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

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

This research investigates the possible benefit of considering the non-hydrostaticity of the pressure distribution in fluid flows. We propose the Boussinesq equations deduced in this work in Cartesian coordinate system for two spatial dimensions, applied to classical flows. For the numerical solution, the Galerkin Discontinuous and Runge-Kutta methods were used, respectively for the discretization in space and time. A review of the necessary mathematical and computational themes is presented. The Boussinesq equations were applied to twodimensional rupture flows of a cylindrical dam with two downstream flow conditions and to a supercritical flow in a contraction, where shock waves occur. The non-hydrostatic correction terms were calculated from spatial and temporal derivatives of the velocity components. The Fourier Series was adopted to allow the determination of a spatial surface interpolating the velocity values, from which the derivatives were determined. The results obtained with the consideration of non-hydrostatic pressure distribution Boussinesq equations present similar results to those obtained with the adoption of Shallow Water Equations, but with a greater computational effort.

Estudo da equação de Boussinesq em duas dimensões horizontais

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

This work proposes a fiber Bragg gratings (FBG) sensor for simultaneous measurements of water level, velocity, and temperature in open channels. The optical fibers are placed on a rectangular aluminum rod attached to the bottom of the channel. The water flow causes a distributed load and a characteristic bend in the rod, depending on the water level and velocity. A set of four FBGs are used to monitor the rod’s bending profile. Taking the wavelength difference of two FBGs makes it possible to eliminate the temperature effect and obtain a system of nonlinear equations dependent on velocity and water level. Velocities from 0.1 ms to 0.9 m/s and water levels from 0.4 m to 0.7 m were experimentally evaluated. A semi-empirical model was obtained to describe the wavelength shift, and finally the water level and velocity are calculated. The results show that the proposed system can determine the water level, mean velocity and temperature with satisfactory accuracy. Artificial neural networks are used as a second and more straightforward approach to determine the water level and velocity from the measured wavelengths, considering mean values and standard deviation. With this method, the maximum absolute deviations were 0.02 m/s and 3.1 cm, for the mean velocity and water level, respectively.

Self-Temperature Compensated Water Level and Velocity Sensor Based on Fiber Bragg Gratings

Ao se conceber uma obra de arte especial (OAE), muitos aspectos devem ser levados em consideracao em seu projeto, construcao e manutencao O pior cenario para essa estrutura e seu colapso, pois assim ela pode isolar pessoas e areas, aumentar custos e tempo de viagens, atrasar a entrega de bens e servicos Uma das maiores causas das quedas de pontes sao falhas nos pilares devido aos efeitos de erosao gerados pelo escoamento do rio Esse enfoque na manutencao dessas estruturas muito vezes nao e levado em consideracao, por isso e necessario incrementa-la em um modelo de avaliacao de obras de arte especiais que ja avalie a estrutura como um todo Esse artigo propoe a analise utilizando o modelo desenvolvido pelo Escritorio Modelo de Engenharia Civil (EMEA) da Universidade Federal do Parana (UFPR) Trabalho publicado no 2o Simposio Paranaense de Patologia das Construcoes

/pdf/modelo-matematico-de-ranqueamento-avaliacao-da-erosao-nos-4bcqzf244i.pdf

Modelo matemático de ranqueamento: avaliação da erosão nos pilares de pontes

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

André Luiz Tonso Fabiani

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

Self-Temperature Compensated Water Level and Velocity Sensor Based on Fiber Bragg Gratings

Modelo matemático de ranqueamento: avaliação da erosão nos pilares de pontes

Aplicação da Equação de Boussinesq em duas dimensões