One- and two-dimensional standing pressure waves and one-dimensional travelling pulses using the US-NRC nuclear systems analysis code TRACE

Abstract

A variety of nuclear system transients can lead to rapid and large local pressure changes that propagate along the hydraulic system at the speed of sound, both in single phase and in two-phase fluids. Because of the relevance for safety issues, nuclear system codes like TRACE need to be assessed with respect to their capabilities to predict pressure wave behaviour. Therefore, we have analyzed the propagation of pressure waves in one-dimensional and two-dimensional configurations, i.e. a pipe and a slab, filled with liquid water. The pressure waves are driven by one-sided pressure boundary conditions, in the one-dimensional case of harmonic or Gaussian shape and in the two-dimensional case also of harmonic shape. The selected harmonic pressure boundary conditions lead to standing pressure waves, while using the Gaussian shape boundary conditions one-dimensional pressure pulses are injected and propagate through the pipe. The agreement of the TRACE results with the analytical solutions are, in general very good to good for the one-dimensional cases with respect to the pressure maxima and a small difference is only obtained in the wave speed. At the resonance frequencies of the one-dimensional standing waves, the code is tested to the extreme and shows that enforcing small time step sizes is crucial for the performance of the code. Non-linear effects are observed in the code results for the large amplitudes encountered at the closest neighborhood of the resonances, where the analytical linear standing wave solution diverges and the linear approximation is outside of its validity range. Also for these non-linear standing waves TRACE yields qualitatively physically correct behaviour as the pressure amplitudes are limited and a plateau is reached. For the one-dimensional pressure pulse of Gaussian shape the change of pulse amplitude and shape was analyzed in a longer system. The maximum amplitude of the pulse is slightly reduced as the pulse travels along the pipe. The effect of numerical diffusion on leading and trailing fronts is slightly asymmetric due to the donor-cell approach used in the numerical integration scheme of TRACE. The accuracy of the code is not negatively influenced by the reflections of the pulse at the boundaries of the pipe. As for the standing waves, the accuracy of the travelling pulse solution calculated by TRACE is negatively affected when the time steps are too large, while the effects of the spatial discretization are rather minor. For the case of two-dimensional standing waves in a slab, a lowest spatial harmonic generated with one wave node in the direction parallel to the pressure driving boundary is considered. TRACE results show an overall good agreement with the linear analytical solution. This good agreement includes for low to medium excitation frequencies the damping properties of the skin effect perpendicular to the pressure boundary, which does not exist in one-dimensional pressure wave propagation, the transition to a harmonic shape of the wave also perpendicular to the pressure boundary and the frequency dependence of the resonance spectrum for further increased frequencies with the rapid changes of the wavelengths encountered. Effects of the model set-up and code limitations with respect to the two-dimensional TRACE model set-up using a TRACE VESSEL component in connection with pressure boundary conditions are discussed, in particular with respect to the underestimation of the damping in the skin effect frequency range and the numerical damping for higher frequencies. All in all, the TRACE code is able to calculate one- and two-dimensional pressure wave propagation in liquid water, when an appropriate spatio-temporal numerical discretization is chosen.