Molecular Motion in Isotropic Turbulence

Transcription

1 Moleular Moion in Isoropi Turbulene Jing Fan, Jian-Zheng Jiang, and Fei Fei Laboraory of High Temperaure Gas Dynamis, Insiue of Mehanis Chinese Aademy of Sienes, Being 9, China Absra Moleular moion in urbulene is a major fundamenal and engineering problem, and he lassial parile approahes are diffiul o solve i beause of a wide separaion of emporal and spaial sales beween hermal moion and urbulen fluuaion A hybrid mehod named as raer moleules in oninuum (TMIC) has been proposed o address he issues Using he TMIC mehod, moleular moion in isoropi urbulene was invesigaed, and he Reynolds number based on he Taylor miro-sale was abou 68 The TMIC alulaion shows ha raer moleules are dominaed by he hermal moion when ime is omparable o he mean ollision ime, and by he urbulen fluuaion when ime is omparable o he inegral ime sale of urbulene Beween he wo ime sales, a mied mode prevails for whih boh hermal moion and urbulen fluuaion are imporan The orresponding urbulen diffusion and visosiy oeffiiens 3 3 obained by TMIC are abou m se and 75 m se, respeively, wo orders larger han hose due o hermal moion Keywords: isoropi urbulene, raer moleule, TMIC, DS, DSMC PACS: 5+y, 477Gs I ITRODUCTIO Turbulen flows prevail in indusrial proesses and naural environmen ha grealy enhanes he mass, momenum and energy ranspor of moleules Rihard Feynman desribes urbulene as "he mos imporan unsolved problem of lassial physis" Compared o enormous sudies on urbulene from he oninuum viewpoin, he ounerpar a moleular level was muh less The laer is sraighforward in priniple using parile approahes suh as he moleular dynamis (MD) mehod or he dire simulaion Mone Carlo (DSMC) mehod, bu i is no so easy in praie A big diffiuly arises from a wide separaion of emporal and spaial sales beween moleules and urbulenes I is eremely ime-onsuming for he parile approahes o resolve urbulen flows direly wih he mean free pah ( λ m ) and mean ollision ime ( τ ) Even for a simples ase, ie homogeneous isoropi urbulene a relaively low Reynolds numbers, he MD or DSMC simulaion is beyond he apabiliies of urren superompuers A hybrid mehod named as raer moleules in oninuum (TMIC) is proposed o solve he kind of issues For moleular moion in isoropi urbulene ineresed here, he dire numerial simulaion (DS) mehod is firsly employed o solve he urbulen flow field Then a number of raer moleules are pu in he ompuaional domain of DS and raked When a raer moleule moves ino a ell of DS, he number densiy of is surrounding moleules an be obained direly from he DS soluion, and heir veloiies are assigned from a loal equilibrium disribuion whose mean veloiy is equal o he urbulen fluuaing veloiy in he ell obained by DS For gaseous medium, he raer moleule an ollide wih is surrounding moleules following a proedure similar o DSMC [] The feaures of moleular moion in he urbulene are obained hrough saisial analysis of he rajeories of he raer moleules In he presen paper, he TMIC mehod is firsly esed in raking raer moleules in a hree-dimensional bo wih periodi boundary ondiions, and he ranspor oeffiiens obained from he rajeories of he raer moleules based on he Einsein relaion and he Green-Kubo formulas are ompared o he heoreial values e, TMIC is applied o isoropi urbulene, and he mean square displaemen and veloiy variaion of raer moleules are ompared wih he lassial heory of urbulene Furher, he diffusion and visosiy oeffiiens of he isoropi urbulene are deermined from he rajeories of he raer moleules

2 II TEST OF TMIC Consider argon gas in he sandard ondiions (73K & am) and a res The ompuaional domain is a periodi bo wih he side lengh of λ m, and i is uniformly divided ino ells The hard-sphere model is employed o desribe he ineraion beween raer moleules and heir surrounding moleules Iniially, fory raer moleules are randomly pu in eah ell For eah ime sep of 3τ, a raer moleule moves and ollides wih is surrounding moleules assigned from a Mawellian disribuion in he sandard sae To es TMIC, he ranspor oeffiiens are alulaed hrough saisial averaging he rajeories of all he raer moleules as follows Aording o he Einsein relaion [], he diffusion oeffiien is proporional o he mean square displaemen of raer moleules when he ime inerval >> τ, ie {, (), (), (), () j j j j 3, j() 3, j() } D = Δr = 6 6 j= + + () Aording o he Green-Kubo formulas [], he visosiy and hermal onduiviy oeffiiens have he following epressions μ = J () ( ), J d () κ = H () ( ), i Hi d (3) where ( ) J () = m, H () = mu 5kT, u = = d d, (4) i, k j, k i i, k i, k i i i k= k= m is he moleular mass, k is he Bolzmann onsan, V and T are he volume and emperaure of he bo, respeively In he TMIC mehod, he orrelaion of a raer moleule before and afer a ollision is aken ino aoun, bu he orrelaion beween he raer moleule and is ollision parner is negleed I an be proven ha erain orreion faors are neessary for TMIC alulaing he visosiy and hermal onduiviy based on Eqs (3) and (4), and hey are equal o 4/3 and 59/3, respeively, under he hard-sphere model As shown in Fig, he ranspor oeffiiens obained by TMIC are onsisen wih he free moleular prediion when << τ, and agree well wih he Chapman-Enskog heory [3] when >> τ 5E E-5 E- E-5 E E- 7 - D(m /se) 9E-6 6E-6 μ (m - s) 5E-5 E-5 κ (WK - se - ) E- 5E-3 3E-6 5E E E FIGURE Transpor oeffiiens alulaed by TMIC Lef: diffusion; ener: visosiy; righ: hermal onduiviy TABLE Comparison of ranspor oeffiiens of argon gas in he sandard ondiions Transpor oeffiien TMIC Einsein Green-Kubo Chapman-Enskog Theory [3] D (m se) μ (m - se) κ (WK - se - ) - 7 6

3 III MOLECULAR MOTIO I ISOTROPIC TURBULECE Consider an isoropi urbulene of argon gas a he sandard ondiions The ompuaional domain is a periodi bo whose side lengh is π, and is he inegral lengh sale of urbulene The Reynolds number based on ( Re = vrms ν ) is 35, where v rms is he roo mean square (rms) veloiy of he urbulen fluuaions The urbulen Mah number ( Ma = v a ) akes a value of 7, and herefore he inompressible avier-sokes rms equaions are approimaely valid ha are numerially solved by a pseudosperal mehod for fored urbulene 3 widely used in lieraures [4, 5] The ompuaional domain is uniformly divided ino 8 ells for DS, and he 3 ime sep is se o be T, wih T = vrms Four differen sales are summarized in Table If we inrodue he Knudsen number based on he Kolmogorov sale, i is abou 6 The Reynolds number based on he Taylor mirosale ( Re = vrmsλ ν ) is abou The ell size and ime sep of DS are 39 m and 6 se, abou 6λ m and 6τ, respeively TABLE Comparison of four differen sales in isoropi urbulene The inegral sales are deermined as follows: vrms from he values of Ma and he sound speed of argon gas a he sandard ondiion, and = Re ν vrms The mean energy dissipaion rae ε = v l = m s, and he Taylor mirosale λ = 5νv ε The Kolmogorov sales rms are derived from 3 / 4 ( ν / ε), v = The mos probable speed of hermal moion m = kt m, λm = ( n σ T) and τ 9λ m m under he hard-sphere model wih he ollision ross seion σ T Sale Inegral Taylor Kolmogorov Moleule Lengh (m) Veloiy (m/s) Time (se) = = ( εν ) 4, and τ ( ν ε) 3 Mean Square Displaemen The displaemen of a raer moleule in he direion an be wrien as Δ () = [ () () + v ] d (5) If he veloiies of he hermal moion and urbulen fluuaion, () and v (), are irrelevan, hen ' ' ' Δ = [ ( ) ( )] ( ) ( ) τ + v τ τ + v τ dτdτ = Δ m + Δ (6) The firs erm on RHS has he following ea soluions in free moleular and oninuum limis [] ' ' m << τ Δ m = ( ) ( ), τ τ dτdτ = (7) D >> τ The seond erm on RHS arises from urbulen fluuaion has he epressions similar o (7), bu he haraerisi ime beomes insead of τ [6] T 3 ' ' vrms << T Δ () = v ( ) ( ) τ v τ dτdτ = vrmst 3 >> T Subsiuion of (7) and (8) ino (6), ogeher wih a kinei relaion D = 3 m τ 4, yields ( 3m + vrms) < < τ Δ = ( 9mτ+ vrms) τ<< << T ( 9mτ + 4vrmsT ) > > T rms λ (8) (9)

4 Figure shows he mean square displaemen of all raer moleules in isoropi urbulene I agrees well wih Eq (9) a he differen ime sales -3 Tubulene <Δ >/(m s - ) -4-5 Free moleule -6 TMIC Thermal moion & urbulene mied FIGURE Variaion of he mean square displaemen of raer moleules in isoropi urbulene ( 3 Mean Square Veloiy Variaion Re = λ 68 ) wih ime The omponen of he mean square hermal veloiy variaion of a raer moleule an be wrien as [ ] ( + ) ( ) = ( + ) + ( ) ( + ) ( ) () In an equilibrium sae, we have ζ ( + ) ( ) = ( ) e, ( ) = ( + ) = 5m () where ζ = kt ( md) = ατ, and α = 3 Subsiuion of () ino () yields [ ] ( + ) ( ) = m( e α τ ) () Aording o he Kolmogorov s small-sale similariy heory [6], he omponen of he seond-order auoorrelaion funion of urbulen fluuaion veloiy has an epression as follows [ v ( ) v ( )] ( τ ) 5 5 Aε ν << Av << τ i + i = = C ε << << T Cv τ << << Tl are onsans where A and C The veloiy of a raer moleule due o hermal moion and urbulen fluuaion veloiies are assumed irrelevan, hen we have α τ ( ) ( τ ) α τ ( ) τ m e + Av << τ Δ u = [ i( + ) i( ) ] + [ vi( + ) vi( ) ] = (4) m e + C v τ << << T Figure 3 shows he mean square veloiy variaion of all raer moleules in isoropi urbulene I is in good agreemen wih Eq (4) when A = 5, and C = See Ref [7] for a deailed disussion on he values of A and IV TURBULET DIFFUSIO AD VISCOSITY COEFFICIETS The Einsein relaion () is no only applied o hermal moion, bu also o urbulen fluuaion Based on he relaion, he diffusion oeffiien of isoropi urbulene an be obained from he mean square displaemen of 3 raer moleules in he long ime limi From Fig, i is abou m se when Re = λ 68 This value is wo 5 orders larger han he diffusion oeffiien due o hermal moion ( 43 m se ) The Green-Kubo formula for visosiy is eended o apply for isoropi urbulene Under he assumpion ha he hermal moion and urbulen fluuaion veloiies of raer moleules o be irrelevan, Eq (3) is simplified as (3) C

5 <ΔV >(m /s ) TMIC Thermal moion Turbulen fluuaion FIGURE 3 Variaion of he mean square veloiy of raer moleules in isoropi urbulene ( where Re = λ 68 ) wih ime μ = μ + μ, (5) m m m m m J J d J mi, kj, k k = μ = J () ( ), ( ) J d J mvi, kvj, k k = μ = () ( ), ( ) =, (6) = (7) In he presen DS alulaion of fored urbulene, he values of kinei energy are fied for he wave number below 4 Suh a ommon reamen leads o an anisoropi effe on he large sale Eq (7) is modified as follows o redue he effe μ = C () (), () () (), C d C = J J (8) wih M J () = J () (9), k M k = Figure 4 shows he evoluion of visosiy oeffiiens as well as C () in isoropi urbulene The mean value 3 of he visosiy oeffiiens due o urbulen fluuaions is abou 75 m se when Re = λ 68 I is wo orders 5 larger han he hermal visosiy oeffiien of argon gas ( m se ) Beause of he anisoropy of he fored urbulen on he large sale, he urbulen visosiy oeffiiens are somewha differen in he hree direions E- 45E-6 35E-6 μ (m - s) 8E-3 4E-3 μ y C() (kg m -4 s - ) 5E-6 5E-6 C y C yz C z μ yz μ z 5E-7 E+ E-6 4E-6 6E-6 8E-6 E-5-5E-7 E-6 4E-6 6E-6 8E-6 E-5 FIGURE 4 Variaion of he visosiy oeffiiens and heir orrelaion funions in isoropi urbulene ( Re = λ 68 ) wih ime

6 V DISCUSSIOS The TMIC mehod provides a ool o observe and undersand moleular moion in urbulene For an isoropi urbulene ase sudied here, ypially shown by Fig and Eq (9), raer moleules are dominaed by he hermal moion when ime is omparable o he mean ollision ime, and by he urbulen fluuaion when ime is omparable o he inegral ime sale of urbulene Beween he wo ime sales, a mied mode prevails for whih boh hermal moion and urbulen fluuaion are imporan Compared wih he noion of fluid elemen widely used in fluid mehanis, moleules are a realiy, and he ineraion beween hem, a leas for gases in a room sae, are learly undersood Figure and Eq (9) sugges an analogy beween urbulen fluuaion and hermal moion: he roo mean square veloiy, inegral lengh and ime, in he viewpoin of diffusion, orrespond o he hermal veloiy, mean free pah and mean ollision ime, respeively A leas for isoropi urbulene, suh an analogy has a rue physial meaning More ompliaed urbulen flows are omposed by "eddies" of differen sizes A asade proess was suggesed by Rihardson [8]: Big whirls have lile whirls, whih feed on heir veloiy; lile whirls have smaller whirls, an so on o he visosiy In his way, he energy is passed down from he large sales of he moion o smaller sales unil reahing a suffiienly small lengh sale suh ha he visosiy of he fluid an effeively dissipae he kinei energy ino inernal energy To a erain een, a universal piure for moleular moion in he asade proess is equivalen o he losure problem of urbulene in oninuum mehanis This is a very imporan issue ha needs more effors in fuure ACKOWLEDGMETS This work was suppored by he aional aural Siene Foundaion of China under Gran o 96 The auhors hank Professor X L Li who provided a DS ode and valuable disussion on isoropi urbulene REFERECES G A Bird, Moleular Gas Dynamis and he Dire Simulaion of Gas Flows, Oford: Clarendon Press, 994 DAMquarrie, Saisial Mehanis, ew York: Harper & Row, 973, pp S Chapman and T G Cowling, The Mahemaial Theory of on-uniform Gases, 3 rd ed, Cambridge: Cambridge Univ Press, 97 4 S A Orszag and G S Paerson, Phys Rev Le, 8, 76 (97) 5 T Ishihara, T Gooh, Y Kaneka, Ann Rev Fluid Meh, 4,6-8 (9) 6 A S Monin, A M Yaglom, Saisial Fluid Mehanis, Cambridge: MIT Press, P K Yeung, Annu Rev Fluid Meh, 34, 5 4 () 8 LF Rihardson, Wheher Prediion by umerial Proess, Cambridge: Cambridge Univ Press, 9