Session A31: CFD: Particle and Immersed Boundary Methods

Towards numerical consistency and conservation for SPH approximations

Typical conservative Smoothed particle hydrodynamics (SPH) approximations introduce two errors: smoothing error is due to smoothing of the gradient by an integration associated with a kernel function; integration error due to approximating of the integration by summation over all particles within the kernel support. The integration error leads to violation of zero-order consistency, i.e., the inability to reproduce a constant field. We show that partition of unity is the condition under which the conservative SPH approximation achieves both consistency and convergence. The condition can be met by relaxing a particle distribution under a constant pressure field and invariant particle volume. The resulting particle distribution is very similar to those observed for liquid molecules. We further show that with two different typical kernel functions the SPH approximation satisfying the partition of unity property is able to achieve very high-order of the integration error for random particle locations. The background pressure used in a weakly compressible SPH simulation implies a self-relaxation mechanism, which explains that convergence with respect to increasing particle numbers could be obtained in SPH simulations, although not predicted by previous numerical analysis. Furthermore, by relating the integration error to the background pressure, we explain why the previously proposed transport-velocity formulation of SPH is able to achieve unprecedented accuracy and stability.   

Developing a weakly compressible smoothed particle hydrodynamics model for biological flows

Smoothed Particle Hydrodynamics (SPH) is a meshless particle method originally developed for astrophysics applications in 1977. Over the years, limitations of the original formulations have been addressed by different groups to extend the domain of SPH application. In biologically relevant internal flows, two of the several challenges still facing SPH are 1) treatment of inlet, outlet, and no slip boundary conditions and 2) treatment of second derivatives present in the viscous terms. In this work, we develop a 2D weakly compressible SPH (WCSPH) for simulating viscous internal flows which incorporates some of the recent advancements made by groups in the above two areas. The method is validated against several analytical and experimental benchmark solutions for both steady and unsteady laminar flows. In particular, the 2013 U.S. Food and Drug Administration benchmark test case for medical devices -- steady forward flow through a nozzle with a sudden contraction and conical diffuser -- is simulated for different Reynolds numbers in the laminar region and results are validated against the published experimental and CFD datasets.   

Momentum exchange in multiphase dispersed flow: a statistical estimator based on multilevel Monte Carlo

Eulerian-Eulerian models for multiphase dispersed flow are commonly derived by means of ensemble (or spatial) averaging. They are therefore based on quantities defined over statistical (or spatial) ensemble of particle configurations. However momentum exchange correlations (e.g., drag, lift) are known (and can be defined deterministically) only for the dilute (isolated spheres) and dense (Ergun) limits. Furthermore it is well known that the overall results are often very sensitive to the correlation chosen and to the closure approximations for fluctuations, strongly limiting the predictive capability of the models. In this work the forces acting on random array of spheres and other granular objects have been studied with a novel statistical approach based on multilevel Monte Carlo. Direct Numerical Simulations are used to resolve the flow around the spheres and both the numerical and statistical error are controlled accurately. Mean and fluctuations of the momentum exchange terms can be characterized to derive new correlations for drag and lift in dense poly-dispersed flows that are statistically robust.   

A Particle-Particle Collision Model for Smoothed Profile Method

Smoothed Profile Method (SPM) is a type of continuous forcing approach that adds the particles to the fluid using a forcing. The fluid-structure interaction is through a diffuse interface which avoids sudden transition from solid to fluid. The SPM simulation as a monolithic approach uses an indicator function field in the whole domain based on the distance from each particle's boundary where the possible particle-particle interaction can occur. A soft sphere potential based on the indicator function field has been defined to add an artificial pressure to the flow pressure in the potential overlapping regions. Thus, a repulsion force is obtained to avoid overlapping. Study of two particles which impulsively start moving in an initially uniform flow shows that the particle in the wake of the other one will have less acceleration leading to frequent collisions. Various Reynolds numbers and initial distances have been chosen to test the robustness of the method. Study of Drafting-Kissing Tumbling of two cylindrical particles shows a deviation from the benchmarks due to lack of rotation modeling. The method is shown to be accurate enough for simulating particle-particle collision and can easily be extended for particle-wall modeling and for non-spherical particles.   

An Immersed Boundary Method for Rigid Bodies

The traditional immersed boundary (IB) method is a very flexible method for coupling elastic structures to fluid flow. When rigid bodies are modeled using an IB approach, a penalty method is usually employed to approximately enforce the rigidity of the body; this requires small time step sizes and leads to difficult-to-control errors in the solution. We develop a method that exactly enforces a rigidity constraint by solving a linear system coupling a standard semi-implicit discretization of the fluid equations with a rigidity constraint. We develop a preconditioned iterative solver that combines an approximate multigrid solver for the fluid problem with an approximate direct solver for the Schur complement system. We demonstrate the efficiency and study the accuracy of the method on several test problems for both zero and finite Reynolds numbers.   

An immersed interface vortex particle-mesh solver

An immersed interface-enabled vortex particle-mesh (VPM) solver is presented for the simulation of 2-D incompressible viscous flows, in the framework of external aerodynamics. Considering the simulation of free vortical flows, such as wakes and jets, vortex particle-mesh methods already provide a valuable alternative to standard CFD methods, thanks to the interesting numerical properties arising from its Lagrangian nature. Yet, accounting for solid bodies remains challenging, despite the extensive research efforts that have been made for several decades. The present immersed interface approach aims at improving the consistency and the accuracy of one very common technique (based on Lighthill's model) for the enforcement of the no-slip condition at the wall in vortex methods. Targeting a sharp treatment of the wall calls for substantial modifications at all computational levels of the VPM solver. More specifically, the solution of the underlying Poisson equation, the computation of the diffusion term and the particle-mesh interpolation are adapted accordingly and the spatial accuracy is assessed. The immersed interface VPM solver is subsequently validated on the simulation of some challenging impulsively started flows, such as the flow past a cylinder and that past an airfoil.   

A Newton-Krylov method with approximate Jacobian for implicit solution of Navier-Stokes on staggered overset-curvilinear grids with immersed boundaries

Time step-size restrictions and low convergence rates are major bottle necks for implicit solution of the Navier-Stokes in simulations involving complex geometries with moving boundaries. Newton-Krylov method (NKM) is a combination of a Newton-type method for super-linearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations, which can theoretically address both bottle necks. The efficiency of this method vastly depends on the Jacobian forming scheme e.g. automatic differentiation is very expensive and Jacobian-free methods slow down as the mesh is refined. A novel, computationally efficient analytical Jacobian for NKM was developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered curvilinear grids with immersed boundaries. The NKM was validated and verified against Taylor-Green vortex and pulsatile flow in a 90 degree bend and efficiently handles complex geometries such as an intracranial aneurysm with multiple overset grids, pulsatile inlet flow and immersed boundaries. The NKM method is shown to be more efficient than the semi-implicit Runge-Kutta methods and Jabobian-free Newton-Krylov methods. We believe NKM can be applied to many CFD techniques to decrease the computational cost.   

An immersed boundary method for imposing solid wall conditions in lattice Boltzmann solvers for single- and multi-component fluid flows

In this work, one proposes an immersed boundary-lattice Boltzmann coupled algorithm to solve single- and multi-component fluid flows, in the presence of fixed or moving solid boundaries. The prescribed motion of immersed boundaries is imposed by adding a body force term in the lattice Boltzmann model, which is obtained from the macroscopic fluid velocity definition interpolated at the Lagrangian solid points. Numerical validation test cases show that the proposed numerical solver is second-order accurate. Furthermore, the Shan-Chen's lattice Boltzmann model is applied for multi-component fluid flows, and a special focus is given to the treatment of different wetting properties of fixed walls. The capability of the new numerical solver is finally evaluated by simulating a cluster of moving cilia in a two-component fluid flow.