Numerical investigations of unsteady critical flow conditions over an obstacle using three models

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This paper focuses on the temporal and spatial evolutions of critical flow under unsteady rapidly varied flow conditions, which is defined based on different viewpoints under steady gradually varied conditions in the classical fluid mechanics theory. So far, the fundamental theory of critical flow is largely unknown for unsteady rapidly varied flow conditions. Taking a Gauss-type weir as an example, this work investigates comprehensively and comparatively the influences of unsteadiness, non-hydrostatic pressure, and non-uniform velocity on the critical flow at the weir crest using three models: (i) the unsteady vertically averaged hydrostatic Shallow Water Equations (SWE); (ii) the unsteady vertically averaged non-hydrostatic Serre–Green–Naghdi Equations (SGNE); and (iii) the unsteady vertical velocity-resolved non-hydrostatic Reynolds-averaged Navier–Stokes Equations (RANSE). The results demonstrate that, for all three models, the critical points defined by the minimum specific energy, minimum specific momentum, and Froude number equals to unity are generally different and do not occur at the weir crest, except for those of the SWE for steady flow conditions. We found that unsteadiness and non-hydrostatic pressure have significant effects on critical flow, whereas non-uniform velocity has a weak effect. Crest discharge hydrographs revealed that the SGNE and the RANSE are good modeling options, whereas the SWE are not reliable when the shock wave approaches the weir crest. The discharge calculated by the water depth of the weir crest in an unsteady process based on the critical depth–discharge relationship is more accurate than the discharge calculated by SWE.