Curvilinear Coordinates

Objective: The objective of the presented paper is establishing the multi-relaxation-time lattice Boltzmann
method (MRT-LBM) for solving 2D flow equations transformed in a curvilinear coordinate system.
Method: Using the complete transformation approach – which includes transformation of both dependent
and independent variables between the physical and computational domain – corresponding forms
of the equilibrium function and of the force term for the 2D Navier–Stokes equations and the shallow
water equations have been derived. The physical flow domain of arbitrary geometry in the horizontal
plane, is covered with adequate curvilinear mesh, while the calculation procedure is carried out in the
D2Q9 square lattice, applying the basic form of the boundary condition method on water-solid and open
boundaries as well.
Test cases: The method is tested using four different examples: Couette flow in a straight inclined
channel, Taylor–Couette flow between two cylinders, a non-prismatic channel in a 180 bend, and a segment
of irrigation channel with a parabolic cross section in a 90 bend. In the cases of the bent channels,
previously available velocity measurements have been used for validation of the model. In addition, the
procedure employs a mathematical model based on traditional CFD procedures.
Results: The remarkable agreement between the results obtained by the proposed model and the
corresponding analytical values and measurements shows that the presented curvilinear form of the
LBM is capable of solving very complex environmental problems, maintaining the order of accuracy,
simplicity and efficiency of the basic LBM.

Estuaries are key areas in between land and ocean which play a major role in the spreading of continental runoff drained by large watershed. This study focused on the Loire Estuary and its adjacent bays (i.e. Bourgneuf bay and Mor-Braz sea) all located in the north-east side of the bay of Biscay. It is influenced by the large tidal wave that propagates upstream the mouth on more than a 100 km, by highly mid-latitude meteorological forcing that may not only induced High variability in the circulation drivers but also on the river runoffs that may vary from 1 to 10 from early spring to late summer. This High variability is studied thanks to numerical simulation and tools dedicated to describe the circulation with synthetic index such as transit time and mean age of water. The approach lies on a numerical model discretized on a structure grid which constraints have been relaxed to better fit the fractal coastal line using non orthogonal grid cells. The optimal coordinate framework (co or contra-variant) have been discussed, and implemented within a pre-existing code (i.e. MARS-3D). This tools was validated with test cases and implemented on a domain with a particular complex geometry. The numerical simulations catch very accurately the dynamic of this large plume at least as it is described by available in situ observations. This numerical solution allowed to exhibit the main path of water masses through the area and from place to place and their variability according to the main forcings.

Using 3D conformal geometric algebra, smoothly deformable curved volumes of space can be constructed with simple expressions, purely in terms of tangents to the space. Rationalization is achieved through continuous transformation of spheres, which are principal contact surfaces of the space. The resulting triply orthogonal curvilinear coordinate system enables lattice bending deformations within a hexahedral volume. An inverse mapping is used to connect the coordinate systems of neighboring volumes. The development of these parameterizations offers a new but foundational tool for computer aided design of machine parts, animated 3D characters, or freeform architecture.