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Details of Grant 

EPSRC Reference: EP/N022548/1
Title: Statistics of fluid turbulence and operational calculus
Principal Investigator: Ohkitani, Professor K
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics and Statistics
Organisation: University of Sheffield
Scheme: Standard Research
Starts: 01 April 2016 Ends: 31 March 2020 Value (£): 312,976
EPSRC Research Topic Classifications:
Continuum Mechanics Non-linear Systems Mathematics
Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
23 Nov 2015 EPSRC Mathematics Prioritisation Panel Meeting November 2015 Announced
Summary on Grant Application Form
*Main objectives of this project are

(1) to introduce and test new methods of finding statistical solutions based on

a simple application of symbolic calculus, to be followed by a comparison with


(2) to carry out a detailed study of hypoviscous fluid equations with critical blowup

criteria and to asssess the effects of non-locality on the characteristics of turbulence by direct

numerical simulations of turbulence with finite energy.


The problem of fluid turbulence deals with phenomena ubiquitous in our daily

life, yet its complete understanding, let alone its control, is way far beyond

our current capability. Turbulence is also regarded as a big open problem in

classical physics. While a suitable description of turbulence should be

inevitably statistical in nature, a satisfactory theory has not yet been developed.

Even for deterministic solutions of the fluid dynamical equations,

there is still room for smooth solutions to break down. Such a potential blow up is

connected with statistical solutions, as it can trigger a transition from

deterministic to statistical descriptions.


(1) A complete statistical description of the Navier-Stokes turbulence is

given by the Hopf characteristic functional, which satisfies the celebrated

Hopf functional differential equation (FDE).

In spite of previous attempts, it remains difficult to find a method of its solutions.

Recently, the PI has discovered a method of converting the FDE into a

functional integral equation, thereby generating a systematic approximation

series. The method lies in the introduction of an exponential operator with

a functional derivative as its exponent. Our aim is to put the approximation

into practice.

Another key is use of first integrals method developed by Vishik and

Fursikov in 70's. It resolves the notorious unclosedness of turbulence

statistics. Its proper understanding, as yet obtained, provides a practical

way of handling Hopf equation.

(2) The Navier-Stokes equations satisfy invariance under scaling transforms.

Recently, it has been proven that some critical norms serve as blowup criteria,

i.e. they become unbounded upon formation of singularities. Hence showing

how difficult it is to have unbounded critical norms, theoretically and

numerically, will be a promising way to improve our understanding the

Navier-Stokes regularity.

We will also carry out direct numerical simulations of turbulence in the whole space

and compare the results with those of periodic boundary conditions.


Both topics will find applications in physical or engineering areas,

because they are related with statistics of turbulence and (near-)singular


Techniques of operational calculus will prove helpful to the general audience

to make Differential Equations approachable. We will prepare a documentation

on the methods in the spirit of the British tradition and make freely available

on the web, while gathering queries from the public audience.

Key Findings
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Date Materialised
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Organisation Website: http://www.shef.ac.uk