1. Turbulent Scaling in the Viscous Sublayer
SOLITONS, COLLAPSES AND TURBULENCE
VI-TH INTERNATIONAL CONFERENCE, NOVOIBIRSK, RUSSIA, 4-8 JUNE 2012
Turbulent Scaling in the Viscous Sublayer
Dmitrii Ph. Sikovsky
Novosibirsk State University, Laboratory for Nonlinear Wave Processes,
Kutateladze Institute of Thermophysics, Laboratory for Fundamentals of Energy Technologies

2. Turbulence in Channel Friction velocity Governing Parameters:
Reynolds Number:

3. Derivation of Logarithmic Law of the Wall:
Recent debate concerning the mean-velocity profile of a turbulent wall-bounded flow
Derivation of Logarithmic Law of the Wall:
Similarity hypothesis: Th. von Karman (1930)
Mixing-length hypothesis: L.Prandtl (1932)
Dimensional reasoning: L.D.Landau (1944)
Questions and Doubts
Nikuradse, J., Gesetzmabigkeiten der turbulenten Stromung in glatten Rohren. VDI-Fortschritt-Heft, No. 356.
Nunner, W., Warmeubergang und Druckabfall inrauhen Rohre. VDI-Forschungsheft, No. 455.
Simpson RL. Characteristics of turbulent boundary layersat low Reynolds numbers with and without transpiration. J Fluid Mech 1970;42:769–802.
Malkus WVR. Turbulent velocity profiles from stability criteria. J Fluid Mech 1979;90:401–14.
Barenblatt GI. Similarity, self-similarity, and intermediate hypothesis. New York: Plenum; 1979.
Long RR, Chen T-C. Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 1981;105:19–59.
Wei T, Willmarth WW. Reynolds-number effects on the structure of a turbulent channel flow. J Fluid Mech 1989; 204:57–95.
George WK, Castillo L, Knecht P. The zero-pressure gradient turbulent boundary layer revisited. In: Reed XB, Patterson GK, Zakin JL, editors. Thirteenth symposium on turbulence. Rolla, MO: University of Missouri; 1992.
Bradshaw P. Turbulence: the chief outstanding difficulty of our subject. Exp Fluids 1994;16:203–16.
Purushothaman K. Reynolds number effects and the momentum flux in turbulent boundary layers. PhD dissertation, Department of Mechanical Engineering, Yale University, New Haven, CT, 1993.
Smith RW. Effect of Reynolds number on the structure of turbulent boundary layers. PhD thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1994.
Afzal N. Power law and logarithmic law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers. Acta Mech 2001;151:195–216.
Buschmann M.H., Gad-el-Hak M. Recent developments in scaling of wall-bounded flows. Prog. Aero. Sci :419-67

4. Plan of the presentation
Classical approach from the viewpoint of method of matched asymptotic expansions
Deficiencies of classical scaling
New scaling for the viscous sublayer

5. Formulation of the Problem
Reynolds averaged equation
for mean momentum transport:
Boundary conditions:
at the wall,
at the centerline,
Viscosity n is a small parameter at higher derivative.
Singular perturbation problem! (K.Yajnik, 1970)

6. Matched Asymptotic Expansions
M.Van Dyke. Perturbation Methods in Fluid Mechanics.N.Y.:Acad. Press, 1964.
Outer asymptotic expansion (K.Yajnik, 1970)
in the outer layer
Viscosity does not affect Reynolds shear stress in the leading order of approximation!
Extension to mean velocity, whole Reynolds stress tensor and higher-order moments of velocity fluctuations leads to Townsend’s Reynolds Number Similarity Principle (A.Townsend, 1956)
does not satisfy the wall boudary condition

7. Classical Scaling: Outer Layer
Since in the near-wall region
then it is reasonable to assume that
Outer layer expansions

8. Classical Scaling: Inner Layer
Rescale the variables in equation so that the viscous stress will be of the same order as Reynolds stress If
then the stretched varible is
This can be derived from classical similarity hypothesis (Monin,Yaglom, 1965; Kader, Yaglom, 1980): friction velocity is a scale for the velocity fluctuations and mean velocity differences
Inner expansions: Law of the Wall

9. Overlap Layer: Matching
Introduce
Van Dyke Matching Principle: The m-term inner expansion of the n-term outer expansion = the n-term outer expansion of the m-term inner expansion
Matching leads to Pexider’s functional equation
Solution is Logarithmic Law of the Wall
From experimental data:
Log layer observed in the range
Von Karman constant:
Friction Law:

10. Weakness of Classical Scaling
Usage of friction velocity as a velocity scale for the outer flow
Influence of the outer scales on the statistics in the viscous sublayer
Possible incomplete similarity
DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

11. Alternative Theories: Power Law or Log Law?
G.I.Barenblatt: No Reynolds number similarity
W.K.George: Reynolds number dependence of
N.Afzal: No Reynolds number similarity in the overlap layer

12. Experimental Results: SuperPipe Data?
McKeon et al., 2004 (JFM):
Logarithmic law is found
in the range
Power law is found in the range

13. Revisiting the Classical Scaling in the Viscous Sublayer
There can be a continuum of possible scalings in the viscous sublayer. Viscous sublayer thickness can be chosen arbitrarily, if the mean velocity scale is chosen as
Consider the following exact conditions at the wall for the derivatives of mean velocity and Reynolds shear stress following from Navier-Stokes equation (Monin, Yaglom, 1975)
where

14. Revisiting the Classical Scaling in the Viscous Sublayer
New scale for viscous sublayer thickness
instead of
Introduce the new variables
Mean momentum transport:
Boundary conditions at the wall:
Universal scaling of mean velocity
and Reynolds shear stress:

15. Comparison with DNS Data: Mean Velocity and Reynolds Shear Stress
Comparison of classical (left) and new (right) scaling of Reynolds shear stresses and mean velocity (inserts). Curves – DNS data in the range , dashed lines mark off the rough border of viscous sublayer.

16. Comparison with DNS Data: Rms. Streamwise velocity
Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

17. Comparison with DNS Data: Rms. Normal Velocity
Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

18. Comparison with DNS Data: Rms. Spanwise Velocity
Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

19. Overlap Layer: Matching
By introducing the new variables
this equations transformed into the Vincze’s functional equation
General solution is

20. Power Law with Non-Universal Exponent
ZPG TBL
Circular pipe
Plane channel
(Cenedese et al., 1998)
Reynolds number dependence of parameter

21. Conclusion
The new inner scaling is proposed, in which the new relevant parameter – the third-order wall-normal derivative of the Reynolds shear stress at the wall is added. Proposed scaling gives the better collapse for both the mean velocity and Reynolds stresses profiles in the viscous sublayer at different Reynolds numbers and types of flow.
It is found that the matching condition for the mean velocity in the overlap region is equivalent to Vincze’s functional equation and has the solution in the form of power law. At finite Reynolds numbers the power law exponent is not universal and may be dependent not only of Reynolds number but of the flow type.