ILASS Americas 8h Annual Conference on Liquid Aomiaion and Spray Sysems, Irvine, CA, May 5 DNS of Turbulen Primary Aomiaion Using a Level Se/Vorex Shee Mehod M. Herrmann Cener for Turbulence Research Sanford Universiy, Sanford, CA 9435 Absrac Modeling he primary aomiaion of urbulen liquid jes and shees sill remains an unsolved problem. Considering numerical grid resoluions ypical for Large Eddy Simulaions, he large scale liquid srucures during primary breakup are well resolved, whereas all small scale srucures are unresolved and hus require modeling. To help in he derivaion of such subgrid models, DNS of he urbulen breakup of surfaces have been performed using he Level Se/Vorex Shee mehod. To address he volume/mass errors inheren in he sandard level se approach, a Refined Level Se Grid mehod is presened. LES filering echniques consisen wih he general scaling symmeries of he level se equaion are discussed and applied o he DNS resuls. The influence of key parameers on he urbulen breakup, like he Weber number and he urbulen Weber number are explored, and a closure for he urbulen producion erm of he subgrid phase inerface lengh scale is proposed. Corresponding Auhor
Inroducion The aomiaion process of liquid jes and shees is usually divided ino wo consecuive seps: he primary and he secondary breakup. During primary breakup, he liquid je or shee exhibis large scale coheren srucures ha inerac wih he gas phase and break up ino boh large and small scale drops. During secondary breakup, hese drops break up ino ever smaller drops ha finally may evaporae. Usually, he aomiaion process occurs in a urbulen environmen, involving a wide range of ime and lengh scales. Given oday s compuaional resources, he direc numerical simulaion (DNS) of he urbulen breakup process as a whole, resolving all physical processes, is impossible, excep for some very simple configuraions. Insead, models describing he physics of he aomiaion process have o be employed. Various models have already been developed for he secondary breakup process. There, i can be assumed ha he characerisic lengh scale l of he drops is much smaller han he available grid resoluion x and ha he liquid volume fracion in each grid cell Θ l is small, see Fig.. Furhermore, assuming simple geomerical shapes of he individual drops, like spheres or ellipsoids, he ineracion beween hese drops and he surrounding fluid can be aken ino accoun. Saisical models describing he secondary breakup process in urbulen environmens can hus be derived [,, 3, 4, 5]. However, he above assumpions do no hold rue for he primary breakup process. Here, he urbulen liquid fluid ineracs wih he surrounding urbulen gas-phase on scales larger han x, resuling in highly complex inerface dynamics and individual grid cells ha can be fully immersed in he liquid phase, compare Fig.. An explici reamen of he phase inerface and is dynamics is herefore required. To his end, we propose o follow in essence a Large Eddy Simulaion (LES) ype approach: all inerface dynamics and physical processes occurring on scales larger han he available grid resoluion x shall be fully resolved and all dynamics and processes occurring on subgrid scales shall be modeled. The resuling approach is called Large Surface Srucure (LSS) model [6]. In order o develop he necessary LSS subgrid models and o analye he dynamics of he phase inerface during urbulen primary breakup, a hreedimensional Eulerian level se/vorex (LSVS) has been developed [6, 7]. The LSVS mehod consiues a promising framework for he derivaion of he LSS subgrid closure models, since i conains explici lo- primary breakup ` / x > Θ l = O() `/ x < Θ l < secondary breakup `/ x << Θ l << Figure. Breakup of a liquid je. cal source erms for each individual physical process ha occurs a he phase inerface. The resuls of a number of DNS of urbulen, iniially plane phase inerfaces using he LSVS mehod are presened in his paper. The influence of some key parameers, as he Weber number, he urbulen Weber number, and he urbulen inensiy on he saisics of he phase inerface geomery are sudied and some iniial closure proposiions for he urbulen source erm of he lengh scale equaion are given. This paper is divided ino four pars. Firs, he LSVS mehod is briefly summaried. Second, he numerical mehods employed o solve he level LSVS mehod are discussed. There, a so called Refined Level Se Grid (RLSG) approach is presened, ha boh addresses some of he shorcomings of he sandard level se approach and provides a naural inerface o subsequen secondary breakup models. Third, he specific requiremens for LES filers of phase inerfaces are discussed and wo filering approaches are oulined and heir respecive resuls presened. Fourh, he DNS resuls of he breakup of urbulen phase inerfaces are presened and some iniial modeling aemps of he urbulen source erms of he lengh scale equaion of he LSS model are presened. Finally, a summary is presened. The Level Se/Vorex Shee Mehod In he following, he level se/vorex shee mehod for wo phase inerface dynamics is briefly summaried. A deailed descripion of he mehod can be found in [6, 7, 8]. Assuming inviscid fluids, he phase inerface consiues a vorex shee wih he ranspor equaion for he normalied vorex shee srengh η given by
η + u η = n [(η n) u] +n [( u n) η] (A + ) + (n κ) We +An a. () Here, u is he velociy vecor, n is he phase inerface normal vecor, We is he Weber number, κ is he phase inerface mean curvaure, A = (ρ ρ )/ρ + ρ ) is he Awood number, and a is he average acceleraion of fluid and fluid a he phase inerface. Equaion () conains on he righ hand side local source erms describing he effec of sreching, surface ension forces and differences in fluid densiy. The locaion x f of he phase inerface is described by a level se scalar G wih G(x f, ) = G = cons, () a he phase inerface, G(x, ) > G in fluid, and G(x, ) < G in fluid, see Fig. Differeniaing Eq. () wih respec o ime hen yields he level se equaion, G + u G =. (3) Then he phase inerface geomerical properies can be deermined from G as n = G G, κ = n. (4) Noe ha he level se equaion is valid only a he locaion of he phase inerface, i.e. x f. This implies ha any ransformaion of he form G = f(g), wih G / G >, (5) does no and should no aler he soluion o Eq. (3) [9]. As will be seen laer, his scaling symmery has imporan implicaions on defining appropriae filer funcions. For reasons of numerical accuracy, however, G away from he phase inerface will be se o a disance funcion in he following, G =, (6) G G and, since Eq. () is also defined and valid only a he locaion of he phase inerface iself, he vorex shee srengh η will be se consan in he phase inerface normal direcion, η G =. (7) y fluid fluid x n G< G = G> Figure. Phase inerface definiion. Equaions () and (3) are coupled by he velociy u. For he DNS resuls of urbulen phase inerfaces presened in his paper, i will be assumed ha u has wo, independen conribuions, u = u V S + u, (8) consising of he vorex shee induced velociy u V S and an independen urbulen flucuaion velociy u ha is impressed ono he phase inerface. This formulaion implies ha he phase inerface has no impac on he urbulen flucuaion velociy field. However, he phase inerface iself is direcly deformed by u via Eq. (3) and reacs o he addiional imposed srain via Eq. (). Thus, sricly speaking, he quesion ha will be addressed in he simulaions presened in his paper, is how a phase inerface reacs o predefined, impressed velociy flucuaions ha are due o a developed urbulen velociy field. The phase inerface induced velociy u V S is calculaed by inroducing he vecor poenial ψ, ψ = ω. (9) wih he voriciy vecor ω calculaed following a vorex-in-cell ype approach [, ] ω(x) = () η(x )δ(x x )δ (G(x ) G ) G(x ) dx, V where δ is he dela-funcion. Then, u V S can be calculaed from u V S (x) = δ(x x ) ( ψ) dx. () V Numerically, he dela funcions in Eqs. () and () have o be approximaed by a smoohed version. Here a formulaion due o Peskin [] is employed, { [ ( x )] + cos : x ε δ ε (x) = ε ε () : x > ε
Approximaing he dela funcions hus, he voriciy, heoreically locaed solely on he inerface, is in effec spread ou ono he neighboring grid nodes, hereby prescribing a consan, non-ero local shear layer hickness. Hence, his approach is similar o he vorex-in-cell mehod ha spreads he voriciy of Lagrangian vorex paricles o heir surrounding grid nodes [, ] and does in fac mimic cerain effecs of viscosiy [7]. 3 Numerical mehods Numerically, Eqs. () and (3) are solved in a narrow band [3] by a 5 h -order WENO scheme [4] using a 3 rd -order TVD Runge-Kua ime discreiaion [5]. The reiniialiaion of G, Eq. (6), is solved by he ieraive procedure oulined in [3] and [6]. The redisribuion of η, Eq. (7), is solved by a Fas Marching Mehod [7, 8, 9]. The ineresed reader is referred o [6, 7, 8] for a deailed descripion of he numerical mehods employed in he level se/vorex shee mehod and a summary of he domain decomposiion paralleliaion approach used. 3. Refined Level Se Grid mehod Tracking inerfaces by a sandard level se approach [] unavoidably inroduces volume/mass errors ha are proporional in sie o he employed numerical grid sie. To avoid hese errors, in principle, wo differen approaches can be followed. One can correc he level se soluion using an inerface racking mehod ha eiher inherenly preserves he volume, as for example he volume of fluid mehod [,, 3], or a leas preserves he volume wih higher accuracy han he level se mehod alone, like for example marker paricles [4]. Alernaively, one can reduce he mass error by refining he underlying numerical grid. However, refinemen mehods available in lieraure do solve boh he level se and he flow describing equaions on one single grid, ha is eiher acive and defined only in a narrow band around he inerface, as is he case in he narrow band approach [3], or adapively refined by means of nesed recangular grids [5], local anisoropic Caresian meshes [6], or ocrees [7]. The advanage of he narrow band approach is ha i reains a simple caresian grid on which he PDEs can be solved fas and accurae, however he angenial and normal resoluion a he inerface is ypically consan. Adapive mesh refinemen approaches on he oher hand allow for variable resoluion of he inerface, hus minimiing he number of compuaional nodes. The disadvanage of hese mehods are however heir higher complexiy and numerical cos, especially wih respec o domain decomposiion paralleliaion. α T,G η-grid α T,G G G = G η = x G-grid G x Figure 3. Refined level se grid definiion. In he presen case of performing hreedimensional DNS of he primary breakup process, i can be assumed ha a refined grid is necessary in large porions of he phase inerface, favoring a narrow band mehodology. However, unlike he sandard narrow band approach [3], here, a second, independen grid shall be inroduced on which only he level se equaion and is associaed PDEs, Eqs. (3), (6), and (7), are o be solved wihin a narrow band, see Fig. 3. This Refined Level Se Grid (RLSG) approach has hree disinc advanages. Firs, he level se grid (G-grid) can be refined independenly from he base grid (η-grid), hus ensuring grid convergence wih respec o he phase inerface represenaion. Second, due o he increased resoluion on he G-grid, enhanced sub-η-grid breakup models can be defined o allow for he physically correc capure of he micro scale breakup process. Third, separae liquid srucures well resolved on he G-grid can be of subgrid sie wih respec o he η-grid. These drops consiue spray drops for a secondary breakup model defined on he η-grid. They can hus be removed from he G-grid and insered in he η- grid spray model, providing a naural inerface o spray models. In he following, he RLSG mehod is described in more deail. Le x be he cell sie of he equidisan Caresian grid on which he η-equaion, Eq. (), is solved. The level se equaion, Eq. (3), is hen solved on a narrow band consising of equidisan Caresian grid cells of sie G x, G x = x/n G, (3) where n G is he grid refinemen facor. The widh of he narrow band α T,G is chosen in such a way ha enough cells are presen o allow for he evaluaion of he 5 h -order WENO sencil during a single CFL-
limied ime sep on he η-grid. To fulfill he CFLcrierion on he G-grid, subcyling ypically has o be employed. This resuls in a widh of he narrow band of { α T,G = 9 G x : n G 4 (.5n G + 3) G x : n G > 4, (4) see Fig. 3. All oher narrow band widhs described in [7] are defined accordingly. The coupling of he η-grid and he G-grid is wo-fold. Firs, he level se scalar field solved on he finer G-grid has o be ransferred o he η-grid. Le G G be he level se scalar defined on he G-grid and Gη be he level se scalar defined on he η-grid. Then, remembering ha any G is defined as he disance funcion away from he inerface, he embedded inerface G G = G can be viewed as a higher order approximaion of he inerface as defined by Gη = G. This implies ha a he same node locaion, he value of Gη should be exacly equal o he value of G G, since boh values describe he disance o he same inerface geomery. I is imporan o noe ha his coupling does no consiue a filering operaion from a finer o a coarser grid. Here, he goal is raher o make use of a higher order approximaion of he inerface o eliminae numerical errors on he coarser grid. In pracice, Gη is deermined from G G on all η-grid nodes ha are direcly adjacen o he Gη = G inerface. All oher Gη values up o a cerain disance away from he inerface are hen reconsruced using he Fas Marching Mehod. Second, he velociy u is iniially only defined on he η-grid. To solve Eq. (3), u has o be ransferred o he G-grid. While conservaive inerpolaion echniques can be use [8], here simple rilinear inerpolaion is employed. By solving he level se equaion separae from he η-equaion on a refined grid, he RLSG mehod also allows for a differen approach in calculaing he source erms in Eq. (). These could sill be evaluaed using Gη [7]. However, o make full use of he available geomery informaion on he G-grid, hese source erms S(xη) defined on he η-grid can now also be evaluaed using G G on he G-grid and hen surface averaged ono he η-grid. This process is a hree-sep procedure: firs, all source erms on he righ-hand side of Eq. () are evaluaed on he G-grid. Then, hese erms are redisribued in he inerface normal direcion on he G-grid by solving Eq. (7) using he Fas Marching Mehod. Finally, he surface inegraion is performed by evaluaing Γη S(x Γ G )dx ΓG S(xη) = Γη dx = Γ G Vη S(x G)δ(G G (x G )) G G (x G ) dx G Vη δ(g G(x G )) G G (x G ) dx G, (5) where Γη is he par of he G G = G inerface ha lies wihin he η-cell locaed a xη and Vη is he volume of ha η-grid cell. The inegraion above is performed on he G-grid. As menioned above, one of he advanages of he RLSG approach is ha due o he increased G-grid resoluion, enhanced micro-breakup models can be employed. The micro-breakup model inheren in he level se approach is ha if wo inerface segmens have a normal disance less han G x, breakup is iniiaed auomaically since G is a single valued scalar. Since he RLSG mehod allows for he independen refinemen of G x, heoreically, refinemen is possible up o a poin where inra-molecular forces ha iniiae he physical micro-scale breakup, become dominan. These forces hen resul in an addiional velociy erm in he G-equaion, ha is a funcion of he minimum disance beween inerfaces, i.e. G. However, in he simulaion resuls presened in his paper, he simple micro-breakup model ha breakup occurs as soon as wo inerface segmens are a normal disance of less han G x apar, is employed. As noed above, he RLSG mehod also consiues a naural inerface o secondary breakup models describing he generaed liquid spray. All separaed liquid srucures wih volume less han Vη can be direcly removed from he G-grid and inroduced ino an η-grid spray model as a liquid drop. Idenificaion of separaed G-grid srucures can be achieved by recursive search algorihms on he G-grid. An alernaive approach is o compare he inerface geomery on he G-grid wih ha on he η-grid or any oher grid of inermediae resoluion. The difference in liquid volume of he differen grid represenaions hen yields he subgrid (spray)- mass on he η-grid. In he simulaions presened in his paper, however, he coupling o a secondary breakup-model on he η-grid is no ye implemened and all small scale srucures remain on he G-grid. RLSG Example: Zalesak s disk The solid body roaion of a noched circle, also known as Zalesak s disk [9], is one of he sandard es problems for evaluaing he accuracy of level se mehods. A disk of radius.5, noch widh.5,
.95.95.75.55.3.5.7.95.95.75.55.3.5.7.95 A/A s/s no correcion.54.7859 paricle correcion.995 - RLSG n G =.9973.9736 RLSG n G = 4.99987.98694 RLSG n G = 8.9999.99368.75.75.75 Table. Normalie disk area A/A and inerface lengh s/s afer one full roaion of Zalesak s disk.55.3.5.7.55.55.3.5.7.3.5.7 Figure 4. Inerface shape afer one full roaion of Zalesak s disk. Solid line denoes numerical soluion and dash-doed line is exac soluion. From op lef o boom righ: no correcion mehod, paricle correcion mehod, RLSG mehod n G =, n G = 4, and n G = 8..5 A/A.4.3...99 3 4 5 6 7 s/s.5.95.9.85.8.75 3 4 5 6 7 Figure 5. Normalied area A/A (lef) and inerface lengh s/s (righ) during one full roaion of Zalesak s disk. Paricle correcion mehod (solid line), no correcion mehod (open box), RLSG mehod n G = (open circle), n G = 4 (solid box), and n G = 8 (solid circle). and noch heigh.5 is placed in a box a (.5,.75). The velociy field is given by u(x, ) = (.5 y, x.5) T. (6) Figure 4 shows he shape of he inerface a = afer one full roaion of he disk using no correcion scheme, he paricle correcion mehod, and he RLSG mehod wih varying n G. Obviously, using no correcion mehod a all causes he noch heigh o decrease subsanially and he lower sharp corners o become significanly rounded. This in urn increases he area A/A of he disk and decreases he lengh of he inerface s/s considerably, as shown in Fig. 5. Using he paricle correcion mehod [4] improves he resuls significanly. However a sligh asymmery occurs. The area of he disk decreases slighly, see Tab., while showing noiceable flucuaions over ime, Fig. 5. These are due o he local, non-coninuous correcion sep of he paricle correcion mehod. Employing he RLSG mehod wih n G = resuls in markedly improved resuls as compared o using no correcion mehod a all. The area of he disk is preserved beer han in he case of he paricle correcion mehod. However, his is due o wo errors canceling each oher: one a he sharp corners leading o an area decrease and he oher a he op of he noch leading o an area increase, see Fig. 4. This cancelaion of errors resuls in an area decrease of only.8%. The oal lengh of he inerface, on he oher hand decreases by abou.7 %. Successively refining he G-grid coninues o improve hese resuls. The inerface shape obained wih n G = 4 is already superior o ha of he paricle correcion mehod. For n G = 8 almos no difference beween he exac soluion and he numerical resul can be discerned, see Fig. 4, and 99.99% of he disk s area and 99.4 % of he inerface lengh is preserved, see Tab.. The resuls of his es case indicae, ha he RLSG mehod wih n G 4 performs comparable, if no superior, o he paricle correcion mehod wih respec o area preservaion, while mainaining symmery and avoiding any flucuaions inroduced by he correcion sep of he paricle correcion mehod [7, 8]. RLSG Example: Turbulen Surface Breakup In his example, he performance of he RLSG mehod wih respec o he urbulen breakup of a surface is analyed. The parameers of he case chosen, case c, are summaried in Tab.. Noe ha due o he lack of any surface ension forces and no molecular force based micro breakup model, his case consiues an exreme case in ha he smalles
> > 6 A/A 5 4 3 6 A/A 5 4 3..4.6.8.8 l.6.6..4.6.8.8 l.4.4....4.6.8..4.6.8 Figure 6. Surface area A/A (op) and flucuaion lengh scale l on G-grid (lef) and η-grid (righ) for case c and varying levels of level se grid refinemen, n G = (open boxes), n G = (solid boxes), n G = 4 (open circles), and n G = 8 (solid circles). inerface scales become ever smaller wih ime. Thus grid convergence can only be achieved up o a cerain poin in ime, before he inerface scales become smaller han he available G-grid resoluion. Figure 6 shows he surface area of he phase inerface as a funcion of ime, boh on he G-grid and on he η-grid for increasing levels of G-grid refinemen. As can be seen, grid convergence is achieved up o abou.5. From here on, finer G-grids would be required o adequaely resolve he fine scale inerface srucures. Figure 6 also shows he evoluion of he inerface flucuaion lengh scale l, defined laer by Eq. (). Here, grid convergence is achieved for he inerface geomery on he η-grid up o abou. Figure 7 shows a snapsho of he phase inerface a = 3. as defined on he G-grid and on he η- grid. For he mos par, he inerface is sreched ino hin shees resuling from counerroaing vorices folding and sreching fluid segmens ino he opposing fluid. Thin filamen srucures are visible ha are generaed by wo differen mechanisms. Some are due o he merging of hin liquid shees, if he shee hickness is no longer suppored by he G-grid, i.e. he shee hickness becomes less han G x. However, analysis of he emporal evoluion of he inerface geomery indicaes ha some fila- Figure 7. Phase inerface geomery for v =, We = a = 3.: G-grid wih n G = 4 (op) and η-grid (boom). men srucures are he resul of vorices pinching ou par of he surface direcly ino elongaed fingers. Comparing he phase inerface defined by G G o ha defined by Gη shows ha mos of he hin shees and filamens sill resolved and suppored on he G-grid are broken up ino small scale drops on he η-grid. This is due o he inheren micro-scale breakup model of he η-grid. Noe ha he above menioned coupling of he RLSG mehod o a secondary breakup spray model would be required o ensure conservaion of local fluid mass. Globally, he mass of each of he fluids is saisically conserved because of he inheren saisical symmery of he problem. However, he global surface area, as depiced in Fig. 6, is sensiive o he loss of unresolved fluid elemens, resuling in he lower surface area on he η-grid as compared o he G-grid and explaining
also he difficuly in achieving grid converged resoluions. Coupling o a secondary breakup spray model will alleviae hese problems. 4 LES Filers for Level Se Scalars As poined ou by Oberlack e al. [9], adherence o he symmery properies of he level se equaion while performing any ype of filering or averaging operaion is crucial. Especially he general scaling symmery of he level se equaion (Eq. 5) excludes he applicaion of volume based filer kernels commonly used in LES direcly on G, since Ĝ becomes a funcion of G G values away from he inerface and any ransformaion according o Eq. (5) will change he posiion of he filered inerface [9]. Insead, Oberlack e al. [9] propose an averaging procedure for ensemble averages based on firs parameeriing he G = G iso-surface by an orhogonal surface aached coordinae sysem Λ = (λ, µ) T, (7) and hen ensemble averaging he surface coordinae x f a consan Λ yielding he ensemble mean posiion of he surface. Based on his formulaion, Pisch [3, 3] inroduced a LES spaial filering procedure for urbulen premixed flames. This procedure will be applied here for urbulen phase inerfaces. Le Λ be a surface parameeriaion of he phase inerface, hen a spaial filer H(Λ Λ ) can be defined as { H(Λ Λ /AH (Λ) : x V ) = H (8) : x / V H where V H is he volume of a cube wih edge lengh cenered a x f (Λ), and A H (Λ) is he unfilered phase inerface surface area inside ha cube (Fig. 8). Then, he filered mean posiion of he phase inerface x f (Λ) can be calculaed from x f (Λ) = H(Λ Λ )x f (Λ )dλ (9) V H Leing Ǧ = G coincide wih he posiion of he filered mean phase inerface x f, he level se scalar Ǧ can be defined in analogy o Eq. (). A characerisic lengh scale l of he subfiler phase inerface flucuaions can be defined as or alernaively as l(λ) = x f (Λ) x f (Λ) () l(λ) = ň (x f (Λ) x f (Λ)), () where ň is he normal o he filered phase inerface. The second definiion is used in he subsequen analysis of he DNS resuls. Then he subfiler phase Λ Λ x f (Λ) l x f (Λ) > filered fron unfilered fron Figure 8. Definiion of surface based filer. inerface variance l can be calculaed by applying he filer funcion, resuling in l = H(Λ Λ )l (Λ )dλ. () V H Noe ha in he following l is used as a shorhand for l. In he numerical implemenaion of he filering operaions (9) and (), he unfilered phase inerface given by G = G is firs riangulaed by a marching cubes algorihm [3, 33]. Then, he filering inegraions reduce o a summaion of all riangles and riangle pars ha fall wihin he filer volume V H. While he filering procedure oulined above is consisen wih he general scaling symmeries of he level se equaion, is main drawback is ha i lacks one of he key desired characerisics of a spaial LES filer, namely ha i removes all geomerical scales in physical space smaller han he filer sie. The reason for his is he fac ha he filer operaes in surface coordinae space Λ. Even if a projecion filer can be devised and applied o x f (Λ), hereby removing all scales smaller han he filerwidh in Λ-space, reransformaion o physical space can reinroduce scales of arbirary small sie due o he non-linear mapping beween x f and Λ. This feaure is clearly visible in Fig. 9, where he phase inerface geomery for case c a =.5 is filered wih successively larger filer sies. Iniially rounded srucures like fingers or hick shees conrac and form ever hinner srucures, see for example he hinning of he verical srucure near x =, y = for increasing filer sies. Even for large filer sies spanning half of he compuaional domain, =, small scale srucures in physical space in he form of hin shees persis. For =, he filering operaion generaes a fla shee as expeced, due o he periodiciy of in he simulaion. An alernaive approach o filering he phase inerface while saisfying he scaling symmeries of
- x y Figure 9. Filered phase inerface geomery a =.5, case c, =, = /64, = /3, = /6, = /8, = /4, = /, =, = (op lef o boom righ). he level se equaion is o apply a Heavyside ransformaion o he level se scalar field, φ(x, ) = H(G(x, ) G ), (3) where H is he Heavyside funcion. Noe ha φ is now invarian o he general symmery ransformaion, Eq. (5), and sandard LES volume filers can be applied o φ, φ(x, ) = H φ(x, )dx. (4) V Here, a simple normalied op-ha volume filer wih widh will be used for H. For >, he above filering operaion will yield φ as smoohly varying beween and. In principle, any iso-surface wih consan value φ = φ, < φ <, can now be idenified as he posiion of he mean phase inerface. However, choosing φ =.5 guaranees ha a planar unfilered surface remains a he same posiion, no maer how large he filer sie. Finally, seing Ǧ(x, ) = φ(x, ) φ + G (5) defines he posiion of he filered mean phase inerface in analogy o Eq. (). Figure shows he phase inerface a =.5 for case c filered wih he Heavyside filer. While no much difference wih respec o he surface based filer (Fig. 9), can be discerned for small filer sies, he Heavyside filer does no inroduce any new small scale srucures and ends o remove surface srucures smaller han he filer sie. From his sandpoin, he Heavyside filer is he superior filer for LES applicaions. However, he definiion of a subfiler lengh scale
- x y Figure. Heavyside filered phase inerface geomery a =.5, case c, =, = /64, = /3, = /6, = /8, = /4, = /, =, = (op lef o boom righ). l is no longer sraighforward. Due o he lack of a surface parameeriaion, he quesion arises beween which poins on he filered and unfilered surface does l measure he disance? Using he local minimum disance of he filered o he unfilered surface does no yield a subfiler flucuaion lengh scale as simple examples of sinusoidal surfaces can demonsrae. Employing a sraighforward definiion of l based on a φ, φ (x, ) = H V ( φ(x, ) φ(x ), ) dx (6) resuls in he undesired propery ha φ is proporional o for he example of a planar unfilered surface, whereas l should be ero in ha case. This will be he case if l is calculaed from l(x, ) = H (φ(x, ) ˇφ(x, ) ) dx /Ǎ, V wih (7) ˇφ(x, ) = H(Ǧ(x, ) G ), (8) and Ǎ he filered phase inerface surface area inside he filer volume V. However, Eq. (7) defines l based on he unfilered volume of fluid inside he filered volume of fluid and vice versa. Alhough consisen wih he scaling symmery for Ǧ, Eq. (7) is hus no a good measure for l. An alernaive approach o explici filering o deermine he posiion of he filered phase inerface and is subfiler flucuaion lengh scale is o consider he subfiler PDF of locaing he phase in-
erface a a given poin x wihin he filer volume. This approach follows he one oulined by Peers [34] for ensemble averaging of premixed urbulen flames. The firs and second momens of he subfiler PDF hen define he filered inerface posiion and is variance. Incidenally, for =, he subfiler PDF mehod and he surface based filer, Eqs. (9) and (), give he exac same resuls for he filered inerface posiion and is variance 5 DNS Resuls The urbulen breakup of seven differen iniially planar phase inerfaces wih Awood number A = and dela funcion spreading parameer ε = 4/64 have been simulaed. Table summaries characerisic parameers and numbers for each of he seven runs ha can be spil ino wo main groups. The firs se analyes he pure urbulen breakup wihou any addiional shear ( v = ), whereas he second se includes he effec of a global shear field ( v = ). Simulaions were performed on a domain of sie 4, periodic in he x- and y-direcion and wih prescribed angenial y-velociy a he -direcion boundary of ± v/. The η-grid was discreied by 64x64x8 equidisan caresian cells and a n G = 4 refined G-grid was employed in all simualions. A = he planar phase inerface was locaed a =. The urbulen flucuaion velociies were provided by he flow solver CDP, solving he incompressible Navier Sokes equaions for forced homogenous isoropic urbulence on he η-grid wih u RMS =. and a Taylor microscale Reynolds number of Re λ = 8.45. case v u RMS / v We We # runs c - c6 c7.5 c8. c6. 8.94 c7. 4.8.5 c8. 8.6. Table. Condiions for DNS runs. Figure shows he evoluion of he phase inerface geomery a hree differen imes for he case of no exernal shear flow, i.e. v =, and decreasing urbulen Weber number We. As already discussed above, if no surface ension forces are presen, successively smaller inerface srucures are creaed ha highly corrugae he phase inerface since no sabiliing mechanism is presen. Inroducing surface ension forces resuls in a markedly differen inerface evoluion. Since surface ension forces in his case iniially represen a resoraion force ha aims o smooh he surface back o is original planar shape, he developmen of small scale srucures is inhibied, consisen wih he qualiaive predicions of linear heory. A laer imes, he onse of surface ension assised insabiliies and breakup, i.e. Rayleigh breakup, can be observed in some isolaed insances. Increasing he surface ension forces by reducing We furher sabilies he phase inerface, unil for We =. he inerface sars o oscillae on he lengh scale of he box. In his case he urbulen flucuaion velociies seem o ransfer energy ino he surface ension force generaed oscillaions. I should be poined ou, however, ha his effec migh be simply due o he chosen iniial velociy field condiions. A number of differen DNS runs wih varying iniial urbulen velociy fields would have o be ensemble averaged o draw definiive conclusions. Figure shows he probabiliy densiy funcion of locaing he phase inerface a f, whereas Fig. 3 depics he evoluion of he normalied phase inerface surface area A/A and he flucuaion lengh scale l. Two separae DNS runs have been ensemble averaged in he case of We =, while all oher daa is based on a single DNS run, see Tab.. Due o he ensemble averaging, P ( f ) for We = is almos symmeric, whereas all oher pdfs show some degree of asymmery. This indicaes ha ensemble averaging of more han one DNS run would be required o achieve he expeced symmeric pdfs. In he We = case, a Gaussian pdf provides an excellen fi o he calculaed pdf. For all oher cases, a Gaussian funcion appears o be a good approximaion as well, however, ensemble averaging of more han one DNS run is required o subsaniae his observaion. Analysis of he evoluion of he flucuaion lengh scale l and he normalied surface area A/A show he impac of surface ension as a resoraive force. As expeced, boh quaniies are reduced by decreasing We, however, he impac on he flucuaion lengh scale is no as large as on he surface area. This indicaes, ha while surface elemens can sill be moved a considerable disance away from heir iniial posiion in he -direcion, he phase inerface iself is much smooher and less corrugaed resuling in less surface area. This conclusion is consisen wih he acual phase inerface geomery depiced in Fig.. Furhermore, he observed oscillaions of he phase inerface in he We =. case can be clearly discerned in boh l and A/A. Alhough for laer imes, he pdfs for he We =
- - 3 x 4 3 4 y Figure. Phase inerface geomery, v = and We = (firs row), We = (second row), We =.5 (hird row), We =. (fourh row) a =.5, =., and =. (lef o righ). case visually seem o collapse, he evoluion of l depics a coninuing growh, see Fig. 4 showing a longer ime evoluion up o = 3. Since in he periodic case analyed here, he filered phase inerface for = is a fla surface locaed a ẑ f = for all imes, a modeling equaion for l can easily be derived. Since Eq. () resuls in l(λ, ) = f (Λ, ) f (Λ, ), (9) aking he square, filering and hen differeniaing wih respec o ime yields d l d = ŵ, (3) where w = w V S + w is he velociy componen in he -direcion and for early imes f = w. In he case of no shear ( v = ) and We =, he vorex shee srengh is η(x, ) =, see Eq. (), and hus u V S = for all ime. Thus, he righ hand side of Eq. (3) reduces o a urbulen producion erm only. Since for homogeneous isoropic urbulence
> > P( ) f.5 P( ) f.5 P( ) f.5 3 P( ) f.5.5.5.5.5.5-6 -4-4 6-6 -4-4 6 P( ) P( ) f f.8.8-6 -4-4 6 P( ) f.8-6 -4-4 6 P( ) f.8.6.6.6.6.4.4.4.4.... -6-4 - 4 6-6 -4-4 6-6 -4-4 6-6 -4-4 6 Figure. Probabiliy densiy funcion P ( f ) for v = and We =, We =, We =.5, We =. (from lef o righ) a =.5 (op, solid circles), =. (op, open circles), =.5 (boom, solid circles), =. (boom, open circles), =.5 (boom, solid boxes), and = 3. (boom, open boxes)..5 l A/A l A/A.75.5.5.5.5.5.5.. 3 4 Figure 3. Flucuaion lengh scale l (lef) and surface area A/A (righ) on G-grid for v = and We = (solid circles), We = (open circles), We =.5 (solid boxes), and We =. (open boxes). Figure 4. Flucuaion lengh scale l (lef) and surface area A/A (righ) on G-grid for case c. Dashed and doed lines denoe fis o he daa, l and A/A e (dashed), and l and A/A e / (doed). w u RMS, Eq. (3) resuls in l (3) for early imes. Noe ha l is used as a shor hand for l. The emporal evoluion of l according o Eq. (3) is shown as a dashed line in Fig. 4. I fis he calculaed daa very well. For longer imes, however, he above assumpions are no longer valid. Insead, a derivaion similar o he one performed for he lengh scale equaion in he case of urbulen premixed flames [3] can be performed. Muliplying Eq. () by ň and hen differeniaing wih respec o ime yields dlň d = dx f d d x f d = (3) u V S + u (u V S + u ) = u. Then muliplying Eq. (3) by lň and applying he filer yields d l d = ň lu, (33) where he righ hand side consiues a urbulen
- - 3 x 4 3 4 y Figure 5. Phase inerface geomery, v = and We = (firs row), We = (second row), We =.5 (hird row), We =. (fourh row) a =.5, =., and =. (lef o righ). producion erm ha can be modeled using a gradien ranspor assumpion resuling in d l d = D,l, (34) where D,l is he urbulen ranspor coefficien of l [3]. Equaion (34) hus yields l (35) for laer imes. Noe ha due o he muliplicaion of Eq. (3) by l, he above derivaion is no valid for early imes, where iniially l =. The emporal evoluion of l according o Eq. (35) is shown as a doed line in Fig. 4. Again, he fi o he compued daa is good, alhough more daa would be required o fully verify Eq. (35). The righ hand side of Fig. 4 shows fis o he evoluion of he surface area. For early imes, he surface area grows exponenially wih A/A e, (36) whereas for laer imes A/A e /. (37) However, i should be noed ha for laer imes he values of A migh be underprediced due o numerical diffusion and he incorrec merging of phase inerfaces close o each oher, as he level se grid refinemen sudy depiced in Fig. 6 indicaes. Thus, Eq. (37) should be considered preliminary. Figure 5 shows he evoluion of he phase inerface geomery in he case of an exising shear layer and increasing surface ension forces. A decrease in Weber number again resuls in a reducion of small scale surface srucures. However, due o he presen shear, Kelvin-Helmhol insabiliies are formed ha resul in an increase in -direcion surface displacemen as compared o he v = case. Also, some elongaed fingers are formed exending in he sheared flow direcion. Alhough no acual breakup of hese fingers ino smaller drops has been observed, mos likely due o he lack of sufficien
> P( ) f.5 P( ) f.5 P( ) f.5.5.5.5-6 -4-4 6 P( ) f.8.6.4. -6-4 - 4 6-6 -4-4 6 P( ) f.8.6.4. -6-4 - 4 6-6 -4-4 6 Figure 6. Probabiliy densiy funcion P ( f ) for v = and We =, We =.5, We =. (from lef o righ) a =.5 (op, solid circles), =. (op, open circles), =.5 (boom, solid circles), =. (boom, open circles), =.5 (boom, solid boxes), and = 3. (boom, open boxes). ime, he onse of a Rayleigh-ype insabiliy can be seen. Figure 6 shows he probabiliy densiy funcion of locaing he phase inerface a f, whereas Fig. 7 depics he evoluion of he normalied phase inerface surface area A/A and he flucuaion lengh scale l. Compared o he cases wihou shear, he pdfs are quie similar for he early ime of =.5. However, quickly, he pdfs generally become more broader exhibiing increased, bu small values a heir borders for laer imes. Especially for We =., he pdf a =. is significanly broader han in he non-shear case. For laer imes, he pdfs for he We =. and We =.5 case seem o collapse in he cenral par, bu coninue o spread and grow in he ouside pars. This indicaes ha he phase inerface in he cenral par of he shear layer migh reach a saisically saionary sae and only fingers and shees proruding furher ino he opposing fluid coninue o grow, reminiscen of he Kelvin-Helmhol insabiliy. This rend of coninued growh is also eviden in he evoluion of l and A/A ha boh coninue o grow wih ime and do no show he amoun of flaening observed in he non-shear case. Also, boh he We =. and We =.5 case exhibi similar evoluions of l, whereas heir A/A values remain significanly differen. This indicaes ha We =.5 should exhibi noiceably less surface wrinkling in he cenral par of he shear layer, while he surface area of elemens.5 l.75.5.5.5.5 A/A.5.5 Figure 7. Flucuaion lengh scale l (lef) and surface area A/A (righ) on G-grid for v = and We = (open circles), We =.5 (solid boxes), and We =. (open boxes). furher away from he filered mean surface a = should be roughly he same in boh cases. This conclusion is suppored by he acual phase inerface geomeries depiced in Fig. 5. The same argumen applies o he evoluion of he We =. case, where basically one wavelengh of sie equal o he compuaional box grows, resuling in relaively high values of l wih small values of he acual surface area. 6 Summary Direc numerical simulaions of he breakup of urbulen phase inerfaces have been performed using a level se/vorex shee mehod. To minimie
volume/mass errors inherenly presen in he sandard level se mehod, a Refined Level Se Grid (RLSG) approach has been presened, ha allows for grid convergence sudies wih respec o he phase inerface represenaion, provides a framework for deailed micro-breakup modeling, and consiues a naural inerface o subsequen secondary breakup models, i.e. spray models. Differen filering sraegies have been discussed ha adhere o he general scaling symmeries of he level se equaion. The geomeric evoluion and saisics of several differen urbulen phase inerface have been presened. In he case of no surface ension and no shear, models for he producion erm of he subgrid flucuaion lengh scale have been presened boh in he shor erm and long erm limi. Acknowledgmens The auhor would like o hank Hein Pisch and Venkaramanan Raman for numerous helpful discussions. Furhermore, he suppor of he Deparmen of Energy s ASC program is graefully acknowledged. References [] P. J. O Rourke. PhD hesis, Princeon Universiy, 98. 53-T. [] P. J. O Rourke and A. A. Amsden. The TAB mehod for numerical calculaions of spray drople breakup. Technical Repor 8789, SAE Technical Paper, 987. [3] R. D. Rei. Aom. Spray Tech., 3:39 337, 987. [4] R. D. Rei and R. Diwakar. Srucure of high pressure fuel sprays. Technical Repor 87598, SAE Technical Paper, 987. [5] F. X. Tanner. SAE Transacions: J. of Engines, 6(3):7 4, 997. [6] M. Herrmann. Modeling primary brekaup: A hree-dimensional Eulerian level se / vorex shee mehod for wo-phase inerface dynamics. Annual Research Briefs-3, pp. 85 96. Cener for Turbulence Research, Sanford, CA, 3. [7] M. Herrmann. J. Compu. Phys., 3():539 57, 5. [8] M. Herrmann. In C. Presser and B. Helenbrook, ediors, ILASS Americas 4, 7h Annual Conference on Liquid Aomiaion and Spray Sysems, NIST Special Publicaion 6, 4. [9] M. Oberlack, H. Wenel, and N. Peers. Combus. Theory Modelling, 5:363 383,. [] J. P. Chrisiansen. J. Compu. Phys., 3:363 379, 973. [] G.-H. Coe and P. D. Koumousakos. Vorex Mehods. Cambridge Universiy Press, Cambridge,. [] C. S. Peskin. J. Compu. Phys., 5: 5, 977. [3] D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang. J. Compu. Phys., 55:4 438, 999. [4] G.-S. Jiang and D. Peng. SIAM J. Sci. Compu., (6):6 43,. [5] C.-W. Shu and S. Osher. J. Compu. Phys., 77:439 47, 989. [6] M. Sussman, P. Smereka, and S. Osher. J. Compu. Phys., 9:46, 994. [7] J. A. Sehian. Proc. Nal. Acad. Sci. USA, 93:59 595, 996. [8] D. Adalseinsson and J. A. Sehian. J. Compu. Phys., 48:, 999. [9] M. Herrmann. A domain decomposiion paralleliaion of he Fas Marching Mehod. Annual Research Briefs-3, pp. 3 6. Cener for Turbulence Research, Sanford, CA, 3. [] S. Osher and J. A. Sehian. J. Compu. Phys., 79: 49, 988. [] A. Bourlioux. Sixh Inernaional Symposium on Compuaional Fluid Dynamics, volume IV, pp. 5, Lake Tahoe, NV, Sepember 4-8 995. [] M. Sussman and E. Faemi. SIAM J. Sci. Compu., (4):65 9, 999. [3] S. P. van der Pijl, A. Segal, C. Vuik, and P. Wesseling. A mass-conserving level-se mehod for modeling muli-phase flows. submied o In. J. Numer. Meh. Fluids, 4. [4] D. Enrigh, R. Fedkiw, J. Feriger, and I. Michell. J. Compu. Phys., 83:83 6,. [5] M. Sussman, A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome. J. Compu. Phys., 48:8 4, 999.
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