[Paper Review] On the behavior of homogeneous, isotropic and stationary turbulence

Massimo Tessarotto, Claudio Asci|arXiv (Cornell University)|Mar 7, 2010

Statistical Mechanics and Entropy5 references3 citations

TL;DR

This paper investigates homogeneous, isotropic, and stationary turbulence (HIST) using inverse kinetic theory (IKT) to derive the statistical behavior of incompressible Navier-Stokes fluids. It proves that under IKT, HIST uniquely requires the one-point velocity probability density function (PDF) to be an isotropic Gaussian, establishing a rigorous statistical foundation for HIST with no alternative functional forms possible under the given symmetries.

ABSTRACT

The recent development of a statistical model for incompressible Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004-2008) poses the problem of searching for particular realizations of the theory which may be relevant for extit{the statistical description of turbulence} and in particular for the so-called extit{homogeneous, isotropic}and extit{stationary} turbulence (HIST). Here the problem is set in terms of the $1-$point velocity probability density function (PDF) which determines a complete IKT-statistical model for NS fluids. This raises the interesting question of identifying the statistical assumptions under which a Gaussian PDF can be achieved in such a context. In this paper it is proven that for the IKT statistical model, HIST requires necessarily that $f_{1}$ must be SIED (namely extit{stationary}, extit{isotropic} and extit{% everywhere-defined}). This implies, in turn, that the functional form of the PDF is uniquely prescribed at all times. In particular, it is found that necessarily the PDF must coincide with an isotropic Gaussian distribution. The conclusion is relevant for the investigation of the so-called homogenous, isotropic and stationary turbulence.

Motivation & Objective

• To determine the statistical conditions under which homogeneous, isotropic, and stationary turbulence (HIST) can be realized in incompressible Navier-Stokes fluids.
• To investigate the role of inverse kinetic theory (IKT) in modeling HIST and its implications for the one-point velocity PDF.
• To identify the necessary symmetry and functional constraints that uniquely prescribe the form of the velocity PDF in HIST.
• To establish whether a Gaussian PDF is the only possible solution under HIST conditions within the IKT framework.

Proposed method

• The study employs inverse kinetic theory (IKT) to construct a complete statistical model for incompressible Navier-Stokes fluids based on the one-point velocity PDF, denoted $ f_1 $.
• It defines HIST in terms of the PDF $ f_1 $ being stationary, isotropic, and everywhere-defined (SIED), imposing strict symmetry constraints.
• The analysis uses stochastic representations with hidden variables $ \mathbf{\alpha} $ and stochastic averaging $ \langle \cdot \rangle_{\mathbf{\alpha}} $ to model fluid fields and their statistical behavior.
• The paper derives the functional form of $ f_1 $ by requiring it to be invariant under spatial translations and rotations, and time-independent, leading to a unique solution.
• It applies measure-theoretic and functional analytic tools to ensure the PDF is well-defined and integrable across the phase space.
• The derivation relies on the requirement that the PDF must be invariant under all spatial and temporal symmetries characteristic of HIST, leading to a unique Gaussian form.

Experimental results

Research questions

• RQ1 Under what conditions does the one-point velocity PDF in an IKT-based statistical model of Navier-Stokes fluids reduce to a Gaussian distribution?
• RQ2 Is the isotropic Gaussian PDF the only possible solution for homogeneous, isotropic, and stationary turbulence (HIST) within the IKT framework?
• RQ3 What symmetries must the velocity PDF satisfy to qualify as HIST, and how do these constrain its functional form?
• RQ4 Can the HIST condition be fully characterized by requiring the PDF to be stationary, isotropic, and everywhere-defined (SIED)?
• RQ5 Does the IKT approach uniquely prescribe the velocity PDF for HIST, or are other functional forms possible under the same symmetries?

Key findings

• The one-point velocity PDF $ f_1 $ must be SIED (stationary, isotropic, and everywhere-defined) for HIST to be realized in the IKT framework.
• Under the SIED condition, the functional form of $ f_1 $ is uniquely prescribed at all times, leaving no freedom in its shape.
• The unique solution for $ f_1 $ is an isotropic Gaussian distribution, meaning no other PDF form can satisfy the HIST conditions in this model.
• The result implies that within the IKT statistical model, HIST is mathematically equivalent to a Gaussian velocity distribution with isotropic variance.
• The derivation confirms that the statistical description of HIST is fully determined by symmetry constraints, with no additional assumptions needed.
• The conclusion provides a rigorous statistical foundation for HIST, showing that Gaussianity is not an assumption but a necessary outcome of the symmetries.