# I’m Kristen Karman-Shoemake and This Is How I Mesh — Another Fine Mesh

I was born in Arlington, TX and spent most of my childhood in and around the Fort Worth area. When I was in high school, my family moved to Chattanooga, TN. I then went on to UTK for my undergraduate degree. I didn’t really know what I wanted to do with my life and ended […]

Kristen and I went to graduate school together along with her husband Lawton.

# The Evolution of Computational Aerospace from 1968-2018 — Another Fine Mesh

Sometimes you write something that ends up on the cutting room floor (as my film-loving friends might say). Such is the case with an article I was asked to write for the 50th anniversary of the Society of Flight Test Engineers in 2018. Alas, plans change and the article went unused. I thought it turned […]

# 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.  Non-circular pipes have their own rules as well, which related to the “circular equivalent” of a cross-section.
• For round objects in free-flow condition, such as cylinders or spheres, the diameter is pretty straightfoward.
• For airfoils the rules are more complicated and are described in Wind Tunnel Testing.
• For “odd shaped” objects, it’s a matter of judgement and convention to determine the diameter.

### 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.

### Some Examples

Let’s take a couple of examples.  The first one comes from the valve loss coefficient example in our discussion on flow metering.  We have a 1.48″ diameter pipe with water flowing through it at 15.9 gallons per minute.  What is the Reynolds Number?

From that example, we computed that the average velocity in the pipe was 35.38 in/sec.  From the monograph Variation in Viscosity, the kinematic viscosity of water is $\nu=10.877\times10^{-6}\frac{ft^{2}}{sec}$.  Using the formula above, by direct substitution (and a quick units conversion) we have

$R_{e}=\frac{VD}{\nu}=\frac{35.58\frac{in}{sec}\times1.48\,in}{10.877\times10^{-6}\frac{ft^{2}}{sec}\times\left(12\frac{in}{ft}\right)^{2}}=33,431$

As another example, consider a USGA standard golf ball, which is 1.68 in. in diameter.  If it flies through still air at 50 mph at an altitude of 600′, on a standard day what is the Reynolds Number?

Working with US units, one of the trickiest decisions we have to make in fluid mechanics (and sometimes in solid mechanics too) is whether to work in feet or inches.  The previous example did so in inches, converting the kinematic viscosity to $\frac{in^2}{sec}$.  For this one we’ll do it in feet.  Which one you pick is up to you; the one you use is the one where error is least likely.  (For very small values of distance, the significant digits problem comes into play.)

In this case we convert the diameter 1.68 in = 0.14 ft and 50 mph = 73.3 ft/sec.  Using the kinematic viscosity for air (again from Variation in Viscosity) $\nu=161.6\times10^{-6}\frac{ft^{2}}{sec}$, we substitute and solve

$R_{e}=\frac{VD}{\nu}=\frac{73.33\frac{ft}{sec}\times0.14\,ft}{161.6\times10^{-6}\frac{ft^{2}}{sec}}=63,531$

Using this online calculator yields a slightly different result, probably because of a difference in the viscosity value.

### 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?

# Indicator Devices and Cards for Vulcan Hammers

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.

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.

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.