We present a reduction-consistent and thermodynamically consistent formulation and an associated numerical algorithm for simulating the dynamics of an isothermal mixture consisting of N (N⩾2) ...immiscible incompressible fluids with different physical properties (densities, viscosities, and pair-wise surface tensions). By reduction consistency we refer to the property that if only a set of M (1⩽M⩽N−1) fluids are present in the system then the N-phase governing equations and boundary conditions will exactly reduce to those for the corresponding M-phase system. By thermodynamic consistency we refer to the property that the formulation honors the thermodynamic principles. Our N-phase formulation is developed based on a more general method that allows for the systematic construction of reduction-consistent formulations, and the method suggests the existence of many possible forms of reduction-consistent and thermodynamically consistent N-phase formulations. Extensive numerical experiments have been presented for flow problems involving multiple fluid components and large density ratios and large viscosity ratios, and the simulation results are compared with the physical theories or the available physical solutions. The comparisons demonstrate that our method produces physically accurate results for this class of problems.
We present an effective method for simulating wall-bounded multiphase flows consisting of N (N⩾2) immiscible incompressible fluids with different densities, viscosities and pairwise surface tensions. ...The N-phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work, and it is developed by considering the reduction consistency if some of the fluid components were absent from the system. We propose an N-phase contact-angle boundary condition that is reduction consistent between N phases and M phases (2⩽M⩽N−1). We also present a numerical algorithm for solving the N-phase governing equations together with the contact-angle boundary conditions developed herein. Extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid-wall boundaries to investigate the wettability effects with multiple types of contact angles. In particular, we compare simulation results with the de Gennes theory for the contact-angle effects on the liquid drop spreading on wall surfaces, and demonstrate that our method produces physically accurate results.
We present an efficient algorithm within the phase field framework for simulating the motion of a mixture of N (N⩾2) immiscible incompressible fluids, with possibly very different physical properties ...such as densities, viscosities, and pairwise surface tensions. The algorithm employs a physical formulation for the N-phase system that honors the conservations of mass and momentum and the second law of thermodynamics. We present a method for uniquely determining the mixing energy density coefficients involved in the N-phase model based on the pairwise surface tensions among the N fluids. Our numerical algorithm has several attractive properties that make it computationally very efficient: (i) it has completely de-coupled the computations for different flow variables, and has also completely de-coupled the computations for the (N−1) phase field functions; (ii) the algorithm only requires the solution of linear algebraic systems after discretization, and no nonlinear algebraic solve is needed; (iii) for each flow variable the linear algebraic system involves only constant and time-independent coefficient matrices, which can be pre-computed during pre-processing, despite the variable density and variable viscosity of the N-phase mixture; (iv) within a time step the semi-discretized system involves only individual de-coupled Helmholtz-type (including Poisson) equations, despite the strongly-coupled phase–field system of fourth spatial order at the continuum level; (v) the algorithm is suitable for large density contrasts and large viscosity contrasts among the N fluids. Extensive numerical experiments have been presented for several problems involving multiple fluid phases, large density contrasts and large viscosity contrasts. In particular, we compare our simulations with the de Gennes theory, and demonstrate that our method produces physically accurate results for multiple fluid phases. We also demonstrate the significant and sometimes dramatic effects of the gravity, density ratios, pairwise surface tensions, and drop sizes on the N-phase configurations and dynamics. The numerical results show that the method developed herein is capable of dealing with N-phase systems with large density ratios, large viscosity ratios, and pairwise surface tensions, and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces.
We present a new energy-stable open boundary condition, and an associated numerical algorithm, for simulating incompressible flows with outflow/open boundaries. This open boundary condition ensures ...the energy stability of the system, even when strong vortices or backflows occur at the outflow boundary. Under certain situations it can be reduced to a form that can be analogized to the usual convective boundary condition. One prominent feature of this boundary condition is that it provides a control over the velocity on the outflow/open boundary. This is not available with the other energy-stable open boundary conditions from previous works. Our numerical algorithm treats the proposed open boundary condition based on a rotational velocity-correction type strategy. It gives rise to a Robin-type condition for the discrete pressure and a Robin-type condition for the discrete velocity on the outflow/open boundary, respectively at the pressure and the velocity sub-steps. We present extensive numerical experiments on a canonical wake flow and a jet flow in open domain to test the effectiveness and performance of the method developed herein. Simulation results are compared with the experimental data as well as with other previous simulations to demonstrate the accuracy of the current method. Long-time simulations are performed for a range of Reynolds numbers, at which strong vortices and backflows occur at the outflow/open boundaries. The results show that our method is effective in overcoming the backflow instability, and that it allows for the vortices to discharge from the domain in a fairly natural fashion even at high Reynolds numbers.
► Method is for wall-bounded incompressible two-phase flows with large density ratios. ► Method can handle both dynamic and static contact angle boundary conditions. ► Method reformulates ...Cahn–Hilliard problem into decoupled Helmholtz equations. ► Method de-couples computations for all flow variables. ► Method involves only linear systems with precomputable constant coefficient matrices.
We present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall-bounded flows of two immiscible incompressible fluids with large density ratios. First, we develop an algorithm for imposing the dynamic contact angle boundary conditions to the Cahn–Hilliard equation. Our algorithm consists of two components: (i) we ignore the boundary conditions and transform the Cahn–Hilliard equation into two nominally de-coupled Helmholtz type equations; (ii) we treat the dynamic contact angle boundary conditions in such a manner that the two Helmholtz-type equations are truly de-coupled. Then, we combine this algorithm, together with a scheme for variable-density Navier–Stokes equations we developed recently, to form an efficient method for the coupled system of Navier–Stokes and Cahn–Hilliard equations for contact line problems involving large density ratios. The overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions. It is endowed with several attractive features that make the method very efficient. In particular, computations for all flow variables are completely decoupled. The resultant linear algebraic systems after discretization for all flow variables involve only constant and time-independent coefficient matrices, which can be pre-computed during pre-processing, even though the coupled Navier–Stokes/Cahn–Hilliard system involves variable density and variable viscosity. Ample numerical simulations of wall-bounded air/water two-phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact-angle conditions involving large density ratios.
We present an effective outflow boundary condition, and an associated numerical algorithm, within the phase-field framework for dealing with two-phase outflows or open boundaries. The set of ...two-phase outflow boundary conditions for the phase-field and flow variables are designed to prevent the un-controlled growth in the total energy of the two-phase system, even in situations where strong backflows or vortices may be present at the outflow boundaries. We also present an additional boundary condition for the phase field function, which together with the usual Dirichlet condition can work effectively as the phase-field inflow conditions. The numerical algorithm for dealing with these boundary conditions is developed on top of a strategy for de-coupling the computations of all flow variables and for overcoming the performance bottleneck caused by variable coefficient matrices associated with variable density/viscosity. The algorithm contains special constructions, for treating the variable dynamic viscosity in the outflow boundary condition, and for preventing a numerical locking at the outflow boundaries for time-dependent problems. Extensive numerical tests with incompressible two-phase flows involving inflow and outflow boundaries demonstrate that, the two-phase outflow boundary conditions and the numerical algorithm developed herein allow for the fluid interface and the two-phase flow to pass through the outflow or open boundaries in a smooth and seamless fashion, and that our method produces stable simulations when large density ratios and large viscosity ratios are involved and when strong backflows are present at the outflow boundaries.
We present a family of physical formulations, and a numerical algorithm, based on a class of general order parameters for simulating the motion of a mixture of N (N⩾2) immiscible incompressible ...fluids with given densities, dynamic viscosities, and pairwise surface tensions. The N-phase formulations stem from a phase field model we developed in a recent work based on the conservations of mass/momentum, and the second law of thermodynamics. The introduction of general order parameters leads to an extremely strongly-coupled system of (N−1) phase field equations. On the other hand, the general form enables one to compute the N-phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions. We show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties. Our numerical algorithm reformulates the (N−1) strongly-coupled phase field equations for general order parameters into 2(N−1) Helmholtz-type equations that are completely de-coupled from one another. This leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters. We demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids. In particular, by comparing simulation results with the Langmuir–de Gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases.
Advances in sequencing technology and bioinformatics have greatly enhanced our ability to understand the human microbiome. Over the last decade, a growing body of literature has linked nutrition and ...the environment to the microbiome and is now thought to be an important contributor to overall health. This paper reviews the literature from the past 10 years to highlight the influence of environmental factors such as diet, early life adversity and stress in shaping and modifying our microbiome towards health and disease. The review shows that many factors such as the mode of delivery, breast milk, stress, diet and medications can greatly influence the development of our gut microbiome and potentially make us more prone to certain diseases. By incorporating environmental factors into models that study the microbiome in the setting of health and disease, may provide a better understanding of disease and potentially new areas of treatment. To highlight this, we will additionally explore the role of the environment and the microbiome in the development of obesity and functional bowel disorders.
We present several time integration algorithms of second-order accuracy that are numerically simple and effective for nonlinear elastodynamic problems. These algorithms are based on a general ...four-step scheme that has a resemblance to the backward differentiation formulas. We also present an extension to the composite strategy of the Bathe method. Appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity, and less dissipative members of each algorithm have been identified. We demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions, and test these algorithms with several three-dimensional nonlinear elastodynamic problems involving large deformations and rotations, employing St. Venant-Kirchhoff and compressible Neo-Hookean hyperelastic material models. These tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well-known instability.
We investigate the dynamical and statistical features of turbulent Taylor–Couette flow (for a radius ratio 0.5) through three-dimensional direct numerical simulations (DNS) at Reynolds numbers ...ranging from 1000 to 8000. We show that in three-dimensional space the Görtler vortices are randomly distributed in banded regions on the wall, concentrating at the outflow boundaries of Taylor vortex cells, which spread over the entirecylinder surface with increasing Reynolds number. Görtler vortices cause streaky structures that form herringbone-like patterns near the wall. For the Reynolds numbers studied here, the average axial spacing of the streaks is approximately 100 viscous wall units, and the average tilting angle ranges from 16° to 20°. Simulationresults have been compared to the experimental data in the literature, and the flow dynamics and statistics are discussed in detail.
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