Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Benard convection in glycerol
Kazuyasu Sugiyama, Enrico Calzavarini, Siegfried Grossmann, Detlef Lohse
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t epl draft Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-B´enard convection in glycerol
Kazuyasu Sugiyama , Enrico Calzavarini , Siegfried Grossmann and Detlef Lohse Physics of Fluids group, Department of Applied Physics, J. M. Burgers Centre for Fluid Dynamics, and Impact-,MESA-, and BMTI-Institutes, University of Twente, P. O. Box 217, 7500 AE Enschede, The Netherlands, Fachbereich Physik der Philipps-Universitaet, Renthof 6, D-35032 Marburg, Germany
PACS – Turbulent convective heat transfer
PACS – Buoyancy-driven instabilities
PACS – Direct numerical simulations
Abstract. - We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensionalRayleigh-B´enard flow in glycerol, which shows a dramatic change in the viscosity with temper-ature. The results are presented both as functions of the Rayleigh number Ra up to 10 (forfixed temperature difference ∆ between the top and bottom plates) and as functions of ∆ (“non-Oberbeck-Boussinesqness” or “NOBness”) up to 50K (for fixed Ra ). For this large NOBness thecenter temperature T c is more than 5K larger than the arithmetic mean temperature T m betweentop and bottom plate and only weakly depends on Ra. To physically account for the NOB devia-tions of the Nusselt numbers from its Oberbeck-Boussinesq values, we apply the decomposition of Nu NOB /Nu OB into the product of two effects, namely first the change in the sum of the top andbottom thermal BL thicknesses, and second the shift of the center temperature T c as comparedto T m . While for water the origin of the Nu deviation is totally dominated by the second effect(cf. Ahlers et al. J. Fluid Mech. , 409 (2006)) for glycerol the first effect is dominating, inspite of the large increase of T c as compared to T m . Introduction. –
In most theoretical and numer-ical studies on Rayleigh-B´enard (RB) convection, theOberbeck-Boussinesq (OB) approximation [1, 2] is em-ployed, i.e., the fluid material properties are assumed tobe independent of temperature T except for the density inthe buoyancy term which is taken to be linear in T . Theproblem has two control parameters, namely the Rayleighnumber Ra = βgL ∆ / ( κν ) (here β is the thermal ex-pansion coefficient, g the gravitational acceleration, L theheight, ∆ the temperature difference between bottom andtop plates, κ the thermal diffusivity, and ν the kinematicviscosity), and the Prandtl number P r = ν/κ . For the OBcase the mean temperature profile shows top-bottom sym-metry. However, in real fluids, if ∆ is large, this symmetryno longer holds due to the temperature dependences ofthe material properties. Thus, for given fluid, ∆ appearsas an additional control parameter, which characterizesthe deviations from OB conditions, leading to so calledNon-Oberbeck-Boussinesq (NOB) effects. The NOB sig-natures can be quantified by (i) a shift T c − T m of thebulk (or center) temperature T c from the arithmetic mean temperature T m between the bottom and top plates) and(ii) by the ratio of the Nusselt numbers N u
NOB /N u OB inthe NOB and OB cases, which deviates from one. Bothquantities have been measured in the large Ra regime forhelium [3], glycerol [4], ethane [5], and water [6] as func-tions of the NOB-ness ∆.As shown in Ahlers et al. [6] the Nusselt number ratio N u
NOB /N u OB can be connected to T c by the identity N u
NOB
N u OB = 2 λ slOB λ slt + λ slb · κ t ∆ t + κ b ∆ b κ m ∆ =: F λ · F ∆ . (1)Here the labels on material properties indicate the tem-perature at which they are taken, e.g. κ t = κ ( T t ) etc.∆ t = T c − T t and ∆ b = T b − T c denote the temperaturedrops over the top and bottom thermal boundary layers,and λ slt and λ slb indicate their thicknesses, based on thetemperature slopes at the top and bottom plates, respec-tively. λ slOB is the thermal BL thickness in the OB case,both at top and at bottom. The factor F ∆ can be calcu-lated from the temperature dependences of the materialproperties immediately, once T c is known. Remarkably,p-1. Sugiyama et al. Ahlers et al. [6] experimentally found that for water F λ ≈ . (2)This has been confirmed by numerical simulations of 2DNOB Rayleigh-B´enard convection in ref. [7]. If the relationeq.(2) holds, the Nusselt number ratio Nu NOB Nu OB already fol-lows from the center temperature T c , which for water canbe calculated within a generalized boundary layer theoryintroduced in ref. [6].The objective of this letter is to answer the appar-ently important question whether the relation F λ ≈ λ slb + λ slt ≈ λ slOB , is more generally valid, i.e.,if it holds for other liquids too. We therefore have per-formed (two-dimensional) NOB simulations with glycerolas the working fluid. For glycerol the kinematic viscositydramatically depends on temperature, i.e., one should ex-pect large changes of the boundary layer thicknesses at topand bottom. For instance, ν decreases from 1759 mm / sto 52 . / s if the temperature increases from 15 o C to65 o C. Another advantage of considering glycerol is the ex-istence of experimental data for the center temperature(see ref. [4]) for comparison (but not for the Nusselt num-ber modification). Our main result will be that the sum ofthe boundary layer widths is indeed changed under NOBconditions, i.e., relation (2) does not hold for glycerol. Itsvalidity for water thus turns out to be coincidental, due tothe specific temperature dependences of its material pa-rameters for the chosen temperatures in the experimentsof ref. [6].Note that the fluid flow in glycerol is very different fromthat in water at the same Ra . Due to glycerol’s hugePrandtl number of about P r ≈ Ra c and the loss ofspatial coherence in the flow is much more extended thanfor water or air, whose Prandtl numbers are of order one.While in air and in water this transitions range extendsto about Ra ≈ · to 10 , only beyond which thereis turbulent convection in the bulk, this range extends tomuch larger Ra in glycerol, namely to Rayleigh numbersof order Ra ≈ . Since the numerical calculations coverthe range up to Ra ≈ only, all results refer to a fluidflow still having coherent structures.To quantify these statements we use an averaged Kol-mogorov length η K as a measure for the scale of coherentstructures in the flow, more precisely ℓ coh = 10 η K = 10 ( ν /ε u ) / . (3)Here ε u is the volume average of the energy dissipationrate of the flow for which the well-known exact relation ε u = ν L − P r − Ra ( N u −
1) holds. With this we obtain ℓ coh /L = 10 P r / ( Ra ( N u − − / (4)as an estimate for a volume-averaged relative coherencelength. Taking N u ( Ra, P r ) from the unified theory of −3 −2 −1 l c oh / L Ra −3 −2 −1 l c oh / L Ra Pr=
Pr= Pr= . DNS-OB
Fig. 1: (color online) The coherence length ℓ coh in multiplesof the cell size L versus Ra number for three fluids. For airand water it is of order 0 . Ra ≈ to Ra ≈ . Forglycerol this is reached much later; turbulent heat convection isexpected beyond Ra ≈ only. Gray shaded region indicatesthe developed turbulent regime. Lines are derived from theunified theory of ref. [8], symbols correspond to the Rayleighnumbers of the present OB numerical simulations. refs. [8], one thus obtains an estimate of the coherencelength as a function of Ra and P r from eq. (4), see figure1. The main features of the coherence length are (i) itspronounced explicit dependence on
P r (the implicit de-pendence via
N u is only weak). It is by about a factor √ Ra -dependence is approximately ℓ coh ∝ Ra − . . General description of numerical simulation. –
We numerically solve the incompressible ( ∂ i u i = 0)Navier-Stokes equations ρ m ( ∂ t u i + u j ∂ j u i ) = − ∂ i p + ∂ j ( η ( ∂ j u i + ∂ i u j ))+ g ( ρ m − ρ ) δ i , (5)and the heat-transfer equation ρ m c p,m ( ∂ t T + u j ∂ j T ) = ∂ j (Λ ∂ j T ) . (6)The temperature dependence of the dynamic viscosity η ( T ), the heat conductivity Λ( T ), and the density ρ areexperimentally known for glycerol. They are given in theappendix of reference [6]. As justified in that reference,we can assume the isobaric specific heat capacity c p andthe density ρ in the time derivatives of the material pa-rameters to be constant at their values ρ m and c p,m at thearithmetic mean temperature T m . We vary the Rayleighnumber Ra up to 10 and the level of the NOBness ∆ upto 50K.The container is two-dimensional (2D, no y -dependence), has height L , and aspect ratio 1. Theflow is wall-bounded, i.e., we use no-slip boundary condi-tions at all solid boundaries: u i = 0 at the top ( z = L )and bottom ( z = 0) plates as well as on the side walls x = 0 and x = L . For the temperature at the side wallsp-2on-Oberbeck-Boussinesq effects in Rayleigh-B´enard convectionheat-insulating conditions are employed and T b − T t = ∆is the temperature drop across the whole cell. TheRayleigh number is defined with the material parameterstaken at the mean temperature T m , i.e., Ra = β m gL ∆ ν m κ m .The arithmetic mean temperature is fixed at T m = 40 o C.We vary the Rayleigh number by varying the height L ofthe box, while the NOBness is changed by varying thetemperature drop ∆. Note that in the buoyancy termin eq.(5) the full temperature dependence of the densityis taken into account, rather than employing the linearapproximation ρ ( T ) − ρ m = ρ m β ( T − T m ) only. (Never-theless, the Rayleigh number is defined as usual with thelinear expansion coefficient of the density with respectto temperature, taken at T m , namely β m = − ρ m dρdT | T m .)The Prandtl number is defined as P r = ν m /κ m ; forglycerol at the chosen temperature T m its value is P r = 2495. The basic equations are directly solved onthe two-dimensional domain by means of the fourth-orderfinite difference method. For a detailed description of thesimulation method as well as its validations, see ref. [7].One may worry if two-dimensional simulations are suf-ficient to reflect the dynamics of the three-dimensionalRB convection. For convection under OB conditions thispoint has been analyzed in detail in ref. [9] and earlierin refs. [10–13]. The conclusion is that for
P r ≥ T c − T m or N u
NOB /N u OB . Results and discussions. –
Large scale flow dynamics and temperature snapshots.
In the steady flow regime (
Ra < . · ) a single large-scale circulation role develops, which however disappearsin the unsteady flow regime ( Ra > . · ) and does notreappear up to the largest accessible value Ra = 10 ofthe present study. Even if we start the simulation with anartificial single roll, the large-scale circulation disappearsin the course of time and then isolated plumes (as shownin figure 2) dominate the flow. This feature holds for bothcases, OB and NOB, and is qualitatively different from theobservations in 2D (OB and NOB) simulations in water(see ref. [7]). We attribute this to the much larger spatialcorrelations in glycerol as addressed above. Note that inexperiment (ref. [4]) for larger Ra = 2 . · a large-scale 3D circulation role has been observed for glycerol.The different behavior between the present DNS and theexperiment could either be due to the smaller Ra or to thetwo-dimensionality in the simulation.Typical temperature snapshots are shown in Figure 2.As observed in experiments, refs. [4, 6], the NOB convec-tion is characterized by an enhancement of the bulk tem- x/L z / L x/L z / L x/L z / L x/L z / L (cid:1)(cid:0)(cid:3)(cid:2) (cid:4)(cid:5) o C Fig. 2: (color online) Snapshots of the velocity and temper-ature fields for Ra = 10 at T m = 40 o C. The upper panelcorresponds to the OB case ( T -independent material param-eters), the lower one to the NOB case, both with ∆ = 40K.The temperature color scheme is the same in both cases. Inthe NOB case a strong temperature enhancement of the centeris clearly visible. perature T c , and a top-bottom asymmetry of the thermalBL thicknesses. Due to the large variation of the glycerolviscosity (the viscosity ratio reaches as much as ν t /ν b ≈ T c as compared to the water case. Mean temperature profiles and center temperature.
Toquantify the enhancement of the bulk temperature T c , thetemperature profiles for Ra = 10 are shown in Fig. 3.Again, a strong asymmetry between top and bottom is ob-served: Due to the more mobile bottom plumes the centertemperature T c is significantly larger than T m .p-3. Sugiyama et al. ( T − T t ) / ∆ z/L NOB (∆ = K ) OB Fig. 3: (color online) Mean temperature profiles for glycerol at Ra = 10 in the OB case (dashed) and in the NOB case with∆ = 40 K (solid). (In both cases T m = 40 o C, same Ra , same∆, but T -independent (OB) or T -dependent (NOB) materialparameters, respectively.) In the NOB case the strong temper-ature enhancement of the center temperature T c by about 5 K becomes visible (relative shift ≈ . To demonstrate this the center temperature shift T c − T m (normalized by ∆) as function of the Rayleigh number Ra and of the NOBness ∆ is shown in Figure 4. Exceptfor small Rayleigh numbers just above onset of convectionand in a region around Ra ≈ · just above the onset ofunsteady motion, the bulk temperature shift ( T c − T m ) / ∆is rather independent of Ra . The tiny increase between Ra = 10 and 10 however is beyond the statistical error-bars. For comparison, the prediction of the NOB BL the-ory given in ref. [6] and the shift for the non-convectivestate (i.e., purely conductive heat transport, driven by thetemperature gradient only) are shown. Though the NOBBL theory from ref. [6] is not applicable here due to thelack of a large scale wind, it gives the correct qualitativetrend for the shift ( T c − T m ) / ∆. We also included ex-perimental data measured at Ra = 2 . · (taken fromref. [4]) in an aspect ratio 1 cylindrical container. Thoughfor that case a large scale convection role has been ob-served, the agreement with the 2D numerical simulationsis reasonable. Nusselt number.
The key question on NOB effects is:How do they affect the heat flux, i.e., the Nusselt num-ber? For water we could address this question within anextended BL theory, cf. ref. [6], but only thanks to the ex-act relation eq.(1) and the experimental input F λ ≈
1, seerelation (2), because then only F ∆ is needed to calculatethe NOB deviations in the Nusselt number ratio, and F ∆ is accessible within the extended BL theory, since it fol-lows directly from T c . But here, with glycerol as workingfluid, we find that F λ ≈ not hold, as demonstratedin Fig. 5. In contrast to water, for glycerol the main ∆-dependence of N u
NOB /N u OB = F λ · F ∆ is due to the∆-dependence of F λ while the factor F ∆ is basically 1 for ( T c − T m ) / ∆ Ra BL theoryNo convection
Steady ChaoticNOB (∆ = K ) ( T c − T m ) / ∆ ∆ [K] BL theoryNo convection
Experiment
Ra= Ra= Ra= Ra= Ra= Fig. 4: (color online) Relative deviation ( T c − T m ) / ∆ of thecenter temperature T c from the arithmetic mean temperature T m for glycerol versus Ra at fixed ∆ = 40 K (upper) and versusthe NOBness ∆ at fixed T m = 40 o C and various values of Ra (lower). The experimental data points (denoted by × ) aremeasured at Ra = 2 . · and were taken from ref. [4]. Forcomparison the T c shift obtained from BL theory (upper solidlines) and for the case of no convection (lower solid lines) arealso plotted. all ∆. This qualitative difference between glycerol and wa-ter in the origin of the Nusselt number modification alsomeans that the experimental finding F λ ≈ T m = 40 K and Ra in the range of 10 − , see ref. [6],is merely accidental and not a general feature of the RBflow under NOB conditions.Both NOB responses, the shift of the center temper-ature T c and thus ∆ b = ∆ t as well as the shift of theBL thicknesses λ slb,t , are determined by the full nonlineardynamics, in glycerol as well as in water. The T c -shift inglycerol is even larger ( ≈ . ≈ λ slb,t -shifts. But the differences inthe temperature drops ∆ b,t enter via F ∆ ; here they areweighed with the explicit temperature dependence of thematerial parameter κ ( T ). Since the thermal diffusivitychanges only minutely in glycerol, κ b,t /κ m − ≈ ± . F ∆ stays near F ∆ ≈ T c re-sponse, cf. Figs. 5,6. This does not happen in F λ ; herethe full changes of λ slb,t enter. Because of the very strongp-4on-Oberbeck-Boussinesq effects in Rayleigh-B´enard convection F λ · F ∆ , F λ , F ∆ ∆ [K] (a) F λ ·F ∆ ( =Nu NOB /Nu OB ) F λ F ∆ BL theory F ∆ F λ · F ∆ , F λ , F ∆ ∆ [K] (b) F λ ·F ∆ ( =Nu NOB /Nu OB ) F λ F ∆ BL theory F ∆ F λ · F ∆ , F λ , F ∆ ∆ [K] (c) F λ ·F ∆ ( =Nu NOB /Nu OB ) F λ F ∆ BL theory F ∆ Fig. 5: (color online) Nusselt number ratio Nu NOB /Nu OB = F λ · F ∆ together with its contributing factors F λ and F ∆ versus∆ for fixed Rayleigh numbers. (a) Ra = 10 , (b) Ra = 10 ,and (c) Ra = 10 . As always, the working liquid is glycerol at T m = 40 o C. The dashed lines correspond to F ∆ resulting fromthe NOB BL theory of ref. [6]. and in particular nonlinear temperature dependence of ν the thicknesses of the BLs change significantly and alsoquite differently in magnitude at the bottom and the topBLs, because p ν t /ν b ≈ . T -dependence of ν ( T ). Therefore the sum λ slb + λ slt nolonger is equal to 2 λ slOB . For water, instead, the domi-nantly linear λ slb,t -NOB modifications are opposite in signand nearly cancel in the sum of the NOB thicknesses, giv-ing λ slb + λ slt ≈ λ slOB or F λ ≈
1. Thus in glycerol wehave F ∆ ≈ F λ ,while in water it is F λ ≈ N u are determined dominantly by F ∆ (which is given by the N u NO B / N u O B Ra Steady Chaotic F λ , F ∆ Ra Steady Chaotic
BL theory F ∆ F λ F ∆ Fig. 6: (color online) The Nusselt number ratio Nu NOB /Nu OB = F λ · F ∆ (upper) and the constitutingfactors F λ and F ∆ individually (lower) versus Ra for fixedNOBness ∆ = 40 K (glycerol at T m = 40 o C). Note thedramatic NOB effect at Ra ≈ · ; this happens still inthe pattern forming range, far below the turbulent high Ra region. We are not aware of its experimental verification. temperature shift alone).Figure 6 shows that the dependences of N u
NOB /N u OB and F λ on the Rayleigh number Ra are non-monotonous.We consider this as due to the nontrivial evolution of var-ious coherent flow patterns with increasing Ra . In par-ticular, as shown in figure 5, for Ra = 10 the function F λ (∆) shows a qualitatively opposite behavior to that forwater, namely F λ increases with increasing ∆ and reachesas large a value as 1 .
017 at ∆ = 50 K . A consequence ofour finding is that in general N u
NOB /N u OB = F λ · F ∆ cannot be calculated within the extended BL theory in-troduced in ref. [6], even if a large-scale wind has formed:Within BL theory only the factor F ∆ can be calculatedbut not the factor F λ , for which in general one cannotassume F λ ≈ N u number itselfas a function of Ra , see Fig. 7, both for the OB and theNOB case. The inset shows the local scaling exponents.When applying the unifying theory of refs. [8], it is 0.306at Ra = 10 and P r = 2500, consistent with our numericalfindings. This local slope practically does not change inthe NOB case.p-5. Sugiyama et al. N u Ra NOB (∆ = K ) OBExperiment d l og ( N u ) / d l og ( R a ) Ra Fig. 7: (color online) The Nusselt number Nu for glycerol ver-sus Ra under OB (dashed line) and NOB (solid line) condi-tions. In both cases T m = 40 o C and ∆ = 40K. OB is pro-vided by keeping the material parameters artificially constantwith T . We have also included the available data from ref. [4].Logarithmic slope d log( Nu ) / d log( Ra ) is plotted in the insetand the line corresponding to the exponent 0 .
297 measured inref. [4] is also shown.
Summary and conclusions. –
In summary, for glyc-erol both the center temperature T c and the Nusselt num-ber N u of the 2D numerical simulations are in good agree-ment with the available experimental data of ref. [4]. Theexperimental finding by Ahlers et al. [6] of a ”thermal-BL-thickness sum rule” for water, F λ ≈ λ slb + λ slt ≈ λ slOB ,is shown to be incidental and seems due to the specific tem-perature dependence of the material parameters of waterat 40 o C. Apparently this cannot be generalized to otherfluids (or other mean temperatures), as our analysis ofRB convection in glycerol has shown. While for water theNusselt number modification
N u
NOB /N u OB is due to themodified temperature drops over the BLs, represented by F ∆ , as shown in refs. [6, 7], for glycerol it is governed bythe variation of the BL thicknesses, namely by F λ . Thiscan be attributed to the strong and nonlinear temperaturedependence of ν ( T ). ∗ ∗ ∗ Acknowledgment:
We thank Guenter Ahlers and FranciscoFontenele Araujo for many fruitfull discussions over thelast years. The work in Twente is part of the researchprogram of FOM, which is financially supported by NWOand SGn acknowledges support by FOM.
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