Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Wittschieber, Stefan; Demkowicz, Leszek; Behr, Marek

doi:10.1016/j.jnnfm.2022.104838

Cited by 11 publications

(4 citation statements)

References 36 publications

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“…The improvement in residual convergence is remarkable when the tangent of the body-force term is taken into consideration for all Ra and numbers. This fact was also observed in [55] for non-Newtonian fluids, when the whole Jacobian is computed with finite differences.…”

Section: Relevance Of the Body-force Term Within The Full Navier-stok...supporting

confidence: 63%

Simulation of Low-Speed Buoyant Flows with a Stabilized Compressible/Incompressible Formulation: The Full Navier–Stokes Approach versus the Boussinesq Model

Hauke

Lanzarote

2022

Algorithms

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This paper compares two strategies to compute buoyancy-driven flows using stabilized methods. Both formulations are based on a unified approach for solving compressible and incompressible flows, which solves the continuity, momentum, and total energy equations in a coupled entropy-consistent way. The first approach introduces the variable density thermodynamics of the liquid or gas without any artificial buoyancy terms, i.e., without applying any approximate models into the Navier–Stokes equations. Furthermore, this formulation holds for flows driven by high temperature differences. Further advantages of this formulation are seen in the fact that it conserves the total energy and it lacks the incompressibility inconsistencies due to volume changes induced by temperature variations. The second strategy uses the Boussinesq approximation to account for temperature-driven forces. This method models the thermal terms in the momentum equation through a temperature-dependent nonlinear source term. Computer examples show that the thermodynamic approach, which does not introduce any artificial terms into the Navier–Stokes equations, is conceptually simpler and, with the incompressible stabilization matrix, attains similar residual convergence with iteration count to methods based on the Boussinesq approximation. For the Boussinesq model, the SUPG and SGS methods are compared, displaying very similar computational behavior. Finally, the VMS a posteriori error estimator is applied to adapt the mesh, helping to achieve better accuracy for the same number of degrees of freedom.

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Section: Relevance Of the Body-force Term Within The Full Navier-stok...supporting

confidence: 63%

Simulation of Low-Speed Buoyant Flows with a Stabilized Compressible/Incompressible Formulation: The Full Navier–Stokes Approach versus the Boussinesq Model

Hauke

Lanzarote

2022

Algorithms

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“…The stress profile for We = 0.3 is in excellent agreement with the stationary results by Westervoß et al 3 Notice that the pressure seen in the figure is relative to the mean outlet pressure p outlet , which is zero in our case because 𝝉 = 0. Table 8 compares the results obtained in this work with those reported by Wittschieber et al, 53 where SUPG stabilization was used. The drag coefficients show excellent agreement for the range of Weissenberg numbers considered here.…”

Section: Creeping Flow Past a Cylindermentioning

confidence: 64%

“…For a more quantitative comparison, we compute the cylinder's drag coefficient

{C}_D

{C}_D=\frac{2}{\nu_0U}{\int}_0^{\pi}\left(2\beta {\nu}_0{\nabla}^{\mathrm{s}}\boldsymbol{u}+\boldsymbol{\sigma} -p\mathbf{I}\right):\left(-\boldsymbol{n}\otimes {\boldsymbol{e}}_1\right)\mathrm{d}s\kern0.3em ,\kern0.60em \mathrm{with}\kern0.60em {\boldsymbol{e}}_1=\left(\begin{array}{c}1\\ {}0\end{array}\right)\kern0.3em .\kern0.5em

Table 8 compares the results obtained in this work with those reported by Wittschieber et al, 53 where SUPG stabilization was used. The drag coefficients show excellent agreement for the range of Weissenberg numbers considered here.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Consistent splitting schemes for incompressible viscoelastic flow problems

Pacheco

Castillo

2023

Numerical Meth Engineering

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Viscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining the Navier–Stokes system with a dynamic tensorial equation, increasing mathematical and computational demands. Hence, fractional‐step methods decoupling the calculation of the flow quantities are an attractive option. In consistent fractional‐step schemes, the splitting of the equations is derived from the continuous level, so that neither mass nor momentum balance is sacrificed. Thus, no corrections or velocity projections are needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees the absence of both numerical boundary layers and splitting errors, enabling high‐order accuracy in space and time. This article introduces the first consistent splitting methods for incompressible flows of viscoelastic fluids, for which arbitrary constitutive laws are allowed. We present the formulation and the algorithm, along with various numerical examples testing their accuracy. First‐, second‐ and third‐order backward‐differentiation schemes are numerically tested, and optimal convergence is confirmed for several spatial and temporal discretizations. Furthermore, good numerical stability is verified in challenging benchmark problems, including steady and time‐dependent solutions.

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