In the present work, the fluid flow and heat transfer inside an integrated solar collector installed on an advanced fac¸ade are investigated. According to Gray and Giorgini [1], the use of the Boussinesq approximation can be considered valid for variations of thermosphysical properties up to 10 % with respect to the mean value. In the configuration under study, there is a variation of about 20 % in the dynamic viscosity and 15 % in the thermal expansion coefficient.
Thus, the main objective of this work is to analyse the validity of the Boussinesq approximation for the turbulent natural convection flow of water in a rectangular
parallelepiped tank. The significance of the
Boussinesq effects is studied comparatively by means of detailed DNS simulations.Peer ReviewedPostprint (author’s final draft

Kizildag, Deniz

Oliva Llena, Asensio

Rodríguez Pérez, Ivette María

English

In the present work, the fluid flow and heat transfer inside an integrated solar collector installed on an advanced fac¸ade are investigated. According to Gray and Giorgini [1], the use of the Boussinesq approximation can be considered valid for variations of thermosphysical properties up to 10 % with respect to the mean value. In the configuration under study, there is a variation of about 20 % in the dynamic viscosity and 15 % in the thermal expansion coefficient.

Thus, the main objective of this work is to analyse the validity of the Boussinesq approximation for the turbulent natural convection flow of water in a rectangular

parallelepiped tank. The significance of the

Boussinesq effects is studied comparatively by means of detailed DNS simulations.Peer Reviewe

UPCommons

On the validity of the Boussinesq approximation in a tall differentially heated cavity with water

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7th International Conference on Computational Heat and Mass Transfer408ON THE VALIDITY OF THE BOUSSINESQ APPROXIMATION IN A TALLDIFFERENTIALLY HEATED CAVITY WITH WATERD. Kizildag, I. Rodrı´guez, A. OlivaCentre Tecnolo`gic de Transfere`ncia de Calor (CTTC)Universitat Polite`cnica de Catalunya (UPC)ETSEIAT, Colom 11, 08222, Terrassa, Barcelona, SpainFax: +34 93 739 89 20 E-mail: cttc@cttc.upc.eduABSTRACTIn the present work, the fluid flow and heat trans-fer inside an integrated solar collector installed onan advanced fac¸ade are investigated. According toGray and Giorgini [1], the use of the Boussinesq ap-proximation can be considered valid for variations ofthermosphysical properties up to 10 % with respectto the mean value. In the configuration under study,there is a variation of about 20 % in the dynamic vis-cosity and 15 % in the thermal expansion coefficient.Thus, the main objective of this work is to analysethe validity of the Boussinesq approximation for theturbulent natural convection flow of water in a rect-angular parallelepiped tank. The significance of theBoussinesq effects is studied comparatively by meansof detailed DNS simulations.INTRODUCTIONThe natural convection flow within enclosures has at-tracted the attention of many researchers due to itspotential to model numerous applications of engineer-ing interest, such as cooling of electronic devices,air flow in buildings, heat transfer in solar collectors,among others. The natural convection studies corre-sponding to the parallelepipedic enclosures can beclassified into two elementary classes: i) heating froma horizontal wall (heating from below); ii) heatingfrom a vertical wall. The characteristic example ofthe former case is the Rayleigh-Be´nard flow, howeverthis work will only focus on the cavities heated fromthe side. This configuration is referred commonly asthe differentially heated cavity.Although the differentially heated cavity configurationrepresents a simple geometry, the flow gets complexfor sufficiently large Rayleigh numbers [2]. The flowundergoes a gradual transition to a chaotic state as theRayleigh number reaches a critical value. For the situ-ation studied in this work, both laminar, transitional,and turbulent zones are expected to coexist within thedomain. Generally the core of the cavity together withthe upstream part of the vertical boundary layers re-main laminar while at some point in the downstreampart of the vertical boundary layers, turbulent fluc-tuations become significant. It is a challenging taskto detect this phenomenon [3]. Another importantissue is the stratification phenomenon taking placein the core of the cavity. It is one of the basic openproblems of this configuration. Comparisons betweennumerical and experimental studies give quite differ-ent results, which may be justified by the thermalradiation effects [4].The vast majority of the performed work in this fieldcorresponds to air-filled cavities (see [3] for a detailedpreview). If the working fluid is water, obtaining solu-tions for the governing equations gets even more com-plicated, as the boundary layer becomes thinner thanfor air at the same conditions. As a consequence, thereis an increasing demand for excessively fine grids inspace and time for solving the three-dimensional andtime dependent flow, in order to capture the smallestscales of the turbulent flow.Additional to the issues explained above, when inves-tigating the fluid behaviour in real working conditions,7th International Conference on Computational Heat and Mass Transferthe validity of the Boussinesq approximation has to bequestioned. According to Gray and Giorgini [1], theuse of the Boussinesq approximation can be consid-ered valid for variations of thermosphysical propertiesup to 10% with respect to the mean value.The main objective of this work is to analyze the turbu-lent natural convection flow of water in a rectangularparallelepiped tank. This configuration corresponds toan integrated solar collector installed on an advancedfac¸ade. The aspect ratio is Γ = 6.68. The workingconditions of the particular design yield a Rayleighnumber of Ra= 2.2×1011 and a Prandtl number ofPr = 3.44. Long-term accurate statistical data bymeans of a DNS simulation is presented, investigat-ing the characteristics of the turbulent flow within thedifferentially heated cavity. Some results of first andsecond order turbulent statistics are also presented.Based on a previous work [5] this works aims at ob-taining more details about the Boussinesq effects inturbulent natural convection flow. The actual workingconditions of the prototype point out a variation ofabout 20 % in the dynamic viscosity and 15 % in thethermal expansion coefficient. Thus, it is of interest toanalyze some features of turbulent natural convectionflow including Boussinesq effects.                                    HWTTh cgx1x2Figure 1: Geometry of the differentially heated cavityCASE UNDER STUDYThe adopted geometry considered in this work isshown in Figure 1. This geometry models the par-allelepiped tank of an integrated solar collector. Theheight of the tank (H), and the width (W) are 0.735mand 0.11m respectively, resulting in an aspect ratio ofΓ= 6.68.Taking into consideration the real working conditionsof the studied prototype, the temperatures at the hotand cold vertical isothermal walls are set to 57 ◦C and47 ◦C respectively.DESCRIPTION OF THE MATHEMATI-CAL AND NUMERICAL METHODThe Navier-Stokes and continuity equations can bewritten asMu = 0 (1)∂u∂ t+C(u)u+νDu+ρ−1Gp+ f = 0 (2)∂T∂ t+C(u)T+ρ−1Cp−1kD(T) = 0 (3)where u ∈ R3m and p ∈ Rm are the velocity vectorand pressure, respectively (here m applies for the totalnumber of control volumes (CV) of the discretised do-main), ν is the kinematic viscosity and ρ the density.f is the body force f = β (T0−Tm)g. Convective anddiffusive operators in the momentum equation for thevelocity field are given by C(u) = (u ·∇) ∈ R3m×3m,D = ∇2 ∈ R3m×3m respectively. Gradient and diver-gence (of a vector) operators are given by G = ∇ ∈R3m×3m and M= ∇· ∈ Rm×3m respectively.In bouyancy driven flows a common approach is toconsider constant thermophysical properties of thefluid, with the exception of the density variations thatare only taken into account in the bouyancy forces,i.e., the so-called Oberbeck-Boussinesq approxima-tion. Thus, the temperature dependency of density islinearized in the bouyancy force as:ρ(T ) = ρm−ρmβm(T−Tm) (4)Here Tm =(Th+Tc)/2 is the mean value of the temper-atures of the cold and hot walls. Those assumptions7th International Conference on Computational Heat and Mass Transferhave its own implications. First, continuity equation istreated in its incompressible form, neglecting acousticphenomena, which in the case of liquids has no majorimplications. However, for liquids, deviations fromthe aforementioned hypothesis are mainly due to vis-cosity variations, as the viscosity strongly decreaseswith the temperature increase.When considering the Boussinesq effects in our work,the following are assumed:• temperature dependent thermophysical proper-ties• density variations are only taken into accountin the bouyancy term• the temperature dependence of the density islinearized as:ρ(T ) = ρm−ρ ′(T ) (5)Under the above assumptions, equations 1-3 read:Mu = 0 (6)∂u∂ t+C(u)u+D(ν(T )u)+ρ−1m Ĝp−ρ−1m ρ ′(T )g = 0 (7)∂T∂ t+C(u)T+ρ−1m Cp−1m D(k(T )T) = 0 (8)The temperature dependencies for ν(T ), k(T ) andρ(T ) are taken from Furukawa.[6]Considering the reference scales for length, time,velocity, temperature and dynamic pressure as H,(H2/α)Ra−0.5, (α/H)Ra0.5, Th−Tc, ρ(α/H2)Ra, re-spectively, Non-Oberbeck-Boussinesq thermal con-vection in the cavity is governed by the non-dimensional quantities: Ra= (gβm∆Tre fH3Prm)/ν2m,Prm = νm/αm, Γ and the non-dimensional thermo-physical properties: ν∗ = ν(T )/νm; k∗ = k(T )/km;ρ∗ = ρm−ρ(T )ρmβm∆Tre f =ρ ′(T )ρmβm∆Tre f .The governing equations are discretized on a collo-cated unstructured grid arrangement, by means ofsecond-order spectro-consistent schemes [7]. Suchdiscretization preserves the symmetry properties ofthe continuous differential operators, i.e., the conser-vation properties are held if, the convective term isdiscretized by a skew-symmetric operator and the dif-fusive term is approximated by a symmetric, positive-definite coefficient matrix. These properties ensureboth, stability and conservation of the global kinetic-energy balance on any grid. Energy transport is alsodiscretized by means of a spectro-consistent scheme.An explicit third-order Gear-like scheme [8] based ona fractional step method is used for time-advancementalgorithm, except for the pressure gradient where afirst-order backward Euler scheme is used.Collocated meshes do not conserve kinetic energywhen fractional step method is used [9, 10]. Thesource of these errors are interpolation schemes andinconsistency in the pressure field, in order to en-sure mass conservation. While the first is eliminatedthrough the use of consevative schemes, the latterequals to εke = (p˜c)∗Mc(Gc−Gs)p˜c. Felten and Lund[10] showed that pressure errors are of the order ofO(4x24t). However, these errors do not have sig-nificant impact on the grid resolutions and time-stepsused in DNS.Boundary conditions and fluid properties: For thevelocities, no-slip condition is applied on all the walls.In the spanwise direction (when applicable) periodicboundary condition is used.Isothermal vertical walls are assumed. Left verticalwall is at 57 ◦C and right vertical wall is at 47 ◦C. Inthe top and bottom confining walls, Neumann bound-ary condition (∂T/∂n= 0) is applied.Except for the Non-Boussinesq calculations, the fluidproperties for water are calculated at the average tem-perature of (Th + Tc)/2 = 52◦C. Rayleigh numberbased on the height of the cavity is Ra = ρβ (Th−Tc)H3/να = 2.2× 1011 and the Prandtl number isPr = ν/α = 3.44, being ν is the kinematic viscosityand α thermal diffusivity.In Figure 2 the temperature dependencies of densityand kinematic viscosity are shown for the workingtemperatures of the prototype.7th International Conference on Computational Heat and Mass Transfer 950 960 970 980 990 1000 320  322  324  326  328  330 0.00046 0.00048 0.0005 0.00052 0.00054 0.00056 0.00058density ρ (kg/m3)viscosity µ (Pa • s)temperature (K)densityviscosityFigure 2: Variable thermophysical propertiesGeometric discretization: The smallest scales atthe hot and cold walls are imposed by viscous andthermal boundary layers, while grid size at the bulkmust be lesser than Kolmogorov scale. For the Prandtlnumber in our case, the thermal boundary layer is thin-ner than the viscous boundary layer as δt ∼ h/Ra0.25and δv ∼ Pr0.5δt [11]. The meshes shown in Table 1are used for our preliminary studies.Table 1: Space and time meshes used in the test cases.mesh m2 m3 m4Nx 258 314 535Ny 770 940 1871∆xmin 6.80×10−5 6.80×10−5 4.08×10−5∆ymin 1.30×10−3 1.06×10−3 5.34×10−4∆t 2.54×10−5 2.54×10−5 9.15×10−6PRELIMINARY RESULTS AND CONCLU-SIONSAs the numerical effort to carry out the present sim-ulations is too large, all the calculations here pre-sented are restricted to two-dimensional (2D) simu-lations. Although 2D calculations might affect thefluid dynamics, some of the characteristics of the flowor the Boussinesq effects can still be captured un-der this assumption. It has been shown earlier byTrias et al. [3] for a differentially heated cavity forRayleigh numbers up to 1010 and by Schalzl et al.[12] for Rayleigh-Benard convection, that in generalas a rough approach to capture the general featuresof the flow and especially boundary layer profilesand Nusselt numbers, 2D simulations can be a goodapproximation.In the present work, the results obtained for some ofthe mentioned meshes are presented, while results forother meshes are being calculated. In Table 2, a briefresume of the studied cases are shown.Table 2: √ calculated cases; × pending cases.mesh Boussinesq Non-Boussinesq(BSQ) (NBSQ)m2√ √m3√ ×m4√ ×In Figures 3 and 4 y-direction velocity profiles atmid height of the cavity are given. Boussinesq ap-proximation solutions are symmetric in the vicinity ofhot and cold walls, while the symmetry is broken inNon-Boussinesq solution. Even though the availableNon-Boussinesq solution corresponds to the coarsestmesh level, the significant differences in y-directionvelocities cannot be attributed to spatial discretizationsince the Boussinesq approximation solution for thesame mesh gives acceptable results with respect toDNS solution.xv0 0.01 0.0200.050.10.150.20.25BSQ m2BSQ m3BSQ m4NBSQ m2Figure 3: y-direction velocity profile at mid height ofthe cavity in the vicinity of hot wallIn the vicinity of the hot wall (see Figure 3), the Non-Boussinesq solution underestimates the peak valueby approximately 35%. On the other hand, in thevicinity of the cold wall, Non-Boussinesq solutionoverestimates the absolute peak value by about 14 %.Morever, the peak values occur at different positions,7th International Conference on Computational Heat and Mass Transferindicating differences in boundary layer thicknesses.For Non-Boussinesq solution, in hot wall the peakvalue corresponds to xmax = 0.00217 while for thefinest mesh with Boussinesq approximation the peakoccurs atxmax = 0.0019. In the cold wall, the valuesare 0.14768 and 0.147775 respectively.xv0.13 0.135 0.14 0.145 0.15-0.25-0.2-0.15-0.1-0.050BSQ m2BSQ m3BSQ m4NBSQ m2Figure 4: y-direction velocity profile at mid height ofthe cavity in the vicinity of cold wallxT0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.4-0.200.20.4BSQ m2BSQ m3BSQ m4NBSQ m2Figure 5: Temperature profiles at mid height of thecavityAs the core temperature is concerned, Non-Boussinesq solution predicts a significantly greatertemperature (TNBSQc > TBSQc ). This effect can be ob-served in Figure 5. In Boussinesq solution the temper-ature at the core of the cavity corresponds to the meantemperature of the case, however in Non-Boussinesqcase this symmetry is broken. 0 200 400 600 800 1000 1200 1400 1600 1800 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1NuyNBSQ m2BSQ m2BSQ m4Figure 6: Local Nusselt number profile in hot wall 0 200 400 600 800 1000 1200 1400 1600 1800 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1NuyNBSQ m2BSQ m2BSQ m4Figure 7: Local Nusselt number profile in cold wallIn Figure 6, the local Nusselt number distribution forboth Boussinesq and Non-Boussinesq solutions aregiven for the coarsest mesh levels, and the referenceDNS solution using Boussinesq approximation (BSQm4). This hot wall graph indicates different transitionpoints. For Non-Boussinesq solution, the transitionpoint is observed around y= 0.35 while as the Boussi-nesq approximation is adopted, the instabilities arenot significant till approximately y= 0.46. These re-sults show that the transition point estimation presentssignificant deviations, at least for the coarsest mesh.The average Nusselt numbers are shown in Table 3with respect to the finest Boussinesq solution.As we focus on the cold wall, the differences betweenBoussinesq and Non-Boussinesq solution in termsof transition point are less pronounced (see Figure6). However, it is important to note that for the Non-Boussinesq solution the transition point corresponds7th International Conference on Computational Heat and Mass Transferto y= 0.60, which clearly indicates the lack of sym-metry in this case.In Figures 8 and 9 the representative instantaneousisotherms corresponding to t = 450 and t = 500 areshown. Note that for the Non-Boussinesq solution,the core of the cavity is warmer than the Boussinesqsolutions. Additionally, for this case the instabilitiesare visible in the vicinity of the hot wall by 1/3 ofthe cavity, while in the case of the cold wall, we canroughly say that the instabilities are ejected to thecore after 2/3 of the cavity length from the leadingedge. However, both Boussinesq approximation solu-tions present symmetric behaviour as the instabilitiesare concerned. In the BSQ m4 solution, the temper-ature drop in the top and bottom region occurs in asmaller region, which in turn makes relatively largerthe mixing region in the core.Table 3: Average Nusselt numberscase Average Nusselt ErrorNBSQ m2 259.47 16.7 %BSQ m2 244.24 9.8 %BSQ m4 222.27 -Figure 8: Representative instantaneous isotherms fort=450; (left) NBSQ m2 , (middle) BSQ m2, (right)BSQ m4.Figure 9: Representative instantaneous isotherms fort=500; (left) NBSQ m2 , (middle) BSQ m2, (right)BSQ m4. 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16v’v’yBSQ m2BSQ m3BSQ m4Figure 10: Reynolds stress (v’v’)With regard to second order statistics, the prelimi-nary results show significant differences (e.g. seeFigure 10 ), which could be due to the used meshlevel and time-integration period. As the solutionsof the pending cases become available with sufficienttime-integration period, more light could be shed onthe remaining topics of interest, like stratification,detailed turbulent statistics, heat transfer and fluiddynamics.7th International Conference on Computational Heat and Mass TransferACKNOWLEDGEMENTSThis work has been by financially supported by theMinisterio de Educacio´n y Ciencia, Secretarı´a de Es-tado de Universidades e Investigacio´n, Spain (ref.ENE2009-07689 and ENE2009-09496).REFERENCES[1] D. D. Gray and A. Giorgini. The validity ofthe boussinesq approximation for liquids andgases. International Journal of Heat and MassTransfer, 19(5):545–551, 1976.[2] D.G. Barhaghi and L. Davidson. Natural con-vection boundary layer in a 5:1 cavity. Physicsof Fluids, 19:125106, 2007.[3] X. Trias, M. Soria, A. Oliva, and C. D. Pe´rez-Segarra. Direct Numerical simulations of twoand three dimensional turbulent natural convec-tion flows in a differentialy heated cavity ofaspect ratio 4. Journal of Fluid Mechanics,586:259–293, 2007.[4] S. Xin, J. Salat, P. Joubert, A. Sergent, andP. Le Quere. Three dimensional numerical sim-ulations of turbulent natural convection in anair-filled differentially heated cavity. In Pro-ceedings of the 13th International Heat TransferConference, 2006.[5] D. Kizildag, J. Ventosa, I. Rodriguez, andA. Oliva. Non-Oberbeck-Boussinesq NaturalConvection in a Tall Differentially Heated Cav-ity. In Proceedings of the Fifth European Con-ference on Computational Fluid Dynamics EC-COMAS CFD 2010, pages 1–13, 2010.[6] M. Furukawa. Practical Expressions for Thermo-dynamic and Transport Properties of CommonlyUsed Fluids. Journal of Thermophysics andHeat Transfer, 5(4):524–531, 1991.[7] R. W. C. P. Verstappen and A. E. P. Veldman.Symmetry-preserving discretization of turbu-lent flow. Journal of Engineering Mathematics,187(1):343–368, 2003.[8] G. M. Fishpool and M. A. Leschziner. Stabil-ity bounds for fractional-step schemes for theNavier-Stokes equations at high Reynolds num-ber. Computers and Fluids, 38(6):1289–1298,2009.[9] Y. Morinishi, T.S. Lund, O.V. Vasilyev, andP. Moin. Fully conservative higher order fi-nite difference schemes for incompressible flow.Journal of Computational Physics, 143(1):90–124, 1998.[10] F. N. Felten and T. S. Lund. Kinetic Energy Con-servation Issues Associated with the CollocatedMesh Scheme for Incompressible Flow. Jour-nal of Computational Physics, 215(2):465–484,2006.[11] J. Patterson and J. Imberger. Unsteady naturalconvection in a rectangular cavity. J. FluidsMech., 100:65–86, 1980.[12] J. Schmalzl, M. Breuer, and U. Hansen. Onthe validity of two-dimensional numerical ap-proaches to time-dependent thermal convection.Europhys. Lett., 67(3):390–396, 2004.

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On the validity of the Boussinesq approximation in a tall differentially heated cavity with water

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