Introduction. by a source term ( ) 242 / Vol. XXVIII, No. 2, April-June 2006 ABCM. Denise Maria V. Martinez et al

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1 ρ ρ Dense Mara V. Martnez et al Dense Mara V. Martnez Departamento de Matemátca Fundação Unversdade Federal do Ro Grande Ro Grande, RS, Brazl Edth Beatrz C. Schettn Insttuto de Pesqusas Hdráulcas Unversdade Federal do Ro Grande do Sul Cx. Postal Porto Alegre, RS, Brazl Jorge Hugo Slvestrn Senor Member, ABCM Departamento de Eng. Mecânca e Mecatrônca Pontfíca Unv. Católca do Ro Grande do Sul Porto Alegre, RS, Brazl The nfluence of Stable Stratfcaton on the Transton to Turbulence n a Temporal Mxng Layer The transton to turbulence n a stably stratfed flow s a problem of consderable nterest n flud dynamcs wth applcatons n both geophyscal scences and engneerng. Ths transton s controlled by competton between the vertcal shear of the base flow and the buoyancy forces due to the densty stratfcaton. The present work nvestgates numercally the effect of stable stratfcaton on the development of a Kelvn-Helmholtz (KH) nstablty and the formaton of streamwse vortces, whch are developed after the saturaton of the prmary bllows of KH. The Drect Numercal Smulaton (DNS) technque was used to solve the complete Naver-Stokes equatons, n the Boussnesq approxmaton. Numercal tests were done wth dfferent Rchardson numbers and forced ntal condtons for velocty fluctuatons. The results showed that hgh stratfcaton nhbts the parng process, reduces the buoyancy flux, weakens the vertcal motons, decreases the thckness of the mxng layer and affects the formaton of streamwse vortces. The three-dmensonal results demonstrated that the streamwse vortces are clearly formed n non-stratfed cases. In the stratfed cases, on the other hand, the streamwse vortces are weakened, due to the streamwse densty gradent, whch decrease the levels of vortcty n the bllows of KH, whle ncrease n the brad zone. Keywords: Stable stratfcaton, drect numercal smulaton, mxng layer, buoyancy force, transton to turbulence Introducton From a fundamental pont of vew, the transton to turbulence n a stably stratfed flow s nterestng because t plays a key role n better understandng the mxng and dynamcal processes n envronmental or ndustral problems. The dynamcs of the stably stratfed mxng layer s controlled by a competton between the vertcal shear of the base flow and the buoyancy forces due to the densty stratfcaton. The buoyancy effects act reducng the growth rate of perturbaton and delayng the transton to turbulence, whle the shear supples knetc energy to the flow. The evoluton of such flows s commonly studed n terms of a model problem: the mxng layer. The stably stratfed mxng layer develops at the nterface of two parallel streams of flud movng horzontally at dfferent veloctes and havng dfferent denstes, the upper stream beng lghter than the lower one. Mles (196 and Howard (196, based on a lnear stablty analyss, showed that for the Kelvn-Helmholtz (KH) nstablty to occur n stratfed mxng layer from an nfntely small dsturbance, the Rchardson number must be less than 0.5 somewhere wthn the flow. Ths frst nstablty that occurs n the mxng layer s due to the nflectonal nature of the velocty profle (Mchalke, 1964): the vortex sheet ntally created s lnearly unstable and rolls up to form the prmary bllows of KH. The second stage of the transton n the mxng layer occurs due to the formaton of streamwse vortces (known as ''rbs'' vortces), whch develop after the prmary bllows of KH. The transton to turbulence n the non-stratfed mxng layer has been wdely studed n the last two decades (Lasheras and Cho, 1988; Rogers and Moser, 199; Comte et al., 199; Moser and Rogers, 1993; Comte et al., 1998). 1 Perrehumbert and Wdnall (198) found that a perodc array of Stuart vortces, wth a smlar confguraton of KH vortces, s Presented at ETT th Sprng School on Transton and Turbulence September 7 th - October 1 st, 004, Porto Alegre, RS, Brazl. Paper accepted: May, 005. Techncal Edtor: Arsteu da Slvera Neto. unstable to dfferent three-dmensonal (3D) dsturbances. They demonstrated that ths unstable mode s characterzed by spanwse oscllaton n phase wth prmary bllows of KH. Ths nstablty, called translatve, s well known n the lterature as beng the responsble for the begnnng of the three-dmensonalty n the nonstratfed mxng layer and, consequently, for the formaton of streamwse vortces. The most unstable spanwse wavelength (λ y ) for a translatve nstablty was found to be /3 of the separaton between KH vortces (streamwse wavelength, λ x ). Metcalfe et al. (1987) studed through Drect Numercal Smulaton (DNS) the evoluton of a non-stratfed mxng layer (R=0.0) and nvestgated a secondary nstablty nduced by dfferent ntal velocty condtons. They showed that the streamwse vortces are characterstc structures of 3D mxng layers and ts generaton depends strongly on the ntal condton. In stratfed mxng layers, the 3D process s more complex than n non-stratfed mxng layers. Ths fact s due to the greater number of secondary nstabltes that propagate n the flow. The nstabltes that develop n a 3D stably stratfed mxng layer may be dvded n two groups: one that grows wthn the vortex core and the other that develops n the regon between the cores (the brads). Wthn the cores two types of nstabltes are found. The one dscovered by Perrehumbert and Wdnall (198) that does not depend upon the buoyancy effects and the gravtatonal convectve nstablty that s drven by buoyancy effects. The other that develops n the regon between the cores was predcted by Klaassen and Pelter (199 and verfed by Schowalter et al. (1994) n laboratory experments. The nstablty that grows n the brads s called secondary shear nstablty (Caulfeld and Pelter, 1994). The gravtatonal convectve nstablty and the secondary shear nstablty that are restrcted to a stratfed mxng layer are caused by the streamwse densty gradent mposed by the buoyancy force. The gravtatonal convectve nstablty makes unstable the sub layers of densty generated durng the roll-up of the KH bllows. The secondary shear nstablty nduced by the streamwse densty gradent develops n the brad regon between two Kelvn- Helmholtz vortces. The presence of secondary shear nstablty n a stratfed mxng layer s due to baroclnc vortcty generaton gven g 0 x, whch concentrates the vortcty n by a source term ( ) 4 / Vol. XXVIII, No., Aprl-June 006 ABCM

2 ν ω The nfluence of Stable Stratfcaton on the Transton to... the baroclnc layer (Staquet, 1995). Thus, the densty gradent source term contrbutes as an extra mechansm for the generaton or destructon of local vortcty by means of the baroclnc torque (Caulfeld and Pelter, 000; Cortes et al., 1998). The Rchardson number effects on the secondary shear nstablty has been studed n some detal by Klaassen and Pelter (199 by performng stablty analyses n the lnear range. The strongest growth-rate for ths knd of nstablty was found to occur between R = 0.08 and R = 0.1, and to approach zero growth rate at R = 0., whch s close to the crtcal nvscd value, R = 0.5, for the prmary KH nstablty (Mles, 196. They found that the most unstable spanwse wavelength s smaller than the streamwse one, for R = 0.0 to 0.04, whereas for hgher stratfcaton the wavelength does not vary sgnfcantly. Schowalter et al. (1994) nvestgated expermentally the threedmensonalzaton of a free shear layer for dfferent Rchardson numbers, wth forced condton. They notced that the streamwse vortces were weakened or renforced by stratfcaton, dependng on ther locaton. On the top the KH bllows, when unstably stratfed regons were formed (heavy flud above, ( ρ z < 0) ), strong streamwse vortcty was measured, whle n the brads the rbs were weak. The present work nvestgates the nature of transton to turbulence n a stably stratfed temporal mxng layer through the drect numercal smulaton (DNS). The man objectve of ths work s to analyze the nfluence of a stable stratfcaton on the development of the KH nstablty and n the formaton of streamwse vortces. In ths study t s shown that the stratfcaton affects the formaton of streamwse vortces, even when forced condtons are used. The DNS technque s used to solve the complete Naver-Stokes equatons, n the Boussnesq approxmaton. Drect numercal smulatons are excellent nstruments for the nvestgaton of the dynamcs of a stably stratfed mxng layer, snce they solve entrely all the spatal and temporal scales of the flow. In the followng sectons frst the governng equatons and the numercal method are presented. Then, code valdaton test cases are presented comparng the DNS results wth results from the lnear stablty theory. The followng secton descrbes the results from the two-dmensonal and three-dmensonal smulatons, for dfferent Rchardson numbers (R = 0; 0.1; 0.), showng the evoluton of the mxng layer, wth the formaton of baroclnc layer and of the streamwse vortces, respectvely. Fnally the conclusons and fnal comments are presented. Nomenclature R = Rchardson number. Re = Reynolds number. R = Rato of the ntal vortcty thckness to the densty thckness. U = Reference velocty. P = Pressure. t = Tme. u = Velocty feld. x, y, z = Streamwse, spanwse and vertcal coordnate drectons. u, v, w = Streamwse, spanwse and vertcal velocty components. Pr = Prandtl number. L x, L y, L z = Streamwse, spanwse and vertcal doman length. g = Acceleraton due to gravty. n x, n y, n z = Number of ponts n the streamwse, spanwse and vertcal drectons. Greek Symbols ρ (x, y z, t) = Densty or actve scalar. ρ 0 = Reference densty. ρ = Densty dfference across the shear layer. δ = Intal vortcty thckness. δ /U = Advectve tme scale. δ d = Densty thckness. α a = The most amplfed wavenumber. λ a = The most unstable wavelength. λ I = Fundamental wavelength. λ x, λ y = Streamwse wavelength, spanwse wavelength. ν = Knematc vscosty. Formulaton and Numercal Method Governng Equatons The equatons that govern the flud moton are the Naver- Stokes equatons wth the Boussnesq approxmaton, n a Cartesan frame of reference R = ( 0; x, y,z ). The momentum equaton for the velocty feld u, wth components ( u,v,w ), s gven by: u 1 = P u R ρ z + u, ( t Re where P = p + ( ρ u / ) s the modfed pressure feld. The contnuty equaton s:. u = 0, () and the transport equaton s: ρ + Uδ 1 ( u ) ρ = ρ. (3) t RePr The varables used n the above equatons are non-dmensonal. There are two non-dmensonal relevant parameters: the Reynolds and the Rchardson numbers. The Reynolds number, based on the half velocty dfference across the shear layer (U) and on the ntal vortcty thckness ( δ ), s defned by: Re =, (4) ρ δ where U δ =. (5) ( du / dz ) max The Rchardson number s defned by: R = (6) 0 U In these equatons, the scales of length, velocty and densty are such that δ =1, U =1 and ρ = 1/R. In ths manner, Re = 1/ν and R = g /ρ 0. g R ρ J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 006 by ABCM Aprl-June 006, Vol. XXVIII, No. / 43

3 α ω, ρ ββ Dense Mara V. Martnez et al Intal and Boundary Condtons The ntal condtons are defned n terms of the velocty and densty felds as n Corcos and Sherman (1976) and Patnak et al. (1976). At the ntal tme (t = 0) the densty feld s ρ(x, y, z,t =0)= ρ 0 +ρ (z). In the present case, no densty fluctuaton s supermposed upon ρ (z) at t = 0. The ntal profles of velocty and densty, at t = 0 are: π z u( z, t = 0) = U erf (7) δ 1 π R z ρ( z, t = 0) = erf (8) R δ Wth the objectve to promote the development of the KH nstablty and to unchan the formaton of the KH bllows, a feld of perturbatons was added to the basc velocty profle. Ths feld s composed by two waves correspondng to the most amplfed wave number (α a ) and ts frth sub-harmonc (α a /), and a snusodal wave n spanwse drecton. The assocated most unstable wavelength gven by the lnear stablty theory s approxmately λ a = 7 δ where 1 the most amplfed wave number α a =π/λ a beng δ, Mchalke, (1964). These perturbatons promote, respectvely, the development of the KH nstablty, the parng process and the formaton of streamwse vortces. The boundary condtons for the temporal mxng layer are: -perodc: used n the streamwse (x) and spanwse (y) drectons; and -free-slp: used n the vertcal drecton (z). Ths condton mposes the followng restrctons: u z = v z = 0 and w = 0 for z = ± L z. Numercal Method Equatons ( to (3) are solved numercally, n the doman shown n Fg. 1, usng a sxth-order compact fnte dfference scheme (Lele, 199) to evaluate spatal dervatves. Fgure 1. Schematc vew of the doman. The compact schemes are mplct ones that relate the value of the dervatve n a pont to the value of the dervatve n the neghborng ponts. For the spatal dscretzaton consderng a unform mesh, where the ndependent varable for each node s ξ = ( ξ, 1 N and ξ = x, y or z, the functon values for the nodes are f = f ( ξ ) and the frst dervatve f ' = f ' ( ξ ) s gven by: f a α f ' ' ' 1 + f + α f + 1 = + 1 f 1 f f + b +. ξ 4 ξ The second dervatve s gven by: f a α f " " " 1 + f + α f + 1 = + 1 f + f 1 f f b ξ 4 ξ f. (9) (10) The sxth order s obtaned wth the set of parameters (Lele, 199): α =, a =, b = for Eq. (9), α =, a =, b = for Eq. (10) Equatons (9) and (10) are vald for the three spatal drectons (x, y, z) n all the mesh ponts. The tme ntegraton s performed wth a thrd-order low-storage Runge-Kutta method (Wllamson, 1980). The ntegraton of Eq. ( at tmes t n and t (n+ s performed through three fractonal tme steps ( 0 ) ( n ) ( 3 ) ( n+ p = 0, 1,, where u = u and u = u, α p ( F u ( p + u ( p ) = t, (1 p ) + β F ( p + ( p + p u ( p +1 ) = 0, ( where 1 F = u R z + u, (13) Re t(n+ (p+ 1 = P dt (14) t t n β and α p, β p, are coeffcents to each fractonal step p, gven by Wllamson (1980): 8 0 =, 0 = 0; =, 1 = ; =, =. 4 1 The Eq. (1 can be splt nto two steps, 44 / Vol. XXVIII, No., Aprl-June 006 ABCM

4 The nfluence of Stable Stratfcaton on the Transton to... * ( p ) u u = t α ( p ) pf + β ( p pf, (15) ( p+ * u u ( p+ =, (16) t In ths conventonal fractonal method, step p+1 s obtaned by solvng the Posson equaton. Thus, the ncompressblty condton s ensured as follows, ( p+ u =. (17) t N3 wth the numercal results of Hazel (197), shown n Tab.1, gves good agreement for the temporal growth rates the stably stratfed mxng layer. Table 1. Comparson of amplfcaton rate wth the reference value or dfferent grds. Amplfcaton rate R N N N Ref. Value Hazel (197) More detals about the numercal code can be found n Lardeau et al. (00) and Slvestrn et al. (00). Equaton (3) s solved n the same way as Eq. ( by makng, ρ ( p+ t where ρ ( p ) = α ( p ) pg + 1 G = ( u. ) ρ + ρ. RePr β ( p+ pg, (18) Code Verfcaton - Amplfcaton Rate In order to valdate the numercal code the evoluton of a small dsturbance was consdered n a D doman. The results were compared wth the lnear stablty theory, where the dsturbance s descrbed by the Taylor-Goldsten equaton (Hazel, 197). The computatonal doman used s a square of sde L =7 δ, 1 correspondng to the most amplfed wave number α a = δ gven by the lnear stablty theory. The Reynolds number s 300, the Prandtl number s 1 and the Rchardson number tested are 0.0, 0.1 and 0., respectvely. The ntal ampltude of the perturbaton was 10-6 U. As the sze doman s 7 δ only the development of a sngle KH bllow happened, excludng the possblty of parng process. Tests wth dfferent computatonal grds of n x nz ponts were done (see Tab.. In these tests the vertcal dffuson term, correspondng to the streamwse velocty was canceled. Ths dffuson ncreases the wdth of the shear layer durng the smulaton and mples a varaton n tme of the base flow, nducng varaton of the amplfcaton rate (Mederos et al., 00). As expected, the grd sze has a great nfluence over the amplfcaton rate. In the test wth a computatonal grd of N = ponts, t was notced that the streamwse resoluton nterferes n the evoluton of the wave ampltude (stratfed case), when comparng wth grd N1 (see Tab.. Thus, for the stratfed case (R = 0. the grd N showed a decrease of the amplfcaton rate due to the ncrease n the vertcal resoluton, wth a error of 7% n relaton to the reference value, whereas for R = 0. there s an ncrease n the amplfcaton rate. Probably ths occurs because the streamwse densty gradent s not beng well solved. Fgure shows the tme evoluton of the ampltude for dfferent Rchardson numbers (0; 0.1; 0.) obtaned from the smulaton wth the grd N3 = ponts. Clearly, there s a regon of exponental amplfcaton, whch corresponds to the regme governed by the lnear theory. In ths test, the errors found are of - 0.3% for R = 0, 3.5% for R = 0.1 and 10.% for R = 0.. The comparson of the smulaton Fgura. Ampltude evoluton for smulaton N3. Two-Dmensonal Vsualzatons the Formaton of the Baroclnc Layer In the followng secton, results from D smulatons are presented. Here the computatonal doman s ( Lx, Lz) = (8δ,8δ ), wth a computatonal grd of ponts, Re = 300 and R = 0; 0.1; 0.. The ampltude of the perturbaton supermposed upon the basc velocty profle, for the fundamental and the subharmonc mode, s 1%U and 0.1% U, respectvely. Frstly, the evoluton of a non-stratfed mxng layer s analyzed. The ntal vortcty, whch s modulated by a small perturbaton, progressvely accumulates n the cores of the KH bllows (Fg. 3). These cores are unstable to perturbaton of wavenumber equal to α a /. The growth of ths subharmonc perturbaton leads to the parng of the two vortces (Fg. 3c and Fg. 3d). Fgure 3. Spanwse vortcty felds, for a non-stratfed case, R = 0, at tmes: 9.5; 19.03; 8.55 and J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 006 by ABCM Aprl-June 006, Vol. XXVIII, No. / 45

5 ω. ω ω ρ Dense Mara V. Martnez et al In the non-stratfed case, R = 0, t s observed that the vortcty remanng n the core s substantally greater than that n the brads. On the contrary, when the stratfcaton s hgher ( R = 0.) the vortcty s ncreased n the brads and the baroclnc layer s vsble, as can be seen n Fg. 5d. Fgure 3. (Contnued). In a mxng layer, the prmary KH vortces are not totally solated from each other, but they are rather connected by a thn brad of spanwse vortcty. In a stably stratfed mxng layer, these thn vortcty layers are straned n between the KH vortces and ntensfed by the buoyancy effects (Fg. 4b and Fg. 5b). As the development of the KH nstablty on a stably stratfed mxng layer proceeds, a streamwse densty gradent occurs n between the KH vortces, at the locaton where the brads form. Ths streamwse densty gradent (correspondng to the spanwse component of the baroclnc torque, n the Boussnesq approxmaton) xω zω feeds the brad wth vortcty and forms the baroclnc layer. Ths layer s assocated wth a strong densty gradent and vortcty feld. The baroclnc layer, as t s referred to n Staquet (1995), does not exst n an nonstratfed flow. It forms under the acton of buoyancy effects and strans between the KH vortces. The component of the baroclnc torque along the spanwse drecton s showed below n the vortcty equaton, y y y 1 y y + u + w = R + + (19) t x z x Re The densty gradent source term contrbute as an extra mechansm for the generaton or destructon of local vortcty by means of the baroclnc torque. Ths property yelds a dfferent behavor of the stratfed mxng layer as opposed to ts nonstratfed counterpart. It follows that vortcty can locally ncrease beyond ts maxmum value for the stratfed case. Two secondary nstabltes are propagated n the stratfed mxng layer caused by the streamwse densty gradent: the gravtatonal convectve nstablty and the secondary shear nstablty. The frst s found wthn unstable regons of the KH core, whch consst of heavy and lght flud wrapped n a spral roll. Wherever heavy flud s found on top of lght flud, the gravtatonal convectve nstablty amplfes due to buoyancy forces. Durng the roll-up of KH bllows, heavy and lght flud are brought together from the outer-sde of the mxng layer towards the center of mxng layer, yeldng a strong densty gradent there. In ths way, the secondary shear nstablty nduced by the streamwse densty gradent concentrates the vortcty n the baroclnc layer. The baroclnc layer s observed to occur n our smulatons (Fg. 4d and Fg. 5d). Fgure 4 and 5 show the development of the Kelvn-Helmholtz nstablty, n the stratfed mxng layer, for the same characterstc tmes. These pctures show that when the Rchardson number s ncreased the parng process s nhbted and the brad vortcty exceeds the core's one. The stable stratfcaton, through streamwse varatons of the densty feld, weakens the vertcal motons and reduces the buoyancy flux, as t can be observed n Fg. 6. Fgure 4. Spanwse vortcty felds, stratfed case, R = 0.1, n tmes: 9.5; 19.03; 8.55 and Fgure 5. Spanwse vortcty felds, stratfed case, R = 0., n tmes: 9.5; 19.03; 8.55 and / Vol. XXVIII, No., Aprl-June 006 ABCM

6 The nfluence of Stable Stratfcaton on the Transton to... a decrease of the overall flud entranment, a reducton the mxng process and a decrease of the thckness of the mxng layer. Fgure 7 shows the tme evoluton of the thckness of the mxng layer for dfferent Rchardson numbers. To quantfy the effect that a stable stratfcaton (buoyancy forces) has on the development of KH nstablty, the tme evoluton of the knetc energy was calculeted. The knetc energy per unt area s gven by L 1 z K = LxLz Lx ( u u ) ( w' ) ) dxdz + 0 0, (0) where the ensemble average of the streamwse velocty s u 1 Lx = u dx. ( Lx 0 Fgure 8 shows the evoluton of the knetc energy for dfferent Rchardson numbers. The peak of the maxmum knetc energy s attaned for the frst parng (non-stratfed and stratfed cases), and then t oscllates and ncreases near the second parng for the nonstratfed case. It seems clear from Fg. 8 that the maxmum KH vortcty ampltude s lmted by the presence of a strong stable stratfcaton (R 0. n the flow and that the tme at whch the maxmum energy s reached s consderably delayed when the stratfcaton s ncreased. These characterstcs are consstent wth the expectaton that the wave needs to execute work aganst the gravtatonal potental n order to grow. Three-Dmensonal Vsualzatons - The Formaton of Streamwse Vortces The computatonal doman s a paralleleppedc box, of sde 14 δ along the longtudnal (x) and vertcal (z) drectons and of sde 10.5δ along the spanwse (y) drecton. All the three-dmensonal tests were carred out wth the resoluton (18, 96, 19) along the (x, y, z) drectons, respectvely. The spanwse length of the doman, L x, s /3 of L y. Ths choce was made to force the most amplfed mode n the spanwse drecton predcted by Perrehumbert and Wdnall (198). The parameters used n the tests are the Reynolds number, Re = 00, the Rchardson number, R = 0 (non-stratfed case), 0.1 and 0. (strong stratfcaton) and Prandtl number, Pr = 1. Fgure 6. Buoyancy flux, ρ w, at dfferent tmes, Re = 300. R =0; R = 0.1; R = 0.. The spreadng of the buoyancy flux ρ w (Fg. 6) s due to the buoyancy effects. In the non-stratfed case, there s no buoyancy force and the densty s a perfect passve scalar (Fg. 6a). At tme t = 8.55 the vortces are dslocated n relaton to the z-center of the doman (Fg. 6b), producng negatve values of the ρ w. Fgure 6b and Fg. 6c show that there s a strong decrease of the buoyancy flux caused by ncreasng the Rchardson number. Ths drastc reducton of the buoyancy flux as the Rchardson number ncreases causes the reducton of flud entranment nto KH vortces and delays parng process of the large structures. Therefore, the stable stratfcaton has a stablzng effect on the growth of the prmary KH nstablty (Fg. 4 and Fg. 5), resultng n Fgure 7. The effect of the bulk Rchardson number on the evoluton of the thckness of the mxng layer, Re = 300. J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 006 by ABCM Aprl-June 006, Vol. XXVIII, No. / 47

7 ω. Dense Mara V. Martnez et al R = 0 R = 0.1 Fgure 8. The effect of the bulk Rchardson number on the evoluton of the wave knetc energy, Re = 300. Three-dmensonal vsualzatons are presented for 3 dfferent Rchardson numbers for the same nstant of tme, wth sosurfaces of vortcty modulus ω = 0. 5ω, where ω = U / δ. The ampltude of the perturbaton (forced) supermposed upon the basc velocty profle, for the fundamental, the subharmonc and spanwse mode s 1%U, 0.1%U and 0.1%U, respectvely. The ntal condton s the same for all the numercal tests. At tme t = 9.5 only the non-stratfed mxng layer (R = 0) was unstable, showng the formaton of two longtudnal wavelength, characterzng the Kelvn-Helmholtz nstablty (Fg. 9). At tme t = 19.0, for the non-stratfed case, R = 0, there are two prmary bllows KH (spanwse vortces), due to the KH nstablty and fundamental mode, whereas for the stratfed case o roll-up occurs (R = 0. and D nstablty s formed n the vortcty layer (R = 0.). Ths occurs because the stratfcaton affects the vertcal movements, dmnshng the ntensty of the oscllatons and reducng the knetc energy, thus delayng the formaton of the KH structures, as t can be verfed n the D smulaton above and n Martnez et al. (004). Fgure 10 shows the results obtaned for R = 0; 0.1; 0. at tmes and In the non-stratfed case, at tme t = 76.19, streamwse vortces are clearly formed. The two pars of counterrotatng streamwse vortces appear between the tmes 47.6 and 57.14, one par for each fundamental wavelength. The streamwse vortces, or rbs, show a hgh degree of coherence and are extend wth nearly the same vortcal ntensty over the complete brad regon. In the stratfed cases, on the other hand, the secondary nstablty domnates the three-dmensonalzaton of the layer and the streamwse vortces are now seen to be less developed over the complete doman. Ths s due to streamwse densty gradent that decreases the levels of vortcty n the KH bllow whle ncreases t n the brad zone. Ths effect s easer to see n hghly stratfed flows, R = 0., than n "mldly" stratfed flow, R = 0.1. Fgure 10 shows the translatve nstablty actng on the resultant vortex of the parng process, whereas for R = 0. the flow s stll two-dmensonal. The three-dmensonalzaton of the mxng layer s domnated by the ntal forcng condton of the sub harmonc and spanwse modes. As sub harmonc mode grows before the spanwse mode, the formaton of the streamwse vortces s delayed. The stratfcaton also delays the parng process and development of the translatve nstablty. Ths fact can be observed when comparng the non-stratfed case wth the stratfed one n Fg. 9 and 10. R = 0. Fgure 9. Isosurfaces of vortcty modulus at t = 9.5 and 19.0, Re = 00, = 0.5 Fgure 11 shows the sosurface of vortcty modulus at tme and 114.9, for the two stratfed cases consdered. In the test wth R = 0.1 a deformaton n the structure of the streamwse vortces s observed. Ths deformaton, probably caused by baroclnc effects, can be orgnated by the translatve nstablty occurrng n the regon between the bllows and not n ts core, as n the non-stratfed case (Caulfeld and Pelter, 000). For R = 0. the stablzng effect of the stratfcaton becomes strongly vsble n the reducton of the ampltude of the 3D dsturbances. It s an ndcaton of the qualtatve varaton of the dynamcs of the flow, where buoyancy forces nhbt the growth of the turbulent knetc energy, manly the vertcal speed. Ths effect can be observed n Fg. 1, whch ndcates that when the Rchardson number ncreases the buoyancy flux decreases. To be able to have the best possble pcture of the vortces spatal structure the Q crteron was used (Dubef and Delcayre, 000; Jeong and Hussan, 1995). It s well known that regons of hgh vortcty often correspond to coherent structure locatons, or even sheared zones wthout any structures. The Q crteron s defned by: 1 u u Q j 1 1 = = Ω S = P, () x j u where, j = 1,,3 and rate. Ω s the rotaton rate and S the stran 48 / Vol. XXVIII, No., Aprl-June 006 ABCM

8 The nfluence of Stable Stratfcaton on the Transton to... R = 0 R = 0.1 R = 0. Fgure 10. Isosurfaces of vortcty modulus at t = and 76.16, Re = 00, ω= 0.5. R = 0.1 R = 0. Fgure 1. Buoyancy flux 38.09; ρ' w', at R = 0; 0.1; 0., Re = 00, at tme. Vortcty modulus vsualzaton emphaszes ntermedate scale vortces, though t may obscure the large structures. The postve Q values occur n flow regons where local rotaton s predomnant, especally n regons assocated wth the vortex cores. In the Q crteron, a balance between the rotaton rate and stran rate was done. Ths mples that postve Q sosurfaces solate areas where the strength of rotaton overcomes the stran. In Fg. 13, t can be observed a comparson between sosurfaces of vortcty modulus and Q-sosurfaces for the non-stratfed case, at tme t = 76.19, and for the stratfed cases at tmes t = (R = 0. and t = (R = 0.). Fgure 14 shows spanwse cross-sectonal plots of the vortcty modulus for a doman of two fundamental streamwse wavelengths (λ I ), for one characterstc tme and wth forced ntal condtons. The longtudnal varaton of the doman vares s 0 < x < 14δ. At t = 76.19, streamwse vortces are clearly formed n a non-stratfed mxng layer whle for the stratfed cases only some concentraton of vortcty may be dentfed, manly for R = 0.1 at the center of the mxng layer. Fgure 11. Isosurfaces of vortcty modulus at t = and 114.9, Re = 00, ω= 0.7. J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 006 by ABCM Aprl-June 006, Vol. XXVIII, No. / 49

9 Dense Mara V. Martnez et al R = 0 R = 0.1 R = 0. (e) (f) Fgure 13. Isosurfaces of ω=.5, R = 0; Q sosurfaces Q = 1, R = 0; Isosurfaces of ω= 0.7, R = 0.1; Q sosurfaces Q = 0., R = 0.1; (e) Isosurfaces of ω= 0.7, R = 0.; (f) Q sosurfaces Q = 0.08, R = 0.. At a later tme, t=104.77, Fg. 15 shows ntense streamwse vortces for the two stratfed cases consdered. Whle ther vortcty s lower than the non-stratfed case, the formaton mechansm seems to be related to the secondary shear nstablty. Comparng the spanwse cross-sectonal plots n Fg.14a for an non-stratfed flow wth those from Fg.14b and Fg.14c, we can remark how the entranment process sgnfcantly nfluences the stratfcaton: the vertcal extent of the actve regon, roughly correspondng to that occuped by entraned flud, has narrowed. Ths s also apparent from the development of the sze of the mxng layer for dfferent R, where entranment s progressvely suppressed wth ncreasng stratfcaton. Therefore, n stratfed cases the longtudnal structures are confned n a shorter vertcal length that for an non-stratfed case, and they are not developed over the whole vertcal doman. (e) Fgure 14. Spanwse cross-sectonal plots of vortcty modulus at x equal ;` 4; 6; 8 and (e) 10 δ, at tme t = / Vol. XXVIII, No., Aprl-June 006 ABCM

10 The nfluence of Stable Stratfcaton on the Transton to... R = 0.1 R = 0. energy between the KH vortces and the flow, lmts the maxmum KH wave ampltude and reduces the buoyancy flux. The 3D smulatons, wth forced condtons, showed that streamwse vortces are clearly formed n a non-stratfed mxng layer. The streamwse vortces, or rbs, showed a hgh degree of coherence and are spread wth nearly the same vortcal ntensty over the complete brad regon. On the other hand, n the stratfed cases the secondary shear nstablty domnates the three-dmensonalzaton of the layer. Ths s due to the streamwse densty gradent whch decreases the levels of vortcty n the KH bllow whle ncreases n the brad zone. Ths effect s easer to be seen n hghly stratfed flows, R = 0., than n a "mddly" stratfed flow, R = 0.1. Acknowledgments The frst author acknowledges the fnancal support receved from CAPES-PICDT. (e) Fgure 15. Spanwse cross-sectonal plots of vortcty modulus at x equal ;` 4; 6; (d )8 and (e) 10 δ, at tme t = Conclusons The purpose of the present study was to nvestgate numercally the effect of the stable stratfcaton (buoyancy forces) on the development of the KH nstablty and the formaton of streamwse vortces, whch are developed after the saturaton of the prmary bllows of KH. The D smulatons showed that hgher stratfcaton ncreasngly nhbts the parng process, reduces the exchange of References Caulfeld, C. P., and Pelter, W. R Three-dmensonalzaton of the stratfed mxng layer. Phys. Fluds, Vol.6, pp Caulfeld, C. P., and Pelter, W. R The anatomy of the mxng transton n homogenuos and stratfed free shear layers. J. Flud Mech., Vol.413, pp Comte, P., Leseur, M., and Lamballas, E Large and small-scale strrng of vortcty and a passve scalar n a 3-d temporal mxng layer. Phys. Fluds A, Vol.4,pp Comte, P., Slvestrn, J. H., and Bérgou, P Streamwse vortces n Large Eddy Smulaton of mxng layers. Eur. J. Mech. - B/Fluds, Vol.17, pp Corcos, G. M., and Sherman, F. S Vortcty concentraton and the dynamcs of unstable free shear layers. J. Flud Mech., Vol.73, part, pp Cortes, A. B., Yadgaroglu, G., and Banerjee, S Numercal nvestgaton of the formaton of three-dmensonal structures n stablystratfed mxng layers. Phys. Fluds, Vol.10, pp Dubef, Y., and Delcayre, F On Coherent-vortex dentfcaton n turbulence. J. Turbulence, Vol.1, n.11, pp.1-. Hazel, P Numercal studes of the stablty of nvscd stratfed shear flows. J. Flud Mech., Vol.51, pp Howard, L. N Note on a paper of John W. Mles. J. Flud Mech., Vol.10, pp Jeong, J., and Hussan, F On the dentfcaton of a vortex. J. Flud Mech., Vol.85, pp Klaassen, G. P., and Pelter, W. R The nfluence of stratfcaton on secondary nstablty n free shear layers. J. Flud Mech., Vol.7, pp Lardeau, S., Lamballas, E., and Bonnet, J. P. 00. Drect Numercal Smulatons of a jet controlled by flud njecton. J. Turbulence, Vol.3 Lasheras, J. C., and Cho, H Three-dmensonal nstablty of a plane free shear layer: an expermental study of the formaton and evoluton of streamwse vortces. J. Flud Mech., Vol.188, pp Lele, S. K Compact fnte dfference schemes wth spectral-lke resoluton. J. Comp. Phys., Vol.103, Martnez, D. M. V., Schettn, E. B. C., and Slvestrn, J. H. 004 Transton to turbulence n a stable stratfed temporal mxng layer through drect numercal smulaton. In: Proceedngs of the 10th Congress of Thermal Scences and Engneerng, ABCM. Mederos, M. A. F., Slvestrn, J. H., and Mendonça, M. T. 00. Usng lnear and non lnear stablty theory for evaluatng code accuracy. In: Proceedngs of the III Escola de Prmavera de Transção e Turbulênca, ABCM. Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S., and Rley, J. J Secondary nstablty of a temporally growng mxng layer. J. Flud Mech., Vol.184, pp Mchalke, A On the nvscd nstablty of the hyberbolc tangent velocty profle. J. Flud Mech., Vol.19, pp Mles, J. W On the Stablty of heterogeneous shear flows. J. Flud Mech., Vol.10, pp Moser, R., and Rogers, M. M The three-dmensonal evoluton of a plane mxng layer: parng and transton to turbulence. J. Flud Mech., Vol.47, pp J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 006 by ABCM Aprl-June 006, Vol. XXVIII, No. / 51

11 Dense Mara V. Martnez et al Patnak, P. C., Sherman, F. S., and Corcos, G. M A numercal smulaton of Kelvn-Helmholtz waves of fnte ampltude. J. Flud Mech., Vol.73, Perrehumbert, R. T., and Wdnall, S. E The two and three dmensonal Instabltes of a spatally perodc shear flows. J. Flud Mech., Vol.114, Rogers, M. M., and Moser, R The three-dmensonal evoluton of a plane mxng layer: the Kelvn-Helmholtz rollup. J. Flud Mech., Vol.43, Showalter, D. G., Atta, C. W. Van, and Lasheras, J. C A study of streamwse vortex struture n a stratfed shear layer. J. Flud Mech., Vol.81, pp Slvestrn, J. H., and Lamballas, E. 00. Drect numercal smulatons of wakes wth vrtual cylnders. Int. J. Comp. Flud Dyn., Vol.16, n.4, pp Staquet, C Two-dmensonal secondary nstabltes n a strongly stratfed shear layer. J. Flud Mech., Vol.96, pp Wllamson, J. H Low-storage Runge-Kutta schemes. J. Comp. Phys., Vol.35, pp / Vol. XXVIII, No., Aprl-June 006 ABCM