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I am developing a project for the subject of fluid mechanics and I had a question. The project we want to develop involves using wind and a small hill as a giant wind tunnel. The hill is filled with wheat and when there is wind, the wheat is bent and you can see a black spot propagating through the field. I thought that if the speed of those spots where measured, you could have the velocity field close to the surface for the whole hill. Then we could apply the equations to determine if the velocity field of a horizontal wind with that obstacle corresponds to the experimental results.

I assume that the propagation of the black spots is the speed of wind in that small region. The main question I have is regarding how wind behaves. Could I consider that the entering wind has the same speed in the vertical axis ?. Worded in another way, is wind locally uniform ? The hill is surrounded by flat ground, so there are not any obstacles surrounding it.

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  • $\begingroup$ You can be pretty certain the wind is not uniform. Assume a mild 2 m/s wind, a 30m scale, and then you get a Reynolds number of 4.2 million - it will be turbulent. I assume those spots are vortices, and they might have velocities different from the wind. $\endgroup$ Apr 12, 2022 at 10:10

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I like the idea and the ambition of your question, however, there are some difficulties.

Firstly, and most importantly, as @Anders Sandberg rightly commentated, the flow is indeed turbulent, so there is no uniformity we can hope for (neither in space nor in time). What could be a goal is to measure statistics of the flow, long-term averages etc.

Some more details:

  • The black spot you mention might be what is know as a "cat's paw", a turbulent phenomenon that is often seen in dark or rippled spots sweeping over a water surface. (https://www.youtube.com/watch?v=WzCajzzfu-k) (An interesting reference not just for this aspect might be R.S. Scorer's "Environmental Aerodynamics".)
  • There is no uniformity of velocity in the vertical direction: Over a flat surface the velocity profile increases as the log of the vertical coordinate. (https://en.wikipedia.org/wiki/Law_of_the_wall) Boundary conditions for this also depend on surface roughness - i.e. a wheat field has a different influence on this profile as compared to bare ground.
  • The hill changes the above-mentioned logarithmic profile and in part diverts the flow to the sides, around it (depending on shape, height etc. etc. For more detail - but probably overkill for your project description - there are some papers by Jackson&Hunt, mid- to late seventies, on a perturbative calculation of that flow profile.).
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  • $\begingroup$ I will therefore do a long measurement of the cat's paw and work with the average of the measurement. The second solution in this thread says that the speed of those spots will be close to the air speed. I will try to find the relationship between both and infer the speed of air from those spots. Hopefully it is possible $\endgroup$
    – Drn
    Apr 12, 2022 at 14:55
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These spots you see are the waves generated on the plant canopy due to the coherent turbulent structures in the wind field. This waves are also called honami and they are mostly associated with the sweep and ejection type of structures. A sweep event means a higher momentum air moving downwards. In general, the convective (advective) velocities are not identical to the local flow velocity, but will be close. Remember, that sweep structures are higher-momentum regions and will be somewhat faster. Ejections, on the other hand, will be somewhat slower.

They have been studied for a long time. See, for example, Finnigan (1979) Turbulence in waving wheat. A simple mathematical model was proposed by Finnigan & Mulhearn (1978)

The wind profile in the atmospheric surface layer (the lowest part of the atmospheric boundary layer) can most often be modeled as logarithmic as the link to the law-of-the-wall given by kricheli shows. However, the law-of-the wall proper is for smooth surfaces. The rough wall profiles are usually formulated using equivalent sand paper roughness engineering, but boundary layer meteorology uses a different description.

Firstly you have to shift your zero plane (the zero-plane-displacement parameter d) and the in real atmosphere there will be departures from the logarithmic wind profile due to stability according to the Monin-Obukhov similarity theory.

The roughness length is an important parameter for the surface layer velocity profile. Unfortunately, it is not a constant and will definitely depend on the velocity itself because of the interaction of the flow with the plant canopy.

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  • $\begingroup$ I will then use the article on convective velocities to try and infer the speed of air from the speed of the black spots. Regarding the fact that the surface is not smooth, you mention that I have to shift the zero plane and a few other things. Is it still possible to have a theoretical model that predicts the way air moves around the hill using the links you have sent? Or do I have to look into additional sources ? Also thank you for your time, same to kricheli. $\endgroup$
    – Drn
    Apr 12, 2022 at 15:03
  • $\begingroup$ @Drn A flow around a hill is a huge topic and those links do not touch it at all, they are purely about the structures in a wheat field. Flows around terrain can be complicated and there are hundreds of articles about that and some books too. There are some realistic flow-around-a-hill test cases described at wemep.readthedocs.io/en/latest/windconditions/… But I am afraid it requires some extensive prior knowledge of geophysical fluid dynamics. The references include the experimental campaigns and computational studies reproducing the cases. $\endgroup$ Apr 12, 2022 at 15:09
  • $\begingroup$ I understand. However, do you think that with a wide enough hill, approximating it to a infinite slope could be enough to explain the basis of the phenomenon ? My interest is experimentally proving with this project that the way a fluid changes its velocity field corresponds with the predictions made by the theory taught in class. So maybe with this approximation the goal can be obtained. In this case it wouldnt be a hill, but a slope. $\endgroup$
    – Drn
    Apr 12, 2022 at 17:21

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