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# Conservation laws in discrete geometry

## Abstract

The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. Here, we describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- Contributing Org.:
- ,Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA; ,Theoretical Design Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

- OSTI Identifier:
- 1525837

- Report Number(s):
- LA-UR-18-23542

Journal ID: ISSN 1941-4897

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Geometric Mechanics

- Additional Journal Information:
- Journal Volume: 11; Journal Issue: 2; Journal ID: ISSN 1941-4897

- Publisher:
- American Institute of Mathematical Sciences (AIMS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; finite scale; conservation laws

### Citation Formats

```
Margolin, Len G., and Baty, Roy S. Conservation laws in discrete geometry. United States: N. p., 2019.
Web. doi:10.3934/jgm.2019010.
```

```
Margolin, Len G., & Baty, Roy S. Conservation laws in discrete geometry. United States. doi:10.3934/jgm.2019010.
```

```
Margolin, Len G., and Baty, Roy S. Sat .
"Conservation laws in discrete geometry". United States. doi:10.3934/jgm.2019010.
```

```
@article{osti_1525837,
```

title = {Conservation laws in discrete geometry},

author = {Margolin, Len G. and Baty, Roy S.},

abstractNote = {The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. Here, we describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.},

doi = {10.3934/jgm.2019010},

journal = {Journal of Geometric Mechanics},

number = 2,

volume = 11,

place = {United States},

year = {2019},

month = {6}

}

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