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 Ben January 8, 2007 22:15

Questions regarding strouhal number

Hi,

Strouhal number is defined as fL/U where L=characteristic length, f=frequency of the vortex shedding, U=velocity.

Some definition states simply that f=frequency of oscillation. Is it because of the "lock-in" effect? Hence frequency of oscillation ~ shedding frequency.

Does strouhal number has the same usage as Reynolds number? In other words, for 2 types of flow past a fixed cylinder, if their Re matches, their cl/cd will match. Similarly, for oscillating cylinder, if their St and Re matches, will their cl/cd match too?

But if they have different frequency (and different velocity to give the same St),shouldn't the plot of cl/cd vs time give 2 different shapes, each varying at its own frequency?

Hope someone can clear this doubt. Thanks

 ganesh January 9, 2007 02:40

Re: Questions regarding strouhal number

Dear Ben,

The floowing are my thoughts on your question. For an oscillating cylinder case, the unsteadiness is induced by the body motion so the frequency of motion is the governing parameter which is non-dimensionalised to giv the Strouhal number. To be more correct, I believe it must be called reduced frequeny as in aerolastic problems, but they do the same job: Non-dimensionalise the frequency. Strouhal number often quoted in laminar vortex shedding does it for shedding frequency(there is no forcing here, the shedding and unsteadiness is purely due to flow instability), while in an oscillating cylinder case the unsteadiness arises from the body motion, and therefore charcterised in the St or reduced frequency.

The idea of using non-dimensional parameters is to obtain a family of solutions depending on given parameter(s). So if two oscillating cylinders has the same St, but different U and f, their Cl vs t response woul not be the same, and the different frequencies could easily be seen, but on a non-dimensional time scale the responses merge on to a single curve, since the non-dimensioanl frequency or St is the same. Thus, a family of solutions can be described by using St. This is the same principle behind obtaining self-similar solutions and also the case of Cl/Cd vs Re you also mentioned.

Hope this helps

Regards,

Ganesh

 Paul January 9, 2007 10:33

Re: Questions regarding strouhal number

Hello Ben

You need to do a Laplace transform or Nyquist/Fresnel diagram of to present your results once you have a frequency.

The frequency can be a different response depending on your inlet frequency

Vin = A.exp(iw - kt)

Vout = B(w).exp(i.f(w) - kt)

Paul

 Ben January 9, 2007 11:54

Re: Questions regarding strouhal number

Thank you Ganesh! I now understand the 1st question.

As for the 2nd question, I don't really understand. For Re no., if the 2 cases are at the same Re, their cl/cd'll be the same. Are you saying that if I were to plot mean Cl vs St, I'll get the same curve for the 2 different cylinder?

Or put it this way, I do a simulation of an oscillating cylinder at Re=100 and St=0.5. f=0.5, L=1, U=1. I get the Cl/Cd vs time. If there is another simulation which runs at the same Re/St but now f=2, L=1, U=4, Is there any way I can predict how the cl/cd vs time will be like?

Thanks alot!

 ganesh January 9, 2007 15:45

Re: Questions regarding strouhal number

Dear Ben,

You have already mentioned that for a viscous steady flow, Cl = f(Re). Thus for a fixed Re, Cl is known. Flows past two cylinders with the same Re ( despite different diamters and flow velocities of the same fluid, say) will give the same value of Cl and Cd. On similar lines in an unsteady flow, with unsteadiness induced by some forcing/instability, we have Cl = f(Re,k) or f(Re,St), where k and St are essentially similar but distinguished as Reduced frequency and Strouhal number to differentiate the nature of unsteadiness. To quote an example, for the range 50<Re<200, the vortex shedding past a circular cylinder is laminar and St= g(Re), given by Williamson's correlation. Thus a known Re gives a unique St. If flow past two different configurations are considered with same Re, we get the same St and consequently the same Cl/Cd. Note that the mean qunatities would be predicted same, but the response against time would depend on the dimensional frequency. The fact that the Cl for flow past two different cylinders is same with Re is a consequece of non-dimensionalisation; you could not have predicted the Cl on the cylinder based on its diameter alone. Similarly, the Cl vs t response is different for the two cases you have quoted simply because you are playing in a dimenisonal scale, which gives the actually frequency of oscillation, but the non-dimensionalisation would give the same St. The reduced frequency is the ratio of time taken by the fluid particle to traverse in flow to the time period of the forcing. In the examples you have quoted, the forcing frequency increases as U increases, thus the ratio of the times are preserved and you end up in the same St. This gives a self similar solution since as mentioned before Cl= f(Re,St).

N.B.: A simple example is that cos wt and cos 2wt do have the same maxima and minima, but trace different curves with different frequencies.

Hope I have clarified the point.

Regards,

Ganesh

 ganesh January 9, 2007 15:48

Re: Questions regarding strouhal number

Dear Ben,

In the last line, "..is that cos wt and cos 2wt do have the same maxima and minima,..", I mean that the maximum and minimum values attained by the functions are same, though not at the same time. Further, the averaged value of these functions are both zero over an integral number of time periods, inspite of them being different functions in time.

Regards,

Ganesh

 Mani January 9, 2007 18:03

Re: Questions regarding strouhal number

Actually, the answer to the first question isn't quite as easy as presented by ganesh. Sure enough, in case of the stationary cylinder, the only Strouhal number is the non-dimensional frequency of vortex shedding. The oscillating cylinder, however, is generally governed by two distinct (but interacting) oscillatory phenomena: Cylinder motion and vortex shedding. In general, those two will have different frequencies, and you would therefore define two different non-dimensional frequencies. The structural non-dimensional frequency is usually referred to as "reduced frequency" and is a non-dimensionalized structural eigen-frequency, or natural frequency. The Strouhal number is strictly used for oscillatory fluid flow, and therefore still refers to the vortex shedding frequency (not to the structural frequency). Depending on combination of inertia, stiffness, and forcing, the vortex shedding may be forced to occur at the frequency of motion. Only when this "lock-in" occurs are the two frequencies identical. This is a very interesting case which is often studied, but is in no way the "general" case. Bottom line: Strouhal number always refers to flow oscillation, regardless if it's identical to the motion frequency or not.

 Ben January 15, 2007 20:32

Re: Questions regarding strouhal number

Hi,

thanks but I am still not too sure how to obtain the cl/cd for unsteady case running at different frequency although they 've the same Re & St.

Do I have to perform Laplace transform or Nyquist/Fresnel diagram as suggested by Paul?

I don't know how to do that. Is there any book to recommend how this can be done?

Thanks!

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