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

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¨ 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

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¨ Most reactor system analysis

codes use the 1st-order upwind scheme such as TRACE, RELAP , COBRA, etc.

¨ While very robust, 1st-order

upwinding leads to excessive numerical diffusion (damping).

¨ 2nd-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

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Mass: Momentum: Energy:

Finite-Volume Method on Staggered Grid

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Nonlinear Flux Limiters

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Theorem: Any TVD numerical method is monotonicity preserving

2nd-order TVD Region TVD Region

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 Φ(r) r

Limiter Functions

MUSCL1.5 OSPRE Van Albada ENO 𝜒(𝑠)=𝑠 𝜒(𝑠)=1

where

Desired Properties of Flux Limiters

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¨ Φ(1) = 1. This is a necessary requirement for 2nd-

• rder 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 2nd-order TVD region

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Linear Advection:

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Linear Advection:

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Implementation of L-W in TRACE

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¨ Implemented in the mass and energy conservation

equations.

¨ The momentum equation in TRACE is in non-

conservative form, in which 2nd-order central difference is used for the convection term.

¨ Time integration scheme is semi-implicit

BWR Single-CHAN Model

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5.5 6 6.5 7 7.5 8 8.5 45 50 55 60 65 70 Mass Flow Rate (kg/s) Time (s)

Upwind

Reference fine mesh dt = 0.0189s dt = 0.01s dt = 0.005s

5.5 6 6.5 7 7.5 8 8.5 45 50 55 60 65 70 Mass Flow Rate (kg/s) Time (s)

C-D with Flux Limiter VA

Reference fine mesh dt = 0.0189s dt = 0.01s dt = 0.005s

5.5 6 6.5 7 7.5 8 8.5 45 50 55 60 65 70 Mass Flow Rate (kg/s) Time (s)

L-W with Flux Limiter VA

Reference fine mesh dt = 0.0189s dt = 0.01s dt = 0.005s

Observations and Remarks

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¨ This study has shown the superior performance of the L-W

scheme (with a flux limiter) as compared with the C-D scheme: 2nd-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 2nd-Order High-Resolution Methods Are Important for Reactor T-H?

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¨ 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

• n 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

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¨ Implementation of high-resolution schemes in

COBRA-TF (CASL).

¨ Locally adaptive time stepping schemes for two-

phase flow simulation.

¨ 2nd-order implicit methods for two-phase flow. ¨ Development of new acceleration schemes for

neutron transport calculations (NEUP).

References

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¨

• 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, 3rd edition, Springer- Verlag, 2009.

Thank You!

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