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turbulence in 2D
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Hi all,
I have read on the book A first course of Turbulence Tenneke that in two-dimensional flow fluctuations that characterize turbulence cannot mantain themselfes. After reading this statement I have few questions: 1) In the book there are some examples of pure shear flows homogeneous in the x1 and x3 plane. Is this case (attached figure 3) one of the cases where fluctuations that characterize turbulence cannot mantain themselfes? If not, maybe I am not getting proprerly the definitiono of two-dimensional flows, can you giving me the right definition? 2) When in the book it is explained the Reynolds stresses and vortex stretching (figure 2 attached), the book states “Two-dimensional eddies (velocity fluctuations without a component normal to the x1, x2 plane) may on occasion have appreciable Reynolds stress, but the mean shear tends to rotate and strain them in such a way that they would lose their capacity for extracting e fom the mean flow rather quickly”. I did not understand why this happens to the two dimensional eddies and which is the definition of two-dimensional eddies (as far as I know all the eddies are placed on a plane). Thank you in advace |
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2D turbulence is a special case of the more general framework. First of all, it is a model used in geophysical flows where the characteristic integral lenght scale in the plane is greater than the extension in the third dimension. You can have a read in a dedicated section of the Lesieur's textbook and here https://www.annualreviews.org/doi/pd...-120710-101240 The more relevant issue is that the vorticity equation in 2D does not allow the presence of the action of the stretching term. This latter is responsible of the transformation of vortical structures from large to smaller lenght scale, a fact that is at the basis of the inertial energy cascade. As a consequence, in 2D you see only the component of the vorticity normal to the plane. Vortices can merge each other, producing larger vortical structures, a fact that is somehow representative of an inverse energy cascade: energy is transferred from smaller to large structures. Also the slope of the intertial energy range changes. |
Can you answer directly to the questions? I cannot understand you
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If you look at the first question I ve asked
1) In the book there are some examples of pure shear flows homogeneous in the x1 and x3 plane. Is this case (attached figure 3) one of the cases where fluctuations that characterize turbulence cannot mantain themselfes? If not, maybe I am not getting proprerly the definitiono of two-dimensional flows, can you giving me the right definition? I am asking about pure shear flow. I thank you for helping me, but I cannot see where is your answer to this questions. |
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If you see figure 2, there is a case of pure shear flow. Is that case a three-dimensional flow or a two-dimensional flow?
In this case, can the fluctuations that characterize turbulence mantain themselves? |
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"We have seen that molecular transport can be interpreted fairly easily in terms of the parameters of molecular motion. It is very tempting to apply a similar heuristic treatment to turbulent transport. We again use a pure shear flow as a basis for our discussion. " While Fig.3 discuss the case of the molecular viscosity that produces the 2D shear, the fig.2 analyses the general 3D Reynolds stresses. That is the average of the product of the 3D fluctuations, a term that appears in the decomposition of the non-linear convection of momentum in case of turbulence. Furthermore, it is stated that the energy to substain the 3D fluctuations is drained my the mean flow ("The interaction between eddies and the mean flow described here is essentially three dimensional"). In 2D turbulence, the way in which the fluctuations are energized is very different owing to the absence of the stretching. Small eddies disappear merging in large eddy. That means you have more energy in the mean flow and less in the fluctuations. If you want to study the turbulence I suggest to use more modern textbooks, such as that of Pope. |
Ok I have understand. I know, I have seen a lot of paper that cite Pope, but unforunately my advisor asked me to study on the Tenneke's book, since -he told me- that this could be more entry level for me.
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I also recommend Tennekes & Lumley for beginners.
Eddies in general are 3D with velocity fluctuations in x1,x2, and x3. The definition of 2D eddy is what you quoted: Quote:
You need to distinguish between mean flow and fluctuations about the mean flow. You need to be specific on whether you are saying that the mean part of the flow is 2D/3D and whether the turbulent part is 2D/3D (the turbulent part should always be 3D). The mean flow can look like a 2D flow (having 0 average velocity component in the third direction) but can still have velocity fluctuations in the third direction. To prevent a velocity fluctuation in the 3rd direction requires some divine intervention and so they don't really exist. A 2D eddy is a 2D fluctuation, which doesn't mean the same thing as a 2D "flow." |
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In my opinion, a modern textbook on turbulence is also required for an entry-level reader. Then, concerning the concept of 2D turbulence, be careful in the distinction of the previous approach from the 2D model that you can often read about some problems after using the Reynolds averaging. This is quite a different concept. |
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Sorry if I am coming back with an other question, but I have still a doubt. In this text it is cited, in the chapter that concerns vorticity dynamics, that (as shown in the figure attached) if the flow is entirely in the x1, x2 plane, this flow cannot turn or stretch the vorticy vector.
Does it mean that also in the pure shear flow the vortex stretching is zero? I am confused, is it possible that, being vortex stretching zero, the turbulence can mantain itself? |
You need to recalibrate the way you talk about flows now that you know something about turbulence. You keep saying "pure shear." I am concerned you are thinking of a laminar flows. Talking about turbulence in the same sentence would be paradoxical. A "pure shear flow" means a laminar flow to a lot of people (at least to me).
Let's say you have flow between parallel plates with one plate moving at some velocity and is a turbulent shear flow (the laminar case is boring). Let the plate be really really long in the 3rd direction (100 km or something). This is still a 3D flow and there is still vortex stretching even though the geometry is equivalent to a 2D one. Now suppose the hand of God somehow magically forces the third velocity component everywhere and all the time, now it is (by divine intervention) a 2D flow and no more is there vortex stretching because there is no vortex stretching in a 2D flow. |
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