An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations

George Bourantas, Benjamin Zwick, Grand Joldes, Vassilios C. Loukopoulos, Angus Tavner, Adam Wittek, Karol Miller

Research output: Contribution to journalArticle

Abstract

We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.
Original languageEnglish
Article number164
Pages (from-to)1-32
JournalFluids
Volume4
Issue number3
DOIs
Publication statusPublished - 3 Sep 2019

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Navier Stokes equations
Geometry
Computational efficiency
Vorticity
Ducts
Interpolation
Derivatives

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title = "An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations",
abstract = "We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.",
keywords = "transient incompressible Navier-Stokes; meshless point collocation method; stream function-vorticity formulation; strong form; explicit time integration",
author = "George Bourantas and Benjamin Zwick and Grand Joldes and Loukopoulos, {Vassilios C.} and Angus Tavner and Adam Wittek and Karol Miller",
year = "2019",
month = "9",
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An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations. / Bourantas, George; Zwick, Benjamin; Joldes, Grand; Loukopoulos, Vassilios C.; Tavner, Angus; Wittek, Adam; Miller, Karol.

In: Fluids, Vol. 4, No. 3, 164, 03.09.2019, p. 1-32.

Research output: Contribution to journalArticle

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T1 - An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations

AU - Bourantas, George

AU - Zwick, Benjamin

AU - Joldes, Grand

AU - Loukopoulos, Vassilios C.

AU - Tavner, Angus

AU - Wittek, Adam

AU - Miller, Karol

PY - 2019/9/3

Y1 - 2019/9/3

N2 - We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.

AB - We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.

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