3' H {3'55 Date 0-7639 L: 117:"- Michiganfir L:;:*:;;‘;;"; I This is to certify that the thesis entitled A ”56005 DIFFUSION M0051. 1:0,? V0,? TEX ~IN—CéZZ. CAL CUM 770/v5 presented by Kue Pan has been accepted towards fulfillment of the requirements for MAS TERA; Kczgwédggree in flaw/wed ENGIAEkPIA/q wag/fl Major professor 2/ 21/86 MS U is an Affirmative Action/Equal Opportunity Institution IV1£31_J RETURNING MATERIALS: Place in book drop to LIBRARIES remove this checkout from —c—- your record. FINES wiH be charged if book is returned after the date stamped beIow. A‘VISOOUS DIFFUSION IDDEL FOR VDIIEIPIN-CELL CALCULAIIONS By [no Pen A THESIS Submitted to liehigen Stete university in pertiel fulfill-ent of the requirenente for degree of EASTER OF SCIENCE Depertnent of lechenicel Engineering 1986 ABSTRACT A.VISCOUS DIFFUSION MODEL FOR VOIIEIPIN-CELL CALCULATIONS By (no Pen A.Vorter-In-Cell technique hee been need to etudy verioue vortex ring evolutione in two dineneione. The noleculer diffusion prooeee ie nodeled by uein; e eonbineticn of e nodified vorticity gredient epprorinetion with e nine-gridrpoint. coneervetive redietribution filter. To exenine the eocurecy end reliebility of the elgorithn. both one end two-dineneionel teet ceeee were investigeted. Reeulte ere conpered with theoreticel eolutione. Furthernore. the cepehility to einulete the vortex ring/loving well interectione with thie technique ie evelueted by conperin; the celeuleted outconee with erperinentel reeulte vie etebility nepe quentitetively end flow vieueliretione quelitetively. A diecueeion of the no-elip boundery condition ie eleo included. To My Perente ii ACINWLENEIENTS I would like to erpreee ly greet epprecietion to ny perente for their eupport during ny edueetionel eereer. Thenk you. I would like to thenk ny nejor profeeeor. Dr. Rioherd I. Bertholc-ew. for his guidence end generoue eeeietence throughout this reeeerch progren. I eleo gretefully eeknowledge the other nenhere on ly theeie col-ittee. Profeeeor R. B. Feloo end Profeeeor C. 1. 'eng. for their fruitful diecueeione end edwice. In eddition. I wieh to thenk liee I. Cline. Ir. C. Chu end Ir. J'. Cheng for help end friendehip they heve provided. Thenke ere eleo due Linee. ly wife. who geve no e eon. Tin. during this etudy. iii TRBLE OF CONTENTS LIST OF Tum OOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO LIST 0’ “Guns O0......O...O0.000COOOOOOOOOCOOOOOOCOO0.0.0.0000... MENU-Am“ 0.0.0.0....0......OOOOOOOOOOOOOOOOIO0.0.0...0.0.000... CHAPTER 1 5 6 INTRODUCTION ............................................... TUE TUEOR! ................................................. 2.1 Governing For-uletion for Inviecid Fluide ............. 2.2 Viecoue Diffueion Ibdel ............................... 2.3 Procedure for Shnuletion of VIC ....................... BENCH EARRING IODELS AND NUIERICAL PARAMETERS .............. 3.1 One-Dineneionel Teeting lodel -- Stoke'e Leyer ....... 3.2 Two-Dineneionel Teeting Model -- Free vortex Peir .... 3.3 Nunericel Perenetere of vortex Ring/loving 'ell Interectione .......................................... RESULTS .................................................... 4.1 Stebility lepe ........................................ 4.2 vector Plote .......................................... DISCUSSION ................................................. mNQIUSIONS O0.0.00.0...0.0...0.0000000000000000000000COO... LIST OF mas .0...0.......0...COOOOOOOOOOOCOOOOOOOOOO00...... mm OOOOOOOOOOOOOOOO0....OOOOOOOOOOOCOOOOOOOOIO0.0000000000000000 FIGURES O0.0.0.0...OOOOOOOOOOOOOOCOOOIOOOCOO...OOOOOOOOOOOOOOOOOOOO iv Pege v vi ix 14 17 17 21 25 28 28 28 30 ‘34 36 38 39 LIST OF TABIES TABLE Pege l The Conetent Coefficients of Queei-Geueeien R.d1‘tribntiou Faction 0.0...OOOCOOOOOOOOOOOOOOOOOOO0.0.0.... 38 Figure 2.1 2.2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 LIST OF FIGURES Sketch of One - Dinensionel Verticity Concentretion Profile on Illustreting Diffusioannduced Velocity .......... The Geo-etry end Notetion of the Nine-Grid-Point Filter ..... Sketch Illustretes the ley to Pertition Vortioity Field of Boundery Leyer. (This is en Exenple of Three lerkers ’0: unit un‘th 0f "11) .OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Conperison of Exeot end Celculeted Velocity Profiles for Stoke's Leyer et Ax/8-1/6.4 Ihioh is Used es Initiel Condition for The Celculetion ....................... Conperison of Exect end Celculeted Vorticity Profiles for Stoke's Leyer et Axle-l/6.4 'hich is Used es Initiel Condition for The Celouletion ....................... The Dependence Between Reletive Error end Spetiel Resolution for 'ell Velocity Celculetious (i.e. uc lo et y-O) Using Quesi-Geussien Filter. hi‘tlvc Errop(n0810.lUV)-1 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee Reletive Error vs. Teaporel Resolution for Diffusion Process Sinuletion 7t Isl} Leyer. Fixed Ax/6.-ll6.4 to: P-8/6.-(as/64)‘ ‘. 21 ’ .ndz OCOO...OOOOOOIOOOOOOOOOOOOO Reletive Error vs. Tenporel Resolution for Diffusion Process Sinuletion 7t lel} Leyer. Fixed Ax/6.-l/3.2 for pus/c,-(68/64)‘ ’. ‘ ’ ‘ndz OOOOOIOOOOIOOIOOOOO00...... Reletive Error vs. Spetiel Resolution for Diffusion Process Sinuletion 7t Isl} Leyer. Fixed At/Tho.2875 for P-e/a,-(68164)‘ ‘. 21 ’ ‘ndz OOOOOOOOOOOOOOOOOOO00...... Discrete Vortioes Arrenged to Sinulete Line Vortex. (“1. 1‘ 'n an.p1. Of $12) OOOOOOOOOOOOOOOOOOOOOCO...00.... vi Pege 39 4O 41 42 43 44 45 46 47 48 3.9 3.10 3.11 3.12 3.13 4.1 4.2 4.3 4.4 4.5 Couperisons of Vortioity Distributions of Vortex Ring for N-l. N-6 per Lobe end Theoreticel Approxinetion of Nexworthy [1977]. The Vortioity is Celculeted by l-Ovlax-aulay‘............................................... Tine-Dependent Redius of Gyretion Celculeted Using N-6 per Lobe for Pure Diffusion end Uydrodynenic Notion with Diffusion. The Streight Line is the Theoreticel Solution Given by Equetion (25) ................. Tine-Dependent Rediue of Gyretion Celouleted Using N-60 per Lobe for Pure Diffusion end Eydrodynenic Notion with Diffusion. The Streight Line is the Theoreticel Solution Given by Equetion (25) ................. Tine-Dependent Redius of Gyretion Celouleted Using N-500 per Lobe for Pure Diffusion end Eydrodynenie lotion with Diffusion. The Streight Line is the Theoreticel Solution Given by Equetion (25) ................. The Gecnetricel Arrengenent of Vortex Ring/loving 'ell Interections for Vortex-In-Cell Nunerieel Sinuletions ..... Stebility Neps of Ccnputetions with/without Diffusion Nodel for 1-15. Cupered with Experinentel Result of Lieng[1984]. The Error Bers for Celculeted Outcones ere 22.5 seconds ................................... Co-puter Genereted Vortioity Notion Pictures for en Unsteble Interection which the Line Peir Ingests 'ell Leyer Fluid es it "Creshes" on the Cell ................ Cclputer Genereted Vortioity Notion Pictures for en Unsteble Interection which the Line Peir Ingests 'ell Leyer Fluid es it Turns Awey fro. the Well ............. Ccnputer Genereted Vortioity Notion Pictures for e Steble Interection which the Line Peir Leeves the "11 'ithont Br.‘k1n‘up O.....IOOOOOOOOOOO0.0.0....0.0.00... Cuputer Genereted Vortioity Notion Pictures for e Steble Interection in the Absence of Diffusion Process. "Photos" Correspond to The Sene Tine Step Nunber es Figure 4.6 lhich The Initiel Conditions lore Set Identicelly to This Cese ................ vii 49 50 51 52 53 54 55 56 57 58 4.6 5.1 Computer Genereted Vortioity Notion Pictures for e Steble Interection in the Presence of Diffusion Process. "Photos” Correspond to The Sene Tine Step Nunber es Figure 4.5 Ihich The Initiel Conditions Vere Set Identicelly to This Cese ................ 59 Caperieone of Stebility Neps for 7-15'. Celculetions without Diffusion for 6 Nerkers per Lobe were Cerried Out by Bertholo-ew .................... 60 viii ;= I: «a 5! ft 9 «e 41 4 NOIENCLATURE Ring die-eter. the distence between two centers of vortex lobes Frection. 0 < f < l Circuletion Number of cells in x.y direction Anguler inpulse Nulber of vortex nerkers used in oonputer celculetion Distence fron vortex center The distence. r. of which oonteins lOOf’i vorticity Reynolds number. pUrDr/u Effective redius of vortex lobe. r..’. Squere of the redius of gyretion Initiel S et tine to Tine Elepsed tine to spreed out vorticity to distence R. Deley tine or duretion of well novenent Discrete tine step length in conputer celculetion Rotetionel tine pereneter for well leyer. 8.]U‘ Rotetionel tine pereneter for vortex lobe. (2nR.)'/G Convection velocity conponent in x.y direction Diffusion-induced velocity component in x.y direction Reel velocity conponent in x.y direction -9 U-u+5.‘V-v+3 Velocity vector Ring speed 'ell speed Longitudinel direction ix Ax.Ay Cell size in x.y direction y Direction nornel to x Vortioity u Dynenic viscosity 9 Fluid density 6 Iell leyer thickness defined by u3/U191/200 et tine t. 4(ut/p)'-‘ b, Initiel well leyer thickness. 8 et tine t. 7 Initiel engle of epproeoh n Sinilerity verieble. yl(4utlp)°°‘ W’ Streen function 0 Quesi-Geuseien redistribution function CHAPTER 1 INTRODUCTION Nowedeys the conplexity of neny flows of technologicel interest Iekes treditionel nunericel lethods such es finite difference/finite elenent lore difficult to treet problens then ever. One of the reesons for this is the constreints set by the conputer: the nenory end CPU tine needed to resolve the flow feetures is excessive due to the existence of e wide renge of length end tine sceles. This hes led to the develop-ent of new conputetionel techniques such es discrete vortex nethods. A problen.which illustretes this point is the topic of this thesis -- the evolution of e vortex linepeir when it cones into proxinity to e loving well. This is e very con-on phencnenon. A well-known exenple is the treiling vortices produced by e julbo eircreft deperting from the runwey. A nunber of experinents heve been perforned by Hervey end Perry [1971]. Berker end Crow [1977] end neny others to investigete the behevior of vortex peirs in the presence of the ground. A theoreticel epproxinetion for the epproeoh of e vortex peir et right engles to e plene snrfeoe in the ebsence of viscous effects wes cerried out by Seffhen [1979]. Although the reeson for the observed rebounding wee not clerified. Peece end Riley [1983] pointed out thet the rebounding of e vortex peir from s plene boundery is essentielly e viscous phenonenon. This is e strong ergunent in fever of incorporeting viscous effects into e discrete vortex nethod when e vortex ring/well interections is to be correctly sinuleted. Ie will return to this point below. 2 The motivetion for studying vortex rings comes from Felco [1977] who hes postuleted thet the turbulence production_nechenisn neer e well is releted to interections between vortex-ring-like eddies end the viscous subleyer. A. series of experimentel investigetions were eleo performed to study this point of view [Feloo. 1978-1982]. In the present work. the study of the vortex ring/loving well interections will be modeled in two dimensions using e vortex linepeir end e nunericel elgorithm‘beeed upon e discrete vortex nethod. The sinuletion will incorporete e viscous diffusion model. This effort will eleo illustrete the oepebility of the technique end will ellow for the deteiled exploretion of the physics for the flow field. A review of the discrete vortex nethod would heve to stert with the hend celculetions nede by Roeenheed [1931] to study e two-dineneionel vortex sheet. In the celculetione. he wee eble to use only e few vortices. However. this wee the first ettempt to eilulete e flow by e discrete vortex nethod. With the evolution of digitel conputere over the decedee. further refinenent of vortex techniques heve nede possible computetions which follows hundreds or even thousends of vortices. Accurete tine integretion. schemes heve eleo been used to reduce computetionel errors. There ere two different weys to eveluete the velocity field induced by the vorticity distribution. One is e grid-free nethod which is besed upon the use of Green's function to compute the velocity by directly summing the velocity field of the individuel discrete vortex elements. The other method celculetes the streem function defined on en Eulerien grid by solving the Poisson's equetion efter the vorticity hes been spreed onto the nesh nodes. The desired velocity is hence obteined by 3 interpoletion: this is termed the Vortex-In-Cell or VIC method. Applicetione which illustrete the use of the Green's function were cerried out by loore end Seffmen [1971]. However. meny of these eerly ccmputetions feiled to represent the physicel phenomene of the flows beceuse point vortices ere too singuler to yield reelietic vorticity distributions. Chorin [1973] first introduced the concept of e finite core to evoid the singulerity of e point vortex end hence overceme this problem. Chorin [1978] eleo used the segments of vortex sheets neer bounderiee insteed of point vortices es the computetionel elements in studying e pietonrcylinder flow. The primery dieedventege of the Green's function formuletion is thet the computetionel effort increesee es the number of point vortices squered. To overcome this dieedventege. the VIC technique wee developed by Christieneen [1973; which he celled ”Cloud-In-Cell"] end wee epplied to the investigetion of the motion for e continuous hydrodynemic fluid. This grid-dependent method wee eleo implemented by Beker [1979]. Neng end Thomson [1978] end meny others in eimnleting flows. An excellent review of these methods end their epplioetions wee done by Leonerd [1980] who else provided edditionel comments contresting the ebility of the two epproechee to setisfy oonservetion principles. Since both vortex models heve the ebility to numericelly investigete verious flows. the selection of suiteble technique is understood to be extremely importent to evoid eny inefficiency end/or ineccurecy. The edventeges end disedventeges of eech method should be cerefully considered when discrete vortex methods ere ebout to be used. Teke for exemple the Green's function scheme which is better suited for high Reynolds number (turbulent) internel flows where the totel number 4 of vortices cen be meinteined et e smell number. The Vortex-In-Cell technique is preferred to treet e problem if e very lerge number of vortex elements ere used (such es externel flows). As mentioned eerlier. viscosity pleye e key role in the rebounding ‘mechenine for the ring/well interections. The viscous effects must be eccounted for in the present celculetions. of course. Two steps ere required to simulete the viscous effects in the computetions since the vorticity diffusion term eppeers in vorticity trensport equetion. The two steps ere es follows. For ell time steps. 1. Simulete the vorticity dispersion in the flow field. 2. Stmulete vorticity generetion et the boundery to meintein the no-slip condition et the well. The former cen be included into either the Green's function or VIC method by edding e rendom welk component or e diffusion induced velocity to the convection velocity of individuel vortex element. Chorin [1973] first published the rendom-welk epproeoh for discrete vortex celculetions to reproduce the noleculer diffusion process in e stochestic sense. A test of this idee hed been performed by lilinexro end Seffimen [1977]. who found thet the number of vortex elements nuet be lerge compered with the Reynolds number if the diffusion ceused by viscosity is to be imiteted correctly. (There hed been. some questions reised by Iilinerro end Seffimen concerning the rete of convergence: they speculeted the rete to be such thet the number of vortex blobs N wee proportionel to the Reynolds number. Roberts [1985] hes shown thet in ectuelly. cereful initielieetion of the initiel distribution reduces this estimete to N"/’Re'1l'.) For the seke of simplicity. Riye et el. [1982] hed every discrete vortex endowed with e pro-chosen constent trensverse velocity without regerding to its locetion when the seperetion bubble over e two-dimensiouel blunt flet plete wee simuleted. This epproximetion yields e bodily displecement of the sheer leyer towerds the trensverse direction. 'e believe thet this model might not be necesserily e bed epproximetion to reproduce the viscous effect in some speciel types of flow. but obviously this will not be well-suited in generel. Ashurst [1977] used en elternetive scheme to epproximete the vorticity diffusion by edding e smell. but growing. length to the interection distence r (core sire) et eech time step. The time-dependent core sire represented the decey of the strength due to viscous "spreeding" of eech individuel point vortex. which he referred to es "eging". The results of the eging celculetions for e two-dimensionel mixing leyer ere in good egreement with experimentel dete. However. the scheme is restricted since it cen not be incorporeted into the VIC technique. This is beceuse the Vortex-In-Cell technique uses e spreeding function to redistribute the vorticity from the Legrengien merkers to the grid for the Poisson solver. This redistribution function cen be interpreted es en "effective sire" of the vortex merker. end is set to e pro-determined constent multiple of the grid length. It hes elso been shown thet "eging" does not necesserily converge for long time simuletion [Greengerd. 1985]. A deterministic diffusion nodel designed for the VIC computetions is introduced in the present study. The elgorithm for this model is besed upon using e combinetion of e modified vorticity gredient epproximetion.with e nine-grid-point. coneervetive redistribution filter to silulete the viscous spreeding of vorticity in the flow field. The computetions hes been done on e microcomputer in the Turbulence Structure Leboretory of the Depertment of lechenicel Engineering et Nichigen Stete University. The solution technique for solving the Poisson's equetion wes chosen eccording to the cepecity of the microcomputer. In Chepter two the elgorithms of inviscid end viscous models ere formuleted. In Chepter three. two prelininery test ceses ere exemined to check the eccurecy end reliebility of the diffusion model. The celculeted results of vortex linepeir/moving well interections ere ocmpered with the experimentel results of Lieng [1984] in Chepter four through the use of stebility meps end vector plots. A discussion of the no-slip boundery condition will eleo be included in Chepter five. CHAPTER 2 THE THEORY In the present enslysis. s Vortex-In-Cell technique is used to model the ring-well interections. The msin reeson for choosing s VIC method is e mstter of economy. The cost of computer time end memory required to updste the informstion for N point vortices would become prohibitive if s Green's function epproeoh were used. This is beceuse the computetionel work increeses es N3 for the Green's function epproeoh. but linesrly for VIC. There ere spproximetely 1.300 vortex merkers needed to numericelly simulete the growing boundery leyer of the moving belt for the problem to be studied here. By edspting the VIC model. we ney ensure the ccmputstion of the motion of such e lerge number of vortices will be et reesonsble cost. In this thesis we will be concerned primerily with idesl fluids which ere purely two-dimensionsl end incompressible (uniform density). The besic principle of the VIC scheme is eimilsr to thet of the "Cloud-In-Cell" method: the detsils ere explsined in this chspter. 2.1 Governing Formuletion for Inviscid Fluids The Vortex-In-Cell epprosch implemented here is s mixed Eulerien-Legrengien description of fluid flow in which the governing formulstion of motion is obteined from the vector curl of the Nevier-Stokes equetion. For s two-dimensionel incompressible fluid the vorticity trensport equetion becomes DQIDt - (u/p)A§ . (1) where D/Dt denotes e totel derivstive. g is the vorticity. u is the dynsmic viscosity. 9 is the fluid density end A is the Lsplsce operstion. Note thet in two dimensions the term representing the stretching end rotstion of vorticity. which is e three-dimensionsl process. is identicslly zero. Consider first the flow of en inviscid fluid (i.e. n - 0). Equstion (1) reduces to Euler's equetion nelnt - o . (2) Before equetion (2) is epplied. the vorticity must be discretised by pertitioning the vorticity into N point vortices. Then 1: 2" c (3) I ' C 1-1 1 N G a G , (3b) E131 i where G. the totel circuletion of vorticity. setisfies the definition : c-j gal-j 5-6. (4) A C where V'is the velocity vector end C represents e curve enclosing the sres A. From equetion (2). one finds thet the vorticity is totslly conserved in s two-dimensionsl flow field from the Legrengien point of view. This will simplify the updsting of vortex merkers in the Legrengien coordinste by simply moving these merkers without chsnging their strength. Furthermore. one mey express the velocity in terms of s streem function ‘P. which setisfies s Poisson's equetion with the vorticity es the source function. If u. v represent the convection velocity components in x. y directions respectively. the formuletions tske the forms Aw- -: , (5) u - aW/ay . (6.) v - 4m. , (6b) Note thet the boundery conditions of streem function must be initislieed before the interior streem function cen be solved by equetion (5). So fer. the epprosch will epproximete the inviscid motion of equetion (1). Nevertheless. we must never forget thet the genersl subject of fluid motion encompssses both s viscous diffusion process end bulk fluid motion. The letter .which results from en inviscid convection process. wee described by equetions (2). (5)snd (6). The former. the viscous pert of motion. will be explored end modeled in next section. 2.2 Viscous Diffusion Iodel Rsther then using the stochestic rendom-wslk epproeoh [Chorin. 1973]. e deterministic epproximetion will be introduced to model the moleculsr diffusion process. The idee of the diffusion model comes from Fick's first lew of diffusion. which ststes thet the mess diffusion rete 10 of e component is proportionel to the concentretion grediente. The lew cen.be formuleted in the form Fx - ~DOCIax . (7s) where F denotes the mess flux per unit sres. D is the diffusivity end C is the mess concentretion of the component. A eimilsr expression could else be written for the diffusion in y direction. Two other lews which ere enslogous to equetion (7e) ere the viscous-sheer equetion end the Fourier lew of hest conduction. i.e. ‘U . [tall/3y p (7b) qx - ~k3T/8x . (7c) Notice thet the physics of equetions (7) ere slso eimilsr. Equstion (7b) represents the trensport of momentum scross fluid leyers csused by the velocity gredient. equetion (7c) describes the trensport of hest by tempersture gredients end equetion (7s) governs the mess trensport by mess concentretion gredients. The three lews illustreted ebove. especislly Fick's first lew. give us e heuristic ides thet the vorticity trensport must occur es s result of diffusion when s vorticity gredient is present. Ue will now modify end edspt this idee to s diffusion model suiteble for vortex celculetions. Consider e simple one-dimensionel vorticity distribution shown in Figure 2.1. The concentretion of vorticity is greeter on the right side of this curve then on the left side. A higher concentretion 11 cen be interpreted es more. discretixed vortex merkers (in unit strength) per unit length. Thus more merkers will move from right to left scross the imsginsry plene (shown by the dsshed line in Figure 2.1) beceuse of moleculer processes scting upon the unbelsnced concentretion. This results in s net vorticity trensport from, the region of high concentretion to the region of low concentretion. In short. the hypothesis of the vorticity diffusion illustretes thet the rete of vorticity dispersion from s region of concentreted vorticity to en outer non-vorticsl region is proportionel to the locel concentretion/vorticity gredient. For the cese of pure diffusion. e "diffusion-induced velocity” for e point vortex.might be formuleted es s - -(u/P§.)(a§lax) . (8e) 7 - -(u/P§e)(3§/ay) . (8b) The coefficient of proportionelity between the vorticity gredient end the fictitious diffusion velocity is now dependent on s messure of the locel vorticity §.. In the flows of interest in this discussion. the only possibility thet the locel vorticity cen be zero is if there ere no vortex merkers present. However. in the celculetions. there is no need to semple the vorticity field st thet locetion if there ere no vortex merkers present. Thus. it mey sppesr there is e singulerity in this fonmuletion. operstionslly. this is not the cese. (The singulerity. which will blow up the epproximetion. msy sppesr physicelly in the centrel line. where the vorticity gredient is not zero end the 12 vorticity itself is zero. of some enti-symmetricsl vorticity structure. Nevertheless. this cen be technicelly elimineted from the computetions.) Accordingly. the locel vorticity §.. which is by no meene s zero vslue in the celculetions. will be repleced by the notetion c for our studies here. By using the diffusion theory of vorticity it is essy to numericelly predict the dispersion retes of point vortices by simply evelueting the vorticity gredients on eech mesh node. Equstions (8) hence will epproximete the viscous pert of the equetions of motion (1). The method of frectionsl steps then stetes thet the totel velocity moving the merkers is the sum of the convected velocity . which is described by equetions (6). end the fictitious diffusion-induced velocity. which is governed by equetions (8). Thus U - u + 3 . (9e) <1 V - v + (9b) where the U. V denote the "reel" velocity components. in x. y directions respectively. of e vortex msrker. Now the epproximetion will setisfy the solution of equetion (1) st ell times in the interior of the computetionel domein. It is interesting to note thet these results could heve been derived using the Nevier-Stokee equetion. From the oonservetion of mess for the incompressible two-dimensionel flow. we heve an/ax + av/ay - 0 or v-V - o . (10) 13 where ‘7 denotes the gredient operetor. The vorticity trensport equetion mey be written in en elternetive form es am: +V°(§V) - (n/pm . (11) if the continuity. equetion (10). hes been edded to equetion (1). Similsrly the Euler's equetion mey be written in the form : atlat + v-tfi'i) - o . (12) After resrrsnging the eppropriete terms in equetion (11). we obtsin aglat + V-[u‘v’ - (gr/puma] - o . (13) This result proves. by being compered with equetion (12). thet the viscous diffusion terms cen be interpreted es s contribution to the velocity components of the vortex merkers in x end y directions. which we celled "diffusion-induced velocities" end is defined in equetions (8). In order to displsce these discrete point vortices st eech time step. e Predictor-Corrector epproximetion is used which is e second-order eccuretc time integretion scheme. The elgorithm for the Predictor-Corrector is of the form x(t+At) . x(t) + 0.5At[U(t)+U(t+At)] , (14s) y(t+At) - y(t) + 0.5At[V(t)+V(t+At)] . (14b) 14 where At is e unit time step length. A nine-grid-point filter which wee preposed by Bertholomew. Iy msjor Professor. is eleo introduced in the celculetions es the spreeding mechenism. This improved technique distributes vorticity to the nine nesrest neighbor mesh nodes rether then using sres weighting to the surrounding four (es used by Christiensen [1973]). This effort will minimise the eliesing conditions thet normslly sppesr on the mesh cells [Beker. 1979]. Figure 2.2 provides the geometry end notetion of the Quesi-Geussien redistribution scheme. which is given by Mme) - (Aa'+sa‘+c¢.‘+ca'+n) -(Ap‘+np‘+ca‘+cp‘+n) for o s («Hal .4 1.5 . (15) O(c.8) - 0 otherwise . where the constents A.B....E ere tebuleted in Thble 1. end a. B ere the nonrdimensionel ~dietences from vortex msrker to the grid points. The meximum error occurs st u-B-0.4 end is on the order of 10" with double precision. This scheme conserves the totel vorticity both locslly end globelly. end hence setisfies en importent feeture of oonservetion principles. Note thet the interpolsted velocities eleo use the ssme nine-weighting fsctors. The testing of viscous diffusion model used in conjunction with this redistribution scheme will be illustreted in Chepter three. 2.3 Procedure for Simuletion of VIC The elgorithms governing the simuletion heve been formuleted in previous sections. Equetion (5) will be eesily solved by using e 15 Ccmpect Non-iteretive Poisson Solver [Bunemen. 1969]. Although the technique tekes more CPU time. we chose it es the Poisson Solver due to the limitetion of memory cepecity of the microcomputer. In this section the procedure for the VIC celculetions will be described es follows. The celculetions will follow the thirteen steps in order : 1. 2. 4. Determine the computetionel domein end grid size. Determine boundery conditions -- initiete streem function on boundery nodes of the computetionel domein. Psrtition vorticity field -- decide on the strength of vortex merkers end discretise the continuous vorticity field into point vortices. Spreed vorticity -- redistribute the msrker's vorticity onto its nesrest nine mesh points by using equetion (15). Eveluste viscous effects -- oelculste end normeliee vorticity gredients (ll§)(3¢/ax). (1/§)(8§/8y) on grid points by using e centered difference scheme. Then eveluete equetions (8). Solve Poisson's equetion.-- use Dunemen's Poisson Solver to eveluete the streem function distribution on interior grid points. Eveluste inviscid motion -- obtein the velocity field st grid points by solving equetions (6) using centered difference. Obtein the "reel” velocity field -- sum up the diffusionrinduced velocity end inviscid convection motion on eech node es defined by equetions (9). Celculete the velocities of vortex merkers -- interpolete velocities beck to vortex merkers else using the seme weighting filter. 16 10. Displece vortex merkers -- use equetions (14). the Predictor-Corrector integretion scheme. es the time merching elgorithm to move point vortices. 11. Bounce vortices beck into flow field (if necessery) -- if e vortex merker crosses the well st the end of s time step. it will be bounced beck into the fluid: i.e. this merker will be returned from (‘i"7i) to (x1.yi) . where the well is essumed to be st y-O. 12. If desired. one cen generete new vorticity (if necessery) -- creste new vortex merkers nesr the well to meintein the no-elip boundery condition if the tengentiel velocity does not vsnish st the solid well. This perticulsr step wee not implemented in the discussion which follows. 13. Repeet steps 4 through 12 for eech time step. The procedure shown ebove mey be repested es long es needed. Stetieticel infonmetion such es Reynolds stress. fluctusticn quentities. ... cen be obteined by putting numericel probes in the computetionel flow field. CHAPTER 3 BENCH EARRING NDDELS AND NUNERICAL PARAIETERS In the chepter we will present two preliminery test ceses in order to obtein e quentitstive eveluetion of the effects of the epproximetions nede in equetions (8). Both testing models employed here ere simple end heve known enslyticslly solutions. Stoke's leyer. the first problem. describing the trensport of sheer leyer csused by viscosity towerds the trensverse direction. which is en one-dimensionel diffusion process. The second testing model is s two-dineneionel free vortex peir. The growth rete of redius of gyretion of the vortex peir provides the informstion of viscous diffusion rete in two dimensions. To numericelly investigete the viscous effects of these testing flows. the viscous diffusion model is incorporeted into the Vortex-In-Cell celculetions. The eccurecy end reliebility of the slgorithm.of the diffusion model will be evelueted by compering the results of test runs with the theoreticel expressions. The terget epplicstion. the sinulstione of two-dimensionel vortex ring/moving well interections. which ere governed by some numericel perencters will eleo be illustreted in the end of this chepter. 3.1 Ome-Dinensionel Testing Nodel -- Stoke's Leyer Now we consider en initielly ststionsry fluid which is bounded by s loving flet well with e constent speed U'. This is s very common flow phenomenon thet hes e tine-dependent boundery leyer which is formed on the well by the effect of fluid viscosity. The governing formuletion of 17 It: 1’) be it Vic he 18 motion is described by Stoke's exect solution of the Nevier-Stokes equetion. which tekes the form an“ 1”,, - 1),,[1 - erf(n)] . (16) where erf(n) denotes the error function of e similerity verisble n-y/(4pt/p)'°‘. Hy meene of equetion (16) end the definition of vorticity. we will be eble to numericelly simulete the Stoke's leyer. Assume the well is st y-O end impulsively sterted to move with e constent speed st t-O. A (I x H) on3 computetionel domein is chosen end defined by (I x I) grids. where I is the width of the rectenguler domein in x-coordinetc. H is the height in y direction end I. J ere integers such thet I-IVAx. J-HIAy. Note thet. for simplicity. we slweys use squeres to refine the domein (i.e. Ax-Ay). Now we mey determine the totel circuletion in the Stoke's leyer per unit length Ax of the well. It is given by G Iinterior V d- U (17) - - s - x . t0tR1 '.1 1 'A It is cleer thet one hes to plece e number of vortex elements. ssy N. neer the well per unit length such thet eech of the elements hes strength D‘Ax/N. The wey to distribute these discrete vortices is slso besed upon the definition of circuletion. A schemstic representetion for the exemple of N-3 for this concept is shown in Figure 3.1 end the trensverse locetions of these vortices cen be presented in equetion form licrt ltyc: 5'45] Inc: 01:: cits the reg: with liye 'ht: 'Dpl this in: 1... Nu teli 1243 19 7.(t.N) - 0.55.:1“[1 - (2m-1)/2N] . m - 1....N (18) where the inverse error function erf-1[erf(u)]-n end 6 is the boundery leyer thickness st time t. which is defined by u3/U‘pllZOO; i.e. 8-4(utlp)"'. Note thet eech grid box in the well leyer hes st leest one merker per cell. end thet the number of merkers st e given spetisl locetion is proportionel to the locel vorticity; i.e. the merkers overlsp. The vorticity field of the computetionel domein cen be obteined efter the N-I vortex merkers heve been distributed. In order to solve the Poisson's equetion (5) by Dunemen's solver. the initiel conditions of streem function on the bounderiee ere eleo required. This cen be geined by integreting the exect solution (16) with respect to y. Hence the exect etreen.function field of Stoke's leyer st time t mey be steted es ¢%.11 leyer ' I 1‘well leyer‘ty s ‘ w[Y-erfc(u) + 2(ut/np)°"(l-cxp'n )] . (19) where erfc(u)-1-erf(n) is the complimentery error function. He will epply Abrenowitx end Stegun's retionel epproximetion [1964] to eveluete this function. The error is on the order of 10". Now we mey initielixe the celculetions by using the vorticity distribution in the interior end the exect boundery condition of streem function from equetion (19). Before further investigetion. we ere going to eveluete the reliebility of the epproeoh illustreted ebove. Let Ax-Ay-O.5cm. 1-41-123. 10-10. U'-10cm/sec. u/p-10-'cm'/sec. t-64sec (1... Ax/6-1/6.4) 20 end the numericel probes pleced et x-16cn. Figure 3.2 shows the comperison of exect end celculeted velocity profiles. Figure 3.3 shows the comperison of vorticity profiles st the sene condition. Obviously. the outcomes of celculetions ere in good sgreement with the theoreticel curves except very user the well. This is beceuse the redistribution filter smooths shsrp peeks such es the peek in the vorticity distribution of the exect solution. There ere spproximetely sn.-11I error which occurs st y-O in the celculeted velocity profile end -10§ error in the vorticity profile. In feet. these reletive errors ere strongly dependent on the retio Ax/b. which is shown in Figure 3.4. Fortunetely. this error could be pertislly reduced. when the retio Axle is eppropristely smell. by forcing the x-component velocity st eech well node to be U;. This correction will be used in ell of the well leyer sinuletions sfterwerds. To further investigete the effect of nolsculer viscosity which is supposed to diffuse. or spreed out. concentretions of vorticity user the well. different lengths of time step heve been studied st the sene rotstionel time psremeter T55./U&-1l23 where 6. is the well leyer thickness st time t.. Simulsticns ere sterting from t. end ending st t. For eech time step length. celculetions heve been.mede for three ceses: 6/5,-(6s/e4)'-‘. 2'-‘ end 2. Figure 3.5 shows the dependence between reletive error end temporsl resolution st the sene spetisl resolution Ar/8,-1l6.4. The reletive error defined here tekes the form N scum. Error - llN 2P1 [(y- (20) — 1 ' celc./yiexec.) ] 21 where y‘exec. is evelueted using equetion (18). As expected. the error increeses se At increeses. However. there is no noticeeble chenge when the retio At/T is smeller then 1. Figure 3.6 slso shows the seme feeture st constent Axlb,-ll3.2. The optinu time step length hence cen be determined eccording to these two figures end other fsctore such es the expense of CPU time. Figure 3.7 shows the reletive errors with different spetisl resolutions st fixed nondimensionel time step length At/ThO.2875. The ultimete resolution thet could be exemined wee Axlb,-l/12.8 due to the memory constreint of the microcomputer. It shows thet the higher spetisl resolution yields the higher sccurecy es predicted. In other words. the growing well leyer cen.be successfully sinulsted by epplying our viscous diffusion.model es long es the resolution perencters At/T end Ax/b. ere chosen moderetely smell. The celculeted results ere quite good especislly for the cese of short time simuletion 8/8.-(68/64)"‘ which corresponds to 4 seconds of reel time. This will setisfy the needs of our finel epplicstion since the interecting time for vortex ring end moving well is elweys shorter then 5 seconds of reel time. 3.2 Tbo-Dimensicnsl Testing Iodel -- Free Vortex Peir The decey of e two-dimensionel vortex peir in the ebsence of e solid well hes slso been studied using our viscous diffusion model to investigete the sccurecy of the epproeoh. Consider e single line vortex heving position (x..y.) end strength Gpv' In poler coordinstes. the velocity component v9 is given es Y9(r.t) - (Gpvl2rr)[l - .rp(-pr‘/4ut)] . (21) 22 The circle of redius r. ” which oonteins 995 of the vorticity cen be shown. using equetion (4). to be e.s r..., - 2[(ut/p)ln[Grvl(va-0.99Ghv)]] . (22) N Now we use N vortex merkers with strength G1 so thet 697.21-161 to represent the single line vortex of initiel effective redius r -R.. O.ss I. 8 Thu. t.-pR. [4“.1nIGrVI(Grv 0.99G’v)]]- is the elepsed time to spreed out vorticity to this effective size. From equetion (22). we cen infer thet the redius rt of s circle which possesses some vortex merkers with totel strength fG'pv cen be formuleted es rf - 2[(ut./p)ln[G’vl(G’v-fGrv)]]..‘ .5 - e.[-o.21715-1r(1-r)]' . (23) where f is s frection. This formuletion will be used to distribute vortex merkers in the effective circle. Since it wes derived from en exect representetion (21) of e line vortex. the reelietic vorticity field ought to be conserved efter being discretised. A schemstic disgrsm.illustrsting the errengement for N-12 is shown in Figure 3.8. Consider the N vortex merkers. the i-th merker is pleced st (‘i'yi)- The squere of the redius of gyretion S is defined by s - -21./G , (24) N where I‘- -O.5§i 161[(xi-x.)7+(y1-y.)z] is the enguler impulse with IOSPOOt t0 (I..y.). In e viscous unbounded fluid. 8 grows linesrly with 23 time. The exect solution tekes the form 3 I S. + 4pt/p , (25) where S. is the initiel squere of redius of gyretion st time t.. By meene of this exect solution. we will be eble to eveluete the outcomes of simuletions leter on. The streem function field induced by the single line vortex in en unbounded fluid cen be obteined by the formuls ‘Pr "y. - -(cp,/4r)1r[(r-r.)’ + (y-y.)‘] . (25) The evolutions of e vortex peir which encompssses two line vortices with equel megnitude of strength Gpv but opposite sign could be numericelly studied by plscing N vortex merkers et eech lobe using equetion (23). If ”r the distence between the two centers of lobes. and “r the speed of the vortex peir. ere known. we mey eveluete the strength for eech lobe. (upper) _ . 2 GpV (10"!) (t) ZEUrDr ( 7) The initiel velues of streem function on bounderiee cen be celculeted using equetion (26) by superposition Wvortex peir " ‘I’xu,yu "’ 59,1,“ . (28) '11." (luau) end (x1.y1) ere the centers of upper lobe end lower lobe 24 respectively. Figure 3.9 shows the vorticity distributions through upper lobe for vortex rings sinulsted using N-l end N-6 st Dr-3R,-5.3Ax. Ccmpered with en exponentisl distribution typified by eech’r [Nsxworthy. 1977]. Apperently. better sgreement for lerger N. To exemine the velidetion of the viscous diffusion model. two ceses. (i) pure diffusion end (ii) hydrodynemics plus diffusion. heve been investigeted with the sene initiel condition. we set Dr-3R,. I-2Ji64. Ax-Ay-O.25cm end choose the velues of Dr' Ur such thet Ax/Rg-1/4 end Reynolds number Re-pUiDr/n-lOOO. A The initiel rotstionel time for eech lobe is T;-(2xR.)'IG’v. It wee found thet the results did not heve eny noticeeble chenge when the unit time step lengths At were chosen to be less then Tr/60. Ccmputstions heve been done with this vslue of At for N-6. N-60 end N-SOO. The totel simuletion time is 3T1" Figure 3.10 shows the normslired squere of the redius of gyretion 8/3. es s function of tITr for both ceses (i) end (ii) with Nh6. 8 used here is the eversge vslue of upper end lower lobes. The exect solution (25) is slso shown for comperison. The reletive error is ebout -25$ efter three rotetions. Figure 3.11 end Figure 3.12 show the results for N-6O end N-500 respectively. The size of the reletive error is ebout -8§ for N-60 end -4§ for N-5OO et integretion time 3T} in cese(i). Apperently the number of vortex merkers N must be lerge enough. ssy the order of 10' in this problem. to yield the reelietic vortex structure end hence reproduce the viscous diffusion. However. the repid improvement in sccurecy from N-6 to N-6O should eleo be noted. In order to seve CPU time end memory 25 specs. the secrifice of e little sccurecy is slweys being considered. As enexemple. N-60 is eccuretc enough to be used insteed of N-500 to resolve the vortex structure. It cen be concluded thet the spproximetions of vorticity diffusion in two dimensions by using our viscous model ere in good sgreement with the theoreticel results. but the resolution of the vortex structure must be in noderstely high order. 3.3 Numericel Peremeters of Vortex Ring/loving Uell Interections The two-dimensionel vortex ring/moving well interections ere essentielly the interections between line-vortex peirs end e moving well. Figure 3.13 shows the gecmetricel errengement of this flow field for VIC simuletions. Some numericel perencters which governs this problem ere introduced es follows. 7 : the initiel engle of epproeoh. Ur/U' : the retio of ring speed to well speed. 5./Dr : the retio of well leyer thickness to ring diemeter. Since the vortex ring/moving well interections cen heve evolutions which sre‘ either steble or unsteble. the governing perencters illustreted ebove could be releted es the representetion 6 I , . ( .ID1.2,1.“1“l Function( 7 UiIU§ ) (29) Io define e steble interection es one in which the vortex ring turns ewey from. the well without ingesting well leyer fluid. An unsteble interection ingests well leyer fluid before/es the ring turning ewey 26 from the well. These definitions were developed by Lieng [1984]. The criticsl lines which distinguish the stebility end instebility of evolutions cen be scquired by systemeticelly verying these releted perencters. The elgorithms for simuleting Stoke's well leyer end line-vortex peir by discrete vortices heve been fully described in Sections 3.1 end 3.2 respectively. The streem functions on bounderiee of computetionel domein ere the only velues thet heve to be determined to initielise the celculetions. However. we know thet the streem function field induced by e line vortex in the presence of solid well cen be obteined by modifying equetion (26) to give (w) VI (30) xe'Ye . wke'Ye - W30-35 ° Consequently. the streem function of this problem cen be obteined from equetions (19) end (30) by superposition. Then ’ (31) . (w) (w ‘Pring/moving well W xu'yu + ‘i’ x1,y1 + stll leyer ' This epproech for the boundery conditions hes been discussed in deteil by Beker [1979] . The computetions heve been done for 7-15. with one merker per lobe for the line-vortex peir end 10 merkers per unit length of well for Stoke's leyer. We set I-4J-128. Ax-Ay-0.5cm. At-1/80sec. Ur-64.4cnlsec. Dr-2.65cm end the initiel height of linepeir is st y-6.23cm. Tho perencters, 6./Dr end Ur/U‘. renge from 0.9 to 2 end 0.4 to 0.8 respectively. 27 Simulsticns with/without viscous diffusion model heve been cerried out st the seme initiel conditions. No new vorticity hes been genereted during celculetions. This meene thet the no-elip condition st well mdght not to be setisfied ell the time. The inportence of viscous diffusion model will be evelueted by competing the celculeted outcomes with experimentel results vie stebility meps end flow visuelixetions in next chepter. RESULTS An L81 11/23 microcomputer hes been cherged with ell the efforts of computetions. The outputs will be presented. quentitetively by meene of stebility meps end quelitetively vie computer genereted vector plots. in the following sections. 4.1 Stebility Reps Figure 4.1 shows the stebility mnp ccmpsrisons for 7-15'. which uses deley time t. es the unit of y coordinste. The meps of numericel simuletions with/without viscous diffusion model ere computed to within $2.5 second error bers. It is found thet the celculetions ere in better sgreement for higher deley times then for lower deley times when compered with the experimentel stebility mep. 4.2 Voctor Plots Figures 4.2 to 4.6 show the computer genereted vector plots of vortex linepeir/moving well interections with different initiel conditions. These plots indicete the evolutions of linepeirs when they come closer to the well. Both the linepeirs end well ere moving from left to right. Only those vortex merkers which lie inside the grephics displey domein (indiceted in Figure 3.13) ere shown. The vectors represent the instenteneous velocities of vortex merkers. The outline eround the linepeir stends for the concentreted vorticity region of ring lobes. which is for reference only. 28 29 Figure 4.2 sequence showing the ingestion of well leyer fluid into the lower lobe es the linepeir "creshes" right on the well. This is the type of strong breekup where the linepeir did not turn ewey from the well. Figure 4.3 shows s linepeir breekup efter ingesting well leyer fluid es it moves ewey from the well. This type of breekup is different from thet in Figure 4.2. Figure 4.4 indicetes the type of survivel of linepeir which lesves the well without bresking up. Note thet the numericel simuletions from Figure 4.2 to 4.4 were ell done with the viscous diffusion model. Figures 4.5 end 4.6 show the evolutions of liftup of linepeirs for celculetions were done without the diffusion model end done with diffusion model respectively. All the initiel conditions of these two figures were set identicslly. but the letter included the vorticity diffusion process while the former did not. Apperently we cen find. by observetion. thet the viscosity mekes the linepeir turn ewey from the well eerlier end eleo smooths the concentreted well vorticity. Discussion of the overell results will be in next chepter. CHAPTER 5 DISCUSSION The results presented in this work demonstrete thet it is possible to simulete the experimentel vortex ring/moving well interections. when the well leyer is thick. by e two-dimensionel linepeir cslculetion. Although the deteiled compsrisons of stebility meps ere not in good sgreement. end es expected the sgreement becomes poor es three-dimensionel effects begin to doninste the interection. In view of the feet thet the two-dimensionel simuletions heve shown the ebility. by meene of vector plots. to reproduce the verious flow phenomene of the steble/unsteble evolutions of the three-dimensionsl ring/well interections. we cen infer thet the overell stebility/instebility of the vortex ring/moving well interections is edequstely described by two-dimensionel mechenisms. This interpretetion indicetes the inportence of two-dimensiouel effects in this flow field. Novertheless. the stretching of the vortex tube element end other three-dimensionel effects ere importent in determining deteils of the flow such es the mixing process user the well. Reynolds stress. fluctustions end etc. which will be significently effected by stretching/reorientetion of the vortex structure. In the vortex ring/moving well interections. the three-dimensionel effects become more importsnt when the rings come closer to the well. This is beceuse the rings. when they neer the well. will be under the influence of vortex stretching which mey csuse. for exemple. the smplifying of well leyer fluid ingestion end hence breeking up fester. 30 31 For the cese of thin well leyer thickness. the ring cen eesily epproech the well so thet the subsequent evolution of the ring is domineted by the effects of three-dimensionel nechenisms. This is the reeson.why the two-dimensionel celculetions feiled to yield good results in the thin well leyer thickness region which corresponds to the short deley time region in Figure 4.1. Since the vortex rings were roughly spproximeted by the line-vortex peirs in the computetions. s question srose -- would the overell outcomes be improved if more vortex merkers were employed. insteed of one merker per lobe. to represent the structure of ring lobes? Usrtholcmew clerifies this by using six vortex.nsrkers per ring lobe to re-cslculste the ring/well interections in the ebsence of viscous diffusion process. The efforts yield e slightly better result in the stebility mep which is shown in Figure 5.1. but the sgreement in the short deley time region is still poor. This result enhences our belief thet the three-dimensionel effects in the short deley time region ere the msjor effects responsible for the poor comperison of the numericel simuletions in this region. Unfortunetely the viscosity seems not very importent in the two-dimensionel vortex peir/moving well interections since there is no difference. within $2.5 seconds error here. between the stebility meps for celculetions done with/without the viscous diffusion model. However. one will find out thet the effects of viscosity in the flow field mey not be so triviel when the computer genereted vector plots in Figures 4.5 end 4.6 ere cerefully compered. These two figures indicete thet the computetions with the viscous diffusion model heve successfully crested two physicel phenomene which heve never been shown in the 32 inviscid celculetions. One is the vorticity gredients heve been smoothed vie the diffusion process of vortex merkers. 'which cen be eesily seen by the growth of boundery leyer thickness for instence. The other is the vortex ring. in s steble interection. turns ewey from the well strongly through the influence of viscosity. This feeture proves thet the viscosity does plsy e significent role in the rebounding mechenism. es mentioned eerlier in Chepter one. of the ring/well interections. In other words. we cen expect thet the criticsl line in the stebility mep for computetions with the diffusion model will be shown lower in some degree then thet in en inviscid celculetions if finer numericel investigetions ere mede; i.e. reduce the sire of error here to. for exemple. $0.5 seconds end see the differences in the viscous/inviscid stebility meps. Nevertheless. e huge time is required to do so under the current fscility which needs spproximetely 26 hours of execution time to run for just one ring/well interection. Due to the time limitetion. we do not refine the celculetions to echieve more precise results. The no-slip condition is elso en importent cherecteristic of viscous effects st the well surfsce. Since our viscous diffusion model is only designed for simuleting the moleculer diffusion process in the flow interior. no perticulsr step wes employed to meintein the no-slip boundery condition which might not be setisfied ell the time during the celculetions. This. of course. introduced some error which mey not be negligible in the overell results. To meintein the no-slip condition. new vorticity should be genereted if the tengentiel velocity is found not venish st the well during the simuletions. In feet. the elgorithm described in Section 3.1 hed been celled to creste new point vortices in 33 the vicinity of the surfece to setisfy the no-slip boundery condition. This ettenpt. however. feiled to represent the reslity of the boundery leyer since too meny week vortices hed been genereted very close to the well. This gives us e lesson thet cere must be teken to evoid spurious results if the new vorticity creetion process et the solid surfece end the subsequent motions of these vortices ere ebout to be correctly modeled. Although we decided not to include this step in the celculetions. we believe thet the use of finite vortex sheets es the newly crested elements could be e better wey to model the creetion process st the well. but there is no herd evidence on this mstter end the question is open. There ere eleo meny other error sources such es the truncetion error. the sccumuleted round-off error. ... heve not been mentioned. In generel. the error contributions by these sources ere reletively smell end not well understood. CHAPTER 6 CONCLUSIONS Numericel simuletions of the vortex ring/moving well interections heve been done with some success using the two-dimensionel Vortex-Ianell technique. In these celculetions the effect of moleculer viscosity diffusion wee spproximeted by edding e deterministic velocity component for eech discrete vortex. This "diffusion-induced velocity" wee defined in terms of e modified gredient epproximetion which wss obteined from.e trensformetion of the vorticity trensport equetion. we heve tested the viscous diffusion model on both one end two-dimensionsl time-developing flows end echieved setisfectory results. The computer-genereted vector plots eleo show thet the use of the diffusion model chenged the locel deteils of the rebounding phenomenon in the ring/well epplicstion. but did not sppesr in the overell stebility meps within $2.5 second error bers. More computer time is needed to resolve this peredox. From the precticsl point of view. we feel thet this model is e simple end effective scheme to represent the vorticity diffusion process in the flow interior. The quelitetive feetures of the ring/well interections ere well reproduced by this numericel method. but close quentitstive comperisons with experiments indicete thet in the reel flow the three-dimensionel deformetion of vortex filements becomes significent in the thin well leyer zone of stebility mep. so thet the VIC technique in two dimensions cen poorly simulete there. In order to further improve the celculeted outcomes. the inclusion of three-dimensionel flow representetions 34 35 eppeers to be required. With this. the viscous diffusion would be more importent due to the thinner well leyers encountered by the ring. However. we do not heve the meene to extend the computetions to three dimensions under the current computing oepebility. On the LSI 11/23 microcomputer it took 311.5 seconds to edvence the positions of 1282 point vortices for eech time step. Approximstely 40! of this time wes teken up by executing Bunemen's Double-Cycle-Reduction routine. This compect Poisson solver is not quite efficient in speed. but the big edventege is its memory requirement fits smell computers. Future work will try to include follows in the celculetions when the VA! 11I750 computer is svsileble : (1) Flow mechsnisms in three dimensions. (2) Fest-Fourier-Trensform routine es the Poisson solver. (3) Suiteble model for treeting no-slip boundery condition. LIST OF REFERENCES 10. 11. 12. 13, LIST OF REFERENGS Abremowite. N.. Stegun. I. A. 1964. Hendbook of lethemeticel Functions. Pegs-299 Ashurst. I. T. 1977. Numericel simuletion of turbulent mixing leyers vis vortex dynemics. Sendie Leb. Rep. SAND77-8612 Beker. G. R. 1979. The "Cloud in Cell" technique epplied to the roll up of vortex sheets. I. Ccmput. Phys. 31:76-95 Berker. S. 1.. Crew. S. C. 1977. The motion of two-dimensionel vortex peirs in s ground effect. I. Fluid Nech. 82:659-671 Duemen. 0. 1969. Cupsct non-iteretive Poisson solver. Stenford Univ.. Celif. Inst. for Plesme Reseerch. SU-IPR-294 UNCLAS Chorin. A. J'. 1973. Nunericsl study of slightly viscous flow. I. Fluid Nech. 57:785-796 Chorin. A. J. 1978. Vortex sheet epproximetion of boundery leyers. I. Ccmput. Phys. 27:428-442 Christiensen. J'. P. 1973. Numericel simuletion of hydrodynemics by the method of point vortices. J’. Ccmput. Phys. 13:363-379 Felco. R. E. 1977. Coherent motions in the outer region of turbulent boundery leyers. Phys. of Fluids 20:124-132 Felco. R. E. 1978. The role of outer flow coherent motions in the production of turbulence neer e well. AFOSR/Leigh Felco. R. E. 1982. A synthesis end model of turbulence structure in the well region. ImlTIIUTAI Conference on the Structure of Turbulence in Heet end less Trensfcr Greengerd. C. 1985. NOTE - The core spreeding vortex method spproximetes the wrong equetion. J'. Ccmput. Phys. 61:345-348 Hervey. J. L. Perry. F. I. 1971. Flowfield produced by treiling 36 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 37 vortices in the vicinity of the ground. AIAA.J. 9:1659-1660 Riye. l.. Seseki. R.. Arie. I. 1982. Discrete-vortex simuletion of s turbulent sepsretion bubble. I. Fluid loch. 120:219-244 Leonerd. A. 1980. Vortex methods for flow simuletions. I. Cuput. Phys. 37 : 289-335 Lieng. S. I. 1984. Vortex ring/loving well interections. Thesis. Nechsnicsl Engr. Dept. of lich. St. Univ. Nexworthy. T. 1977. Some experimentel studies of vortex rings. 1. Fluid lech. 81:465-495 lilineseo. F.. Seffmsn. P. G. 1977. The celculetion of lerge Reynolds number two-dimensionel flow using discrete vortices with rendom welk. J. Comput. Phys. 23:380-392 loore. D. I.. Seffimsn. P. G. 1971. Structure of s line vortex in en imposed etrsin. Aircreft like Turbulence. ed. Olsen. J. H.. Goldburg. A.. Rogers. I. 339-354 Peece. A. 1.. Riley. N. 1983. A viscous vortex peir in ground effect. I. Fluid loch. 129:409-426 Roberts. S. 1985. Accurecy of the rendom vortex method for e problem with nonrsmooth initiel conditions. J. Comput. Phys. 5 8: 29-43 Rosenhesd. L. 1931. The formstion of vortices from s surfece of discontinuity. Proc. R. Soc. London Ser. A 134:170-192 Seffmen. P. G. 1979. The epproech of e vortex peir to e plene surfece in inviscid fluid. 1. Fluid Nech. 92:497-503 Thomson. I. C. 8.. long. I. A. L. 1978. Numericel studies of some nonlinesr hydrodynemic problems by discrete vortex element methods. J. Fluid loch. 84:433-453 TABLE 38 Coefficient Velue Tebulsted in Double Precision A 1.7636684303351 X D—3 B -3.3950617283950 X D-2 C 2.5738536155203 X D-l D -7.229662698413O X D-l E 6.6517857142857 X D-1 Teble 1 The Constsnt Coefficients of Quesi-Geussien Redistribution Function FIGURES 39 . A Higher vorticity. higher concentretion of vortex merker Lower vorticity. lower concentretion of vortex merker Isrkers "Y 3 ¢ -1/§.(3§/3x) Figure 2.1 Sketch of One - Dimensionel Vortioity concentretion Profile on Illustreting Diffusion-Induced Velocity 4O (i-1.j+2) Ax (1.1+2) (i+1.j+2) A! point vortex e I i E (i-l.j+1) E ( 1 .1+1) (i+l.j+l) i b i i : (i-1.j) (1.3) (i+1.j) For node ( i . j ): u - s/Ax B - b/Ay Figure 2.2 The Geometry end Notetion of the Nine-Grid-Point Filter 41 ""‘YE:P""'8‘S’5"""W""'K'J ’1'0.2388 ------------------- (3)-«~04 _ "xx-9:93.59 ...... WWW Figure 3.1 Sketch Illustretes the Fey to Pertition Vortioity Field of Boundery Leyer. (This is en Exemple of Three Nerkers per Unit Length of Well) y DIMENSIONLESS HEIGHT ABOVE THE WALL [E] k) 0.8 0.6 0.4 0.2 0.0 42 O : CALCULATED —— : STOKES SOLUTION 0.0 0.2 0.4 0.6 0.8 1.0 1 l 1 l 1 l 1 l 1 l T T I I I T l I .9 i. _.- O ..r \\\ ._ \\\9 -_ \\.. ..\\\ i. \Q -L \\ -\\ _._ ‘\g\ —L_ -L. \ O\\ ..L O ‘ "\-, 1 I 1 l 1 l 1 1 n1 “ T I r I I I I I V T 0.0 0.2 0.4 0.6 0.8 1.0 U DIMENSIONLESS VELOCITY [U ] W Fizure 3.2 Comperison of Exect end Celculeted Velocity Profiles for Stoke's Leyer st Ax/8-1/6.4 Which is Used es Ini- tiel Condition for The Cslculetion 1.2 0.8 0.6 0.4 0.2 0.0 43 0 : CALCULATED -- : STOKES SOLUTION DIMENSIONLESS HEIGHT ABOVE THE WALL [%] 0.4 0.6 0.8 1.0 1.2 : '1 H. I .L J, : 1.2 “I 1.0 .... 1 0 ...r. 0.8 -- 0.8 I \\\. 0.5 d-- \\o ..— 0.6 .0. \Q\ r- 0.4 -- \‘R‘ -+— 0.4 .... >‘K\\ -... 02 -I— Bx —— 02 .1). O \\ .11.. 0 \ 0.0 1 1r f [L E 4, I I e 0 0 0.0 0.2 0.4 0.6 0.8 1.0 E DIMENSIONLESS VORTICITY [ ? ] w Figure 3.3 Compsrison of Erect end Calculated Vorticity Profiles for Stoke's Leyer st Ax/8=1/6.4 Which is Used es Ini- tiel Condition for The Cslculetion ( RELATIVE ERROR ) x 100 44 0.0 0.3 0.6 0.9 1.2 1.5 —70.0 ‘1 I I I 1 4r 1 I + —-70.0 I ._ —60.0 A- —— —60.0 «>- H)- + —50.0 —— -L —50.0 4)— + -I.— —40.0 -I— -H- —40.0 .L + -_ —30.0 -b H— -30.0 +~ + ~~ —20.0 ~— + «H- —20.0 + .- + I- + —10.0 -I- f -- —10.0 A}? ._ L I 1 LJ 1 I 1 I 1 0.0 I T F T I T r I r 0.0 0.0 0.3 0.6 0.9 1.2 1.5 X RATIO OF GRID SIZE TO WALL LAYER THICKNESS [ A? ] Figure 3.4 The Dependence Between Reletive Error end Spetiel Resolution for Well Velocity Celculetions (i.e. “celc. st y=0) Using Quesi-Geussien Filter. Reletive /U )-1 w Error=(uca1°. ( RELATIVE ERROR ) X 100 45 -#— : P=2.0 —0— : P=1.4142 —+— : P=1.0308 0.0 3.0 6.0 9.0 12.0 I 1 l l L l l l l l I "20.0 I I I I I r I I l I T ‘20.0 {I- «r- -15.0 -I- -II- -15.0 ,/* /’// ./ .9. 1/ A}— /’//' z // -1o.o -- / ,4) --p- --10.0 / //" / I,/ / ,/ dr /// ,/// ‘- / —5.o -- /_/' / -- -5.o / / -/+ d .../x ‘LV/ “L L ' L J l l L l 1 L J l 0.0 l T I I l I I I f I I 0-0 0.0 3.0 6.0 9.0 12.0 At DIMENSIONLESS TIME STEP LENGTH [ T ] Figure 3.5 Reletive Error vs. Temporal Resolution for Diffusion Process Simulation/at Well Lsyer. Fired Ax/5°=l/6.4 for p=5/5.=(68/64)‘ ’. 21 ’ and 2 ( RELATIVE ERROR ) x 100 46 -... : P=2.0 -O— : P=I.4142 -+- : R=1.0308 0.0 3.0 5.0 9.0 12.0 I l I l l l l l l l 1 —20.0 I j I I I I T I ‘20-0 1 I l ,/ // .‘// "I” , ’0/ " /,., // -15.0 -— ,,/ 4— —15.0 / ///o w ’ /R / ~r / // / -10.0 -""' // "" -10.0 /i://9’/’ «L— l‘ / JI— / /+ M/ / / ”/ /. ’l/ “PW/”’0’ .0- o.o I + I I I I I I [L I I 0.0 0.0 3.0 5.0 9.0 12.0 DIMENSIONLESS TIME STEP LENGTH [Aft ] Figure 3.6 Reletive Error vs. Temporsl Resolution for Diffusion Process Simulation st ls}l Leyer. Fixed Ax/6.=l/3.2 for Pas/5.-(63/64)‘/’. 2‘ ’ end 2 ( RELKHVE ERROR ) X 100 -30.0 -25.0 -20.0 -15.0 -10.0 0.0 Figure 3.7 Reletive Error vs. Simulation/at for p-5/s,=(6s/64)1 ’. 47 -:— P=2.0 —O— : P=1.4142 —+— : P=I.O308 0.00 0.30 0.60 0.90 1.20 1.50 1 SJ 4 1 .J 1 1 l I L I l j I I I I j I r T1 I I r ~I- «I- // -w- /’ ..E. /, JI- // .1}— J_ / d— -- --I—- -L ._ -—II- -+— I / #+ “I L_E l l I J l l J I l L I I r I I l I f r I l I I 0.00 0.30 0.60 0.90 1.20 1.50 Ax RATIO OF ORIO SIZE TO WALL LAYER THICKNESS ( E ) I ‘0 Process We 1 Layer. 21}: end 2 Spetisl Resolution for Diffusion Fixed At/T=O.2875 -30.0 -25.0 -20.0 —15.0 -10.0 -5.0 0.0 48 yo f1 = 4/24. rf1 0.1993o f. = 4/12 + 8/24. rf: 0.4881:o Figure 3.8 Discrete Vortices Arranged to Simulate Line Vortex. (This is an Exemple of N312) I [5. 3° NORMALIZED VORTICITY 49 —1- : ONE PER LOBE —6- : SIX PER LOBE -M- : MAXWORTHY 0.0 0.6 0.8 1.0 1 L 1 1 E 1.0 I I r I I I 0 ~- «I- e -- 0.8 -L ._ 0.6 -- —I- 0 6 __ .0 E 0-4 "“'" -I- 0.4 1- .L 0.2 -- \ “\\ -— 0.2 "E \\ N ‘I' \\ \ . 00 L 1 1 l 1 Egg 1 ‘ 00 - I T I I T l I I o 0.0 0.2 0.4 0.5 0.8 1.0 I DIMENSIONLESS DISTANCE FROM VORTEX CENTER I: HO ] Figure 3.9 Comparisons of Vorticity Distributions of Vortex Ring for N-l. N-6 per Lobe and Theoretical Approximation of Maxworthy [1977]. The Vorticity is Calculated by §=avlax-au/ay [i] ? NORMALIZED VORTICITY 49 -1- : ONE PER LOBE -6- : SIX PER LOBE -M— : MAXWORTHY 0.0 0.2 0.4 0.6 0.8 1.0 1 l L I 1 I l L L 1.0 r r If I *T I I I I 1 O 1- + h —— 0.8 0.6 -P- HP 06 A... ._ I} 0.4 -- __. 0.4 . \\ .11. \ K\ .1— 0.2 """- \\\‘ \\\\ -- 0.2 ., -.\ N f .\\ \\\ 't 00 L l I L 1 M. L m 00 ' I T I r I I I I o 0.6 0.8 1.0 0.0 0.2 0.4 l’ DIMENSIONLESS DISTANCE FROM VORTEX CENTER [ EC ] Figure 3.9 Comparisons of Vorticity Distributions of Vortex Ring for N-l, N-6 per Lobe and Theoretical Approximation of Maxworthy [1977]. The Vorticity is Calculated by {=3v/Bx-au/ay NORMAUZED SQUARE OF RMNUS OF GYRAHON [ 50 o : HYDRODYNAMICS PLUS DIFFUSION + : PURE DIFFUSION 0 1 2 3 l l l l l l l l 1 l I 2000 I I I I j I I r I I F 2-00 T + L 1.50 —— /,—— 1.50 ' ’/,/»/””/ I “I" / "L' 000003931 QQQQQQQQQQ+++++ LOO-m OQQQQQQQQ -I—‘l.00 I - 0.50 -- -- 0.50 .L ._ O 00 1 I I l l 1 l I l l J 0 DO - I I I I I I I r T I I ' 0 1 2 3 t NUMBER OF TURNS I: ? ] I Figure 3.10 Time-Dependent Radius of Gyration Calculated Using N-6 per Lobe for Pure Diffusion and Hydrodynamic Motion with Diffusion. The Straight Line is the Theoretical Solution Given by Equation (25) NORMAUZED SQUARE OF RMNUS OF GYRAHON [ 50 0 : HYDRODYNAMICS PLUS DIFFUSION + : PURE DIFFUSION 0 1 2 3 l J L l l 1 I l 1 1 I 2000 I I I l I I I I I I I 2-OO 4~ 5P 1.50 J— ,—r- 1.50 our. .... 0 0 In 0 0 0 0 0 g + + + @QQQQQ999999999+++++ 1.00 - 9 Q g a .5- 1.00 0.50 -- -1. 0.50 -L S- 0.00 I I I I I I I I I I I 0.00 0 1 2 3 t NUMBER OF TURNS [ ? ] I' Figure 3.10 Time-Dependent Radius of Gyration Calculated Using N-G per Lobe for Pure Diffusion and Hydrodynamic Motion with Diffusion. The Straight Line is the Theoretical Solution Given by Equation (25) 51 O : HYDRODYNAMICS PLUS DIFFUSION + : PURE DIFFUSION f—fi ° 0 1 2 3 2.00 FIS‘LfIIfi‘IIIFIIITI 2.00 -- “I- 1.50 1- ‘II' 1.50 ~/6/’ O 9 NORMALIZED SQUARE OF RADIUS OF GYRATION [ g 1.00 -M w- 1.00 .. I 0.50 -I- —L- 0.50 l l l L l l L L l I I I I 0000 W l— I T fi fi T wt T j r '1 I T 0'00 0 1 2 3 Figure 3.11 Time-Dependent Radius of Gyration Calculated Using N-60 per Lobe for Pure Diffusion and Hydrodynamic lotion with Diffusion. The Straight Line is the Theoretical Solution Given by Equation (25) NORMAUZED SQUARE OF RNNUS OF GYRAHON ( 52 0 : HYDRODYNAMICS PLUS DIFFUSION + : PURE DIFFUSION 0 1 2 3 2.00 I I I I I I I I I I I 2.00 .... «I 1.50 -I- -+- 1.50 06"” 0’6+ + *I' + + + My??? -P I Q’?’ 1.00 «M d- 1.00 0.50 "I" -r- 0.50 3- .L 0.00 I I I— I I I I— I I R I 0.00 O I 2 3 t NUMBER OF TURNS ( — I Tl' Figure 3.12 Time-Dependent Radius of Gyration Calculated Using NISOO per Lobe for Pure Diffusion and Hydrodynamic Motion with Diffusion. The Straight Line is the Theoretical Solution Given by Equation (25) 53 muomesuaamm uaomuosdz g—ooInHIuouuo> uou acomuoauoauu "not unm>oaxuqmm noduo> uo uuoaomnauu< HaomuuoEOoc ugh mu.m ounmwm Eu an Eco "" ...‘o--. _ I _ . _ 2:05- EVE? w .. Emmmmmam- .. .0 once _ _ .....2. fl 3,. .5 7 a. u _ _ ENE—OD . REMMfiMHESm _ .ccozflaano r1 IQflvfll I I I I I I I L .89 [NELJKY"11hflEE I to] sec. A 160- '- I 120' '- .5 <3 <3 I I 00 CD I I 60.- 40-- 54 0 i Experiment i =1marker per lobe for calculations (Liang) with/without diffusion model Survival \ I \__,I Ingestion of wall layer fluid leading to breakup I I9-% I : I I > 0.9 1.0 (£4 (£5 (£0 117 lJ(18 SPEED RATIO l —' l Uw Figure 4.1 Stability Maps of Computations with/without Diffusion Model for 1-15. Compared with Experimental Result of Liang[1984]. The Error Bars for Calculated Outcomes are $2.5 sec. 55 Ea... N ..Z . ... . . s _ . . ._ _. , .. .. . . a” . I s w A"? . . _ .. .. .... a, :. ... . u .r: . ... L5. - f... :5 p 5 : .. i / a . . Ma‘ \ 9.! WWI / fl .. ... a. \IIIIIJIIIII A \IIIIIII/IH/u. . \ I d ~ .I 3‘. A /I.\.I .....3/I / ..., 2 . . ._ .., u g _ H “I“ H “MM . .. ....» ..va . ... ”an": . .xnr:« ..SJMJ .. .. .. 2.3. ...L . may“; ._ .22. a ...z. :a .w. 3:“. . .. {a} w“ ..wmw AJWfl~ . .Ia . a . . .. .sz: .. . JILL: . . . :gzaadaa .1.1 # -..... n ‘ .... .__ . . ...: ... .1 .U. _ . ..u“ 1 - . In"; : ‘ . .s...‘ ~ 2.1. . a ”at? a. . . ... for Unstable Interaction which the Line Pair Ingests Wall Layer Fluid as it "Crashes” on the Wall Figure 4.2 Computer Generated Vorticity Motion Pictures 56 qt ‘_.--o ~_-o-o ‘ s .0 ‘ . .. .32.: f“ . . . . . lazuhafi. for Unstable Interaction which the Line Pair Ingests Wall Layer Fluid as it Turns Away from the Wall Figure 4.3 Computer Generated Vorticity Motion Pictures M e ‘ ° 0 a . . e , . e ' e e '. ' I ’ I ) " ' v I a " x a .. o ”i a II, 0 ‘ ‘ .h- ‘d‘ ’ '1': ’ ’l ‘ \ ‘ ”gm “fa—4‘ ’/ I” I \ ‘ \\ ‘ h #43” ’l'a' ’ l O ‘ a.“ a. 52" ‘0’ ' ‘ ’ . ’ s ‘ ... ‘ “ ” al' p"; \ ‘ \~“~.-~ ”’ a- u ‘ . ‘ ““ for a Figure 4.4 Computer Generated Vorticity Motion Pictures Stable Interaction which the Line Pair Leaves the 'all without Breaking Up 58 . for Interaction in the Absence of Diffusion Pro- Figure 4.5 Computer Generated Vorticity Motion Pictures Stable Time Step 4.6 Which The Initial Conditions Same to The c 8 I C 8 .1 d n n 0 P o s t a r v. r 1 «W01 :8 no "8.1 sit an t e .0 II I u t 0 :3 .0 8.00 8h: 0 O o W ‘3 =- - . I I s ‘ ‘ ... ~ — -.. .— " " f a ' . ‘ I F..—.... .:£'—" ‘1'": :‘a‘LCr - " j ’ ' ' ' . . . - ‘ f "--"-—- :; =c;nu»fi”':, " 2 I o, . 5 ~ - ~ ... a: 4.. 9 f. _. F1..-’. - ‘ . . - e ‘s ‘ s u ... 3".“ 'wm’fi 5i:- -- ‘1... - _. ..—- ‘ _ "‘ bah ,n W _ a; -. ‘ Figure 4.6 Computer Generated Vorticity lotion Pictures for a Stable Interaction in the Presence of Diffusion Pro- cess. ”Photos" Correspond to The Same Time Step Number as Figure 4.5 Which The Initial Conditions Were Set Identically to This Case 60 o : Experiment i = 1 marker per lobe '8 =6markers per lobe 160- - * Survival \ 2.9 0' \~ ”I 0140- - (D 0 w 120.. / .a C3 CD I I DELAY TIME 92° *\ E/ * Ingestion of 40-1- wa'l Iayer "Hid leading to breakup 20“ W I I I I I I In» 0 0.4 0.5 0.6 0.7 U 0.8 0.9 1.0 SPEED RATIO 1 —! I Uw Figure 5.1 Comparisons of Stability Maps for 7-15.. Calculations without Diffusion for 6 Markers per Lobe were Carried Out by Bartholomew