On The Use Of The Periodicity Condition In Cross-flow Tube
EPJ Web of Conferences 92, 0200 5 (2015)DOI: 10.1051/epjconf/ 201 5 9 20200 5 C Owned by the authors, published by EDP Sciences, 2015On the use of the periodicity condition in cross-flow tubeBoumedienne Beladjine1, Mohammed Aounallah 2, a, Mustapha Belkadi 2, Lahouari Aadjlout 2, Omar Imine 31Laboratory of Energy and Naval Propulsion, University of Sciences and Technology of Oran-Mohamed Boudiaf,P. O. 1505 El-Mnaouar Oran Algeria.2Laboratory of Aero-Hydrodynamics Naval, University of Sciences and Technology of Oran-Mohamed Boudiaf,P. O. 1505 El-Mnaouar Oran Algeria.3Laboratory of Aeronautics and Propulsive Systems, University of Sciences and Technology of Oran-MohamedBoudiaf, P. O. 1505 El-Mnaouar Oran Algeria.Abstract. This paper presents the results of measurements and numerical predictions of turbulent cross-flowthrough an in-line 7 7 bundle configuration with a constant transverse and longitudinal pitch-to-diameter ratioof 1.44. The experiments are conducted to measure the pressure around tubes, using DPS differential pressurescanner with air flow, in square channel at a Reynolds number of 35000 based on the gap velocity and the tubediameter. The commercial ANSYS FLUENT is used to solve the unsteady Reynolds–Averaged Navier–Stokes(RANS) equations. The primary aim of the present study is to search for a turbulent model that could serve asan engineering design tool at a relatively low computational cost. The performances of the Spalart-Allmaras,the RNG k- , the Shear Stress Transport k- and the second moment closure RSM models are evaluated bycomparing their simulation results against experimental data. The second objective is to verify the validity ofthe periodicity assumption taken account in the most previous numerical works by considering the filled bundlegeometry. The CFD results show that the Spalart-Allmaras model on the fine mesh are comparable to theexperiments while the periodicity statement did not produce consistently the flow behavior in the 7 7 tubebundle configuration.1 IntroductionFlow in tube bundles have many important industrialapplications and have been studied both experimentallyand through numerical simulations. The ability tocorrectly predict complex turbulent flows is fundamentalto the design of many fluid engineering systems. In spiteof the great number of scientific contributions related tothe cross-flow turbulent tube bundle, few numericalworks were carried out to investigate the dynamic filed inthe entire tube bundle. The majority of the studiesconsider only a partial domain enclosed with periodicboundary conditions.Launder and Massey [1] obtain the flow field for astaggered tube bundle at Reynolds number up to 1000applying periodic boundary conditions under theassumption of fully developed flow. The authorsdemonstrate the same pattern of the velocity filed inevery spatial period and propose an arithmetical law forthe pressure difference. Simonin and Barcouda [2]perform LDA measurements and numerical studies of theturbulent cross-flow in a tube bundle with different pitchratio; they visualize a small recirculation zone in thewakes of tubes with fairly high turbulence intensityavalues and highest length of the recirculation zone behindthe first row. Numerically, they report a good agreementfor the mean velocities and a significant discrepancy inthe prediction of the turbulent kinetic energy with the k- model. Sebag et al. [3] use non-standard coefficients forReynolds Stress Transport Models (RSTM) andparadoxically the standard k– model returns reasonablepredictions of mean velocities and global level ofturbulent kinetic energy. They note that is purely bychance thanks to the erroneous overproduction of kineticenergy on the impinging side of the tubes while itsmaximum and minimum locations are wrong. Meyer [4]uses the LDA technique to measure 2D mean velocitiesand Reynolds stresses in two successive rows in themiddle of tube bundle. The measurements are comparedwith the numerical predictions using the standard k– model and the second moment closure, both models givepoor predictions of the Reynolds stresses. Zukauskas andUlinskas [5] present some experimental results on thecharacteristics of the flow and the heat transfer around asingle tube and cross-flow tube bundle. They establishthat, in the sub-critical flow regime, the pressure drop in atube bundle decreases with Reynolds number but beyonda critical Reynolds number it increases. They also reportCorresponding author: aounallah 2000@yahoo.fr ! " # ! " Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20159202005
EPJ Web of Conferencescorrelations for the total pressure drop and heat transfercoefficients for a wide range of Reynolds numbers andseveral tube bundle configurations. Rollet-Miet et al. [6]use the LES technique and the k– model to predict theturbulent flow as produced by the experiment of Simoninand Barcouda [2]. Both LES and k– models calculate themean velocity profiles reasonably well while k– modelgives poor predictions of the Reynolds stresses in thewake region inconsistency to the LES results.Benhamadouche et al. [7] compute the cross-flow in astaggered tube bundle with LES and a transient ReynoldsStress Transport Models (RSTM) in 2D and 3Dconfigurations with two levels of grid refinement. It isshown that the LES results on the fine mesh arecomparable to the DNS and the experiment andreasonable agreement is still achieved with a coarsemesh. The RSTM produces also satisfactory results in 3Dbut showed no advantage over the LES when the grid iscoarse. The 2D RSTM, which produced strong vortexshedding, is found to be physically unreasonable.Moulinec et al. [8] use the DNS technique for a Reynoldsnumber of 6000 to predict the flow field. The Navier–Stokes equations are discretized on a staggered grid withthe finite volume approach. Overall, they report that theirpredictions are in good agreement with the experiment.Paul et al. [9] use the low-Reynolds number k– , the k– and the -based second moment closure models to predictthe flow in the typical section of the experiment ofSimonin and Barcouda [2]. They report that the k– andthe k– models give better predictions of the mean flowin the region just before and after the tube than thesecond moment closure (LRR-IP) model. Moreover, thislatter does not produce consistently good predictions ofthe Reynolds stresses. Benhamadouche et al. [10]examine the influence of the pitch ratio (P/D 1.44 andP/D 1.75) for a cross-flow 3D square in-line tubebundle by means of LES with periodic boundaryconditions and using the in-house EDF code Saturne andCD-Adapco code STAR CCM. It is shown that a highaccuracy is needed for pressure gradient calculation toobtain a physical solution. Drag and lift coefficients havereasonable values, but their rms values are very sensitiveto the extrusion length. Mizushima and Suehiro [11]investigate the transition of flow past only two tubes intandem arrangement, which can be considered as a verysimple model of tube bundle, and found a hysteresis inthe amplitude of velocity in the oscillatory flow. Theyexplain that the hysteresis comes from a bending ofsolution branch due to non-linear interactions of the wakewith the downstream tube, which leads to two stableoscillatory solution branches connected to each otherthrough an unstable solution branch. Paul et al. [12]achieve measurements in a staggered tube bundle withfixed pitch ratio at sub-critical Reynolds number usingPIV technique. Their results show that the flow exhibitsstrong Reynolds number dependence in the developingregion, however no significant effect is observed in thespatially periodic region. Imran [13] studies the turbulentflow through square in-line arrays using various RANSand LES simulations. Several flow parameters such asvelocity profiles, pressure distributions, instantaneousand RMS lift and drag coefficients and Reynolds stressesare compared to the available experimental data. Theauthor found an unexpected asymmetric behavior withflow switching directions. Yahiaoui et al. [14] investigateexperimentally the turbulent flow through an in–line tubebundle formed by 7 7 tubes arranged with a constanttransverse and longitudinal pitch ratio of 1.44. Thepressure distribution on the tube surface indicates thestagnation pressure region located somewhere around 45ºsimilarly to the LES results of Imran [13].The first aim of the present study is to performdetailed assessment of the ability of the commercial CFDcode, ANSYS Fluent, together with four differentturbulence models in their unsteady state formulations,Spalart- Allmaras, RNG k– , SST k– , and RSM secondmoment closure, to reproduce the mean flow in the inline tube bundle. The good agreement of the numericalresults with the experimental measurements associatedwith a relatively low computational cost referee the bettermodel. As mentioned previously, most of the numericalinvestigations have treated this problem with the periodicboundary conditions assumption to reduce the grid sizeand consequently the CPU time. Here, the entire tubebundle configuration formed by the whole 49 tubes isconsidered to verify the validity of the periodicitycondition. The turbulent flow through the tube bundle isassumed as a two-dimensional case although it isgenerally characterized by significant 3D flow structuresand highly fluctuating wake regions with vortex sheddingfrom the time when Yahiaoui et al. [14] confirm the sametrend of the pressure coefficient, the drag and liftcoefficients distributions for different transversalpositions along the tubes.2 Experimental setup and measurementprocedureThe experiments are carried out in a closed-circuit airtunnel with a test section of 460 mm wide, 460 mm deepand 1000 mm long. It is driven by an A.C motor and axialflow fan that forces air around the circuit and produces amaximum velocity of 60 m/s. The test section is madewith a clear Plexiglas to facilitate visualization. Thebundle contains 7 rows and 7 arrays of PVC tubes ofouter diameter of 40 mm arranged in square in-linearrays. Half tubes are also fixed along the top and bottomwalls of the test model to simulate infinite tube bundleand minimize the wall boundary layers. The pitch ratio isrespected in both direction; longitudinal and transverse(P/D T/D 1.44). The deep-to-diameter ratio is equalto 11.5, larger than 7.2 used by Balabani et al. [15], whoconsider measurements at the mid-span to be free fromthe end effects. The air with an approach velocity, Uinf 3.13 m/s enters to the test section uniformly giving aReynolds number of 35 000 based on the tube diameterand the gap bulk velocity Ugap 10.25 m/s. Figure 1shows the cross-sectional view of the tube bundle modelwith the definition of geometrical parameters. Forconvention, each tube is coded by a number; Tube 23 islocated in the second line from the bottom and the thirdcolumn from the right. The angle measurement is similarto counter clockwise direction and starts from the inletfree stream direction.02005-p.2
EFM 2014simulations. The two dimensional assumption is validatedby Paul et al. [12] who clearly demonstrate the 2D flowin the mid-plane of the channel. The flow is assumed tobe unsteady and turbulent. The incompressible fluid isNewtonian with constant density and dynamic viscosity . The body force due to buoyancy is neglected. Theequations governing the development of the flow throughthe tube bundle are as follows: U j(1) 0 x j U j t P ρ U jU i x j xi x j()·§ U i μ ρ ui u j (2) x j¹ where ui u j is the Reynolds stress tensor. The mean flowFigure 1. Cross-flow through the in-line square tube bundle.The pressure measurements are done using the TE44DPS differential pressure scanner. The latter pressurescanning box allowed sequential selection of up to 20pressure tapings. The display unit links to a computer,loaded with DATASLIM software for data analysis andlogging of results. Figure 2 shows the pressure coefficientmeasurement deviation for the pivot tube 44. It can beseen that the uncertainty analysis of the presentmeasurement is insignificant since the uncertainly in themean pressure at 95% confidence level is estimated to be 4 % and the standard deviation is only of the order of1.6%.pressure is P and the mean velocity Ui. The boundaryconditions for the computational domain are defined asfollows. At the inlet, an average upstream value of themean velocity, Uinf 3.13 m/s fixed from measurementsand a turbulence intensity of 4% are specified, theturbulent quantities are estimated using the turbulenceintensity and the hydraulic diameter of tube. The no-slipboundary condition is prescribed on the tube surfaces, thebottom and top walls. At the exit, a pressure-outletcondition is specified with a reference value of zero.Several turbulence models are used; the one equationmodel developed by Spalart-Allmaras [16], the twoequation of renormalization group k- developedby Yakhot and Orszag [17], the shear stress transport(SST) k- model developed by Menter [18] and thesecond moment closure Reynolds Stress Modeldeveloped by Launder et al. [19].4 Numerical 60Azimuthal angle ( )Figure 2. Pressure coefficient measurement deviation for thetube 44.3 Numerical modelingThe geometry of the 2D domain representing the typicalregion in the in-line bundle with the tube diameter D, thetransverse pitch P and the longitudinal pitch T is shownin Figure 1. The same number of tubes in theexperimental investigation is considered for theThe coupled conservation equations are resolvednumerically using the commercial code ANSYS Fluent6.3.26 with the segregated solver and the finite volumemethod approach of Patankar [20]. The partial differentialequations are discretized on a grid where all the variablesare collocated. The second-order upwind scheme is usedin the momentum and the turbulence transportdiscretization while the standard scheme is employed forthe pressure equation. The SIMPLE algorithm is used todeal with the pressure–velocity coupling between themomentum and the continuity equations. Thecomputations are done with double precision and thesolution is achieved when all residuals of the discretizedequations fall down 10-5. The relaxation parameters areadjusted for each simulation in order to accelerateconvergence. The computational domain is filled with anon-uniform hybrid grid with a very fine spacing near thetube walls, as needed for accurately resolving the steepgradients in the thin boundary layer. Mesh-independencetests are performed to investigate the influence of gridrefinement on the solution. Figure 3 shows a partial viewof the computational domain with the coarse and the finemesh.02005-p.3
EPJ Web of Conferencesa)b)section, only the grid independence for the SpalartAllmaras model is presented. As will be seen later, thissimple one-equation model could be a very promisingtool for numerical simulation of complex turbulent flowsthrough the tube bundle compared to the two-equationmodels assumed eddy–viscosity and the second momentclosure model. Qualitatively, all the grids tested give thesame trend of the pressure coefficient. The fine meshgives more neighboring value to the experimental results[14] and the LES of Imran [13], particularly in the rangebetween 120 - 240 , while the deficit of the coarse andthe medium grids is obviously located among 75 and180 . Surprisingly, the coarse grid surmounts the othersgrids by the good capture of the high pressure around thestagnation point behind the angle of 45 where themedium mesh underestimates dramatically the pressuredistribution. It is clearly seen that the experimentalmeasurement forecasts a constant pressure distributionfrom the angle 240 until 300 in opposition to thenumerical predictions which predict a decrease then anincrease. From the two thirds of the tube, all the gridstried are unable to predict the physical behavior of thepressure distribution even the LES. In overall and on thecomplete circumference of the tube, the fine mesh can bekept as the suitable one that reproduces neighboringpressure distribution to the experiment results.Coarse meshFine meshc)Exp. [14]LES [13]coarsemediumfine1,0Figure 3. A partial view of the computational domain.0,5Cp5 ValidationsMesh-independence tests are performed to investigate theinfluence of grid refinement on the solution. Table 1provides some details of the three grids tested includingthe total number of nodes and cells, the number of nodesalong the longitudinal, the transversal directions and onthe tube surface respectively. The boundary layer regionaround tubes is formed by 8 layers with an expansionfactor of 1.1. Many numerical tests have been carried outin this section to choose the suitable grid that provides theclosest results to the experimental measurement.Table1. Grids details used for mesh-independence tests.GridcoarsemediumfineNodesCellsNxNyNtube51 060159 318236 37778 512271 312413 5082905287519619225652104130Figure 4 shows the computed pressure coefficientdistribution around the tube 33 (row: 3, column: 3) interms of azimuthal angle. In overall, the pressure curveshows an asymmetric distribution with a fairly highpressure region on the windward side and a low pressureregion on the leeward side. This asymmetry is createdfrom the deflection of the flow through array. In this0,0-0,504590135180225270315360azimuthal angle ( )Figure 4. Grid independence tests of the Spalart-Allmarasmodel.Figure 5 illustrates the ability of the turbulencemodels used to resolve the flow characteristics using thefine grid and namely the pressure distribution along thetube 33 surface. Once again, the asymmetric pressuredistribution due to the flow direction switching isobserved. The stagnation point is at an angle of 45degrees, as expected in the literature. Lam and Fang [21]found the location of the high pressure in the angle rangeof 20-60 degrees. The adverse pressure region on theleeward side which corresponds to an angle range from80 to 300 degrees with a sharp dip at 90 degrees, leads toan almost no zero net lift force. In comparison with theexperimental data of Yahiaoui [14] and the numericalLES computation conducted by Imran [13], the generaltendency of the pressure coefficient is set up by all the02005-p.4
EFM 2014RANS models tested with some quantitativediscrepancies. As common remarks, all the turbulencemodels underestimate the pressure peak at the stagnationpoint and none can establish the null pressure gradient inthe angle range between 240 and 300 even the LES. Astrong decrease and increase of pressure is observed inthis area. Inconsistently, the simple one equation model;Spalarat-Allmaras agree enough well with experiment inspite of the advance position of the stagnation point. TheRNG k- model is reasonably good but it dramaticallyunderestimates the value of the stagnation point andoverestimates the pressure at the angle of 90 . Obviously,the shear stress model k- SST seems to be somewhatfailed due to the underestimation of the pressurecoefficient at the angle of 45 and exclusively to the overassessment of the pressure in the range 90 - 180 . As forthe RSM model, it can be refereed less good than theother models. This latter underestimates the pressurecoefficient on the majority surface of the tube and alsoover predicts the pressure at the angle 90 . With that, itshould not forget the warrant of the additionalcomputational expense. Finally, it is important to note thepoor quality of the numerical models tested even the LESin the region 240 -300 , where the experimentalmeasurements attest a null pressure gradient.Exp. [14]LES [13]S-ARNG K-εKω imuthal angle( )Figure 5. Pressure coefficient distributions along the tube 33.6 Results and discussionFigure 6 shows the streamlines across the tube bundleobtained with the Spalart-Allmaras model discretized onthe fine grid at t 15 s. This figure ascertains the nonresemblance of the flow behavior in the space for the twodifferent times, the flow pattern around any tube is notsimilar to it adjacent. It is apparent from both contours,that the periodicity boundary condition is not valid tosimulate the cross-flow in tube bundle. The random andthe chaotic characters of the flow are obviously seen onthe full computational domain. Following the flowdirection, the streamlines are nearly horizontal in thepassages trough the tubes. Different vortices withdifferent size and opposite in rotational direction areshaped backstream of some tubes. It is also interesting tonote that the flow pattern in the first row is qualitativelysimilar to that an isolated single cross-flow. The flowbecomes oscillatory downstream the last array and largevortices with different rotation direction tend to beformed. After the wake region, the streamlines becomesregular and igure 6. Flow pattern in the tube bundle.In order to check the validity of the periodicitycondition, a particular attention is allocated to thepressure coefficient predictions of the central horizontalline and the middle vertical column tubes. Igarashi [22]classify the flow pattern into three types according to themaximum value of the pressure coefficient and itspositions. The three situations are designated as W, R andJ hereinafter, respectively. The situation withoutreattachment of the shear layer, in which the Cp value hasno maximum, belongs to the first type. The situation withreattachment of the shear layer, in which the Cp value hasa maximum, belongs to the second type. The situation ofrolling up of the shear layer in front region of thecylinder, in which the Cp value has maximum at 0 angle,belongs to the third type.Figure 7 reveals the pressure distribution comparisonin the fourth vertical column. As can be seen, the sametrend of the pressure coefficient along the border tubes isestablished by both experimental measurements andnumerical predictions. The reattachment of the shearlayer is experimentally located in the range 40 -50 andfairly found advanced with the numerical computation(25 -35 ). This remark is also valid for the separation ofthe shear layer indicated by the lower pressure coefficientaround the angle 100 . The major quantitativediscrepancy of the numerical approach compared to theexperiment is well realized in the angle range of 330 360 , it represents a second reattachment. Unless thislatter statement, it can be concluded that the flow showsunexpected symmetric behavior in the transversedirection.02005-p.5
EPJ Web of 033014243444546474Numerical results30150018021060Numerical 647300270270Figure 7. Pressure coefficient distribution in the 4th column.Figure 8. Pressure coefficient distribution in the 4th row.Following the horizontal middle row and contrary tothe vertical array, notable qualitative and quantitativedifferences exist between both methodologies as can beshown in figure 8 with the only similarity of the firsttube, where the pressure action appears fairly symmetricand resemble to an isolated cylinder. Numerically, theprofiles of the successive tubes are almost the same; twostagnation points indicating the reattachment of the shearlayer are predicted around 25 and 315 , whileexperimental measurement explored switched differentpressure distributions. The measurements show that thecylinders are more solicited close to the 45 angle withdifferent intensity except tube 43 which has it stagnationpoint at 325 . The flow shows unexpected asymmetricbehavior with flow switching directions. So, it can beconclude that no periodic pattern occurs in this direction.modelled and tested by four models to resort the best onethat expresses the closest pressure distribution to theexperimental data.The numerical results confirm that with carefullydesigned grid resolution and distribution, the SpalartAllmaras model can predict rather well the characteristicsof turbulent flow past the tube bundle in spite of theadvance location of the stagnation point. The flow patternobtained numerically inside the full computationaldomain is characterized by a totally variance in spacesanctioning the use the periodicity boundary condition.The number and the size of the vortices formeddownstream tubes are markedly unalike and disappearaltogether the last tube. It is confirmed by the significantdifference of the pressure distribution curves on thehorizontally and vertically central tubes. The resultsreveal also the instantaneous pressure and velocitymagnitude fields and pressure which describe thecomplex flow characteristics in tube bundles. Thisimplies that further measurements and numerical testsshould be conducted in the future to more comprehendsuch complex flow.ConclusionIn the present study, Experimental and numericalinvestigations of the turbulent flow through an in-line7 7 bundle configuration with a constant transverse andlongitudinal pitch-to-diameter ratio of 1.44 were carriedout. The investigation covers the validity of the periodicboundary conditions with the appropriate turbulencemodel. The DPS differential pressure scanner techniqueis used to perform detailed measurements. Turbulence is02005-p.6
EFM 2014References1.B. E. Launder and T. H. Massey, J. Heat Transfer,vol. 100, (1978)2. O. Simonin and M. Barcouda, Fourth InternationalSymposium on Applications of Laser Anemometryto Fluid Mechanics. Lisbon, Portugal, (1988)3. S. Sebag, V. Maupu, D. Laurence, TSF8, (1991)4. K.E. Meyer, Experimental and numericalinvestigation of turbulent flow and heat transfer instaggered tube Bundles. PhD thesis, TechnicalUniversity of Denmark, (1994)5. A. Zukauskas and R. Ulinskas, Banks of plain andfinned tubes. In: Hewitt, G.F. (Ed.), Heat ExchangerDesign Handbook. HEDH, New York, 2.2.4-1–2.2.4-17, (1998)6. P. Rollet-Miet, D. Laurence, J. Ferziger, Int. J. HeatFluid Flow 20, (1999)7. S. Benhamadouche and D. Laurence, Int. J. HeatFluid Flow 24, (2003)8. C. Moulinec, M. J. B. M. Pourquie, B. J. Boersma, F.T. M. Nieuwstadt, Int. J. Comput. Fluid Dyn. 18 (1),(2004)9. S. S. Paul, S. J. Ormiston, M. F. Tachie, Proceedingsof the 12th Annual Conference of the CFD Societyof Canada, Heat Transfer 1, Ottawa, Canada, (2004)10. S. Benhamadouche, D. Laurence, N. Jarrin, I. Afgan,C. Moulinec, NURETH-11(Nuclear ReactorThermal-Hyd.), (2005)11. J. Mizushima, N. Suehiro, Phys. Fluids 17, (2005)12. S. S. Paul, M. F. Tachie, S. J. Ormiston, Int. J. HeatFluid Flow 28, (2007)13. I. Afgan, Large Eddy Simulation of cylindricalBodies incorporating unstructured finite volumemesh, PhD thesis, University of Manchester, (2007)14. T. Yahiaoui, L. Adjlout, O. Imine, O. Imine,Mechanika, (2010)15. S. Balabani, M. Yianneskis, Proc. IMechE Part C, J.Mech. Eng. Sci. 210, (1996)16. P. Spalart and S. Allmaras, Technical Report AIAA92-0439, American Institute of Aeronautics andAstronautics, (1992)17. V. Yakhot and S. A. Orszag, Journal of ScientificComputing, 1(1),(1986)18. F. R. Menter, AIAA J. 32 (8), (1994)19. B. E. Launder, T. H. Massey, Trans. ASME J. HeatTransfer 100 (4), (1978)20. S.V. Patankar, Numerical Heat Transfer and FluidFlow, Hemisphere (McGraw-Hill), New York,(1980)21. K. Lam, X. Fang, J. of Fluids and Structures, 9.(1995)22. T. Igarashi, Bulletin of JSME, Vol.29, N 249,(1988)02005-p.7
the periodicity assumption taken account in the most previous numerical works by considering the filled bundle geometry. The CFD results show that the Spalart-Allmaras model on the fine mesh are comparable to the experiments while the periodicity statement did not produce consistentl
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