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
1
Turbulence in Channel
U ( y)  ?
dP  w
Ue ,  ,
 ,
dx 
Governing
Parameters: U e , v ,
cf 
2 w
?
2
U e
u nv m wk  ( y )  ?
Friction velocity
, 
Reynolds Number:
Re 
w
v 
,

U e

,  
v 

,
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., 1932. Gesetzmabigkeiten der turbulenten Stromung 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 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. 2007. 42:419-67
Plan of the presentation
1. Classical approach from the viewpoint of
method of matched asymptotic expansions
2. Deficiencies of classical scaling
3. New scaling for the viscous sublayer
Formulation of the Problem
Reynolds averaged equation
for mean momentum transport:
Boundary conditions:
at the wall,
y  0:
at the centerline,
y  :
v2
d 2U d uv 
1 dP
 2 


dy
dx 
dy
dU v2
U  0, uv   0,

dy 
dU
U  U e , uv   0,
0
dy
dU
 y
uv   v2 1   
dy
 
 0
Viscosity  is a small parameter at higher derivative.
Singular perturbation problem! (K.Yajnik, 1970)
Matched Asymptotic Expansions
M.Van Dyke. Perturbation Methods in Fluid Mechanics.N.Y.:Acad. Press, 1964.
dU
 y
uv   v 1   
dy
 
2
uv    0   1  O( 2 )
U  U 0  U1  O( 2 )
Outer asymptotic expansion (K.Yajnik, 1970)
in the outer layer
y   O(1)
y

 0  v 1  
 
U0  ?
2
dU
dy
U1  ?
1  
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)
 0 does not satisfy the wall boudary condition
y  0
 0  v2
Classical Scaling: Outer Layer
Since in the near-wall region
uv  ~ v2 , then it is reasonable to assume that
u , v  O (v )
U e  U ( y )  O (v )
U 0  U e  v F ( )  O(v  1 )
y 
Outer layer expansions
U0
 1   F ( )  O( 1 )
Ue
  v Ue  c f 2
uv 
1 dF
1

1





O
(Re
)

2
d
v
Classical Scaling: Inner Layer
Rescale the variables in equation so that the viscous
stress will be of the same order as Reynolds stress
If U ( y )  O (v ), then the stretched varible is
uv  ~ 
y
y 
l
dU
~ v2
dy
l 

v
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
U
 U   f ( y )  O( 1 )
v
uv 
df
1

1


O
(

 )
2
dy
v
Overlap Layer: Matching
y
y
Overlap layer
   0

l
Introduce f  ( y )  lim f ( y ) F0 ( )  lim F ( )
y 
l  y  
 0
y 
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:
f ( y )   1 ln y  B
Log layer observed in the range
Von Karman constant:
f (  )   1 (  )  F0 ( )
50  y  0.1 
  0.4,
B5
Friction Law:
 1 (  )   1 ln    D  O( 1 )
F0 ( )   1 ln  B  D
Weakness of Classical Scaling
1.
2.
3.
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)
Alternative Theories:
Power Law or Log Law?
G.I.Barenblatt: No Reynolds number similarity
3  5 
f  ( y ) 
y
2
3

2ln Re
W.K.George: Reynolds number dependence of
, B
N.Afzal: No Reynolds number similarity in the overlap layer
dF0 ( )
df ( y )
y

dy
d

Ci y  Co

ln(Co Ci )

ln  
Experimental Results: SuperPipe Data
?
McKeon et al., 2004 (JFM):
Logarithmic law is found
in the range
600  y  0.12 
Power law is found in the range 50  y  300
1
U 
ln y  5.6
0.421
U   8.48 y0.142 ~ y1 7
Revisiting the Classical Scaling in
the Viscous Sublayer
dU
uv  ~ 
dy
~ v2
There can be a continuum of possible scalings in the viscous sublayer.
Viscous sublayer thickness l can be chosen arbitrarily, if the mean
velocity scale is chosen as
v2l
U 

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)
d n uv 
 0, n  0  2
n
dy
dU v2

dy 
where
d 2U v2

2
dy

d 3 uv 
 6
3
dy
d 3U
0
3
dy
d 4U
6


dy 4

  0.5 (u / y )( 2 w / yz )
y 0
Revisiting the Classical Scaling in
the Viscous Sublayer
 uv  ~ y 3
New scale for viscous sublayer thickness
l  v2 3  1 3
instead of
l3 ~ v2
l   v
Introduce the new variables
y  y1 3v2 3 U  1 3U v8 3    uv  / v2
Mean momentum transport:
  1
dU
 O( y  )
dy
d n 
Boundary conditions at the wall:
 0, n  0  2
n
dy
d 2U l
dU
d 3U
U  0
1
0
 0
3
2
dy
dy
dy

Universal scaling of mean velocity
and Reynolds shear stress:
U  U ( y )
d 3 
6
3
dy
d 4U
 6
4
dy
     ( y )
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 180    2000 , dashed lines mark off the
rough border of viscous sublayer.
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)
180    2000
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)
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)
Overlap Layer: Matching
U e  uo F    v8 3 1 1 3 f ( y )
By introducing the new variables
  1 3v2 3 H  1 3v8 3U e G  1 3v8 3uo
this equations transformed into the Vincze’s functional equation
H ( )  G( ) F    f ( )
General solution is
f ( y )  A
y  1



G ( )  
C
 1
F ( )  A
D

  1
H ( )  A
 C  D

Power Law with Non-Universal Exponent
ln(uo 1 3 )

ln(  1 3 )
ZPG TBL
Circular pipe
Plane channel
   4 105  1 2
(Cenedese et al., 1998)
Reynolds number dependence of parameter  
Conclusion
1) 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.
2) 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.