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Three-Layer Non-hydrostatic Staggered Scheme for Free Surface Flow

Published online by Cambridge University Press:  31 January 2018

Ade C. Bayu*
Affiliation:
Industrial & Financial Mathematics Research Group, Faculty of Mathematics & Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung, 40132, Indonesia
S. R. Pudjaprasetya*
Affiliation:
Industrial & Financial Mathematics Research Group, Faculty of Mathematics & Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung, 40132, Indonesia
I. Magdalena*
Affiliation:
Industrial & Financial Mathematics Research Group, Faculty of Mathematics & Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung, 40132, Indonesia
*Corresponding
*Corresponding author. Email addresses:ade_zildji@yahoo.co.id (A. C. Bayu), sr_pudjap@math.itb.ac.id (S. R. Pudjaprasetya), ikha.magdalena@math.itb.ac.id (I. Magdalena)
*Corresponding author. Email addresses:ade_zildji@yahoo.co.id (A. C. Bayu), sr_pudjap@math.itb.ac.id (S. R. Pudjaprasetya), ikha.magdalena@math.itb.ac.id (I. Magdalena)
*Corresponding author. Email addresses:ade_zildji@yahoo.co.id (A. C. Bayu), sr_pudjap@math.itb.ac.id (S. R. Pudjaprasetya), ikha.magdalena@math.itb.ac.id (I. Magdalena)
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Abstract

In this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

MSC classification

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Hunter, J. K., An introduction to the incompressible Euler equations [Lecture note], 2006, Retrieved from http://www.math.ucdavis.edu/∼hunter/notes/euler.pdf.Google Scholar
[2] Kim, D. H. and Lynett, P. J., Dispersive and nonhydrostatic pressure effects at the front of surge, J. Hydraul. Eng., 137 (2011), 754765.CrossRefGoogle Scholar
[3] Kämpf, J., Advanced Ocean Modelling Using Open-Source Software, Springer-Verlag, Berlin Heidelbert, 2010.Google Scholar
[4] Pudjaprasetya, S. R., Magdalena, I., and Tjandra, S. S., A nonhydrostatic two-layer staggered scheme for transient waves due to anti-symmetric seabed thrust, J. Earthquake and Tsunami, 11 (2017), 1740002.CrossRefGoogle Scholar
[5] Soares-Frazão, S. and Guinot, V., A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels, Int. J. Numer. Meth. Fluids, 58 (2008), 237261.CrossRefGoogle Scholar
[6] Soares-Frazão, S. and Zech, Y., Undular bores and secondary waves–experiments and hybrid finite-volume modeling, J. Hydr. Research, 40 (2002), 3343.CrossRefGoogle Scholar
[7] Stelling, G. and Duinmeijer, S. P. A., A staggered conservative scheme for every Froude number in rapidly varied shallow water flows, Int. J. Numer. Meth. Fluids, 43 (2003), 13291354.CrossRefGoogle Scholar
[8] Stelling, G. and Zijlema, M., An accurate and efficient finite-difference algorithm for nonhydrostatic free surface flow with application to wave propagation, Int. J. Numer. Meth. Fluids, 43 (2003), 123.CrossRefGoogle Scholar

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Three-Layer Non-hydrostatic Staggered Scheme for Free Surface Flow
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