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Details of Grant 

EPSRC Reference: EP/F009267/1
Title: Novel numerical approaches to fundamental problems of fluid dynamics
Principal Investigator: Ohkitani, Professor K
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Applied Mathematics
Organisation: University of Sheffield
Scheme: Standard Research
Starts: 01 September 2007 Ends: 31 August 2011 Value (£): 164,023
EPSRC Research Topic Classifications:
Continuum Mechanics Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
06 Jun 2007 Mathematics Prioritisation Panel (Science) Announced
Summary on Grant Application Form
*ObjectivesThe aim is to carry out computational studies which reveal salientmathematical features on fundamental fluid mechanics, and which can sustainapplications in physical or engineering areas.*MotivationsRegarding the big problems of the Navier-Stokes regularity/singularityissues, recently there have been some progress in mathematical analyses.In many cases, they take the forms of some criteria under which flows can beshown to remain regular. At the moment, it is not known that these criteriaare actually met, that is, it is impossible to show that they are satisfied,by working directly with the first principles. Here arises the need fornumerical computations of fluid mechanics. However, those people who performthose mathematical analyses do not know very well computational techniquesthey can turn to. On the other hand, people who are engaged in numericalmethods in fluid mechanics are not always updated with recent mathematicalprogress.We intend to fill in this gap by the current project which is interdisciplinarybetween mathematics and numerical simulations.*Strategy(1) To distill relevant useful approaches from ongoing mathematical analyseson fluid mechanics. This is important, as many of mathematical theorems existon their own footing and are not suitable for numerical implementations.(2) To actually implement them into numerical methods.(3) To interpret the outcome and seek further progress.For (1), more specifically, we will investigate the following 3 topics(a)Flows with Clebsch potentials,(b)Viscous flows with modified dissipativity,(c)Numerical estimation on Caffarelli-Kohn-Nirenberg criteria.For (2) and (3), details are given in the attached project description.*BenefitsIf the project is successful, it will find positive feedbacks on physicaland engineering problems on fluid mechanics. For example, in connection withthe Navier-Stokes regularity the problem, growth in total enstrophy is crucial.Conventional studies on its growth (inequalities which bound from above), arerelated to third-order moments of velocity. By performing a similar analysisat the forth-order, we can compare how the result of mathematical analysisoverestimates that of a closure theory and that of direct numericalsimulations.In the case of the closure theory, regularity may be ensured mainly byquasi-normality (quasi-Gaussianity) of the velocity distribution, rather thannonlinearity depletion due to the presence of vortex structure. By seekingthe conditions under which the growth in the enstrophy limited, we aim to geta more sophisticated statistical closure of turbulence.The filtering techniques are used in handling weak solutions of theNavier-Stokes equations. Recent mathematical theory reveals that filtering(that is, smoothing) advection velocity serves as dissipativity effectivelyin the case of 1D Burgers equation. This idea deserves attention in that itmay open a possibility of an alternative method of simulating high Reynoldsnumber flows. It may provide Large Eddy Simulations algorithms with somemathematical underpinnings.*Other issuesA novel aspect is to combine recently developed, advanced mathematical theorywith state-of-the-art numerical simulations.The outcome will be disseminated through presentations at internationalmeetings and journal publications.Because the numerical work, that is, coding of programs, data managing andpost-processing requires lots of effort, PhD studentship will be requestedin the second year. In the first year PI will start the project withthe topic (a) while preparing for basic codes to be used with topics (b) and(c).In the second year we are planning to invite visiting researchers to discussissues in their expertise which are closely related with the project.
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Organisation Website: http://www.shef.ac.uk