Application of Grid Convergence Index Estimation for Uncertainty Quantification in V&V of Multidimensional Thermal-Hydraulic Simulation

Transcription

1 The ASME Verification and Validation Symposium (V&V05) May 3-5, Las Vegas, Nevada, USA Application of Grid Convergence Index Estimation for Uncertainty Quantification in V&V of Multidimensional Thermal-Hydraulic Simulation Masaaki Tanaka Japan Atomic Energy Agency (JAEA)

2 High Cycle Thermal Fatigue around CIP at Bottom of UIS in JSFR Thermal fatigue caused by thermal mixing (thermal striping) between - Hot sodium from fuel subassemblies and - Cold sodium from control rod channels and blanket fuel subassemblies Upper Guide Tube Flow-hole st Baffle Plate Control Rod Driving Shaft Core Instruments Plate (CIP) (Top view image around a control rod channel) Hot Sodium Hot Sodium Cold Sodium Target areas related to thermal fatigue Cold Sodium Fuel Subassemblies Primary Control Rod (PCR) Channel Blanket Fuel Fuel Subassembly Subassemblies Backup Control Rod (BCR) Channel

3 Approach for Extrapolated Estimation of High Cycle Thermal Fatigue in JSFR 3 In extrapolated estimation of high cycle thermal fatigue in JSFR, - Confirmation of consistency of two methods to ensure the credibility. () Analytical estimation method based on the experimental data - Suitable for the design work and it is easy to handle - Difficulties to integrate and generalize experimental data by sub-scale experiments in view point of scaling effect () Direct estimation method using numerical simulations - Direct estimation from thermal-hydraulics to structural response - Requirement of comprehensive procedure for V&V and prediction Implementation of V&V (establishment of practical procedure) Appropriate arrangement of experiments for V&V Uncertainty quantification for each examination in V&V and integration of uncertainties for prediction Prediction (extrapolation) for the JSFR

4 VUP (Verification and Validation plus Uncertainty quantification and Prediction) 4

5 Conceptual Model for Numerical Estimation of High Cycle Thermal Fatigue around CIP 5 Thermal-hydraulics in upper plenum by RANS method (AQUA / CFD) - External flow for blanket fuels - Internal flow for control rods UIS Interal flow External flow Boundary conditions for local analysis Unsteady thermal-hydraulics and heat conduction in structure in local areas around - Control rod (CR) channels - Blanket fuel (BF) subassemblies by fluid-structure thermal interaction simulation code (MUGTHES) Temperature information in structure 3Estimation of structural integrity by thermal stress analysis (FINAS) Whole upper plenum analysis (for CRs) (for BFs) Local analysis 3Thermal stress analysis

6 Verification and Validation Processes in VUP 6 (Code) Verification: Uncertainty quantification(uq) by GCI estimation Comparing with the theoretical results of governing equations -step Validation: Step-) Fundamental Validation (functions of the solution verification and a part of validation processes): UQ by GCI estimation (a) Fundamental problems (FPs): including fundamental phenomena (b) Separated effect tests (SETs): including elemental phenomena related to the target issue Step-) Validation: Application of AVM (MAVM) method (c) Component effect tests (CETs): simulating subcomponent system including significant phenomena in the target issue (d) Integrated effect tests (IETs): simulating component system or subsystem of the target plant

7 Uncertainty Quantification Method by GCI Estimation ~ Establishment of Reference Method in VUP ~ <Integrated program for GCI estimation> INPUT: Data-sets along the traverse line of the reference results and more than 3 numerical results at different mesh arrangement. () Pretreatment: Setting up the numerical results for the estimation, Ref.) using spline interpolation of the numerical results to set at the position of the reference result, or Opt.) choosing the numerical result at the closest position to the reference result. () GCI estimation: - - ASME V&V 0 method (Roache s method) - Modified ASME V&V 0 method (based on the Roache s method) - Global average method - Roy s modification method T. S. Phillips and C. J. Roy, 49th AIAA Aerospace Sciences, AIAA Eca s least square version GCI - SLS-GCI (Simplified Least Square version GCI) Least square version GCI modified from Eca s GCI method OUTPUT: Uncertainty in the domain and detailed information at each position 7

8 Uncertainty Quantification (GCI estimation) ~ Modified ASME V&V 0 method ~ 8 (More than) three calculations with different mesh number (N k : N >N >N 3 ) (Better to keep a constant mesh increasing rate) r = r 3 = N/ N3 N/ N, Prediction of convergence rate p. ~ p ln[ ( ) ( ) ( ~ )] ( )) ( ε3) = ( f3) ( f) 3 q p ln r = ε ε +,, ( ε) = ( f) ( f) q ( ) [ ~ ( ) ( )] p ~ p p = ln r s r s ~, = sign( ε ) ( ε ) p = max( p, p~ ) - Modification: (p l =.0) p ( f ) r ( f ) ( f ) c p ( u ) = F ( ε ) ( r ) G p [ ] ( r ) = f S f f f c f= p = F f r S c 3Prediction of extrapolation solution (f c ) 4GCI estimation (safety coefficient: F S ) 3 l [ ] s 3 (ASME: F S =.5) Fs=.5 in Fs=3.0 in p l : Minimum order of accuracy p f : Maximum order of accuracy p l < p p f p p l or p f < p ( p f =.0)

9 Uncertainty Quantification (GCI estimation) ~ SLS-GCI (Simplified Least Square version GCI) ~ Proposal of Least Square (LS) method modified from Eca s LS method Without udgment of convergence condition (as the first step of Eca s method) - Dispersed results around the theoretical curve - Limited data to udge the convergence condition Solve equation on (α, p, f c ) by the LS method ~ f = f + α h Estimations of the GCI using estimation values f and the error in the curve ) k c p l ( = ) < p ( ) p k ~ Fs f u =, u GCI p s= fk fc+ α( hk) ( h h) n k= ~ f ~ n p [ { }] F s =.5 F s =3.0 p ( ) < p p ( = l = f p > p f (= ) ) 9 ) 0 p p ( = ), = π l = < l ~ 3.0 f π u GCI = ( h h), u s= [ f k { f c+ αhk} ] n k= ~ f n 3) p 0 or p is not converged. u = 0 GCI, (GCI can not be defined) u s = n n ( f k f c) k=, f = n c f k n k=

10 Estimation of Combined Standard Uncertainty in Domain Combined standard uncertainty (C e =) in the domain by root sum square (RSS) method. 0 UG U S = Ce + ε c + U.5 e U GCI as 95% uncertainty (σ) is modified to σ value with expansion factor of.5 - Mean GCI (U G ) by root mean square (RMS) U ε U c = = = m [ ( ) ] G u G m = m [ ( f ) ( ) ] c f 0 m = m ( ) e u e m =, ( f0) ( f) max (m:number of reference positions) - Mean difference (ε c ) between the reference (f 0 ) and the extrapolation (f c ) values - Mean error (U S ) in the LS δ max = u e = n [ n { + ( ) }] p fk fc α hk k= - The maximum difference (δ max ) between the reference (f 0 ) and the result of f

11 Application of GCI Estimation Methods in Verification and Fundamental Validation Processes in VUP Uncertainty quantification by GCI method is performed in verification and fundamental validation processes. For Verification Couette-Poiseuille Flow: - Theoretical results of N-S equation without uncertainty - Verification of coupling functions of velocity and pressure fields at adverse /even /positive pressure gradient condition For Fundamental validation with Fundamental Problem and SET Back-Facing Step Flow as a Fundamental Problem: - Denman s Experimental result at Re=9 - Influence of set of mesh density conditions on the uncertainty WATLON experiment as a SET: - Water experiment in T-unction piping system - Typical problem for thermal mixing in turbulence conditions

12 Uncertainty Quantification in Verification ~ Couette-poiseuille Flow Problem ~ <Couette-Poiseuille Flow in Prallel Cannel at Re=0> (U 0 =0.00 m/s, H= 0.0 m, Water at 0 o C) y H dp y u( y) = U0 + H µ dx H Case: U m =0 Case: U m =0.5U 0 Case3: U m =U 0 (-dp/dx)<0 (-dp/dx)=0 (-dp/dx)>0 y H <Mesh arrangements> - Ny (Vertical direction) for H: 0, 0, 40 (Ref.), 80 - Nx (Depth direction) for H: 49 (Periodic condition) - Nz (Axial direction) for 5H: 0(Periodic condition) Ny=0 Ny=0 Ny=40 Ny=80

13 Uncertainty Quantification in Verification ~ Couette-poiseuille Flow Problem ~ 3 Data U G ε c U e U S Case- (U m =0, (-dp/dx)<0) ASME Modified ASME GA Roy's GCI SLS-GCI Eca's LS GCI Case- (U m =0.5U 0, (-dp/dx)=0) ASME 0 N/A N/A - N/A Modified ASME GA Roy's GCI SLS-GCI Eca's LS GCI 0 N/A N/A - N/A Case-3 (U m =U 0, (-dp/dx)>0) ASME Modified ASME GA Roy's GCI SLS-GCI Eca's LS GCI

14 Uncertainty Quantification n Fundamenal Validatin ~ Back-Facing Step Flow Problems ~ 4 - Denham s Experimental result at Re=9 - Height of downstream duct: 3H (H: Step height) - Uncertainty of inlet boundary condition potentially exists in the simulation. Axial velocity profile measured at (3/4)H upstream from the step edge was set at 4H upstream in the simulation. <Mesh arrangements> - Vertical direction: (parameter for GCI) Ny (for H) = 7, 36, 48 (r=4/3) - Depth direction: (Periodic condition) - Axial direction: Nx (for 4H) = 3 Nz (for 6H (4H for upstream)) = 35 0H (Step edge) 4H

15 Uncertainty Quantification n Fundamenal Validatin ~ Back-Facing Step Flow Problems ~ 5 0H (Step edge) U G ε c H U G ε c U S U S 4H U G ε c 8H U G ε c U S U S

16 Uncertainty Quantification in Fundamental Validation ~ Outline of T-pipe Water Experiment (WATLON)~ 6 (Wall Jet) (Impinging Jet) Wall et case Main pipe Branch pipe Inner Diameter: 0.5 m(=d m ) 0.05 m(=d b ) Inlet Temperature: 48 (=T m ) 33 (=T b ) Mean Velocity:.46 m/s(=w m ).0m/s(=V b ) Reynolds number: 3.8x x0 4 Measured data: - Velocity components by PIV - Fluid temperature by T/Cs Mm (kg.m/s) Mr = Mm/Mb 0.35<Mr<.35 ( :Deflecting et) Mr < 0.35 ( :Impinging et) Flow-pattern map M b (kg.m/s ) Mr >.35 ( :Wall et) Movable thermo-couples (T/Cs) tree D m Jet direction 噴流の向き m M m m Support T/Cs tree Mm b b M m b b W D b Mm m D b Main pipe flow: M m =ρ m Branch pipe flow: M = ρ πd b b U ( D D ) W m b ( ) V b V b W m 4 b m

17 Uncertainty Quantification in Fundamental Validation ~ Mesh Arrangements for WATLON Simulation ~ 7 Case-M: Reference mesh Case-C: Coarse mesh For GCI estimation Case-F: Fine mesh Radial length of boundary cell on the wall: 0.5mm δ Case N z δ c =D m /N x, total (z<+.5d m ) D m /N y h k for GCI [(δ z )x(δ c ) ] /3 F 60,000.5 mm.5 mm h =.5 mm M 6, mm 3.75 mm h = 3.57 mm C 9, mm 5.77 mm h 3 = 5.8 mm * 0 seconds transient calculation (0,000 data) for estimation ) r =.43 ) r 3 =.45

18 Uncertainty Quantification in Fundamental Validation ~ Typical Numerical Results at 0.5Dm Downstream ~ Magnitude of error bar equals to that of uncertainty of Us in the next slide. (on the symmetric cross section) (near main pipe surface (mm inside)) 8 Time average Fluctuation intensity Axial velocity Fluid temperature

19 Uncertainty Quantification in Fundamental Validation ~ For Numerical Results at 0.5Dm Downstream ~ - Mod. ASME V&V 0 still over-estimates the uncertainty. - SLS-GCI can stably use for the uncertainty quantification. 9 Data (0.5D m ) ε max Mod.-ASME SLS-GCI U G ε c U S U G U e ε c U S T T' W W' T ( mm) T (3 mm) T ( mm) T (3 mm) (Uncertainty quantification using the results applying the spline interpolation)

20 Conclusions 0 Introduction of a procedure called VUP (V&V plus Uncertainty quantification and Prediction) for a numerical estimation method for thermal mixing phenomena in a sodium-cooled fast reactors. Specification of the reference GCI estimation method in the VUP that is used in the verification and the fundamental validation processes Proposal of two modified methods, - Modified method from Roache s GCI based on the ASME V&V-0 guideline - SLS-GCI (Simplified Least Square version GCI estimation method) Examinations of the GCI estimation methods through the numerical simulations: - in Verification; Theoretical results of Navier-Stokes equation of Couette-Poiseuille Flows as a common benchmark problem - in Fundamental Validation (Fundamental Problem and Separated Effect Test): FP: Laminar flow problem over a backward-facing step SET: Water experiment of a T-pipe in JAEA called WATLON Through the examinations, the SLS-GCI is defined as a reference method for GCI estimation method in VUP