Validity of Spalding in y+ range
 

Hi,

As I have found in the literature the Spalding (and Reichardt) law seems to be valid until y+=2000.

https://www.sciencedirect.com/scienc...15X?via%3Dihub

In other publications I have seen that measured data depicts from Spalding profile starting from y+=300.

Does anyone have a more clear image of a generic range of y+ where Spalding law is valid? Is it completely dependable on any specific conditions?

Thanks

y+ of 1000 is near the edge of the boundary layer, almost the freestream.

I think that it does not include the wake component part of the profile though. But regarding my question is there any more complete answer?

"All models are wrong, some are useful"

Could you please elaborate more on this?

Can you give a complete question first?

Spalding curve fitted turbulent pipe data from Laufer at a Reynolds number of 500 000 and some data at 50 000. And whereas the data provided was y+<70, Spalding was concerned with asymptotic behavior at y+>25

Depending on how you define "validity" Spalding's curve is therefore valid only for y+ of 0 to 25.5, and only for the circular pipe of Laufer. Laufer's data was done with a real gas whereas Spalding's curve-fitting does not recover the correct compressibility effects that you would expect in the perturbation parameter for a compressible gas. Yet even further it does not recover the correct powers of growrth. Furtherfurthermore, Spalding himself noted there is nothing special about the coefficients he obtained. Spalding's curve therefore is valid just about nowhere.

So, what use is Spalding's law to you? Why are you concerned with what happens at y+ of 2000, which is the least universal scale and is problem dependent? If you plot the u+ vs y+ for a pipe, you will clearly see that the profile plateaus in a Reynolds number dependent fashion. And then if you change the geometry to a square pipe, it changes. Do you even have a pipe or is this a Spaceship?

Have you read Spalding's paper? I would suggest you start there.

I want to do a fit with the Spalding profile to data that simulate atmospheric boundary layer.

Generally the questions refers to ABL, sorry for not including that.

So your question is nothing to do with Spalding's work but really how to curve fit a velocity profile with a 4th order polynomial? Then just do it.

There are limits of course to how high of a y+ you can go and match real data. y+ is the scaling parameter only for the inner layer and it should be expected that curve fits using only y+ will not work in the outer layer, because the scaling parameter for the outer layer is y/delta and not y+. Boundary layer problems are multiscale problems. So I guess the answer to your question is curve-fits of y+ are valid throughout the inner layer, wherever it happens to stop. The y+ where the inner region becomes the outer region is flow dependent.

As far as I know Spalding profile (like Musker) can be used in the atmospheric boundary layer except pipe flows.

https://www.egr.msu.edu/tmual/Papers...ar_EIF2008.pdf

Okay so in the outer region these universal laws could not provide good fit because it is dependent on y/δ, that sounds reasonable. I guess in the inner region which includes:

1.viscous sublayer (linear+buffer) AND
2. the overlap region (aka logarithmic region)

the fit could be good for ABL near smooth wall.

When you say flow dependent you have some specific flow parameters in mind?

Yes. The Reynolds number.

If you have actual flow data you can just plot u+ vs y+ and see where the boundary is between inner and outer region. It will be easy to identify the outer layer between it's freakin flat in the outer layer. But there will be a gradual transition and not a sharp boundary (kind of like the buffer layer between linear and log layer). Do this at two different conditions and you quickly see it's not always the exact same y+ number where the inner layer terminates.

There is tons of work beyond Spalding's if you are intersted in the entire boundary layer. Take a look at Tennekes and Lumley for a generalized law of the wall which includes y/delta as a perturbation parameter. It is also well known that Spalding profile (if you use the 0.4 and 0.1108 of Spalding) is wrong for flows with pressure gradients. That doesn't mean you can't curve fit the inner region with y+, it just means that Spalding's specific tuned coefficients are the incorrect ones. Mixed asymptotic matching was also done earlier in the work of von Karman and predecessors (Nikuradse and Prandtl). All Spalding did was provide an easy to use curve fit.


Again, your question is really not about Spalding's work but where does the log law of the wall end? That should have been the question.

Is it so obvious when the outer region begins?

In these data (Oesterlund) I am not so sure that one could visually define it.

https://we.tl/t-BIwHc6szdw

As long as Spalding in smooth wall boundary layers is valid more or less in the inner region I think yes I should answer your question

Generally thanks a lot for the references you have provided and the breakdown of your thinking pattern, appreciate it

Atmospheric boundary layers are not smooth walls... You have trees and buildings and... stuff. "The constants 0.4 and 0.1108 are not to be regarded as sacrosant." Spalding, 1961 in his own words


That being said, inner layers are mathematically defined (in MAE principle) as the region where the inner scaling works (and that's why we call it the inner layer). It applies to other branches of physics as well. It is a characteristic of multiscale physical problems. For fluid flows this is due to competing viscous effects and inertial effects.

True, but in certain circumstances (for example for a certain roughness Reynolds number) they can behave more like a smooth wall BL meaning that roughness will not affect much the physical description of the problem. In other situations (where roughness is significant enough) one should use different relations in order to place importance on the roughness effects (for example Hama's work).

Yes. In several works, though where roughness is not critical I have seen numerous times to use these exact values stating that it is for smooth boundary layers.

But of course every problem depending on its nature could demand different values for κ and B (or e^(κ*Β) )

As mentioned by LuckyTran, the question really is about the validity of the inner law, which is Re dependent. A standard, probably not universal or even accurate, requirement is y/delta <0.1. Then you also need y+ > 30-50 (again, more common than probably accurate) to have a log law. Finally, you need to remember that the buffer layer part is mostly arbitrary for all the y+ insensitive laws that cover it, it is just a possible interpolation between the linear and log range.

Is it correct to apply Spalding when you have y/delta < 0.1 but your y+ is, at most, 50? Probably not, even if results will not be too out.

Thank you for your answer, too.

By reading this, 2 questions come in mind.

1. How can I clearly determine the end of the inner region or the start of my outer region for dataset that include measured velocities at different heights(without covering the full thickness of BL)


2. For a relatively smooth BL or generally for flows even in pipes which would be the minimum assumptions/requirements to apply a legit Spalding profile?

Let me rephrase the question a bit.

Given that I have velocity data only in the inner region, how can I determine where the inner region stops?

You can't. The inner region has no info about what is happening in the outer region except for the wall shear stress. You need more info, a priori knowledge.

But why this obsession with Spalding's profile? Spalding himself found multiple fits. Whatever you are trying to accomplish in your line of work, I need you to understand (just like Spalding did) that this one curve-fit is not "42". It is simply one possible curve fit of one guy's data. I recommend you to look at more generalized/unified wall functions that do go into the outer region. See Lumley's work for example.