563 ME - Intermediate Fluid Dynamics - Su
Lecture 36 - Turbulence: more on scaling, and the Reynolds stress
In the last class we looked at the Kolmogorov similarity hypotheses, which expressed the idea that the smaller scales of turbulence approach a universal state for high enough Reynolds number. (The Reynolds number we're interested in is Re = u0 l0 /, where u0 is a characteristic velocity of the flow large scale, and l0 is a characteristic large-scale flow dimension.) That is, while the large scales will obviously be different, being set by the different flow boundary conditions, etc., the statistics of the small scales of the flow will be the same between flows if the Reynolds numbers are high enough. Also, if you look at the smallest turbulent scales, the turbulence at those scales will be isotropic, meaning that all of the spatial directions will look the same. Based on Kolmogorov's hypotheses, we can also extimate the size of the smallest turbulence length, time and velocity scales. From Kolmogorov's first similarity hypothesis, the universal form of the smallest turbulence scales is dependent only on the viscosity, and the rate of energy dissipation, . The units of these terms are = length2 velocity2 length2 , = = time time time3 (1)
where we'll note that is really the specific rate of energy dissipation, i.e. the rate per unit mass. From these, we can form a unique length scale, , time scale, , and velocity scale, u , as: = 3 1/2 = u = ()1/4 .
1/4
(2)
These are collective called the Kolmogorov scales, though if you see reference to `the Kolmogorov scale,' usually this means just the length scale, . From the scales (2), we can form the Reynolds number Re = u = 1, (3)
which tells us that at the Kolmogorov scales, inertial effects and viscous effects are of roughly equal importance. This is consistent with the idea that the the scales (2) represent the smallest scales of turbulence, and with the notion of the energy cascade we also discussed earlier. Remember that we said that in a turbulent flow, energy is transferred without dissipation to successively smaller scales until, at the smallest scales, viscosity becomes important and energy is dissipated. The scales defined by (2) give us useful insight, but in reality, the dissipation is very difficult to measure experimentally. It would be more convenient if we could express the Kolmogorov scales in terms of large-scale quantities. In particular, we'd like to express , and u in terms of the large-scale length and velocity scales l0 and u0 . To do this, we remember from our prior discussion (Lecture 34) that the dissipation rate scales with l0 and u0 as u3 0 , l0 (4)
which told us that the energy dissipated by a turbulent flow is proportional to the energy that's
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put into the cascade process at the large scales. Using (4) in (2), we get the relations Re-3/4 l0 u Re-1/4 u0 Re-1/2 (l0 /u0 )
(5)
where Re = u0 l0 /. From (5), it is clear that as the Reynolds number the increases, disparity between the large and small scales increases. For example, if we take a given flow system and bump up the Reynolds number by increasing the velocity, the small length scale will get smaller even as the large scale l0 stays the same.
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The Reynolds stress
u i = ui + ui ,
Recall from Lecture 34 that if we decompose the velocity field in turbulent flows as (6)
and also write p = p +p, then the evolution equation for ui becomes ui ui 1 p =- + + uj t xj xi
2
(7)
ui -
u i uj xj
,
(8)
which we compare to the Navier-Stokes equation for the raw ui : ui ui 1 p =- + + uj t xj xi
2
ui .
(9)
The difference is obviously the last term in (8), which we'll write as Rij , where Rij = ui uj . xj (10)
(Note: calling this term Rij is not standard notation.) It goes without saying that this term Rij /xj is significant. From (8) and (9), the raw field ui and the mean field ui would be identical except for Rij /xj . This means that all of the disorder that a turbulent flow manifests can be traced to Rij /xj . This is a pretty strong statement. (Consider Fig. 1, which shows the scalar concentration of a turbulent water jet. (The scalar in this case is a laser fluorescent dye in the jet fluid.) The mean scalar field is nice and smooth, basically Gaussian in profile. All of the disorder that you see is due to the scalar equivalent of the term Rij /xj .) What exactly does this Rij /xj represent? From our discussion of the Navier-Stokes equations, remember that (ignoring gravity) Dui 1 Tij , = Dt xj (11)
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Figure 1: The scalar concentration of a turbulent water jet. where Tij was the stress tensor, given by Tij = -pij + uj ui + xi xj . (12)
Supposing that we're thinking about a small fluid element, what (11) and (12) tell us is that the acceleration of the fluid element is caused by pressure forces acting on the element, and by the transfer of momentum between the element and the surrounding fluid. Now let's look at (8). With (11) and (12) in mind, rewrite (8) as ~ D ui 1 Tij , = Dt xj where ~ Tij = - p ij + uj ui + xi xj - u i uj , (14) (13)
~ and Tij is, effectively, the stress tensor that's relevant for the mean velocity field, ui (why can't we call this Tij ?). If we're again considering some fluid element, (13) and (14) describe the mean acceleration of the fluid element; and, looking at the terms in (14), the first term tells us that there's acceleration due to the mean pressure, the second term tells us that there's acceleration due to transfer of mean momentum, and the third term tells us that there's acceleration due to momentum transfer by the fluctuating velocity field. This last point is very significant, because one of the major features of turbulent flows is that they show increased transport of quantities like momentum, scalar concentration etc. In other words, the disorder and small-scale activity in turbulent flows makes it easier for quantities like momentum and scalar concentration to mix.
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