The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Ski jump spillways are frequently implemented to dissipate energy from high-speed flows. The general feature of this structure is to transform the spillway flow into a free jet up to a location where the impact of the jet creates a plunge pool, representing an area for potential erosion phenomena. In the present investigation, several tests with different ski jump bucket angles are executed numerically by means of the OpenFOAM® digital library, taking advantage of the Reynolds-averaged Navier–Stokes equations (RANS) approach. The results are compared to those obtained experimentally by other authors as related to the jet length and shape, obtaining physical insights into the jet characteristics. Particular attention is given to the maximum pressure head at the tailwater. Simple equations are proposed to predict the maximum dynamic pressure head acting on the tailwater, as dependent upon the Froude number, and the maximum pressure head on the bucket. Results of this study provide useful suggestions for the design of ski jump spillways in dam construction.

https://www.mdpi.com/2311-5521/5/4/168/pdf

Flow Pressure Behavior Downstream of Ski Jumps

The traditional approach for the hydrodynamic characterization of the flow down stepped spillways is through physical modeling, which is susceptible to scale effects and has limitations related to experimental apparatus, laboratory space and the spatial discretization of data collection. Computational fluid dynamics (CFD) is an important tool for hydrodynamic analysis because, if used properly, it presents great potential for application in hydraulics. In this work, CFD was used to model the skimming flow down a stepped spillway to investigate the effects of possible pressure measurement errors due to uncertainties in the position of the sensors within the steps. The numerical model was validated through literature velocity profiles and pressure experimental data. The results showed that the best values of water fraction (α) to define free surface are α = 0.30 in the nonaerated region and α = 0.10 in the aerated region. Statistical parameters were calculated using experimental data to estimate extreme pressures. These parameters and the simulation results were used to determine that the extreme maximum and minimum pressures occur, respectively, in the region of 0.81 < x/l < 0.98, in the horizontal faces, and in the region of 0.93 < y/h < 0.98, in the vertical faces.

/pdf/combination-of-experimental-and-numerical-approaches-to-3oxxjxvwfq.pdf

Combination of experimental and numerical approaches to determine the main characteristics of skimming flow in stepped spillways

New formulation of the two-dimensional steep-slope shallow water equations. Part I: Theory and analysis

This study constructs an optimization model to address ski-jump energy dissipation problems in different reservoir environments. The multiobjective genetic algorithm is herein applied as the calculation method. In the process of changing the ski-jump flow model, the runoff of ski-jump flow is changed (especially the abnormal flip bucket). Water flow might go from having a single falling point to multiple falling points. Optimization is only performed after the stabilization of the fall of the water because the fall of the water from the starting point to the ground is not stable over a short period. Only a stable flow can reduce the computation time for optimization. The optimal overflow width and height of the flip bucket were calculated through optimal computation, which can minimize the scouring force of the water flow as it falls to the ground. The results obtained provide theoretical references for practical engineering and reduce the potential safety hazards. The energy in flip buckets following the optimization of the water flow can be fully dissipated, creating an ecosystem where water can flow unobstructed. Guiding a water flow to water-deficient areas is of great significance in ensuring the long-term protection of environments and ecosystems.

/pdf/research-on-the-optimization-design-of-abnormal-flip-buckets-3azkzfgur4.pdf

Research on the Optimization Design of Abnormal Flip Buckets

A basic hypothesis adopted for theoretical formulation of fluid flows is the hydrostatic pressure distribution. However, many researchers have pointed out that this simplification can lead to errors, in cases such as dam break flow. Discrepancy between computational solution and the experiment is attributed to the pressure distribution. These findings are not new, but it is not presented any formulation in the literature that considers the non-hydrostatic pressure distribution in 2D flow. This article deduces the Boussinesq Equations as an evolution of the Shallow Water Equations with the hypothesis of non-hydrostatic pressure distribution in the vertical direction. XYZ Orthogonal Cartesian System is used, considering the influence of channel bed slope and head losses of flow. It is presented the non-hydrostatical correction in the Boussinesq equation in two dimension using Fourier series. The solution uses Runge-Kutta Discontinuous Galerkin Method and the formulation is applied to a cylindrical dam-break.

/pdf/two-dimensional-boussinesq-equations-applied-to-channel-10qjzdhn2n.pdf

Two-dimensional Boussinesq equations applied to channel flows: deducing and applying the equations

Flaring Piers and Slit Buckets are structures that can improve the process of energy dissipation and have been implanted in Chinese spillways for at least four decades. These structures basically narrow the passages of the outflow through the spillway, promoting changes in the effluent jets, reducing their erosion potential. However, the literature available on the subject is very limited, and the benefits are disseminated only qualitatively. In order to learn more about how these structures perform, a test program on a hydraulic model was carried out, using a spillway originally designed with a conventional ski jump. The results indicate that there is a correlation between the angle of deflection of the walls that narrow the chute and the depth of the scour hole formed downstream. One of the alternative tests had scour depth 60% smaller than that generated by the spillway without the device, indicating that they can be promising solutions.

/pdf/reduction-of-the-erosive-effects-of-effluent-jets-from-2prhpovip6.pdf

Reduction of the erosive effects of effluent jets from spillways by contractions in the flow

Usually spillways are controls of their discharge, which features are well known. However, the number of low-drop spillways grows each day. In this kind of spillway, the crest loses a great deal of its discharge capability, as it happened on HPP Belo Monte´s spillway. It´s shape had to be altered and the final solution, a non-conventional one, was only found after long studies carried out in a physical model with the goal of achieving the required discharge capability. The difficulties encountered led to the question whether a mathematical model could help optimize the path to the solution, since there are only few information regarding the validation of such tools. Therefore, this article searched to evaluate the capability of a mathematical modeling tool, FLOW 3D®, in properly reproducing the characteristics of the complex flow through such spillways. To fulfill this goal, a case-study was done with HPP Belo Monte´s spillway on which gate controlled operation was also analyzed. Discharge capacity, flow line and pressure distribution from both models were compared. The results of the free-flow simulations indicated a good representation of the flow by the mathematical model. But bigger differences in discharge capability for the gate-controlled simulations were found, mainly regarding small openings. The comparisons of pressure distribution showed a very good similarity between the models. The results indicate that the mathematical model is able to satisfactorily reproduce both types of flows since appropriate measures are taken regarding mesh quality to ensure the convergence of the simulation.

/pdf/modelagem-computacional-tridimensional-de-um-vertedouro-de-1uuicgelg6.pdf

Modelagem Computacional Tridimensional de um Vertedouro de Baixa Queda

Este trabalho apresenta a discretizacao adotada para o termo de correcao da nao hidrostaticidade nas equacoes de Boussinesq em duas dimensoes, utilizando Series de Fourier. A solucao utiliza o Metodo de Runge-Kutta Galerkin Descontinuo. A formulacao e aplicada a ruptura de uma barragem cilindrica. Os resultados obtidos mostram que esta consideracao da nao hidrostaticidade nas Equacoes de Aguas Rasas conduz a resultados mais assimetricos que os obtidos com a consideracao das Equacoes de Aguas Rasas convencional, que leva em conta uma distribuicao hidrostatica de pressoes.

/pdf/aplicacao-da-equacao-de-boussinesq-em-duas-dimensoes-53nx51pp0i.pdf

José Junji Ota

Papers

Two-dimensional Boussinesq equations applied to channel flows: deducing and applying the equations

Reduction of the erosive effects of effluent jets from spillways by contractions in the flow

Modelagem Computacional Tridimensional de um Vertedouro de Baixa Queda

Aplicação da Equação de Boussinesq em duas dimensões