A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model

Abstract

The gravity-driven spreading of one fluid in contact with another fluid is of key importance to a range of topics. These phenomena are commonly described by the two-layer shallow-water equations (SWE). When one layer is significantly deeper than the other, it is common to approximate the system with the much simpler one-layer SWE. It has been assumed that this approximation is invalid near shocks, and one has applied additional front conditions to correct the shock speed. In this paper, we prove mathematically that an effective one-layer model can be derived from the two-layer equations that correctly captures the behaviour of shocks and contact discontinuities without additional closure relations. The result shows that simplification to an effective one-layer model is justified mathematically and can be made without additional knowledge of the shock behaviour. The shock speed in the proposed model is consistent with empirical models and identical to front conditions that have been found theoretically by e.g. von Kármán and by Benjamin. This suggests that the breakdown of the SWE in the vicinity of shocks is less severe than previously thought. We further investigate the applicability of the SW framework to shocks by studying one-dimensional lock-exchange/-release. We derive expressions for the Froude number that are in good agreement with the widely employed expression by Benjamin. The equations are solved numerically to illustrate how quickly the proposed model converges to solutions of the full two-layer SWE. We also compare numerical results from the model with results from experiments, and find good agreement. arXiv:1902.04648v3 [physics.flu-dyn]

A combined fluid-dynamic and thermodynamic model to predict the onset of rapid phase transitions in LNG spills

“…This follows naturally from the present formulation of the SWE Eq. ( 1), as discussed in [34], At t = 30 s, the pool is a monotonically decreasing function of r until it vanishes at about r = 20 m due to evaporation. Once the spill rate is reduced to zero, the LNG thickness starts to reduce at the origin and outwards, as shown in Figure 7b.…”

“…Both ρ l and T bub are calculated from the EoS, see Section 2.5. As shown by Fyhn et al [34], the water-LNG and LNG-air interface positions are given by…”

“…Here we assumed that u LE = n • u LE , which from the Rankine-Hugoniot condition implies that u LE = 2g eff h LE , as shown in Ref. [34]. If we assume that u[r(t a )] and h[r(t a )] are well approximated by the steady state solutions and t b − t a is sufficiently small so that we may neglect the contribution from the integral, we can write the time from initial spill to a possible RPT as…”

“…surface increases. This in turn affects the spreading rate that depends on the leading-edge height [34]. Because of the coupling between spreading rate and vapor volume, the flow model must include the dynamics of the vapor mixed into the LNG.…”

“…[37,38]. In this work, we use a form of the singlelayer SWE derived and analysed by Fyhn et al [34]. This form of the SWE has the benefit of being simple while still capturing essential features of the two-layer equations for liquid-on-liquid spreading.…”