# Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk

## Abstract

An efficient and accurate algorithm for solving the two-dimensional (2D) incompressible Navier-Stokes equations on a disk with no-slip boundary conditions is described. The vorticity-stream function formulation of these equations is used, and spatially the vorticity and stream functions are expressed as Fourier-Chebyshev expansions. The Poisson and Helmholtz equations which arise from the implicit-explicit time marching scheme are solved as banded systems using a postconditioned spectral {tau}-method. The polar coordinate singularity is handled by expanding fields radically over the entire diameter using a parity modified Chebyshev series and building partial regularity into the vorticity. The no-slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint. Significant grains in run times were realized by parallelizing the code in message passage interface (MPI).

- Authors:

- Publication Date:

- Research Org.:
- Los Alamos National Lab., NM (US)

- OSTI Identifier:
- 20015657

- Resource Type:
- Journal Article

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 21; Journal Issue: 1; Other Information: PBD: Sep 1999; Journal ID: ISSN 1064-8275

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; NAVIER-STOKES EQUATIONS; ALGORITHMS; TWO-DIMENSIONAL CALCULATIONS; BOUNDARY CONDITIONS; SINGULARITY; CALCULATION METHODS

### Citation Formats

```
Torres, D.J., and Coutsias, E.A.
```*Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk*. United States: N. p., 1999.
Web. doi:10.1137/S1064827597330157.

```
Torres, D.J., & Coutsias, E.A.
```*Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk*. United States. doi:10.1137/S1064827597330157.

```
Torres, D.J., and Coutsias, E.A. Wed .
"Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk". United States. doi:10.1137/S1064827597330157.
```

```
@article{osti_20015657,
```

title = {Pseudospectral solution of the two-dimensional Navier-Stokes equations in a disk},

author = {Torres, D.J. and Coutsias, E.A.},

abstractNote = {An efficient and accurate algorithm for solving the two-dimensional (2D) incompressible Navier-Stokes equations on a disk with no-slip boundary conditions is described. The vorticity-stream function formulation of these equations is used, and spatially the vorticity and stream functions are expressed as Fourier-Chebyshev expansions. The Poisson and Helmholtz equations which arise from the implicit-explicit time marching scheme are solved as banded systems using a postconditioned spectral {tau}-method. The polar coordinate singularity is handled by expanding fields radically over the entire diameter using a parity modified Chebyshev series and building partial regularity into the vorticity. The no-slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint. Significant grains in run times were realized by parallelizing the code in message passage interface (MPI).},

doi = {10.1137/S1064827597330157},

journal = {SIAM Journal on Scientific Computing},

issn = {1064-8275},

number = 1,

volume = 21,

place = {United States},

year = {1999},

month = {9}

}