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## Theory of Fluid Dynamics Part 3: Pressure

By Joe Supercool

You've backed up for still more? I assume you have read Parts 1 and 2 as otherwise this Part 3 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. Think Stanwell Tops, Torrey Pines, or up past General Cornpone on Mount Clarence.

We have a hill. It is outside of Esperance, in Western Australia, which is one beautiful town, about as far from anywhere as you'll ever get. It looks out over the Southern Ocean, so wind-farms and slope soaring are “de rigeur”.

The wind coming off the Southern Ocean has travelled a few thousand kilometres over water to get to our hill. So it is pretty smooth, no turbulence from trees, just the occasional wake turbulence from a passing Albatross or Cape Barren Goose, or blast of hot air from Horace, the farting Sea Lion.

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.

End reprise. So we have the streamlines, coming off the sea and climbing over the ridge. We also have the speed of the air as it runs along the streamlines. 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.

For some mad reason he has it the other way around, ie, getting velocity from pressure. In our work here, we have started with the flow flux and derived the velocity in the streamlines: I guess if you were measuring airspeed using a pitot tube his equation would have made more sense.

I’ll see if I can trick MS Word into letting me write this equation out.

We have:

V = wind-speed, m/s
^ means raise to the power of

y = the ratio of specific heats at constant pressure and constant temperature, which has the value of about 1.4. y is the Greek letter “gamma”

Po = the pressure in the air when the wind is not blowing

po = the air density when the wind is not blowing: units are Kg/m^3 = 1.28. That’s right, one cubic metre of air weighs 1.28 Kg! ? is the Greek letter “rho”

P = the air pressure in the flowing air, MKS units are Kg/m^2

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, where 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".

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

Also of interest is the pressure at the maximum wind speed. This occurs just a little back from the brow of the ridge. At that point, the pressure is at a minimum. Right where you want to stand while piloting your slope soarer!