Turbulent Scaling in the Viscous Sublayer

1. Turbulent Scaling in the Viscous Sublayer SOLITONS, COLLAPSES AND TURBULENCE VI-TH INTERNATIONAL CONFERENCE, NOVOIBIRSK, RUSSIA, 4-8 JUNE 2012 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. Recent debate concerning the mean-velocityprofile 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., 1932. Gesetzmabigkeiten der turbulentenStromung in glatten Rohren. VDI-Fortschritt-Heft,No. 356. Nunner, W., 1956. 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 stabilitycriteria. J Fluid Mech 1979;90:401–14. Barenblatt GI. Similarity, self-similarity, and intermediatehypothesis. New York: Plenum; 1979. Long RR, Chen T-C. Experimental evidence for theexistence of the ‘mesolayer’ in turbulent systems. J. FluidMech. 1981;105:19–59. Wei T, Willmarth WW. Reynolds-number effects on thestructure of a turbulent channel flow. J Fluid Mech 1989;204:57–95. George WK, Castillo L, Knecht P. The zero-pressuregradient turbulent boundary layer revisited. In: Reed XB,Patterson GK, Zakin JL, editors.Thirteenth symposium onturbulence. Rolla, MO: University of Missouri; 1992. Bradshaw P. Turbulence: the chief outstanding difficulty ofour subject. Exp Fluids 1994;16:203–16. Purushothaman K. Reynolds number effects and themomentum flux in turbulent boundary layers. PhDdissertation, Department of Mechanical Engineering, YaleUniversity, 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 inturbulent boundary-layer flow: equivalent relations at largeReynolds numbers. Acta Mech2001;151:195–216. Buschmann M.H., Gad-el-Hak M. Recent developments in scaling of wall-bounded flows. Prog. Aero. Sci.2007. 42:419-67

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 Overlap layer 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.