[Paper Review] Multi-scale turbulence modeling and maximum information principle. Part 1

Limin Tao|arXiv (Cornell University)|Sep 9, 2010

Probabilistic and Robust Engineering Design9 references3 citations

TL;DR

This paper proposes a multi-scale turbulence modeling framework for incompressible Newtonian fluids using the maximum information principle to derive an unbiased probability density function for turbulent fluctuations across spatial scales. By maximizing information under equality and inequality constraints—particularly those derived from the Cauchy-Schwarz inequality—it closes the model up to fourth-order correlations, with computationally tractable alternatives like the trace and determinant of the covariance matrix used to circumvent the intractability of direct information maximization, yielding a closed, information-theoretically grounded model applicable to homogeneous turbulence with promising initial results.

ABSTRACT

We discuss averaged turbulence modeling of multi-scales of length for an incompressible Newtonian fluid, with the help of the maximum information principle. We suppose that there exists a function basis to decompose the turbulent fluctuations in a flow of our concern into the components associated with various spatial scales and that there is a probability density function $\pdf$ of these fluctuation components. The unbiased form for $\pdf$ is determined and the turbulence model is closed, with the multi-scale correlations up to the fourth order, through maximizing the information under the constraints of equality and inequality for that flow. Due to the computational difficulty to maximize the information, a closely related but simple alternative objective is sought, like the determinant or the trace of the second order correlations of the turbulent flow. Some preliminary results and implications from the application to homogeneous turbulence are presented. Some issues yet to be resolved are indicated.

Motivation & Objective

• To develop a statistically consistent, multi-scale turbulence model for incompressible Newtonian fluids that resolves correlations across spatial scales.
• To address the closure problem in multi-scale turbulence modeling by applying the maximum information principle to determine an unbiased probability density function for fluctuation components.
• To overcome computational intractability in maximizing information by seeking simpler, related objective functions such as the trace and determinant of the covariance matrix.
• To test the feasibility and physical relevance of the model in homogeneous turbulence, a canonical test case for multi-scale modeling.
• To identify and resolve fundamental challenges in extending the framework to inhomogeneous, wall-bounded, and complex flows.

Proposed method

• Assumes a function basis decomposes turbulent velocity and pressure fluctuations into components associated with distinct spatial scales.
• Formulates statistical constraints in both wave number and physical spaces, including equality constraints from the Navier-Stokes equations and inequality constraints such as the Cauchy-Schwarz inequality.
• Applies the maximum information principle to derive the unbiased form of the joint probability density function f of fluctuation components under these constraints.
• Replaces the intractable maximization of information with computationally feasible alternatives: the trace (total fluctuation energy) and determinant (non-Gaussianity) of the second-order correlation matrix βij.
• Derives a closed turbulence model including up to fourth-order correlations, with no additional free parameters beyond the chosen objective function.
• Validates the model in the special case of homogeneous turbulence, using spectral and statistical methods to assess consistency with DNS and experimental data.

Experimental results

Research questions

• RQ1 Can the maximum information principle be effectively applied to multi-scale turbulence modeling to produce an unbiased, closed model for fluctuation statistics?
• RQ2 How can the intractable computational problem of maximizing information be addressed while preserving physical consistency?
• RQ3 What are the implications of using the trace and determinant of the covariance matrix as proxy objectives for information maximization?
• RQ4 Does the resulting model produce physically meaningful predictions in the context of homogeneous turbulence?
• RQ5 What are the key challenges in extending this framework to inhomogeneous and wall-bounded turbulent flows?

Key findings

• The maximum information principle successfully yields a closed multi-scale turbulence model up to fourth-order correlations by determining the unbiased form of the joint probability density function under physical and statistical constraints.
• The model is structurally minimal, containing no additional coefficients beyond the chosen objective function, and is closed without requiring empirical modeling of higher-order terms.
• Preliminary results for homogeneous turbulence show compatibility with direct numerical simulation (DNS) and experimental data, particularly in the case of homogeneous shear turbulence.
• The use of the trace of the covariance matrix as an objective function is physically justified as it corresponds to the total fluctuation energy in the flow.
• The determinant of the covariance matrix is a suitable alternative for capturing non-Gaussian behavior in slightly non-Gaussian flows.
• Significant challenges remain in extending the framework to inhomogeneous flows, including proper treatment of wave number locality, high-wave-number isotropy, and the formulation of consistent inequality constraints in physical space.