Flow around an airfoil

This is one of the earlier projects I worked on, but it’s a fascinating one, this process was done and will be done when before prototyping an airfoil profile for wings on airplanes. Airplanes went through a rapid development process from World War I until after World War II, it was an arms race and each side wanted to make sure their planes were better than the enemy’s planes. Many different wing types and profiles were produced and each one of them served a different purpose. The plane either had to be agile, create more lift to carry more load or roll better. Bending moments had to be taken into consideration when deciding the wings’ length and materials, it’s a really complex system. In the past (and I mean early 20th century) drag forces and aerodynamic coefficients were less used in the design but the trial and error method was more dominant. A lot of different countries and different people were involved in the development of flying machines, independently and in groups and in governments and so on.

Equation 1 is a classical aerodynamics formula that represents the force generated due to drag. I like to think about drag as the thing that’s pushing you back when you cycle really fast down a hill, but the best intuitive example is when you put your hand out the window while in a car. If you are in the habit of playing with air flow outside the window of your car you would notice that at low speeds the drag isn’t very powerful, but if you put your hand outside the window while the car is going 90 km\h you know it can get harsh. This is mainly due to the square relationship between Drag force and velocity.

 

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Equation 1 – Force of drag

where:
v -Velocity (normal)
ρ – Density of the fluid
Cd – Drag coefficient
A – Area in front of the fluid

The drag coefficient is dictated by the projection of the object and is usually an empirical value. The values can be seen in the list below.

 

2675e0e869aa5afcf4ea44bd4908acb8248a8a76_large.jpg

 

 

As always we start geometry. The first figure is a bit misleading and seems irrelevant, but if you look closely there’s a tiny hole near the origin point

Geometry 1
Figure 1 – Surface geometry

When performing this type of analysis we have to allow for enough excess surface area outside of the geometry we are sampling, this is mostly done so that end cases are also met. If it wasn’t done like this, the flow at y-infinity would not be realistic. the total length of the airfoil is about a meter, and we have over 20 times in each of the normal directions out of the airfoil.

 

Geometry 2
Figure 2 – Airfoil close up

Something to note is that the profile is very streamlined, meaning it doesn’t favor the lift direction in any particular direction. Or what I like to call a “neutral” airfoil.

 

Mesh 1
Figure 3 – Close up of the mesh around the airfoil

I had a lecturer from FEM course who said “a pretty mesh will fly” meaning when the mesh is visually pleasing, the simulation based on the mesh, will be reliable which in turn will generate reliable results for the plane or wing or whatever’s the case in point. I stress this in every post I have, a good mesh is essential for reliable results.

 

Mesh 2
Figure 4 – Zoomed out view of the mesh

A slightly bigger picture of the mesh around the airfoil, I tried to make the mesh as parallel to the airfoil as possible, the right-hand half of the mesh is just a grid that each element gets larger the further it is from the center

 

 

Mesh 3
Figure 5 – The y-infinity mesh

Mesh of the whole domain.

Let’s have a look at the flashy parts. The velocity  inlet is at a 10 degrees offset and 1 [m/s] from the whole semi-circle on the left

In the figure above we see the velocity streamlines, as expected the air hits the airfoil and creates the difference in velocity (as well as pressure, we know these are inversely related). Another thing to notice – the flow is laminar and there is no turbulence at these air speeds, also in the rest of the trail that can be seen in the next figure. I’ve also added the scale for the streamlines.

 

Slightly zoomed out depiction of the velocity profile around the airfoil. It’s nice to see the trail left by the airfoil all the way to y-infinity

 

The velocity profile of the whole domain

 

Next up, we look at the pressure profile

What I first notice is that the high-velocity focus is inverse to the high-pressure focus point, and this is how lift is generated – the air flows and the low pressure “pulls” the wing up, while the high pressure “pushes” the wing up

Pressure profile zoomed out

Pressure profile – whole domain

 

 

Conclusion

  • I’ve learned a lot about how to mesh and how a mesh should look before you run a calculation. Of course, a strong machine would help with the trial and error of how to properly set up a mesh.
  • This was an interesting simulation to run because this is effectively a wind-tunnel experiment that I can run without owning a wind-tunnel.
  • I think that in the future I’ll revisit a few more airfoil profiles so I can do a comparative study
  • Laminar flow is very important for consistent lift, I’d like to do a case study of different reynold’s numbers to research the end cases of this particular airfoil
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