# A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates

## Abstract

A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in curvilinear coordinates is presented. The non-staggered-grid method originally developed by C. M. Rhie and W. L. Chow for steady state problems is extended to compute unsteady flows. In the computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the mid-point on their corresponding cell faces. The momentum equations are integrated semi-implicitly by the approximate factorization technique. The intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure Poisson equation, which is solved iteratively with a multigrid method. The compatibility condition of the pressure Poisson equation is satisfied in the same manner as in a staggered-grid method; mass conservation can be satisfied to machine accuracy. The pressure boundary condition is derived from the momentum equations. Solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method.

- Authors:

- (Stanford Univ., CA (United States))

- Publication Date:

- OSTI Identifier:
- 7075984

- Alternate Identifier(s):
- OSTI ID: 7075984

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics; (United States)

- Additional Journal Information:
- Journal Volume: 114:1; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; INCOMPRESSIBLE FLOW; COMPUTERIZED SIMULATION; NAVIER-STOKES EQUATIONS; NUMERICAL SOLUTION; CURVILINEAR COORDINATES; MESH GENERATION; COORDINATES; DIFFERENTIAL EQUATIONS; EQUATIONS; FLUID FLOW; PARTIAL DIFFERENTIAL EQUATIONS; SIMULATION 420400* -- Engineering-- Heat Transfer & Fluid Flow; 990200 -- Mathematics & Computers

### Citation Formats

```
Zang, Yan, Street, R.L., and Koseff, J.R.
```*A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates*. United States: N. p., 1994.
Web. doi:10.1006/jcph.1994.1146.

```
Zang, Yan, Street, R.L., & Koseff, J.R.
```*A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates*. United States. doi:10.1006/jcph.1994.1146.

```
Zang, Yan, Street, R.L., and Koseff, J.R. Thu .
"A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates". United States. doi:10.1006/jcph.1994.1146.
```

```
@article{osti_7075984,
```

title = {A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates},

author = {Zang, Yan and Street, R.L. and Koseff, J.R.},

abstractNote = {A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in curvilinear coordinates is presented. The non-staggered-grid method originally developed by C. M. Rhie and W. L. Chow for steady state problems is extended to compute unsteady flows. In the computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the mid-point on their corresponding cell faces. The momentum equations are integrated semi-implicitly by the approximate factorization technique. The intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure Poisson equation, which is solved iteratively with a multigrid method. The compatibility condition of the pressure Poisson equation is satisfied in the same manner as in a staggered-grid method; mass conservation can be satisfied to machine accuracy. The pressure boundary condition is derived from the momentum equations. Solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method.},

doi = {10.1006/jcph.1994.1146},

journal = {Journal of Computational Physics; (United States)},

issn = {0021-9991},

number = ,

volume = 114:1,

place = {United States},

year = {1994},

month = {9}

}