As Prof. Andriy mentioned it depends on the what problem you are looking to solve. Researchers use different dimensionless length depends on their convenient such as displacement and momentum thickness, ratios of distance with diameter and length. However your ultimate challenge lies in resolving the boundary layer by choosing the appropriate y plus value where the boundary layer thickness will be used.

Thank you very much for your answer, Prof. Andriy. Different problems have different choices. So would you recommend any references or sites about how to chose the dimensionless length for  specifical problem? And I can get a more detailed understanding.

Thanks a lot for you detailed answer, Mr.Murugan. Indeed, appropriate y plus is vital for resolving the boundary layer.

Concerning DNS, consider the example of the time evolving boundary layer in a plane channel.

Usually, the equations are normalized in terms of the reference velocity u_tau=sqrt(tau_wall/rho). As a consequence, the resolved velocity is the non-dimensional u+ and the forcing pressure gradient is simply the unity. Similarily, the lenghts are normalized in terms of a reference lenght, for example the heigh of the channel H=2*delta. Consequently, the characteristic Reynolds numebr is Re_tau=u_tau*delta/ni.  Then you can see  y* = y/delta= y/(Re_tau*ni/u_tau) = (u_tau*y/ni)/Re_tau = y+/Re_tau.

Time is made non-dimensional using u_tau and delta. All other statistical quantities are made non-dimensional using such reference values.

It depends which sublayer (outer, inertial, or inner) we are talking about. This is because each sublayer has its own distinct characteristic length scale to be used for normalization purposes.

Hi Shijie,

I agree with Filippo about DNS characteristic design lengths. I would like to add a few points to the current discussion.

Answering your main question very quickly: If you consider a real engineering problem, it will be almost for sure turbulent. Meaning that you will have a multi-scale problem, from the largest (boundary layer thickness d99) scales to the smallest scales where dissipation occurs (at kolmogorov scale), measured in wall-units. Therefore, if you want to design a simulation you need to resolve all your scales. You will design your computational box to fit a few of the largest scales (for example a spanwise length of 8*d99) and to capture the smallest one, of the order of the thickness of vortex (around 10 kolmogorov+). You will find in the literature people talking about Re_tau numbers, that is the thickness of the TBL in wall units, giving you an estimate of the scale disparity (approx Re_tau/100+; where 100+ is the characteristic spanwise length of the energetic  buffer layer streaks)

The characteristic lengths that governs a "canonical" boundary layer (zero-pressure gradient, incompressible) is a field of ongoing research. Much have been talk about  which are the characteristic [length/time] scales of the velocity or pressure fluctuations, of the very large scale motions (VLS), or of the different part of the energy spectra for the streamwise velocity fluctuations (Psi_uu^+), etc...

Sticking strictly to lengths in BL, to say that to the standard momentum and displacement thickness, you will find in the literature lengths like the Rota-Clauser thickness, the Taylor microscale (a blend between inner and outer lengths), the Kolmogorov scale (characteristic length of the thickness of vorticity), or the characteristic length scale of a thermal boundary layers in high compressible flows. 

You can get a flavor of this topic in the review of Jimenez & Hoyas (2008) "Turbulent fluctuations above the buffer layer of wall-bounded flows",  and also in Sillero et al. PoF (2013) "One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to δ+≈2000". Those papers are about how turbulent quantities scales or not across a boundary layer. 

Finally, if you want to check how to numerically design a DNS grid to resolve your problem, you will find an ample discussion in the second chapter of my thesis.

Best

http://torroja.dmt.upm.es/pubs/2008/JimHoyas_JFM08.pdf

http://torroja.dmt.upm.es/pubs/2013/sillero_jimenez_moser_PoF13.pdf

http://torroja.dmt.upm.es/pubs/2014/thesis_sillero.pdf

Thank you very much Mr. Juan A ! Your answer is of great help to me. I will learn the further information from the papers you recommonded.

Hello,

Since your question is  which reference length is appropriate for the description of dimensionless length of the domain, the answer is the size of the largest eddies, which is the boundary-layer thickness, as Juan Silero said.

For other details, other length-scales can be more appropriate.

Thank you Mr.Alexandre Silva Lopes. Yep, I finally use the  boundary-layer thickness as the reference length of the domain.

I have figured out this question with the help of scholars upstairs and refering to some text books.   Actually, this question is mainly concerning the so-called inner and outer scales, two views developed over the past century for describeing the wall flow struture .  wall unit, calculated from the inner variables (friction velocity and kinematic viscosity), is approriate for the characterization of the flow physics very near the wall surface. Boundary layer thickness, momentum  thickness and  displacement thickness are approriate for scaling the bulk or large scale motion away from the wall.

When you use wall units you can see in more detail what happens near the wall. This in semi-log scale.