A Second-Order Accurate Pressure-Correction Scheme for Viscous Incompressible Flow

Kan, Jiangming

doi:10.1137/0907059

Cited by 654 publications

(407 citation statements)

References 10 publications

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“…In addition to the cell-center velocities (u i ), the face-center velocities, U i , are computed. The equations are integrated in time using the fractional step method of Van-Kan [46] which consists of three sub-steps. In the first substep of this method, a modified momentum equation is solved and an intermediate velocity u* obtained.…”

Section: Governing Equations and Discretization Schemementioning

confidence: 99%

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

Mittal

Dong

Bozkurttas

et al. 2008

Journal of Computational Physics

1,014

768

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A sharp interface immersed boundary method for simulating incompressible viscous flow past threedimensional immersed bodies is described. The method employs a multi-dimensional ghost-cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three-dimensional, stationary, moving and/or deforming bodies. The complex immersed surfaces are represented by grids consisting of unstructured triangular elements; while the flow is computed on non-uniform Cartesian grids. The paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries. Simulations of a number of canonical two-and three-dimensional flows are used to verify the accuracy and fidelity of the solver over a range of Reynolds numbers. Flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems. Finally two cases inspired from biology with highly complex three-dimensional bodies are simulated in order to demonstrate the versatility of the method.

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Section: Governing Equations and Discretization Schemementioning

confidence: 99%

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

Mittal

Dong

Bozkurttas

et al. 2008

Journal of Computational Physics

1,014

768

View full text Add to dashboard Cite

show abstract

“…We employ a fractional step method similar to that proposed in [22] to integrate the governing equations in time. First, intermediate volume fluxes V q(*) that do not satisfy continuity are calculated by solving implicitly (see below for details) the following momentum equations at the surface centers (we shall denote this step as the momentum step): (21) where V q(n) represents the solution at time step n, and G q is the curvilinear gradient operator expressed as in Eq.…”

Section: Time Integration Schemementioning

confidence: 99%

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

Sotiropoulos

2007

Journal of Computational Physics

358

348

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A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

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“…Stability criteria for the non-stationary convection-diffusion equation often result immediately in stability conditions for numerical methods to compute nonstationary incompressible flows, if the widely used pressure-correction method (Chorin (1968), van Kan (1986) is applied. This is because for the velocity prediction step something very close to the convection-diffusion equation is solved, and the stability analysis boils down to the stability analysis for the convection-diffusion equation, which is the topic of this paper.…”

Section: Introductionmentioning

confidence: 99%

von Neumann stability conditions for the convection-diffusion eqation

Wesseling

1996

IMA J Numer Anal

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A method is presented to easily derive von Neumann stability conditions for a wide variety of time discretization schemes for the convection-diffusion equation. Spatial discretization is by the (c-scheme or the fourth-order central scheme. The use of the method is illustrated by application to multistep, Runge-Kutta and implicit-explicit methods, such as are in current use for flow computations, and for which, with few exceptions, no sufficient von Neumann stability results are available.

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A Second-Order Accurate Pressure-Correction Scheme for Viscous Incompressible Flow

Cited by 654 publications

References 10 publications

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

von Neumann stability conditions for the convection-diffusion eqation

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