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Significance Statement
The detailed numerical modelling of morphological processes such as the development of bed forms (ripples, dune and breaker bars) or scour around structures with CFD seem to lack behind the developments in the capabilities for numerical modelling of pure hydrodynamics, e.g. river flow and wave loads on structures. This is likely to be explained by the time scale of the morphological processes that are much larger than the hydrodynamic time scale. This means that the modelling of morphological processes has largely been restricted to structured computational meshes.
The advances in numerical techniques does continuously spread to morphodynamic simulations, but subtle details such as mass conservation of the sand have not yet been offered much attention. It is possible to analytical derived that previous works did not conserve mass. A consequence is that it is difficult to analyse the results of for instance the scour around a bridge pier, if 10% of the volume of the scour hole is missing due to numerical sink terms (or sources). Since the mass error can have both signs, it makes it difficult to analyse physical processes separately from numerical errors. Especially for morphodynamic systems, where there is a tight and highly non-linear coupling between the hydrodynamics and the evolving morphology.
In the present work, it was shown that the use of unstructured or non-equidistantly discretised computational meshes results in a strict requirement for the treatment of the mass conservation of the sediment. A simple approach to conserve the mass of the sediment on three dimensional, polyhedral meshes was suggestions. This approach is mass conserving to machine precision and it is shown mathematically that it simplifies to an inverse linear interpolation scheme for two dimensional simulations. Subsequently, it was proven that the inverse interpolation scheme is a unique, local interpolation scheme that preserves the mass of the sediment.
A mass conserving numerical model is the first step towards a more reliable interpretation of the numerical results, and this work is the first step towards a generally mass conserving morphodynamic model. Many challenges, however, await. These are related to physical problems with for instance graded sediment (multiple fractions) and non-erodible surfaces.
Journal Reference
International Journal for Numerical Methods in Fluids, 2015, Volume 78, Issue 4, Pages 189–256.
Niels Gjøl Jacobsen
[expand title=”Show Affiliations”]- Department of Coastal Structures and Waves, Deltares, Boussinesqweg 1, 2629 HV Delft, The Netherlands
- Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé 2800 Kgs. Lyngby, Denmark
Abstract
Computational morphodynamics in finite volume methods are based on the evaluation of the rate of bed level change in the vertices on the deforming bed. With the use of finite volume methods on collocated (unstructured) grids, the rate of bed level change needs to be interpolated from the mesh faces to the vertices. First, this work reviews two methods based on a vectorial shape of the bed evolution equation (no scalar contributions from storage, erosion and deposition) in terms of their mass conserving properties. Second, a method that allows for scalar contributions in the bed evolution equation (the Exner equation) is proposed for general, unstructured meshes, and an analytical derivation for the simple one-dimensional problem on a non-equidistantly discretised grid is considered. The solution is compared with the general two-dimensional formulation. The two-dimensional formulation leads to the formulation of a geometric sand sliding routine on unstructured grids. The newly proposed interpolation method and the sand sliding routine are tested, and mass conservation of the sediment is considered with special emphasis on the effect of the solution accuracy for the suspended sediment transport. Discussions on other interpolation methods and their mass conserving properties are given with a special focus of the distance weighted interpolation method directly available and easily applied in OpenFOAM. Furthermore, effects from horizontal displacements of the vertices, explicit filtering of the evolving bed and morphological acceleration on global mass conservation, are discussed. Copyright © 2015 John Wiley & Sons, Ltd.
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