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So I have a flat plate with a surface of 7sqm and an aspect ratio of 12 with air blowing straight at it (like a wing) at a speed of 50km/h. I would like to manually calculate the thickness of the laminar boundary layer at the trailing edge of the plank.

How do I do that?

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Look up the Blasius solution for the laminar boundary layer.

There are three measures of boundary layer thickness -- the plain thickness, the momentum thickness, and the displacement thickness. There are equations for all three available.

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