Une nouvelle méthode smoothed particle hydrodynamics : simulation des interfaces immergées et de la dynamique Brownienne des molécules avec des interactions hydrodynamiques

In this thesis we develop a new smoothed particle hydrodynamics (SPH) method suitable for solving the incompressible Navier-Stokes equations, even with singular forces. Singular source terms are handled in a manner similar to that in the immersed boundary (IB) method of Peskin (2002) or in the method of regularized Stokeslets (Cortez, 2001). The numerical scheme implements a second-order pressure-free  projection method due to Kim and Moin (1985) and completely obviates the difficulties that may be faced in prescribing Neumann pressure boundary conditions. We    present two variants of this approach, one Langrangian which is commonly used and one Eulerian, simply because we consider that the SPH particles are quadrature points  on which the fluid properties are calculated, therefore, these points can be kept fixed in time.   The proposed SPH method is first tested on the planar start-up Poiseuille problem and a detailed error analysis is performed. For this problem, the results are similar  whether the SPH particles are free to move or fixed on a regular grid.   Our hybrid SPH-IB method is then used to calculate the dynamics of a stretched immersed elastic membrane. The membrane is represented by a cubic spline along which the  tension in the membrane is computed and transmitted to the surrounding fluid. The Navier-Stokes equations with singular force due to the membrane are then solved to  determine the velocity of the fluid in which the membrane is immersed. The fluid velocity thus obtained is interpolated on the interface, to determine its displacement. We discuss the advantages, in this problem, of fixing the SPH particles, rather than allowing them to move with the fluid.  A new coupled Brownian dynamics-SPH method for the computation of confined flows of non-dilute polymer solutions with full hydrodynamic  interaction and excluded volume forces is next presented. The starting point for the algorithm is the system of coupled Langevin equations for polymer and solvent (CLEPS)  (see Oono and Freed (1981) and Öttinger and Rabin (1989), for example) describing, in the present case, the microscopic dynamics of a flowing polymer  solution with a bead-spring representation of the macromolecules.   Numerical tests of some two-dimensional channel  flows reveal that use of a second-order projection scheme coupled with fixed SPH quadrature points leads to second-order velocity convergence and almost second-order  pressure convergence, provided that the solution is sufficiently smooth. In the case of large-scale dumbbell and bead-spring chain calculations, an appropriate scaling  of the number of grid points as a function of the number of beads N ensures, in the absence of excluded volume forces, that the cost of our algorithm is O(N) flops.  Finally, we begin calculations in three dimensions with our SPH model. To this end, we solve in three dimensions the problem of Poiseuille flow between two infinite and  parallel plates and the problem of Poiseuille flow in a rectangular infinitely long duct. In addition, we carry out three dimensional computations of confined flows of  non-dilute polymer solutions with full hydrodynamic interaction and excluded volume forces.

Keyphrases: Smoothed particle hydrodynamics (SPH), Incompressible flow, Projection method, Singular force, Immersed boundary, Brownian dynamics, Hydrodynamic interactions, Fluctuating hydrodynamics, Coupled Langevin equations, Confined flows