Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

Comminal, Raphaël; Spangenberg, Jon; Hattel, Jesper Henri

doi:10.1016/j.jnnfm.2015.05.003

Cited by 55 publications

(77 citation statements)

References 76 publications

(117 reference statements)

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“…It also alleviates timestep restrictions and numerical errors due to the pressure-velocity decoupling. Moreover, it avoids possible competing effects between the pressure and the trace of the stress tensor τ in the momentum balance, that are believed to degrade the iterative convergence of segregated solvers [54].…”

Section: Streamfunction Flow Formulationmentioning

confidence: 99%

“…2. The streamfunction equations, and the log-conformation equations (for the viscoelastic fluids), are discretized with the implicit numerical schemes described in Section 3.2; see also [54,55] for more details. The streamfunction and the log-conformation tensors are computed sequentially with successive substitution iterations, until iterative convergence is reached.…”

Section: Overview Of the Algorithmmentioning

confidence: 99%

“…We have essentially used the same discretization scheme as the one proposed in [54], for single-phase viscoelastic fluid flows. In order to discretize the streamfunction equations, we first discretize the momentum equations (3) and the curl operators in Eqs.…”

Section: Discretization Of the Governing Equationsmentioning

confidence: 99%

“…However, the stream-tube method remains difficult to use in flows that contain recirculations [53]. In previous works [54,55], we have shown that a pressure-free pure streamfunction formulation [56][57][58][59][60] can also be used to simulate two-dimensional internal viscoelastic flows. The absence of the pressure variable removes the decoupling errors and the time-step size restrictions due to the pressure correction in the classical velocity-pressure algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…The absence of the pressure variable removes the decoupling errors and the time-step size restrictions due to the pressure correction in the classical velocity-pressure algorithms. The numerical scheme presented in [54,55] used the log-conformation representation of the viscoelastic constitutive models, proposed by Fattal and Kupferman [61,62], to tackle the numerical instabilities occurring when the magnitude of the elastic stresses exceeds a critical value. Moreover, a preliminary coupling of the streamfunction formulation with the VOF method was presented in [63], but only for the case of Newtonian two-phase flows.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

Comminal

Pimenta

Hattel

et al. 2018

Journal of Non-Newtonian Fluid Mechanics

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. Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods. Journal of Non-Newtonian Fluid Mechanics, 252, 1-18. DOI: 10.1016/j.jnnfm.2017 A B S T R A C TWe present an Eulerian free-surface flow solver for incompressible pseudoplastic and viscoelastic non-Newtonian fluids. The free-surface flow solver is based on the streamfunction flow formulation and the volume-of-fluid method. The streamfunction solver computes the vector potential of a solenoidal velocity field, which ensures by construction the mass conservation of the solution, and removes the pressure unknown. Pseudoplastic liquids are modelled with a Carreau model. The viscoelastic fluids are governed by differential constitutive models reformulated with the log-conformation approach, in order to preserve the positive-definiteness of the conformation tensor, and to circumvent the high Weissenberg number problem. The volume fraction of the fluid is advected with a geometric conservative unsplit scheme that preserves a sharp interface representation. For the sake of comparison, we also implemented an algebraic advection scheme for the liquid volume fraction. The proposed numerical method is tested by simulating the planar extrudate swell with the Carreau, Oldroyd-B and Giesekus constitutive models. The swell ratio of the extrudates are compared with the data available in the literature, as well as with numerical simulations performed with the open-source rheoTool toolbox in OpenFOAM ® . While the simulations of the generalized Newtonian fluids achieved mesh independence for all the methods tested, the flow simulations of the viscoelastic fluids are more sensitive to mesh refinement and the choice of numerical scheme. Moreover, the simulations of Oldroyd-B fluid flows above a critical Weissenberg number are prone to artificial surface instabilities. These numerical artifacts are due to discretization errors within the Eulerian surface-capturing method. However, the numerical issues arise from the stress singularity at the die exit corner, and the unphysical predictions of the Oldroyd-B model in the skin layer of the extrudate after the die exit, where large extensional deformations occur.

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Section: Streamfunction Flow Formulationmentioning

confidence: 99%

Section: Overview Of the Algorithmmentioning

confidence: 99%

Section: Discretization Of the Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

Comminal

Pimenta

Hattel

et al. 2018

Journal of Non-Newtonian Fluid Mechanics

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Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

Gao

Ouyang

Zhou

2017

Numerical Methods in Fluids

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Summary In this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.

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A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

Ata

Şahin

2019

Numerical Methods in Fluids

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Summary A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face‐centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence‐free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large‐scale algebraic linear equations are solved in a fully coupled manner using the one‐ and two‐level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid‐driven cavity and backward facing step benchmark problems in two and three dimensions. The lid‐driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.

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Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

Cited by 55 publications

References 76 publications

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

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