Maximum power from a turbine farm in shallow water

AbstractThe maximum power that can be obtained from a confined array of turbines in steady or tidal flows is considered using the two-dimensional shallow-water equations and representing the turbine farm by a uniform local increase in friction within a circle. Analytical results supported by dimensional reasoning and numerical solutions show that the maximum power depends on the dominant term in the momentum equation for flows perturbed on the scale of the farm. If friction dominates in the basic flow, the maximum power is a fraction (half for linear friction and 0.75 for quadratic friction) of the dissipation within the circle in the undisturbed state; if the advective terms dominate, the maximum power is a fraction of the undisturbed kinetic energy flux into the front of the turbine farm; if the acceleration dominates, the maximum power is similar to that for the linear frictional case, but with the friction coefficient replaced by twice the tidal frequency.

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Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

Symmetry ◽

10.3390/sym12030345 ◽

2020 ◽

Vol 12 (3) ◽

pp. 345

Author(s):

Sudi Mungkasi ◽

Stephen Gwyn Roberts

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Numerical Solutions ◽

Adaptive Mesh ◽

Finite Volume Methods ◽

Water Type ◽

Two Dimensional ◽

Water Flows ◽

Shallow Water Flows ◽

Smoothness Indicators

This paper proposes some formulations of weak local residuals of shallow-water-type equations, namely, one-, one-and-a-half-, and two-dimensional shallow water equations. Smooth parts of numerical solutions have small absolute values of weak local residuals. Rougher parts of numerical solutions have larger absolute values of weak local residuals. This behaviour enables the weak local residuals to detect parts of numerical solutions which are smooth and rough (non-smooth). Weak local residuals that we formulate are implemented successfully as refinement or coarsening indicators for adaptive mesh finite volume methods used to solve shallow water equations.

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SEMI-IMPLICIT NUMERICAL SCHEMA IN SHALLOW WATER EQUATION

Jurnal Natural ◽

10.24815/jn.v17i2.7998 ◽

2017 ◽

Vol 17 (2) ◽

pp. 102

Author(s):

Safwandi Safwandi ◽

Syamsul Rizal ◽

Tarmizi Tarmizi

Keyword(s):

New York ◽

Shallow Water ◽

Shallow Water Equations ◽

Numerical Solutions ◽

Numerical Models ◽

Shallow Water Equation ◽

Two Dimensional ◽

Numerical Schemes ◽

Spectral Difference ◽

Spectral Difference Method

Abstract. A two-dimensional shallow water equation integrated on depth water based on finite differential methods. Numerical solutions with different methods consist of explicit, implicit and semi-implicit schemes. Different methods of shallow water equations expressed in numerical schemes. For bottom-friction is described in semi-implicitly. This scheme will be more flexible for initial values and boundary conditions when compared to the explicit schemes. Keywords: 2D numerical models, shallow water equations, explicit and semi-implicit schema.Reference Hassan, H. S., Ramadan, K. T., Hanna, S. N. 2010. Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method, Scie.Res,App.Math. (1):104-117. Omer, S, Kursat, K. 2011. High-Order Accurate Spectral Difference Method For Shallow Water Equations. IJRRAS6. Vol. 6. No. 1. Kampf, J. 2009. Ocean Modelling for Beginners. Springer Heidelberg Dordrecht. London, New York. Wang, Z. L., Geng, Y. F. 2013. Two-Dimensional Shallow Water Equations with Porosity and Their Numerical scheme on Unstructured Grids. J. Water Science and Engineering. Vol. 6, No. 1, 91-105. Saiduzzaman, Sobuj. 2013. Comparison of Numerical Schemes for Shallow Water Equation. Global J. of Sci. Fron. Res. Math. and Dec. Sci. Vol. 13 (4). Sari, C. I., Surbakti, H., Fauziyah., Pola Sebaran Salinatas dengan Model Numerik Dua Dimensi di Muara Sungai Musi. Maspari J. Vol. 5 (2): 104-110. Bunya, B., Westerink, J. J. dan Shinobu, Y. 2004. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids 2005 (47): 1451–1468.

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Two-Dimensional Shallow-Water Model with Porosity for Urban Flood Modeling

Proceedings ◽

10.3390/proceedings2201307 ◽

2018 ◽

Vol 2 (20) ◽

pp. 1307

Author(s):

Malika Benslimane ◽

Saâdia Benmamar ◽

André Paquier

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Urban Areas ◽

Large Scale ◽

Momentum Equation ◽

Water Model ◽

Finite Volume Scheme ◽

Two Dimensional ◽

Flood Modeling ◽

Urban Flood

In the world, floods are at the forefront of natural hazard. Urban areas are often at risk of flooding and just as often unprepared for management. Flood modeling is nowadays a very important topic in the theme of water, it inevitably involves the numerical resolution of the shallow water equations derived from the Navier Stocks equations governing flows. Two-dimensional shallow water models with porosity appear as an interesting path for the large-scale modeling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two-dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. The developed solution method consists in solving the two-dimensional shallow water equations with porosity via a finite volume scheme solving the conservative form of the equations which can be reduced to a calculation of flux through an edge, a problem that can be approached by a one-dimensional problem in the normal direction at the edge (Riemann problem).

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MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

Proceedings of International Conference "Managinag risks to coastal regions and communities in a changinag world" (EMECS'11 - SeaCoasts XXVI) ◽

10.31519/conferencearticle_5b1b94407213d3.12732916 ◽

2017 ◽

Author(s):

Xiao-Hua Zhu ◽

Ze-Nan Zhu ◽

Xinyu Guo ◽

...

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Mean Amplitude ◽

Momentum Equation ◽

Tidal Currents ◽

Acoustic Tomography ◽

Mean Values ◽

Residual Currents ◽

Water Depths ◽

Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

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Modelling Weirs in Two-Dimensional Shallow Water Models

Water ◽

10.3390/w13162152 ◽

2021 ◽

Vol 13 (16) ◽

pp. 2152

Author(s):

Gonzalo García-Alén ◽

Olalla García-Fonte ◽

Luis Cea ◽

Luís Pena ◽

Jerónimo Puertas

Keyword(s):

Experimental Data ◽

Shallow Water ◽

Shallow Water Equations ◽

Water Model ◽

Two Dimensional ◽

Shallow Water Model ◽

Experimental Dataset ◽

River Hydraulics ◽

Shallow Water Models ◽

Water Models

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.

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