SOLVING TWO-PHASE FLOW TRANSPORT EQUATIONS USING THE LAX-WENDROFF - PowerPoint PPT Presentation

SOLVING TWO-PHASE FLOW TRANSPORT EQUATIONS USING THE LAX-WENDROFF SCHEME Dean Wang, UMass Lowell John Mahaffy and Joseph Staudenmeier, NRC June 10, 2015 ANS Annual Meeting, San Antonio, TX

Outline 2 ¨ Introduction and Background ¤ First-order numerical methods for reactor T-H ¤ High-resolution numerical methods for reactor T-H (Wang 2012 and Wang et al 2013) ¤ First-order in time and second-order in space ¨ Second-Order Lax-Wendroff Scheme (Lax and Wendroff 1962) ¨ Concluding Remarks

Numerical Diffusion 3 ¨ Most reactor system analysis codes use the 1 st -order upwind scheme such as TRACE, RELAP , COBRA, etc. ¨ While very robust, 1 st -order upwinding leads to excessive numerical diffusion (damping). ¨ 2 nd -order methods can effectively reduce numerical diffusion, but often produce spurious oscillations and slow down convergence ¨ It has to be carefully treated for reactor analysis, particularly for BWR stability analysis and boron tracking

Two-Phase Flow Transport Equations 4 Mass: Momentum: Energy:

Finite-Volume Method on Staggered Grid 5

Nonlinear Flux Limiters 6 Limiter Functions 2.5 MUSCL1.5 2 nd -order OSPRE TVD Region 2 Van Albada 𝜒(𝑠)=𝑠 ENO 1.5 Φ( r) 1 𝜒(𝑠)=1 TVD Region 0.5 0 0 0.5 1 1.5 2 2.5 3 r where Theorem: Any TVD numerical method is monotonicity preserving

Desired Properties of Flux Limiters 7 ¨ Φ(1) = 1. This is a necessary requirement for 2 nd - order accuracy on smooth solutions ¨ Φ (r)/r = Φ (1/r). This symmetric property ensures that a flux-limiter has the same actions on forward and backward gradients ¨ Φ (r) is located in the 2 nd -order TVD region 7

Linear Advection: 8

Linear Advection: 9

Implementation of L-W in TRACE 10 ¨ Implemented in the mass and energy conservation equations. ¨ The momentum equation in TRACE is in non- conservative form, in which 2 nd -order central difference is used for the convection term. ¨ Time integration scheme is semi-implicit

BWR Single-CHAN Model 11 Upwind 8.5 Reference fine mesh dt = 0.0189s 8 dt = 0.01s Mass Flow Rate (kg/s) dt = 0.005s 7.5 7 6.5 6 5.5 45 50 55 60 65 70 Time (s) L-W with Flux Limiter VA C-D with Flux Limiter VA Reference fine mesh 8.5 Reference fine mesh 8.5 dt = 0.0189s dt = 0.0189s 8 8 dt = 0.01s Mass Flow Rate (kg/s) dt = 0.01s Mass Flow Rate (kg/s) dt = 0.005s dt = 0.005s 7.5 7.5 7 7 6.5 6.5 6 6 5.5 5.5 45 50 55 60 65 70 45 50 55 60 65 70 Time (s) Time (s)

Observations and Remarks 12 ¨ This study has shown the superior performance of the L-W scheme (with a flux limiter) as compared with the C-D scheme: 2 nd -order accuracy in BOTH time and space . ¨ L-W can speed up reactor simulation by using a relatively large time step up to the CFL limit. ¨ In addition, L-W is easy to implement and does not incur significant computational cost. ¨ L-W can effectively improve the numerical solution of two- phase flow transport equations and passive scalar transport in the fluid. ¨ While the tests in this paper are all 1D, it can apply for 2D and 3D problems.

Why 2 nd -Order High-Resolution Methods Are Important for Reactor T-H? 13 ¨ Provide a solution to the important and long-lasting numerical diffusion issue in the 1d system T-H analysis codes ¨ They have a promising application in BWR stability analysis. ¤ In practice we used to develop a non-uniform nodalization to mitigate numerical diffusion. ¤ With these HRMs, we can achieve high numerical accuracy on a uniform coarse nodalization. ¨ Boron Tracking and Core Void Prediction ¨ These high-resolution methods are now officially released with TRACE V5.0p4

Current and Future Work 14 ¨ Implementation of high-resolution schemes in COBRA-TF (CASL). ¨ Locally adaptive time stepping schemes for two- phase flow simulation. ¨ 2 nd -order implicit methods for two-phase flow. ¨ Development of new acceleration schemes for neutron transport calculations (NEUP).

References 15 D. Wang, “Reduce Numerical Diffusion in TRACE Using the High-Resolution Numerical ¨ Method ENO,” Trans. AM. Nucl. Soc.107, 2012 D. Wang, et al., “Implementation and Assessment of High-Resolution Numerical Methods in ¨ TRACE,” Nuclear Engineering and Design 263 (2013) 327-341. P.D. Lax and B. Wendroff, “Systems of Conservation Laws,” Pure Appl. Math., 13: 217- ¨ 237, 1960. TRACE V5.840 Theory Manual, U.S. Nuclear Regulatory Commission, 2013. ¨ R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems , Cambridge University Press, ¨ 2002. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics , 3 rd edition, Springer- ¨ Verlag, 2009.

16 Thank You!