A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is uncondition- ally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the weak coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry

We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak-strong uniqueness property of the weak solutions is provided as well

Interaction of a vortex induced by a rotating cylinder with a plane

In this article,we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado

On the Instabilities and Transitions of the Western Boundary Current

We study the stability and dynamic transitions of the western boundary currents in a rectangular closed basin. By reducing the infinite dynamical system to a finite dimensional one via center manifold reduction, we derive a non-dimensional transition number that determines the types of dynamical transition. We show by careful numerical evaluation of the transition number that both continuous transitions (supercritical Hopf bifurcation) and catastrophic transitions (subcritical Hopf bifurcation) can happen at the critical Reynolds number, depending on the aspect ratio and stratification. The regions separating the continuous and catastrophic transitions are delineated on the parameter plane

Error Estimate of a Decoupled Numerical Scheme for the Cahn-Hilliard-Stokes-Darcy System

We analyze a fully discrete finite element numerical scheme for the Cahn-Hilliard-Stokes-Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn-Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn-Hilliard-Stokes-Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229-255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the ℓ ∞(0, T; H1) ∩2 (0, T; H2) norm for the phase variables, as well as in the ℓ ∞ (0, T; H1) ∩ ℓ2 (0, T; H2) norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete ℓ2 (0;T; H3) bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo-Nirenberg inequality in the finite element setting

A Decoupled Unconditionally Stable Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System

We propose a novel decoupled unconditionally stable numerical scheme for the simulation of two-phase flow in a Hele-Shaw cell which is governed by the Cahn-Hilliard-Hele-Shaw system (CHHS) with variable viscosity. The temporal discretization of the Cahn-Hilliard equation is based on a convex-splitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operator-splitting strategy. Thus the computation of the nonlinear Cahn-Hilliard equation is completely decoupled from the update of pressure. Finally, a pressure-stabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme

A Linear Second-Order in Time Unconditionally Energy Stable Finite Element Scheme for a Cahn-Hilliard Phase-Field Model for Two-Phase Incompressible Flow of Variable Densities

We propose a novel second-order BDF time stepping method of variable time step sizes combined with a classical residual-based stabilized finite element spatial discretization using the Streameline-Upwind Petrov-Galerkin (SUPG)/pressure stabilization Petrov-Galerkin (PSPG)/grad-div stabilization for solving the phase-field model for two-phase incompressible flow of different densities and viscosities in the advection dominated regime. In the case of uniform time step size and without extra stabilization, the scheme is shown to satisfy a discrete energy law. Benchmark test of the Rayleigh-Taylor instability under high Reynolds number and Péclect number demonstrates that the scheme captures details of the instability comparable to results in the literature by schemes based on sharp-interface models