EPSRC Reference: 
EP/N022548/1 
Title: 
Statistics of fluid turbulence and operational calculus 
Principal Investigator: 
Ohkitani, Professor K 
Other Investigators: 

Researcher CoInvestigators: 

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 
Nonlinear Systems Mathematics 
Numerical Analysis 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

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
numerics,
(2) to carry out a detailed study of hypoviscous fluid equations with critical blowup
criteria and to asssess the effects of nonlocality on the characteristics of turbulence by direct
numerical simulations of turbulence with finite energy.
*Motivations
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.
*Strategy
(1) A complete statistical description of the NavierStokes 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 NavierStokes 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
NavierStokes 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.
*Benefits
Both topics will find applications in physical or engineering areas,
because they are related with statistics of turbulence and (near)singular
structures.
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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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


Sectors submitted by the Researcher 
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Project URL: 

Further Information: 

Organisation Website: 
http://www.shef.ac.uk 