A multi-grid finite-volume method for free-surface flows

In recent years significant advances have been made in computational fluid dynamics applied to free-surface flow calculations. The flow pattern of the aforementioned open channels either natural or technical is highly complex. There exist classes of free-surface flow problems, which can adequately be described in the context of depth-averaged 2D mathematical models. These simplified representations of 3D flows are justified where turbulent mixing, due to bottom roughness, effectively generates a uniform velocity distribution over the depth of the flow field. For free-surface flows in complex geometry it is convenient to make predictions using non-orthogonal boundary fitted computational meshes.

Numerous publications were reported for 2D free-surface flow simulation, among them Soulis J. [16] developed an explicit finite-volume numerical technique, with transformed grid, to simulate subcritical and supercritical depth-averaged freesurface flows. Molls T. et al. [8] derived a depth-averaged open channel flow model while Molls T. et al [9] applied an alternative direction implicit scheme and the MacCormack explicit scheme, both second order accurate, to numerically simulate 2D flows near spur-dikes. Yulistiyanto Β. et al [19] solved the continuity and momentum equations for 2D horizontal flow by vertical depth-integration and the differential equations were also evaluated using the MacCormack scheme. Lien H. et al [6] presented a 2D depth-averaged model for simulating flow pattern in channel bends using an orthogonal curvilinear coordinate system to efficiently simulate the flow field with irregular boundaries. The two-step splitoperator approach consisting of the dispersion step and the propagation step with the staggered grid was used to numerically solve the flow governing equations. Molls Τ. et al [10] numerically simulated supercritical flow in a channel with a wavy sidewall by solving the 2D depth-averaged equations using two different second-order accurate finite-difference schemes, an implicit model that uses an alternating direction implicit technique to solve the governing equations and an explicit model employing the MacCormack two-step predictor-corrector scheme. Liu M. et al [7] presented an unsteady 2D depth-averaged flow model to simulate the bend-flow field by transforming the governing system of differential equations into an equivalent system applied over a square-grid network in order to overcome the difficulties and inaccuracies associated with the determination of characteristics near the flow boundaries. The MacCormack two-step explicit scheme with second-order accuracy was used for the solution of the transformed system of equations. Papanicolaou Α. et al [11] performed a sensitivity analysis to examine the predictive capability of a 2D hydrodynamic model, a finite-element surface water modeling system to adequately describe the flow characteristics around emergent bendway weir structures. Chen Υ. et al [1] presented the water stage prediction-correction method, based on the theory of characteristics to couple numerical models in the boundaryconnected way for shallow-water flows. An 1D–2D coupled numerical model was established, which incorporates the artificial porosity method capable of treating wetting and drying.