Two- and three-dimensional analysis of supercritical flow in bends

Document Type : Research Article


Civil Eng. Dept., School of Eng., Ferdowsi University of Mashhad Mashhad, Iran


Introduction: Channel bends are sometimes unavoidable due to project conditions or land topography. However, oblique cross waves are a distinct feature of supercritical flow in bends. These waves continue for a long distance downstream and increase the height of water considerably. Initially, the complex behavior of supercritical flow in bends was studied by hydraulic models in the laboratory. Later on, numerical models were found inexpensive tools to investigate flow patterns and explain features that may not even be possible to measure. In this article, the supercritical flow in a rectangular horizontal channel of 90º bend is studied with different ratios of radius to channel width (rc/b) using two- and three-dimensional numerical models. Water surface profiles are then compared with the data that were obtained from our experimental bend models. It is proved that three-dimensional models are more successful in predicting the flow profile, peak, and location of waves at the outer wall bend.
Methodology: In this study, Flow3D was used for the three-dimensional simulation of flow patterns. This software had a wide variety of applications and capabilities. The user could enter information to select different models to provide a range of flow phenomena. Flow3D integrated the Navier-Stokes equations (N-S) with finite volume method (FVM), with different mesh configurations, suitable for complex geometries. The k-ε turbulence model was used to close the N-S partial differential equations. The volume of fluid (VOF) method was used to model the free surface boundary. Additional boundary conditions for supercritical flow in bends included constant depth and velocity at the inflow section and no-slip or zero velocity conditions at the floor and solid walls. The Roe2D model was used for the simulation of two-dimensional shallow water equations. This model was able to capture discontinuities such as shock waves in supercritical flow. A triangular mesh was used for the space discretization, and a minmod slope limiter was implemented to control oscillations. Experiments were performed in the curved channel of the hydraulic laboratory of Ferdowsi University of Mashhad. This rectangular channel was horizontal, 40 cm in width, and the walls and floor were made of transparent plexiglass sheets. A straight channel, 1.8 m length, was installed before the bend to ensure flow development length. At the end of this channel was the 90º channel bend with internal and external radii of 40 and 80 cm, respectively. The channel width could be changed by adding interior walls; thereby, the ratio of rc/b might be changed accordingly.
Results and Discussion: Several experiments were run in the curved channels with widths of 15, 20, 30, and 40 cm and different radius of curvature to channel width (rc/b). The flow rate and water depth were measured, and thereby, the approach Froude number Fro was calculated. New experimental equations were obtained to calculate the maximum flow depth and location of the first wave’s crest along the outer wall in terms of the approach Froud number and the geometric specification of the bend. For each experiment, the corresponding two- and three-dimensional computer models were performed too. The three-dimensional model was well able to estimate the behavior of the supercritical flow, including the depth and position of wave crest at the outer wall of the bend. As Fro increased or rc/b decreased, the wave peak increased and moved downstream. However, the two-dimensional model had acceptable accuracy only for low values of Fr0 < 3. the assumption of hydrostatic pressure in depth-averaged 2D models was not applicable to supercritical bend flows. For flows with low Fro, the vertical acceleration might be ignored; however, as Fro increased, it became significant within the bend, and its negligence led to large errors in computations. In flows with high Fro, the maximum vertical acceleration occurred at the beginning of the bend (minimum depth point), and the minimum occurred at the wave crest. At high Fro, the vertical acceleration was downward, causing the hydrodynamic pressure to become less than the corresponding hydrostatic pressure.
Conclusions: The three-dimensional model of Flow3D is a suitable tool for the simulation of high-velocity supercritical flows in bends in comparison with the two-dimensional depth-averaged model of shallow water equation of Roe2D. By examining the pressure distribution and vertical acceleration in numerical models, it may be concluded that the basic assumption in the extraction of shallow water equations, namely the hydrostatic pressure distribution, is not admissible, especially at high Froude numbers. Moreover, the effect of vertical acceleration of water particles has a great effect on the estimation of wave crest depth and its position in the bend. .


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  • Receive Date: 22 September 2021
  • Revise Date: 28 November 2021
  • Accept Date: 01 December 2021
  • First Publish Date: 01 December 2021