I was thinking about what should be the characteristic length taken to compute the Reynolds number for a real world problem. Let's say that I have a model of a city (several buildings with different shapes, width, height, etc...) where I'm trying to simulate the flow or the air around it. In this case, what's the "characteristic length"?

Note that even for a slow flow we could have a very high Reynolds number since the air viscosity is about 10^-5. In this sense, should a flow of air at 0.1 m/s be considered turbulent?! (I'm not only thinking about the Reynolds number, in this case, I'm just using my feelings...)

BTW, this "characteristic length", for me, only has sense when we are representing scales such those employed to represent airplanes models in wind tunnels. In this case we can reduce or enlarge the model and keep the Reynolds constant since we know, a priory, what "characteristic length" must be followed.

thanks for any discussion.

In practice the Reynolds numbers are so high for these types of problem that the flow is usually assumed to be considered Reynolds number independent (look at the roughness Reynolds number); i.e. the only diffusion is from the turbulence model.

You should think about the purpose of using the Reynolds number.

Non-dimensional parameters are used primarily because they make it very convenient to categorize flow problems. It's easy to compare an airplane model with the full scale airplane, if you describe the flow by non-dimensional Mach number, Reynolds number, Strouhal number, and so on... ... but don't forget that a direct comparison will only be meaningful if one geometry is a scaled version of the other (i.e. if they are "similar" in the geometrical sense).

You can compare a small building with a large building of the same shape, but does it make sense to compare New York City with Seattle? If the differences are complex, a single Reynolds number is not a useful parameter for categorization or comparison.

Do you need to define a Reynolds number to compute the flow around a complex geometry. No, of course not. Viscosity and density (for incompressible flow) will have to be defined.

The choice of a length scale is arbitrary any way. Some length scales just seem more reasonable than others, for simple geometries. Define the Reynolds number to serve a purpose. What do you want to do with the Reynolds number? If you can answer that question you'll find a proper way to defined it.

Are you simply looking to set the right switch on your turbulence model? Has your turbulence model been designed for cities or flat plates? How well does it cope with an array of flat plates with different sizes?

Ok, Tom & Mani confirmed my suspicious... Reynolds number in complex flows has no (or few) sense.

I haven't seen such kind of discussion in any fluid mechanics book or paper I've already read. I think that Reynolds number and other dimensionless numbers are "over-used" by some researchers. It's not hard to find CFD articles where the NS equations are presented in a dimensionless form, even when the paper is about some complex flow. There are those cases where the author simply doesn't present what's the "characteristic lenght" he's using in his experiments (and it's not so obvious for some problems).

As I said, the Reynolds number is very useful to define scales. As Mani pointed, when we're reproducing smaller or larger scales we must have some reference in order to be assured if our model is being well represented or not. Note that when dealing with computational tools, we are normally using the true scale. The cost of a CAD or a mesh will be equal regardless the dimension we're using --imagine how expensive would be a wind tunnel to simulate a true scaled airplane ;-)

In fact, this doubt came to my mind after looking at the air viscosity (too small). I thought that even for air flowing in a very low velocity I would have a turbulent flow cause the Reynolds number that would be high -- This assumption was the source of my doubt: "Would be the Reynolds number a good reference for this kind of flow?! As you answered, not!"

Answering Mani that asked me about what I want to do with the Reynolds number in this case.

I didn't want to do nothing ;-) I was just trying to realize how turbulent would be a real world flow related to air. Is it measurable?

Thanks for the discussion

Renato.

Dear Renato Probably you must refer to atmospheric boundary layer literature, i mention that in our turbulence cource there wrere some material about this subject so i recommend refer to : A First Course in Turbulence by H. Tennekes,JL Lumley, compact and plentifull book.

also i think that your lenght scale is tickness of atmospheric boundary layer which is function of air velosity, turbulence is realy complex phenomena and has several categories, there is not doubt that turbulence flow is city has different structure in comparison with other turbulence regime.

"also i think that your lenght scale is tickness of atmospheric boundary layer which is function of air velosity,.."

Most meteorologists/oceanographers would disagree with this - using only the viscosity of the fluid the Ekman layer would be less than 5m thick! The atmospheric boundary layer can vary from a few hundred metres to well over 1km depending on the flow conditions (i.e. stratification). The only way viscosity plays a role in this instance is that without it there would be no turbulence.

The main differences between atmospheric turbulence and that encountered in more industrial type applications is (i) the length scales are far greater - this increase far exceeds any reduction in velocity when estimating a Reynolds number, (ii) density stratification/buoyancy is significant within the boundary layer and (iii) the flow is always (effectively) fully turbulent and so there is no need to consider the lamiar and transitional regimes.

In atmosphic flows it is the roughness Reynolds number that is crucial due to the fact that the surfaces generally vary from the transitionally rough to the fully rough regime. Basically it is usually assumed that there is no laminar sub-layer and so laminar viscosity is insignificant.

Hi Renato,

Please be careful drawing conclusions too fast about the (un)importance of the Reynolds number or any dimensionless number.

Renato: "I think that Reynolds number and other dimensionless numbers are "over-used" by some researchers. It's not hard to find CFD articles where the NS equations are presented in a dimensionless form, even when the paper is about some complex flow."

It's true that the Reynoldsnumber (and Mach and Froude and many others) are being used to compare scaled models with the real world, but it says much more than that. The fact that the flow is complex or simple doesn't have anything to do with it. The Reynoldsnumber is certainly not over used, people express the equations in these ways in order to find out which term is important and which one is not and, next, in order to find a proper scheme to solve for the flow. By these means researchers try to prevent that they'll spend CPU time on something that has less influence than the numerical accuracy of their machine. It also gives you an insight in the parameters that you could use to trigger your design.

It's very dangerous to do things in dimensional form, because you may find 10 different empirical relations that in non-dimensional form happen to be the same curve.

Renato: "As I said, the Reynolds number is very useful to define scales. As Mani pointed, when we're reproducing smaller or larger scales we must have some reference in order to be assured if our model is being well represented or not. Note that when dealing with computational tools, we are normally using the true scale. The cost of a CAD or a mesh will be equal regardless the dimension we're using --imagine how expensive would be a wind tunnel to simulate a true scaled airplane ;-)"

I hope I answered that by now (see above). I'd rather state that the Reynolds number is under used You could also define your own dimensionless number, for example, Renato=(smallest grid cell size)/(largest builing size) in order to know what kind of turbulence you're able to capture with your CFD model (it's just an example, I don't want to start a discussion on this)

Renato: "In fact, this doubt came to my mind after looking at the air viscosity (too small). I thought that even for air flowing in a very low velocity I would have a turbulent flow cause the Reynolds number that would be high -- This assumption was the source of my doubt: "Would be the Reynolds number a good reference for this kind of flow?! As you answered, not!"

Yes, it is! You have to choose a proper length scale! That length scale depends on the physical phenomena that you want to study. If you want to study the formation of eddies of the same size as one building you'll need another characteristic length scale than when you want to study the boundary layer turbulence on the roof of one of those buildings. Compare it with flow over an airfoil where the characteristic length is the chord and not the roughness parameter of the biggest scratch in the surface, simply because people don't want to study the molecular formation of turbulence (of course there're some who do).

Gerrit: "Please be careful drawing conclusions too fast about the (un)importance of the Reynolds number or any dimensionless number."

Ok, I see your point. I'm not against the use of dimensionless number and forms. I recognize its importance. I only think that in ** some cases ** they are used without any specific purpose. Who already lost some time trying to understand what characteristic length an author used to define his experiment know what I mean. In these cases I'd rather see the model dimensions and fluid properties than a dimensionless number lacking information.

Gerrit: "people express the equations in these ways in order to find out which term is important and which one is not"

Ok, It's another point. No doubt dimensionless forms have its importance but in ** some cases ** we don't need to used them. We don't need to write the NS equation in a dimensionless form (using the Reynolds number for example) to understand how diffusive or convective a flow is. We know the terms and which one is responsible for the computational difficulties. In this case, we only need to define 3 properties (density, velocity and viscosity) but when using the dimensionless form with the Reynolds number we need to know also the characteristic length.

"We know the terms and which one is responsible for the computational difficulties. In this case, we only need to define 3 properties (density, velocity and viscosity) but when using the dimensionless form with the Reynolds number we need to know also the characteristic length."

If you are using an explicit time marching scheme you will find that you are actually calculating nondimensional numbers in order to ensure stability of the scheme; e.g. the local grid Reynolds number restricts your timestep just as the Courant number does.

As an extreme example of Gerrit's point consider flow over the Alps. How would you model this in a windtunnel which allows you to control the flow conditions? You're definely not going to build a full size model! By understanding the controlling nondimensional numbers you can attempt scale down the atmospheric flow conditions to a size where a windtunnel experiment is feasible.

As another example consider the above windtunnel simulation. What happens if I double the windspeed and strength of the stratification? If I perform the experiment and CFD simualtion I've just wasted my time because the results are approximately a rescaling of the original experiment (The Froude numbers are the same - although some care has to be taking with the Reynolds number and Monin-Obukhov length).

By the way, is there any LES model or some other CFD approach using FEM and/or FVM specific to deal with environmental and complex flows?

ps.: I changed the subject since most of the CFD problems are, in fact, "real world" problems. Flow in pipes, for example, is a real world application where Reynolds number is well defined ;-)

My comments were not meant to make you outright disregard non-dimensional parameters. I was trying to emphasize that you need to know what you are looking for.

I agree with Gerrit and Tom on the importance of Reynolds number for scaling (suppose you get a good shape representation of the Alpes), and also for determining dominant terms in the non-dimensional equations.

The problem with a complex configuration (complex geometry, not just complex flow), is that a single Reynolds number may not be sufficient to describe your problem in detail.

Let's stay with the city example and say you have a small wind velocity. You decide to take the largest possible scale, i.e. the size of the city, and get a very large Reynolds number. Does that mean the flow is turbulent? Much of it will be, yes! Just imagine wind blowing over a street that stretches through much of the city. Certainly the Reynolds number gives you a sense of instability on the scale that you're looking at.

Does that mean the flow over every building is turbulent? Not necessarily. You'd need a different Reynolds number to judge on that, based on a characteristic scale of the building.

Does the flow over the antenna of the TV tower show turbulent vortex shedding? That, again, will be determined by the local Reynolds number based on the antenna's diameter and the local flow, regardless of the size of the city.

My point is, your choice of Reynolds number will depend on your focus in the complex problem. If the large scale is all you care about, then that's what you take as a characteristic scale.

Likewise, the question which term in the equations is dominant cannot be answered generally for a flow of multiple scales. It will be different in each corner of the city, and you'll get a different picture depending on the scale of your interest.

interesting... Maybe looking at the maximum and minimum values would give us an idea of this kind of flow. For example, computing the cell based Reynolds number (Reynolds number computed at the scale of the grid cells) and filtering out the maximum and minimum values would give us an idea about where we have turbulent/laminar flows.

Citing a case: I saw a simulation of gas toxic dispersion in an US city (I guess was Seatle). This simulation employed an academic code in a very sophisticated system including a model built with satellite images and a very fine mesh. Maybe, depending on the scale taken account for the mesh and turbulence model, the results of this guy would be completely different. How can we say if this simulation has some sense or not?! If we put other parameters, such as the wind properties (direction, velocity, etc..) which we have only statistical data, this problem becomes much more complex.

NOTE: please, I'm in favor of dimensionless numbers when they are well employed and defined in a text! I was just talking about when they are misused or over-used.

Well it would be surprising if there were not any LES of enviromental flows - LES originated in meteorology (Smagorinsky was an atmospheric scientist).

Have a look at some of the national meteorology websites; e.g

http://www.ncar.ucar.edu/ , http://www.metoffice.gov.uk/ or http://www.mmm.ucar.edu

should be a start. Also look at the journal "Boundary-layer Meteorology" which covers research on flow over hills and urban areas as well as other things.

Hope this is of some relevance to your question, Tom.

I've computed the contamination of a town (approx. 200 buildings) by chemicals due to the terrorist attack. I used LES. Re was the approx. size of an eddy over a building. LES SGS was a Larey averaged equations, and it worked fine for a model (of a scale 1:100 in the lab) and for real world. But i think that flow properties depend more on Nu and Gr.