From what you describe there are two different length scales associated with this problem. The one ascociated with flow through the porous medium (the 'bed packed with some objects'), and the second ascociated with the flow around the 'bed'. I will assume you are talking about the flow through the porous 'bed'.
Normally the characteristic dimension or length scale for internal flows is taken to be the hydraulic diameter. This is defined to be four times the cross-sectional area (of the fluid), divided by the wetted perimeter. However, for such things as 'pebble beds' etc. the Reynolds number is defined differently.
For flow of fluid through a bed of approximately spherical particles of diameter D in contact, if the voidage (fraction of the bed not filled with particles) is ε and the superficial velocity V (that is, the fluid velocity through the bed as if the spheres/objects were not present) then a Reynolds number can be defined as:
$$Re = \frac{\rho V D}{\mu(1 - \epsilon)}$$
Laminar conditions apply up to Re = 10, fully turbulent from 2000 (Wikipedia). There are more advanced formulas for this, and they work in a variety of regimes; from not-so-packed beds, to very packed-beds, also with a variaty of pebble/object shapes.
Many experiments have been done on convective and radiative heat transfer in pebble bed nuclear reactors and other such heat exchangers. I am sure you should be able to find some journal papers on this stuff along with the standard correlations you need for your particular flow.
For the convective heat transfer coefficient for this flow however, you should be using the Nusselt Number which is a measure of the ratio of convective to conductive heat transfer a solid-fluid boundary.
I hope this helps.
I think @Killercam is right, I'll try to explain the same thing a little more elaborately.
Firstly. in the case considered, since the fluid and the cylinder is chosen, increase in velocity directly translates to increase in the Reynolds number as $R_e = \frac{\rho V D}{\mu}$.
Before considering flow in the range $250 < R_e < 2\times 10^5$ , lets first observe what happens in the region where the viscous force dominates over inertial forces i.e $R_e <<1$. The fluid slowly "crawls" over the surface of the cylinder. There are 2 stagnation points on the leftmost and rightmost parts of the cylinder.
From the solution for inviscid flow over cylinder (superposition) we can note that tangential velocity is maximum at mid-section and decreases as and decreases as we proceed "downhill"
This can be extended to viscous flow and two important things are to be noted here:
- The shear stress is maximum at the mid-section, which is implicit as a higher velocity gradient is created because of the larger value of tangential velocity.
- The static pressure starts to increase after the mid-section. i.e the pressure is increasing in the direction of flow $\frac{\partial P}{\partial x}>0$ which is called as an adverse pressure gradient
As $R_e$ is increased (i.e velocity is increased), the inertial forces start to dominate over viscous forces.The flow velocity is zero at the surface and the particles very close to the boundary have a very low momentum since they experience very strong viscous forces. On the right part of the cylinder, the fluid particles close to the cylinder not only experience strong viscous force, but also adverse pressure gradient which eventually forces the fluid particles to stop/reversed, causing the neighboring particles o move away from the surface. This is called as flow separation. It results in the creation of a free shear layer which ultimately rolls up to form a vortex.
@Killercam said:
The velocity of the flow divided by the diameter of the cylinder is the typical crossing time of the fluid, hence is directly related to the frequency of the observed oscillations for a specific Reynolds number.
After a vortex is shed, the fluid particles behind have to undergo the same process i.e it takes the same distance to come to rest and then causing the neighboring particles to separate. Since this distance is a small part of the cylinder, $dist \propto D$ and hence the time interval between 2 vortices shed from the same side(top/bottom) of the cylinder $time \propto \frac{dist}{V}$ i.e $time \propto \frac{D}{V}$.
The time interval is exactly the time period of vortex shedding and hence the frequency of shedding is
$f=\frac{1}{T}\propto \frac{V}{D}$
Best Answer
So how this is done is a bit of a black art, much like how you choose what to use for other non-dimensional numbers in fluids (like Reynolds number). But you sort of have it backwards. You don't want to compute $St$ to find the shedding frequency; $St$ is good if you want to compare flows over different conditions but want to show the physics is the same, or it's good if you were given the number, velocity, and length scale from somebody and need to compute the frequency.
The best way, if you can, to compute the shedding frequency is to take a time signal of your velocity and compute the FFT of it and look for spikes in low-frequency ranges. Depending on the frequencies of the vortices, you may not have the appropriate temporal resolution. For instance, if the vortex is shedding at 1 Hz (1 vortex per second), then you need to take at least 2 samples per second due to frequency aliasing.
Assuming you have a frequency computed, your next choice is the velocity. Again, there is some art in choosing the correct one. The best choice here is the wind-speed ahead of the mountain/cliff if you have that data (or outside of the shed vortices behind the mountain so you are getting a "free-stream" velocity). Any velocity you choose from inside the vortex shedding would have to be averaged to remove the vortex effects. But it's unlikely you'll have steady data and so your average will be a moving one, not very helpful.
Lastly is the choice of length. Cylinders choose the diameter typically; bluff bodies or backward facing steps choose the step height. So for a cliff, the height of the cliff is a good number. Same could be said for a mountain.
Ultimately though, what you choose doesn't matter so long as you choose it explicitly and consistently! So if you want the length scale to be the height of the cliff, always use that one when comparing $St$ from various conditions. Same goes for the other terms; if you are publishing, please be very clear about what you are choosing. It's very frustrating to try and compare results with incomplete information and makes it very suspicious when authors don't list their reference scales and only present the resulting non-dimensional number!