[cfdnewb123]

Turbulence is characterized by a range of scales. However, when defining velocity scales to flows like turbulent jet/shear layer/channel flow, they only have one velocity scales while turbulent wake has two velocity scales. How do we go about assigning velocity scales to different flow types?

[LuckyTran]

Shear flows are self-similar about the velocity difference and so you generally don't need to take about absolute velocity scales. But for any boundary layer problem (shear flows, channel flows, etc.), you have at least the bulk velocity scale and the friction velocity. So it is not correct that there is only one velocity scale.

[cfdnewb123]

If you look at Tennekes and Lumley, Chapter 4, it shows that turbulent jets and mixing layers only have one velocity scales while turbulent wake has two. This is confusing to me because how do we decide if a flow has one or more velocity scales? Velocity scale is like the order of magnitude and since the velocity of the flow crosswise is continuous (e.g., from zero to centerline maximum velocity), shouldn't they all just have one velocity scale?

[LuckyTran]

You don't need to decide, you need to recognize.

You can just look at the governing equation. Each term has an associated physical phenomenon. For fluid flows that means Navier-Stokes which has three terms: the unsteady inertial term, convective term, a diffusion term, and a pressure term. Each has a velocity scale (even the pressure term has an acoustic speed for the compressibility waves).

For problems that exhibit similarity, the scales become relatable to one another and the number of independent scales that you need to define the problem becomes less. The infamous scaling parameters are things like your Reynolds number and so on. Actually, the existence/presence of multiple scales is what gives rise to boundary layers.

Finally, if you zoom all the way into the boundary layer and neglect the freestream solution then you will of course only see the diffusion scale and not the freestream scale.

[cfdnewb123]

In that case, does this means that the total number of scales is limited to the number of types of variables (e.g., diffusion, convection, unsteadiness, bulk viscosity/pressure) in the Navier-Stokes equation?
(The boundary layer is a very good example!)

But what about free shear flow like wake, jet and mixing layer. They all have the same equations of udu/dx+vdu/dy = -d(u'v')/dy. It make sense that turbulent wake has two velocity scales (e.g., diffusion by fluctuating velocity and freestream) but why does jet and mixing layer have one velocity scale only? For instance, mixing layer should have two velocity scales instead of one since it has diffusion scale and relative velocity between two freestreams. As for jet, it should have one velocity scale for diffusion and another for centerline velocity.

[LuckyTran]

udu/dx+vdu/dy = -d(u'v')/dy is what you get after you do the order of magnitude analysis and find out that molecular diffusion does not transport much momentum and then drop that term from the momentum equation. It should make sense that when you assume there is no diffusion, that corresponding velocity scale does not show up in your bookkeeping. In reality, the scale still exists.

Free shear flows have a natural boundary condition whereas wall bounded flows have a wall boundary condition. Near walls, the velocity and length scales are limited (because fluid cannot flow thru a wall) and so the Reynolds stresses cannot carry all the momentum everywhere like in free shear flows. If you move away from walls, you can also do the same trick and drop the diffusion term for wall bounded flows as well. It is more obvious that diffusion plays a very significant role in wall bounded flows and less obvious that diffusion plays a tiny role in free shear flows.

Even for 1 flow, you have different sizes for the scales at different locations in the flow. But for any location in the flow, you should look to identify the scales corresponding to each process. They have scales. You simplify the equations after doing the order of magnitude analysis, not before.