Direct Numerical Simulation of Particles-Bubbles Collisions Kernel in Homogeneous Isotropic Turbulence

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1 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n Hassan E. Fayed and Saad A. Ragab* Department of Engneerng Scence and Mechancs Vrgna Tech, Blacksburg, VA Receved: January, Accepted: June Abstract Partcles and bubbles suspended n homogeneous sotropc turbulence are tracked and ther collsons frequency s determned as a functon of partcle Stokes number. The carrer phase velocty fluctuatons are determned by Drect Numercal Smulatons (). The effects of the dspersed phases on the carrer phase are neglected. Partcles and bubbles of szes on the order of Kolmogorov length scale are treated as pont masses. In addton to Stokes drag, the pressure gradent n the carrer phase and addedmass forces are also ncluded. Equatons of moton of dspersed phases are ntegrated smultaneously wth the equatons of the carrer phase usng the same tme steppng scheme. The collson model used here allows overlap of partcles and bubbles. Smulatons for three turbulence Reynolds numbers Re λ = 7, 77, and 9 have been performed. Collsons kernel, radal relatve velocty, and radal dstrbuton functon found by are compared to theoretcal models over a range of partcle Stokes number. Comparsons are made wth Zachk et al. [] model, whch s applcable to heavy partcles, and Zachk et al. [] model whch s vald for an arbtrary Stokes number. Zachk et al. [] s essentally a model for the radal relatve velocty, and for the purpose of computng the collson kernel, t assumes the radal dstrbuton functon to be one. In general, good agreement between and Zachk et al. models s obtaned for radal relatve velocty for both partcle-partcle and partcle-bubble collsons. The results show that around Stokes number of unty partcles of the same group undergo expected preferental concentraton whle partcles and bubbles are segregated. The segregaton behavor of partcles and bubbles leads to a radal dstrbuton functon that s less than one. Exstng theoretcal models do not account for effects of ths segregaton behavor of partcles and bubbles on the radal dstrbuton functon. 7. INTRODUCTION Collsons frequency of dspersed phases suspended n a turbulent carrer phase s a decdng factor n many ndustral processes such as mnerals flotaton, lquds purfcaton and nuclear power generaton. In ths paper, we are nterested n collsons frequency of sold partcles and gas bubbles n a turbulent lqud moton. The partcle and bubble szes are on the order of the small scales of turbulent fluctuatons, and hence those scales have strong effects on the moton of colldng partcles. Rogallo and Mon [], among others, have demonstrated the vablty of drect numercal smulatons () for reproducng expermentally obtaned data for many turbulent flows. In the present paper, turbulent fluctuatons of the carrer phase have been determned by. Two well known theoretcal models for collsons frequency are due to Saffman and Turner [] and Abrahamson []. Saffman and Turner model s vald for colldng partcles whose response tmes are small n comparson to Kolmogorov tme scale of the turbulence, that s to say, n the lmt of zero Stokes number. Abrahamson model s applcable for cases where nerta effects of the colldng partcles are domnant, whch s the case for very hgh Stokes number. However, most of *Correspondng author. Emal: ragab@vt.edu

2 8 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n the ndustral applcatons nvolve partcles of fnte Stokes number. Applcatons of the prevously mentoned collsons models for practcal Stokes numbers ether overestmate or underestmate the collsons frequency. Consderable efforts have been expended to develop a more general collson model that s applcable n the entre range of partcles Stokes number. Wllam and Crane [] developed a model for the relatve moton of two partcles or droplets of an ntermedate Stokes number n gaseous turbulent flow. The added mass effect experenced by the partcles movng through the carrer phase were neglected and Stokes drag law was used. Yuu [] derved a model for the fluctuatng relatve velocty of two nertal partcles. Ths model takes nto consderaton the added mass effect on the partcles moton as well as the turbulent shear effect. Krus and Kusters [8] found that both models of Wllam and Crane [] and Yuu [] are not applcable for partcle szes on the order of the nertal subrange of turbulence and the models fal to approach the zero Stokes lmt accordng to Saffman and Turner [] model. The approaches of Wllam and Crane [] for large partcles and that of Yuu [] for small partcles have been ncluded n one model by Krus and Kusters [8] to formulate a unversal collsons model. Ths new model s based on a very low mass loadng, sotropc turbulence, Stokes drag law and for partcles that are larger than the mean free path of the flud. Drect numercal smulaton of sotropc turbulent flow enables drect calculatons of collsons frequency and proved to be very useful for valdaton of theoretcal models. Sundaram and Collns [] demonstrated that collsons frequency of mono-dspersed nertal partcles suspended n dynamc sotropc turbulent flow s a functon of partcle Stokes number and bounded from below by Saffman and Turner [] model and from above by Abrahamson model []. These results have been confrmed by Zhou et al. []. Wang et al. [9] studed the collsons frequency of mono-dspersed partcles n a frozen sotropc turbulent flow. Collsons frequency from are compared wth the results of Krus and Kusters [8]. It s noted that ths model underpredcts the collsons frequency for partcles Stokes number greater than unty [Wang et al. [9]]. All of the smulatons conducted by Wang et al. [9] and Zhou et al. [7] allow overlap of partcles durng collsons. Whereas, Sundaram and Collns [] consdered hard sphere perfectly elastc collsons. Two mechansms contrbute to collsons rate between two dfferent speces of partcles n turbulent flows: the transport mechansm and the accumulaton effects. The accumulaton effects are accounted for by the radal dstrbuton functon at contact and the transport effects are accounted for by the mean value of the partcles radal relatve velocty []. Zhou et al. [7] extended ther smulatons of partculate sotropc turbulent flows of mono-dspersed nertal partcles to b-dspersed partcles. They derved a new model called eddy partcle nteracton (EPI) for hgh nerta partcles. The flud velocty n ths model was treated as a Monte-Carlo stochastc process wth a fxed eddy lfetme as a Gaussan. The developed EPI model can predct the collson kernel for partcle response tme n the order of eddy lfe tme scale. The model does not gve reasonable predcton for collsons kernel near Stokes number of unty. In calculatons partcles trajectores are calculated by solvng the equaton of moton for each partcle where drag s the leadng order term. Another way to descrbe the dspersed phase moton n a turbulent flow theoretcally s to solve the knetc equaton for the partcle velocty probablty densty functon. Zachk and Pershukov [] solved a stochastc equaton of Langevn type to a statstcal descrpton of a partcle ensemble. Ths model takes nto account both the nteracton between the partcles and flud fluctuatons and partcle-partcle collsons effects. Zachk and Alpchenkov [] developed two theoretcal models to predct the effects of clusterng of nertal partcles n n-homogeneous and homogeneous turbulent flows. The frst model s based on a one-pont PDF of partcle velocty dstrbuton n Gaussan random feld. Ths model predcts ntensty of fluctuatons and concentraton of partcles n flat channel and round ppe flow. The second model s based on two-pont PDF of partcle velocty dstrbuton n random Gaussan feld. Ths model predcts relatve velocty of partcle pars and ther radal dstrbuton velocty n homogeneous sotropc turbulence. Zachk et al. [] have developed a model to predct the collsons frequency between two dfferent nerta partcles. Ther approach nvolves solvng a system of three nonlnear ordnary dfferental equatons that descrbe the radal dstrbuton functon, longtudnal and transverse velocty structure functons. Mean radal relatve velocty of Journal of Computatonal Multphase Flows

3 Hassan E. Fayed and Saad A. Ragab 9 the partcles, radal dstrbuton functon and collson kernel of the model show qualtatve agreement wth results by Zhou [7] that were obtaned for partcle trackng n a frozen turbulent flow feld. Fayed and Ragab [] also studed the collsons of mono- and b-dspersed phases n a frozen turbulent flow feld wth ten dfferent flow feld realzatons nstead of the same turbulent flow feld wth dfferent random ntal postons for the partcles. Ther results show that large uncertantes exst n the collsons kernel for both mono- and b-dspersed phases especally around partcles Stokes number of one. Consequently, dynamc (.e. tme evolvng) turbulence smulatons are hghly needed for relable valdaton of theoretcal models that predcts collsons frequency. The objectve of ths paper s to provde data for the collson kernel of partcles and bubbles n tme evolvng homogeneous sotropc turbulnece, and to compare wth theoretcal models by Zachk el al. [] and []. Ths paper s organzed as follows: n Secton () numercal methods for the carrer phase and moton of the dspersed phases are presented. Methods for drect calculatons and estmaton of collsons kernel, radal dstrbuton functon and radal relatve velocty are presented n Secton (). A summary of theoretcal collsons rate models s gven n Secton (). results and comparsons wth theoretcal models are presented n Secton (), and summary and conclusons are gven n Secton ().. NUMERICAL METHODS.. Carrer Phase Model - The three phase flow s composed of a carrer phase, whch s treated as an ncompressble flow (lqud), and two dspersed phases whch are sold partcles and gas bubbles. The lqud flow s a homogeneous sotropc turbulence n a perodc cube of sde π. The carrer phase s governed by the ncompressble Naver-Stokes equatons.. r u = r u + r r = + r + r u. u p ν u F f t ρ f () () Heren u s the nstantaneous velocty feld, ρ f s the densty of the carrer phase and p(x, t) s the pressure feld. Vscous effects dsspate the turbulent knetc energy, and n the absence of external forcng the turbulence wll decay. Therefore, a forcng term F s added to the RHS of Eq () to mantan statstcally statonary turbulent flow feld. The forcng vector s constructed n such a way as to make t dvergence free, hence.f =. The Fourer-Galerkn approach [] has been used to rewrte Naver-Stokes equatons n the Fourer space r r k.ˆ u r k = () () where the term ˆ C = ( u. u) k k () s the convectve nonlnear term that s treated by a pseudo-spectral method and /-rule for dealasng. The pressure s determned by projectng Eq () onto the wavenumber vector, and then usng the contnuty equaton, Eq (). The Eq () can be rewrtten as Volume Number

4 7 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n () Followng Rogallo [] we use an ntegraton factor e ν f k (t t n ) to ntegrate the vscous term n Eq () analytcally. Therefore, the lmtaton on the tme step for stablty s determned by the convectve term. A low-storage thrd-order Runge-Kutta method [] s used for tme advancement. Statstcally steady turbulence s acheved by forcng a range of low wavenumbers, < k K f. The method of forcng s gven by Eswaran and Pope []. In ths method, the acceleraton of all Fourer modes n the forced range s augmented by random (n tme) components, whch are gven by Ornsten-Uhlenbeck (OU) stochastc processes (e.g. Gllespe []). The OU process s defned by ts standard devaton σ and relaxaton tme scale τ. In summary the forcng term n Eq () takes the form r r r r ˆ ˆ.ˆ r Fr br k kbr k = k k r k (7) where where n denotes the tme step, and Rˆ k s a complex-valued vector whose components are normally dstrbuted (Gaussan) random numbers wth zero mean and unt varance (e.g. Press et al. []). The forcng term s updated at the begnnng of each tme step usng Eqs (7) and (8), but t s held constant durng the sub-steps of the tme ntegraton scheme. For the constructon of the forcng term n the spectral doman, t s mportant to satsfy r r r r ˆ ( ) ˆ * Fr k = Fr ( k ). k k (8) D energy spectrum s calculated from the Fourer modes by summng the energy over modes N n sphercal shells of rad, where N k s the maxmum possble wave number, and radal n thckness dk n. The D nstantaneous energy spectrum s then averaged over tme. Turbulent velocty r.m.s (u ) and turbulent dsspaton rate (ε) are calculated from the energy spectrum k max u q = E() k dk (9) ε = ν f k max kekdk () () where ν f s the flud knematc vscosty. The Lagrangan ntegral length scale L f s calculated as L f π = u k max Ek () dk k () Journal of Computatonal Multphase Flows

5 Hassan E. Fayed and Saad A. Ragab 7 The product of k max η should be greater than. to guarantee that the smallest turbulent scales (.e. Kolmogorov mcroscales) are well resolved. The computatonal doman s a cube of sde B. We ntroduce a reference length L r = B/π, and a reference velocty V r. Snce the flow s ncompressble, these two reference values are suffcent to non-dmensonalze all dependent (velocty) and ndependent varables (tme and space coordnates). The nput parameters that defne the forced sotropc turbulence are: number of grd nodes n each space drecton, N, range of forced Fourer modes, K f, knematc vscosty, ν f, standard devaton, σ, and tme scale, τ, of Ornsten-Uhlenbeck stochastc process. All of these parameters are made non-dmensonal by L r and V r. Dfferent smulaton parameters are summarzed n Table (). The spectra have been valdated by comparsons wth spectra obtaned by Eswaran and Pope [] for Re λ =. (n nterest of space, the comparsons are not shown here). Statstcally steady carrer phase turbulence fluctuatons have been establshed frst before trackng the dspersed phases. Afterwards, the equatons of moton of partcles and bubbles are advanced smultaneously wth the carrer phase equatons for an addtonal N t tme steps (Table ()), and collsons statstcs are collected. The D energy and dsspaton spectra for three Reynolds numbers Re λ = 7, 77, and 9 are shown n Fgures (a c). In the three smulatons presented here the maxmum resolved wavenumbers are k max η =.,., and.9. The spectra show that dsspatve scales are well resolved... Dspersed Phase Model A system of sold partcles and gaseous bubbles are tracked by solvng Newton s equaton of moton. No lmtatons are mposed on the partcle densty. Partcles and bubbles are treated as pont masses and Stokes drag law s used. The added mass and pressure gradent terms are accounted for n the equatons of moton. r r r dv () t u V() t r r f Du f Du = r ρ ρ dv t + + C () A dt τ ρ Dt ρ Dt dt r dx () t r = V() t dt () () where X (t) and V (t ) are the nstantaneous poston and velocty of partcle ( = p for sold partcles, and = b for bubbles). u and Du /Dt are the carrer phase velocty and acceleraton Volume Number Table. Isotropc Turbulence Smulaton Parameters. Smulaton A B C grd sze N 8 knematc vscosty ν f.7.. OU process σ.9.. OU process τ... forced wavenumbers K f tme step t... turbulence Reynolds number Re λ Kolmogorov length scale η.8..9 dsspaton rate ε velocty fluctuaton (rms) u Taylor mcroscale λ.7..9 ntegral length scale L f...98 resolved wavenumbers k max η...9 number of tme steps for collson statstcs N t

6 7 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n Energy Dsspaton E (K), ε (K) K Λ (a) Re λ = 7, N = Energy Dsspaton E (K), ε (K) E (K), ε (K) K η (b) Re λ = 77, N = 8 Energy Dsspaton K η (c) Re λ = 9, N = Fgure. D energy and dsspaton spectra. Journal of Computatonal Multphase Flows

7 Hassan E. Fayed and Saad A. Ragab 7 evaluated at the nstantaneous partcle center. On the rght sde of Eq (), the frst term s the Stokes drag, the second term s the effect of pressure gradent n the carrer phase, and the thrd term s the effect of added (vrtual) mass. The lft and Basset (or hstory) force are neglected n ths study bascally to be consstent wth the collson frequency theoretcal models. There s no dffculty n ncludng the lft force as t depends on the local vortcty vector and shear that are readly avalable n the course of smulaton. The Basset force s mportant for gas bubbles n hgh frequency lqud turbulence as supported by expermental data (e.g. L Espèrance et al. [9]). Hjelmfelt and Mockros [7] showed analytcally that the added mass force, pressure gradent, and Basset terms are all mportant for bubble moton n lqud turbulence. Ther results also showed that neglectng only the Basset force can stll produce reasonable ampltude and phase of bubble moton. However, the effects of the Basset force on the mean moton of partcles and bubbles may not be sgnfcant as shown theoretcally by Ahmad and Goldschmdt []. Incluson of the Basset force n partcle trackng requres hgh computng resources especally when large numbers of partcles and bubbles are used as n ths study. Van Hnsberg and Boonkkamp [] have developed an effcent method for computng the Basset force that mght be helpful n future studes of the effects of such a force on collson frequency. Assumng sphercal partcles, we take C A =.. τ = ρ d /8µ s the partcle response tme. Rearrangng terms, we rewrte Eq () as where r r r dv () t u V() t A Du r = + dt * τ Dt τ * τ +C A ρ f ρ () () and A = ρ + C / ρ A + C f A () For gaseous bubbles n lquds ρ b /ρ f, and hence Eqs () and () gve τ * b = C d b A 8ν f (7) and A b = + /C A (8) Thus for sphercal bubbles wth C A =., we get A b =. If we neglect the effects of pressure gradent then A b =. On the other hand, for heavy partcles (ρ p /ρ f ), we get A p. Partclepartcle and partcle-bubble nteractons durng collsons are neglected. The partcle response tme defned by τ = ρ d /8µ f s a measure of the partcle nerta. Partcle Stokes number s the rato of the partcle response tme to the Kolmogorov tme scale St = τ τ k (9) In Eq () flud velocty and acceleraton are needed at the nstantaneous partcles centers. Snce we have spectral representaton of the flud velocty, ts dervatves can be calculated accurately at the grd nodes. To attan hgh-order spatal accuracy for the flud velocty, an ncomplete cubc Hermte nterpolaton s appled n the physcal space. Ths nterpolaton requres the flud velocty Volume Number

8 7 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n Table. Partcles Parameters. Smulaton I II III IV Turbulence Reynolds number Re λ Number of partcles (P ) N p Number of bubbles (or partcles P ) N b Partcle dameter d p /η Bubble dameter d b /η components and ther spatal frst-order dervatves at the eght corners of the cell contanng the partcle center. In addton to the values of a velocty component there are three spatal dervatves at each corner, and hence there are degrees of freedom for the velocty nterpolaton. As a byproduct of usng the /-rule for de-alasng, the velocty and ts gradent become avalable on a fner grd (N/) n the physcal space. The flud velocty s nterpolated on the fner grd. For the acceleraton nterpolaton, Du /Dt, a tr-lnear nterpolaton s used on the orgnal grd (N ). Table () summarzes the dfferent partcle parameters used. All smulatons (I) to (IV) were conducted n an evolvng turbulent flow feld. For each smulaton, partcle and bubble dameters are fxed and the partcle Stokes number s vared by changng the partcle response tme, whch allows us to study the effects of Stokes number on radal dstrbuton functon, radal relatve velocty and collson kernel. The effect of bubble sze s studed n ths paper through the comparson of smulaton (II) and (III).. COLLISIONS FREQUENCY CALCULATION.. Collsons Kernel Geometrc partcle-partcle and partcle-bubble collsons are consdered n ths paper where the partcles and bubbles szes are on the order of Kolmogrov scale. Two dfferent methods for calculatng collsons kernel of mono-dsperse and b-dsperse phases are used and compared. The frst method calculates the collsons rate by drect countng as recommended by Sundaram et al. []. As depcted n fg () three dfferent types of collsons are shown. In type (I) collsons, the two partcles are completely separated at the begnnng of the tme step and overlapped at the end of the tme step. Some other partcles may be separated at the begnnng and end of the tme step but overlap durng the tme step. Ths type of collsons s denoted as type (II). In the thrd type of collsons, the two partcles are overlapped at the begnnng and the end of the tme step but separated at some nstant durng the tme step. It s clear from these three dfferent scenaros that a collson event s consdered f the two partcles were separated before gettng overlapped, whch means that f these two partcles reman overlapped durng the tme step no collson s consdered. Wang et al. [7] assumed that the t(n) t(n ) Type I Type II Type III No collson Fgure. Collson types. Journal of Computatonal Multphase Flows

9 Hassan E. Fayed and Saad A. Ragab 7 second and thrd types of collsons can be neglected snce the tme step s small enough. However, n ths paper, only type (III) s neglected. Type (I) s drectly computed based on the current and former postons of the partcles to check for overlap and non-overlap at the begnnng and end of the tme step. For those partcles that were separated at the end and begnnng of the tme step, type (II) of collsons s checked. Based on the ntal relatve poston and radal relatve velocty of the colldng partcles, the tme requred for collson s estmated by solvng Eq () () where ro and v o are the ntal partcle poston and velocty vectors at the begnnng of the tme step. d = (d + d j )/ s the collson radus of the two colldng partcles. A collson event s recorded f the soluton of Eq () for t c s postve and less than or equal to the computatonal tme step ( t c t), Sundaram et al. []. Collson frequency s defned as the number of (partcle-partcle or partcle-bubble) collsons per unt tme per unt volume. Drect countng of partcles-partcles collsons nvolves O(N p ) operatons. Consderng the hgh number of partcles N p ths process s computatonally expensve. Instead, the computatonal doman s dvded nto smaller sub-domans such that the search for partcles encounterng a certan partcle wll be n ths smaller doman. Accordng to Sundaram and Collns [] the savngs n computaton tme are proportonal to the number of sub-domans of the system of partcles. In our study we used sub-domans. The collsons count are averaged over tme and normalzed by the number densty of the colldng speces to fnd the collsons kernel as shown n Eq () and Eq (). Collsons kernel for partcle-partcle and partcle-bubble calculated by drect countng can be wrtten as, β pp = N& pp nn p p () βpb = N pb nn p b () where N. pp and N. pb are the spatal average of collson frequency and n P and n b are the number densty of the colldng partcles and bubbles. Two man statstcal varables contrbute to the collson kernel; namely the average radal relatve velocty of the two colldng speces w r (d) and radal dstrbuton functon at contact g pp or g pb. As noted by Wang et al. [8] and Wang et al. [9], average radal relatve velocty of the two colldng speces w r (d) s a measure of the transport effects, and radal dstrbuton functon at contact g pp or g pb s a measure of the accumulaton effects. The second method for estmatng collsons kernel uses these two statstcal varables. Followng Wang et al. [8] and Wang et al. [9] a sphercal formulaton s used whch s more accurate than the cylndrcal formulaton. The estmated collson kernel s calculated accordng to Eq () for partcle-partcle and Eq () for partcle-bubble collsons. The followng subsectons dscuss the calculaton of RDF and radal relatve velocty of the colldng speces. β ( est.) = πd w ( d) g pp r pp () β = πd w () d g pb( est.) r pb () Volume Number

10 7 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n.. Radal Dstrbuton Functon and Radal Relatve Velocty Radal dstrbuton functon (RDF) at contact s a measure of the average colldng partcles concentraton. Number of colldng pars are determned wthn a sphercal shell as recommended by Wang et al. [9]. RDF s the rato of spatal average of number of pars at contact per unt volume of the sphercal shell (n lm ) to the average number of possble pars per unt volume of the computatonal box N N l m where partcles are assumed to be unformly dstrbuted wthn the Vbox computatonal doman. g lm n lm () d =, l =, and m=, N N l V box m () where (d = (d m + d l )/) s the collson radus. Index () denotes partcles and () denotes bubbles. Repeated ndces mply a mono-dsperse case. N l and N m, s the total number of partcles and bubbles n the computatonal doman, respectvely. Wang et al. [9] used sphercal shell of radal thckness r to be d δ r d and < < + δ δ chosen to be %. Each tme step, total number of pars that exsts wthn the sphercal shells s montored then averaged over the total number of tme steps. Eq () s rewrtten as g lm N V ()= d V N N N lm total pars box shell t l m () where (N t ) s the total number of tme steps used to count for the total number of pars at contact, and (V box = (π) ). In ths study, the two colldng partcles are not necessarly of equal szes as gven n Table. Partcles and bubble are allowed to overlap whch s consstent wth exstng theoretcal models. Under ths condton, the calculatons of RDF and ( w r (d) ) are senstve to the shell nner and outer rad. It s found that the use of a sphercal shell wth radal dstance r n the range d δ r d underpredcts the radal dstrbuton functon. Ths s due to some < < + δ partcles or bubbles that contrbute to the collsons kernel n the drect countng are beng overlapped whle ther centers fall outsde the shell, and hence they do not contrbute to the radal dstrbuton functon. Instead, we use a sphercal shell of radus n the range mn ( r, r r d to capture the contrbuton of such partcles and obtan a more p p)< < δ + realstc values for the radal dstrbuton functon (RDF). However, the radal relatve velocty ( w r (d) ) has to be calculated wthn a sphercal shell as recommended by Wang et al. [9] as t counts for the mass flux of the colldng partcles as computed n Eq () and Eq () at the collson radus. In ths paper we used δ = %. The rato of the statstcally estmated kernel to the drect computed kernel are depcted n fgures a d, the rato s close to one except for Stokes number near unty where the devaton from unty s around %.. THEORETICAL MODELS FOR COLLISIONS KERNEL Theoretcal models for collsons frequency of two groups of partcles suspended n homogeneous sotropc turbulence have been developed by many authors. The partcle nerta s quantfed by the Journal of Computatonal Multphase Flows

11 Hassan E. Fayed and Saad A. Ragab Partcle-partcle Partcle-bubble. Partcle-partcle Partcle-bubble β est./β β est./β (a) Re λ = 7, N = Partcle-partcle Partcle-bubble β est./β β est./β.8... (b) Re λ = 77, N = 8, smulaton (II) 8 Partcle-partcle Partcle-bubble stp (c) Re λ = 77, N = 8, smulaton (III) (d) Re λ = 9, N = Fgure. Rato of estmated collson kernel to drectly counted collson kernel. rato of the partcle response tme τ to the Lagrangan ntegral tmescale T L of the carrer flud. Accordng to Zachck et al. [], when the added mass force s consdered, the effectve partcle response tme τ * = τ +C A ρ f ρ (7) should be used nstead of τ. Thus, the partcle nerta s quantfed by Ω = τ * /T L (8) The collson kernel models presented n ths paper can be wrtten n non-dmensonal form β ud d d ρ ρ = f λ η, η, ρ, ρ,re f f (9) Volume Number

12 78 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n where u =( u k u k /) / s the turbulence ntensty, η = (ν f /ε)/ s the Kolmogorov mcroscale, d = (d + d )/ and Re λ = u λ/ν f s the Reynolds number based on the Taylor mcroscale λ = (ν f u /ε) /. It s mportant to note that the collson kernel depends on four ndependent propertes of the colldng partcles; namely ther szes relatve to Kolmogorov mcroscale and denstes relatve to the carrer phase densty. Stokes number defned by St ρ = 8 ρ f d η () may be ntroduced as a combnaton of densty and sze but, n general, t s not suffcent to reduce the number of ndependent varables from four. Isotropc homogeneous turbulent fluctuatons of carrer phase are also characterzed by three ndependent parameters, namely u, η and Re λ. Alternatvely, we can use u, ε, and ν f. The non-dmensonal collson kernel s related to the non-dmensonal mean radal velocty of colldng partcles by () Zachk et al. s [] model provdes a value for g j (d), however, all other models presented here assume that g j (d) =. For the latter models the non-dmensonal collson kernel s equal to the non-dmensonal mean radal relatve velocty, () Therefore, assessment of dfferent models should be based on comparson wth drect numercal smulatons for the radal relatve velocty nstead of the collson kernel... Saffman-Turner Model For zero-nerta partcles (Ω =Ω = ), the collson kernel s gven by Saffman and Turner [] model β ST π = 8 ε ν f / d () and n non-dmensonal form β ST ud π = 8 / Re λ d η () where d = (d + d )/. Assumng g =, we obtan a non-dmensonal mean radal relatve velocty wr() d u ST = 8π π / d Re λ η () Journal of Computatonal Multphase Flows

13 Hassan E. Fayed and Saad A. Ragab 79 For partcles-bubbles collson, we have d = (d p + d b )/... Zachk-Smonn-Alpchenkov Model- Zachk et al. [] presented two statstcal models for predctng collsons kernel of bdsperse heavy partcles n homogeneous sotropc turbulence. In the present paper we wll use the second model and refer t as Zachk- model. Strctly speakng, applcaton of ths model should be lmted to heavy small partcles. Its applcaton to the case of partcle-bubble collsons may be questonable, but wll be presented here only for reference. The collson kernel β Z s gven by β 8πS = u Z pll ud () d / g j () d () where S pll s the longtudnal component of the partcle velocty structure functon evaluated at the collson radus d = (d + d )/. For ths model the non-dmensonal mean radal relatve velocty s (7) To compute the collsons kernel, a two-pont boundary-value problem governed by three coupled nonlnear ordnary-dfferental equatons must be solved for partcle velocty structure functons and radal dstrbuton functon, Zachk et al. []. The system of three equatons are replaced by an equvalent system of fve frst-order equatons that are solved usng MATLAB. To execute the model and obtan the non-dmensonal kernel and relatve velocty, we must specfy four parameters, namely the partcles Stokes numbers St and St, the collson radus d/η, and Taylor mcro-scale Reynolds number Re λ of the carrer phase. The partcle densty does not appear explctly n the model; ts effects appear only through the partcle Stokes number... Zachk-Smonn-Alpchenkov Model- Zachk et al. [] developed a statstcal model for the collson kernel of b-dsperse partcles. The model apples to arbtrary values of densty rato and partcle szes. They defned a partcle-to-flud densty parameter A = ρ + C / ρ A + C f A (8) In Zachk et al. [], no dstncton between A and A s made, and hence the parameter A appears unsubscrpted n ther formulas. In the present paper, we are nterested n the collsons of sold partcles wth ar bubbles n lqud, and hence the value of A p s sgnfcantly dfferent from that of A b. Thus the formulas presented here take nto consderaton the dfferent values of the partcle-to-flud densty parameter for the two groups of colldng partcles. For ths model the nondmensonal mean radal relatve velocty s (9) Volume Number v A z u = Ω + Ω z Ω Ω ζ = ξ ξ Fd () / () ()

14 8 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n Ω + AΩ + z ξ = Ω A Ω z Ω Ω [( + + )( + + z )] / where C =, and z = τ T /T L. Zachk et al. [] provded T L τ T ( Re + = λ ) ε 7 ν f λ = Re a / d / Fd () = η exp ( d / η) ( C) / ( d / η) + ( Re / C) λ / 7 a = + Re + Re ε ν f λ λ / / / () () () () (). THEORETICAL MODELS ASSESSMENT.. Radal Dstrbuton Functons It s well known that preferental concentraton of nertal partcles occurs for Stokes numbers near unty ( Wang et al. [7], Zhou at al. [7]). Due to centrfugal acceleratons, partcles accumulate n hgh-stran regons between vortex cores. On the contrary, gas bubbles n lqud tend to mgrate nto vortex cores. Hence partcles-bubbles segregaton s possble, and t can result n radal dstrbuton functons beng less than one. As an example, heavy partcles and gas bubbles suspended n the feld of a steady column vortex wll be completely segregated, where partcles wll be expelled from the vortex core whle bubbles wll accumulate n the vortex core. In ths deal stuaton, the radal dstrbuton functon at collson for partcle-bubble collsons wll be nearly zero. Thus, partcles-bubbles segregaton has an mportant effect on the partcle-bubble collsons rate. To demonstrate ths phenomenon, we consder collsons of sold partcles (mono-dsperse) and sold partcles wth gaseous bubbles (b-dsperse) at turbulence Reynolds numbers Re λ = 7, Re λ = 77 and Re λ = 9, as presented n Table () and Table (). To llustrate the accumulaton of the monodsperse and b-dsperse phases, snapshots of partcles and bubbles are pcked for Re λ = 9. The partcle Stokes number s St p =.9. Partcles postons whose centers fall n a thn slce of thckness d b that s parallel to the xy plane of the computatonal box are projected onto the mdplane of that slce. The projected centers and enstrophy contours are shown Fg (a), where the partcles are shown by small whte crcles. The red sles are regons of hgh vortcty whle the blue sles are regons of hgh stran rate. The fgure demonstrates the well known preferental concentraton (accumulaton) of partcles at Stokes number near unty. Partcles accumulate n the sles of hgh stran rate between vortex cores. Bubbles postons whose centers also fall n the same slce are projected on the same plane at the same nstant of tme and are shown n Fg (b). We recall that the bubble dameter n ths smulaton s d b =.88η. The bubbles tend to concentrate n regons of hgh vortcty n and around vortex cores. The relatve postons of partcles and bubbles n the slce are shown n Fg (c), they are partally segregated. The segregaton s enhanced by the pressure gradent term that forces the bubbles toward the low pressure regons. Comparsons between and theoretcal models wll be presented next. The collson results are obtaned by fxng the bubble and partcle dameters as gven n Table (), and varyng the partcle densty. The partcle Stokes number, St p, s then computed from Eq (). The same procedure s appled when computng collson kernel, RDF and radal relatve velocty usng the Journal of Computatonal Multphase Flows

15 8 Hassan E. Fayed and Saad A. Ragab y x (a) Partcles accumulaton n hgh stran regons y x (b) Bubbles accumulaton near hgh vortcty regons y x (c) Partcles-bubbles segregaton Fgure. A snapshot of partcles-bubbles segregaton, Stp =.9, Reλ = 9, N =. Volume Number

16 8 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n 9 Zachk et al. 7 RDF, g pp 8 (a) Re λ = 7, N = 9 Zachk et al. 7 RDF, g pp 8 (b) Re λ = 77, N = 8, smulaton (III) 9 Zachk et al. 7 RDF, g pp 8 8 (a) Re λ = 9, N = Fgure. RDF comparson of and Zachk et al. [] model for mono-dsperse partcle- partcle collsons. Journal of Computatonal Multphase Flows

17 Hassan E. Fayed and Saad A. Ragab 8 theoretcal models by Zachk et al. [] and []. We note that applcaton of the former model s lmted to heavy small partcles. Its applcaton to the case of partcle-bubble wll be presented here only for reference. Radal dstrbuton functons for mono-dsperse (partcles-partcles), g pp, and bdsperse (partcles-bubbles), g pb, are shown n Fg a-c and a-d. For mono-dsperse collsons, the radal dstrbuton functon, g pp, s always greater than one, and as a result of preferental concentraton t has a narrow peak near Stokes number of one. It approaches one for very small and very hgh Stokes numbers. The only collson model that predcts RDF for mono-and b-dsperse phases s Zachk et al. [] model. Comparson between the RDF calculated by and that predcted by Zachk et al. s [] model qualtatvely agree. Ths model predcts varaton of RDF wth partcle Stokes number for dfferent Re λ smlar to that by, however, the peak of the RDF s around Stokes number equal to St p =.8 nstead of one. Smulaton (II) s not shown because the results are dentcal to smulaton (III) for the mono-dsperse case. Zachk et al. [] model has been appled to predct the RDF for the case of b-dsperse partclebubble collsons. Fg a d shows the comparson between the RDF by and ths model. Pressure gradent and added mass forces enhance the segregaton of nertal partcles and bubbles. Ths mples that the RDF s less than unty as evdent from these fgures. Added mass and pressure gradent terms n the equaton of moton of partcles are ncluded n. However, Zachk et al. [] model predcts RDF for ths case of partcle- bubble to be approxmately equal to one. Ths behavor of the model s due to neglectng the effects of pressure gradent and added mass forces n the model. These terms are more sgnfcant for large bubbles. In smulaton (II) the bubble dameter s. η whereas for smulaton (III) t s. η. As evdent n Fg b c the model, n whch those effects are neglected, RDF, g pb.. RDF, g pb... Zachk et al. 8 St p. Zachk et al. 8 St p (a) Re λ = 7, N = (b) Re λ = 77, N = 8, smulaton (II) RDF, g pb. RDF, g pb.... Zachk et al.. Zachk et al St p St p (c) Re λ = 77, N = 8, smulaton (III) (d) Re λ = 9, N = Fgure. RDF comparson of and Zachk et al. [] model for b-dsperse partclebubble collsons. Volume Number

18 8 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n shows better agreement wth for the smaller bubble dameter. Zachk et al. [] take nto consderaton the effects of these two forces n fndng the relatve radal velocty, however, t does not provde a model for the RDF; whch s assumed to be one. Hence, theoretcal kernels for partcles-bubbles collsons that assume unty for the radal dstrbuton functon stand to overestmate the collsons kernel... Radal Relatve Velocty w r (d) As shown by Eqs () and (), the non-dmensonal collson kernel β/πu d and mean radal relatve velocty w r (d) /u are equal only f the radal dstrbuton functon g j (d) s unty. For models that assume the radal dstrbuton functon to be one, the assessment should be based on comparsons of the mean radal relatve velocty w r (d) /u rather than the collson kernel β/πu d. Zachk et al. [] model has a restrcton on the partcle sze to be on the order of the smallest turbulent length scale whle Zachk et al. [] model s developed to be applcable for an arbtrary partcle sze. Fg (7a 7c) depcts the comparson between w r (d) calculated from and that predcted from the two models, Zachk et al. [] and [] for mono-dsperse collsons. Radal relatve velocty w r (d) from and Zachk et al. [] model asymptotcally approach the value of w r (d) calculated from Saffman-Turner [] model at very small partcles Stokes number. Both radal relatve velocty w r (d) from calculatons and Zachk et al. [] have very good agreement over a wde range of partcle Stokes number. However, Zachk et al. [] model overpredcts the radal relatve velocty w r (d) of mono-dsperse (partcle-partcle) collsons..8. Zachk et al. Zachk et al..8. Zachk et al. Zachk et al. < W r pp >/u. < W r pp >/u St p St p (a) Re λ = 7, N = (b) Re λ = 77, N = 8, smulaton (III).8. Zachk et al. Zachk et al. < W r pp >/u St p (c) Re λ = 9, N = Fgure 7. w r (d) comparson of, Zachk et al. [] and Zachk et al. [] models for mono-dsperse partcle-partcle collsons. Journal of Computatonal Multphase Flows

19 Hassan E. Fayed and Saad A. Ragab 8.8 Zachk et al. Zachk et al..8 Zachk et al. Zachk et al. < W r pb >/u.. < W r pb >/u St p St p (a) Re λ = 7, Ν = (b) Re λ = 77, Ν = 8, smulaton (II).8 Zachk et al. Zachk et al..8 Zachk et al. Zachk et al. < W r pb >/u.. < W r pb >/u St p St p (c) Re λ = 77, N = 8, smulaton (III) (d) Re λ = 9, Ν = Fgure 8. w r (d) comparson of, Zachk et al. [] and Zachk et al. [] models for b-dsperse partcle-bubble collsons. The radal relatve velocty of the b-dspersed phase (partcle-bubble) for dfferent Reynolds numbers are compared and shown n Fg (8a 8d). These fgures show good agreement between radal relatve velocty w r (d) calculated by and both models developed by Zachk et al. [] and []. Agan, the comparson s remarkably good for smulaton (II) wth smaller bubble dameter. Overall, Zachk et al. [] s n better agreement wth for small Stokes number snce t has a better representaton of the effects of the partcle sze... Collson Kernel Snce Zachk et al. [] model predcts both RDF and w r (d ), comparsons between and ths model for collsons kernel normalzed by Saffman-Turner [] are shown n Fg (9a 9c) for mono-dsperse partcle-partcle collsons. Collsons kernel calculated from and ths model approach collsons kernel calculated from Saffman-Turner [] n the lmt of very small Stokes number. The agreement between collsons kernel calculated from and ths theoretcal model s qualtatvely good. Despte the very good agreement for radal relatve velocty w r (d ) between and ths model; the shft of the maxmum value n the RDF s responsble for these dfferences between collsons kernel calculated from and ths theoretcal model. For the case of b-dsperse (partcle-bubble) the segregaton behavor has a sgnfcant effect on the comparson between calculatons and Zachk et al. [] model predcton as shown n Fg (a d). It s shown n these fgures that Zachk et al. [] model Volume Number

20 8 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n 8 Zachk et al. 8 ββ/ st 8 ββ/ st 8 Zachk et al. 8 (a) Re λ = 7, N = 8 (b) Re λ = 77, N = 8, smulaton (III) 8 Zachk et al. ββ/ st (c) Re λ = 9, N = Fgure 9. Collsons kernel comparson of and Zachk et al. [] model for monodsperse partcle-partcle collsons. Zachk et al. Zachk et al. β /(u' d ) β /(u' d ) 8 8 (a) Re λ = 7, N = (b) Re λ = 77, N = 8, smulaton (II) Zachk et al. Zachk et al. β /(u' d ) β /(u' d ) 8 (c) Re λ = 77, N = 8, smulaton (III) 8 8 (d) Re λ = 9, N = Fgure. Collsons kernel comparson of and Zachk et al. [] model for bdsperse partcle-bubble collsons. Journal of Computatonal Multphase Flows

21 Hassan E. Fayed and Saad A. Ragab 87 over predcts the collsons kernel because t does not consder partcle-bubble segregaton or over predct RDF although t has a very good agreement wth for radal relatve velocty w r (d).. SUMMARY AND CONCLUSIONS Partcles and bubbles suspended n homogeneous sotropc turbulence are tracked and ther collsons frequency s determned as functon of partcle Stokes number. The carrer phase velocty fluctuatons are determned by Drect Numercal Smulatons usng a pseudo- spectral method. The equatons of moton of dspersed phases account for Stokes drag, added-mass force, and pressure gradent n the carrer phase. The equatons of moton are ntegrated smultaneously wth the forced Naver-Stokes equatons of the carrer phase usng a thrd-order Runge-Kutta tme steppng scheme. Smulatons for turbulence Reynolds numbers Re λ = 7, 77 and 9 have been performed, and the computed collsons kernels are used to evaluate theoretcal models developed by Zachk et al. [] and Zachk et al. []. The present results revealed the mportance of accumulaton effects on the collsons kernel of partcles and bubbles. In we observed that bubbles, as they mgrate to low pressure vortex cores, tend to be segregated from heavy partcles. Ths segregaton leads to reduced collsons frequency between partcles and bubbles. We found that the radal dstrbuton functon at collson s less than one for the case of partcle-bubble collsons. Most theoretcal models assume such a functon to be one, and hence they stand to over predct the collson frequency especally for partcle Stokes number near one for whch partcles accumulate n the hgh stran regons. Excellent agreements between and Zachk et al. model [] for mono-dsperse (partclepartcle) and b-dsperse (partcle-bubble) radal relatve velocty are obtaned. Even though the radal relatve velocty s well predcted by the model, the radal dstrbuton functon show only qualtatve agreement wth. The model s applcable only to heavy small partcles, and f appled to the case of b-dsperse (partcle-bubble) collsons, the predcted radal dstrbuton functon s nearly one whereas predcts lower than one values due to partcle-bubble segregaton. A more accurate theoretcal model for the radal dstrbuton functon for b-dsperse partcle-bubble collsons s needed. The model should consder the partcle-bubble segregaton. Zachk et al. [] accounts for both pressure gradent n the carrer phase and added-mass force, and t provdes only expressons for the radal relatve velocty. Excellent agreement between and the model predctons are obtaned for the b-dsperse partcle-bubble collsons, but only qualtatve agreement s obtaned for the mono-dsperse partcle-partcle collsons. ACKNOWLEDGMENT Ths work has been supported by FLSmdth Mnerals, Salt Lake Cty, Utah. REFERENCES [] J. Abrahamson. Collson rate of small partcles n a vgorously turbulent flud. Chemcal Engneerng Scence, :7 79, 97. [] G. Ahmad and V.W. Goldschmdt. Moton of partcles n a turbulent flud-the Basset hstory term. Journal of Appled Mechancs, Trans. ASME, E8:, 97. [] C. Canuto, M.Y. Hussan, A. Quarteron, and T.A. Zang. Spectral Methods. Evoluton to Complex Geometres and Applcatons to Flud Dynamcs. Sprnger, New York, 7. [] V. Eswaran and S.B. Pope. An examnaton of forcng n drect numercal smulatons of turbulence. J. Computer and Fluds, No. :7 78, 988. [] D.T. Gllespe. Exact numercal soluton of the Ornsten-Uhlenbeck process and ts ntegral. J. Physcal REVIEW, :8 9, 99. [] H.Fayed and S. Ragab. Collsons frequency of partcles and bubbles suspended n homogeneous sotropc turbulence. AIAA th Aerospace Scences Meetng Nashvlle- TN, January. [7] A.T. Hjelmfelt and L.F. Mockros. Moton of dscrete partcles n a turbulent flud. Appl. Sc. Res. A, :9, 9. [8] F.E. Krus and K.A. Kusters. The collson rate of partcles n turbulent flow. Chemcal Engneerng Communcatons, 8:, 997. Volume Number

22 88 Drect Numercal Smulaton of Partcles-Bubbles Collsons Kernel n [9] D. L Espèrance, J.D. Trolnger, C.F.M. Combra, and R.H. Rangel. Partcle response to low-reynolds-number oscllaton of a flud n mcrogravty. AIAA J.,, No. :,. [] W.H. Press, S.A. Teukolsky, W.T. Vetterlng, and B.P. Flannery. Numercal Recpes n Fortran, nd Edton. Cambrdge Unversty Press, 99. [] R.S. Rogallo. An ILLIAC program for the numercal smulaton of homogeneous, n- compressble turbulence. Techncal Report TM-7, NASA, 977. [] R.S. Rogallo and P. Mon. Numercal smulatons of turbulent flows. Annual Revew of Flud Mechancs, :99 7, 98. [] P.G. Saffman and T.S. Turner. On the collson of drops n turbulent clouds. Journal of Flud Mechancs, :, 9. [] S. Sundaram and L.R. Collns. Numercal consderatons n smulatng a turbulent suspenson of fnte-volume partcles. J. Comput. Phys., :7, 99. [] S. Sundaram and L.R. Collns. Collson statstcs n an sotropc partcle-laden turbulent suspenson. Part.. Drect numercal smulaton. J. Flud Mech., :7 9, 997. [] M. A. T. van Hnsberg, J. H. M. ten Thje Boonkkamp, and H. J. H. Clercx. An effcent, second order method for the approxmaton of the basset hstory force. J. Comput. Physcs, 8: 78,. [7] L. Wang, A.S. Wexler, and Y. Zhou. On the collson rate of small partcles n sotropc turbulence.. zero-nerta case. Phys. Fluds, : 7, 998. [8] L. Wang, A.S. Wexler, and Y. Zhou. Statstcal mechancal descrptons of turbulent coagulaton. Phys. Fluds, :7, 998. [9] L. Wang, A.S. Wexler, and Y. Zhou. Statstcal mechancal descrptons and modelng of turbulent collson of nertal partcles. J. Flud Mechancs, :7,. [] J.J.E. Wllams and R.I. Crane. Partcle collson rate n turbulent flow. Internatonal J. Multphase Flow, 9:, 98. [] S. Yuu. Collson rate of small partcles n a homogeneous and sotropc turbulence. AIChe Journal, ():8 87, 98. [] L. I. Zachk, O. Smonn, and V. M. Alpchenkov. Collson rates of bdsperse nertal partcles n sotropc turbulence. Physcs of Fluds, 8 No. :,. [] L. I. Zachk, O. Smonn, and V. M. Alpchenkov. Turbulent collson rates of arbtrary- densty partcles. Int. J. Heat and Mass Transfer, :,. [] L.I. Zachk and V.M. Alpchenkov. Statstcal models of clusterng partcles n wall and sotropc turbulent flows. Hgh Temperature, No. :,. [] L.I. Zachk and V. Pershukov. Modelng of partcle moton n a turbulent flow wth allowance for collsons. J. Flud Dynamcs, No. :9, 99. [] Y. Zhou, A.S. Wexler, and L. Wang. On the collson rate of small partcles n sotropc turbulence. II. Fnte nerta case. Physcs of Fluds, No. :, 998. [7] Y. Zhou, A.S. Wexler, and L. Wang. Modellng turbulent collson of bdsperse nertal partcles. J. Flud Mech., :77,. Journal of Computatonal Multphase Flows