Control over multiscale mixing in broadband-forced turbulence

Transcription

1 Conrol over muliscale mixing in broadband-forced urbulence Arkadiusz K. Kuczaj and Bernard J. Geurs, Deparmen of Applied Mahemaics, J.M. Burgers Cener for Fluid Mechanics, Universiy of Twene, P.O. Box 7, 75 AE Enschede, The Neherlands Deparmen of Applied Physics, Eindhoven Universiy of Technology, P.O. Box 53, 56 MB Eindhoven, The Neherlands The effecs of explici flow modulaion on he dispersion of a passive scalar field are sudied. Broadband forcing is applied o homogeneous isoropic urbulence o modulae he energy cascading and aler he kineic energy specrum. Consequenly, a manipulaion of urbulen flow can be achieved over an exended range of scales beyond he direcly forced ones. This modifies ranspor processes and influences he physical-space urbulen mixing of a passive scalar field. We invesigae by direc numerical simulaion he sirringefficiency associaed wih urbulence modified by forcing. This is quanified by monioring he surface-area and wrinkling of a level-se of he passive scalar field. We consider differen forcing o manipulae he qualiy and rae of mixing. The insananeous mixing efficiency measured in erms of surface-area or wrinkling is found o increase when addiional energy is inroduced a he smaller scales. The increased inensiy of small scales significanly influences he small-scale mixing characerisics depiced by wrinkling, while he forcing of large scales primarily affecs he surface-area. Evaluaion of geomerical saisics in broadband-forced urbulence indicaes ha he self-amplificaion process of voriciy and srain is diminished. This leads generally o smaller exremal values of he velociy gradiens bu higher average values as a resul of he compeiion beween he naural cascading processes and he explici small-scales forcing. Inroducion Turbulen mixing of embedded scalar fields is imporan in a diverse range of fluid mechanics problems, from process-engineering, environmenal issues o non-premixed combusion. The efficiency of mixing is governed by a number of aspecs. Nowadays, growing compuaional capabiliies allow he deerminaion of saisics of urbulen flows a quie high Reynolds and Schmid numbers []. Simulaneously, he engineering approach is direced owards conrol

2 LARGE-SCALE FORCING FORCING Arkadiusz K. Kuczaj and Bernard J. Geurs of mixing by modulaion of he driving velociy fields []. Recenly, he use of muliscale forcing mehods was proposed o model urbulen flows ha are disurbed a various spaial scales [3] as may arise in case of flows hrough complex geomerical srucures such as meal foams, or fores canopies. These numerical experimens indicae ha urbulen mixing properies, e.g., expressed by surface-area and surface-wrinkling growh-parameers of scalar level-ses, can be significanly influenced by exernal agiaion. We consider he incompressible Navier Sokes equaions wih broadband forcing working as a complex sirrer in which a specrum of lengh-scales is simulaneously perurbed. Tradiionally, only large-scale forcing was included in a simulaion. This induces an average flow of energy oward smaller scales. The addiional forcing in a high wavenumber band agiaes a specific range of spaial scales as depiced in Fig. (a). We focus on he conrol over basic mixing-properies ha may be obained from such explici broadband forcing and concenrae on he consequences (i) for he ime needed o reach a perfecly-mixed sae and (ii) he accumulaed large- and small-scale mixing. E(k) k -5/ 3 ε /3 k 5/3 E(k) 3 k (a) kη (b) Fig.. (a) Broadband forcing in specral space. (b) Time-averaged compensaed energy specrum for he large-scale (K,: solid) and addiional broadband (K 7,: dashed) forced urbulence, where ε is he energy-dissipaion rae and η - Kolmogorov scale. We invesigae he dispersion of srongly localized iniial scalar concenraions. Direc numerical simulaion of he forced urbulence shows ha he maximal surface-area and wrinkling as well as he ime a which such a maximum is achieved can be conrolled by variaion of forcing parameers. The ime-inegraed surface-area and wrinkling are indicaors of he accumulaed effec. The simulaions show ha a small Schmid numbers, a higher emphasis on small-scale flow agiaion yields a significan increase in he mixing.

3 Conrol over muliscale mixing in broadband-forced urbulence 3 Compuaional flow model We solve he incompressible Navier Sokes equaions in specral space: ( + Re k ) u α (k,) = M αβγ p+q=k u β(p,)u γ (q,) + F α (k,), () where u α (k,) is he velociy field coefficien a wavevecor k (k = k ) and ime, and Re is he compuaional Reynolds number. The ensor M αβγ = (k β D αγ +k γ D αβ )/(ı), where D αβ = δ αβ k α k β /k, accouns for he pressure and incompressibiliy effecs. We adoped a forcing procedure moivaed by flow hrough a fracal gaske []. This paricular forcing has a consan energy inpu rae ε w for he enire sysem: F α (k,) = ε w k β k K m,p u(k,) k β e α(k,). () Each forced band K m,p (m p) consiss of p m+ adjacen spherical shells S n = π(n /)/L b < k π(n + /)/L b : m n p. Here L b is he size of he compuaional domain. We force he firs shell K, wih a consan energy injecion rae ε w, and a single high-k band K m,p wih ε w,. The vecor e(k, ) = u(k, )/ u(k, ) + ık u(k, )/( k u(k, ) ) has he general form proposed in [] and he complexiy of he sirring objec is parameerized by he exponen β = D f relaed o he fracal dimension D f. The scalar concenraion C evolves in a velociy field v by: C(x,) + (v(x,) )C(x,) = (Re Sc) C(x,), (3) where Sc is he Schmid number. We adop a level-se inegraion mehod o quanify basic mixing-properies of he evolving scalar fields [5]. Geomeric properies of a level-se S(a,) = {x R 3 C(x,) = a} may be evaluaed by inegraing a corresponding densiy funcion g over his se. In fac, we have: I g (a,) = da g(x,) = dx δ(c(x, ) a) C(x, ) g(x, ) () S(a,) V where he volume V is he flow-domain. Seing g(x,) = we may deermine he surface-area A of S. In case g(x,) = n(x,), where n(x,) = C(x, )/ C(x, ) is a uni normal vecor, we can deermine he wrinkling W of S. We focus on he evoluion of he surface-area and wrinkling, monioring he insananeous value as well as he accumulaed effec: ϑ Z (a,) = I Z(a,) I Z (a,) ; ζ Z(a,) = ϑ Z (a,τ)dτ ; Z {A, W } (5) By deermining ϑ A and ϑ W we may quanify he rae a which surfacearea and wrinkling develop, he maximal values ha are obained and he ime-scale a which hese are achieved. The cumulaive measures ζ A (a,) and ζ W (a,) express he oal surface-area and wrinkling ha has developed in he course of ime.

4 Arkadiusz K. Kuczaj and Bernard J. Geurs 3 Mixing efficiency The broadband forcing is observed o modify he kineic energy specrum in a srongly non-local manner as seen in Fig. (b). This illusraes he deviaions from he classical Kolmogorov picure ha is characerized by a 5/3 slope in he specral energy disribuion (for more deails see [3]) ϑa.5 ϑw (a) (b) ζa ζw (c).5.5 (d) Fig.. Evoluion of passive scalar dispersion parameers: a) surface-area ϑ A, b) wrinkling ϑ W, c) accumulaed surface-area ζ A, d) accumulaed wrinkling ζ W. Large-scale forcing K, wih ε w =.5 and addiional forcing in he band K 5,8 a ε w, =,.3,.5,.6 (, solid, dash, dash-doed). To esablish he influence of forcing on mixing properies we simulaed he spreading of a passive racer a Schmid number Sc =.7 wih urbulence a R λ 5 (Re = 6). The simulaions sared from a spherical racer disribuion C of radius r = 3/6 scaled o be beween and and he level-se a = / was considered. The resoluion requiremens were saisfacorily fulfilled: k max η ranges from.3 o 3.5 using a resoluion in he range grid-cells. Here k max is he highes wave-number ha is resolved in he simulaion. For he passive scalar hese resoluions correspond o k max η OC in he

5 Conrol over muliscale mixing in broadband-forced urbulence 5 range from 3 o.5, where η OC is he Obukhov-Corrsin scale [6]. The characerizaion of he mixing-efficiency was based on an ensemble of simulaions, each saring from an independen fully-developed realizaion of he velociy field. Individual velociy fields were separaed by wo eddy-urnover imes. In Fig. we compare he insananeous and accumulaed mixing properies for a number of forcing parameers. We include large-scale forcing in K, a ε w, =.5 as well as forcing of he band K 5,8 a various ε w,. We observe ha he large-scale forcing mainly governs he developmen of he surfacearea, while forcing in he second band has a larger influence on he wrinkling. An increase in he srengh of he forcing in K 5,8 leads o a sligh increase in ϑ A and a considerable reducion in he ime a which ϑ A reaches is maximum. The final cumulaive surface area, however, decreases wih increasing ε w,. In conras, an increase in ε w, quie srongly influences he insananeous wrinkling; he maximal value increases and he ime of maximal mixing decreases. The cumulaive effec on W increases noably wih increasing ε w,. Similar resuls were obained when forcing he band K 3,6 insead. pdf z 3 z pdf z.5.5 cos(ω,z i) (a)..5.5 cos(ω, W) (b) Fig. 3. PDFs of he cosine of he angle beween voriciy ω and (a) he eigenvecors z i of he rae of srain ensor, and (b) he vorex sreching vecor W, for largescale forcing K, a ε w, =.5 and addiional broadband forcing wih energy inpu ε w, =,.5,.5 o he second band K 7, (,, ). Evaluaion of geomerical saisics [7] shows ha broadband forcing considerably changes he general characerisics of urbulence. The selfamplificaion process of voriciy and srain is diminished and he saisical flow-srucure alered. This can be seen in Fig. 3 where he pdfs of he alignmen beween he voriciy ω = u and eigenvecors of he rae of srain and he vorex sreching vecor are ploed for various forcing-srenghs in he second band. Increased forcing of he small scales leads o less pronounced alignmen.

6 6 Arkadiusz K. Kuczaj and Bernard J. Geurs Conclusions Forcing mehods agiaing a flow in a wide range of scales induce significan differences compared o he case obained classically in which only he large scales are forced. In his sudy we devoed aenion o a recenly proposed muliscale forcing ha models a flow under he influence of an addiional perurbaion by a complex sirrer []. We performed numerical simulaions of he dispersion of a passive scalar field in a urbulen flow ha is driven by such forcing. By monioring global properies of level-ses of he evolving passive scalar we could quanify he modificaion of he mixing ha resuls from broadband forcing. I was found ha broadband forcing causes addiional producion of smaller scales in he flow. This is direcly responsible for he enhancemen of wrinkling. In conras, he surface-area of a level-se of he racer is found o be mainly governed by convecive sweeping by he larger scales in he flow. Hence, he surface-area is conrolled o a greaer exen by he energy injeced a he larges scales. The addiional energy inroduced by forcing a small scales compees wih processes ha govern Kolmogorov-ype urbulence, e.g., expressed by he selfamplificaion of voriciy and srain and vorex sreching. The forcing also changes he srucure of urbulence; i modifies he alignmen of voriciy wih eigenvecors of he rae of srain ensor. Fuure sudy will be devoed o clarifying his role of he small-scale forcing by evaluaing he geomerical saisics of he urbulen flow as funcion of forcing parameers. Acknowledgmens This work is par of he FOM research program on urbulen flow. AKK would like o hank Arkady Tsinober (Imperial College, London) for many fruiful commens regarding geomerical saisics in urbulence. References. P.K. Yeung, D.A. Donzis, and K.R. Sreenivasan. High-Reynolds-number simulaion of urbulen mixing. PoF, 7:873, 5.. M. Kearney. Engineered fracal cascades for fluid conrol applicaions. In Proc. of Fracals in Engineering Conference, Arcachon, France. INRIA, June, A.K. Kuczaj and B.J. Geurs. Mixing in manipulaed urbulence. JoT, o appear.. B. Mazzi and J.C. Vassilicos. Fracal generaed urbulence. JFM, 5:65,. 5. B.J. Geurs. Mixing efficiency in urbulen shear layers. JoT, :,. 6. T. Waanabe and T. Gooh. Saisics of a passive scalar in homogeneous urbulence. New J. Phys., 6():,. 7. A. Tsinober and B. Galani. Exploraory numerical experimens on he differences beween genuine and passive urbulence. PoF, 5():35 353, 3.