Numerical Simulation of Turbulent Buoyant Helium Plume by Algebraic Turbulent Mass Flux Model

Transcription

1 Numerical Simulation of Turbulent Buoyant Helium Plume by Algebraic Turbulent Mass Flux Model Hitoshi Sugiyama 1),Naoto Kato 1), Masahiro Ouchi 1) 1) Graduate School of Engineering,Utsunomiya University Saori Mogami ) ) Fuji Electric FA Components & Systems Co., Ltd Atsuhiko Terada 3), Ryutaro Hino 3) 3) Japan Atomic Energy Agency Graduate school of engineering, Utsunomiya University

2 Purpose Driving force of buoyancy is classified into two kinds of forces. One is driving force generated by temperature and the other one is produced by different density. Buoyant force for different density is the target in this study. The aim of this research is to propose the turbulent buoyant model composed of algebraic Reynolds stress and algebraic turbulent mass flux models to predict turbulent flow with buoyant force. Numerical target is the experimental data of turbulent buoyant helium plume measured by O Hern et al. to confirm the validation of the presented anisotropic turbulent buoyant model. Graduate school of engineering, Utsunomiya University

3 Basic governing equations for turbulent flow Reynolds-averaged Navier-Stokes (RANS) equation Ui Ui 1 P 1 U C i Uk uu i j t xk xi x k Re x Fr k Reynolds-averaged mass equation Reynolds stress C C 1 C U k i t xk xk Re Sc x uc k Uref Dref Uref Re, Fr, Sc gd i ref Turbulent mass flux Buoyant force for different density Basic governing equations are expressed as dimensionless form because we need not to consider scaling effect. Scale is considered indirectly as dimensionless parameters. Reynolds stress and turbulent mass flux are obtained by solving transport equations in order to predict correctly anisotropic turbulent flow. D Re : Reynolds number Fr : Froud number Sc : Schmidt number Graduate school of engineering, Utsunomiya University 3

4 Reynolds stress and turbulent mass flux Transport equation of Reynolds stress Production term for buoyancy uu i j uu i j U j Ui p ui u j U k uiuk u juk giu jc g juic t xk xk xk x j xi convection term Pressure-strain term uiu j p uiu j uu i juk jkui iku j xk xk xkxk diffusion term Transport equation of turbulent mass flux uc i uc i C U p c U k uiu j u jc g i c t xk x j x j xi convection term pressure-mass gradient term Mass fluctuation for buoyancy p cui uu i jc cij ( ) x j x jx j diffusion term Rodi s approximation is applied to convection and diffusion terms to save computational time, but it is true that this approximation can not express more exactly relationship between parameters than differencing equation form. Pressure-stain and pressure-mass gradient terms play an important role to redistribute turbulent energy and turbulent mass flux, respectively. Graduate school of engineering, Utsunomiya University 4

5 Modeling for pressure-strain term and model constants ij,1 ji,1 C 1 u u k k 3 i j ij ij, ji, ij,3 ji,3 ij, w ji, w C 8 U U j X i 8C ( D ij Pk ij ) 11 3 i ( Pij Pk ij ) k 11 3 X j C 3 P 3 P ij, c c ij / w / w / C C C f L X C C C f L X * ' * ' * ' f L X w U j U i U k U k Pij u iu k u ju k, D ij u iu k u ju k, X k X k X i X i U P u u, P g u c g u c, P g u c, k k k l ij, c i j j i c i i X l 3/4 3/ / w / w f L X C k X * * C C 1 * C 1 C C 3 ' C Graduate school of engineering, Utsunomiya University 5

6 Modeling of pressure-mass gradient term and model constants ic,1 u iu j C 1 c u ic C 1 c ij u jc k k k 3 U i U m C c u m c C c u m c X X ic, m C 3 c g i c ic,3 ic, w C C 1 C f L / X * 1 c 1 c 1 c, w w C C 1 C f L / X * 1 c 1 c 1 c, w w C C 1 C f L / X * c c c, w w C C 1 C f L / X * c c c, w w i c u k c 4 c 1 C C k X 3/4 3/ / w / f L X C k X k w * * C 1c 1c C * C c C * c C3c C C C 4 c 1 cw, cw, Graduate school of engineering, Utsunomiya University 6

7 Modeling of turbulent energy and dissipation and model constants Transport equation of turbulent energy Dk k k U c u u uu g uc Dt X Re X X kj i s k j i k i i k j k Transport equation of turbulent dissipation D kj k U i c ukuj c1uu i k c3 giuc i c Dt X k Re X j k X k Cs C C 1 C C Diffusion terms of turbulent energy and turbulent dissipation are modeled by Daly-Harlow model. Since there is no information about model constant C 3ε in turbulent dissipation, it was set to 0.5 as a result of trial and error calculations. Graduate school of engineering, Utsunomiya University 7

8 Momentum-dominated buoyant jets and buoyancy-dominated plumes Momentum-dominated buoyant jets When the helium is discharged at relatively high speed, it moves due to convection, and appearance of a significant buoyancy effect is limited to the region quite far downstream from the jet outlet. Subbarao, E.R., Cantwell, B.J.:Investigation of a co-flowing buoyant jet: experiments on the effect of Reynolds number and Richardson number, J.Fluid Mech., 45, pp (199) Panchapakesan, N.R.,Lumley, J.L.:Turbulence measurements in axisymmetric jets of air and helium.part.helium jet, J. Fluid Mech., 46, pp.5-47 (1993) Buoyancy-dominated plumes When the jet outlet speed is extremely low, the propulsion force of the flow is dominated by the buoyancy arising from the difference in density between the helium and the surrounding fluid. Large scale is needed in order to form a turbulence field. O Hern, T.J., Weckman, E.J., Gerhart, A.L., Tieszen, S.R., Schefer, R.W.: Experimental study of a turbulent buoyant helium plume, J. Fluid Mech., 544, pp (005) Graduate school of engineering, Utsunomiya University 8

9 Calculation target O Hern et al. Experimental study of a turbulent buoyant helium plume, J. Fluid Mech.,544(005) Measured parameter Velocity Helium concentration Reynolds stresses Experimental condition Discharge outlet velocity : 0.35 m/s Helium discharge concentration : 100% Reynolds number : Re=38 Froude number : Fr= Characteristic phenomena Accelerated flow Rapid diffusion of Helium Periodic velocity fluctuation due to Raleigh-Taylor instability Graduate school of engineering, Utsunomiya University 9

10 Calculation grids layout and coordinate system Calculation grids are set to 68, 11 and 11 along X 1, X and X 3 coordinates, respectively. Graduate school of engineering, Utsunomiya University 10

11 Calculation condition of discharge outlet velocity Calculation condition Discharge outlet velocity : 0.1 m/s Helium discharge concentration : 100% Froude number : Fr= m/s O Hern reported discharge outlet velocity is 0.35 m/s, but the position corresponding to 0.35 m/s was around m from the discharge outlet, and they measured the 0. m/s isoline at around 0.01 m. From this perspective, in this analysis, the discharge outlet velocity was set to what is thought to be a more realistic 0.1 m/s. Graduate school of engineering, Utsunomiya University 11

12 Comparison of streamwise velocity All calculated and experimental results are normalized by discharge outlet velocity. Graduate school of engineering, Utsunomiya University 1

13 Comparison of horizontal velocity u k in region of U 3 X 0 u k in region of U 3 X 0 Graduate school of engineering, Utsunomiya University 13

14 Time history of streamwise velocity 0 Prediction -0.4 U 1 /Ur t /(D/U r) Periodic velocity variation was reported to arise from Rayleigh-Taylor instability theory where high-density fluid descends and low-density fluid rises. Calculated periodic velocity variation is 1.78 Hz, which is slightly different from the measured value 1.34 Hz. Graduate school of engineering, Utsunomiya University 14

15 Animation of streamwise velocity vectors Graduate school of engineering, Utsunomiya University 15

16 Comparison of helium concentration Graduate school of engineering, Utsunomiya University 16

17 Animation of helium concentration Graduate school of engineering, Utsunomiya University 17

18 Comparison of turbulent energy Graduate school of engineering, Utsunomiya University 18

19 Comparison of streamwise normal stress u 1 Big discrepancy is found at streamwise normal stress. Later slides show whether which results are more reasonable, or not. Graduate school of engineering, Utsunomiya University 19

20 Comparison of horizontal normal stress u In the analysis, although the isoline distribution is different, the experimental values were quantitatively predicted. It is also pointed out that calculated horizontal normal stress is greater than streamwise normal stress in accelerated flow region. Graduate school of engineering, Utsunomiya University 0

21 Comparison of shear stress uu 1 Experimental isolines of shear stress show the different sign region. Calculation also predicts well this different sign region of shear stress. Graduate school of engineering, Utsunomiya University 1

22 u 1 u Relation between and Boussinesq s eddy-viscosity model U U i j uu i j t k ij X j X i 3 U 1 U 1 u1 u u 3 u1 t k t X1 3 X1 3 u u 3 U 1 u1 3 t X1 U1 u1 u 3 t u u3 Axisymmetric flow X1 In the case of accelerated flow U, 1 / X1 0 u 1 is smaller than u. In the case of decelerated flow U / X, is greater than. u u Calculated result of 1 shows that is smaller than u in accelerated flow. On the other hand, experimental result of is not satisfied with this relation. u 1 u 1 u Graduate school of engineering, Utsunomiya University

23 Mechanism for decrease of and increase of u Buoyancy part of pressure-strain term, i.e., redistribution term C P P 3 ij,3 ji,3 3 ij, c c ij P g u c g u c, P g u c g u c ij, c i j j i c i i 1 1 u 1 u 1 u u 3 P ij, c P c 0 0 C 3 P, P 3 4 C P ij c c ij C 3 C c P 3 c P 3 c u The normal stresses and u 3 increase by the value C3P c /3 through the redistribution term from u 1 and u 1 decreases by distribution of the value 4 C3P c /3to the horizontal normal stress. Graduate school of engineering, Utsunomiya University 3

24 Relation between u and turbulent energy Boussinesq s eddy-viscosity model uu U U k 3 i j i j t ij X j X i k U u t X 3 k u k in region of U 3 X 0 This region corresponds to upper part of helium plume According to Boussinesq s eddy-viscosity model, horizontal normal stress nearly equals to k/ 3 in region of U / X 0. Calculated result is satisfied with this relation. On the other hand, the experimental result is not satisfied with this relation. Judging from several considerations, calculated results seem to be more reasonable than the experimental data. Graduate school of engineering, Utsunomiya University 4

25 Conclusions Algebraic Reynolds stress and algebraic turbulent mass flux models were proposed to predict anisotropic turbulent flow with buoyant force due to density differences. The proposed anisotropic turbulent model predicted characteristic features that are accelerated flow, rapid diffusion of helium and periodic velocity fluctuation due to Raleigh-Taylor instability. Judging from the comparison of Reynolds stresses, the presented algebraic turbulent models underestimated especially for streamwise normal stress of experimental data. This underestimation of streamwise normal stress was caused by buoyant part of pressure-strain correlation term. However, this underestimation in accelerated flow was consistent with the theoretical consideration derived from Boussinesq s eddy-viscosity model. Graduate school of engineering, Utsunomiya University 5

26 Action plans from now and future In order to achieve high-precision of calculation, it is necessary to study continuously the modeling of pressure-strain, pressure-temperature and pressure-mass gradient terms. Next target is to calculate the turbulent flow with heat and mass transfer for actual unclear reactor in severe accident by using the presented anisotropic algebraic turbulent model. The last goal is to propose algebraic Reynolds stress, heat flux and mass flux models which are able to predict reasonably the turbulent flow with heat and mass transfer in actual fields. Graduate school of engineering, Utsunomiya University 6

27 Acknowledgement This numerical study of turbulent flow with buoyant force was performed in the Advanced Nuclear Hydrogen Safety Research Program entrusted by Agency for Natural Resources and Energy of Ministry of Economy, Trade and Industry (METI). Graduate school of engineering, Utsunomiya University 7

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