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I am working on a project involving Von Karman vortices coming off of a mountain. I was able to calculate the shedding frequency (thanks to tpg2114 in a prior question), but now find it necessary to calculate the wind speed of these Von Karman vortices. In about three days of searching I only came up with one article which mentioned how to obtain expected wind speeds from von Karman vortices (here if you would like to know, mostly with the sixth section of this being pertinent), through the use of the formula $V = \frac{0.72k}{2\pi r}$ where $V$ is the highest tangential velocity in the vortex and $r$ is the radius length at which the high speed occurs.

Based upon the information which I have and the formulas from this article, I can find the radius if I know what the speed at which the vortices are traveling. The $k$ in this formula however is the tricky part as it is called the "strength of line vortex in an infinite medium", but never numerically defined.

So here are my three questions in order from simplest to most difficult:

  1. Is there an easy way to calculate the vortex speed? This article defines it as by the identity $$4\pi\frac{a}{h}\frac{v_v}{v_a}\left(1- \frac{v_v}{v_a}\right) = 1$$ with $v_v$ being vortex velocity and $v_a$ being air velocity, but when this is solved yield a quadratic with two solutions, thus being not very helpful.

  2. What is a mathematical definition of $k$ so that I can get a value for it knowing the air speed (averaged at ~10 m/sec), shedding frequency, and mountain dimensions (height ~1000m)?

  3. Is this even the right direction to go into if I am attempting to approximate the influence that Von Karman vortices will have on wind speed measurements? If not, what should I do in order to figure this out?

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If you're looking for the average wind speed, you might use the Strouhal number: $St=\frac{f L}{v}$

where $f$ is the shedding frequency, $L$ is a characteristic length scale, and $v$ is velocity. You can find charts comparing Strouhal numbers to Reynolds numbers ($Re=\frac{vL}{\nu}$). I realize that both depend on velocity (v), but you know all the other variables ($L$, $\nu$, $f$). An iterative approach will quickly converge on a velocity:

  1. Guess A velocity
  2. Calculate $Re$
  3. Use chart to find $St$
  4. Solve for velocity
  5. Use new velocity to repeat steps 2-5 until you converge on an answer.

if you're looking for other components ($u$) of velocity due to the shed vortices, i'm sure there are papers that will help you once you determine the average velocity $v$.

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