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https://www.reddit.com/r/askscience/comments/4gkwao/if_i_could_very_accurately_measure_the/

I am not a member of Reddit, but if I was this would be my answer:

In theory, if you could measure the temperature field perfectly, you would have enough information to know some of the terms in the energy portion of the N-S equation. NASA provides a good picture:

https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html So essentially, you'd be able to know the E terms (using cp*T), the q terms (the heat flux), but you could not determine the shear stress, or the velocity components, u,v,w, and the pressure field unless you assumed P=rho*R*T.

This leaves a few cases:

If you had a body with no separation, you might be able to measure the heat generated in the boundary layer and compute the velocity profile. For simple cases like the flat plate you can even find exact solutions. From this information you could compute the viscous drag

If you assume P=rho*R*T and neglect viscosity You can figure out the magnitude of the velocity field, since you simply have the advective acceleration of the flow at that point equaling the pressure gradient.

If you a assume 1 or 2 dimensional flow

There might be some special cases where you can get an exact solution.

So overall:

It isn't practical

It can be done only in limited cases

For difficult and interesting problems, the answer is not without more information.