Rheology of dense Suspensions : development and implementation of New Numerical methods, taking close range interactions into account – RheoSuNN

The project is based on two existing codes developed by members of the project: a fluid/particle solver (CAFES) and a contact solver (SCoPI). In order to reach our goal, we first want to integrate a friction model in SCoPI. Indeed, it has been shown that taking into account the friction during contacts is essential to obtain physical macroscopic results. For this purpose, we implement a contact model, within the framework of the non-smooth convex analysis, and having the advantage of being numerically stable.
The next task is dedicated to the coupling of the two codes, as well as to their optimization. The coupling of the codes is indeed essential to consider the effects of contacts on the fluid. In parallel, we design new methods to allow the consideration of the effects of the fluid between close particles (lubrication), for various shapes of particles. These methods are based on the asymptotic development of the solution in the gap between close particles when their distance goes to zero.

This project leads to highly coupled fluid/particle problems to be solved in an implicit way, in presence of singularities. It needs fine numerical expertise to be treated in an efficient way. We plan to deliver a code including the implicit methods developed in the project. In order to run simulations for real physical configurations, it is crucial to develop a numerical code which is optimised and able to run with good scalability on the national and European HPC infrastructures proposed by GENCI and PRACE. This code will allow to reach more precise and dense simulations than the existing ones and will enable the study of open questions about the rheology of suspension.

In order to implement friction in the code SCoPI, we based our work on the non-smooth contact algorithm developed in the thesis of Hugo Martin (IPGP/LJLL). We extended this algorithm to the case of non-spherical particles and it is currently being implemented. We implement it for a family of particle shapes, subset of super-ellipsoids. In order to take into account the friction, it has been necessary to implement the rotations of the particles, which had not been done before in the code (since useless in the case of normal contacts and in the absence of force moments). We also take advantage of these modifications to optimize the inputs/outputs of the code and to set up a visualization adapted to the new type of particles.

In parallel, we have developed a new method to take into account the effects of the fluid between close particles without having to finely mesh the gap between the particles (which would be too expensive from a computational point of view). We have proposed a method called «singular/regular field decomposition« in which the singular field is assumed to be known and the regular field can be computed numerically on coarse meshes. The singular field is computed analytically and is adapted to numerical simulations. We have shown that the remaining field is regular and can be computed on coarse meshes. The method has been implemented and validated in two dimensions, on fitted meshes, with FreeFem++. This work has been submitted. Following this, we worked with Fabien Vergnet (as part of a post-doctorate funded by the ANR) to adapt the method to fluid/particle solvers based on fictitious domain methods. We have started to implement it in the CAFES solver. To do so, it is necessary to extend the singular field in the particles without losing the regularity of the rest. This work is in progress.

The implementation of friction and the extension of the decomposition method to fictitious domain methods are in progress. The coupling of the contact and the fluid solvers will then be performed. Validation tests and first rheological studies are planned. The code obtained at the end of this project will make it possible to carry out new rheological studies, not accessible with the current codes, such as the study of the interphase stress or that of suspensions of non-spherical particles while taking into account lubrication and its effect on the fluid field.

A. Lefebvre-Lepot et F. Nabet, Numerical simulation of rigid particles in Stokes flow: lubrication correction for any (regular) shape of particles, soumis (2020)
hal.archives-ouvertes.fr/hal-02433849v1

RheoSuNN project brings together researchers in the domains of mathematics, numerical analysis, HPC and mechanics, interested in numerical simulations of dense suspensions of rigid particles embedded in a Stokes fluid. The scientific program of this project is based on the following objective: achieve a numerical rheological study in order to understand the interphase stress contribution to the total fluid stress. The last year of the project is dedicated to this task. It consists in a challenging study, for which no result is available (neither experimental, nor theoretical, nor numerical). The key point to make a breakthrough in the understanding of the interphase stress is to compute the velocity and pressure fields in the whole fluid domain for a dense suspension, to model carefully the multi-body lubrication and contact interactions and their feedback on the flow. From a computational point of view, it leads to suspension simulations with density around 50%, containing up to about one million of suspended particles. The corresponding mesh size for the direct fluid solver is around 500 x 2 000 x 10 000. Despite the wide range of available techniques, there is a great need of designing new mathematical models and numerical methods to achieve this numerical study.

The project is based on two existing codes developed by members of the project: a fluid/particle solver (CAFES) and a contact solver (SCoPI). To reach the target, the project first two years are dedicated to the two codes coupling, together with their optimisation and the design of new numerical methods to be able to take into account stiff close interactions and their effects on the whole flow, for any shape of particles. This leads to highly coupled fluid/particle problems to be solved in an implicit way, in presence of singularities. This problem need fine numerical expertise to be treated in an efficient way. We plan to deliver a code including the implicit methods developed in the project. In order to run simulations for real physical configurations, it is crucial to develop a numerical code which is optimised and able to run with good scalability on the national and European HPC infrastructures proposed by GENCI and PRACE. This code will allow to reach more precise and dense simulations than the existing ones and will enable the study of open questions about the rheology of suspension.

The code we develop allows to deal with non-spherical particles (more precisely with superellipsoids) while taking lubrication into account with retroaction on the flow, which is not possible with the other existing codes. As a consequence, even though we focus in this project on the interphase stress, it opens the opportunity for numerous other interesting and original numerical rheological studies of non-spherical particles suspensions (like rods, fibres or micro-swimmers).

Moreover, taking in a very fine way lubrication phenomenon into account, it can be a reference code, providing precise results to study new rheological phenomenon or to compute reference solutions to be compared to other methods. It can also be a first step towards direct numerical simulation of suspensions of particles embedded in non-Newtonian fluids or of active suspensions which are two very active domains of research.