A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium

In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution

A Multigrid Multilevel Monte Carlo Method for Stokes–Darcy Model with Random Hydraulic Conductivity and Beavers–Joseph Condition

A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between these two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions

Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. © 2018 Society for Industrial and Applied Mathematics

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method

A Domain Decomposition Method for the Steady-State Navier-Stokes-Darcy Model with Beavers-Joseph Interface Condition

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In this paper, the well-posedness of the Navier-Stokes-Darcy model with Beavers-Joseph condition is analyzed by using a branch of nonsingular solutions. By following the idea in [Y. Cao et al., Numer. Math., 117 (2011), pp. 601-629], the three physical interface conditions are utilized together to construct the Robin-type boundary conditions on the interface and decouple the two physics which are described by Navier-Stokes and Darcy equations, respectively. Then the corresponding multiphysics DDM is proposed and analyzed. Three numerical experiments using finite elements are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis

Topology Optimization of Convective Cooling System Designs

This research investigates an approach to finding the optimal geometry of convective cooling system structures for the enhancement of cooling performance. To predict the cooling effect of convective heat transfer, flow analysis is performed by solving the Brinkman-penalized Navier-Stokes equation, and the temperature profile is obtained from the homogenized thermal-transport equation. For accurate and cost-effective analysis, stabilized finite element methods (FEM) and the adjoint sensitivity method for the multiphysics system are implemented. Several stabilization methods with different definitions of their stabilization tensors and the Newton-Raphson iteration method are introduced to solve the governing equations. This study investigates numerical instabilities, such as velocity and pressure oscillation at the fluid-solid interfaces, which result from the fact that the non body-conforming mesh for the topology optimization method fails to capture the sharp change in velocity gradient with a high Reynolds number flow. These oscillations are not problematic at the system analysis level, but prevent the design from converging to an optimized shape at the design optimization level, creating element-scale cavities near the solid boundaries. Several stabilization methods are examined for their ability to alleviate the instabilities. The Galerkin/least-square method produces less oscillation in most cases but it is insufficient in resolving the convergence issue. The density and sensitivity filters do not effectively suppress the cavities at the design optimization level, while a move-limit scheme easily prevents this instability without significant increase in computational cost. The topology optimization method is applied to the convective cooling system design, by using the same configuration that was successfully used in designing the Navier-Stokes flow system. The main design purpose is to design a flow channel to maximize cooling efficiency. A numerical issue concerning the behavior of the Brinkman penalization is presented with example designs. The optimizer frequently ignores the Brinkman penalization and creates infeasible designs. To resolve this issue, a multi-objective function that also minimizes pressure drop is suggested. As design examples, 2D and 3D cooling channels are designed by the multi-objective function, and the effect of Reynolds and Prandtl number change is discussed.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91462/1/kjun_1.pd

ParMooN - a modernized program package based on mapped finite elements

{\sc ParMooN} is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of {\sc ParMooN}. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library {\sc PETSc}. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.Comment: partly supported by European Union (EU), Horizon 2020, Marie Sk{\l}odowska-Curie Innovative Training Networks (ITN-EID), MIMESIS, grant number 67571