Oliva, A.

Soria, M.

Trias, F.X.

Verstappen, R.W.C.P.

English

Dissertations of the University of Groningen

   University of GroningenParameter-Free Symmetry-Preserving Regularization Modelling of Turbulent NaturalConvection FlowsTrias, F.X.; Verstappen, R.W.C.P.; Soria, M.; Oliva, A.Published in:Turbulence and InteractionsIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2010Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Trias, F. X., Verstappen, R. W. C. P., Soria, M., & Oliva, A. (2010). Parameter-Free Symmetry-PreservingRegularization Modelling of Turbulent Natural Convection Flows. In Turbulence and Interactions Universityof Groningen, Johann Bernoulli Institute for Mathematics and Computer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Parameter-Free Symmetry-PreservingRegularization Modelling of Turbulent NaturalConvection FlowsF.X. Trias, R.W.C.P. Verstappen, M. Soria, and A. OlivaAbstract. Since direct numerical simulations of natural convection flows cannotbe performed at high Ra-numbers, a dynamically less complex mathematical for-mulation is sought. In the quest for such a formulation, we consider regularizations(smooth approximations) of the nonlinearity. The regularization method basicallyalters the convective terms to reduce the production of small scales of motion bymeans of vortex stretching. In doing so, we propose to preserve the symmetry andconservation properties of the convective terms exactly. This requirement yields anovel class of regularizations that restrain the convective production of smaller andsmaller scales of motion by means of vortex stretching in an unconditional stablemanner, meaning that the velocity cannot blow up in the energy-norm (in 2D also:enstrophy-norm). The numerical algorithm used to solve the governing equationspreserves the symmetry and conservation properties too. The regularization modelis successfully tested for a 3D natural convection flow in air-filled (Pr = 0.71) dif-ferentially heated cavity of height aspect ratio 4 at Ra = 1010 and 1011. Moreover,a method to dynamically determine the regularization parameter (local filter length)is also proposed and tested.1 IntroductionTurbulent natural convection in enclosed cavities has been the subject of numer-ous studies over the past decades. They model many engineering applications suchF.X. Trias · M. Soria · A. OlivaCentre Tecnològic de Transferència de Calor, ETSEIAT, Technical University of Catalonia,C/Colom 11, 08222 Terrassa, Spaine-mail: cttc@cttc.upc.eduF.X. Trias · R.W.C.P. VerstappenInstitute of Mathematics and Computing Science, University of Groningen, P.O. Box 407,9700 AK Groningen, The Netherlandse-mail: R.W.C.P.Verstappen@rug.nlM. Deville, T.-H. Lê, and P. Sagaut (Eds.): Turbulence and Interactions, NNFM 110, pp. 369–375.springerlink.com c© Springer-Verlag Berlin Heidelberg 2010370 F.X. Trias et al.as ventilation of rooms, cooling of electronics devices or air flow in buildings.Simultaneously, this configuration has served as prototype for the development ofnumerical algorithms. Therefore, an accurate prediction of the flow structure andthe heat transfer in such configuration is of great interest. However, due to the com-plex behaviour exhibit (see figure 1), and despite the great effort devoted, an ac-curate turbulence modelling of this configuration remains as a great challenge. Onthe other hand, direct numerical simulation (DNS) has become an essential tool togive new insights into the physics of turbulence and provide indispensable data forthe development of better turbulence models. However, at high Ra-numbers, DNSsimulations are not feasible.In this context, a dynamically less complex mathematical formulation is sought.In the quest for such a formulation, we consider regularizations (smooth approx-imations) of the nonlinear convective term. The first outstanding approach in thisdirection goes back to Leray [1]; the Navier-Stokes-α model also forms an exampleof regularization modelling (see [2], for instance). The regularization methods basi-cally alter the convective terms to reduce the production of small scales of motion.In doing so, we proposed to preserve the symmetry and conservation properties ofthe convective terms exactly [3]. This requirement yielded a family of symmetry-preserving regularization models [3]: a novel class of regularizations that restrainthe convective production of smaller and smaller scales of motion in an uncondi-tional stable manner, meaning that the velocity cannot blow up in the energy-norm(in 2D also: enstrophy-norm). In this work, a method to dynamically determine theregularization parameter (local filter length) with the requirement that the vortexstretching must be stopped at the scale set by the grid is also proposed and tested.The numerical algorithm used to solve the governing equations preserves the conser-vation properties too [4] and is therefore well-suited to test the proposed simulationmodel.Fig. 1 Left: several instantaneous temperature fields at Ra = 1011 and Pr = 0.71 (air). Right:zoom around the point where the vertical boundary layer becomes totally disrupted and largeeddies are ejectedParameter-Free Symmetry-Preserving Regularization Modelling 3712 C4-Regularization ModellingWe restrict ourselves to the C4 approximation (see [3], for details): theconvective term in the Navier-Stokes equations is then replaced by the followingO(ε4)-accurate smooth approximation C4(u,v) given byC4(u,v) = C (u,v)+C (u,v′)+C (u′,v) (1)where C (u,v) = (u ·∇)v represents the convective operator. Note that here a primeindicates the residual of the filter, e.g. u′ = u− u, which can be explicitly evalu-ated, and (·) represents a normalised self-adjoint linear filter with filter length ε .Therefore, the governing equations result to∂t u +C4(u,u) =1ReΔu−∇p + f ; ∇ ·u = 0 (2)Note that the C4 approximation is also a skew-symmetric operator like the originalconvective operator. Hence, the same inviscid invariants -kinetic energy, enstrophyin 2D and helicity- that the original Navier-Stokes equations are kept for the new setof partial differential equations to be solved (2).Since now the performance of the C4 approximation has been successfully testedfor a turbulent channel flow [3] and an air-filled (Pr = 0.71) differentially heatedcavity (DHC) of height aspect ratio A = 4 at Ra-numbers (based on the cavityheight) 1010 [5, 6] and 1011 [7] by means of direct comparison with the DNS re-sults [8, 9, 10]. In the latter cases, the filter width ε was treated as a parameter.Then, the value of ε was prescribed in advanced. Figure 2 displays illustrative re-sults for the aforesaid DHC problem at Ra = 1010 as a function of ε/h ratio. Theseresults show that the numerical solution is weakly dependent for sufficiently largevalues of ε . Similar behaviour has also been observed for the other aforementionedconfigurations.1001051101151201251300 0.5 1 1.5 2 2.5 3Nusseltε / hDNSC4  (  8x17x40)C4  (16x34x80)Parameter-free C4  (  8x17x40)Parameter-free C4  (16x34x80)0.120.140.160.180.20.220.240 0.5 1 1.5 2 2.5 3(u3)maxε / hDNSC4  (  8x17x40)C4  (16x34x80)Parameter-free C4  (  8x17x40)Parameter-free C4  (16x34x80)Fig. 2 The overall averaged Nusselt number (left) and the maximum of the averaged verticalvelocity at the horizontal mid-height plane (right) as a function of the ratio of the filter lengthε to the average grid width h372 F.X. Trias et al.3 On the Dynamic Determination of the Filter LengthIn the present work, we propose to determine ε dynamically with the requirementthat the vortex stretching must be stopped at the scale set by the grid. The ideabehind is to modify convective operator sufficiently to guarantee that the followinginequality is holdλk(ε)−νk2 ≤ 0 (3)where λk(ε) = ωk · C4(ω ,u)/(ωk · ωk) is the Rayleigh quotient of the vortex-stretching at the grid scale, k = π/h. In practice, the value of λk has been boundedby the largest (positive) eigenvalue of the straintensor S,λk(ε) ≤ f4,k (ε) λmax (S) (4)For the C4-approximation, the damping function, 0 < f4,k ≤ 1, at the highest fre-quency is given by 3ĝ2k(ε)− 2ĝ3k(ε) (see [3], for details), where 0 < ĝk(ε) ≤ 1is the transfer function of the linear filter. Therefore, it suffices that the followinginequality be locally hold3ĝ2k(ε)−2ĝ3k(ε) ≤νk2λmax (S)−→ ĝk(ε) −→ ε (5)to guarantee that the vortex-stretching mechanism be stopped at the smallest scaleset by the mesh.Table 1 Description of meshes: the spanwise (Nx), the wall-normal horizontal (Ny), and thevertical (Nz) resolutions for tested cases. Further details about the meshing can be found in [8]DNS RM1 RM2Ra = 1010 128×190×462 16×34×80 8×17×40Ra = 1011 128×682×1278 12×45×85 8×30×564 Results for a Turbulent Differentially Heated CavityThe performance of the proposed method to dynamically determine the regulariza-tion parameter ε of the C4 approximation has also been tested for the aforesaid DHCproblem. Again two very coarse meshes (see table 1) have been solved. In this work,a fourth-order accurate Gaussian filter [11] has been chosen. In 1D it becomesui =ε4 −4ε21152(ui+2 + ui−2)+16ε2 − ε4288(ui+1 + ui−1)+ε4 −20ε2 + 192192ui (6)where the filter length, ε , need not be restricted to integers multiples of the gridwidth. The boundary conditions that supplement the Navier-Stokes equations arealso applied to (6).Parameter-Free Symmetry-Preserving Regularization Modelling 373First results displayed in figure 2 exhibit the great potential of this method. Notethat in the parameter-free C4 cases, the results do not depend on ε/h. At least asgood results as the optimal ε/h ratio determined by trial-and-error procedure arebeing obtained.4.1 Grid Dependence AnalysisA reliable modelling of turbulence at (very) coarse grids is a great challenge. Thecoarser the grid, more convincing model quality is perceived. However, it mighthappen that solution be strongly dependant on meshing parameters and thus someparticular combinations could ’accidentally’ provide good results. An example ofthis behaviour can be found in [12] for a turbulent channel flow. In order to elucidatethis point, the same DHC problem has been solved on a series of 50 randomly gen-erated meshes: with (Nx,Ny,Nz)-values limited by those given by meshes RM1 andRM2 (see table 1), i.e. 8 ≤ Nx ≤ 16, 17 ≤ Ny ≤ 34, and 40 ≤ Nz ≤ 80. Results forthe overall Nusselt and the centerline stratification values are displayed in figure 3.At first sight, we can observe that the parameter-free C4 modelling predicts goodresults irrespective of the meshing whereas very poor and dispersed results are ob-tained when model is switched off. Specially significant is the fairly good predictionof the stratification (note the big dispersion obtained without model).0204060801001201400 0.2 0.4 0.6 0.8 1 1.2avg(Nu)StratDNSNo modelParameter-free C4Fig. 3 The overall Nusselt number and the centerline stratification for randomly generatedgrids4.2 Performance at Higher Ra-NumbersThe performance of the parameter-free C4-regularization model has also been testedat a higher Ra-number (Ra = 1011). Results for two coarse meshes (see table 1) aredisplayed in figure 4 and compared with the DNS reference results [9, 10]. Al-though in this case the Ra-number is one order of magnitude higher and the mesh374 F.X. Trias et al.1801902002102202302402502602702800 0.5 1 1.5 2 2.5 3Nusseltε / hDNSC4  (  8x30x56)C4  (12x45x85)Parameter-free C4  (  8x30x56)Parameter-free C4  (12x45x85)0.30.350.40.450.50.550.60.650.70.750 0.5 1 1.5 2 2.5 3(u3)maxε / hDNSC4  (  8x30x56)C4  (12x45x85)Parameter-free C4  (  8x30x56)Parameter-free C4  (12x45x85)Fig. 4 The overall averaged Nusselt number (left) and the maximum of the averaged verticalvelocity at the horizontal mid-height plane (right) as a function of the ratio of the filter lengthε to the average grid width h at Ra = 1011resolutions are kept almost the same respect to the case at Ra = 1010, the proposedmethod is able to predict reasonable well the mean flow values.5 ConclusionsThe performance of the proposed method to dynamically determine the regulariza-tion parameter ε of the C4 approximation has been tested for an air-filled DHCproblem of aspect ratio 4 and Rayleigh numbers 1010 and 1011. Results presentedhere exhibit the great potential of the proposed method as new turbulence model: theparameter-free C4 approach is able to predict at least as good results as the optimalε/h ratio determined by trial-and-error procedure.In conclusion, the proposed method constitutes a parameter-free turbulencemodel. Moreover, since no ad hoc phenomenological arguments that cannot be for-mally derived for the Navier-Stokes equations are used it suggest that this methodshould be valid for other configurations. Nevertheless, more simulations for a widevariety of cases and meshes will be necessary to confirm these conclusions.AcknowledgementsThis work has been financially supported by the Ministerio de Educación yCiencia, Spain, (Project: “Estudio y parametrización de fenómenos de transferéncia de calory dinámica de fluidos involucrados en diseños de arquitectura bioclimática”. Contract/grantnumber ENE2006-14247) and a postdoctoral fellowship Beatriu de Pinós (2006 BP-A 10075)by the Generalitat de Catalunya.Parameter-Free Symmetry-Preserving Regularization Modelling 375References1. Leray, J.: Sur le movement d’un liquide visqueaux emplissant l’espace. ActaMathematica 63, 193–248 (1934)2. Geurts, B.J., Holm, D.D.: Regularization modeling for large-eddy simulation. Physics ofFluids 16, L13–L16 (2003)3. Verstappen, R.: On restraining the production of small scales of motion in a turbulentchannel flow. Computers and Fluids 37, 887–897 (2008)4. Verstappen, R.W.C.P., Veldman, A.E.P.: Symmetry-Preserving Discretization of Turbu-lent Flow. Journal of Computational Physics 187, 343–368 (2003)5. Trias, F.X., Soria, M., Oliva, A., Verstappen, R.W.C.P.: Regularization models for thesimulation of turbulence in a differentially heated cavity. In: Proceedings of the EuropeanComputational Fluid Dynamics Conference (ECCOMAS CFD 2006), Egmond aan Zee,The Netherlands (September 2006)6. Trias, F.X., Verstappen, R., Soria, M., Oliva, A.: Symmetry-preserving regularizationmodelling of a turbulent differentially heated cavity. In: LES in Science and Technology,Poznan, Poland (April 2008)7. Trias, F.X., Verstappen, R.W.C.P., Soria, M., Gorobets, A., Oliva, A.: Regularizationmodelling of a turbulent differentially heated cavity at Ra = 1011. In: 5th EuropeanThermal-Sciences Conference, EUROTHERM 2008, Eindhoven, The Netherlands (May2008)8. Trias, F.X., Soria, M., Oliva, A., Pérez-Segarra, C.D.: Direct numerical simulations oftwo- and three-dimensional turbulent natural convection flows in a differentially heatedcavity of aspect ratio 4. Journal of Fluid Mechanics 586, 259–293 (2007)9. Trias, F.X., Gorobets, A., Soria, M., Oliva, A.: Direct numerical simulation of a differ-entially heated cavity of aspect ratio 4 with Ra-number up to 1011 - Part I: Numericalmethods and time-averaged flow. International Journal of Heat and Mass Transfer 53,674–683 (2010)10. Trias, F.X., Gorobets, A., Soria, M., Oliva, A.: Direct numerical simulation of a differen-tially heated cavity of aspect ratio 4 with Ra-number up to 1011 - Part II: Heat transfer andflow dynamics. International Journal of Heat and Mass Transfer 31, 1195–1220 (1999)11. Sagaut, P., Grohens, R.: Discrete Filters for Large Eddy Simulations. International Jour-nal for Numerical Methods in Fluids 31, 1195–1220 (1999)12. Meyers, J., Sagaut, P.: Is plane-channel flow a friendly case for the testing of large-eddysimulation subgrid-scale models? Physics of Fluids 19, 048105 (2007)

Parameter-Free Symmetry-Preserving Regularization Modelling of Turbulent Natural Convection Flows

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Parameter-Free Symmetry-Preserving Regularization Modelling of Turbulent Natural Convection Flows

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