Leray – On the Motion of A Viscous Liquid Filling Space

I am currently studying the above paper to get some understanding concerning the mathematical analysis of NS.
This is not going to be easy. Somehow I will need to translate the mathematical jargon into engineering concepts that I can get my mind around.

Leray Phase 1:

The first part of the paper defines a number of relations related to the Schwarz inequality.

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Navier Stokes, Turbulence, Etc.

For the past eight years I have been fascinated with the Navier Stokes problem as described by the Clay Mathematics Institute as one of its Millenium prizes. It along with seven other problems are deemed difficult enough that a million dollar prize will be awarded to the individual(s) that can solve them. The one on the Poincaire conjecture was recently solved by mathematician, Grigori Perelman. The link to the problems is here:
Clay Problems

Of the seven original problems, Navier Stokes (NS), is closest to my own experience and work.

As a chemical engineer, fluid flow problems come up frequently. In fact, most engineering problems, in my field, have some aspect of fluid flow. Like other engineers; however, we are divorced from the pure theoretical aspects of fluid flow even before we get our first engineering degree. Navier Stokes and Bernoulli’s equations go out the window. They are replaced with the Moody diagram, Colebrook/White equation, Darcy’s equation, etc. All based on empirical data. We do not question them.  We just use them.

Engineering empirical equations have a characteristic appearance that come about from their derivation. Most relate dimensionless numbers (Reynolds number, e.g.) with “tell-tale” non-integer exponents. With the “tell-tale” meaning that the equation is not derived from theory. In most cases, these empirical equations are straight forward to solve.  This, of course, gives a great endorsement to dimension analysis in its ability to reduce complex phenomena to easily manageable, and surprisingly accurate, engineering formulas.

In the case of fluid flow; however, this is something going on so complex, so bizarre that even dimensional analysis is humbled. The usual dimensionless quantities in fluid flow (I will speak only in terms of flow in conduits since this is what I know, not airplane wings, e.g.) are the Reynolds numbers , the friction factor (Darcy or Fanning), and relative roughness. The relationship of these three quantities is depicted very well in the Moody diagram – see link. Moody Diagram(For those that prefer equations,  Colebrook -White, Colebrook EquationThe relative roughness (on the right side of the diagram) is the degree of roughness on the inside wall of the conduit relative to the conduit diameter. The Reynolds number (at the bottom) is a measure of the degree of mixing/turbulence occurring inside the conduit. The friction factor (on the left) indicates the conduit’s resistance to flow and is usually the key parameter engineer’s desire in using the diagram. Upon examining this diagram, one minor peculiarity is that the velocity is embedded twice — in the Reynold’s number and the friction factor. This makes for some minor hassles in that the velocity cannot be solved for explicitly from knowing the other quantities. An iteration process is needed which is can be started by guessing the friction factor, solving for the Reynold’s number, and then using the diagram to fetch a new friction factor and repeating the process until there is convergence at the correct point on the diagram. The most striking oddity in the diagram, and one that has fascinated me for many years, is that even though it is a log/log plot of two carefully chosen dimensionless quantities, the curves are still highly unlinear. Also, at Reynolds number, 2000 – 10000, there is a huge undefined gap where laminar flow transitions to turbulent flow.

Oddities in Moody’s by list:

1. Turbulent curves by roughness high unlinear for decreasing Re number.
2. Turbulence curves look as though they “should” asymptotically approach the laminar
line but do not. Instead, they just halt, and the laminar line picks up at a significantly lower friction factor.

To most engineers, the oddities in the Moody diagram do not warrant a second notice. For many years though I have, in my spare time, have toyed with the quantities in the Moody diagram, to gain some understanding to it’s irregularity. My first thoughts were why isn’t there a simpler diagram? Why hasn’t anyone find a more explicit aesthetically logical curve? It turns out that there are methods other than Moody’s / Colebrook. A few years back, the flagship publication of the American Institute of Chemical Engineers, “Chemical Engineering Progress”, had an article touting a method to solve the parameters in a totally explicit fashion. The general idea behind this is to use a dimensionless number (I believe the Karman number) in lieu of the Reynold’s number to eliminate the velocity at the bottom. Although, initially fascinated, I wasn’t convinced that it was an improvement of the problem.

Eventually I gravitated toward improving the Colebrook equation. Somehow it just wasn’t elegant. Somewhere, deep down, was an elegant, possible complex though, equation that covered the laminar and turbulent parts of the diagram equally well without any ugly breaks (the transition region) that appear in the Moody diagram.

(to be continued)

P.S. Reynolds number is currently only empirically derived — as far as I know.

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Some of my past posts on engineering topics.


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PhDs Want a Break?


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