# A new two-dimensional, limited, zone-centered tensor artificial viscosity

## Abstract

It is the goal of this paper to describe a fully multidimensional, limited discretization viscosity. The authors prefer discretization to artificial because it arises naturally from discretizing the momentum equation of fluid dynamics. By this is meant that the velocity-dependent stress tensor is not arbitrary, as has been assumed since von Neumann`s paper of 1952, but unavoidable, if a method of weak solution is used to solve the Euler equations. This weak solution method--due to Harten, Lax and van Leer [1985]--shows that Q arises in order to numerically conserve momentum, when the equation is integrated over a finite space and time interval. Understanding the ramifications of this insight lead to the construction of a tensor Q that is the sum of a number of dyadic terms (one per dimension). Thus, the k- and l-directed viscosities of k-l mesh codes is to be understood as the two dyadic terms corresponding to the two eigenvalues of the strain-rate tensor in 2-D (where the assumption had not been explicitly stated that the faces of the control volume--on which the stresses push--are orthogonal or parallel to the eigenvectors of the strain-rate). Calculating velocity jumps within zones to higher order and using limiters is donemore »

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 555532

- Report Number(s):
- LA-UR-97-3237; CONF-9709141-

ON: DE98001316; TRN: AHC29802%%122

- DOE Contract Number:
- W-7405-ENG-36

- Resource Type:
- Conference

- Resource Relation:
- Conference: 5. joint Russian-American computational mathematics conference, Albuquerque, NM (United States), 2-5 Sep 1997; Other Information: PBD: [1997]

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; 66 PHYSICS; HYDRODYNAMICS; TWO-DIMENSIONAL CALCULATIONS; TENSORS; MESH GENERATION; DIFFERENTIAL EQUATIONS; SHOCK WAVES; COMPUTER CODES; EIGENVALUES; EIGENVECTORS; COMPUTER CALCULATIONS

### Citation Formats

```
Clover, M, and Cranfill, C.
```*A new two-dimensional, limited, zone-centered tensor artificial viscosity*. United States: N. p., 1997.
Web.

```
Clover, M, & Cranfill, C.
```*A new two-dimensional, limited, zone-centered tensor artificial viscosity*. United States.

```
Clover, M, and Cranfill, C. Mon .
"A new two-dimensional, limited, zone-centered tensor artificial viscosity". United States. https://www.osti.gov/servlets/purl/555532.
```

```
@article{osti_555532,
```

title = {A new two-dimensional, limited, zone-centered tensor artificial viscosity},

author = {Clover, M and Cranfill, C},

abstractNote = {It is the goal of this paper to describe a fully multidimensional, limited discretization viscosity. The authors prefer discretization to artificial because it arises naturally from discretizing the momentum equation of fluid dynamics. By this is meant that the velocity-dependent stress tensor is not arbitrary, as has been assumed since von Neumann`s paper of 1952, but unavoidable, if a method of weak solution is used to solve the Euler equations. This weak solution method--due to Harten, Lax and van Leer [1985]--shows that Q arises in order to numerically conserve momentum, when the equation is integrated over a finite space and time interval. Understanding the ramifications of this insight lead to the construction of a tensor Q that is the sum of a number of dyadic terms (one per dimension). Thus, the k- and l-directed viscosities of k-l mesh codes is to be understood as the two dyadic terms corresponding to the two eigenvalues of the strain-rate tensor in 2-D (where the assumption had not been explicitly stated that the faces of the control volume--on which the stresses push--are orthogonal or parallel to the eigenvectors of the strain-rate). Calculating velocity jumps within zones to higher order and using limiters is done with the same degree of ad hoc rigor as in Godunov codes. The structure of this paper is to reprise certain results from 1-d analysis of a momentum equation, in order to highlight certain lessons that, when understood, enable one to easily generalize to higher dimension. The authors will also present some results on analytic test problems that this Q produces in a Free-Lagrange hydro code.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1997},

month = {12}

}