- Research
- Open Access
Analytical solution to a hydrodynamic model in an open uniform reservoir
- Kaboon Thongtha^{1, 2}Email author and
- Jaipong Kasemsuwan^{1}
https://doi.org/10.1186/s13662-017-1205-3
© The Author(s) 2017
- Received: 1 February 2017
- Accepted: 12 May 2017
- Published: 25 May 2017
Abstract
The hydrodynamic model is used to determine the water wave flow. In this research, a nondimensional form of a two-dimensional hydrodynamic model with generalized boundary condition \(g(x,t)\) and initial conditions for describing the elevation of water wave in an open uniform reservoir is proposed. The separation of variables method with mathematical induction is employed to find an analytical solution to the model. An example of flow calculations in an open uniform reservoir is also demonstrated.
Keywords
- separation of variables method
- hydrodynamic model
- open uniform reservoir
- water elevation
MSC
- 35L05
- 35L04
- 35L40
1 Introduction
In [1] and [2], the finite element method was used to solve the water pollution models. In literature, several mathematical models need the data of water flow, while the velocity and elevation of water flow are provided by the hydrodynamic model. In [3], the finite difference method was used to solve the hydrodynamic model with constant coefficients in the closed uniform reservoir.
In [4], an analytical solution to the hydrodynamic model in a closed uniform reservoir was proposed. In [5], the Lax-Wendroff finite difference method was also proposed to approximate the water elevation and water flow velocity. However, the analytical solution to the hydrodynamic model in an open reservoir has not been considered.
2 A nondimensional form of a hydrodynamic model
2.1 Initial and boundary conditions for an open uniform reservoir
The initial conditions of Eqs. (10)-(12) are assumed to be \(u=0\), \(v=0\), \(d=f_{1}(x,y)\) and \(\frac{\partial d}{\partial t}=f_{2}(x,y)\). The boundary conditions of the model in an open uniform reservoir are assumed to be \(v=0\), \(\frac{\partial u}{\partial x}=0\) at the planes \(x=0\) and \(x=1\). \(u=0\), \(\frac{\partial v}{\partial y}=0\) at the planes \(y=0\). \(u=0\), \(\frac{\partial v}{\partial y}=0\), \(d(x,1,t)=g(x,t)\) and \(\frac{\partial{d}}{\partial{y}}=0\) at the planes \(y=1\). \(d=0\) on \(\partial\Omega\setminus\{(x,y)\in\partial\Omega:y=1\}\).
3 An analytical solution
4 Application to open uniform reservoir
Water elevation \(\pmb{\zeta(x,y,t)}\) (m) at \(\pmb{t=2}\) hrs 50 min
y ∖ x | 320 | 640 | 960 | 1,280 | 1,600 | 1,920 | 2,240 | 2,560 | 2,880 |
---|---|---|---|---|---|---|---|---|---|
320 | 0.1729 | 0.2620 | 0.3174 | 0.3352 | 0.3177 | 0.2716 | 0.2055 | 0.1272 | 0.0430 |
640 | 0.2835 | 0.4295 | 0.5203 | 0.5495 | 0.5209 | 0.4453 | 0.3368 | 0.2085 | 0.0704 |
960 | 0.3871 | 0.5864 | 0.7104 | 0.7503 | 0.7112 | 0.6080 | 0.4599 | 0.2847 | 0.0962 |
1,280 | 0.4811 | 0.7289 | 0.8830 | 0.9326 | 0.8840 | 0.7557 | 0.5716 | 0.3538 | 0.1195 |
1,600 | 0.5633 | 0.8534 | 1.0338 | 1.0919 | 1.0350 | 0.8848 | 0.6692 | 0.4143 | 0.1400 |
1,920 | 0.6316 | 0.9570 | 1.1592 | 1.2243 | 1.1605 | 0.9922 | 0.7504 | 0.4646 | 0.1570 |
2,240 | 0.6844 | 1.0369 | 1.2561 | 1.3266 | 1.2575 | 1.0751 | 0.8131 | 0.5034 | 0.1701 |
2,560 | 0.7203 | 1.0913 | 1.3220 | 1.3963 | 1.3235 | 1.1315 | 0.8558 | 0.5298 | 0.1790 |
2,880 | 0.7385 | 1.1189 | 1.3554 | 1.4315 | 1.3569 | 1.1600 | 0.8774 | 0.5432 | 0.1835 |
5 Conclusions
A two-dimensional hydrodynamic model for describing the elevation of water wave in an open uniform reservoir is derived and presented. The separation of variables method is employed to find an analytical solution to the model.
Declarations
Acknowledgements
This paper is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The authors greatly appreciate valuable comments received from the reviewers.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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