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Plane channel flowWall roughness
Conclusion
Fluidmechanik turbulenter StromungenTurbulent flows
Markus Uhlmann
Institut fur Hydromechanik
www.ifh.uni-karlsruhe.de/people/uhlmann
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Plane channel flowWall roughness
Conclusion
Summary of last lecture
Lecture 5 – The scales of turbulent motion
I The turbulent energy cascadeI hierarchy of eddies, downward transfer of energy
I dissipation determined by large scales,performed by small scales
I Kolmogorov’s theoryI building block of turbulence research
I valuable results for small scales (e.g. κ−5/3 spectrum)
I BUT: Problem of non-universality of large scales
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Plane channel flowWall roughness
Conclusion
LECTURE 6
Wall-bounded shear flows
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Plane channel flowWall roughness
Conclusion
Questions to be answered in the present lecture
What is the general structure of wall-bounded flows?
How does the presence of a solid boundary affect the turbulentmotion?
What is the effect of wall roughness?
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Plane channel flowWall roughness
Conclusion
Examples of tubulent wall-bounded flows (1)
Rivers
also: man-made canals, some ocean currents
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Plane channel flowWall roughness
Conclusion
Examples of tubulent wall-bounded flows (2)
Vehicle aerodynamics
Citroen DS, ONERA NASA aerodynamic truck
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Plane channel flowWall roughness
Conclusion
Examples of tubulent wall-bounded flows (3)
Atmospheric boundary layer:
storm clouds
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Plane channel flowWall roughness
Conclusion
Examples of tubulent wall-bounded flows (4)
Internal flows:
pipeline systems
water turbine
also: internal combustion engines, compressors, pumps,blood flow, . . .
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Plane channel flowWall roughness
Conclusion
Canonical wall-bounded flows
Idealized configurations
I plane channel flow
I pipe flow
I flat-plate boundary layer
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
h=2d
b z
y
flow
(a)
L
z
y
(c)
U0
flowy
rR=dD
(b)
x
x
x
Figure 7.1: Sketch of (a) channel flow (b) pipe flow and (c) flat-plate
boundary layer.
1
Pope “Turbulent flows”
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Plane channel flow configuration
2hy,v
x,uz,w
I statistically stationary & fully developed (far from inlet)
→ homogeneous in x- & z-direction
I definition of “bulk velocity”: ub ≡∫ 2h
0 〈u〉dy/(2h)
→ bulk Reynolds number: Reb ≡ ubh/ν
I Reynolds at centerline: Re0 ≡ U0h/ν with U0 = 〈u〉(y =h)
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Mean flow equations
Continuity equation (considering x-z-homogeneity)
���∂x〈u〉+ ∂y 〈v〉+���∂z〈w〉 = 0
⇒ mean wall-normal velocity 〈v〉(y) = 0, since 〈v〉 = 0 at walls
Mean wall-normal momentum equation
1
ρ∂y 〈p〉+ ∂y 〈v ′v ′〉 = 0
⇒ mean axial pressure gradient is constant: ∂x〈p〉 = cst.
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Mean flow equations (2)
Mean streamwise momentum equation
1
ρ∂x〈p〉+ ∂y 〈u′v ′〉 = ν∂yy 〈u〉
I define total shear stress: τ ≡ ρν∂y 〈u〉 − ρ〈u′v ′〉
⇒ varies linearly: τ(y) = τw (1− y
h) where τw ≡ τ(y = 0)
I integral balance yields: ∂x〈p〉 = −τw/h
In laminar flow we obtain:
I 〈u〉(y) = τwρν
y2
(2− y
h
), ub = 2
3 U0
I friction factor: cf ≡ τwρu2
b/2= 6
Reb 12 / 39
Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Mean velocity profile
DNS data of Jimenez et al.
〈u〉U0
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 Re ↑
laminar
y/h
visc
ous
stre
ss/τ
w
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Re ↑
laminar
y/h
I mean velocity profiles depend upon Reynolds number(contrary to free shear flows!)
I viscous stress ρν∂y 〈u〉 is always important near the wall
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Characteristic scales in the near-wall region
Viscosity and wall shear-stress are important parameters
I define friction velocity and viscous length scale:
uτ ≡√τwρ, δν ≡ ν
√ρ
τw=
ν
uτ
→ friction Reynolds number: Reτ ≡ uτh
ν=
h
δνI wall-distance in viscous lengths (“wall units”):
y + ≡ y
δν=
uτy
ν
→ y + equivalent to a local Reynolds number
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Importance of viscous and turbulent effects
Data shows near the wall:
I stress curves collapse whenplotted against y +
I viscous contributiondecreases with y + st
ress
con
trib
uti
on
DNS data of Jimenez et al, Reτ = 180 . . . 2000
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
−ρ〈u′v ′〉τ(y)
ρντ(y)∂y 〈u〉
y +
I y + < 5: viscous sublayer – Reynolds stress negligible
I y + < 50: viscous wall-region – direct effect of viscosity
I y + > 50: outer layer – no direct viscous effect
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Scaling of the mean velocity profile
Selecting the relevant parameters – dimensional considerations
I channel flow is fully determined by: ρ, ν, h, uτ
I wall-distance y is the only relevant coordinate
→ two non-dimensional numbers (y/h & Reτ or y/h & y +)
⇒ we can write the ansatz:d〈u〉dy
=uτy· Φ(
y +,y
h
)I Φ is a general non-dimensional function
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
The inner layer – the law of the wall
For y/h� 1 velocity profile is independent of h
I limy/h→0
Φ(
y +,y
h
)= ΦI
(y +)
→ this yields:du+
dy +=
1
y +· ΦI
(y +), with: u+ ≡ 〈u〉/uτ
I integration gives: u+ = fw (y +) “law of the wall”
I experiments show:
fw (y +) is universal function (BL, channel, pipe . . .)
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Mean flow behavior derived from the law of the wall
Viscous sublayer (y+ < 5)
I Taylor expansion around y =0:
u+ = y + “linear law”
Logarithmic layer (y+ > 30)
I for large y +, the functionΦI (y +) becomes constant:
u+ =1
κlog(y +) + B “log-law”
I κ ≈ 0.41, B ≈ 5.2
u+
——DNS by Jimenez et al, Reτ = 180 . . . 2000
0 5 10 15 200
5
10
15
y +
u+
100
101
102
103
0
5
10
15
20
y +
• exp. Wei & Willmarth Reτ = 1655
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Outer layer – velocity defect law
For y+ > 50: velocity profile independent of viscosity
I limy/h→∞
Φ(
y +,y
h
)= ΦO
(y
h
)→ this yields:
du
dy=
uτy· ΦO
(y
h
)I integration from y to h:
U0 − 〈u〉uτ
= FD
(y
h
)I note: function FD is not
universal!(different in boundary layer)
U0−〈u〉
uτ
——DNS by Jimenez et al., Reτ = 2000
10−1
100
0
2
4
6
8
10
y/h
for comparison: – – – –log-law
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Summary of regions in near-wall flow
I inner layer: 〈u〉 independent of U0
and h
I viscous wall region: viscouscontribution of shear stresssignificant (y+ < 50)
I viscous sublayer: Reynolds shearstress negligible
I buffer layer: region between viscoussublayer and log-region
I logarithmic region: the log-law holds
I outer layer: direct effects of viscosityon 〈u〉 negligible
I overlap region: overlap betweeninner and outer layers
y
h
Reb
(from Pope “Turbulent flows”)
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
The friction law for channel flow
I friction factor:cf ≡ τw
ρu2b/2
= 2u2τ
u2b
I transition occurs aroundReb = 1000
I integrating log-law:
Reτ ≈ 0.166 Re0.88b
cf
– – – – laminar (cf = 6/Reb), • experiments (Dean 1978)
—— turbulent approx. (cf = 0.055Re−0.24b
)
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
102 103 104 105 1060.000
0.002
0.004
0.006
0.008
0.010
Re
cf
Figure 7.10: Skin friction coefficient cf ≡ τw/(12ρU 2
0 ) against Reynolds
number (Re = 2Uδ/ν) for channel flow: symbols, experimental
data compiled by Dean (1978); solid line, from Eq. (7.55); dashed
line, laminar friction law cf = 16/(3Re).
9
2Reb
I lengthscale ratio h/δν =Reτ increases almost linearly with Reb
→ very small viscous scales at high Reynolds numbers
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Profiles of the Reynolds stress components
Observations from exp. &DNS
I peak TKE in buffer layer
→ signatures of coherentstructures (cf. lecture 7)
I anisotropy bij
I strong changes inviscous wall region
I plateau in log-region
I towards isotropy in coreflow
〈u′i u′j 〉u2τ
DNS by Jimenez et al., Reτ = 2000
0 50 100 150 200−2
0
2
4
6
8
〈u′u′〉
〈w′w′〉
〈v′v′〉 〈u′v′〉
y+
bij
0 400 800 1200 1600 2000
0 0.2 0.4 0.6 0.8 1
−0.4
−0.2
0
0.2
0.4
b11
b33
b22b12
y/h
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Near-wall asymptotics of velocity fluctuations
Taylor expansion for small wall-distance y :
u′(y) = �a1 + b1 y + c1 y2 + . . .
v ′(y) = �a2 +��b2 y + c2 y2 + . . .
w ′(y) = �a3 + b3 y + c3 y2 + . . . ⇒〈u′u′〉 = 〈b2
1〉 y2 + . . .
〈v ′v ′〉 = 〈c22 〉 y4 + . . .
〈w ′w ′〉 = 〈b23〉 y2 + . . .
〈u′v ′〉 = 〈b1c2〉 y3 + . . .
I no-slip, impermeability: a1 = a2 = a3 = 0
I continuity: ∂y v ′(y = 0) = b2 = 0
I taking mean of products of Taylor series:
⇒ Reynolds-stress components have different near-wall slopesO(y 2), O(y 3), O(y 4)
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Near-wall asymptotics of Reynolds-stresses
I 〈u′u′〉 = O(y 2)
I 〈v ′v ′〉 = O(y 4)
I 〈w ′w ′〉 = O(y 2)
I 〈u′v ′〉 = O(y 3)
〈u′i u′j 〉u2τ
DNS by Jimenez et al., Reτ = 950
10−4
10−3
10−2
10−10
10−5
100
〈u′u′〉〈w′w′〉
〈u′v′〉
〈v′v′〉
y/h
I this knowledge can be used for building consistent models
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Turbulent kinetic energy bugdet
Transport equation for statistically stationary channel flow
P −ε +νd2k
dy2−1
2
d
dy〈v ′u′j u′j 〉 −1
ρ
d
dy〈v ′p′〉 = 0
I II III IV V
I pseudo-dissipationε ≡ ν〈u′i ,ju′i ,j〉
I peak production occurs inbuffer layer
I buffer layer exports energy
I dissipation not zero at wall,balanced by visc. diffusion
bu
dg
et
DNS by Jimenez et al.,Reτ = 2000
0 10 20 30 40 50 60−0.3
−0.2
−0.1
0
0.1
0.2
0.3I
II
III
IV
V
y+
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Length scales in plane channel flow
Observations from DNS by Hoyas & Jimenez (Reτ = 2000)
L(1)αα
h
streamwise integral length scale
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
L11
L22
L33
y/h
L(3)αα
h
spanwise integral length scale
0 0.2 0.4 0.6 0.8 10
0.05
0.1
L33
L22
L11
y/h
I integral scale depends strongly on component and direction
→ turbulence structure highly anisotropic and inhomogeneous
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Plane channel flowWall roughness
Conclusion
Mean flowTurbulence statistics
Length scales in plane channel flow (2)
Definition of isotropic length scale
I L ≡ k3/2
εI no direct physical
interpretation
I but: often used inturbulence models
I slope in log-layer:L/h ∼ 2.8y/h
I in outer region:L/h ≈ 0.8
k3/2
ε
1
h
DNS by Jimenez et al.,Reτ = 2000
y+
0 400 800 1200 1600 2000
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
2.8 yh
y/h
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Plane channel flowWall roughness
Conclusion
Flow over rough walls
I s is characteristic length ofprotrusions
I roughness: s . h/20(else: “obstacles”)
I details of flow in cavities iscomplicated
I consequence: ansatz for meanvelocity modified
d〈u〉dy
=uτy· Φ(
y +,y
h, s+)
s
h
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Plane channel flowWall roughness
Conclusion
Logarithmic law for rough walls
Inner layer (y/h < 0.1):
limy/h→0
Φ(
y +,y
h, s+)
= ΦR
(y +, s+
)I integration for y + � 1:
u+ =1
κlog(y
s
)+ B(s+)
⇒ κ = 0.41 as for smooth wall
⇒ B depends on s+
I “fully rough” for s+ & 70
B
fully rough: B = 8.5
← smooth
s+
(Nikuradse’s data, from Pope “Turbulent flows”)
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Plane channel flowWall roughness
Conclusion
Transitional roughness – influence of shape
Roughness function ∆U+
I equivalent form of log-law:
u+ = 1κ log (y +) + B −∆U+
I with:∆U+ = 1
κ log (s+) + B − B
I ∆U+ measures departurefrom smooth-wall behavior
I measurements show:friction can be decreased!
∆U+
(from Jimenez, Ann. Rev. Fluid Mech. 2004)
10 Nov 2003 22:16 AR AR203-FL36-08.tex AR203-FL36-08.sgm LaTeX2e(2002/01/18)P1: IBD
ROUGH-WALL BOUNDARY LAYERS 183
Figure 3 Roughness function for several transitionally rough surfaces, as a functionof the Reynolds number based on the fully rough equivalent sand roughness.e, uni-form sand (Nikuradse 1933);O, uniform packed spheres (Ligrani & Moffat 1986);N,triangular riblets (Bechert et al. 1997);· · · · · ·, galvanized iron; – – – –, tar-coated castiron; — ·—, wrought-iron (Colebrook 1939); ——, Equation 12.
Bradshaw (2000) revived the question, noting that a minimum transitional heightwas unlikely for sparse roughness because the drag of the roughness elements in ashear is proportional tok2 even in the low Reynolds number limit, and this shouldbe reflected in1U+. In recent years the matter has become topical because some ofthe experiments undertaken to clarify the high Reynolds number behavior of flowsover smooth walls have surfaces that would be hydrodynamically smooth or roughdepending on which criterion is used (Barenblatt & Chorin 1998, Perry et al. 2001).Figure 3 shows that there is no “true” answer, and that each surface has to be treatedindividually.
The solid symbols in Figure 3 correspond to triangular riblets measured byBechert et al. (1997). The drag-reducing property of streamwise-aligned ribletsis a transitional roughness effect (Tani 1988). When they exceedk+ ≈ 10 theyloose effectiveness, and their behavior whenk+ À 1 is that of regulark-surfaces.Their drag-reducing mechanism is reasonably well understood. Luchini, Manzo &Pozzi (1991) showed that, in the limitk+ ¿ 1, the effect of the riblets is to imposean offset for the no-slip boundary condition which is further into the flow for the
s+
◦ Nikuradse sand grainsO Ligrani & Moffat spheresN Bechert et al. riblets
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Plane channel flowWall roughness
Conclusion
Friction law for rough pipes
Considering pipe flow
I friction factor:
cf ≡ τwρu2
b/2=
2u2τ
u2b
I using velocity defect law:
(U0 − ub)/uτ = 3/(2κ)
I using rough-wall log-law:
cf = 2
(1
κlog(R/s) + B(s+)− 3
2κ
)−2
4cf
(Nikuradse’s data, from Pope “Turbulent flows”)
smooth
Reb
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Summary
Main questions of the present lecture
I What is the general structure of wall-bounded flows?I inner layer/outer layer: linear/log-law, defect law
I How does the presence of a solid boundary affect theturbulent motion?
I stronger anisotropy than free shear flows
I wall has selective effect on velocity components
I What is the effect of wall roughness?I shift of the log-law compared to smooth walls
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Additional effects on wall-bounded flows
Important effects which are not treated here:
I imposed streamwise pressure gradient
I surface curvature
I system rotation
I geometrical complexities (secondary flows,. . .)
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Wall turbulence – visualization
Channel flow at Reτ = 590 2hy,v
x,uz,w
I visualizing streamwise velocity fluctuations u′
x-y slice
→ flow direction
z-y slice
⊗ flow direction
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Wall turbulence – visualization (2)
Channel flow at Reτ = 590, wall-parallel planes, u′
x-z slice, wall-distance y+ = 45
→ flow direction
x-z slice, wall-distance y+ = 170
→ flow direction
I typical structures: streamwise velocity “streaks”
→ found in all boundary-layer type flows
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Facts about velocity streaks in the buffer layer
Statistically speaking:
I lateral spacing of streaks:
∆`+z ≈ 100
I how do we know?
⇒ two-point correlations:minimum of Ruu athalf of the streak spacing
=⇒ flow
Ruu
0 100 200
−0.2
0
0.2
0.4
0.6
0.8
1
r+z
(Moser et al. 1999)
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Streamwise vortices
•• Streamwise vortices (ω′x)
`+x ≈ 200
I associated with streaks
(from Jeong et al. 1997)198 J. Jeong, F. Hussain, W. Schoppa and J. Kim
(b)
(a)
(c)
Low-speed streak
Sections in figure 9 (a)–(e)
E
H
SN
G
SP
F
Q4
Q2 B
E
HD
SPU
W
U
W
A
C
θ
θ
x
z
y
x
y
z
(d)
Q2
H
Q4
SP
E
x
z
Top view
y
E H
Q3
Q1
x
x
z
Time
t1
θ1
t2
θ2
t3
θ3
(e)
Figure 10. Conceptual model of an array of CS and their spatial relationship with experimentallyobserved events discussed in the text: (a) top view; (b) side view; (c) structures at cross-section FGin (a); (d) expanded views of structures C and D in (a,b), showing the relative locations of Q1, Q2,Q3, Q4, E and H. A schematic demonstrating the counteracting precession of SN in the (x, z)-planedue to background shear is shown in (e). The arrows in (b) denote the sections of figure 9(a–e).
=⇒ flow
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Complex vortex tangles at different Reynolds numbers
Reτ = 180
Reτ = 1900
(from del Alamo et al. 2006)
(movie)
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Plane channel flowWall roughness
Conclusion
Physical insight from DNSFurther reading
Further reading
I S. Pope, Turbulent flows, 2000→ chapter 7
I H. Schlichting and K. Gersten, Grenzschicht–Theorie, 10thedition, 2006→ part D
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