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## Fluid Dynamics Part 4: Temperature

By Joe Supercool

You've backed up for still more? I assume you have read Parts 1, 2 and 3 as otherwise this Part 4 will be incomprehensible. If not, here is a brief reprise.

So our fluid dynamics problem is to represent the flow of air over a cliff or hill, on some sort of diagram. This diagram will have lines on it, like contours, called "streamlines": also, it will have a picture of a cliff or hill. We have a hill. It looks out over the Southern Ocean, so wind-farms and slope soaring are “de rigeur”.

So now we get a definition. The path the wind follows is a "streamline". The same air remains in the streamline, but it can go faster: its pressure and density can change, but it is still the same streamline, with the same air.

Now it would be interesting to know the pressure distribution as well, but how do we get it?

Fortunately Edward R. C. Miles, a research mathematician at The Johns Hopkins University at Baltimore, USA, 1950, felt the need to publish a book called “Supersonic Aerodynamics”. On page 13 of this book he has the Saint-Venant and Wantzel equation, which is a version of the Bernoulli Equation adapted to adiabatic flow. The beauty of this equation is that it relates pressure to velocity directly: ie, knowing velocity, we can easily calculate the pressure.

Don’t worry about this maths too much, I have entered the equation into my computer and we can see the result below.

For the non-mathematically minded, here is a plot of the streamlines. The colours indicate the pressure bands present in the streamlines. Look carefully and you will find that the pressure at the bottom of the cliff is right up. In fact, at the bottom of the cliff the wind has stopped. This is called the "stagnation point".

What we find is that when the wind-speed is high, the air pressure is low.

OK. Now we can move on. Hands up if your understand any of this. Well, that was risky, because here are the trick questions.

1. Why does the wind blow over the ridge? Why does it not just give up and stop?
2. Some of the air actually speeds up. This means that the kinetic energy of the parcel of air in that region has increased. Where did that energy come from?

Hey, now you wish you hadn’t put your hand up! Never volunteer, never stand out from the mob, and never try to outsmart Supercool!

In my youth, my first car was a Ford Prefect, 1950 model. At only 10 years, it was an old bomb. But I got it for \$100, did 30000 miles in it in 4 years, and figured that was a good deal. The engine was side valve, which meant that the inlet and exhaust valves were beside the cylinders, operated by pushrods from down near the crankshaft: not overhead valves like today. The exhaust valves used to burn out, which meant most times only about 2 cylinders were working, which left about 5 HP for my motoring!

I recall returning one evening from the Central Western C/L champs in Orange. This meant climbing up over the Blue Mountains before coasting down the other side in neutral to get to Sydney. The old girl chugged along at about 10 MPH, slowly but inevitably getting to the top. Do you remember, prof?

With my headlights on, I felt like the head of a comet, with a long tail of other headlights winding away into the depths behind me. I guess I won’t know that thrill again in this lifetime, but the way my 1994 Toyota Camry keeps breaking down, I might yet, but this time it will be the wretched thing burning, by my hand.

Now here is my point. At least the old Ford had a source of energy to get it up over the mountain: but I don’t see that the wind is so supplied, so how does it get over the mountain? Surely the old Ford would stop as soon as I turned off the engine: but the wind seems to make it over anyway. Weird.

Well, lets be clear about this. Air is weird. Everything about air is weird. It never does what you want, and it holds its secrets very close.

The best we can do is invent a sort of simplified air, hoping that perhaps SA (Simplified Air) will behave a bit like real air.

So here we go. We will assign these properties to SA

1. SA is incompressible. This is rubbish, because real air compresses very easily: but even so, for air flowing fairly slowly, say up to 100 MPH, we will not be too far wrong. Don’t believe me, just suspend disbelief!
2. SA is irrotational. Good grief, what does the mean? Well, it has no eddies, and just flows smoothly. That’s fair enough.
3. SA is warm. That is, the molecules in the air are all racing around like Emu chicks frightened by a Dingo. We will measure this property in terms of temperature.

Now consider a parcel of air climbing up over the ridge. We have seen that it speeds up and the pressure drops.

Well, it’s nice that the pressure drops, at least its flowing from a region of higher pressure to one of lower pressure. That figures. But how can its energy increase when it speeds up? Surely the energy is not coming from the ridge, as the ridge is inert. And the energy can’t come from behind, as there is no mode for its transmission forward.

Maybe its pressure energy (whatever that is). After all, the SA starts with high pressure and this falls. Maybe the pressure in the air means that it is compressed, like a spring, and, as it expands, it speeds up the air.

Well no, we already decided that SA is incompressible. Ok, so it sure does have pressure, but being incompressible, it can neither compress nor expand. That’s what the words mean. Can’t be that.

So, all that is left to us as a property of SA is temperature. All those molecules dashing about suggest that there is heat energy available in SA.
So this can all happen if heat energy converts into kinetic energy!

Well, that must mean that where the air speeds up, the temperature must fall. So it really is cold up there on the ridge! And it is warm down there with Horace, as the air is nearly stopped!

Final words of wisdom. Air really is weird. When we need to discuss air in terms of temperature and pressure, we have entered a new world: the study of this new world is called thermodynamics.