On a marching level‐set method for extended discontinuous Galerkin methods for incompressible two‐phase flows: Application to two‐dimensional settings

Smuda, Martin; Kummer, Florian

doi:10.1002/nme.6853

Cited by 9 publications

(10 citation statements)

References 45 publications

(103 reference statements)

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“…Following the XDG discretization in Smuda and Kummer, 19 we propose the following discretization for the moving contact line problem (2), (4) and boundary conditions (6) with the GNBC (7) and (8): Find

(u^{n + 1}, p^{n + 1}) \in 𝕍_{k}^{X} (t^{n + 1})

, such that

\forall (v^{n + 1}, q^{n + 1}) \in 𝕍_{k}^{X} (t^{n + 1})

\begin{align} \frac{1}{d_{0} normalΔ t} m false({boldu}^{n + 1}, {boldu}^{n}, {boldu}^{n prefix- 1}, {boldu}^{n prefix- 2}, {boldv}^{n + 1}, {boldv}^{n}, {boldv}^{n prefix- 1}, {boldv}^{n prefix- 2} false) + c false({boldu}^{n + 1}, {boldu}^{n + 1}, {boldv}^{n + 1} false) + b false(p^{n + 1}, {boldv}^{n + 1} false) prefix- b false(q^{n + 1}, {boldu}^{n + 1} false) \\ prefix- a false({boldu}^{n + 1}, {boldv}^{n + 1} false) prefix- a_{normalS} \end{align}

…”

Section: The Xdg Methods For Moving Contact Line Problemsmentioning

confidence: 99%

“…The evolution algorithm presented in Smuda and Kummer 19 is divided into two stages. In the first stage, the extension velocity is computed monolithic with high‐order accuracy on the cut‐cells

K_{cc}

using the reformulation of the EVP (28) into an elliptic PDE 30 .…”

Section: Interface Evolution and Coupling With Nsementioning

confidence: 99%

“…In context of the XDG method for transient two‐phase flow problems, 19 two level‐set fields are introduced

φ_{evo} \in ℙ_{k} (K_{h}) and φ \in ℙ_{k + 1} (K_{h}) \cap C^{0} (Ω) .

The first one is used for handling the level‐set evolution and is discretized by a standard DG space with the same polynomial degree

k

as the velocity components in

u

. The second one is used for the discretization, since the XDG method requires at least that

φ (x, t) \in C^{0} (Ω)

.…”

Section: Interface Evolution and Coupling With Nsementioning

confidence: 99%

“…The discretization is based on the symmetric interior penalty (SIP) method 18 and the stabilization against small cut‐cells is ensured by cell agglomeration. In Smuda and Kummer, 19 this method is extended to the instationary case, where the XDG adapted moving interface approach by Kummer et al 20 is used for the temporal discretization. A signed‐distance level‐set field is used for the representation of the interface and discretized by a standard DG formulation.…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we adapt the XDG method presented in Smuda and Kummer 19 to allow the direct numerical simulation of moving contact line problems for the two‐dimensional case. For this purpose, we employ the generalized Navier boundary condition for the XDG discretization.…”

Section: Introductionmentioning

confidence: 99%

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The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

Smuda

Kummer

2021

Numerical Methods in Fluids

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In this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.

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(u^{n + 1}, p^{n + 1}) \in 𝕍_{k}^{X} (t^{n + 1})

, such that

\forall (v^{n + 1}, q^{n + 1}) \in 𝕍_{k}^{X} (t^{n + 1})

\begin{align} \frac{1}{d_{0} normalΔ t} m false({boldu}^{n + 1}, {boldu}^{n}, {boldu}^{n prefix- 1}, {boldu}^{n prefix- 2}, {boldv}^{n + 1}, {boldv}^{n}, {boldv}^{n prefix- 1}, {boldv}^{n prefix- 2} false) + c false({boldu}^{n + 1}, {boldu}^{n + 1}, {boldv}^{n + 1} false) + b false(p^{n + 1}, {boldv}^{n + 1} false) prefix- b false(q^{n + 1}, {boldu}^{n + 1} false) \\ prefix- a false({boldu}^{n + 1}, {boldv}^{n + 1} false) prefix- a_{normalS} \end{align}

…”

Section: The Xdg Methods For Moving Contact Line Problemsmentioning

confidence: 99%

K_{cc}

using the reformulation of the EVP (28) into an elliptic PDE 30 .…”

Section: Interface Evolution and Coupling With Nsementioning

confidence: 99%

“…In context of the XDG method for transient two‐phase flow problems, 19 two level‐set fields are introduced

φ_{evo} \in ℙ_{k} (K_{h}) and φ \in ℙ_{k + 1} (K_{h}) \cap C^{0} (Ω) .

The first one is used for handling the level‐set evolution and is discretized by a standard DG space with the same polynomial degree

k

as the velocity components in

u

. The second one is used for the discretization, since the XDG method requires at least that

φ (x, t) \in C^{0} (Ω)

.…”

Section: Interface Evolution and Coupling With Nsementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

Smuda

Kummer

2021

Numerical Methods in Fluids

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Electrochemical Catalytic Synthesis of CH₃OH for In‐situ Resource Utilization

Yang,

Dong,

Zhu

et al. 2023

Chem Eng & Technol

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In space exploration activities, a large amount of materials needs to be carried, which limits the sustainable development of exploration activities. In‐situ resource utilization (ISRU) is an important means to realize resource recycling and continuous space exploration, which converts space resources into oxygen and hydrocarbon fuels. The traditional ISRU in outer space mainly uses high temperature and high pressure to electrolyze water or reduce CO2, having problems such as low conversion efficiency, high energy consumption, and excessive equipment volume. Here, an electrochemical catalytic synthesis technology based on a microfluidic device is proposed, which can convert H2O and CO2 into O2 and organic matter by electrocatalytic method at room temperature and achieve efficient energy and matter conversion. The gas‐liquid mixing and electrochemical reaction were analyzed. A mathematical model of gas‐liquid two‐phase mixing and microfluidic chemical reaction was established. The research results demonstrate the reliability and efficiency of the microfluidic reaction device designed in this paper for ISRU.

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An extended discontinuous Galerkin shock tracking method

Vandergrift,

Kummer

2024

Numerical Methods in Fluids

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In this paper, we introduce a novel high‐order shock tracking method and provide a proof of concept. Our method leverages concepts from implicit shock tracking and extended discontinuous Galerkin methods, primarily designed for solving partial differential equations featuring discontinuities. To address this challenge, we solve a constrained optimization problem aiming at accurately fitting the zero iso‐contour of a level set function to the discontinuities. Additionally, we discuss various robustness measures inspired by both numerical experiments and existing literature. Finally, we showcase the capabilities of our method through a series of two‐dimensional problems, progressively increasing in complexity.

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On a marching level‐set method for extended discontinuous Galerkin methods for incompressible two‐phase flows: Application to two‐dimensional settings

Cited by 9 publications

References 45 publications

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

Electrochemical Catalytic Synthesis of CH₃OH for In‐situ Resource Utilization

An extended discontinuous Galerkin shock tracking method

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On a marching level‐set method for extended discontinuous Galerkin methods for incompressible two‐phase flows: Application to two‐dimensional settings

Cited by 9 publications

References 45 publications

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

Electrochemical Catalytic Synthesis of CH3OH for In‐situ Resource Utilization

An extended discontinuous Galerkin shock tracking method

Contact Info

Product

Resources

About

Electrochemical Catalytic Synthesis of CH₃OH for In‐situ Resource Utilization