Heat and mass fluxes across a density interfaces submitted to a gridgenerated

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1 Heat an mass fluxes across a ensity interfaces submitte to a grigenerate turbulence Pascal DUPONT 1&2 et Hassan PEERHOSSAINI 2 1 LGCGM, EA3913, INSA e Rennes, Campus Beaulieu, Rennes ceex 2 Thermocinétique, UMR6607, Polytech Nantes, BP 50609, Nantes ceex pascal.upont@insa-rennes.fr Abtract : The present stuy was first motivate by the inustrial problem of the Liquefie Natural Gaz storage tank where the stratification between two ifferent LNG layers coul be estroye by the increasing unstable thermal stratification. Then the choice was to reprouce the historical experiments of Turner with a thin stratifie interface separating two homogeneous meia mixe by oscillating gris. Comparisons between thermally or salinity stratifie experiments exhibit the strong epenence of the flux on the molecular iffusivity. A phenomenological moel is propose for these transfer with a clear serial scheme which is ifferent from the previous parallel ones. It gives a clear unerstaning of the influence of the molecular iffusivity on the flux across those ensity interfaces. Résumé : La présence une stratification stable en ensité moifie profonément l évolution es cuves e stockage e Gaz Naturel Liquéfié en inhibant les mouvements verticaux et les transferts e chaleur et e masse inuits. Nous avons simplifié l étue en consiérant le cas une interface fine séparant eux milieux homogènes agités par une turbulence e grille oscillante. Les résultats expérimentaux sont présentés e manière à mettre en exergue l influence u coefficient e iffusion moléculaire et à proposer un moèle phénoménologique évolution e ces stratifications. Key-wors : Heat an mass transfer ; stratifie meium ; thermo-haline interface 1 Introuction Stratification of flui meium arose spontaneously either because of the large scale involve as in geophysical case, or because of the relatively large ensity graient in the inustrial case. The various flui motions mix partially the meium that is constitute by wellmixe layer separate by interfaces where the ensity varies rapily. The most stuie situation is that of the ouble-iffusive natural convection where the flow in inuces by the estabilising flux of heat. In that case, Turner have performe the first an still use atabase of the fluxes against the interface ensity ratio R ρ. Linen (1974) have propose a moel of transfer, in which the fluxes by iffusion an entrainment operate in parallel. The total buoyancy flux is then the sum of two fluxes, an the entrainment flux was base on an empirical equation euce from the results of Turner (1968), in the case of a salt stratification. Harina an Fernano (1989) propose a complex moel phenomenological, in which they suppose that the interface has two bounary layers: one thermal an the other salt (as Veronis). Consiering that the key point is the flux of the stratifying property in relation to the energy either potential with a estabilising graient or mechanical with an external flow, Zellouf et al. (2005) have presente new an accurate results of the flux of a single property across ensity interfaces in a grigenerate turbulence. This force convection case was alreay performe by Turner (1968) who has escribe for the first time the important influence of molecular iffusivity. Turner have mae an attempt to preict the relation between the flux an the Peclet number that gives the 1

2 general philosophy of the present work. Turner (1968) suggeste an entrainment law base entirely on imensional arguments which integrates the Peclet number Pe without test it experimentally: u e α Ri Pe u' Following the work of E an Hopfinger (1986), Mory (1991) evelope also an entrainment moel taking into account the effect of molecular iffusivity with an assumption that mixing an entrainment was performe simultaneously by the same range of turbulent eies. Nevertheless his work was essentially theoretical an comparison with particular experiments ifficult. Almost all the stuies consiere an infinitely thin interface. Breinenthal (1992) consiere a thin interface with a thickness less than the kolmogorov microscale so that the interface cannot introuce another length scale into the entrainment problem. Accoring to Fernano (1989), the entrainment occurs by the impinging eies on the interfacial layer that scour an etach thin elements of flui from it an mix them with the rest of the mixe layer. However, when the ensity step ρ between two stratifie mixe layers is important, the kinetic energy containe in the flui is insufficient to entrain flui straight from the lower layer to the upper layer an vice versa( Ri is high). Then, entrainment can only occur on flui which presents a weaker ensity variation with the mixe layer. This cannot exist only if we consier an interface with a known thickness an continuous ensity profile. This interface thickness is the result of a competition between thickening relate to the mixing an the iffusion within the interface, an the thinning own of this interface by entrainment. Therefore the moel which we propose is a serial moel contrary with other moels that exists, with a step of iffusion in the interface an entrainment near its frontiers. The present imensional evelopment is base on three constant which were ajust by comparison with the previous experiments of Zellouf et al. (2005) 2 Inustrial phenomenon Inustrial processes involving ouble-iffusion are numerous but the case of the LNG storage tank is quite unique with very large imensions more than 50 meters. Then the flows behave as complex as geophysical flows. FIG. 1 LNG storage tank with a stratifie interface between two ifferent LNG. 2

3 The figure 1 raws the case of a LNG tank fille with two LNG ifferent in composition: the lighter LNG in the upper layer is able to release the parietal heating through the free surface by evaporation contrary to the heavier LNG in the lower layer where natural convection is blocke by the ensity graient in the interface. The thickness of this interface was estimate aroun 10 meters an evolves with time. As the temperature of the lower layer increases the ensity ifference ecreases up to a conition of instabilities, which can either inuce a progressive mixing with a global entrainment of the interface or can inuce an abrupt incient calle the roll-over. 3 Experiments As the key point to foresee the mixing of the two previous LNG layers is the integration in time of the heat an mass flux across the interface submitte to a turbulent natural convection, we ecie to extract the most simple situation: an interface stratifie by only one property, either temperature or salt, submitte to a well-known gri generate turbulence in orer to be in a force convection 1D transfer case. 3.1 Experimental apparatus The apparatus evelope uring the PhD work of Yacine Zellouf was previously reporte in Zellouf et al. (2005). motor support frame guie tank tank support filling uct ensimeter Vertical ensity profiles are performe by a travelling sampler pipe carrying flui to a ensimeter for the salt case an a thermocouple at the top of the same pipe for the thermal case. Salinity or ensity are then obtaine through our calibrate formula: ρ( T, S) = ρ ( T, S ). 1 α T T + β S S with T 0 =20 C, S 0 =0%, α=0,00031 C -1 an β =0,0072% -1. ( ) ( ) motor FIG. 1 Experimental apparatus square frame insulating stirring gris sampler pipe raining uct isplacement evice of sampler pipe As the transfer is manage by the turbulent flows above an below the interface we performe extensive PIV measurements in orer to know exactly the length an velocity scales of the turbulent eies at the position of the interface exactly in the mile of the tank. In the following the frequency an stroke of the gri oscillation was fixe an the corresponing turbulent scaling was given in the table Heat an mass flux measurements The evolution of the ensity uring the experiments is reporte in the figure 2 an is quite ientical for the two cases. The interface i not move vertically, its thickness was reuce until the graient vanishe. Note that the time uration of the salt stratifie experiment is larger than for the temperature stratifie one. 3

4 Height h (mm) S 0 lower = 3,91 % S 0 upper = 1,48 % S 0 = 2,43 % t=0s t=10856s t=18971s t=26881s t=33738s t=40402s t=46728s Height h (mm) t = 0s t = 600s t = 1200s t = 2400s t = 3420s t = 3900s t = 4500s Density ρ (g.cm -3 ) Température T ( C) (a) (b) FIG. 2 Evolution of vertical ensity profile in time, (a) in salt stratifie case an (b) in a temperature stratifie case From these profiles it is easy two extract the interfacial flux from a buget in the upper (sup) or lower (inf) layer: Aint ( ρinf ct p inf hinf ) = FA t int ( ϕpertes A) t inf For the heat flux: Aint ( ρsupcptsuphsup ) = FA t int + ( ϕpertes A) t sup For the salt flux: Fs = ( ρinf hinf Sinf ) = ( ρsuphsupssup ) In orer to compare the two fluxes we reuce them to the same imension in term of buoyancy fluxes an plotte them against the relative ensity ifference between the layers in figure 3. The most surprising result is the opposite behaviour with a negative power in the case t t 1E-7 1E-6 9E-8 Salt buoyancy flux βfs (m.s -1 ) 8E-8 7E-8 6E-8 5E-8 4E-8 Heat Buyancy Flux αf T /ρc p (m.s -1 ) 1E-7 3E ρ/ ρ 0 1E (a) (b) FIG. 3 Evolution of the interfacial buoyancy fluxes with the ensity ifference ecreasing in time for the salt case (a) an for the thermal case (b). of salt an positive in the case of temperature. For the same range of relative ensity ifference the buoyancy flux is almost higher in the temperature-stratifie case but the value of these fluxes evolve towars the same value for weak stratification. ρ/ ρ 0 4

5 As a matter of fact the ensity ifference has opposite effect increasing iffusive transfer an ecreasing vertical motion in the interface. It seems that the balance between the two is manage by the value of the molecular iffusion which is the only ifference between the two cases. 3 Phenomenological moel Most of the moels intening to follow the transfer across those interfaces istinguishe two moes of transfer: iffusion in the thickness of the interface an irect flui penetration through the interface (Linen 1974). This assumption was in contraiction to the large value of the global Richarson number Ri= (sg ρ/ρ l')/u which is always large except at the roll-over stage. Thus complete penetration of the interface by flui of one layer is not possible. We ecie to buil an equivalent serial scheme where the transfer has always a iffusive step insie the interface (see figure 4) an entrainment (convective transfer) of flui of the interface by the turbulent eies. The opposite effect of the ensity graient is then clearly ientify with the iffusion insie the interface an the entrainment governe by a local Richarson number (figure 4 an Table 1). This moel are then able to foresee the thickness of the interface which is increase by iffusion an is reuce by entrainment. ρ u upper l u Gri oscillation h int (t) 0 δ iff δw Molecular Diffusion Internal wave δ ρ 0 u u (δ) δ u u 0 Viscous bounary layer Ri c <Ri δ e Ri z ρ lower ρ l z z (a) (b) (c) FIG. 4 Schematic iagram of entrainment mechanism with three successive steps: (a) thickening of the interface through molecular iffusion an internal wave mixing, (b) iffusion of momentum an then (c) entrainment in the layer with a local Richarson number lower than a critical value. In orer to take into account the mixing phenomenon insie the interface, a mixing length δ w ue to the internal wave breaking completes the thickening of the interface by iffusion. This turbulent iffusion insie the interface is assume to be proportional to the internal wave amplitue generate by the turbulent eies insie the interface (table 1). l'=12mm u'=2.5 mm.s -1 t=l /u 2 u' δw = Riw δ = C kt δ = δ 1 + δw g ρ ρ u δρ ( z) l' ( z) ρ l' 1 Ri δu = C νt Ri ( z ) = g << Ri = g c hint δ 2 ρu u' ( z) 2 ρu u' 2 e = δ δ u 4 Ri u TABLE 1 values an efinitions of the moel parameters 5

6 In orer to aapt all the assumptions to the reality of the experiments, the values of C 1, C 2, Ri w an Ri c shoul be optimise. There exists an infinite set of values that coul fit the experiments an which shoul be analyse more systematically in the future. In this work we ecie to take classic values of C 1 =1, Ri w =0.25 an the best fit of the moel on both salt an temperature cases have given C 1 =2.7 an Ri w =42. The comparison between the calculate transfer an the experiments is rawn on figure 5 in term of the entrainment rate versus the transient global Richarson number as Turner have one. The slopes in the loglog scaling exhibit the same power values as Turner foun, -1.5 for the salt an 1 for the temperature case. Entrainment rate u e /u' Salt Moel Experiments u e /u' = C Ri -1,5 Entrainment rate u e /u ' Temperature Moel Experiments u e /u' = C Ri -1 1E-3 1E Richarson number Ri Richarson number Ri FIG. 1 Entrainment rate comparison between experiment an moel 4 Conclusions This very simple moel is able to reprouce the estruction of the stratification in the case of a ensity interface stirre by turbulence in a single iffusive case with two extreme imensionless iffusivity k/ν=4 an 800. It is especially able to reprouce the two opposite effects of the property step X on the X flux: X increases the iffusive flux across the interface but it inhibits the entrainment flux. Now the unerstaning of the molecular iffusivity effect is quite clear an future consieration of ouble or multi-iffusive interfaces will be easy. An other quite simple evelopment shoul be to preict these entrainment for numerous ifferent Peclet number in orer to compare the entrainment rates at the same Richarson number an then to obtain a continuous correlation in the (Ri,Pe) space as Turner (1968) an Mory (1991) have propose. References E X. et Hopfinger E.J.1986, On mixing across an interface in stably stratifie flui. J. Flui Mech. 166, pp Fernano H.J.S. 1989,. Buoyancy transfer across a iffusive interface. J. Flui Mech. 209, Linen, P.F. 1974, A note on the transport across a iffusive interface. Deep-Sea Research 21, Mory M. 1991, A moel of turbulent mixing across a ensity interface incluing the effect of rotation. J. Flui Mech. 223, Turner J.S. 1968, The influence of molecular iffusivity on turbulent entrainment across a ensity interface. J. Flui Mech. 33, Zellouf Y., Dupont P., Peerhossaini H. 2005, Heat an mass fluxes across ensity interfaces in a gri-generate turbulence, J. Heat an Mass Transfer 48,