1. Wall bounded shear flows – INI – September 8-12, 2008 The Turbulent Shear Stress in ZPG Boundary Layers Peter A. Monkewitz, Kapil A. Chauhan & Hassan M. Nagib IIT Illinois Institute of Technology

2.  Introduction  Derivation of the composite Reynolds stress from the “inner” and “outer” fits of the mean velocity profile based on the log-law in the overlap region  The asymptotic location and magnitude of the maximum Reynolds stress  Open issues: the contribution of normal stresses comparison to pipe and channel flows References : Panton, Review of wall turbulence as described by composite expansions, Appl. Mech. Rev. 58, 2005 Monkewitz, Chauhan & Nagib, Self-consistent high-Reynolds number asymptotics for ZPG turbulent boundary layers, Phys. Fluids 19, 2007 Outline

3. The problem

4. withfor inner functions for outer functions “inner” and “outer” fits for the RS from the mean velocity profile

5. Mean velocity profile inner overlap outer

6. Definition of “inner” fit U + inner (y + ) Inner scaling : with  1  for y + >> 1 y + = yu  / ; u  2 =  w / 

7.  = y + dU + /dy +

8. Definition of “Outer” Fit Outer scaling : ; y + /  = >> 1 for  << 1 

10. “Inner” - “Outer” Matching Rotta relation : Overlap region :

11. Rotta relation Coles sin 2 with  = 0.55 (  = 0.41 & B = 5.0 )

12. withfor inner functions for outer functions “inner” and “outer” fits for the RS from the mean velocity profile Integration with respect to y

13. inner RS

14. outer RS

15. maximum RS

17. Open question : The role of normal stresses 0 0.1 0.2   Small effect on U +   Effect on RS apparently relatively minor – needs further study

18. Open question : The role of normal stresses From Philipp Schlatter, …., D. Henningson, 22nd ICTAM, Adelaide, 2008

19. Open question : How to compare with pipe and channel ? from Sreenivasan & Sahay, 1997 Re  * R + ??  =  * U +  R

20.  The mean velocity & RS modeled with 2 layers overlapping in the log-region fit the data well  The location of maximum RS scales on the intermediate variable (y +  ) 1/2 but this does NOT imply a third layer with different physics !  Open questions :  Scaling and influence of normal stresses on RS (appears to be small – as on virtual origin)  Comparison with pipe and channel flow Conclusions

21. Virtual Origin in ZPG : d  /dx = dRe  / dRe x  Re x (Re  )  + …..

22. Virtual Origin with x from virtual origin with nominal x ^

23. 100% O 2 81% O 2 - 19% SF 6 79% O 2 - 21% SF 6 78% O 2 - 22% SF 6

26. « Skin Friction » U + ∞ and Shape Parameter H H(Re  * ) Re  * = H x Re  U + ∞ (Re  ), H(Re  )

27. + ….

29. x from leading edge ^ different measuring stations

30. Virtual Origin in ZPG : d  /dx = dRe  /dRe x + ….

31.  / x from virtual origin   / x from leading edge ^ KTH IIT 0.016 Re x -0.15

32. Open question : PARTIAL contribution to the log law correction at O(1/Re)  Information from experiments/DNS « somewhat diverse »

33. Open questions Total stress and its derivative Spalart’s « plateaus »

34.  Does not look like an universal feature