A coupled finite-volume solver for numerical simulation of electrically-driven flows

Abstract

The accuracy and stability of implicit CFD codes are frequently impaired by the decoupling between variables, which can ultimately lead to numerical divergence.Coupled solvers, which solve all the governing equations simultaneously, have the potential to fix this problem. In this work, we report the implementation of coupled solvers for transient and steady-state electrically-driven flow simulations in the finitevolumes framework. The numerical method, developed in OpenFOAM ® , is generic for Newtonian and viscoelastic fluids and is formulated for the Poisson-Nernst-Planck and Poisson-Boltzmann models. The resulting coupled systems of equations are solved efficiently with PETSc library. The performance of the coupled solvers is assessed in two test cases: induced-charge electroosmosis of a Newtonian fluid around a cylinder; electroosmotic flow of a PTT viscoelastic fluid in a contraction/expansion microchannel. The coupled solvers are more accurate in transient simulations and allow the use of larger time-steps without numerical divergence. For steady-state simulations, the coupled solvers converge in fewer iterations than segregated solvers. Although coupled solvers are much slower in a per time-step basis, the overall speedup factor obtained in this study reached a maximum value of ~100, where the highest factors have been obtained with semi-coupled solvers, which drop some coupling terms between equations. While further research is needed to improve the efficiency of the matrix solving stage, coupled solvers are already superior to segregated solvers in a number of cases.

Numerical Methods for Viscoelastic Fluid Flows

“…These coupled equations prompt the question of whether it is desirable to solve the whole set, or parts of it, in a direct way, thus allowing the solution for u and τ to be obtained simultaneously. While in the past, FEM studies in relatively coarse meshes were often based on such direct methods (e.g., Marchal & Crochet 1987, King et al 1988, there is presently a tendency to apply the same procedures in the FVM context even for much finer meshes (Fernandes et al 2019, Pimenta & Alves 2019).…”

“…Oliveira (2017) reported that the amplification factor for the reduced stress (τ/ f ) formulation with FENE-CR or FENE-P models is much smaller than for the actual stress by a factor of t/λ, thus explaining the reduction in the number of global outer iterations within a time step t (to about three or four iterations). Recently, Pimenta & Alves (2019) showed that although coupled solvers are slower on a per-time-step basis, they can be made significantly faster using semicoupled solvers applied to electrically driven flows of viscoelastic fluids.…”

A Fully Implicit Log-Conformation Tensor Coupled Algorithm for the Solution of Incompressible Non-Isothermal Viscoelastic Flows

“…For the solution of the global system of discretized algebraic equations, it is fundamental that an efficient linear solver is used to obtain the best overall convergence. In this work, the iterative solver Bi-Conjugate Gradient Stabilized (BiCGStab) [ 50 ], combined with an LU preconditioner, was used to retrieve the solution of the global system of discretized algebraic equations (see detailed discussion in Pimenta and Alves [ 30 ]). The initial residual for each iteration is evaluated based on the current values of the field, before solving the block-coupled system.…”

“…Spanjaards et al [ 23 ] performed a 3D transient non-isothermal simulation to predict the extrudate shape of viscoelastic fluids emerging from an asymmetric keyhole-shaped die. However, the current state-of-the-art codes depend on iterative algorithms, such as the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) procedure [ 24 ], which are known to delay the convergence of the problem of interest when compared to monolithic or coupled algorithms [ 25 , 26 , 27 , 28 , 29 , 30 ]. The iterative algorithms, also known as segregated algorithms, are characterized to provide a separate solution of the linear momentum, mass, viscoelastic polymer stress tensor and energy conservation equations, which are then iterated until convergence.…”

Modelling of Droplet Dynamics in Strong Electric Fields

“…An electroquasistatic (EQS) model using a sharp interface approach for the simulation of droplet oscillations in electric fields was described by Songoro et al [28]. Recently, more sophisticated models based on the diffuse-interface description have been introduced, taking into account charge relaxation effects by explicitly solving the charge conservation equation [16,24]. Most of these works, however, focus on the electrokinetics of charged species in electrolytic solvents rather than on capillary droplet dynamics.…”