An Adaptive Viscoelastic Flow Solver with Template Based Mesh

The Eighth International Conference on Computational Fluid Dynamics (ICCFD8), Chengdu, Sichuan,China, July 14-18, 2014

ICCFD8-2014-0429

An Adaptive Viscoelastic Flow Solver with Template Based Mesh Renement Evren ONER1 & Mehmet SAHIN1∗ 1

Faculty of Aeronautics and Astronautics, Astronautical Engineering Department, Istanbul Technical University, Maslak, Istanbul, 34469, TURKEY

Abstract: The parallel large-scale unstructured nite volume method algorithm Visco-Solve has

been extended for dynamic mesh adaptation using a template based mesh adaptation technique with conservative remapping. The dynamic mesh adaptation is carried out by implementing a template based conformal mesh renement/coarsening for quadrilateral and hexahedral elements. A conservative second-order solution remapping technique is used between the source and target meshes in order to transfer solution elds. The numerical algorithm is tested for the classical benchmark problems such as the ow of an OldroydB uid past a conned circular cylinder in a channel. Keywords:

Volume.

Mesh Adaptation, Conservative Remapping, Viscoelastic Flows, Unstructured Finite

1 Introduction The numerical solution of the viscoelastic uid ows are rather challenging in term of accuracy, stability, convergence and required computer power for both steady-state simulations and fully implicit time integration of the incompressible viscoelastic uid ow equations. In the past two decades, considerable eort has been given to the development of robust and stable numerical algorithms. In order to enhance the numerical stability, Perera and Walters [1] introduced the idea of elastic viscous split stress (EVSS) formulation in a nite dierence method. Guenette and Fortin [2] proposed the so-called discrete elastic viscous split stress (DEVSS) method in a mixed FEM implementation. Sun et al. [3] proposed an adaptive viscoelastic stress splitting formulation (AVSS) and its applications: the streamline integration (AVSS/SI) and the streamline upwind Petrov–Galerkin (AVSS/SUPG) methods. Oliveira et al. [4] developed a collocated nite volume method on non-orthogonal grids; the velocity–stress–pressure decoupling was removed by using an interpolation similar to that of Rhie and Chow [5]. Although signicant development has been made for robust and stable numerical algorithms for the solution of viscoelastic uid ows, most numerical methods lose convergence at small or moderate Weissenberg numbers, limiting their applications due to the so-called High Weissenberg Number Problem (HWNP). Recently, a log representation of the conformation tensor was proposed by Fattal and Kupferman [6]. In this approach, the governing constitutive equation is written in terms of the logarithm of the conformation tensor. This representation ensures the positive deniteness of the conformation tensor and captures sharp elastic stress layers which are exponential in nature. However, the accurate prediction of sharp elastic stress layers is highly demanding because they are required to resolve physical features such as geometric singularities, viscoelastic boundary layers and wakes. These features are generally associated with steep gradients in the ow variables, embedded in or adjacent to regions where these ow variables change more smoothly. An adaptive mesh technique takes advantage of these low gradient regions in order to avoid a ne meshing of the entire computational domain. Therefore, mesh adaptation is a useful technique for increasing accuracy at a lower computational cost. Considerable eort has been ∗ Corresponding

author's email: [email protected].

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given to the implementation of adaptive procedures and a posteriori error estimates for nite element and nite volume methods. Mutlu et al. [7] used an error estimates based on velocity gradients and an h− renement based on a Delaunay procedure for an Oldroyd-B and a Phan-ThienlTanner model. Chauviere and Owens [8] used an error indicator in conjunction with a p−adaptive spectral element method to solve the benchmark problem of an Oldroyd-B uid past a sphere in a tube. Roquet and Saramito [9]combined a high order mixed nite element approximation, an anisotropic auto-adaptive mesh procedure, and the augmented Lagrangian method to solve the Bingham uid around a cylinder falling at constant speed between two parallel plates. Guénette et al. [10] combined a nite element method based on log-conformation with an anisotropic adaptive remeshing technique for viscoelastic uid ows. The authors reported mesh convergence up to W e = 0.7 for the classical benchmark problem of an Oldroyd-B uid past a cylinder in a conned channel. The objective of the current study is to carry out adaptive parallel large-scale calculations using the unstructured nite volume algorithm Visco-Solve given in [11] in conjunction with a template based dynamic mesh adaptation technique on conformal unstructured quadrilateral/hexahedral elements in order to solve the viscoelastic uid ows at relatively high Weissenberg numbers. The numerical algorithm based on the side-centered nite volume method where the velocity vector components are dened at the mid-point of each cell face, while the pressure term and the extra stress tensor are dened at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modications in order to enhance the pressure-velocity-stress coupling. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure elds by solution of a generalized Stokes problem as in the work of Caola et al. [12]. The numerical algorithm also uses the log-conformation representation proposed in [6] in order improve the limiting Weissenberg numbers. In here, the Stokes-like system is solved using the FGMRES Krylov iterative method preconditioned with non-nested multigrid method. The present multilevel preconditioner for the Stokes system is essential for parallel scalable viscoelastic ow computations. Because the solution of Stokes ow is a serious bottleneck in performing parallel large-scale viscoelastic numerical simulations. The task in conformal quadrilateral/hexahedral mesh renement is to nd a transition from the rened part to the coarse part of the mesh such that the resulting mesh has no hanging nodes. While it is possible to use point insertion (Delaunay-type algorithms) or advancing-front methods to rene triangular and tetrahedral meshes [15], it is not possible to rene quadrilateral and hexahedral meshes by these methods without creating triangular or tetrahedral elements in the process. The inability to use point insertion and advancing-front type algorithms limits the number of approaches for hexahedral mesh renement and makes it quite dicult to implement. One widely used technique to solve the conformal hexahedral mesh renement problem is the template-based hierarchical renement method proposed by Schneiders in 1996 [16]. In template based renement, various sets of templates that dene possible element divisions are used to avoid hanging nodes and maintain the connectivity between the elements in the coarse and the rened level. Schneiders scheme in its original form has been used by several researchers [20, 18] but its inability to rene concave regions eciently lead to the introduction of new set of templates by other researchers. Garimella proposed new set of templates for an edge based marking scheme along with a coarsening procedure for quadrilateral elements [19]. Zhang and Zhao [20] dened new set of hexahedral templates to overcome the concavity problem of Schneiders' templates [20]. Their approach uses two templates from Schneiders and introduces four new templates. Although their results show that the propagation of renement regions into unrened regions is avoided, there is no detailed description of the templates they used. The works of Ito et al. [21] and Sun et al. [22] are similar to that of Zhang and Zao [20] but their templates are described in detail. The templates dened in the work of Schneiders [16] and Ito et al. [21] are used in this study. The use of an adaptive renement technique requires an interpolation algorithm in order to transfer the solution elds from source mesh to target mesh. For a proper coupling between the ow solver and the renement module, the remapping technique has to be conservative and suciently accurate due to stability restrictions. Various approaches has been proposed for conservative solution remapping in the literature [26, 24, 25, 23]. In the current study, the approach of Menon [27] based on the supermesh algorithm of Farrell et al. [28] are used to transfer the cell centered solutions elds. The approach has been extended for the face/edge centered data and the vertex based data for nite volume formulations.

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2 Governing Equations and Finite Volume Method The governing equations for three-dimensional unsteady ow of an incompressible and isothermal Oldroyd-B uid can be written in dimensionless form as follows: the continuity equation

−∇ · u = 0, the momentum equations

 Re

 ∂u + (u · ∇)u + ∇p = β∇2 u + ∇ · T ∂t

and the constitutive equation for the Oldroyd-B model   ∂T > We + (u · ∇)T − (∇u) · T − T · ∇u = (1 − β)(∇u + ∇u> ) − T. ∂t

(1)

(2)

(3)

In these equations u represents the velocity vector, p is the pressure and T is the extra stress tensor. The dimensionless parameters are the Reynolds number Re, the Weissenberg number W e and the viscosity ratio β . The above governing equations are discretized using the side-centered nite volume method on conformal unstructured quadrilateral/hexahedral meshes as described in detail in [11]. In this approach, the velocity vector components are dened at the mid-point of each cell face, while the pressure term and the extra stress tensor are dened at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modications in order to enhance the pressurevelocity-stress coupling. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure elds by solution of a generalized Stokes problem as in the work of Caola et al. [12]. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method [29] with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system [11]. The present multilevel preconditioner for the Stokes system is essential for parallel scalable viscoelastic ow computations. This is because, as it is well known, one-level methods lead to non-scalable solvers since they cause increase in the number of iterations as the number of sub-domains is increased. However, it is not possible to apply the standard multigrid methods with classical smoothing techniques (e.g., Jacobi, Gauss-Seidel) for the coupled iterative solution of the momentum and continuity equations because of the zero-block in the saddle point problem. In order to avoid the zero-block in the saddle point problem, we use an upper triangular right preconditioner which results in a scaled discrete Laplacian instead of a zero block in the original system. The log-conformation representation proposed in [6] has been implemented in order to simulate the three-dimensional viscoelastic instabilities at high Weissenberg numbers. The implementation of the preconditioned Krylov subspace algorithm, matrix-matrix multiplication and the multilevel preconditioner were carried out using the PETSc software package [13] developed at the Argonne National Laboratories for improving the eciency of the parallel code. For the implementation details of the current algorithm, the reader is referred to the paper by Sahin [11].

3 Template Based Mesh Renement One widely used technique to solve the conformal hexahedral mesh renement problem is the templatebased hierarchical renement method proposed by Schneiders in 1996 [16]. In template-based renement, various sets of templates that dene possible element divisions are used to avoid hanging nodes and maintain the connectivity between the elements in the coarse and the rened level. Renement starts with marking elements to be rened. This renement information is then propagated from elements to nodes and an appropriate template is applied based on the number of marked nodes. Appropriate template for an element is then decided by examining its marked nodes. After the propagation stage, the mesh is coarsened where the target element levels are lower than their current level and rened where the target element levels are higher than their current level. During both coarsening and renement, the target renement levels are adjusted so that there is at most one level dierence between neighboring elements. The one-level dierence rule ensures that the mesh is smoothly graded. In order to perform multi-level renement and coarsening 3

[a]

[b] Figure 1: Two-dimensional renement templates used by Schneiders (1996) [a] and Ito et al. (2009) [b] . on a given mesh, a hieararchical structure of mesh elements needs to be stored. This is achieved by using a tree structure. The algorithm is structured so that dierent template sets dened by dierent researchers can be used. Figure 1 shows renement templates of Schneiders [16] and Ito et. [21] which are implemented in the current work. The three-dimensional templates proposed by Schneiders [16] and Ito et. [21] are also implemented and the templates are provide in Figure 2. However, Schneiders' scheme in its original form has a problem rening concave regions eciently. Because only four templates are proposed and if a hexahedral element has a dierent node marking pattern, it has to be converted to one of the four templates. As a result, renement of concave regions requires renement of the entire domain or most parts of the domain.

[a]

[b] Figure 2: Three-dimensional renement templates used by Schneiders (1996) [a] and Ito et al. (2009) [b] .

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4 Conservative Interpolation The use of adaptive renement algorithm requires an interpolation (remapping) of solution variables from the source mesh to the adapted target mesh. However, the interpolation should be suciently accurate and conservative due to the stability restrictions. In this study, the approach used by Menon and Schmidt [27] which is extension of the supermesh approach proposed by Farrell et al. [28]. In the supermesh approach, a logical mesh (supermesh) is constructed which has the following features:

• Supermesh must contain nodes and preserve the edges of both source and target meshes. • Intersection volume of every cell in the supermesh must be either zero or equal the volume of supermesh cell. The supermesh acts as an indermediate mesh for solution transfer. Figure 3 shows an example of supermesh in 2D. Here (a) is the source mesh, (b) is the target mesh and (c-d) are the construction of source and target meshes from the supermesh. After decomposition of the initial mesh to triangles/tetrahedrals, a method to identify intersecting pairs of elements of source and target meshes is required. Since template based mesh renement is hierarchical in nature, this identication step is straightforward. Once the intersection values are dened, the cell centroid values are interpolated using the least square interpolation. Finally, the supermesh centroid values are summed to construct the target element centroid values. We refer the paper by Menon and Schmidt [27] showing that the second-order interpolation will exactly conserve source and target integral values. Tis approach is used to transfer extra stress tensor components between the source and target meshes. The supermesh construction in 3D is shown in Figure 4. Recently, this approach has been extended to the side-centered data and it has been used to transfer the velocity vector components.

Figure 3: Two-dimensional supermesh construction..

5 Numerical Experiments For this benchmark problem we consider a circular cylinder of radius R positioned symmetrically between two parallel plates separated by a distance 2H . The blockage ratio R/H is set to 0.5 and the computational domain extends a distance 12R upstream and downstream of the cylinder. The dimensionless parameters are the Reynolds number Re = ρhU iR/η , the Weissenberg number W e = λhU i/R and the viscosity ratio β = ηs /η . The physical parameters are the density ρ, the average velocity at the inlet hU i, the relaxation time λ, the zero-shear-rate viscosity of the uid η and the solvent viscosity ηs . The viscosity ratio β is chosen to be 0.59, which is the value used in the benchmarks for the Oldroyd-B uid. The fully developed 5

Figure 4: Three-dimensional supermesh construction. channel velocity prole is imposed at the inow boundary. At the outow the boundary conditions are set to traction-free condition. No-slip velocity boundary conditions are imposed on the cylinder surface and the channel lateral walls. It is known that the problem is very dicult due to the very thin extra stress boundary layer on the cylinder surface and in the wake behind the cylinder. In order to capture the sharp gradients within the viscoelastic wake behind the cylinder (along the channel symmetry line) and the viscoelastic boundary layer over the cylinder surface, a sensor function is used to dene the elements which require mesh renement. The current sensor function based on the undivided dierence of von Mises stress for neighbouring cells. The von Mises stress value is give by 1 q 2 + T 2 + 6T σ = √ (Txx − Tyy )2 + Txx (4) xy yy 2 and the sensor function becomes

f (σi ) = max(|σi − σj |)

(5)

where j is the neighbouring element centroid value. The sensor function values f (σi ) above a certain threshold value are marked for the renement. In the current renement the maximum number of the renement is limited to 4. The several snapshots of the dynamic mesh adaptation and the computed Txx for an Oldroyd-B uid past a conned cylinder are shown in Figure 5 at W e = 0.7. The computational mesh captures the extremely large Txx within the viscoelastic wake behind the cylinder. However, we have problem with the very thin sharp gradients just behind the cylinder which is extremely dicult to capture due to the rather coarse initial mesh used here. Although one can reduce the threshold value, this increases the number of elements drastically.

6 Conclusion and Future Work A parallel unstructured nite volume method algorithm Visco-Solve in conjunction with the dynamic mesh renement and the conservative interpolation has been described for large-scale simulation of viscoelastic uid ows. The numerical algorithm based on the side-centered nite volume method where the velocity 6

[t=5]

[t=10]

[t=15]

[t=50] Figure 5: The dynamic mesh adaptation and the computed Txx for an Oldroyd-B uid past a conned cylinder at Re = 0, W e = 0.7 and β = 0.59. vector components are dened at the mid-point of each cell face, while the pressure term and the extra stress tensor are dened at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modications in order to enhance the pressure-velocitystress coupling. The dynamic mesh adaptation is carried out by implementing a template based conformal mesh renement/coarsening proposed by Schneiders [16] and Ito et. [21] for quadrilateral and hexahedral elements. A conservative second-order solution remapping technique based on Farrell et al. [28] is used between the source and target meshes. Recently, this approach has been extended to the side-centered data and it has been used to transfer the velocity vector components.

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Acknowledgments

The author gratefully acknowledge the use of the computing resources provided by the National Center for High Performance Computing of Turkey (UYBHM) under Grant No. 10752009 and the computing facilities at TUBITAK ULAKBIM, High Performance and Grid Computing Center.

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