Don Liu

Force-coupling method for flows with ellipsoidal particles

The force-coupling method, previously developed for spherical particles suspended in a liquid flow, is extended to ellipsoidal particles. In the limit of Stokes flow, there is an exact correspondence with known analytical results for isolated particles. More generally, the method is shown to provide good approximate results for the particle motion and the flow field both in viscous Stokes flow and at finite Reynolds number. This is demonstrated through comparison between fully resolved direct numerical simulations and results from the numerical implementation of the force-coupling method with a spectral/hp element scheme. The motion of settling ellipsoidal particles and neutrally buoyant particles in a Poiseuille flow are discussed.

Simulations of dynamic self-assembly of paramagnetic microspheres in confined microgeometries

We simulate the dynamic self-assembly of paramagnetic beads in triangular microducts and micropipes. First, we demonstrate, in the absence of any net flow, that depending on the magnetic field, the number of microparticles and the shape of microdomain, different self-assembled structures can be produced. Subsequently, we study the effect of net flow on the stability of the group of chains of beads formed as a function of the pressure drop in a micropipe. Dynamic self-assembly in confined microgeometries provides an exciting possibility for making reconfigurable multi-functional microdevices, and may also suggest new protocols for fabricating three-dimensional microsystems and nanosystems.

Modeling and optimization of colloidal micro-pumps

Manipulating micro-particles is very important in microfluidic applications, such as biomedical flows and self-assembled structures. Here, flows generated by the forced motion of colloidal micro-particles in a microchannel are investigated. The force coupling method combined with the spectral/hp element method is used to numerically simulate the dynamics of the flow, while a penalty method is used to determine the required forces on the particles. The pumping motion is investigated for two specific systems: a peristaltic micro-pump and a gear micro-pump. We verify the accuracy of the simulations and then for each system, we investigate the net flow rate as a function of pump frequency and channel dimension, and present optimization results. The results for the net flow rate are comparable to and within the range of the experimental data.

Large Eddy Simulation of Unidirectional and Wave Flows through Vegetation

AbstractMassively parallel large eddy simulation (LES) experiments were conducted to study the flow fields developed by unidirectional flow over submerged vegetation and wave flow-through emergent vegetation. For the submerged vegetation, vertical profiles of mean and turbulent horizontal and vertical velocities were found to be in good agreement with laboratory experiments. Canopy-averaged bulk drag coefficient calculated from the depth-integrated forces on the cylinders compares well with empirical measurements. For the emergent vegetation, wave-induced drag forces were calculated, and the inertia and pressure-drag coefficients were compared with laboratory experiments, which were found to be in good agreement. Vertical variation of forces and moments about the stem base are presented and compared with a single stem case under a variety of wave conditions and Keulegan-Carpenter (KC) numbers. It is seen that at low KC numbers the effect of the inertia force is significant. The vertical variation of the v...

Improved PHSS iterative methods for solving saddle point problems

An improvement on a generalized preconditioned Hermitian and skew-Hermitian splitting method (GPHSS), originally presented by Pan and Wang (J. Numer. Methods Comput. Appl. 32, 174–182, 2011), for saddle point problems, is proposed in this paper and referred to as IGPHSS for simplicity. After adding a matrix to the coefficient matrix on two sides of first equation of the GPHSS iterative scheme, both the number of required iterations for convergence and the computational time are significantly decreased. The convergence analysis is provided here. As saddle point problems are indefinite systems, the Conjugate Gradient method is unsuitable for them. The IGPHSS is compared with Gauss-Seidel, which requires partial pivoting due to some zero diagonal entries, Uzawa and GPHSS methods. The numerical experiments show that the IGPHSS method is better than the original GPHSS and the other two relevant methods.

Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation

In this paper, we propose a new two-level implicit compact operator method of order two in time (t) and four in space (x) for the solution of time dependent Burgers-Huxley equation with appropriate initial and boundary conditions. The presence of Reynolds number and nonlinear terms in the problem leads to severe difficulties in the numerical approximation. To overcome such difficulties, the method based on operators is constructed. We use only 3-spatial grid points and the obtained tridiagonal nonlinear system has been solved by Newton’s iteration method. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed method. The computed numerical solutions are in good agreement with the exact solutions. We show that the proposed method enables us to obtain high accuracy solution for high Reynolds number.

A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

High-Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation

A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

Modal Spectral Element Solutions to Incompressible Flows over Particles of Complex Shape

This paper develops the virtual identity particles (VIP) model to simulate two-phase flows involving complex-shaped particles. VIP assimilates the high efficiency of the Eulerian method and the convenience of the Lagrangian approach in tracking particles. It uses one fixed Eulerian mesh to compute the fluid field and the Lagrangian description to handle constitutive properties of particles. The interaction between the fluid and complex particles is characterized with source terms in the fluid momentum equations, while the same source terms are computed iteratively from the particulate Lagrangian equations. The advantage of VIP is its economy in modeling a two-phase flow problem almost at the cost of solving only the fluid phase with added source terms. This high efficiency in computational cost makes VIP viable for simulating particulate flows with numerous particles. Owing to the spectral convergence and high resolvability of the modal spectral element method, VIP provides acceptable resolution comparable to DNS but at much reduced computational cost. Simulation results indicate that VIP is promising for investigating flows with complex-shaped particles, especially abundant complex particles.

Spectral Element Simulation of Complex Particulate Flows

This paper uses a mathematical model Virtual Identity Particles, developed by the author, to simulate conjugated motion of complex particles in a fluid. Assimilated the advantages of Eulerian and Lagrangian approaches, this model treats each particle as a variable source term to the fluid and is designed for simulating numerous particles in two-phase flows. The economic formulation in this model is the salient feature. Considering both precision and computational cost, this model maintains an excellent balance between accuracy and efficiency in modeling particulate flows with complex particles. Simulation results demonstrate that this model is viable for investigating complex particulate flows, especially at a moderately high particle number density.