs. The ntroducton of thermal fluctuatng terms leads to the GENERIC stochastc dfferental equatons that contan (reversble and rreversble) determnstc terms, a term that relates the dsspatve (rreversble) dynamcs wth stochastc terms, and a stochastc term (Eq. (38) n Espanol and Revenga, 003) as E S dx L M kb M dt dx x x x, (.1) In the above dx s the stochastc term. The term LE x s the reversble part of the dynamcs, and the second term M S x s defned as the rreversble part. The matrces L s antsymmetrc, and M s symmetrc and postve sem-defnte. The followng condtons must be satsfed by L and M S L x E 0, M 0, (.13) x I L E 0, I M S 0. (.14) x x x x The stochastc term dx satsfes dxdx T k Mdt (.15) B whch s an expresson of the fluctuaton-dsspaton theorem. It s also requred that E I dx 0, dx 0, (.16) x x 48

49 The term k M x B M s gven by s from the stochastc nterpretaton of Ito ˆ process. Therefore, the matrx M T dvdv j dvds j 0 Mj kbdt kbdt T dsdv j dsds j 0 kbdt kbdt (.17) The matrx L s defned by 0 1 j 0 1 L Lj j 0 0 m 1 (.18) The dervatves of the energy and entropy wth respect to the state varables result n P 0 d E S, 0 x mv x 1 T (.19) and the thermal nose s defned as 0 dx d v (.0) ds Eq. (.9) (.11) can be gven n a matrx form as 0 0 P r 0 d d j d ds j d d v v v v 0 j j kbdt j S j k j j k B Bdt v L dt j m M v (.1) j j S v 1 ds ds d j dsds j T v j j kbdt j S j k dt Bdt v 49

50 The fnal result s a set of dscrete partcle equatons that descrbe the determnstc and stochastc dynamcs, referred to as the SDPD equatons (Eq. (63) n Espanol and Revenga, 003). We rewrte below the SDPD equatons for the ndependent varables, v and S n a format that s conducve to numercal mplementaton and allows also drect comparson wth SPH and DPD. They are gven by dr v dt, (1.6) m d v = F dt F dt F, (1.63) C D R TdS EV dt ECdt ER. (1.64) The terms appearng n the momentum equaton (1.63) are broken nto the conservatve, dsspatve and velocty random terms gven by P P j FC F jr j, (.) j d d j F D TT j kb kb 5 Fj 1 j j T j 3 T C C v d j d j TT j k F B kb j 1 5 j T T j C C e e v j 3 dd j j j j, (.3) F R j 1/ TT j 5 F j 8kB dwj T Tj 3 dd j e 1/ I TT j 5 F j 8kB 8 tr d j 3 T Tj 3 dd W j j. (.4) The terms n entropy equaton (1.64) are dvded nto vscous, conductve and random terms. 50

51 1 TT j kb kb Tj kb 5 Fj EV 1 j j T C C j T Tj C 3 d T v j d j F k TT 5 F 5 ' 3 d d m T T 3 d d j B j j ej vj j j j j, (.5) Fj k F B j E T ' T d d C d d, (.6) Cd j j j j j j 1/ TT j 5 F j 8kB dwj 1 T Tj 3 dd j E : ev R 1/ j j j I TT j 5 F j 8kB 8 tr d j 3 T Tj 3 dd j + 4 k TT F 1/ dv j B j j j dd j A state equaton s requred to close the system E W. (.7) E ( N, V, S ). (.8) eq The state equaton provdes the partcle temperature and pressure as T eq E, (.9) S The term j j j eq E P. (.30) V e r r r n the SDPD equatons s a unt vector, vj v v j, Tj T Tj and F F r r W r r. (.31) j j The terms and represent the shear and bulk vscosty; C s the heat capacty at constant volume of partcle and s an extensve property, C cv m where c V s the specfc heat capacty of flud; the Boltzmann constant s k B and s the thermal conductvty of the flud. 51

52 The velocty and entropy random terms contan I the dentty matrx and dw j the traceless symmetrc part of a matrx of ndependents ncrements of the Wener process dw. 1 T I dwj d j d j tr d j W W 3 W. (.3) where the trace s defned as tr dw dw. (.33) j j j For k C 0 the fluctuatng terms reduce to zero and the set of Eqs.(1.6)-(1.64) reduce to a B dscrete SPH-form of the determnstc Naver-Stokes equatons for r, (.9)-(.11). v, and S gven by Eq. Inputs Generate Partcle- Search Grd Load & Index BPs & FPs Intalze FPs Entropy Perform NNPS FPs Away from Boundares Evaluate FP Densty FPs Near SB Perform DVPA Evaluate FP Pressure and C p FPs Near PB Perform PBP Method FPs Away from Boundares =1,N F FP Evaluate FP Forces and S FPs Near SB Perform DVPA FPs Near PB Perform PBP Method No Penetraton Integraton VV for r and v, RK for S Penetrate SB Perform BF Penetrate PB Perform Renjecton Output Partcle Samples Generate Grd Post-process for Partcle & Transport Propertes Post-process for Feld Propertes Fgure 4. General flow chart of SDPD-DV and post-processng. 5

53 . Mathematcal and Computatonal Aspects of the SDPD-DV method In ths secton, we present the mathematcal and computatonal aspects of the SDPD-DV developed n ths work for smulaton of wall-bounded domans. The general flow chart s presented n Fgure 4. The physcal doman s shown n Fgure 5 and s characterzed by arbtrary external and nternal sold-wall boundares, planar perodc nlets and outlets wth equal areas, and the flud regon. A rectangular doman s also constructed to ad n the numercal mplementaton of the partcle search and ncludes the physcal doman as shown n Fgure 5. Fgure 5. Physcal wall-bounded doman wth an exteror wall and an nteror sold body showng the flud partcles (FPs), boundary partcles (BPs), and vrtual partcles (VPs) used n the SDPD- DV. Inlets and outlets are consdered perodc. The large rectangular cells are used for nearest neghbor partcle search (NNPS). 53

54 ..1 Boundary Partcle Loadng and Global Indexng The boundary of the physcal doman s dscretzed wth a surface trangulaton based on a surface length scale small enough to resolve the physcal characterstcs of the surface and much smaller than the smoothng length, h. A total of N B boundary partcles ( BPs) are placed on the vertces of the surface grd as shown n Fgure 5. The BPs poston and velocty are stored n the global partcle lst r ( ) (0) (0), (0), (0) k t rk xk yk zk k 1, N v ( t) v (0) v (0), v (0), v (0) k k kx ky kz B (.34) In addton, the normal vector enterng the flud doman at each BP s evaluated and later used n flud partcle-wall nteracton. For a surface defned mplctly by F( x, y, z) 0 the normal vector s evaluated analytcally by n ( x, y, z ) F( x, y, z ). For an arbtrary sold k k k k k k wall surface the local normal to each surface trangle surroundng a vertex s evaluated as n r r and the closest s assgned to the vertex as shown n Fgure Fgure 6. Normal vector for a BP on a sold boundary... Flud Partcle Loadng and Global Indexng The computatonal volume of the physcal doman s populated wth N ( 0) F t flud partcles ( FPs ) each wth mass m, correspondng to a total mass M F followng Eq.(.1). The FPs are 54

55 assocated wth a global partcle lst wth ndex NB 1, NB NF 1. The FPs are assgned upon ntalzaton wth poston and a velocty as r (0) x (0), y(0), z(0) NB 1, NB NF 1 v (0) v (0), v (0), v (0) x y z (.35) To smplfy FP loadng n cases of complex physcal geometres, a mesh for the flud regon s generated wth the number of tetrahedrons vertces to be equal to the total number of FPs n the doman...3 Nearest Neghbor Partcle Search (NNPS) The evaluaton of the propertes of a flud partcle ( FP ) requre the dentfcaton of the nearest neghborng partcles (NNP) n ts support doman shown n Fgure 7(c). A flud partcle j s consdered as a nearest neghbor of when t s located wthn the smoothng doman of partcle, and therefore r j h, wth determned by the smoothng functon. An all-par NNP search among N F partcles n a doman requres operatons per computatonal cycle. ( N F ) Numerous approaches have been developed for varous types of partcle smulatons that requre such a search. We mplemented n ths work, a NNPS approach that uses concepts from the lnked-cell (Hockney and Eastwook, 1981), (Putz, 1998) and the Verlet-lst algorthm (n't Veld, Plmpton and Grest, 008). The approach uses a partcle-search grd wth sze LX, LY, L Z as shown n Fgure 7(c). Ths search-related grd contans N C rectangular cells generated wth lengths L, L, L max( h). In case of a perodc (physcal) boundary the search-grd face has CX CY CZ to algn wth the physcal boundary, as shown n Fgure 5(a). The steps for the NNPS algorthm are summarzed n Table 1. 55

56 Fgure 7. Rectangular cells used for NNPS n the nteror, near a sold and near a perodc boundary. Table. Nearest Neghborng Partcle Search Algorthm used n SDPD-DV S-1. Update n the global partcle lst the current propertes of each FP N 1, N N 1. B B F S-. For each cell C 1, NC dentfy the BPs and FPs contaned n t. Generate N C cell partcle lsts each wth NFC() t FPs and NBC() t BPs respectvely. S-3. For each cell partcle lst C 1, NC loop through the NFC() t FPs and search all FPs and all BPsn the neghborng 7 cells. For each FP create a lst wth J() t NN FPs and a lst wth the K() t NN BPs Ths partcle-search grd s used also n the mplementaton of perodc boundary condtons. For a FP resdng n a cell wth a perodc boundary face the NNPS generates the cells on the correspondng perodc boundary, as shown on Fgure 7(c). All the FPs n FP s support doman from the correspondng cells are ndexed and become avalable for densty and force evaluaton. 56

57 ..4 Flud Partcle Densty Evaluaton The densty evaluaton procedure depends on the locaton of the FP. The three cases consdered n ths work nclude a FP away from a boundary (Fgure 8(a)), a FP near a sold boundary (Fgure 8(b)) and a FP near a perodc boundary (Fgure 8(c)). (a) FP away from boundares. (b) FP near a sold boundary. (c) FP near a perodc boundary. Fgure 8. Support doman for a flud partcle (FP) for three cases consdered n SDPD-DV. (a) FP Away from Boundares The partcle number densty (number of FPs n unt volume) at the locaton of the FP s defned by the summaton over the neghborng FPs n ts support doman shown n Fgure 8(a) and provdes the thermodynamc volume V, L () t 1 d( t) W r ( t) r j( t), h, 1, N F (.36) V() t j where L t s the number of partcles n FP s support doman. The summaton ncludes the contrbuton from the FP tself. Consequently, the mass densty at the poston r s gven by 57

58 m ( t) [ r ( t)] m d( t) (.37) V() t Ths equaton along wth the normalzaton condton Eq. (.4) satsfes the Lagrangan form of the contnuty equaton and therefore ensures that mass s conserved n a closed or n a perodc doman ( Espanol and Revenga, 003). An alternatve approach could be based on the contnuty equaton followng SPH by Lu and Lu (003). A common choce n SPH and SDPD for the nterpolant s Lucy s (1977) smoothng functon 105 r r W ( r, h) 1 3 1, 3 16h h h 3 (.38) whch, through Eq. (.31) gves F j 315 r 5 1 j. 4h h (.39) (b) FP Near Sold Boundares: Dynamc Vrtual Partcle Allocaton Method When the support doman of a flud partcle falls outsde a boundary (sold or perodc as shown n Fgure 8(b) and Fgure 8(c) respectvely) the smoothng functon approxmaton of Eq. (.36) wll result n error due to the wth the unaccounted (truncated) part of the support doman. Ths boundary truncaton error would lead to ncorrect densty evaluaton and would affect the partcle momentum and entropy evaluaton. Ths ssue has been addressed n SPH (Morrs et al., 1997; Randles and Lbersky, 1996; Lu and Lu, 003; L and Lu, 007) and SDPD (Ltvnov et al., 008; Ban et al., 01; Vazquez-Quesada, 009). Wth the Lucy smoothng functon Eq. (.38) ths occurs when r w h, where r w s the dstance between the partcle and the boundary as shown n Fgure 8(b). It should be noted that due to the summaton densty approach followed n SDPD the densty error does not affect the overall mass conservaton but affects the total volume. 58

59 We descrbe the densty evaluaton method developed n ths work to compensate for the boundary truncaton error. For a FP near a sold wall we developed and mplemented the dynamc vrtual partcle allocaton (DVPA) method, by whch the truncated porton of the support doman s dynamcally flled wth vrtual partcles (VPs). Approaches based on statc and dynamc ghost partcle allocaton appeared n SPH (Morrs et al., 1997; Randles and Lbersky, 1996; Lu and Lu, 003; L and Lu, 007) and SDPD (Ltvnov et al., 008; Ban et al., 01). These vrtual partcles are ncluded n the summaton densty of Eq. (.36) followng the algorthm descrbed n Table 3. The VPs are generated as mrrored mages of the FP and ts neghbors as shown n Fgure 9(a). (a) Support doman for a FP near a sold boundary represented by BPs. (b) Reflecton of a FP near a sold boundary to create a vrtual partcle (VP). Fgure 9. Dynamc vrtual partcle allocaton (DVPA) method used n SDPD-DV for densty and force evaluaton. A vrtual partcle (VP) s generated for a flud partcle (FP) near a sold boundary represented by boundary partcles (BP). 59

60 In case of curved walls, we requre that h ar mn (.40) where, the coeffcent a 0. and Rmn s the smallest radus of the curvature for the sold boundary n FP s support doman. Wth such a constrant t s possble to consder the surface of the wall contaned wthn the smoothng doman of a FP as quas-flat and ts normal vector as locally constant. The algorthm s summarzed n Table 3. Table 3. Dynamc Vrtual Partcle Allocaton and Densty Evaluaton Algorthm used n SDPD-DV. S-1. Loop through each FP. S-. Compute the partcle number densty d usng the FP j neghbors from Eq. (.36) and FP j tself. S-3. Search the nearest-neghbor BP lst to fnd the closest BP k. Compute r ( r r ). n w k k and compare wth the average mnmum dstance l FPs between the FPs. A choce s l mn 0.1mn( lfps) provded by the surface trangulaton. If rw lmn create a vrtual partcle VP wth ts poston r VP reflected across the normal to the surface plane, r r r n. The process s shown n Fg. 4(b). VP w k S-4. If r rvp h add to partcle number densty d the contrbuton due to the VP. S-5. Loop through each of the nearest-neghbors FP j and create a mrrored VP j followng S-3. S-6. If rj r VP h then add to partcle number densty d the contrbuton due to the VP j. 60

61 (c) FP Near Perodc Boundares: Perodc Boundary Partcle Allocaton Method The densty of a FP contanng a perodc boundary wthn ts support doman conssts of contrbutons from the surroundng FPs and the FPs n the truncated part as shown n Fgure 8(c). We mplemented a perodc boundary partcle allocaton (PBPA) method, by whch the truncated porton of the support doman s flled wth coped partcles from the perodc cells dentfed durng the NNPS (Sec...3). The densty of the FP s then obtaned usng Eq. (.36) where the r j for a coped FP s gven by j Px j Py j Pz r x L y L z L j (.41) The values for LPx, LPy, L Pz are determned by the locaton of the perodc boundary at the specfed cell face. If there s only one perodc boundary at a cell face, as s the perodc outlet n Fgure 5, L L, L 0, L 0. In a case of fully perodc boundares, a corner cell has Px X Py Pz three perodc boundares and L L, L L, L L when copyng the FP from the correspondng corner. Px X Py Y Pz Z..5 Flud Partcle Pressure and Temperature Evaluaton The full algorthm of SDPD, whch nclude entropy equaton, needs a closure by equaton of states that provdes pressure P P, S and temperature T T, S gven by Eq. (1.65) and (1.66). Therefore, the nternal energy s requred as a functon of of m, S, and V, n the form E E m, V, S. (.4) For deal monatomc gases we consder the Sackur-Tetrode entropy equaton gven by Tetrode (191) or Laurendeau (005) 61

62 S Nk B V ln 4 me N 3Nh 3 5, (.43) where N s the number of molecules n the system, h s the Planck s constant. For an SDPD flud partcle whch s consdered as a thermodynamc subsystem, followng Eq. (.43), the entropy s gven by where gven by S N k B V 4 ln me N 3Nh 3 5, (.44) N s the number of molecules n FP, wth volume V. Then the nternal energy of FP s E 5 3 3hN 5 S N, V, S exp 4 3 mv 3 Nk B 3, (.45) Therefore, the temperature and pressure of monatomc deal gas are gven by Espanol and Serrano (1999) E T E N, V, S S 3Nk B E P N k T B V V (.46) (.47) And the heat capacty s gven by 3 C NkB (.48) To ntalze the entropy of FP, we follow (.44) by S /3 3NkB mv kb 5 0 ln T /3 0 hn 3 (.49) 6

63 In Chapter 3, we consder also the sothermal SDPD mplementaton, where entropy equaton s decoupled from the system, and temperature s therefore kept constant and ntally assgned to the FPs. Closure n ths case s obtaned by an equaton of state that provdes P P(, T ). For an deal gas flow we use P RT (.50) For a perfect gas wth constant specfc heats, the pressure s defned as (Batchelor, 1967) For an ncompressble flow, we follow Batchelor, (1967) P 1 e (.51) c 0 P 0 1 (.5) where, c s an artfcal sound speed and 0 s the ntal densty. A smlar law s gven by Morrs et al. (1997) and Lu and Lu, (003), P (.53) c The value for c should be low enough to avod usng very small tme steps and hgh enough to be suffcently close to the real flud. As suggested by Morrs et al. (1997) and Lu and Lu (003), c can be estmated by c V V F L max,, L (.54) where, 0 s the relatve densty perturbaton, V 0 s a velocty scale, L 0 s a length scale, 0 s the knematc vscosty, and f o s a body force per unt mass. Several alternatve forms of Eq. (.5) could be found as 63

64 P P0 0 1 (.55) P B 0 1 (.56) P P0 b 0 (.57) where B and b are the coeffcent chosen to keep the densty fluctuaton. In chapter 4, the full algorthm of SDPD-DV wll be tested and valdated...6 Flud Partcle Force Evaluaton The force evaluaton procedure depends on the locaton of the FP. As wth densty we consdered three cases depcted n Fgure 8. (a) FP Away from Boundares For a FP wth a poston away from a boundary, as shown n Fgure 8(a), the forces F C, FD and F R are evaluated drectly from Eq. (.), (.3) and (.4). The summaton s performed over all FPs n the support doman of the FP provde by the NPPS. (b) FP Near Sold Boundares: Dynamc Vrtual Partcle Allocaton When a FP s n the proxmty of a sold boundary ( r w h) a correcton term s requred also n the momentum Eq. (1.63) due to the presence of the wall and truncated doman. Ths correcton term s based only on the local thermodynamc state of the flud and the local geometry of the wall descrbed by the BPs. For force evaluaton, we use the DVPA method outlned for densty n Sec...4(b). and develop addtonal force terms to supplement the SDPD equatons (.)- (.4). 64

65 The thermodynamc propertes of each VP are assgned to be dentcal to the propertes of the correspondng FP that s mrrored, specfcally j _ V, P _ j j V P. (.58) j For each FP n the proxmty of the wall the conservatve F C n Eq. (.) becomes P P j VP FC = Fj rj F C (.59) j1: Nd d j where the conservatve force due to the VPs s VP P P j_ V FC Fj _ Vr j _ V (.60) j1: N d VP d j _ V In the above equatons, N and N V are respectvely the number of FP s and VPs contaned wthn the support doman of a FP. Generally, NV N, F _ j V F and the dentty sgn s j acheved only when the FP s located exactly on the wall ( rwall 0 ). Consstent wth the above methodology we also ntroduce an addtonal term to the dsspatve part of the dynamcs to account for the truncated part of the doman. For each FP n the proxmty of the wall the F D n Eq. (.) becomes F D TT j kb kb 5 Fj = 1 j j1: N T C C j 3 d T v j d j TT k k F 1 5 e e v + F j B B j VP j j j D j1: N T C C j 3 d T j d j (.61) where the dsspatve force due to the VPs n the truncated doman s 65

66 F TT k k 5 F = 1 vj _ V TT j k F B kb j 1 5 j1: NV T T j C C e e v j 3 dd j VP j B B j D j1: N 3 V T T C j C j dd j j _ V j _ V j _ V (.6) For a gven wall velocty v W the velocty of the VPs requred to compute the velocty vector v j _ V s defned as Substtutng n the defnton of the velocty vector v j _ V vw v j, j (.63) v j _ V t become v v v v v j _ V j _ V j v W, j (.64) The above formulaton s suffcent to mpose the non-slp condton at a sold wall boundary. For a planar wall and a FP located on the wall e t t _, n n j V ej ej _ V e j, Eq. (.64) enforces v v W. gven by For a wall wthout heat flux, we assgn the VP temperature equal to the wall temperature T j _ V T (.65) w If the temperature gradent along wall surface s not equal to zero, the T w would be the temperature of the nearest BP of FP j. Whereas f the heat flux through the surface of the wall s not zero, the VP temperature s assgned by T T T (.66) j _ V w j (c) FP Near Perodc Boundares The force evaluaton of a FP contanng a perodc boundary wthn ts support doman follows the approach dscussed n the densty evaluaton (Sec ). The FPs n the truncated part are 66

67 coped from the correspondng perodc cells. The force on the FP s then obtaned usng Eq. (1.63) and Eq. (1.64) where the r j for a coped partcle s gven by x j LPx, y j LPy, z j LPz r x y z (.67) j..7 Integraton of Flud Partcle Poston, Momentum and Entropy Equatons (a) FP Away from Boundares Once the partcle neghbor lst s constructed and the densty, pressure and force s evaluated the ntegraton of moton and momentum Eqs. (1.6)-(1.64) proceeds as ndcated n Fgure 4. The ntegraton scheme used n ths work s an mplementaton of the Velocty-Verlet scheme (Nkunen et al., 003) for momentum equaton, coupled wth Runge-Kutta scheme for entropy equaton, and summarzed by the algorthm n Table 4. The choce for s based on requred accuracy n the fractonal change of the velocty estmate. The requred t follows standard SPH condtons (Morrs et al., 1997; L and Lu, 007) and once choce provdes a posteror check, after the ntegraton has proceeded, h t (.68) v max (b) Near Sold Boundares: Reflectve Bounce Forward Condton In general the addtonal repulsve force Eq. (.59) exerted on the FP from the VPs s not enough to prevent FP penetratng the sold wall. When such event occurs the FP s renserted n the flud doman by a bounce forward reflecton, ths takes place after S-3 n the ntegraton algorthm Table 4. The bounce forward reflecton method s depcted n Fg. 5 and summarzed by the algorthmc steps n Table 5 whch are embedded n S-3 of Table 3. 67

68 (c) FP Near Perodc Boundares In case that a FP penetrates a perodc boundary t s re-njected nto the related perodc cell. Ths operaton follows the bounce-forward method. Table 4. Tme ntegraton n SDPD-DV. Velocty-Verlet s used for the partcle poston and momentum equaton, and Runge-Kutta s used for the entropy equaton. S-1. Calculate n n FC r, T, F n, n, n D r v T, n n FR r, T and E n, n, n V r v T, n n ECd, T E r n, v n, T n. R n n S-. Update t t v v F F F, and calculate m 1/ 11 C D R E dt E dt E ds, then update n 1/ n 1/ n T T f ds n 1/ V Cd R n n n T T T. r, n1 n n1/ S-3. Update r r v. t n1 n1/ n1 n1/ S-4. Calculate FC r, T, FR r, T, and n 1 1/ 1/, n, n ER T n n S-5. Update t v v F F, and T m 1 1/ 11 C R r v. T. n1 n1/ S-6. Calculate n 1 1 1, n, n FD r v T, n 1 1 1, n, n n1 n1 EV r v T and ECd, T r. n n S-7. Update v v F, and calculate m t D ds E dt E dt E, n1 V Cd R n1 n1 n1/ T T T then update n 1 n 1/ 1 n T T f ds. S-8. If v v n1 n1 n1 v, update T T, loop over step 6. n1 n1 Calculate n 1 1 1, n, n FD r v T, n 1 1 1, n, n n1 n1 EV r v T, ECd, T r and go to step S-. 68

69 (a) Poston reflecton of a penetratng FP. (b) Velocty reflecton of a penetratng FP. Fgure 10. Reflectve bounce-forward method n SDPD-DV appled n a case where a flud partcle (FP) penetrates a sold boundary represented by boundary partcles (BP). Table 5. Bounce-forward Method used n SDPD-DV n case of partcle penetraton. S-1. After the FPs poston has been updated to the new tme step n+1, check the sgn of the n scalar product ( P BPP n1 n ), where n n1 P BPP s the vector between the FP n+1 n n1 poston and the closest BP of n poston, n s the wall normal unt. If P BPP n 0, partcle penetrates, and proceed to S-. S-. If FP penetrates the wall, compute the scalar product between the vector PP n n1 and the unt normal wall vector n obtanng the dstance P P n. n n1 l p S-3. Compute the dstance between the wall surface and FP new poston (nsde the wall) l l r. n1 P as p W n 1 n 1 S-4. Reflect the FP nsde the doman and compute ts new poston as P P n. BF l S-5. Impose the velocty component of FP normal to the wall to be opposte n sgn. Ths s acheved by mposng the velocty of FP after the reflecton to be v v v n. n1/ n1/ n1/ BF 69

70 ..8 Partcle Propertes and Transport Coeffcents (a) Self-Dffuson Coeffcent A property of a FP from the SDPD-DV smulaton at a dscrete tme t k t s desgnated as k X r( t), v ( t), S ( t), T ( t) etc. An SDPD-DV output (or sample) conssts therefore of all k partcle propertes X, 1, N F. For steady SDPD-DV smulatons we gather after reachng steady-state, m 1, M ndependent SDPD-DV samples. For unsteady smulatons, a number of M ndvdual runs are performed n order to generate a suffcent number of ndependent samples. Transport coeffcents such as dffusvty, shear vscosty are dynamcal propertes desred from SDPD smulatons. The self-dffuson coeffcent D can be evaluated for a system wth dmensonalty d D from the mean-square dsplacement (MSD) through the Generalzed Ensten formula as shown by Allen and Tldesley (1997) D lm r t r t 0 0 d D (.69) where, s the delay tme nk t, n s the tme nterval of each partcle sample, and K the number of partcle samples n delay tme whch s nt tk 1 tk and tkk tk. The schema s shown n Fgure 11. It can also be evaluated by the Green-Kubo formula through the velocty autocorrelaton functon (VACF) as ntroduced by Allen and Tldesley (1997) 1 D d t t d v 0 v 0 (.70) 0 D The term n the brackets n Eq. (.69) and Eq. (.70) denotes the tme correlaton functon for tme-dependent sgnals A( t), B( t ) (Hale, 1997) 70

71 1 t ( t) A( t0) B( t0 ) lm A( t0) B( t0 ) dto t t (.71) 0 The functon At ( 0) s sampled at t 0 and Bt ( 0 ) after a delay tme. The ntegral s evaluated over many tme orgns t 0 shown n Fgure 11. For a gven number of tme samples M and a number of partcle samples K n delay tmes the tme correlaton can be approxmated as a summaton over M K 1 number of avalable tme orgns (Hale, 1997) MK1 1 ( t) A( t ) B( t ) M 0 0 (.7) K 1 m1 The calculaton can be mproved by averagng over all partcles n each sample, MK1 N 1 P A( t0) B( t0) A( t0) B( t0 ) (.73) M K 1 N m1 1 Then the self-dffuson coeffcent based on MSD s provded by P MK1 N 1 P D lm t0 t0 d t M K 1 N r r (.74) m1 1 D And the self-dffuson coeffcent based on VACF s gven by P D dt t t d MN 1 1 MK NP v v (.75) D P m1 1 The ntegral s approxmated wth a summaton over all the dscrete tme n delay tme shown by nkt M K1 NP 1 D d D MN P tt0 m1 1 vt0 v t0 tt (.76) In ths work, we mplemented the VAC algorthms followng Hale (1997) and the MSD algorthm followng Rapaport (1995). 71

72 Fgure 11. Samplng used nn SDPD-DV for evaluaton of the self-dffuson coeffcent. The delay tme s τ. The total number of samples s M, and K s the number of samples n delay tme. The tme orgns are ndexed from m=1 ~ (M-K+1). (b) Shear Vscosty Another transport coeffcent we studed n ths work s the bulk vscosty. An expresson analogous to the Ensten dffuson Eq. (.69) lead to the vscosty expresson gven by Allen and Tldesley (1987) where 1 lm y x y x 6 kbtv x y m v t0 r t0 m v t0 r t0, (.77) denotes a sum over the three pars of dstnct vector components ( xy, yz, and zx ) x y whch mprove the statstcs. The alternatve Green-Kubo form s gven by Allen and Tldesley (1987) V 3kT B P 0 xy t Pxy dt, (.78) 0 x y where 1 1 P m v v r f V j j xy j xj yj xj yj (.79) 7

73 The second term n P xy can be fulflled wth the force computaton, and for Lennard-John potental the weghtng functon s f r f r r, whch leads to Pxy Pyx. j j j j The numercal mplementaton follows secton..8.(a) as shown n Fgure Instantaneous and Sample-Averaged Flud Feld Propertes For analyss and vsualzaton we construct also Euleran (feld) propertes Vr, t, S r, t, T r, t. To obtan such propertes we frst generate a tetrahedral mesh over the flud doman of nterest. The vertces of the tetrahedron are desgnated as the nodes of the doman, each wth an assgned ndex d 1,.., Gd and coordnates r d ( xd, yd, zd ). An nstantaneous flud property assocated wth a node d at t smoothng functon approxmaton Eq. (.8) shown n Fgure 1, as k t, s obtaned usng the m k W ( rd r ) X X ( r d, t) X ( d) X d (.80) W ( r r ) m m k m k The summaton s extended over all the FPs wthn a set dstance l d from the vertex r d. For unsteady smulatons, we obtan after reachng steady-state the sample-averaged feld property at a node d at t k t, s gven by d k m k d d d m1 X ( r, t) X X (.81) For steady smulatons, we obtan after reachng steady-state the sample-averaged feld property at a node d s gven by m k d d d m1 X ( r ) X X (.8) 73

74 The choce of the Euleran grd sze and thus the nterpolatng dstance l d must be compatble wth the length scale of the phenomena under consderaton. An excessvely coarse (Euleran) grd or nterpolatng dstance could smooth out small scales of nterest. (a) A typcal post-processng grd. (b) Samplng regon near vertex of the grd. Fgure 1. Post-processng grd used for evaluaton of feld (Euleran) nstantaneous and tmeaveraged propertes from SDPD-DV partcle samples. 74

75 3. VERIFICATION, VALIDATION, AND ERROR OF THE ISOTHERMAL SDPD-DV In ths chapter we utlze the sothermal SDPD-DV code and perform verfcaton and valdaton tests that cover the hydrodynamc and mesoscopc regmes. The frst verfcaton of SDPD-DV nvolves comparsons wth analytcal solutons for a body-force drven, transent, Poseulle flow of water between parallel plates of 10-3 m heght. Physcal nsghts on the force method n SDPD-DV are obtaned by calculatng the components of the forces attrbuted to the dynamcally allocated vrtual partcles and compared wth prevous DPD nvestgatons by Altenhoff et al. (007) and Fedosov et al. (008). The second verfcaton test nvolves the transent, Couette flow of water between parallel plates of 10-3 m heght. The thrd test used for verfcaton nvolves the low-reynolds number, ncompressble flow over a cylnder of radus 0.0 m. Ths benchmark test has been used n SPH by Morrs et al. (1997), Vazquez-Quesada and Ellero (01) and n DPD smulatons by Km and Phllps (004). Our SDPD-DV results are compared wth those obtaned from ANSYS FLUENT (FLUENT 6.3.6, help system, ANSYS Inc.). The fnal set of benchmark tests nvolves calculaton of equlbrum states for lquds and gases n mesoscopc domans. Ths extended set of SDPD-DV smulatons evaluates the translatonal temperature for lqud water and gaseous ntrogen, and the self-dffuson coeffcent of lqud water. The SDPD-DV results are compared wth analytcal expressons ntroduced by Ltvnov et al. (009), Brd et al. (007) and experments Holz and Sacco (000). The SDPD- DV lqud water smulatons examne also the scale effects on the self-dffuson coeffcent and the Schmdt number by varyng the mass and sze of the flud partcles (Vazquez-Quesada et al., 009), (Ltvnov et al., 009). The materal from ths chapter can be found n Gatsons et al. (013). 75

76 3.1 Transent Body-Force Drven Planar Poseulle Flow The frst test case nvolves an ncompressble Poseulle flow between two statonary nfnte plane parallel plates as shown n Fgure 13. The velocty profles as predcted by SDPD-DV are compared to theoretcal formulatons. The SDPD-DV densty s plotted along wth the standard devaton. In addton, the boundary forces due to vrtual partcles are evaluated, and the dstrbutons are plotted. Ths test case verfes the ablty of the SDPD-DV to mnmze densty fluctuatons and enforce the no-slp condton on sold boundares Input Condtons and Computatonal Parameters The test nvolves an ncompressble Poseulle flow wth densty across two nfnte parallel, statonary walls wth separaton heght L Z as depcted n Fgure 13. The flud consdered n the SDPD-DV smulatons s H O(l) wth -3 a 1,000 kg m and kg m s. (a) 3D vew (b) X-Z plane D vew. Fgure 13. SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. Physcal doman showng the BPs and FPs. 76

77 Table 6: Input parameters used n SDPD-DV smulatons of transent planar Poseulle flow, transent planar Couette flow, and flow over a cylnder. Input Parameters Case Transent Poseulle Transent Couette Flow Over Cylnder L X (m) L Y (m) L Z (m) R m N/A N/A 0.0 f - (ms ) 10-4 N/A x M F kg T (K) a -3 1,000 1,000 1,000 kg m 1 1 kg m s kg m s -1-1 cv J kg K 4, W m K V -1 xw ms N/A N/A N 9,000 9,000 17,63 F N 8,8 8,8 7,440 B N C h (m) c (ms ) t (s) l mn m l d m M The physcal doman has L X 3 10 m, L Y m, L Z 3 10 m. The total mass of M F kg n the doman s represented by 9,000 flud partcles wth a constant temperature oft 300 K. The sold walls are represented by 8,8 boundary partcles as shown n Fgure 13. Perodc boundary condtons are mposed along the x -axs and y -axs. 77

78 Closure for pressure and densty s obtaned by usng the artfcal compressblty relaton Eq. (.5). Integraton s carred out followng Sec...7 and a constant body force of f 10 m/s 4 s mposed on each flud partcle. The DVPA method (Sec..4(b) and..6(b)) s appled for densty and force evaluaton. Over M 50 smulatons were performed to generate nstantaneous samples used to derve the nstantaneous partcle propertes. Flud propertes were then sampled on a structured grd wth edge length DV are lsted n Table 6 (Gatsons et al., 013). l d 4 10 m. Input parameters n the SDPD Results and Dscusson The classc Poseulle flow s due to an appled pressure gradent whch can be replaced by a body force per unt mass f (ms - ) parallel to the x-axs. The flow starts from rest and develops a velocty profle gven by Morrs et al. (1997) and Papanastasou et al. (1999) f 4 fl Z z ( ) sn exp n 1 Vx z, t z z LZ 3 3 n 1 t n0 n 1 L (3.1) Z LZ The sample-averaged feld densty ( r, t ) and velocty v( r, t) profles are plotted n d Fgure 14 along wth the standard devaton. The SDPD-DV densty ( Z d ) shown n Fgure 14(a) has an error a a of less than 4% n the nteror and less than 5% near the wall 3 boundary. The feld densty averaged over the entre channel, 966 kg/m s used Eq. (3.1) to evaluate the analytcal velocty profle. Comparsons between the analytcal and SDPD-DV velocty profles across the channel s plotted n Fgure 14(b) and show them to be n excellent agreement for all tmes consdered. The densty and velocty results demonstrate the ablty of our DVPA-based densty and force evaluaton method as well as ntegraton algorthm mplemented n SDPD-DV. d 78

79 (a) Densty dstrbuton wth standard devaton. (b) Velocty dstrbuton. Fgure 14. Sample-averaged flud densty ρ(z d ) and velocty Vx(Z d,t) from SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1,000 kg/m 3, T =300 K, η =10-3 kg m -1 s -1, and f x =10-4 m/s. The analytcal profles are plotted for verfcaton (Morrs et al. 1997). To further nvestgate the DVPA method and ts ablty to enforce the no-slp boundary condtons we evaluate the boundary forces due to the vrtual partcles usng Eq. (.60) and Eq. (.6). The dstrbuton of the vrtual partcle force components are shown n Fgure 15. As shown n Fgure 15(a) and Fgure 15(b), the normal components of conservatve and dsspatve VP forces are decreasng as the dstance from sold wall ncreases. The negatve and postve sgn of the conservatve normal vrtual partcle forces shows that they are perpendcular to the plates and pontng to the nner doman. Fgure 15(c) and Fgure 15(d) shows the tangental component of the conservatve and dsspatve vrtual partcle forces. Both contrbute to the mposton of the non-slp condton. Our results show that the forces due to vrtual partcles n 79

80 SDPD-DV have the same qualtatve behavor as the DPD boundary forces [Fg. 4, Altenhoff et al., 007] and to the average wall forces [Fg. 4, Fedosov et al., 008]. (a) Conservatve normal component. (b) Dsspatve normal component. (c) Conservatve tangental component. (d) Dsspatve tangental component. Fgure 15. Forces due to vrtual partcles from SDPD-DV smulatons of transent, body-force drven, planar Poseulle flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1,000 kg/m 3, η =10-3 kg m -1 s -1, and f x =10-4 m/s. 80

81 3. Transent Planar Couette Flow The test nvolves an ncompressble flow across two nfnte parallel walls wth the top wall movng wth a constant velocty as shown n Fgure 16. The velocty profles as predcted by SDPD-DV are compared to theoretcal formulatons. The SDPD-DV densty s plotted along wth the standard devaton. Ths benchmark test s used as further verfcaton of the SDPD-DV and ts ablty to enforce no-slp and no penetraton on a movng sold boundary Input Condtons and Computatonal Parameters The test nvolves an ncompressble flow wth densty across two nfnte parallel walls as depcted n Fgure 13 wth the top wall movng wth a constant velocty V xw.the SDPD-DV smulaton consders H O(l) wth -3 a 1,000 kgm and kg m s. The physcal doman has L X 3 10 m, L Y m, and L Z 3 10 m. The total mass of M F kg n the doman s represented by 9,000 flud partcles wth a constant temperature of T 300 K. The sold walls are represented by 8,8 boundary partcles as shown n Fgure 16 each wth v 5 1 xw Vxw ms. Perodc boundary condtons are mposed along the x-axs and y -axs. Closure for pressure and densty s obtaned by usng the artfcal compressblty relaton Eq. (.5). Input parameters are lsted n Table 6. 81

82 (a) 3D vew (b) X-Z plane D vew. Fgure 16. SDPD-DV smulatons of transent planar Couette flow. Physcal doman showng the BPs and FPs. 3.. Results and Dscusson The flow starts from rest and develops a velocty profle gven by Morrs et al. (1997) and Papanastasou et al. (1999) V x V V, 1 n n n sn z exp t Z n1 L Z LZ (3.) xw xw zt z L n The sample-averaged feld densty ( r, t ) and velocty Vr (, t ) profles are plotted n d Fgure 17(a),(b) along wth the standard devaton. The feld densty shown n Fgure 17(a) has an error a a of less than 4% n the nteror and less than 5% near the wall boundary. The 3 analytcal velocty profle from Eq. (3.) usng the average densty 966 kg/m s plotted n Fgure 17(b). A comparson between the analytcal and SDPD-DV velocty profles across the channel s plotted n Fgure 17(b). The numercal results from SDPD-DV are n very good agreement wth the analytcal soluton for all tmes consdered. d 8

83 (a) Densty dstrbuton wth standard devan. (b) Velocty dstrbuton. Fgure 17. Sample-averaged flud densty ρ(z d )and velocty V x (Z d, t) from SDPD-DV smulatons of transent, planar, Couette flow. The doman has L X =10-3 m, L Y = m, L Z =10-3 m, the flud s H O(l) wth ρ a =1000 kg/m 3, T =300 K, η =10-3 kg m -1 s -1, and V xw =1.5x10-5 m/s. The analytcal profles are plotted for verfcaton (Morrs, Fox and Zhu, 1997). 3.3 Steady Low-Re Incompressble Flow over a Cylnder The thrd benchmark test nvolves the SDPD-DV smulaton of flow over a cylnder. Ths test appeared n several SPH (Morrs et al., 1997; Vazquez-Quesada and Ellero, 01) and DPD (Keaveny et al., 005; Km and Phllps, 004) smulatons. The sample-averaged, steady velocty and pressure felds from SDPD-DV and comparson wth FLUENT are shown. The SDPD-DV flow feld exhbts the antcpated behavor for the low-reynolds number flow over a cylnder (Keaveny et al., 005; Morrs et al., 1997; Vazquez-Quesada and Ellero, 01; Km and Phllps, 004). For further drect quanttatve comparson the SDPD-DV and FLUENT, velocty 3 and pressure felds are supermposed and plotted ( x, y m, z). Addtonal verfcaton 83

84 s obtaned by comparng velocty and pressure feld along contours. Ths test further verfes the ablty of SDPD-DV to smulate curved sold boundares and the mplementaton of the artfcal compressblty method n SDPD-DV Input Condtons and Computatonal Parameters The ncompressble flud of densty s drven by a body-force f x over a cylnder of radus R. Perodc boundary condtons are assumed n the x, y and z drecton as shown n Fgure 18(a). The perodc boundary condtons can be realzed as a flow n an nfnte array of cylnders as depcted n Fgure 18(b). For verfcaton of the SDPD-DV results we also performed smulatons usng FLUENT over a doman shown n Fgure 18(b). The SDPD-DV smulaton consders H O(l) wth -3 a 1,000 kgm and kg m s drven by a body force f x ms. The physcal doman has L 0.1 m, L m, L 0.1 m and the cylnder radus R 0.0 m. The total flud mass X Y Z of M kg n the doman s represented by 17,63 flud partcles wth a constant F temperature of T 300 K. The sold walls of the cylnder are represented by 7,440 boundary partcles as shown n Fgure 18(a). The ntal flud partcle dstrbuton s shown n Fgure 18(a). The Reynolds number Re RV0 based on the cylnder radus and V 0 n the average velocty feld s Re 1. Closure s obtaned by usng the artfcal compressblty relaton Eq. (.5). Usng Eq. (.68) ntegraton s carred out wth t 1 10 s. Steady state was reached and M 100 ndependent samples were used to evaluate flud propertes on a grd wth 3 l d 5 10 m. The nput parameters for the SDPD-DV smulaton are lsted n Table 6. 84

85 (a) (b) (c) Fgure 18. Computatonal domans used n SDPD-DV and FLUENT smulatons of steady, low- Reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s. (a) SDPD-DV doman wth L X =0.1 m, L Y =0.015 m, L Z =0.1 m showng the BPs on the cylnder and FPs n the doman. (b) FLUENT doman L X =0.3 m, L Y =0.015 m, L Z =0.3 m ncludes an array of cylnders to smulate the perodc flow. The nsert shows the extend of the SDPD-DV physcal doman. (c) Planes P1,P,P3 and contours C1,C,C3,C4 used for comparson of SDPD-DV and FLUENT results. In order to acheve perodc boundary condtons n ANSYS FLUENT, we placed 9 cylnders n a lattce as shown n Fgure 18(b). The cylnders were placed apart so that there s no nfluence and smulaton was performed wth 46,13 cells. The physcal doman for the FLUENT smulaton has L 0.1 m, L m, L 0.1 m. The gravtatonal acceleraton X Y Z per unt mass g x n ANSYS FLUENT was equal to the SDPD-DV body force f x. For drect 85

86 comparson we sampled feld propertes on planes P1 (y=0 m), P (y= m), P3 (y=0.015 m) and along contours C1, C, C3 and C4 shown n Fgure 18(c). (a) SDPD-DV (b) FLUENT (a) SDPD-DV (b) FLUENT Fgure 19. Sample-averaged V x (r) and P(r) on y = 0 m, y = m, y = m planes from SDPD-DV smulatons. V x (x,z) and P(r) from FLUENT smulaton. Steady, low-reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s. 86

87 3.3. Results and Dscusson The overall flow feld characterstcs are shown n Fgure 19. The sample-averaged, steady V ( r ) and P( r ) pressure felds from SPDP-BV and V () r and P() r from FLUENT are shown x d d on planes P1, P, P3 parallel to x axs. The SDPD-DV flow feld exhbts the antcpated behavor for the low-reynolds flow over a cylnder (Keaveny et al., 005; Morrs et al., 1997; Vazquez-Quesada and Ellero, 01; Km and Phllps, 004). The SDPD-DV propertes are qualtatvely and quanttatvely smlar to those obtaned from FLUENT. For further drect quanttatve comparson the SDPD-DV and FLUENT velocty and pressure felds are x supermposed and plotted n Fgure 0(a)-(b) on the plane 3 ( x, y m, z). The nonsmooth character of the SDPD-DV pressure contours are due to fluctuatons ntroduced by the artfcal compressblty model appled to ths ncompressble flow. It s mportant to note also that veloctes and pressures consdered are very small and susceptble to numercal perturbatons. Addtonal verfcaton s obtaned by comparng velocty and pressure feld along contours C1, C, C3, and C4 shown n Fgure 0(c)-(d). The pressure comparson along the centerlne C3 n Fgure 0(d) shows the ablty of our sold boundary densty and force evaluaton to accurately predct pressure n the ram and wake sde of the cylnder where flud partcles are mpngng the curved boundary. The overall very good agreement demonstrates the ablty of our SDPD-DV mplementaton to smulate curved sold boundares. 87

88 (a) V X (r) on y= m plane. (b) P(r) on y= m plane. V x 10-5 (m/s) SDPD-DV C1 FLUENT C1 SDPD-DV C FLUENT C P 10-6 (Pa) SDPD-DV C3 FLUENT C3 SDPD-DV C4 FLUENT C Z (m) X (m) (c) V X (r) along C1 and C. (d) P(r) along C3 and C4. Fgure 0. Sample-averaged V x (r) and P(r) on y = m plane and along C1,C,C3,C4 from SDPD-DV smulatons. V x (x,z) and P(r) from FLUENT smulaton. Steady, low-reynolds number water flow wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1. The cylnder has R=0.0m and the body force per unt mass s f x =g x =1.5x10-7 m/s. 88

89 3.4 Equlbrum State and Self-Dffuson Coeffcent The last seres of tests also serve for valdaton and verfcaton snce the SDPD-DV results for the self-dffuson coeffcent are compared wth data and analytcal formulas. The tests also provde the opportunty to examne the scale-dependence of our SDPD-DV mplementaton and compare our results wth those of Ltvnov et al. (009) and Vazquez-Quesada et al. (009). These tests are ntended as a demonstraton of our SDPD-DV to smulate mesoscale flows at equlbrum states Input Condtons and Computatonal Parameters We performed SDPD-DV smulatons of H O(l) wth M F n the range kg, 3 a 1,000 kg/m, kg m s and T 300 K. We performed also SDPD-DV smulatons of N (g) wth M F n the range kg wth 3 a kg/m, 10 kg m s and T 300 K. In order to examne flud partcle scale effects we follow Vazquez-Quesada et al. (009) and assume that the sze of the flud partcle s gven n terms of the SDPD varables as, / 1/3 1/3 D V m (3.3) For comparson, we need also length scales for the real lqud and gas molecules. For a lqud wth vscosty A, temperature T A and assumng that the flud occupes a cubc lattce where the molecules smply touch each other, the approprate scale s gven by Brd et al. (007) R ( V / N ) ( m / ) (3.4) 1/3 1/3 A A A A A where, V A s the molar volume and N A s the Avogadro number, m A s the mass of the molecule A. For a gas a relevant length scale can be consdered the molecular dameter D g. The nput 89

90 parameters and some derved varables from the SDPD-DV smulatons are lsted n Table 7 and Table 8. Table 7. Input and derved parameters n SDPD-DV smulatons of mesoscale flows of lqud water at equlbrum states n rectangular domans wth perodc boundares. Inputs, H O(l), T 300 K Derved SDPD-DV Case L X, Y, Z ( m) m m HO N FP t ( s) h (m) c 1/3 ( V ) ( R ) H 0 D ( m s -1 ) h D ( m s -1 ) , , , , , , , , , , , , , , , , ,

91 Table 8. Input and derved parameters n SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares. Inputs N (g), a kg/m 3, T 300 K Derved SDPD-DV Case L X, Y, Z m mn ( m) N FP t ( s) h (m) c ( V ) D 1/3 N ,371 3, ,505 1, , Results and Dscusson Fgure 1 shows the ( v v ) phase plots for the smallest and largest scales consdered. The x y plots show that the equlbrum states sampled have partcle veloctes that are not scale free. To gan further nsght, we use the flud partcle veloctes and feld mean veloctes obtaned from the SDPD-DV smulatons to evaluate the average translatonal temperature for the entre system. Followng the standard descrpton for the thermal (or random) partcle velocty C ( t) v ( t) V( r, t) (Gombos, 1994) T t N F vx Vx vy Vy vz Vz m 1 m c 3k N 3k B F B (3.5) Table 7 and Table 8 show the rms thermal speeds, c evaluated from the SDPD-DV smulatons. These thermal speeds are nversely related to the flud partcle masses consdered and they are equal to those obtaned wth the equlbrum Maxwellan formula 3 k T / m for B 91

92 the SDPD flud. Fgure (a) shows the evoluton of the translatonal temperature for Case 10. Ths behavor s typcal for all cases consdered and shows that the system quckly reaches equlbrum and the mposed FP temperature of T 300 K. Fgure (b) and Fgure (c) depct the average translatonal temperature for all cases smulated n Table 7 and Table 8. It s clear that for the range of flud partcle masses and szes consdered the average translatonal temperature s scale-free and matches the mposed flud partcle temperature for both the H O(l) and N (g). We also evaluate the self-dffuson coeffcent for the H O(l) smulatons followng the MSD Eq. (.74) and VAC Eq. (.76) methods. Varous analytcal formulas for the selfdffusvty are used for valdaton. The self-dffuson coeffcent s gven by Brd et al. (007) by D AA kt B A ( R ) (3.6) A A Assumng by analogy, that the SDPD lqud has a lattce scale D from Eq. (3.3), then Eq. (3.6) can be expressed as D kt B (3.7) D An addtonal expresson for the self-dffusvty n a SDPD lqud s obtaned followng Ltvnov et al. (009) and usng the smoothng functon (.38) used n ths work. It provdes h k T D (3.8) h B 63m 9

93 Fgure 1. Phase plot v x -v y from SDPD-DV smulatons of H O(l) wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1 and N (g) wth ρ a =1,184 kg/m 3, T = 300 K, η =10-5 kg m -1 s -1. Results show the scale effects of flud partcle sze on velocty. The Schmdt number s then evaluated by Sc D (3.9) 93

94 (a) (b) (c) Fgure. Average translatonal temperature from SDPD-DV smulatons of H O(l) wth ρ a =1,000 kg/m 3, T = 300 K, η =10-3 kg m -1 s -1 and N (g) wth ρ a =1,184 kg/m 3, T = 300 K, η =10-5 kg m -1 s -1. (a) Average translatonal temperature as a functon of tme (Case 1, H O(l)). (b) Average translatonal temperature and standard devaton as a functon of m and D for H O(l). (c) Average translatonal temperature and standard devaton as a functon of m and D for N (g). In Fgure 3(a) we plot the self-dffuson coeffcent for H O(l). The expermental value D H O,H O.5 10 m s 9-1 s from Holz and Sacco (009). Wth 6 m HO kg and 3 HO 1,000kg m then Eq. (3.4) provdes 10 RHO m and the analytcal value D AA s evaluated from Eq. (3.6). The values of D from the MSD Eq. (.74) and VAC Eq. (.76) are also plotted n Fgure 3(a). For comparson we also evaluate D from Eq. (3.7) and h D from Eq. (3.8) usng SDPD-DV results for all the varables enterng these expressons. Fgure 3(a) shows that the self-dffuson coeffcents obtaned from MSD, VAC and the SDPD-lqud formulas are n good agreement been wthn a factor of about 3 of each other. The expermental 94

95 and analytcal value of the self-dffuson coeffcent s approached by SDPD-DV, as the mass of the SDPD partcles approaches asymptotcally the relevant physcal scale. (a) (b) Fgure 3. Self-dffuson coeffcent and Schmdt number from SDPD-DV smulatons of H O(l), ρ a =1,000 kg/m 3, η=10-3 kg m -1 s -1, T=300K for varous szes of the SDPD-DV flud partcles. Expermental and analytcal estmates, as well as analytcal estmates usng SDPD-DV parameters are shown for valdaton and verfcaton. It should be noted, that at the real molecule level the applcablty of the SDPD model becomes tenuous. These results also show that for the flud partcle szes consdered n ths work, the self-dffuson coeffcent from SDPD-DV s not scale-free, a result that corroborates 95

96 earler results of Ltvnov et al. (009) and Vazquez-Quesada et al. (009). The Sc numbers from Eq. (3.9) usng the varous dffuson-coeffcents are plotted n Fgure 3(b). Smlar to prevous results, we show that the Sc number scales wth the mass (or sze) of the SDPD partcles, dentfed n our work wth ether the SDPD-flud sze D through Eq. (3.7) or the smoothng length h through Eq. (3.8). The results show also that wth SDPD-DV we acheve Sc values close to the realstc ones. 96

97 4. VERIFICATION, VALIDATION AND ERROR ANALYSIS OF SDPD- DV In ths chapter, we perform valdaton and verfcaton of SDPD-DV usng the full-set of SDPD Eq. (1.6)-(1.64) as mplemented n the SDPD-DV method dscussed n Chapter. The valdaton and verfcaton ncludes an extensve set of SDPD-DV smulatons of gaseous ntrogen n mesoscopc perodc domans n equlbrum. The self-dffuson coeffcent for N (g) and shear vscosty at equlbrum states are obtaned through the mean-square dsplacement for the range of flud partcle masses (or szes) consdered. Addtonal verfcaton nvolves SDPD-DV smulatons of steady Couette N (g) flow between parallel plates. The top plate s movng at V xw =30m/s and separated by 10-4 m from the bottom statonary plate. The top plate s assgned a constant temperature T 1 =330K and bottom plate T =300K. The SDPD-DV feld velocty and temperature profles are compared wth those obtaned by FLUENT. 4.1 Equlbrum State and Transport Coeffcents We consder frst N (g) systems n thermal equlbrum. The full algorthm of SDPD equatons s consdered wth entropy provded by Eq. (1.64) and closed by Eq. (.44)-(.49) Input Condtons and Computatonal Parameters The smulatons of N (g) were performed n rectangular domans wth LX LY LZ n the range ~ m, wth mass M F n the range kg. The smulatons examne the effects of flud partcle mass m / m and the smoothng length h. N 97

98 Table 9: Input parameters n full-set SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares. Case L XYZ Inputs N (g), a kg/m 3, T 300 K,P 0 = m h t ( 10-6 FP m) mn ( 10-8 m) ( 10-8 m) ( s) N V 1/3 0 ( 10-5 kg m -1 s -1 ) k 0 (W m -1 K -1 )

99 Table 10: Derved parameters n SDPD-DV smulatons of mesoscale flows of N (g) at equlbrum states n rectangular domans wth perodc boundares. Derved SDPD-DV 1/3 Case ( V ) T P D C ( 10-8 m) (kg/m 3 ) (K) (Pa) ( 10-9 m /s) ( 10-5 kg m -1 s -1 )

100 For each m / m we consdered four h s so that the resultng number of flud partcles N wthn the support doman, L, s 50, 100, 00, 360 upon ntalzaton. Perodc boundary condtons are mposed on each sde of the rectangular doman. Upon ntalzaton we set 3 a kg/m, ( 0) 300K T t, 5 ( t 0) kg/m s and ( t 0) 0.06 W/m K. Durng the computaton the transport coeffcents appearng n the SDPD equatons (1.6)-(1.64) are evaluated as functons of partcle temperature T t accordng to power law (Whte, 1974) T () t () t 0 T 0 n (4.1) T t k t k0 T 0 n (4.) where n s the power law coeffcent of the order of 0.7, T0 300K the reference temperature, W m K 1 1 the heat conductvty of N (g) at 0 T, and kgm s s the vscosty of N (g) at T 0. The heat capacty (J/K) s evaluated as Eq. (.48). 3 C NkB (.48) In order to examne flud partcle scale effects we follow Vazquez-Quesada et al (009) dscusson and assume that the sze of the flud partcle s gven n terms of the SDPD varables as, and Table 10. / 1/3 1/3 D V m (4.3) Input and some derved parameters from the SDPD-DV smulatons are shown n Table 9 100

101 4.1. Results and Dscusson Table 10 and Table 11 show that the average densty, temperature T and pressure P exhbt mnmal perturbatons from the nput value for the entre range of parameters consdered. These rms thermal speeds, c evaluated from the SDPD-DV smulatons, are nversely related to the flud partcle masses consdered and they are equal to those obtaned wth the equlbrum Maxwellan formula 3 k T / m for the SDPD flud. B m /m N (g) =118 m /m N (g) =155 m /m N (g) =7544 m /m N (g) = m /m N (g) = Eq. (4.4) Data (Holz and Sacco, 000)) 10-5 Self-Dffusvty D (m /s) Case Case Case 9 Case 13 Case Smoothng Length h (m) Fgure 4: Self-dffuson coeffcent from SDPD-DV smulatons of N (g) for values of h and m /m N(g) used n Cases 1-0. Analytcal estmates are from Eq.(4.4) and data by Holz and Sacco (000). 101

102 We also evaluate the transport coeffcent for the N (g) smulatons followng the MSD Eq. (.74) for self-dffuson coeffcent and Eq.(.77) for vscosty. Valdaton and verfcaton s obtaned by comparson wth expermental (Holz and Sacco, 000) and analytcal values. For a gas wth molecular dameter gven by Brd et al. (007) as D A, molecular mass m A and mass densty A the self-dffusvty s D AA m k T 1. (4.4) 3 A B A DA A The self-dffuson coeffcent of SDPD-DV smulatons from Eq. (.74) for N (g) are plotted n Fgure 4 for varous smoothng lengths h and partcle masses of m. For comparson, the values D AA from Eq. (4.4) are evaluated and plotted n Fgure 4 usng N (g) molecular value. As t shown n Fgure 4, the self-dffusvty of SDPD-DV s not scale free. The smulatons show that D decreases wth ncreasng mass rato. Ths s because that the smaller the flud partcle s, the larger s ts stochastc agtaton. 5 Shear Vscosty kg m -1 s -1 ) SDPD-DV (MSD) t m /m N (g) Fgure 5 : Shear vscosty from SDPD-DV smulatons of N (g) for value of m /m N(g) used n Cases 1-0. Intal nput vscosty η(t=0) s plotted for verfcaton. For a gven mass rato, ncreasng the h, ncreases D, snce that the stochastc agtaton s stronger n the bgger support doman. The values for calculated from the SDPD-DV 10

103 smulatons usng Eq. (.77) are plotted n Fgure 5. We plot for comparson the ntal value ( t 0) whch s also 0. It can be seen that shear vscosty s scale free and s not affected by the choce of partcle mass or the smoothng length. Case 9 Case 10 Case 11 T K T K t r Pa P tt s(s) tt s(s) ts Fgure 6: Effects of tme step on equlbrum temperature, translatonal temperature and pressure from SDPD-DV smulatons of N (g). Results are for Case 9, 10, and 11. (K) T T tr (K) (K) t (s) T t Case 17 N Case m =45 18 Case N m =90 19 N m = t (s) t (s) t (s) t (s) Fgure 7: Effects of tme step on equlbrum temperature, translatonal temperature and PX10 5 (Pa) 5 10 (p ) pressure from SDPD-DV smulatons of N (g). Results are for Case 17, 18, and 19. a P Case 17 Case 18 Case

104 Table 11: Fluctuatons n temperature, densty, pressure and velocty, from SDPD-DV and analytcal expressons. Analytcal Varance Varance n SDPD-DV case T ( ) P V T ( P ) V

105 We examne also the effects of tme step n the SDPD-DV smulatons on the equlbrum characterstcs. In Fgure 6 and Fgure 7, we plot the SDPD-DV temperature T averaged over the entre doman, the average translatonal temperature T t from (3.5), and the average SDPD- DV pressure P for Case 9, 10, 11 and Case 17, 18, and 19. These results show that tme step can lead to a sgnfcant error dependng on the FP mass and smoothng length. The results show also that for larger FPs the stochastc agtaton s smaller. We compute also the varances n temperature, densty, pressure and velocty for each case 1-0 accordng to Eq. (1.5), (1.), (1.3) and (1.7). The results n Table 11 show that SDPD-DV are n very good agreement wth the analytcal values. 4. Steady Planar Thermal Couette FLow In ths secton, we contnue the verfcaton of SDPD-DV by performng smulatons of steady planar non-sothermal Couette flow Input Condtons and Computatonal Parameters The test nvolves an ncompressble flow wth densty 3 a kg/m across two nfnte parallel walls as depcted n Fgure 8(a) wth mposed constant wall temperatures and the top wall s movng wth a constant velocty V xw. The SDPD-DV smulaton consders N (g) wth densty -3 a kg m, ntal vscosty ( t 0) kg m s and ntal heat -1-1 conductvty ( t 0) 0.06 W m K. Closure for pressure and densty s obtaned by usng the Eq.(.47) and (.46). Heat capacty C V s gven by Eq.(.48) wth ntal value C t. 14 ( 0).4 10 J/K 105

106 The physcal doman has 4 L XZ, 1 10 m and L Y m as shown n Fgure 8. Perodc boundary are mposed along the x-axs and y -axs. The flud s represented by 10,890 FPs and total mass M F kg. The upper wall s located at Z m and s 1 assgned a velocty V 30 ms and temperature T 330K. The temperature of the lower 1 xw wall located at Z 0 s set to T 300K. The sold walls are represented by 10,560 BPs. Input condtons used n the SDPD-DV smulaton are shown n Table 1. Table 1: Input parameters used n SDPD-DV non-equlbrum smulatons of Couette flow. Input Parameters Coutte L (m) X L (m) Y L (m) Z - f (ms ) N/A x M kg F Upper T (K) Lower T (K) a kg m 1 1 t kg m s ( t 0) 0 kg m s C ( t 0) J K t W m K 0.06 V -1 xw ms 30 N F N B N C 300 h (m) t (s)

107 For verfcaton we also performed a smulaton usng FLUENT n a doman shown n Fgure 8 and nput parameters shown n Table 1. The wall boundary condtons are assgned to both upper wall and lower wall wth perodc boundary condtons to other boundares. V xw T 1 V xw Wall Boundary T 1 T Perodc Boundary T Wall Boundary (a) SDPD-DV doman wth L x =10-4 m, L y = m, L z =10-4 m showng the BPs on the top and bottom and FPs n the doman. (b) FLUENT doman wth L x =10-4 m, L y = m, L z =10-4 m showng mposed wall boundary condtons and perodc boundary condtons. Fgure 8: Computatonal domans used n SDPD-DV and FLUENT smulatons of steady N (g) Couette flow wth ρ a =1.184 kg/m 3, η 0 = kg m -1 s -1, κ 0 =0.06 W m -1 K -1. The upper wall T 1 = 330 K, lower wall T = 300 K and V xw =30m/s. 4.. Results and Dscusson The overall flow feld characterstcs are shown n Fgure 9. We plot the SDPD-DV sampleaveraged steady temperature T r and velocty V r felds on planes X 0m, x 5 X 510 m, and 4 X 110 m. 107

108 (a) SDPD-DV (b) FLUENT (c) SDPD-DV (d) FLUENT Fgure 9: Sample-averaged V x (r) and T(r) on y = 0 m, y = m, y = m planes from SDPD-DV and FLUENT smulatons of steady state Couette flow. The doman has L X =10-4 m, L Y = m, L Z =10-4 m, upper wall T 1 = 330 K, lower wall T = 300 K, and V xw =30 m/s. The flud s N (g) wth ρ a =1.184 kg/m 3, η 0 = kg m -1 s -1 and, κ 0 =0.06 W m -1 K