An implicit conformation tensor decoupling approach for viscoelastic flow simulation within the monolithic projection framework

Highlights

• Two different pressure-conformation tensor-velocity and conformation tensor-components decoupling procedures are proposed.

• All nonlinear terms in the governing equations are solved without iteration.

• All flow quantities can be resolved without iteration in the present decoupling approach.

• Both linear and nonlinear constitutive equations can be decoupled implicitly by the present procedure.

• Implementation of high-resolution scheme for non-iterative implicit method is presented.

Abstract

The highly nonlinear nature of the system governing equations makes it difficult to simulate viscoelastic flows efficiently. In this paper, an implicit decoupling approach is proposed for the viscoelastic flow simulation with a monolithic projection method. The decoupling approach can be derived from the approximate block factorization at the matrix level, which has been successfully applied into the Newtonian flow simulations. In our work, we extend this approach to decouple the pressure, conformation tensor and velocity from the viscoelastic flow system sequentially. Firstly, the pressure-conformation tensor-velocity decoupling is realized by a two-step approximate factorization. By combining the conformation tensor and velocity together at the first-step factorization, the original efficient pressure Poisson solver for the Newtonian flow simulation can be directly adopted in the present framework for non-Newtonian flow simulation. Secondly, the conformation tensor components are further decoupled from each other by the present component-decoupling approach. Then all quantities, including pressure, conformation tensor and velocity, can be resolved without iteration. Additionally, the accuracy corrector is proposed to access the prior scheme limiter-estimation problem introduced by the implicit high-resolution scheme. We numerically demonstrate that all quantities preserve the second-order accuracy in time and space as expected by the theoretical splitting errors. Finally, several different types of canonical benchmark flow examples are conducted to validate the present solver, including laminar flow, turbulent drag-reducing flow and the rotation of a spherical particle immersed in a sheared viscoelastic fluid.

Introduction

Viscoelastic fluid flows are ubiquitous in natural and industrial areas. Examples of these flows include the complex fluid environment where microorganisms live [1], drilling mud recirculation systems in the process of petroleum exploitation [2], viscoelastic ink jet processes [3] and blood flows through arterial aneurysms [4]. Compared with their Newtonian counterparts, viscoelastic fluid flows exhibit more peculiar features, such as Weissenberg effect (rod climbing), extrudate swelling, laminar secondary flow in non-circular channel, polymer induced turbulent drag-reduction [5], microfluidic chaotic flows introduced by the elastic instability [6], just to name a few. Understanding the mechanisms of the above flow behaviors is important for designing and optimizing the industrial applications involving the viscoelastic flow features. Therefore, from both the fundamental and practical perspectives, developing an accurate and efficient simulation method for viscoelastic flows is a must and substantial importance task, which motivates the present research.

Due to the hyperbolic nature of the constitutive equation and the highly nonlinear coupling between the elastic stress components, how to develop an efficient simulation method for viscoelastic fluid flows has long been seen as one of the major obstacles for computational rheology. There have been many efforts devoted to enhancing the efficiency and robustness of viscoelastic flow simulations. According to the resolution sequence of the system governing equations, the most widely used viscoelastic flow simulation methods can be categorized into two groups: (a) monolithic/coupled-solution methods and (b) segregated/sequential-solution methods.

The viscoelastic flow is a canonical complex system with quantities highly coupled: fluid velocity, pressure and elastic stress are coupled in the momentum equation, and the velocity and elastic stress are interlinked in the constitutive equation. In monolithic-solution methods, the discretization equations governing the coupled system are assembled into a single large sparse matrix, and the system unknowns can be obtained simultaneously based on the large-sparse matrix resolution techniques, such as lower-upper (LU) decomposition or multigrid iterative method [7], [8], [9]. Thus, the monolithic-solution methods are the most straightforward approach to resolve the strongly coupled flow systems, however, these methods generally require a huge computational cost, especially in three-dimensional flow problems [10]. Consequently, monolithic-solution methods are mainly limited into the simple flow simulations on the relatively coarse meshes [11]. Recently, several coupled solvers for viscoelastic flows have been implemented in OpenFOAM [12], [13]. The results indicated that although the monolithic-solution methods are much slower in a single iteration step due to the substantial computational cost of large matrix operations, the overall iteration steps are remarkably reduced for coupled solvers [12]. Furthermore, monolithic-solution methods could ensure the coupling nature between system equations and decrease the explicit contributions [13], thus they can strengthen the numerical stability and allow to adopt a larger time-step to improve the efficiency of the transient flow simulations [12]. Despite such overwhelming advantages, the development of monolithic-solution method is still hindered by the large memory usage and the expensive computation cost. Before using the monolithic or coupled solution methods to resolve the complex viscoelastic flow problems, such as viscoelastic turbulence or multiphase flows, there is much to be tackled, such as the efficient sparse-matrix solving technologies.

As an alternative to the monolithic-solution methods, the segregated-solution methods have been long used in Newtonian and viscoelastic flow simulations. These uncoupled-type methods solve each governing equation separately, thus the system unknowns can be obtained in a specific sequence. The major drawback of these segregated-type methods is that the coupling nature between different flow quantities are weakened, which leads to use a small relaxation factor to solve the equations [13], and increases the computational time, especially for viscoelastic flows. Despite the above shortcomings, the segregated-solution methods are still widely used in complex flow simulations due to their low memory usage. Depending on the solution methods of the uncoupled equations, there are two main types of the segregated-solution methods: iterative-solution type (e.g. SIMPLE-type methods [14]) and non-iterative-solution type algorithms (e.g. projection methods [15] and fractional-step methods [16], [17]).

For iterative segregated-solution methods, there exist abundant works on using this type solvers to resolve viscoelastic flow problems [18], [19], [20], [21], [22]. As indicated by Alves et al. [23], when the interlinks between equations are stiff (e.g. high-Weissenberg number flows), or the nonlinearities are dominant (e.g. high-Reynolds number flows), it might need much more nonlinear iteration steps to obtain a converged solution. Therefore, the efficiency of the iterative segregated-solution methods might be unsatisfactory when simulating complex viscoelastic flows.

As for the other type of the segregated-solution methods, i.e. non-iterative segregated-solution methods, the nonlinear terms in the uncoupled system equations are approximated by the explicit schemes, which make the different unknowns governed by individual equations, so that, the governing equations can be integrated without iteration within a time step. Non-iterative segregated-solution methods have been widely applied in viscoelastic flows considering its high efficiency for unsteady flow calculations. Yu and Kawaguchi [24] used the fractional-step method to simulate the viscoelastic turbulent drag-reducing flow. In their method, the nonlinear terms in the momentum and the constitutive equations were treated explicitly by the Adams–Bashforth scheme (AB2) which is the second-order accuracy in time. Yu et al. [25] decoupled the particulate viscoelastic flow system by a fractional-step time scheme, where the elastic stress, advection, polymer stretching and coiling terms were also discretized by the explicit schemes in time. Esteghamatian and Zaki [26] investigated the flow characteristics of the viscoelastic turbulent channel flow laden with neutrally buoyant spherical particles using a fractional-step algorithm. Similar with the aforementioned research [24], the advection term of the momentum equation was also approximated by the explicit AB2 scheme, while the constitutive equation was resolved by the third-order Runge-Kutta method in their algorithm. Izbassarov et al. [27] simulated the bubbly and particle-laden elastoviscoplastic flows using the projection method. The flow equations were integrated in time with the AB2 scheme. Recently, the constitutive equation represented by log-conformation formulation was implemented in the Smoothed Particle Hydrodynamics (SPH) approach by the projection method [28], and the nonlinear terms were also treated by the explicit Eulerian schemes to avoid iteration when solving system equations.

Although the above explicit non-iterative segregated-solution methods have high efficiency for unsteady flow simulations, the explicit treatment of the nonlinear terms might bring the numerical instabilities into calculation, especially for the complex viscoelastic flows. The computational time step should be restricted by CFL (Courant-Friedrichs-Lewy) number due to stability consideration, which significantly increases the number of computational time steps. In order to improve the calculation robustness and efficiency, some researchers proposed several hybrid methods of fractional-step method and iteration (so-called iterative fractional-step methods), where some nonlinear terms in the governing equations were treated implicitly. Unlike the aforementioned iterative segregated-solution methods, in these hybrid methods, the iteration process was only applied into solving the intermediate velocity equation due to the implicit treatment of convective terms. These iterative fractional-step methods can be found in both Newtonian and non-Newtonian flow simulations. For example, Choi and Moin [29] employed a fractional-step method based on the fully implicit scheme to investigate the effects of computational time steps on the statistical results of Newtonian turbulent channel flow. The Newton-iterative method was adopted to resolve the coupled intermediate velocity components. In the non-Newtonian viscoelastic flow context, D'Avino et al. [30] proposed a two-step decoupling algorithm for simulating the inertial transient viscoelastic flow. In the second step of their method, the Newton-iterative method was utilized to resolve the momentum equation due to the implicit treatment of the convective term. Castillo and Codina [10] designed first, second and third-order fully implicit fractional-step methods for viscoelastic flow simulations at the pure algebraic level. In their methods, an iterative method was applied to linearize the convective term in the momentum equation.

To further improve the calculation efficiency of the iterative fractional-step methods, several strategies were proposed to avoid the nonlinear iteration processes introduced by the fully implicit treatment of nonlinear terms. Rosenfeld [31] suggested a three-level linearization scheme to decouple the momentum equation. Kim et al. [32] proposed a decoupling procedure with the temporal second-order accuracy for the intermediate velocity components based on the matrix approximate factorization method. However, the velocity-components decoupling presented above mainly focused on the Newtonian flows, this component decoupling method was still not widely applied in viscoelastic flow simulations.

To sum up, the related works on the methods of the viscoelastic flow simulation reveal that the development of the viscoelastic flow solvers roughly follows its Newtonian counterpart. Corresponding to the Newtonian solvers, there exist different-type of viscoelastic flow solvers, except for the fully implicit segregated-solution methods without nonlinear iteration, as shown as in Fig. 1. There are two major difficulties limiting the development of these fully implicit non-iterative segregated-solution methods: (a) velocity-elastic stress decoupling and (b) elastic stress/conformation tensor components decoupling. For the first difficulty, the elastic stress-velocity decoupling is still a developing topic of discussion [12]. There are few works on elastic stress-velocity segregation. In [10], the velocity and elastic stress were splitted by the inexact block LU factorization. D'Avino [30] segregated the elastic stress and velocity by reformulating the elastic stress with the current time-step velocity and the previous time-step conformation tensor. For the second difficulty, to the best of our knowledge, there is lack of such procedure for the elastic stress or conformation tensor components decoupling, which now is a part of the present work.

In the present study, an implicit elastic stress/conformation tensor decoupling approach for the viscoelastic flow simulation with the monolithic projection method is proposed at the matrix level. The pressure, elastic stress and velocity are decoupled sequentially by the approximate block LU factorization. Different from Castillo and Codina [10], we will introduce another linearization strategy for nonlinear terms in the momentum and constitutive equation. In the present work, not only the pressure-elastic stress-velocity decoupling is achieved, but also the components of intermediate velocity and conformation tensor can be further decoupled by the proposed procedure. As a result, the decoupled equations are solved directly without iteration, thus the computational cost and memory can be significantly saved.

The work is organized as follows. In Section 2 we introduce the governing equations of viscoelastic flow problems. The numerical discretization schemes and the decoupling procedure for the monolithic projection method are described in detail in Section 3. Numerical results are presented in Section 4, and finally, conclusions are drawn in Section 5.