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A new formulation of Discontinuous Galerkin method for interface capturing approaches
W. Herri, K. Benmoussa, R. Bouydou, D. Yakoubi
TL;DR
Accurate transport of interface-capturing fields is needed to simulate free-surface flows while tracking moving interfaces and sharp fluid-property transitions. The paper reformulates Lesaint–Raviart DG for interface transport and couples it to Navier–Stokes flow with surface tension; the approach is reported stable and accurate on rising-bubble and Rayleigh–Taylor benchmarks.
Problem
Two-fluid simulations need to accurately track moving interfaces while preserving sharp transitions in fluid properties, and interface-transport equations are central to level-set and pseudo-concentration approaches.
Source: §1 · pp. 1–2Method
A reformulated Lesaint–Raviart DG method solves interface transport without explicit upwind element ordering or inflow-boundary identification, alongside Taylor–Hood Navier–Stokes discretization and CSF surface tension.
Source: §1 · p. 2, §2.2 · p. 5Results
The RDG method is reported stable and accurate for both interface representations on rising-bubble and Rayleigh–Taylor benchmarks, with bubble results agreeing well with benchmark quantities.
Source: §4.1.1 · p. 13, §4.1.2 · p. 14, §4.2 · p. 16Takeaways & Limitations
The reported benchmark results support applying the RDG method to other interface-capturing techniques.
Source: §5 · p. 18The authors note that assumptions used to establish mathematical consistency remain restrictive from a practical point of view.
Source: §3.1 · p. 7
Abstract
from arXiv · showhide
In computational fluid dynamics, the numerical simulation of free-surface flows using interface capturing approaches (level set, pseudo-concentration) requires accurate resolution of a transport equation. In the present work, this equation is solved using a new formulation of the Lesaint--Raviart discontinuous Galerkin method, which is particularly suitable for problems without inflow boundaries. The motion is described by the incompressible Navier--Stokes equations with surface tension, discretized using the standard $P2-P1$ Taylor--Hood finite element method in space. Surface tension effects are represented using the so-called Continuum Surface Force model. Finally, the proposed approach is implemented and tested on several benchmarks, including the single rising bubble and the Rayleigh--Taylor instability.
1. Introduction
Two-fluid flow simulations must track a moving interface while preserving sharp changes in fluid properties. The paper motivates a reformulated discontinuous Galerkin method for interface transport and tests it on rising-bubble and Rayleigh–Taylor benchmarks.
1. Introduction
- Accurately tracking moving interfaces while preserving sharp density and viscosity transitions is a central challenge in two-fluid flow simulation.Interface capturing represents the fluids and interface on a fixed domain using a scalar field; the paper focuses on level-set and pseudo-concentration methods.Source: §1 · pp. 1–2
- Standard DG element ordering and inflow-boundary requirements can make the method ill-defined for closed domains or costly to update for unsteady and recirculating flows.The paper motivates a reformulation of Lesaint–Raviart DG that removes explicit upwind ordering and inflow-boundary identification.Source: §1 · p. 2
- The proposed method applies a reformulated Lesaint–Raviart DG scheme to interface transport and couples it with incompressible Navier–Stokes flow and Continuum Surface Force surface tension.The flow equations use a Taylor–Hood finite element method; level-set reinitialization uses a linearized semi-Lagrangian method of characteristics.Source: §1 · p. 2
- The paper evaluates the approach on rising-bubble and Rayleigh–Taylor instability benchmarks.The authors present these problems as verification and validation tests for the proposed approach.Source: §1 · p. 3
2. Modeling of the problem
The model describes two incompressible, immiscible fluids separated by a moving interface, represented by either a level-set function or a pseudo-concentration field. It couples interface transport to Navier–Stokes flow with surface tension.
2. Modeling of the problem
- The modeled domain is partitioned into two fluid subdomains with distinct densities and viscosities, separated by a moving interface.The fluids are treated as incompressible and immiscible, and the interface is assumed smooth.Source: §2 · p. 3
2.1.1. Level set method and its reinitialization equation
- The level-set method represents the interface with the zero contour of a signed-distance field transported by the velocity; reinitialization restores its distance-function behavior.The paper advances reinitialization in pseudo-time with a semi-Lagrangian characteristics method; its advective part uses the characteristic operator and its source term is evaluated explicitly.Source: §2.1.1 · pp. 3–4
2.1.2. Pseudo-concentration method
- Pseudo-concentration uses a continuous marker to distinguish the two fluids, with its 0.5 isoline defining the interface, and transports that marker with the velocity field.The marker is assigned values 0 and 1 in Fluid-1 and Fluid-2, respectively.Source: §2.1.2 · p. 4
2.2. Equations of motion
- The flow equations use a Continuum Surface Force representation of surface tension, with a tensorial formulation that avoids direct curvature computation and treats both interface representations uniformly.The model assumes Newtonian, homogeneous, isothermal, incompressible, immiscible fluids; density and viscosity transitions are defined according to the chosen interface representation.Source: §2.2 · pp. 5–6
3. Numerical method
The section establishes the coupled variational framework and finite-element discretization, then reformulates interface transport to avoid inflow-boundary and element-ordering requirements. It closes with the per-step mesh, flow, and interface update sequence.
3. Numerical method
- The numerical method is set in functional spaces that provide the framework for its variational formulations.The analysis recalls transport-problem existence and regularity results and uses them to establish the coupled problem's mathematical consistency.Source: §3 · p. 6
3.1. Well-posedness of the coupled variational problem
- The coupled problem has an existence result under restrictive assumptions, while the Navier–Stokes variational problem has at least one solution, unique in two dimensions.The cited coupled result gives a local strong solution for sufficiently regular initial data.Source: §3.1 · p. 7
3.2. Discretization of the Navier–Stokes equations
- The Navier–Stokes system uses Taylor–Hood P2–P1 elements and first-order implicit Euler time stepping with semi-implicit convection.The implicit scheme is described as unconditionally stable, and the semi-implicit convective treatment avoids solving a nonlinear problem at every time step.Source: §3.2 · p. 8
3.3. Discretization of the interface equation
- The interface transport uses a Lesaint–Raviart DG formulation recast as a coupled field-and-face-flux system, condensed to a face-based system without element ordering or inflow-boundary identification.The original ordered element solve becomes ill-defined in closed domains or recirculating flows; the reformulation uses static condensation.Source: §3.3 · p. 9
3.4. Summary of the algorithm
- Each time step adapts the mesh, projects the solution fields, updates fluid properties, solves for velocity and pressure, and advances the interface.Level-set simulations add reinitialization to preserve the signed-distance property; pseudo-concentration simulations do not require it.Source: §3.4 · p. 10
4. Numerical experiments
The experiments use rising-bubble and Rayleigh–Taylor benchmarks to evaluate RDG-LS and RDG-PC. Reported comparisons show close benchmark agreement in test case 1 and early test case 2, and matching Rayleigh–Taylor interface evolution.
4.1. Rising bubble in 2D
- The rising-bubble benchmark compares center of mass, rise velocity, and circularity, using a radius-0.25 bubble centered at (0.5, 0.5) in a [0, 1]×[0, 2] domain.The benchmark configuration follows Hysing et al..Source: §4.1 · p. 12
4.1.1. Test case 1
- At t=3, the test-case-1 bubble is nearly spherical with slight base deformation, and its computed outlines are compared with the FreeLIFE benchmark.The comparison includes both RDG-LS and RDG-PC.Source: Figure 5 · p. 13, §4.1.1 · p. 13
- 0.2421 for RDG-LS closely matches the 0.2420 FreeLIFE peak rise velocity, while RDG-PC reaches 0.2409; both reproduce center-of-mass and circularity results closely.The center of mass at t=3 s is 1.0807 for RDG-LS and 1.0778 for RDG-PC, against the benchmark value 1.0798; the circularity difference for RDG-PC is approximately 0.2%.Source: §4.1.1 · pp. 13–14, Figure 6 · p. 14
4.1.2. Test case 2
- In test case 2, benchmark agreement persists to t=1.7, after which results diverge, attributed to sensitivity to numerical approach and mesh.This case has large density and viscosity contrasts and surface tension reduced by approximately 92% relative to test case 1.Source: §4.1.2 · p. 14, Figure 8 · p. 15
4.2. Rayleigh–Taylor instability
- Without surface tension, both methods reproduce the reported Rayleigh–Taylor interface evolution, including late-time Kelvin–Helmholtz mushroom roll-up.The interface evolution compares well with Gao et al. and previously published results.Source: §4.2 · p. 16, Figure 10 · p. 17
- 0.3% maximal relative mass error for RDG-LS without volume correction contrasts with an error on the order of 10^-5 for RDG-PC.The relative mass variation of the heavier fluid is shown for both methods.Source: §4.2 · pp. 16, 18, Figure 11 · p. 18
5. Conclusions
The reformulated DG approach solves interface-advection equations for level-set and pseudo-concentration methods alongside Taylor–Hood Navier–Stokes discretization with CSF surface tension. Across the rising-bubble and Rayleigh–Taylor benchmarks, the authors report stable, accurate results for both interface representations.
5. Conclusions
- The reformulated DG method remains stable and accurate for both level-set and pseudo-concentration representations across the rising-bubble and Rayleigh–Taylor benchmarks.The authors also report that it handles different surface-tension effects and instability phenomena.Source: §5 · p. 18
- The approach combines reformulated DG advection with standard P2−P1 Taylor–Hood discretization of incompressible Navier–Stokes equations and CSF surface tension.Surface tension is integrated into the Navier–Stokes equations as an external force.Source: §5 · p. 18
- The reported benchmark results support applying the RDG method to other interface-capturing techniques, while three-dimensional simulations with high-performance computing remain planned future work.Source: §5 · p. 18