A Canadian Boffin: Balloon Ascent Mystery

Balloon Ascent Mystery · Mar 01, 2006

The last time friends and I launched a high-altitude balloon was for the final flight of the Mark I glider (far too long ago now). On that flight, something odd happened during the balloon ascent that I didn’t mention in the flight’s web entry.

Initially after launch, the balloon pulled the glider up at a good rate, but then as it went through about 6 km altitude (25,000 feet), it seemed to slow. At first I worried that it might be a leak – until the speed increased again after a few minutes. But then as it went higher, the ascent rate dropped further, and stayed down.

It was only weeks later, looked at the telemetry data again, that I noticed there was a clear pattern. You can see the change in ascent rate in the plot for that final flight:

Well, maybe it was something about the weather conditions? Or it really was a leak – that stopped? Then I dimly remembered this had happened before. My curiosity aroused, I hunted down some other glider-launch telemetry sets, and found a similar trend in the other high-altitude launch, and hints of it in the two moderate-altitude ones.

The behavior that I had always expected was a gradual increase in ascent rate, not a decrease. This is because as the balloon goes up, the air density goes down. The balloon expands in volume, but its cross section area doesn’t go up as fast, so its rate of ascent should go up slowly as it rises. That seems evident in the last part of the plot above, but not the start, and for sure not the middle. It slows down abruptly after about 6 km. Why?

So I threw tried to come up with a working hypothesis:

A leak: but one that stops?
Turbulent weather: that would explain the noise in the data, but a 20,000 foot thermal seems unlikely.
Temperature: Helium is very thermally conductive, but as it starts warm at launch and then travels into cooler air; you would actually expect that to cause speeding up, not slowing down.

Then I hit on an idea: the Reynolds number must change as it rises. Reynolds number – Re for short – is a measure of the viscosity to inertia ratio of the fluid flow in question.

Reynolds number varies by:

Re = D * V * density / k

Where V = freestream velocity in m/s, k = kinematic viscosity of air in m²/s, D = diameter of the body in metres, and density is in kg/m³.

The character of a fluid flow can change dramatically as this “scale effect” varies, with flow around a sphere being one of the best-known examples. Basically, at low Reynolds numbers, the boundary layer, which is the thin layer of air in vicious interaction with the surface, just doesn’t have enough oomph to round the corner very far. At a given Re, at some critical length along the body it will separate, causing a large amount of drag.

Now, the sphere as I said is a classic example, and its Coefficient of Drag varies strongly with Reynolds number, in a well known way. Here’s a classic set of experimental data points for it:

In this plot, the blue zone represents the approximate range of Reynolds numbers during the ascent of a small helium weather balloon going from sea level, to say 30 km altitude. Note that as it rises, it increases in size, so higher altitudes are towards the left side of the plot. The most dramatic feature, to my eye, is that severe dip at an Re of about 300,000. That should be noticeable in flight, right?

And so in the grand physics tradition of the spherical cow, I present to you the spherical balloon:

Ok, not quite. But probably closer to spherical as it goes higher, it expands, and its skin tension starts to dominate. A photo on Kaymont’s website would seem to support that idea. So, if the Reynolds number range is correct, and modeling the balloon as a sphere is close to valid, a rapid drop-off in speed at some point isn’t just possible, it’s expected.

The term “Cd” on the Re vs Cd plot is the Coefficient of Drag based on frontal area, and gives drag force via the classic equation:

F = (density / 2) * V² * Cd * Area

Where V = velocity, in meters per second (m/s), Area is in square metres (m²), and density is in kg/m³.

Ok, now we have a model! Well, except that the atmosphere’s density and viscosity doesn’t really vary in a simple way with altitude, and an approximation probably won’t do here. So, I dug up a table of the properties of the ICAO standard atmosphere, and fed it into a Matlab model, along with a table of the Re vs Cd of a sphere, both interpolated.

Running through a range of altitudes from sea level to 20 km, that model spits out the following ascent velocity plots:

Here the blue line is the model without Reynolds number effects, using a fixed Cd of about 0.20, which is typical for a spheroid shape at “normal” speeds and diameter (assuming Re change is minor). Then, the red line is the model where Cd varies with changing Re, assuming a spherical balloon shape. Notice that it has the sharp drop in ascent rate at about the right altitude, followed by a gradual increase again. Quite a dramatic prediction, and very clean (maybe too clean).

The plot for the balloon’s Reynolds number vs Altitude follows:

First, notice the general trend – lower Re as the balloon goes up – the radius increases, but density goes down faster. Then, a sudden drop. Why? Well, according to our spherical-balloon model, at some critical Re the drag will start to increase steeply, slowing the balloon down. Because it is slower, Re has gone down, increasing Cd again, which lowers Re again … slamming on the brakes.

It seems clear that for a high altitude balloon payload in the 2-3kg range, we fall right on the boundary of critical Re part way into the flight. Smaller payloads, having a smaller diameter balloon with lower Re to start, we would expect to transition at slightly lower altitudes than I found. Ditto for a payload with less free lift, and so a slower initial ascent rate. Larger payloads or those with more free lift, we can expect to transition at higher altitudes; large enough and/or fast enough, and you might not see the slowdown at all.

But the real data isn’t that clean – our first plot showed a definite knee at around 6 km altitude, but not a cliff, more a hill. In fact, looking at the other relevant data set, it doesn’t seem that simple at all:

Note, there are two drop offs! One at the expected altitude, followed by a lot of noise, then a sharper, final one at a bit over 10 km altitude, followed by the expected gradual increase in ascent rate.

Well, maybe that was a rocky flight for some reason, wind shear in the jetstream perhaps. That might knock the airflow the balloon around, making the separation point unstable. So, go back to the (much smoother) first data set:

It obviously doesn’t have as sharp a drop off as the model predicts, so it doesn’t perfectly match the behavior of the drag vs Re curve for a sphere. What would its Cd vs Re plot actually look like? Working backwards from that flight data, we get the following:

Looks to me like a heavily softened version of the sphere’s Cd vs Re “knee”. Looking into the well-known variations in the drag of a sphere, it seems the expected effect of simple increase in turbulence or roughness is just a shifting of the plot for a sphere (blue) to the left. But not softening of the knee.

So there are still mysteries here. If anyone reading this article has a good set of GPS altitude vs time data from a balloon flight (and ideally the mass and free-lift at launch), it would be great to run the scripts again, and see if this pattern holds true generally.

You can find my Matlab source files here
if you want to see if I missed anything, or would like to make your own plots. The zip also includes an ICAO 1976 standard atmosphere table, with values for pressure, density, temperature, and viscosity.

Finally, some links to help make sense of the theory above:

An explanation of airflow separation on a sphere with specific application to golf ball dimples.

The classic Cd vs Reynolds number plot for a sphere

An original source for the raw experimental data for a sphere can be found in the paper:
UMAP Module 712, The Drag Force on a Sphere (UMAP Journal, Volume 12, no. 1, Spring 1991, pp. 47-80.), by H. Edward Donley,

This guy about half picked up on the issue

Kaymont’s website might also be useful.

Postscript: Ralph Wallio (W0RPK) has collected ascent rate data here from a wide variety of flights, most of which seem to show the “knee” effect! Other than that though, there’s a whole lot of variability, expecially at low altitudes. And it’s tough to tell where the knee “should be” without knowing the launch mass, from which an estimate of balloon diameter at launch might be had.

* * *

Can you put the raw GPS data from your actual ballon/glider flights on line to donwload? I’d like to look at them on Google Earth, like I have done with the Virgin Atlantic Global flyer here:
http://www.arjam.net/flyer.kml
— Robert J Munro Mar 2, 01:23 AM #
Art, We have been heavily influenced by your past flights, and are currently working on a flight of our own (minus the glider bit). We will collect the Altitude vs. Time and will send any data your way. It will take some time as we are just starting out the design phase.
— Chris Mar 2, 10:15 PM #
Nice to see you’re nerding up the place again… Keep it up! :)
— Alex Mar 3, 01:07 PM #
It occurs to me that a balloon with payload hanging from it looks more like a teardrop than a sphere—is it possible that the better streamlining (lengthening of the boundary layer) is responsible?
— Alex Mar 3, 01:11 PM #
Hi Alex,

Very possible. But really tough to quantify, expecially because the balloon package tends to swing on most (all?) flights.

As it goes up though, the balloon expands, the skin is under more tension and it gets more spherical. So I’d think near the “critical Re” altitude for most flights, it would be close (enough) to a sphere.

But you’re right, that might be the cause of the soft knee in the Cd vs Re curve.
— Art Mar 3, 01:43 PM #
I check this site daily for updates. Thanks! With the Spring just around the corner I can’t wait for more progress. Keep up the good work.
— JC Mar 4, 05:39 PM #
I’ve noticed that type of effect in my radiosonde work, without trying too hard to analyse it. I’m hoping to make a few more radiosonde launches in the next couple of months. Does anyone have any ideas for how I could modify a weather balloon to create a ‘dimpled’ effect and reduce the drag?
— Helen May 3, 05:24 AM #
I am sure it not so important to the overall project, but what is the possibility of observing the ballon during ascent? Maybe a second small and cheap digital camera that you can trigger every minute or so.

Of course, I guess you might be able to calculate the shape it could be at various altitudes. The wonders of math.
— Matt C. Jul 17, 06:29 AM #
Just to check: Did you take into account the change of diameter of the balloon? Increasing in size slowly as it goes up would slow it down gradually.

Btw: I really enjoyed reading about the 4 launches. Fantastic stuff.
— Mike Jul 28, 05:51 AM #
A recent find, for me, is the Wayne State Suntracker series of balloon launches.

http://suntracker.eng.wayne.edu/flights.html#st3

Have a look at their analysis of their balloon ascent data. Their payload weight and balloon size (implied) is available and their data shows the slowing down below 30K and turbulance though 40K
— Warren May 8, 11:15 PM #
458 days since the last update…
Is the project dead?
— Matt C. Jun 2, 09:34 PM #
If you’ll drop me your email address, I’ll send a copy of the slides I used for a presentation at the Grand Plains Super Launch in Grand Island, Nebraska in Aug on Rn and the change in ascent rate.

— Nick Hanks Oct 21, 09:46 PM #

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A Canadian Boffin

It's not quite flying, closer to falling with style.

Balloon Ascent Mystery · Mar 01, 2006