Computers and Fluids 161 (2018) 136–154 Contents lists available at ScienceDirect Computers and Fluids journal homepage: www.elsevier.com/locate/compfuid Comparison of 2D triangular C-grid shallow water models Hamidreza Shirkhani a , Abdolmajid Mohammadian a , Ousmane Seidou a , Hazim Qiblawey b a Department of Civil Engineering, University of Ottawa, CBY D216, 161 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada b Department of Chemical Engineering, Qatar University, P.O. Box 2713, Doha, Qatar a r t i c l e i n f o Article history: Received 6 May 2016 Revised 14 October 2017 Accepted 21 November 2017 Available online 24 November 2017 Keywords: Shallow water equations Dispersion relation analysis Triangular C-grid Source terms a b s t r a c t An ideal two-dimensional (2D) shallow water model should be able to simulate correctly various types of waves including pure gravity and inertia-gravity waves. In this paper, two different triangular C-grid methods are considered, and their dispersion of pure gravity waves, frequencies of inertia-gravity waves and geostrophic balance solutions are investigated. The proposed C-grid methods employ different spa- tial discretization schemes for coupling shallow water equations together with the various reconstruc- tion techniques for tangential velocity estimation. The proposed reconstruction technique for the second method, which is analogous to a hexagonal C-grid scheme, is shown to be energy conservative and sat- isfies the geostrophic balance exactly while it supports the unphysical geostrophic modes for hexagonal C-grid. Because of the importance of the application of 2D shallow water models on fully unstructured grids, particular attention is also given to various types of isosceles triangles that may appear in such grids. For the gravity waves, the results of the phase speed ratio of the computed phase speeds over the analytical one are shown and compared. The non-dimensional frequencies of various modes for inertia- gravity waves are also investigated and compared in terms of being monotonic and isotropic respect to the continuous solution. The analyses demonstrate some advantages of the first method in phase speed behaviour for gravity waves and monotonicity of inertia-gravity dispersion. The results of the dispersion analysis are verified through a number of numerical tests. The first method, which is shown to have a better performance, examined through more numerical tests in presence of various source terms and results confirm its capability. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Two-dimensional (2D) shallow water models have essential im- portance in a wide range of coastal and environmental engineer- ing problems. An ideal 2D shallow water model should be capable of simulating pure gravity and inertia-gravity waves. Several stud- ies were conducted to evaluate the performance of various models for simulating the flow in large-scale shallow water bodies [37,41]. In numerical modelling of shallow water equations, one needs to couple the momentum and continuity equations. In order to do so, there are many possibilities of variables’ placement for a certain choice of grids. Choosing the location of the variables is a delicate problem since it may lead to spurious oscillations in the numer- ical solutions. Mesinger and Arakawa [22] proposed various stag- gered grids. These grids were analysed, and among them, the C- grid was found to be more promising [3,42]. There has been an in- creased trend in using the C-grid approach with different numeri- cal schemes such as finite difference, finite element, and finite vol- E-mail addresses: h.r.shirkhani@uottawa.ca (H. Shirkhani), majid.mohammadian@uOttawa.ca (A. Mohammadian), oseidou@uottawa.ca (O. Seidou), hazim@qu.edu.qa (H. Qiblawey). ume. Moreover, different versions of the C-grid scheme has been proposed and investigated for various types of grids such as rect- angular, triangular and hexagonal [2,10,25,43]. The C-grid spatial discretizations of the shallow-water equations on regular Delau- nay triangulations on the sphere has also been analyzed [7]. The C-grid approach has been widely used in different oceanic models, to name a few, the Princeton Ocean Model [6], MICOM [5], MITgcm [1,21], ROMS [30,31] and UnTRIM [17,18]. A hydrostatic atmospheric dynamical core is developed using triangular C-grids on spherical icosahedral grids as part of numerical weather prediction and cli- mate application models [44,46]. Dispersion relation has been widely used as a useful tool for analysis of various models [2,23,27,28]. The structured rectangu- lar C-grid has been well investigated and documented. Dukowicz [13] obtained the dispersion relation of the rectangular C-grid, for inertia-gravity waves in terms of accuracy. Adcroft et al. [2] per- formed a dispersion relation analysis for the rectangular C-grid and reported spurious modes due to the Coriolis term. They suggested a new treatment by augmenting the C-grid variables using the d- grid ones, and they proposed the CD-grid. Thuburn [38] also re- ported artificial slowing effects for inertia-gravity waves in the nu- merical results of the rectangular C-grid. It was shown that the https://doi.org/10.1016/j.compfluid.2017.11.013 0045-7930/© 2017 Elsevier Ltd. All rights reserved.