The Stability of Linear Diffusion Acceleration Relative to CMFD

Coarse Mesh Finite Difference (CMFD) is a widely-used iterative acceleration method for neutron transport problems in which nonlinear terms are introduced in the derivation of the low-order CMFD diffusion equation. These terms, including the homogenized diffusion coefficient, the current coupling coefficients, and the multiplicative prolongation constant, are subject to numerical instability when a scalar flux estimate becomes sufficiently small or negative. In this paper, we use a suite of contrived problems to demonstrate the susceptibility of CMFD to failure for each of the vulnerable quantities of interest. Our results show that if a scalar flux estimate becomes negative in any portion of phase space, for any iterate, numerical instability can occur. Specifically, the number of outer iterations required for convergence of the CMFD-accelerated transport problem can increase dramatically, or worse, the iteration scheme can diverge. An alternative Linear Diffusion Acceleration (LDA) scheme addresses these issues by explicitly avoiding local nonlinearities. Our numerical results show that the rapid convergence of LDA is unaffected by the very small or negative scalar flux estimates that can adversely affect the performance of CMFD. Therefore, our results demonstrate that LDA is a robust alternative to CMFD for certain sensitive problems in which CMFD can exhibit reduced effectiveness or failure.

Reduced-Order Modelling Applied to the Multigroup Neutron Diffusion Equation Using a Nonlinear Interpolation Method for Control-Rod Movement

Producing high-fidelity real-time simulations of neutron diffusion in a reactor is computationally extremely challenging, due, in part, to multiscale behaviour in energy and space. In many scientific fields, including nuclear modelling, the application of reduced-order modelling can lead to much faster computation times without much loss of accuracy, paving the way for real-time simulation as well as multi-query problems such as uncertainty quantification and data assimilation. This paper compares two reduced-order models that are applied to model the movement of control rods in a fuel assembly for a given temperature profile. The first is a standard approach using proper orthogonal decomposition (POD) to generate global basis functions, and the second, a new method, uses POD but produces global basis functions that are local in the parameter space (associated with the control-rod height). To approximate the eigenvalue problem in reduced space, a novel, nonlinear interpolation is proposed for modelling dependence on the control-rod height. This is seen to improve the accuracy in the predictions of both methods for unseen parameter values by two orders of magnitude for keff and by one order of magnitude for the scalar flux.

Turbulence and Coral Reefs

The interaction of coral reefs, both chemically and physically, with the surrounding seawater is governed, at the smallest scales, by turbulence. Here, we review recent progress in understanding turbulence in the unique setting of coral reefs—how it influences flow and the exchange of mass and momentum both above and within the complex geometry of coral reef canopies. Flow above reefs diverges from canonical rough boundary layers due to their large and highly heterogeneous roughness and the influence of surface waves. Within coral canopies, turbulence is dominated by large coherent structures that transport momentum both into and away from the canopy, but it is also generated at smaller scales as flow is forced to move around branches or blades, creating wakes. Future work interpreting reef-related observations or numerical models should carefully consider the influence that spatial variation has on momentum and scalar flux.