Characteristics of Fiber Suspension Flow in a Turbulent Boundary Layer

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1 Chrcterstcs of Fber Suspenson Flow n Turbulent Boundry Lyer Jnzhong Ln, Suhu Shen, Xoke Ku Deprtment of Mechncs Hngzhou, Zheng CHINA Correspondence to: Jnzhong Ln eml: meczln@publc.zu.edu.cn ABSTRACT The equtons of verged momentum, turbulence knetc energy, turbulence dsspton rte wth the ddtonl term of the fbers, nd the equton of probblty dstrbuton functon for men fber orentton re derved nd solved numerclly for fber suspenson flowng n turbulent boundry lyer. The mthemtcl model nd numercl code re verfed by comprng the numercl results wth the expermentl ones n turbulent chnnel flow. The effects of Reynolds number, fber concentrton nd fber spect-rto on the men velocty profle, turbulent knetc energy, Reynolds stress, turbulent dsspton rte nd eddy vscosty coeffcent re nlyzed. The results show tht the velocty profles become full, nd the turbulent knetc energy, Reynolds stress nd eddy vscosty coeffcent ncrese, whle turbulent dsspton rte decreses, s the Reynolds number, fber concentrton nd fber spect-rto ncrese. The effect of the fber spect-rto on the turbulent propertes s lrger thn tht of the Reynolds number, but smller thn tht of the fber concentrton n the rnge of prmeters consdered n ths pper. INTRODUCTION The flows of turbulent fber suspenson n boundry lyer flow re nvolved n mny prctcl pplctons. The chrcterstcs of the flow process nd fnl product usully depend on the velocty profle nd turbulent propertes of fber suspenson. The turbulence s used to mke hgh qulty products nd good energy svngs n some ndustrl processes. However, there re stll mny unknowns bout the flows of turbulent fber suspenson. In order to understnd the flow chrcterstcs, t s mportnt to know the effect of fbers on the velocty profle, turbulent knetc energy, Reynolds stress, turbulent dsspton rte nd eddy vscosty coeffcent. There hs exsted some lterture delng wth the fber suspenson flowng n the turbulent boundry lyers. Tucker [1] used the rheologcl theores for fber suspensons to exmne the flow n nrrow gps. Order-of- mgntude estmtes were developed for the velocty, stress nd fber-orentton components. The estmtes revel four dstnct flow regmes. Lmttons due to entrnce effects nd strt-up trnsents were dscussed, long wth pplctons to the modelng of necton nd compresson moldng of fber-renforced polymers. Andersson nd Rsmuson [] studed the flow nd trnston of fber suspensons to turbulence n rotry sher tester by Lser Doppler Anemometer (LDA) mesurements. The results show tht the fluctuton veloctes pproched those of sngle-phse flow wth n ncresng rottonl speed, untl they were nerly equl. The velocty profle s lner nd close to the wll even wth fbers present. The presence of the fbers lso flttened the profles, ndctng n ncresed momentum trnsfer. Pettersson nd Rsmuson [] performed detled mesurements of men nd RMS veloctes by LDA of turbulent gs/fber/lqud suspenson n rotry sher tester. Plottng RMS nd men veloctes versus mpeller speed nd power nput ndctes tht both decrese wth ncresng gs nd fber contents. Sun et l. [] used the LVEL model of Phoencs softwre to smulte the turbulent boundry lyer of medum-consstency pulp fber suspensons. The results show tht, from the nner surfce of the ppelne to.9m, the dstrbutng of the velocty s lner cross the wter rng. The surfce of the plug s qute smooth. The concentrton s very hgh wy from the nner surfce, whle t s low on the surfce of the ppelne. Ln et l. [5] derved the equton of probblty dstrbuton functon for Journl of Engneered Fbers nd Fbrcs 17 Volume, Issue 1 1

2 men fber orentton n turbulent boundry lyer, n whch the correlton terms of the fluctutng velocty, fluctutng ngulr velocty wth the fluctutng probblty dstrbuton functon were relted to the grdent of men probblty dstrbuton functon nd the dsperson coeffcents. The fnte-dfference method ws used to solve the equton numerclly. The results show tht the fbers tend to lgn wth the stremlne, nd the fber spect-rto hs sgnfcnt effect on the orentton dstrbuton of fbers, whle the effect of the dstnce from the wll s neglgble. Xu nd Adun [] mesured the velocty profle of fber suspenson flow n rectngulr chnnel by pulsed ultrsonc Doppler velocmetry, nd the effect of fber concentrton nd Reynolds number on the shpe of the velocty profle ws nvestgted. Fve types of flow behvor were observed when the fber concentrton ncreses or the flow rte decreses progressvely. The turbulent velocty profles of fber suspenson cn be descrbed by correlton wth fber concentrton nd Reynolds number s the mn prmeters. It ws found tht the presence of fber n the suspenson wll reduce the turbulence ntensty nd thus reduce the turbulent momentum trnsfer. Schmdt nd Zhu [7] used dgtl mgng nd found tht the mxng lyer n the chnnel becomes less unform wth n ncrese n fber length nd suspenson concentrton nd wth decrese n the pressure drop cross the flow restrcton. Pschkewtz et l. [] clculted the orentton moments nd stresses long the Lgrngn pthlnes n the flow, nd showed tht the more commonly observed smll stress fluctutons ppered to mke the lrgest contrbuton to the fber dsspton of turbulent knetc energy nd thus re responsble for the morty of the drg reducton effect. The lrgest contrbuton to fber dsspton of turbulent knetc energy s mde by smll fluctutons n the spn wse sher stress component. Pettersson et l. [9] performed detled study of the flow behvor n the ner wll regon of pulp suspensons up to.7% usng Lser Doppler Anemometry. Axl men velocty profles show dstnct plug flow nd n ncrese of the plug regon s the flow rte decreses nd fber concentrton ncreses. They ndcted dluton regon t 1- mm from the wll tht s lrger thn the nnulus regon. The dluton regon ncreses wth ncresng flow rte, decresng concentrton when usng longer fbers. Gllssen et l. [1] studed turbulent fber suspenson flow usng drect numercl smulton nd concluded tht the nstntneous stress cnnot be smulted drectly wth the prtcle method due to computer restrctons. The effect of fbers on the turbulent flow s equvlent to n ddtonl Reynolds verged vscosty. Zhng et l. [11] studed the trnsport nd deposton of ngulr fbrous prtcles n turbulent chnnel flows. For dlute suspenson of fbers, one-wy couplng ssumpton ws used n tht the flow crres the fbers, but the couplng effect of the fber on the flow ws neglected. Ensembles of fber trectores nd orenttons were generted nd sttstclly nlyzed. The results were compred wth those for sphercl prtcles nd strght fbers nd ther dfferences were dscussed. Effects of fber sze, spect rto, fber ngle, turbulence ner wll eddes, nd vrous forces were studed. Ln et l. [1] proposed model reltng the trnsltonl nd rottonl trnsport of orentton dstrbuton functon (ODF) of fbers to the grdent of men ODF nd the dsperson coeffcents to derve the men equton for the ODF, nd predcted the ODF of fbers by numerclly solvng the men equton for the ODF together wth the equtons of turbulent boundry lyer flow. The results show tht the most fbers tend to orent to the flow drecton. The fber spect rto nd Reynolds number hve sgnfcnt nd neglgble effects on the orentton dstrbuton of fbers, respectvely. The sher stress of fber suspenson s lrger thn tht of Newtonn solvent nd the frst norml stress dfference s much less thn the sher stress. The exstence of fbers nevtbly ffects the turbulent flow. The nvestgtons mentoned bove dd not nvolve the effect of fbers on the turbulent propertes. In order to understnd the chrcterstcs of turbulent fber suspensons, t s necessry to modfy the equtons of the verged momentum, turbulence knetc energy nd turbulence dsspton rte. Ln et l. [1] derved the modfed equtons of the verged momentum nd the probblty dstrbuton functon for the men fber orentton, nd ppled the equtons to the chnnel flow. The results show tht the flow rte of the fber suspenson s lrge under the sme pressure drop n comprson wth the rte of Newtonn flud. The reltve turbulent ntensty nd the Reynolds stress n the fber suspenson re smller thn those n the Newtonn flow. Gllssen et l. [1] studed fber-nduced drg reducton usng Nver-Stokes equtons supplemented by the fber stress tensor n turbulent chnnel flow. The results were used to Journl of Engneered Fbers nd Fbrcs 1 Volume, Issue 1 1

3 vldte n pproxmte method for clcultng fber stress, n whch the second moment of the orentton dstrbuton s solved. Ln et l. [15] solved the verged momentum equton wth the term of ddtonl stress resultng from fbers n the flow of turbulent fber suspensons flowng through contrcton wth rectngulr cross-secton to get dstrbutons of the men velocty, men pressure, turbulent knetc energy nd turbulent dsspton rte. It s found tht fbers reduce turbulent ntensty nd turbulent dsspton t rte t the centrl lne, but fbers enhnce them over the cross secton t the ext. Fbers hve no effect of restrnt on the turbulence n the contrcton flow. However, the modfed equtons of turbulence knetc energy nd turbulence dsspton rte wth ddtonl terms of the fbers hve not been derved nd solved to determne the effect of fbers on the turbulent propertes. Therefore, n the present study, we derve the modfed equtons of verged momentum, turbulence knetc energy nd turbulence dsspton rte wth ddtonl terms of the fbers, nd pply the equtons to the turbulent boundry lyer flow. We chose the boundry lyer flow becuse t s typcl bounded flow nd thus provdes buldng block for mny prctcl nhomogeneous flows. EQUATIONS OF FIBER SUSPENSIONS For n ncompressble fully developed turbulent fber suspenson flow, the contnuty equton nd momentum equton wth ddtonl fber stress tensor re [1]: u x = (1) vscosty of the suspenson for the dlute regme (nl <<1) [1] ( μ s the vscosty of the suspendng flud, n s the number of fbers per unt volume, l s the fber hlf-length nd r s the fber spect-rto), nd re the second- nd fourth-orentton tensor of fber [17, 1]: () p p p p pdp () k l where p s unt vector prllel to the fber s xs, p s the probblty dstrbuton functon for fber orentton, nd pdps the probblty tht the fber orenttons re locted between p nd p dp. p stsfes the equton of conservton: p u t x p where p s the fber ngulr velocty [19]: k l (5) p p p p p p () where s the vortcty tensor, ( u x u x)/, r 1 r 1 s relted to the spect-rto r. AVERAGED MOMENTUM EQUATION The nstntneous velocty, pressure, tensor of rte of strn nd orentton tensor cn be expressed s men prt plus fluctuton prt: 1 p u u u u t x x x f 1 [ ( I ) ] x () where u s the velocty, p s the pressure, s the knetc vscosty of the suspendng flud, u x u x / s the tensor of rte of strn, f nl lnr s the pprent (7) We substtute Eq. (7) nto Eq. () nd tke the verge. The terms of ' ' nd ' ' re relted to the smll scle vortces whch re sotropc. Snce ' nd ' depend on the rotton ngle of the fber, whle ' depends on the sptl poston of the fber, when the coordnte system turns 1 o, we hve ' ' = ' ' nd Journl of Engneered Fbers nd Fbrcs 19 Volume, Issue 1 1

4 ' ' ' ' becuse of the sotropc behvor, whch mens ' ' nd ' '. Then we cn derve the verged momentum equton wth n ddtonl term of the fbers s follow: 1 P U U U u t x x x uu f 1 I x x ( ). () EQUATION OF THE PROBABILITY DISTRIBUTION FUNCTION FOR MEAN FIBER ORIENTATION We cn obtn nd n Eq. () by tkng verge of Eq. () nd Eq. (): p p pdp (9) p p p p pdp (1) k l Then we should buld the equton for p. Expressng the nstntneous velocty, probblty dstrbuton functon for fber, fber ngulr velocty s men prt plus fluctuton prt: u U u,, p p p. (11) Substtutng Eq. (11) nto Eq. (5) nd tkng the verge, we hve: p ( u ) ( p ) U t x p x p. (1) Smlr to the reltonshp between Reynolds stress nd men velocty grdent, the correltons between fluctutng probblty dstrbuton functon nd fluctutng velocty, fluctutng fber ngulr velocty cn be wrtten s: where nd / x k / p wth k beng the turbulence knetc energy, the turbulence dsspton rte nd the knemtc vscosty []. x nd p re the dsperson coeffcents of lner nd ngulr dsplcement, respectvely. Combnng Eq. (1) nd Eq. (1) we hve: p U +.(1) t x p x p x p In order to get p n Eq. (1) we express the nstntneous qunttes n Eq. () s men prt plus fluctuton prt, nd tke the verge: p p p p p p. (15) k l Substtutng Eq. (15) nto Eq. (1) yelds: (1) For the ncompressble flow we hve = nd, then Eq. (1) cn be wrtten s: (17) whch s the equton of probblty dstrbuton functon for the men fber orentton. EQUATION OF THE TURBULENCE KINETIC ENERGY Up to now there s no turbulent model for turbulent fber suspenson flows; therefore, we extend the turbulent model n the pure flud to fber suspensons. Here we use k-ε model becuse t s vld n the turbulent boundry lyer for pure flud. u = x ; p p (1) x p The Reynolds stress tensor cn be expressed s: - uu n Eq. () r u u uu T( ) k x x Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1 (1)

5 where the eddy vscosty T C k / wth k beng the turbulent knetc energy, the turbulent dsspton rte nd C.9. For solvng Eq. (1) nd Eq. (), k -equton nd -equton wth ddtonl terms of the fbers re gven: k U T k U uu [( ) ] S x x x x k k (19) () u U C uu C x k x k x 1 T [( ) ] S x () where s the vscosty of the suspendng flud, C1 1., C 1.9, k 1. nd 1. [1]. S k nd S re the source terms cused by the exstence of the fbers. As shown n Eq. (), the source term n the momentum equton s: f 1 [ ( I ) ] x (1) Multplyng n component of Eq. (1) wthu m, m component of Eq. (1) wthu n, respectvely, nd then combnng both, we hve f 1 { um [ n ( In) ] x 1 un [ m ( Im) ]}. x () Substtutng Eq. (7) nto Eq. () nd tkng the verge yelds: As shown n Eq. () the verge form of source term s: f 1 x ( I ) () Multplyng n component of Eq. () wthu m, m component of Eq. () wthu n, respectvely, nd then combnng both, we hve f { Um n Un m x x 1 1 In Um Im Un } x x (5) Subtrctng Eq. (5) from Eq. () yelds: f 1 { um n In um x x 1 un m Im un }. () x x Let m=n nd multplyng Eq. () wth 1/ we hve (7) Eq. (7) gves the effect of fbers on the turbulent knetc energy. From Eq. (7) we cn see tht such effect s relted to the pprent vscosty of the suspenson, flud densty, fluctutng velocty, men second- nd fourth-orentton tensor of fber, fluctutng tensor of rte of strn. Journl of Engneered Fbers nd Fbrcs 1 Volume, Issue 1 1

6 EQUATION OF THE TURBULENCE DISSIPATION RATE Dervtng Eq. (1) wth respect to nd then x m multplyng wth u xm, we hve Dervtng Eq. () wth respect to multplyng u xm, we hve x m nd then f u 1 [ ( I ) ] xm xm x f u u { xm xm x xmx xm u u xm xm x xmx xm I I x x x x x x I u u } x x x x x x u u m m m m I m m m m () Subtrctng Eq. () from Eq. (9) yelds: () Substtutng Eq. (7) nto Eq. () nd tkng the verge yelds: f u u S { x xm x m xm xm x u u xm xm x xm x xm I I u u x xm x m xmx x m I u I u } xm xm x xmx xm (1) (9) Effect of fbers on the turbulent dsspton rte s gven by Eq. (1) whch s complex nd relted to the flud vscosty, pprent vscosty of the suspenson, flud densty, fluctutng velocty, men second- nd fourth-orentton tensor of fber, fluctutng tensor of rte of strn. Eq. (), (17), (1), (19), (), (7) nd (1) re the bsc equtons of turbulent fber suspenson flow. TURBULENT BOUNDARY LAYER FLOW AND CORRESPONDING EQUATIONS A fully developed turbulent boundry lyer flow s presented n Fgure 1 where the thck of boundry lyer s δ nd the ngle between the fber s xs nd horzontl s. Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1

7 FIGURE 1. Fully developed turbulent boundry lyer flow nd fber. Applyng Eq. (7) nd (1) to the fully developed turbulent boundry lyer flow we hve: Sk u u u y y y y y v u xyyy u xyyyu v xyyy v y y y y f u xxyy u v xyyy { xxyy 1 u xy u v v xy v v y y y y y v 1 v yy v yyyy v v yy v } y y y y yyyy f xxyy u u u u xxyy S { y y y y y y u u xyyy uv v u xxyy xyyy y y y y y y y xyyy y y y y y y y y yyyy vv vv yyyy v v yyyy y y y y y y y y 1 xy uv uv xy uv xy y y y y y y y y 1 yy vv vv yy vv yy } y y y y y y y y v u v u xyyy u v xyyy () () grdents qute well, nd mproves rpdly s the grd s refned. However, t cn produce undershoots nd overshoots n regons of steep grdents. The SIMPLEC lgorthm enforces mss conservton nd cheves pressure-velocty couplng. The Enhnced Wll Functons gven by Kder [] s used n the ner-wll regon. The numercl smulton s crred out n domn of 1.5 long the x nd y drectons, respectvely. The computton grd s comprsed of x y= =1 grd ponts. The computtonl grd ndependence hs been tested. The fber concentrton s defned s Cf Vf Vs, where V f nd V s re the volume of the fbers nd of the suspenson, respectvely. The fber spect-rto s wrtten s r. The smultons cover rnge of Re=5-75, C f =.-1.% nd r =5-. For ths rnge of fber (sem-dlute regme) concentrton the pprent vscosty of the suspenson s tken s []: nl. ln ln(1/ C f ) p 1 ln(1/ Cf ) ln(1/ Cf ) () RESULTS AND DISCUSSIONS Vldton of Mthemtcl Model nd Numercl Code In order to vldte the mthemtcl model nd numercl code, we solve the bove derved equtons n the turbulent chnnel flow of fber suspenson nd compre the numercl results wth the expermentl ones []. The comprson of men strem-wse velocty, U, s shown n Fgure where b s the chnnel wdth nd Umx s the velocty t the centerlne. We cn see tht the numercl results re generlly consstent wth the expermentl ones. NUMERICAL METHOD AND PARAMETERS A QUICK scheme (Qudrtc Upwnd Interpolton for Convecton Knetcs) nd fnte volume method re used to dscretze nd solve the equtons, respectvely. The QUICK scheme fts prbol between three ponts to pproxmte the veloctes t the cell fces, nd cn be shown to be thrd order ccurte. The QUICK scheme represents the steep FIGURE. Comprson of the numercl nd expermentl results n the turbulent chnnel flow of fber suspenson ( C f =.5% nd r=). Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1

8 Fber Orentton Dstrbuton Functon Addtonl fber stress tensor s relted to the fber orentton dstrbuton functon s shown n Eq. ()-(). The reltonshp between the orentton dstrbuton functons ψ nd the ngle φ ncluded between the fber prncpl xs nd the x-xs for vryng y/δ s shown n Fgure n whch most fbers re orented between o nd o,.e., the fbers tend to orent, on verge, to the flow drecton. The reson s tht the lrge grdent n men velocty ner the wll exerts lrge torque on the fbers, mkng the fbers rotte to stte experencng the smllest torque. (U -U)/u* y/ No fbers :Re=75; fber suspenson: : Re=75; : Re=5; : Re=55; : Re=5 FIGURE. Vrton of the men velocty wth the Reynolds number for C f =.1% nd r=5. FIGURE. Orentton dstrbuton functon of fbers for vryng y/ δ( C f =.5%, Re=75 nd r=5). Effect of the Reynolds Number on the Flow Chrcterstcs The effect of Reynolds number on the men velocty s shown n Fgure where the expermentl result for pure flud [] s lso gven. The experment ws performed wth outflow velocty of 15m/s nd outflow reltve turbulent ntensty of.%. In the fgure U s the outflow velocty nd u* w / s the wll frcton velocty wth w beng the frcton stress on the wll. It cn be seen tht the numercl results re consstent wth the expermentl ones for the pure flud t Reynolds number of 75. The velocty profles show fully developed turbulent flow. Even though the velocty profles become full s the Reynolds number ncreses n the rnge of 5 <Re<75, the Reynolds number hs nsgnfcnt effect on the velocty profle. The vrtons of turbulence knetc energy, Reynolds stress, turbulent dsspton rte nd eddy vscosty coeffcent wth the Reynolds number re shown n Fgures 5,, 7 nd, respectvely. The expermentl results for pure flud [] re lso gven. The numercl results re consstent wth the expermentl ones for the pure flud t Reynolds number of 75. The mgntude of Reynolds number s drectly proportonl to the flow rte when the length scle nd flud vscosty remn constnt. We cn see tht the turbulent knetc energy decreses grdully from the wll to the outflow s Fgure 5. The turbulent knetc energy ncreses wth ncresng the Reynolds number. As shown n Fgure there s mxmum of the Reynolds stress round y/δ=.5. The Reynolds stress ncreses from the wll to y/δ=.5, nd then decreses to zero from y/δ=.5 to the outflow. The Reynolds stress ncreses wth ncresng the Reynolds number. From Fgure 7 t cn be seen tht the turbulent dsspton rte decreses from the wll regon to the outflow. The Reynolds number does not hve n obvous effect on the profle of turbulent dsspton rte. Fgure shows the vrton of eddy vscosty coeffcent wth the Reynolds number. The mxmum of the eddy vscosty coeffcent ppers round y/δ=.. The eddy vscosty coeffcent ncreses from the wll to y/δ=., nd then decreses to zero from y/δ=. to the outflow. The eddy vscosty coeffcent ncreses s the Reynolds number ncreses. At ths rnge of prmeters, the effect of Reynolds number on the flow behvor s bout the sme s tht of Newtonn flud, becuse the presence of fbers hs n nsgnfcnt effect on the turbulent ntensty nd the momentum trnsfer s domnted by turbulence. Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1

9 k/u* 1 1 m /u* y/ No fbers :Re=75; fber suspenson: : Re=75; : Re=5; : Re=55; : Re=5 FIGURE. Vrton of the turbulence knetc energy wth the Reynolds number for C f =.1% nd r=5. u'v'/u* y/ No fbers :Re=75; fber suspenson: : Re=75; : Re=5; : Re=55; : Re=5 FIGURE 5. Vrton of the Reynolds stress wth the Reynolds number for C f =.1% nd r=5. /u* y/ No fbers :Re=75; fber suspenson: : Re=75; : Re=5; : Re=55; : Re=5 FIGURE. Vrton of the turbulent dsspton rte wth the Reynolds number for C f =.1% nd r= y/ No fbers :Re=75; fber suspenson: : Re=75; : Re=5; : Re=55; : Re=5 FIGURE 7. Vrton of the eddy vscosty coeffcent wth the Reynolds number for C f =.1% nd r=5. Effect of the Fber Concentrton on the Flow Chrcterstcs Effect of fber concentrton on the men velocty, turbulence knetc energy, Reynolds stress, turbulent dsspton rte nd eddy vscosty coeffcent for Re=75 nd r =5 re shown n Fgures 9, 1, 11, 1 nd 1, respectvely. The expermentl results [] for pure flud re lso gven. In Fgure 9 the velocty profles become full s the fber concentrton ncreses n the rnge of.1%< C f <1%. Fgure 1 shows tht the turbulent knetc energy ncreses over the boundry lyer wth n ncresng fber concentrton. The exstence of fbers wll hnder the momentum trnsfer through decresng turbulence ntensty nd, on the other hnd, enhnce the momentum trnsfer by provdng sold lnk between dcent flud lyers. The overll momentum trnsfer s determned by the competton of these two effects. The curves n Fgure 11 llustrte tht the vrton of the fber concentrton does not chnge the shpe of the Reynolds stress profle. The mxmum of Reynolds stress s locted round y/δ=.1, nd the Reynolds stress ncreses s the fber concentrton ncreses becuse the prmry effect of ncresng the fber concentrton s to ncrese the mgntude of the fber extr stresses. From Fgure 1 we cn see tht the turbulent dsspton rte s ncresed wth decresng fber concentrton. Ths mens tht the presence of fbers, t ths rnge of concentrton, results n the ncrese of vscosty, whch cuses the decrese of velocty grdent n the wll regon. The turbulent energy equton s blnce between turbulent energy producton nd turbulent dsspton, whch cn be llustrted by comprng Fgure 1 nd Fgure 1,.e., the Journl of Engneered Fbers nd Fbrcs 5 Volume, Issue 1 1

10 turbulent knetc energy ncreses nd turbulent dsspton rte decreses wth n ncresng fber concentrton. Perlekr et l. [5] crred out hgh-resoluton drect numercl smulton of decyng, homogeneous, sotropc turbulence wth polymer ddtves, nd reveled tht the drg reducton corresponds to the reducton n the turbulent dsspton rte. Therefore, the drg reducton cn be nduced by fbers n turbulent boundry lyer, nd the effect of ncresng the fber concentrton s n ncrese n the drg reducton effectveness becuse the drg reducton performnce s closely correlted to resstnce to extensonl motons, nd for fbers the extensonl vscosty s proportonl to the fber concentrton. The effectve vscosty ncreses wth n ncresng fber concentrton, whch results n the ncrese of turbulent knetc energy nd decrese of turbulent dsspton rte. The vrton of the eddy vscosty coeffcent wth the fber concentrton s shown n Fgure 1 from where t cn be seen tht the eddy vscosty coeffcent ncreses wth ncresng the fber concentrton. Generlly, the effect of fber concentrton on the turbulent propertes s lrger thn tht of the Reynolds number. (U -U)/u* y/ : No fbers; fber suspenson: : C f =.1%; : C f =.%; : C f =.7%; : C f =1% FIGURE. Vrton of the men velocty wth fber concentrton for Re=75 nd r =5. k/u* y/ : No fbers; fber suspenson: : C f =.1%; : C f =.%; : C f =.7%; : C f =1% FIGURE 9. Vrton of the turbulence knetc energy wth fber concentrton for Re=75 nd r =5. u'v'/u* y/ : No fbers; fber suspenson: : C f =.1%; : C f =.%; : C f =.7%; : C f =1% FIGURE 1. Vrton of the Reynolds stress wth fber concentrton for Re=75 nd r =5. /u* y/ : No fbers; fber suspenson: : C f =.1%; : C f =.%; : C f =.7%; : C f =1% FIGURE 11. Vrton of the turbulent dsspton rte wth fber concentrton for Re=75 nd r =5. Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1

11 m /u* y/ : No fbers; fber suspenson: : C f =.1%; : C f =.%; : C f =.7%; : C f =1% FIGURE 1. Vrton of the eddy vscosty coeffcent wth fber concentrton for Re=75 nd r =5. Effect of the Fber Aspect-Rto on the Flow Chrcterstcs The dstrbuton of men velocty profle, turbulence knetc energy, Reynolds stress, turbulence dsspton rte nd eddy vscosty coeffcent long the lterl drecton t dfferent fber spect-rtos re shown n Fgures 1, 15, 1, 17 nd 1, respectvely, for Re=75 nd C f =.1%. Chngng the spect rto modfes both the dynmc behvor of the fbers nd the fber extr stress. We cn see tht the fber spect-rto hs scnt effect on the men velocty profle n Fgure 1. The turbulent knetc energy ncreses wth ncresng the fber spect-rto s shown n Fgure 15. It s esy for the fbers wth lrge spect-rto to ncrese the momentum trnsfer through provdng sold lnk between the dcent flud lyers [, 7]. The Reynolds stress, s shown n Fgure 1, lso ncreses wth ncresng the fber spect-rto nd ppers mxmum round y/δ=.5. From Fgure 17 t s found tht the turbulent dsspton rte decreses wth ncresng fber spect-rto. The fbers wth lrge spect-rto lgn more esly long the flow drecton, whch results n the ugment of turbulent dsspton rte. The effect of ncresng the fber spect rto s n ncrese n the drg reducton effectveness, whch s consstent wth the concluson gven by Pschkewtz et l. [] usng drect numercl smulton for turbulent chnnel flow. Ths effect cn be explned by exmnng the behvor of the fbers n the sher flow. As the spect rto s ncresed, the totl extr sher stress t the wll s decresed, enhncng the drg reducton effectveness. Addtonlly, s the spect rto s ncresed, extensonl stresses re ncresed nd ths effect my lso be responsble for the hgher levels of drg reducton. The vrton of eddy vscosty coeffcent wth the fber spect-rto s gven n Fgure 1, we cn see tht the eddy vscosty coeffcent frst ncreses from the wll long the lterl drecton, nd then decreses to zero t the outflow. The eddy vscosty coeffcent enhnces wth the ncrese of the fber spect-rto nd hs mxmum round y/δ=.. Comprng the contrbuton of the Reynolds number, fber concentrton nd fber spect-rto we cn see tht the effect of the fber spect-rto on the turbulent propertes s lrger thn tht of the Reynolds number, but smller thn tht of the fber concentrton n the rnge of prmeters consdered n ths pper. (U -U)/u* y/ : No fbers; :r =5; : r =1; : r =; : r = FIGURE 1. Vrton of the men velocty wth fber spect-rto for Re=75 nd C f =.1%. k/u* y/ : No fbers; :r =5; : r =1; : r =; : r = FIGURE 1. Vrton of the turbulence knetc energy wth fber spect-rto for Re=75 nd C f =.1%. Journl of Engneered Fbers nd Fbrcs 7 Volume, Issue 1 1

12 u'v'/u* y/ : No fbers; :r =5; : r =1; : r =; : r = FIGURE 15. Vrton of the Reynolds stress wth fber spect-rto for Re=75 nd C f =.1%. /u* y/ : No fbers; :r =5; : r =1; : r =; : r = FIGURE 1. Vrton of the turbulent dsspton rte wth fber spect-rto for Re=75 nd C f =.1%. m /u* y/ : No fbers; :r =5; : r =1; : r =; : r = FIGURE 17. Vrton of the eddy vscosty coeffcent wth fber spect-rto for Re=75 nd C f =.1%. CONCLUSIONS The chrcterstcs of fber suspenson flow n turbulent boundry lyer re dfferent from those n the turbulent sngle phse boundry lyer. In order to explore the effects of the Reynolds number, fber concentrton nd fber spect-rto on the men velocty profle, turbulent knetc energy, Reynolds stress, turbulent dsspton rte nd eddy vscosty coeffcent, the equtons of verged momentum, turbulence knetc energy, turbulence dsspton rte wth the ddtonl term of the fbers, nd equton of probblty dstrbuton functon for men fber orentton re derved nd solved numerclly n the fber suspenson flowng n turbulent boundry lyer. In order to vldte the mthemtcl model nd numercl code, the derved equtons re solved n the turbulent chnnel flow of fber suspenson, nd some of the numercl results re consstent wth the vlble expermentl results. The results show tht the velocty profles become full, nd the turbulent knetc energy, Reynolds stress nd eddy vscosty coeffcent ncrese, whle turbulent dsspton rte decreses, s the Reynolds number, fber concentrton nd fber spect-rto ncrese. The effect of the fber spect-rto on the turbulent propertes s lrger thn tht of the Reynolds number, but smller thn tht of the fber concentrton n the rnge of prmeters consdered n ths pper. ACKNOWLEDGEMENT Ths work s supported fnnclly by the reserch grnt from the reserch fund for the Doctorl Progrm of Hgher Educton of Chn (No ) REFERENCES [1] C. L. Tucker, Journl of Non-Newtonn Flud Mechncs, 1991, 9,, 9-. [] S. R, Andersson, A. Rsmuson, AICHE Journl,,,, [] J. Pettersson, A. Rsmuson, Cndn Journl of Chemcl Engneerng,,,, 5-7. [] J. L. Sun, K. F. Chen, Y. Y.Chen, Recent Advnces n Flud Mechncs,, [5] J. Z. Ln, K. Sun K, J. Ln, Chnese Physcs Letters, 5,, 1, [] H. J. J. Xu, C.K. Adun, Interntonl Journl of Multphse Flow, 5, 1,, 1-. Journl of Engneered Fbers nd Fbrcs Volume, Issue 1 1

13 [7] E. A. Schmdt, J Y. Zhu, Tpp Journl, 5,,, 9-1. [] J. S. Pschkewtz, Y. Dubef, E. S. G. Shqfeh, Physcs of Fluds, 5, 17,, 1. [9] A. J. Pettersson, T. Wkstrom, A. Rsmuson, Cndn Journl of Chemcl Engneerng,,,, -. [1] J. J. J. Gllssen, B. J. Boersm, P. H. Mortensen, Physcs of Fluds, 7, 19, 11, [11] H. F. Zhng, G. Ahmd, B. Asghrn, Aerosol Scence nd Technology, 7, 1, 5, [1] J. Z. Ln, K. Sun, W. F. Zhng, Act Mechnc Snc,,,, -5. [1] J. Z. Ln, Z.Y. Go, K. Zhou, Appled Mthemtcs Modelng,,, 9, [1] J. J. J. Gllssen, B. J. Boersm, P. H. Mortensen, Physcs of Fluds, 7, 19,, [15] J. Z. Ln, S. L. Zhng, J. A. Olson, Fbers nd Polymers, 7,,1, -5. [1] G. K. Btchelor, Journl of Flud Mechncs, 1971,, 1-9. [17] E. J. Hnch, L.G. Lel, Journl of Flud Mechncs, 197, 7, 17-. [1] S. G. Advn, C. L. Tucke, J. Rheol., 197, 1,, [19] J. S. Cntr, C. L. Tucker, J. Rheol., 1995, 9,, [] J. A. Olson, Interntonl Journl of Multphse Flow, 1, 7, 1, -1. [1] B.E. Lunder, D.B. Spldng, Comp. Meth. Appl. Mech. Eng., 197,, 9-9. [] B. Kder B, Interntonl Journl of Het Mss Trnsfer, 191,, 9, [] E. S. G. Shqfeh, G. H. Fredrckson, Physcs of Fluds A-Flud Dynmcs, 199,, 1, 7-. [] P. S. Klebnoff, Ntl. Advsory Comm. Aeronut. Tech. Notes, 195, 17, 5-7. [5] P. Perlekr, D. Mtr, R. Pndt, Physcl Revew Letters,, 97, [] J. Z. Ln, X. Sh, Z. S. Yu, Interntonl Journl of Multphse flow,, 9,, [7] J. Z. Ln, X. Sh, Z. J. You, Journl of Aerosol Scence,,, 7, [] J. S. Pschkewtz, Y. Dubef, C. Dmtropoulos, E. Shqfeh, P. Mon, Journl of Flud Mechncs,, 51, AUTHORS ADDRESSES Jnzhong Ln Suhu Shen Xoke Ku Deprtment of Mechncs, Zhed Rod Hngzhou Zheng 17 CHINA Journl of Engneered Fbers nd Fbrcs 9 Volume, Issue 1 1