High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework

Kuan Shuo Liu, Tony Wen Hann Sheu, Yao Hsin Hwang, Khai Ching Ng

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Owing to the fact that the Poisson equation of pressure for incompressible fluid flow is purely elliptic, it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent Moving Particle with embedded Pressure Mesh (MPPM) method. In the current work, a modified MPPM method known as the Mixed Lagrangian–Eulerian (MLE) method is proposed for solving the incompressible Navier–Stokes equations. In the current velocity–pressure formulation, the momentum and continuity equations are approximated on the moving particles (Lagrangian) and the uniform Cartesian grid points, respectively. Meanwhile, the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles, thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error. In the conventional Moving Particle Semi-implicit (MPS) and MPPM methods, numerical accuracies of the Laplacian and gradient operators are strongly dependent on the regularity of the particle distribution. In some implicit schemes, the gradient and Laplacian terms are of second-order and first-order accuracy, respectively. In the current work, the second-order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform Cartesian grids calculated by using the high-order Combined Compact Difference (CCD) scheme. From the numerical results of Laplacian term approximation by using various numerical schemes, it is shown that the new MLE scheme is at least second-order accurate. The proposed Mixed Lagrangian–Eulerian (MLE) method can be easily applied to simulate fluid flow problems ranging from low to high Reynolds number. It is found that the numerical results compare well with the benchmark solutions. Moreover, it is more accurate than the recently proposed MPPM method.

Original languageEnglish
Pages (from-to)77-101
Number of pages25
JournalComputer Methods in Applied Mechanics and Engineering
Volume325
DOIs
Publication statusPublished - 01 Oct 2017

Fingerprint

mesh
Flow of fluids
Momentum
Derivatives
Poisson equation
fluid flow
grids
Numerical methods
Reynolds number
momentum
gradients
incompressible fluids
continuity equation
high Reynolds number
regularity
convection
formulations
operators
approximation
Convection

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

@article{0d1cb194a0df4fcfa279d1b031b75e82,
title = "High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework",
abstract = "Owing to the fact that the Poisson equation of pressure for incompressible fluid flow is purely elliptic, it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent Moving Particle with embedded Pressure Mesh (MPPM) method. In the current work, a modified MPPM method known as the Mixed Lagrangian–Eulerian (MLE) method is proposed for solving the incompressible Navier–Stokes equations. In the current velocity–pressure formulation, the momentum and continuity equations are approximated on the moving particles (Lagrangian) and the uniform Cartesian grid points, respectively. Meanwhile, the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles, thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error. In the conventional Moving Particle Semi-implicit (MPS) and MPPM methods, numerical accuracies of the Laplacian and gradient operators are strongly dependent on the regularity of the particle distribution. In some implicit schemes, the gradient and Laplacian terms are of second-order and first-order accuracy, respectively. In the current work, the second-order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform Cartesian grids calculated by using the high-order Combined Compact Difference (CCD) scheme. From the numerical results of Laplacian term approximation by using various numerical schemes, it is shown that the new MLE scheme is at least second-order accurate. The proposed Mixed Lagrangian–Eulerian (MLE) method can be easily applied to simulate fluid flow problems ranging from low to high Reynolds number. It is found that the numerical results compare well with the benchmark solutions. Moreover, it is more accurate than the recently proposed MPPM method.",
author = "Liu, {Kuan Shuo} and Sheu, {Tony Wen Hann} and Hwang, {Yao Hsin} and Ng, {Khai Ching}",
year = "2017",
month = "10",
day = "1",
doi = "10.1016/j.cma.2017.07.001",
language = "English",
volume = "325",
pages = "77--101",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0374-2830",
publisher = "Elsevier",

}

High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework. / Liu, Kuan Shuo; Sheu, Tony Wen Hann; Hwang, Yao Hsin; Ng, Khai Ching.

In: Computer Methods in Applied Mechanics and Engineering, Vol. 325, 01.10.2017, p. 77-101.

Research output: Contribution to journalArticle

TY - JOUR

T1 - High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework

AU - Liu, Kuan Shuo

AU - Sheu, Tony Wen Hann

AU - Hwang, Yao Hsin

AU - Ng, Khai Ching

PY - 2017/10/1

Y1 - 2017/10/1

N2 - Owing to the fact that the Poisson equation of pressure for incompressible fluid flow is purely elliptic, it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent Moving Particle with embedded Pressure Mesh (MPPM) method. In the current work, a modified MPPM method known as the Mixed Lagrangian–Eulerian (MLE) method is proposed for solving the incompressible Navier–Stokes equations. In the current velocity–pressure formulation, the momentum and continuity equations are approximated on the moving particles (Lagrangian) and the uniform Cartesian grid points, respectively. Meanwhile, the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles, thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error. In the conventional Moving Particle Semi-implicit (MPS) and MPPM methods, numerical accuracies of the Laplacian and gradient operators are strongly dependent on the regularity of the particle distribution. In some implicit schemes, the gradient and Laplacian terms are of second-order and first-order accuracy, respectively. In the current work, the second-order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform Cartesian grids calculated by using the high-order Combined Compact Difference (CCD) scheme. From the numerical results of Laplacian term approximation by using various numerical schemes, it is shown that the new MLE scheme is at least second-order accurate. The proposed Mixed Lagrangian–Eulerian (MLE) method can be easily applied to simulate fluid flow problems ranging from low to high Reynolds number. It is found that the numerical results compare well with the benchmark solutions. Moreover, it is more accurate than the recently proposed MPPM method.

AB - Owing to the fact that the Poisson equation of pressure for incompressible fluid flow is purely elliptic, it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent Moving Particle with embedded Pressure Mesh (MPPM) method. In the current work, a modified MPPM method known as the Mixed Lagrangian–Eulerian (MLE) method is proposed for solving the incompressible Navier–Stokes equations. In the current velocity–pressure formulation, the momentum and continuity equations are approximated on the moving particles (Lagrangian) and the uniform Cartesian grid points, respectively. Meanwhile, the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles, thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error. In the conventional Moving Particle Semi-implicit (MPS) and MPPM methods, numerical accuracies of the Laplacian and gradient operators are strongly dependent on the regularity of the particle distribution. In some implicit schemes, the gradient and Laplacian terms are of second-order and first-order accuracy, respectively. In the current work, the second-order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform Cartesian grids calculated by using the high-order Combined Compact Difference (CCD) scheme. From the numerical results of Laplacian term approximation by using various numerical schemes, it is shown that the new MLE scheme is at least second-order accurate. The proposed Mixed Lagrangian–Eulerian (MLE) method can be easily applied to simulate fluid flow problems ranging from low to high Reynolds number. It is found that the numerical results compare well with the benchmark solutions. Moreover, it is more accurate than the recently proposed MPPM method.

UR - http://www.scopus.com/inward/record.url?scp=85026363837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85026363837&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2017.07.001

DO - 10.1016/j.cma.2017.07.001

M3 - Article

VL - 325

SP - 77

EP - 101

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -