TY - JOUR

T1 - An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

AU - Shirokoff, D.

AU - Rosales, R. R.

N1 - Funding Information:
We would like to thank Matt Ueckermann for many helpful discussions and suggestions. In addition, we greatly appreciate conversations with Alex Marques and J.C. Nave regarding the numerical implementation of the method, and both Alex Chorin and Benjamin Seibold for various comments regarding theoretical considerations. The authors would like to acknowledge the support of the National Science and Engineering Research Council of Canada , the University Carlos 3 de Madrid, Spain, as well as the National Science Foundation . This research was partially supported by an NSERC PGS , and by NSF grant DMS-0813648 .

PY - 2011/9/20

Y1 - 2011/9/20

N2 - Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.

AB - Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.

KW - Incompressible flow

KW - Navier-Stokes

KW - Poisson boundary conditions

KW - Pressure Poisson equation

KW - Projection methods

KW - Staggered grid

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U2 - 10.1016/j.jcp.2011.08.011

DO - 10.1016/j.jcp.2011.08.011

M3 - Article

AN - SCOPUS:80053133014

VL - 230

SP - 8619

EP - 8646

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 23

ER -