Numerical simulation of the viscoelastic flow in a three-dimensional lid-driven cavity using the log-conformation reformulation in OpenFOAM®
Introduction
Simulation of complex viscoelastic flows at high Weissenberg numbers is an outstanding challenge. Fortunately, the last years provided significant progress in developing stable and accurate numerical algorithms [5], [10], [29]. Benchmark problems used in academia to test numerical algorithms include contraction flows [28], [29], flows around spheres [30], [17] and cavity flows [10], [11], [16] besides others.
Most of the work on cavity flows done so far was solely theoretically motivated, which is mainly because remarkably complex flow patterns develop in this very simple geometry [23]. Nevertheless, predicting and understanding the flow inside cavities is also of particular industrial relevance for short-dwell and flexible blade coaters [2]. Numerical simulation of the flow of a Newtonian fluid inside a cavity is straightforward and literature on that is exhausting, e.g. Sheu and Tsai [26], who studied the steady flow topology in a three-dimensional lid-driven cavity with a finite-element method at a Reynolds number of . In contrast, predicting the flow of a viscoelastic fluid in a cavity is demanding and literature about that is few so far. A reason for the little interest may partly be due to the comparatively very low Weissenberg number that can be achieved. For example, Demir [8] studied the transient flow of a viscoelastic fluid governed by the upper convected Criminale–Ericson–Filbey (CEF) equation and the maximum Weissenberg number obtained was only for all Reynolds numbers considered. Similar to others, Demir [8] imposed a uniform velocity at the moving lid, which leads to discontinuities at the two upper corners. This limits the maximum attainable Weissenberg number typically to below 0.2 [13]. More recent works impose regularized boundary conditions in order to have a vanishing velocity and velocity gradient at the two upper corners, see for example Fattal and Kupferman [11]. However, a thin boundary layer close to the lid and a singular point for the conformation tensor at the downstream upper corner remain, which still leave this problem to be difficult and pose an upper limit for the Weissenberg number [16].
Fattal and Kupferman [10], [11] proposed the so-called log-conformation reformulation (LCR), in which a logarithmized evolution equation for the conformation tensor is solved instead of solving the constitutive equation itself. This removes the exponential variation of the stress (and also the conformation tensor) at stagnation points. The new variable (the logarithm of the conformation tensor) can better be approximated by polynomial-based interpolation schemes than the exponentially behaving conformation tensor (or stress) itself. A beneficial side-effect of this technique is that the positive-definiteness of the conformation tensor is always preserved. Fattal and Kupferman tested this technique for a two-dimensional viscoelastic cavity flow of a FENE-CR [10] and Oldroyd-B [11] fluid and they presented stable simulations up to a Weissenberg number of 5. Oscillations in the kinetic energy show the loss of convergence and the onset of a transient flow pattern for Weissenberg numbers above 3. Pan and Hao [25] applied the log-conformation technique in operator-splitting form to their finite-element code and simulated the two-dimensional Stokes flow of a viscoelastic fluid in a cavity up to a Weissenberg number of 3. They solve the logarithmized evolution equation for the conformation tensor on a coarser grid than the velocity, similar as it was done in the work of [10], [11]. This reduces the number of high frequency modes, which, in turn, further stabilizes the solution of the logarithmized equation. A first-order upwind difference scheme was used for discretizing the advection term of the constitutive equation in that work. Upwind differencing, however, is known to be least accurate as it introduces a large amount of numerical diffusion, although this helps to stabilize the solution and significantly increases the maximum achievable Weissenberg number. Oliveira [23] used a finite-volume method to simulate the steady flow and the transient recoil of a FENE-CR fluid with a regularized boundary condition and a comparatively large retardation ratio of 0.79. The maximum Weissenberg number was 10 for creeping flow conditions (). The advection term of the constitutive equation was discretized with the convergent and highly accurate CUBISTA scheme, which is formally of order three [3]. Yapici et al. [33] used a finite-volume method code, in which they also use the upwind scheme for treating the convective term in the constitutive equation. They performed simulations for the flow of an Oldroyd-B fluid in a cavity at different Reynolds numbers and presented results up to a Weissenberg number of 1 for creeping flow conditions.
In this work, the log-conformation reformulation is implemented in the collocated finite-volume based open-source software OpenFOAM®. Results are presented for the flow of an Oldroyd-B fluid in a startup Poiseuille flow and a three-dimensional cavity at creeping flow conditions. The remaining of this work is organized as follows: in Chapter 2 the governing equations are presented and the theory of the log-conformation approach is explained. In the following Chapter 3 the numerical implementation in OpenFOAM® is described. In Chapter 4 the results for the startup of Poiseuille flow and the three-dimensional lid-driven cavity are presented and discussed. Finally, in Chapter 5 this work ends with a summary.
Section snippets
Governing equations
We consider the flow of an incompressible and isothermal viscoelastic fluid, which is governed by the Oldroyd-B constitutive equation [22]. The balance equations are the mass and momentum balancewhere is the velocity, is the density, t is time, p is pressure and is the extra stress tensor, which can be written as the sum of a solvent and polymer contribution
For the solvent contribution the Newtonian law holdswhere is the solvent viscosity. For
Numerical method
In this section we seek to describe the numerical method developed in this work. The above described model is implemented in the software package OpenFOAM® and the developed solver can be thought of a further-development of the initial work by Favero et al. [12] and our continuative paper [14].
Startup Poiseuille flow
First we will validate our numerical algorithm and show the order of convergence with respect to the time-step size and the grid size. Therefore, we consider the startup of a Poiseuille flow of a viscoelastic fluid, for which there is an analytical solution [31]. The sketch of this problem is shown in Fig. 1. We impose periodic boundary conditions on the left and right boundary and discretize the domain with only one cell in x-direction. The length can then be chosen arbitrarily (we use
Summary
In this work, the log-conformation reformulation as proposed by Fattal and Kupferman [10] is adopted and implemented in the finite-volume CFD-software OpenFOAM®. This reformulation allows to extenuate the well-known high Weissenberg number problem, which is omnipresent when pursuing numerical simulations of flows at high elasticities. The implementation is done such as to finally obtain a second-order accurate algorithm in time and space when using an adequate convection scheme such as the
Acknowledgment
Florian Habla gratefully acknowledges the financial support of the TUM Graduate School.
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