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Related Experiment Video

Updated: Sep 25, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

The background method: theory and computations.

Giovanni Fantuzzi¹, Ali Arslan¹, Andrew Wynn¹

¹Department of Aeronautics, Imperial College London, SW7 2AZ London UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|April 25, 2022

Summary

This review details advances in the background method for bounding turbulent flow properties. It covers new theoretical frameworks and computational approaches for rigorous analysis.

Keywords:

auxiliary functions background method bounds semidefinite programming variational methods

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Area of Science:

Fluid dynamics
Mathematical physics
Computational mathematics

Background:

The background method is a key technique for rigorously bounding mean properties of turbulent flows.
Advances in theoretical formulation and numerical implementation are crucial for its application.
Understanding turbulent flow behavior is essential in many scientific and engineering fields.

Purpose of the Study:

To review recent theoretical and numerical advances in the background method.
To present a systematic formulation within an 'auxiliary function' framework.
To explore computational optimization approaches and their interrelations.

Main Methods:

Formulation within a general 'auxiliary function' framework.
Exploitation of flow symmetries and maximum principles.
Semidefinite programming and timestepping approaches for bound optimization.

Main Results:

Systematic formulation of the background method using auxiliary functions.
Demonstration on Rayleigh-Bénard convection.
Connection between semidefinite programming and timestepping via convex duality and low-rank factorization.

Conclusions:

The background method offers a rigorous approach to bounding turbulent flow properties.
New theoretical and computational methods enhance its applicability.
Further numerical analysis is needed to explore open questions.