# von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion

## Abstract

A new approach to obtain the closure formulas for the von Kármán–Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics is characterized by the finite scale separation vector which is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von Kármán–Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, we show that the times of evolution of the developed kinetic energy and temperature spectra are finite quantities which depend on the initial conditions.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22560322

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 368; Journal Issue: Complete; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; CORRELATIONS; EQUATIONS; FLUIDS; KINETIC ENERGY; LAGRANGIAN FUNCTION; LIOUVILLE THEOREM; LYAPUNOV METHOD; TURBULENCE; VECTORS; VELOCITY

### Citation Formats

```
Divitiis, Nicola de, E-mail: n.dedivitiis@gmail.com.
```*von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion*. United States: N. p., 2016.
Web. doi:10.1016/J.AOP.2016.02.010.

```
Divitiis, Nicola de, E-mail: n.dedivitiis@gmail.com.
```*von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion*. United States. doi:10.1016/J.AOP.2016.02.010.

```
Divitiis, Nicola de, E-mail: n.dedivitiis@gmail.com. Sun .
"von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion". United States. doi:10.1016/J.AOP.2016.02.010.
```

```
@article{osti_22560322,
```

title = {von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion},

author = {Divitiis, Nicola de, E-mail: n.dedivitiis@gmail.com},

abstractNote = {A new approach to obtain the closure formulas for the von Kármán–Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics is characterized by the finite scale separation vector which is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von Kármán–Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, we show that the times of evolution of the developed kinetic energy and temperature spectra are finite quantities which depend on the initial conditions.},

doi = {10.1016/J.AOP.2016.02.010},

journal = {Annals of Physics},

issn = {0003-4916},

number = Complete,

volume = 368,

place = {United States},

year = {2016},

month = {5}

}