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New Directions for my Research Program

19 Wednesday Aug 2020

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Today – August 19, 2020 – I restart in EARNEST this blog and Website. Promise!

I last posted  AShortPaperOnTheoreticalTurbulence three years to the day. Since then life happened with weddings and new children in the family. But my research continued at a slow pace.

Over the last few months, I have been greatly encouraged by the work of Lee & Moser at UTexas on high Reynolds Number Channel Flows. Even more persuasive for my purposes is the work of Mortensen & Langtangen at UOslo using Python for DNS studies of both decaying (HIT) and channel (PPF) turbulent flows.

I believe that extrapolations from the Python code will allow me to develop a rational and analytic theory of Turbulence. A tall order to be sure.

Today, I publish https://randomtheoryofturbulence.com/wp-content/uploads/2020/08/isthereasolution.200724r.pdf . This paper proposes that there IS a rational analytic solution to decaying and steady turbulence problems including HIT and PPF. By this we mean that there exist functions by which we can construct a solution to (say) PPF for a given Reynolds number without simulation or resorting to the equations of motion. See the paper for details.

I promise to add to this blog monthly with results and progress.

Adding an Essay on my Research Program

19 Saturday Aug 2017

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Today – August 19, 2017 – I add AShortPaperOnTheoreticalTurbulence which  is 3000 words and attempts to explain the essence of my research program.  I attempt a linear step-by-step discussion of the Wiener Machinery in both Polynomial Functional form and in Quantile Function form.

I expect to add further short papers on related subjects, e.g.  Also, I am completing a re-write of my original thesis with newer tools, fewer errors, better computational results, and (hopefully) greater clarity on this very complex problem.

Restarting the Blog and Website

30 Tuesday May 2017

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Today – May 30, 2017 – I return to adding content to this website after a four year hiatus.  I now have the time to pursue my avocation, namely Theoretical Turbulence using Wiener Functionals and Quantile Functions.

In the last four years I have made some substantial progress in several areas:

  1. Applying Quantile Function in place of Wiener Functionals which is much more efficient analytically and numerically.
  2. Revising Maple and CPP code to yield good results for Plane Poiseuille Flow.
  3. Revising, correcting, unifying, and generally improving the documentation.

I intend to regularly update this website with my latest results.

Comments and constructive criticisms are welcome.

Adding a Paper on Integration Theory.

17 Thursday Oct 2013

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The Wiener Stochastic Convolution Integral – as a scalar Random Variable with a single parameter – is a good start to getting robust tools for studying turbulence.  But the results must be expanded to vector quantities of 4 parameters (3 space and time).  This is a complex task made more difficult by the fact that the Wiener Process (Brownian Motion) is continuous but has unbounded variation.

The first step is to insure that the Stieltjes-form integral with the Wiener Process as integrator actually exists, can be extended to infinite limits, has a derivative, and has appropriate stationary and ergodic properties.  “PrettyGoodIntegrationTheory” (today added to Other Topics) is designed to answer some of the basic existence questions posed by these integrals.

From these results, various chapters in The Revised Thesis (to be added to this blog by and by as revisions are completed) will expand the basic integral to vector form with multiple parameters.  And from this we shall form Wiener’s “Homogeneous Polynomial Functionals” (HPF) and “Orthogonal Polynomial Functionals” (OPF).

These Polynomial Functionals are then used to express velocities (et alia) which are then inserted into the equations of motion.  We then solve these equations of motions either analytically, or by a Galerkin best fit process, of by direct numerical computation.