NUMERICAL INVESTIGATION OF BUOY- ANCY DRIVEN FLOWS IN TIGHT LATTICE FUEL BUNDLES

Transcription

1 Fifth FreeFem workshop on Generic Solver for PDEs: FreeFem++ and its applications NUMERICAL INVESTIGATION OF BUOY- ANCY DRIVEN FLOWS IN TIGHT LATTICE FUEL BUNDLES Paris, December 12 th, 2013 Giuseppe Pitton 1 Hisashi Ninokata Davide Baroli 2 Energy department MOX, Mathematics department 1 giuseppe.pitton@gmail.com 2 davide.baroli@polimi.it

2 .. ACKNOWLEDGEMENTS Gianluigi Rozza All the people in the Cesnef-MOX collaboration: Marco Enrico Ricotti Lelio Luzzi Antonio Cammi Luca Formaggia SISSA-mathLab Energy dep. Energy dep. Energy dep. MOX, Math dep. 2

3 .. INDEX

4 INTRODUCTION

5 .. NEW SODIUM FAST REACTORS Generation-IV International Programme breeder or burner reactors low pitch-diameter ratio natural convection Reactor P (mm) D (mm) P/D BN FFTF Monju Phénix Superphénix S

6 Index.. EXAMPLE: SUPERPHÉNIX Nuclear Engineering Division, Department of Energy, Politecnico di Milano prof. M.E. Ricotti Figure: Detail of fuel assemblies for the Superphénix reactor. 16 6

7 .. NEW SODIUM FAST REACTORS low pitch-diameter ratio natural convection Thermohydraulic consequences: flow oscillations between subchannels increased heat, mass and momentum transfer between subchannels not shown by subchannel analysis codes (COBRA, RELAP, ) require modeling can CFD support subchannel analysis codes? 7

8 .. GOAL can CFD support subchannel analysis codes? Yes, without thermal coupling (Baglietto, Merzari, Ninokata) How about thermal coupling? H. Ninokata, E. Merzari and A. Khakim, Analysis of low Reynolds number turbulent flow phenomena in nuclear fuel pin subassemblies of tight lattice configuration. Nuclear Engineering and Design 239 (2009) 8

9 MATHEMATICAL MODEL

10 .. COMPUTATIONAL DOMAIN Krauss & Meyer experiment: 37-pin rod bundle P/D = 1.06 too expensive T. Krauss and L. Meyer, Experimental investigation of turbulent transport of momentum and energy in a heated rod bundle. Nuclear Engineering and Design 180 (1998) 10

11 .. COMPUTATIONAL DOMAIN Krauss & Meyer experiment: 37-pin rod bundle P/D = 1.06 too expensive simulate a small periodic part T. Krauss and L. Meyer, Experimental investigation of turbulent transport of momentum and energy in a heated rod bundle. Nuclear Engineering and Design 180 (1998) 11

12 .. MATHEMATICAL MODEL The hypotheses Incompressible flow Stokesian flow Boussinesq approximation The equations t u u ( u) (2νD(u)) + p T = gβ(ϑ ϑ 0 ) u = 0 t ϑ + u ϑ α ϑ = 0 + b.c. and i.c. 12

13 .. VARIATIONAL FORM Variational Navier-Stokes Find u H 1 (Ω), u = t on Γ D, p L 2 0 (Ω) such that t > 0, v H 1 0,Γ D (Ω), q L 2 (Ω) m(u, v) + a(u, v) + ĉ(u, u, v) + b(v, p) = F (v) b(u, q) = 0 u(t = 0, Ω) = u 0. Variational forms introduced: a(u, v) = ( v, ν u) b(v, p) = ( v, p) m(u, v) = (v, t u) ĉ(w, u, v) = (v, u ( w)) F (v) = (v, f) + v, d ΓN 13

14 .. VARIATIONAL FORM Variational energy equation Find ϑ H 1 (Ω), ϑ = ϑ D on Γ D, such that { (φ, dt ϑ) + e(ϑ, φ) = φ, α ϑ ΓN φ H 1 0,Γ D (Ω) ϑ(t = 0, Ω) = ϑ 0 where e(ϑ, φ) = ( φ, α ϑ) where d t denotes the total derivative: d t ϑ = t ϑ + u ϑ 14

15 .. WELL-POSEDNESS Before trying to solve a problem, see if it is correctly posed Hadamard definition A problem is well posed if: a solution exists the solution is unique the solution depends continuously on data 15

16 .. WELL-POSEDNESS Proved for energy equation, if u is sufficiently regular (Hille-Yosida). For Navier-Stokes, Caution existence of weak solutions shown uniqueness of weak solutions open problem (proved for small times or small data) regularity of weak solutions open problem (only partial regularity results) 16

17 NUMERICAL METHOD

18 .. GALERKIN PROJECTION Choose two finite dimensional spaces: V h H 1 (Ω) for velocity Q h L 2 (Ω) for pressure and project the continuous solution onto these spaces. How to choose V h and Q h? Many possibilities: Lagrangian elements P 0,., P 4 Discontinuous elements P 0dg,., P 4dg Boundary elements (implemented using P 0edge ) Mortar 18

19 .. GALERKIN PROJECTION Choose two finite dimensional spaces: V h H 1 (Ω) for velocity Q h L 2 (Ω) for pressure and project the continuous solution onto these spaces. How to choose V h and Q h? Many possibilities: Lagrangian elements P 0,., P 4 Discontinuous elements P 0dg,., P 4dg Boundary elements (implemented using P 0edge ) Mortar 19

20 .. TRIANGULATION With bamg and TetGen: 20

21 .. FINITE ELEMENT METHOD Write velocity and pressure as linear combination of the basis functions {ϕ j } and {ψ k } for each element: N u u h = u j ϕ j p h = p k ψ k. j=1 New unknowns: the nodal values {u j } and {p k }. Substituting into variational Navier-Stokes, and projecting on each dof (v h = ϕ j, p h = ψ k ): N p k=1 { m(vh, u h ) + a(v h, u h ) + ĉ(v h, u h, u h ) + b(v h, p h ) = F (v h ) b(u h, q h ) = 0 v h V h, q h Q h. 21

22 .. FINITE ELEMENT METHOD Still a nonlinear problem Implicit Euler in time + Picard linearization For each time step n + 1, solve the problem: u n+1 un t t un ( u n+1 ) (2νD(u n+1 )) p T = gβϑ n u n+1 = 0 ϑ n+1 ϑn t t + un ϑ n+1 α ϑ n+1 = 0. 22

23 .. FINITE ELEMENT METHOD Eventually, a linear algebra problem appeared: [ C n B T ] { U n+1} { G n+1} B 0 P n+1 = 0 Main difficulties saddle point problem pressure locking (incompressibility) non symmetric matrix 23

24 .. FINITE ELEMENT METHOD The algebraic formulation reads: [ C n B T ] { U n+1} { G n+1} = B 0 P n+1 0 Main difficulties saddle point problem bubble-stabilization on velocity pressure locking (incompressibility) add penalization non-symmetric matrix GMRES 24

25 .. TURBULENCE MODEL cannot resolve all the motions' scales (too expensive) solve only the large eddies, and model the small eddies Smagorinsky LES model the unresolved scales with subgrid diffusion: s(v h, u n h )un+1 h = ( v h, 2C 2 S 2 D(u n h ) D(un h ))un+1 h to be added to momentum balance equation similarly for energy balance equation: ( φ h, h) = ( φ h, ν T Pr T ϑ n+1 h ) 25

26 RESULTS

27 .. GENERAL PARAMETERS Main data P/D = 1.06 Re = Gr = 1181 q = Wm 2 Software used: FreeFem++-mpi 27

28 .. A NOTE ON PERIODICITY curved sides: no-slip for velocity and imposed heat flux for energy straight sides: periodic b.c. 28

29 .. A NOTE ON PERIODICITY velocity: fully periodic periodic pressure and temperature are not physical decompose pressure and temperature: p T (x, t) = p H z + p T(x, t) T (x, t) = T H z + T (x, t) and impose periodic b.c. only on the fluctuating part p T, T New equations: t u u u (2νD(u)) + p T = gβ(ϑ ϑ 0 ) p u = 0 t ϑ + u ϑ α ϑ = T H u z H κ 29

30 .. WALL TEMPERATURE ~T (K) φ ( ) Figure: Profile of time averaged wall temperature as computed (left) and from the results of Krauss and Meyer (right). 30

31 .. AVERAGED TEMPERATURE (a) Figure: Time averaged fluctuating temperature along the mid-height slice plane. (b) 31

32 .. TEMPERATURE TIME EVOLUTION ~T ( C) time (s) Oscillation frequency: 117 Hz, in line with experiment 32

33 .. VELOCITY TIME EVOLUTION 1 u x u y 0.8 u z 0.6 Velocity (m/s) Time (s) Oscillation frequency: 117 Hz, in line with experiment 33

34 .. COHERENT STRUCTURES Defined by Zaman and Hussain as Connected, large-scale turbulent fluid mass with a phase correlated velocity over its spatial extent. or, iso-value for the Q-factor: Q = II u = 1 (ΩΩ DD) 2 where Ω = 1 2 ( u ut ) D = 1 2 ( u + ut ) K. B. M. Q. Zaman and A. K. M. F. Hussain, Taylor hypothesis and large-scale coherent structures. Journal of Fluid Mechanics 112 (1981) 34

35 .. HOW TO TREAT THE CONVECTIVE TERM? Problem Matrix RHS Factorization Solution All 2d str A d str R d str L d str A d str R d str L d ustr A d ustr R d ustr L A : u ( u) R : u ( u) L : Du Dt 35

36 .. SCALABILITY 36

37 CONCLUSIONS

38 .. CONCLUSIONS agree quite well with experimental data: oscillations' frequency wall temperature distribution 38

39 .. PERSPECTIVES full fuel bundle simulation domain decomposition improve turbulence modeling (VMS) POD in time and Reduced Basis for optimal control 39

40 R. Sanzio et al., La Scuola di Atene, Musei Vaticani, Roma, 1510?. THANK YOU FOR YOUR ATTENTION QUESTIONS? SUGGESTIONS? NEW IDEAS?

41 R. Sanzio et al., La Scuola di Atene, Musei Vaticani, Roma, 1510?. THANK YOU FOR YOUR ATTENTION QUESTIONS? SUGGESTIONS? NEW IDEAS?