Getting the Reynolds Number Right

One of the most common mistakes I see in my Fluid Mechanics lab course is incorrect computation of the Reynolds number.  This is a brief treatment of this subject.

First, the Reynolds number is defined as

R_e = \frac{VD\rho}{\mu}

or

R_e = \frac{VD}{\nu}

where

  • R_e = Reynolds number (yes, I know the newer textbooks use a different notation, but I’m a dinosaur)
  • V = Average or reference velocity
  • D = “Diameter” or other governing dimension
  • \mu = Dynamic viscosity
  • \rho = Fluid Density
  • \nu = Kinematic viscosity

Let’s go over this variable by variable.

Velocity

The velocity is generally some kind of “average” velocity, which basically makes the fluid velocity through or around the object under analysis uniform.  In the case of closed ducts or pipes, such as are described here, it’s an average velocity, which can be computed by dividing the fluid flow per unit time by the cross-sectional area of the pipe or duct through which the fluid flows.  For Wind Tunnel Testing (actual and virtual, such as in CFD) it’s the “free stream” velocity, or the velocity through which the airfoil or object moves through the fluid.

Diameter

The “diameter” used to compute Reynolds numbers would seem to be clear-cut, but it’s more a matter of convention than anything else.  With straight pipes of uniform diameter, it’s just the diameter of the pipe.  For flow meters such as orifices or venturi meters, it’s customarily the diameter of the incoming pipe, although in the past that wasn’t always the case.  For airfoils the rules are more complicated and are described in Wind Tunnel Testing.

Density

This is simply the density at wherever the velocity is chosen.  For incompressible fluids, that’s pretty simple.  For compressible fluids it’s more complicated.  With Wind Tunnel Testing it’s the free-stream density of the fluid.

Viscosity

This, I think, is where most students get into trouble.  If you use dynamic viscosity and density, it’s easy to get into trouble with the units.  My advice to students is to use the kinematic viscosity \nu whenever possible, which means that we use the second form of the Reynolds number.  The units for this (unless you’re given it in stokes or centistokes, in which cases you’ll need to convert it) are \frac{length^2}{time} , which cancel nicely with the other dimensions.  For water and air, the two fluids we mostly test in my course, the properties are in my monograph Variation in Viscosity, along with a discussion of viscosity in general.

Conclusion

Reynolds numbers are important in fluid mechanics, but they can trip up a new student.  This is some advice to avoid that.

Now the serious question: how many Reynolds numbers do you know?

 

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Indicator Devices and Cards for Vulcan Hammers

vulcanhammer.info

The indicator card, and the devices that produced them, have been around about as long as there have been steam engines.  The basic idea is simple: as the piston of the engine moved, a pressure indicator moved a needle and pen up and down on a paper (usually a rotating drum) and produced what’s called in thermodynamics a pV diagram, shown below.

indicator card An indicator card, taken from A Practical Treatise on the Steam Engine Indicator and Indicator Diagrams by Amice, edited and enlarged by W. Worby Beaumont, 1888. The area of the central region would indicate the energy output of the engine. The displacement is noted on the x-axis and the pressure on the y-axis. The straight lines over the region are probably a method of graphical integration, although even then (before the advent of CAD and numerical integration) a planimeter would be much easier.

The steam engine (or any…

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Proof of Harten’s Lemma re the Convergence of TVNI Finite-Difference Schemes

An academic paper with a rather unusual history is that of the Israeli mathematician Ami Harten’s “High Resolution Schemes for Hyperbolic Conservation Laws.”  First published in the Journal of Computation Physics in 1983, it was republished in 1997 in the same journal, and is often cited with the later date.

At the time of republishing Peter Lax, who earlier had saved the computer from the hippie radicals, made the following statement about this paper in an introduction:

This paper was a landmark; it introduced a new design principle—total variation diminishing schemes—that led, in Harten’s hands, and subsequently in the hands of others, to an efficient, robust, highly accurate class of schemes for shock capturing free of oscillations. The citation index lists 429 references to it, not only in journals of numerical analysis and computational fluid dynamics, but also in journals devoted to mechanical engineering, astronautics, astrophysics, geophysics, nuclear science and technology, spacecraft and rockets, plasma physics, sound and vibration, aerothermodynamics, hydraulics, turbo and jet engines, and computer vision and imaging.

One point in the paper was a lemma concerning the convergence of TVNI (total variation nonincreasing) finite difference schemes.  Concerning the name of these schemes, Lax points out the following:

Harten originally called his schemes variation diminishing, abbreviated TVD; when Osher pointed out the usual meaning of these initials, the name was switched to total variation nonincreasing (TVNI), but was eventually settled on the more euphonious TVD.

The following is an expansion of Harten’s proof of the lemma.

Schemes which are total variation nonincreasing (TVNI) can be characterized as follows:
\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|

We can thus define
TV\left(u^{n}\right)=\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|

TV\left(u^{n+1}\right)=\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|

and substituting
TV\left(u^{n+1}\right)\leq TV\left(u^{n}\right)

Consider the general expression
u_{{j,n+1}}=u_{{j,n}}-C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)

where
C_{{j-1,n}}\geq 0

D_{{j,n}}\geq 0 C_{{j-1,n}}+D_{{j,n}}\leq 1

We should observe that our ultimate goal is to sum these values from negative infinity to positive infinity; thus, we can shift the index at will. The inequalities will still hold but the specific location in space may change. It is also worth noting that the coefficients may themselves change at different points in space.

Let us consider the next spatial step, to wit
u_{{j+1,n+1}}=u_{{j+1,n}}-C_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)

Subtracting the previous spatial step from this yields
u_{{j+1,n+1}}-u_{{j,n+1}}=u_{{j+1,n}}-C_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)-u_{{j,n}}+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)-D_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)

Some rearranging yields
u_{{j+1,n+1}}-u_{{j,n+1}}=\left(u_{{j+1,n}}-u_{{j,n}}\right)\left(1-D_{{j,n}}-C_{{j,n}}\right)+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)

Taking the absolute value of both sides, we have
\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|=\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\left(1-D_{{j,n}}-C_{{j,n}}\right)+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|

At this point we observe that
C_{{j-1,n}}\geq 0
 D_{{j+1,n}}\geq 0
D_{{j,n}}+C_{{j,n}}\leq 1
1-D_{{j,n}}-C_{{j,n}}\geq 0

We can thus limit the absolute values and write the expression as follows:
\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\left|\left(u_{{j,n}}-u_{{j-1,n}}\right)\right|C_{{j-1,n}}+\left|\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|D_{{j+1,n}}

Taking the summation for both sides,
\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j,n}}-u_{{j-1,n}}\right)\right|C_{{j-1,n}}+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|D_{{j+1,n}}

Since, as we observed before, we can shift the indices (as the “centre” of the system is arbitrary with infinite boundaries) we can rewrite the above as follows
\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right| \leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|C_{{j,n}}+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|D_{{j,n}}

in which case
\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right| \leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|

Substituting, we have at last
TV\left(u^{n+1}\right)\leq TV\left(u^{n}\right)

Citation: Harten, A. (1997) “High Resolution Schemes for Hyperbolic Conservation Laws.” Journal of Computational Physics, Vol. 135, pp. 260-278.

Taking the Last Voyage with Newton and Pascal

He’s not widely known outside of the fields he specialised in, but Adhémar Jean Claude Barré de Saint-Venant (1797-1886, usually known in the Anglophone world as simply Saint-Venant) was one of the premier scientists, engineers and mathematicians of the nineteenth century.  His accomplishments were many and include the following:

  • saint-venant-torsionSuccessful derivation of the Navier-Stokes Equations for a viscous flow before Stokes; these equations are the basis of computational fluid dynamics and the analysis of things that fly.
  • Systematisation and development of methods in the theory of elasticity of solids, including his semi-inverse methods for torsion, important in things such as automobile crank shafts.
  • Methods for the analysis of wave mechanics in bars, which we see in many places, from musical instruments to driven foundation piles.

Saint-Venant was born into a royalist, aristocratic, traditionally Roman Catholic family at a time when it was not safe to be any of these: the French Revolution, at that point stumbling from the Reign of Terror to control of France–and most of Europe–by Napoleon Bonaparte.  It was about the latter where Saint-Venant made a statement about himself that got him into trouble with the “new” Europe.  As described in S. Timoshenko’s History of Strength of Materials:

The political events of 1814 had a great effect on Saint-Venant’s career.  In March of that year, the armies of the allies were approaching Paris and the students of the École Polytechnique were mobilized.  On March 30, 1814, they were moving their guns to the Paris fortification when Saint-Venant, who was the first sargeant of the detachment, stepped out from the ranks with the exclamation: “My conscience forbids me to fight for an usurper…” His schoolmates resented that action very much and Saint-Venant was proclaimed a deserter and never allowed to resume his study at the École Polytechnique.

Saint-Venant’s statement of conscience was at once a political and religious statement, and “progressives” of his day didn’t miss either.  The French, then and now innocent of anti-discrimination legislation or sentiments, made his life miserable. The École Polytechnique was and is France’s premier technical institute of higher learning; getting kicked out of it was the equivalent of, say, being expelled from Princeton or MIT.  He worked in the powder industry for nine years, then was admitted to the École des Ponts et Chausées, where his fellow students shunned him.  He graduated first in his class anyway and began his illustrious career in technical things both theoretical and practical.

In spite of his difficulties within France, his reputation outside of her was another matter.  When François Napoleon Moigno wrote his book on statics, he discovered the following:

He (Moigno) wanted the portion on the statics of elastic bodies to be written by an expert in the theory of elasticity, but every time he asked for the collaboration of an English or a German scientist, he was given the same answer: “You have there, close to you, the authority par excellence, M. de Saint-Venant, consult him, listen to him, follow him.” One of them, M. Ettingshausen, added: “Your Academy of Sciences makes a mistake, a great mistake when it does not open its doors to a mathematician who is so highly placed in the opinion of the most competent judges.” In conclusion Moigno observes: “Fatally belittled in France of which he is the purest mathematical glory, M. de Saint-Venant enjoys a reputation in foreign countries which we dare to call grandiose.”

The French finally broke down and admitted Saint-Venant into the Academy of Sciences in 1868.  He continued his work, much of it from his home, up until the time of his death.  When the President of the Academy announced that passing, he made the following statement:

Old age was kind to our great colleague.  He died, advanced in years, without infirmities, occupied up to the last hour with problems which were dear to him and supported in the great passage by the hopes which had supported Pascal and Newton.

Europeans of the time would not have missed the import of the last statement: Pascal and Newton were Christians, and Saint-Venant was being identified with them as one also.  It was also a statement that Saint-Venant, for all of his achievements and interests which have enriched the world, also had an eternal goal as well.

There’s no evidence that Saint-Venant was ostentatious in his faith walk; descriptions of his life show the contrary.  And–shock to today’s atheist–there’s no evidence that it ever impeded the progress of his research or his thought.  As the statistician and eugenicist Karl Pearson, no friend of Christianity, noted:

The more I studied Saint-Venant’s work, the more new directions it seemed to me to open up for original investigation of the most valuable kind. It suggested innumerable unsolved problems in atomic physics, in impact, in plasticity and in a variety of other branches of elasticity, which do not seem beyond solution, and the solution of which if obtained would be of extreme importance. I felt convinced that a study of Saint-Venant’s researches would be a most valuable directive to the several young scientists, whose recent memoirs shew their interest in elasticity as well as their mathematical capacity. Many of the problems raised by Saint-Venant’s suggestive memoirs were quite beyond my powers of analysis, and I recognised that the most useful task I could undertake, was by a careful account of the memoirs themselves to lead the more competent on to their solution.

saint-venant-wave-equation

The biggest impediment he had to face was the blowback from his stand at the École Polytechnique, and that came from his secularist colleagues.  But, when the end came, all of his colleagues knew where he stood, in this life and the next one.

I spend a lot of time on this site and others talking about sea (and sometimes air) voyages.  And I’ve spent most of my career (and all the academic part of same) in the applied sciences.  But when I take my last voyage into eternity, I want to do it in the same hope of Newton and Pascal–and Saint-Venant and Euler for that matter–namely that which comes from following Jesus Christ out of the grave and into eternal life.

And you should too.

Note: I have a more complete account of Saint Venant’s story with a list of references here.

“I Need to Find the Way”

When numerical methods are combined with fluid mechanics, you ultimately get Computational Fluid Dynamics (CFD.)  CFD has been a test of numerical methods for a long time, especially with the solution of the Navier-Stokes equations.  The non-linear nature of these equations has made mating them with CFD a challenging problem.

When you’re first introduced to numerical methods you’re told that there are ways one can predict the stability and consistency of these methods with a given set of physics.  These methods are helpful but things go wrong.  On top of that coding these solutions into a language like C++ or Fortran for parallel processing (or any other computer application) is a challenging business; the endless parade of bugs we see exploited in internet security can be also seen in modelling physics such as fluid flow, just different bugs.  Probably the most important lesson one learns in mastering such an art is that, just because it comes out of a computer, it can’t always be trusted.

That situation confronted a Turkish friend of mine who was working on a particularly sticky CFD coding problem.  One frustrating day he exclaimed, “I need to find the way,” i.e., out of his CFD coding problem.

Finding the way has been a key issue in the Middle East for a long time.  The last night before he was crucified, Jesus Christ had this exchange with his disciples:

“We do not know where you are going, Master,” said Thomas; “so how can we know the way?”

Jesus answered: “I am the Way, and the Truth, and the Life; no one ever comes to the Father except through me. (John 14:5-6 TCNT)

This, then, is a key difference between Jesus Christ and the others.  They knew (or they thought they knew) the way, or they have an idea of what the way is.  But Jesus Christ is the way, something he showed when he rose from the dead shortly thereafter.

My Turkish friend was doubly blessed; not only did he find the solution to his CFD problem, but he also found the way in Jesus Christ, and has been baptised.  You can see a video of another Turk who found the way here.

And for the rest of us…

For more information click here.