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IIT. Illinois Institute. of Technology. 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. Outline. Introduction

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

  2. Outline • Introduction • Derivation of the composite Reynolds stress from the “inner” and “outer” fits of the mean velocity profile based onthe 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 turbulentboundary layers, Phys. Fluids 19, 2007

  3. The problem

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

  5. Mean velocity profile outer overlap inner

  6. Definition of “inner” fit U+inner(y+) with 1  k y+ = yut/n ; ut2 =tw /r Inner scaling : for y+ >> 1

  7. X = y+ dU+/dy+

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

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

  10. Coles sin2 with P = 0.55 ( k = 0.41 & B = 5.0 ) Rotta relation

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

  12. inner RS

  13. outer RS

  14. maximum RS

  15. maximum RS

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

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

  18. Open question :How to compare with pipe and channel ? Red*R+ ?? D = d*U+R from Sreenivasan & Sahay, 1997

  19. Conclusions • 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+h)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

  20. Virtual Origin + …..  in ZPG : dq /dx = dReq /dRex Rex (Req )

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

  22. 81% O2 -19% SF6 78% O2 -22% SF6 100% O2 79% O2 -21% SF6

  23. « Skin Friction »U+∞and Shape Parameter H H(Red*) Red* = H x Req U+∞(Req ), H(Req )

  24. + ….

  25. + ….

  26. ^ x from leading edge different measuring stations

  27. Virtual Origin + …. in ZPG : dq/dx = dReq /dRex

  28. ^ q/x from leading edge 0.016 Rex-0.15 KTH IIT q/x from virtual origin

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

  30. Open questionsTotal stress and its derivativeSpalart’s « plateaus »

  31. Open questionsTotal stress and its derivativeSpalart’s « plateaus »  Does not look like an universal feature

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