Hector D. Ceniceros

Computation of multiphase systems with phase field models

Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.

Numerical Solution of Polymer Self-Consistent Field Theory

We propose efficient pseudospectral numerical schemes for solving the self-consistent, mean-field equations for inhomogeneous polymers. In particular, we introduce a robust class of semi-implicit methods that employ asymptotic small scale information about the nonlocal density operators. The relaxation schemes are further embedded in a multilevel strategy resulting in a method that can cut down the computational cost by an order of magnitude. Three illustrative problems are used to test the numerical methods: (i) the problem of finding the mean chemical potential field for a prescribed inhomogeneous density of homopolymers; (ii) an incompressible melt blend of two chemically distinct homopolymers; and (iii) an incompressible melt of AB diblock copolymers.

Coalescence of two equal-sized deformable drops in an axisymmetric flow

The coalescence of two equal-sized deformable drops in an axisymmetric flow is studied, using a boundary-integral method. An adaptive mesh refinement method is used to resolve the local small-scale dynamics in the gap and to retain a reasonable speed of computation. The thin film dynamics is successfully simulated, with sufficient stability and accuracy, up to a film thickness of O(10−4) times the undeformed drop radius, for a range of capillary numbers, Ca, from O(10−4–10−1) and viscosity ratios from O(0.1–10). The results are compared with experimental results from our earlier studies as well as the simple scaling theory for film drainage. The collisions for time-independent flow simulating head-on collisions in the experimental studies show two distinctively different regimes. At lower capillary numbers, the interfaces of the thin film between the colliding drops remain almost spherical up to the point of film rupture, and the dimensionless drainage time scales as tdG∼Ca. At higher capillary numbers, t...

The effects of surfactants on the formation and evolution of capillary waves

The effects of surface-active agents on the formation and evolution of small capillary ripples developing in the forward front of short water waves is investigated numerically. The capillary waves, believed to have a significant relevance in the process of wave breaking and the onset of turbulence, accompany the initial development of spilling breakers. A novel hybrid numerical methodology is introduced to couple the full two-fluid Navier–Stokes equations with the free boundary motion and with the surfactant dynamics. The hybrid method uses dynamically adaptive front-tracking to accurately represent interfacial quantities and forces and to aid in treating the numerical difficulties associated with surface tension. At the same time the method employs the level set approach to efficiently update the material properties of the flow. It is found that the capillaries are dramatically affected by the presence of surfactants. The capillary region is invariably marked by accumulation of surfactants that reduces l...

A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids

We introduce a method for solving the variable coefficient Poisson equation on non-graded Cartesian grids that yields second order accuracy for the solutions and their gradients. We employ quadtree (in 2D) and octree (in 3D) data structures as an efficient means to represent the Cartesian grid, allowing for constraint-free grid generation. The schemes take advantage of sampling the solution at the nodes (vertices) of each cell. In particular, the discretization at one cells node only uses nodes of two (2D) or three (3D) adjacent cells, producing schemes that are straightforward to implement. Numerical results in two and three spatial dimensions demonstrate supra-convergence in the L∞ norm.

Three-dimensional, fully adaptive simulations of phase-field fluid models

We present an efficient numerical methodology for the 3D computation of incompressible multi-phase flows described by conservative phase-field models. We focus here on the case of density matched fluids with different viscosity (Model H). The numerical method employs adaptive mesh refinements (AMR) in concert with an efficient semi-implicit time discretization strategy and a linear, multi-level multigrid to relax high order stability constraints and to capture the flows disparate scales at optimal cost. Only five linear solvers are needed per time-step. Moreover, all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of AMR in a conservative, phase-field setting. We validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation, Kelvin-Helmholtz instability, and flow-induced drop coalescence.

A nonstiff, adaptive mesh refinement-based method for the Cahn-Hilliard equation

We present a nonstiff, fully adaptive mesh refinement-based method for the Cahn-Hilliard equation. The method is based on a semi-implicit splitting, in which linear leading order terms are extracted and discretized implicitly, combined with a robust adaptive spatial discretization. The fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size. We demonstrate the efficacy of the method with numerical examples. Both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy. Employing this approach, we also identify several stationary solutions of that decomposition on the 2D torus.

Numerical study of Hele-Shaw flow with suction

We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle (cusp) of the zero-viscosity-ratio solution.

Numerical Solutions of the Complex Langevin Equations in Polymer Field Theory

Using a diblock copolymer melt as a model system, we show that complex Langevin (CL) simulations constitute a practical method for sampling the complex weights in field theory models of polymeric f...

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskins lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the systems matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.