# Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model

## Abstract

The well-posedness, convergence and the stability of the two-fluid code has been studied for a long time. Most of the investigations concern the semi-implicit upwind solution scheme for the six equation two-fluid model such as used in RELAP5 3 or TRACE 21. Since the system code, SPACE 2, adopts one more field, a droplet field, it consists of nine equations (3 mass, 3 momentum and 3 energy balance equations) and thus more involved investigations are necessary to confirm the stability and convergence. For this objective, the old issue of the well-posedness, convergence and the stability is revisited and some general guidelines to develop a well-posed numerical multi-fluid model are derived as follows; (1) Hyperbolicity of the corresponding system of partial differential equations is not a necessary condition for the development of a numerical model for multi-phase flow, but whether or not it is hyperbolic can provide guidance relative to initial conditions, boundary conditions, and expected high frequency behavior of the model. (2) A necessary condition for a well-posed numerical model is stability in the von Neumann sense, i.e. growth factor less than 1.0 for the shortest wave-length, 2{Delta}x. (3) The smallest node size used for convergence studies should be ofmore »

- Authors:

- NSSS Engineering and Development, KEPCO EandC, Inc., 150 Deokjin, Yuseong, Daejeon 305-353 (Korea, Republic of)
- School of Nuclear Engineering, Purdue Univ., 400 Central Drive, West Lafayette, IN 47907-2017 (United States)

- Publication Date:

- Research Org.:
- American Nuclear Society, 555 North Kensington Avenue, La Grange Park, IL 60526 (United States)

- OSTI Identifier:
- 22107810

- Resource Type:
- Conference

- Resource Relation:
- Conference: ICAPP '12: 2012 International Congress on Advances in Nuclear Power Plants, Chicago, IL (United States), 24-28 Jun 2012; Other Information: Country of input: France; 22 refs.; Related Information: In: Proceedings of the 2012 International Congress on Advances in Nuclear Power Plants - ICAPP '12| 2799 p.

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 22 GENERAL STUDIES OF NUCLEAR REACTORS; BOUNDARY CONDITIONS; CONVERGENCE; DROPLETS; ENERGY BALANCE; FLOW MODELS; GEOMETRY; GROWTH FACTORS; MULTIPHASE FLOW; NONLINEAR PROBLEMS; NUCLEAR POWER PLANTS; PARTIAL DIFFERENTIAL EQUATIONS; SIMULATION; STABILITY; WAVELENGTHS

### Citation Formats

```
Lee, S. Y., Park, C. E., Hibiki, T., Ishii, M., and Ransom, V. H.
```*Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model*. United States: N. p., 2012.
Web.

```
Lee, S. Y., Park, C. E., Hibiki, T., Ishii, M., & Ransom, V. H.
```*Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model*. United States.

```
Lee, S. Y., Park, C. E., Hibiki, T., Ishii, M., and Ransom, V. H. Sun .
"Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model". United States.
```

```
@article{osti_22107810,
```

title = {Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model},

author = {Lee, S. Y. and Park, C. E. and Hibiki, T. and Ishii, M. and Ransom, V. H.},

abstractNote = {The well-posedness, convergence and the stability of the two-fluid code has been studied for a long time. Most of the investigations concern the semi-implicit upwind solution scheme for the six equation two-fluid model such as used in RELAP5 3 or TRACE 21. Since the system code, SPACE 2, adopts one more field, a droplet field, it consists of nine equations (3 mass, 3 momentum and 3 energy balance equations) and thus more involved investigations are necessary to confirm the stability and convergence. For this objective, the old issue of the well-posedness, convergence and the stability is revisited and some general guidelines to develop a well-posed numerical multi-fluid model are derived as follows; (1) Hyperbolicity of the corresponding system of partial differential equations is not a necessary condition for the development of a numerical model for multi-phase flow, but whether or not it is hyperbolic can provide guidance relative to initial conditions, boundary conditions, and expected high frequency behavior of the model. (2) A necessary condition for a well-posed numerical model is stability in the von Neumann sense, i.e. growth factor less than 1.0 for the shortest wave-length, 2{Delta}x. (3) The smallest node size used for convergence studies should be of the order of the characteristic dimension of the average description, i.e. smaller nodes can be used so long as they do not result in unphysical growth of wave-lengths less than the characteristic dimension. The usual mathematical definition of convergence i.e. the behavior of the solution as the node size approaches zero, is not appropriate for the discrete averaged numerical model, since there is diminished physical meaning to behavior at wavelengths less than the characteristic dimension of the average description. Under these guidelines, dispersion analysis and von Neumann stability analysis are performed for the three field multi-fluid, semi-implicit, upwind numerical model to show that the necessary conditions for well-posedness are met. To study the non-linear stability and the convergence, various runs with the torus problem using the SPACE code are utilized to confirm the frequency cascading effects that augment non-linear stability. A robust mechanism for the flow regime change is also a very important factor for developing non-linearly stable and convergent code. A open pipe flow problem is also simulated to investigate the non-linear stability effects in a flow geometry more typical of real safety code applications. (authors)},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2012},

month = {7}

}