Turbulent flow on Smooth and Rough SurfacesTurbulent flow on Smooth and Rough Surfaces
Alexander Smits
Workshop on Friction: A Grand Challenge at the Interface of Solid and Fluid Mechanics
Montreux, March 13-16, 2008
General observations on turbulent flow
• Turbulent flow is governed by the Navier-Stokes equation• Turbulent flow has no analytical solutions• Turbulent flow can be computed without modeling using
Direct Numerical Simulation (DNS), although boundary conditions for roughness are non-trivial
• DNS is limited to low Reynolds number R+ = O(103), whereas vehicles can have R+ = O(106)
• Higher Reynolds number computations need turbulence models
• Turbulence modeling needs theory and experiments• Modeling effects of roughness requires empirical input• Scaling turbulent flow, for changes in Reynolds number and
surface roughness is a first-order problem
The Moody Diagram for pipe flow
Smooth pipe(Prandtl)
Smooth pipe(Blasius)
Laminar
Increasing roughness k/D
Uey
Turbulent flow over a “smooth” surface(Newtonian, continuum flow)
• No-slip condition produces a boundary layer where the velocity gradients are large
• Turbulent flow over a smooth surface is (at least) a two-scale problem: a balance between viscous stresses ν∂U/ ∂y and turbulent stresses –u’v’
• Very near the surface, the fluid stress is dominated by viscosity because the turbulent stresses must go to zero (no-slip condition) - called the “inner” region
• Away from the wall, the turbulent shear stress quickly becomes much larger than the viscous stress, and it is the dominant stress over almost the entire layer - called the “outer” region
flow
Corke & Nagib
Stress distributions
Near-wall behavior
y+ = yuτ /ν τw is the stress at the solid surface (the “skin” friction),and near the wall it is equal to the total stress in the fluid
Reynolds number R+ = Ruτ /ν
Similarity analysis for pipe flow
Incomplete similarity (in Re) for inner & outer region
Uuτ
= U+ = fyuτ
ν,
Ruτ
ν⎛ ⎝ ⎜ ⎞
⎠ ⎟ = f y+ , R+( )
UCL − U
uo
= gyR
, Ruτ
ν⎛ ⎝ ⎜ ⎞
⎠ ⎟ = g η, R+( )
Complete similarity (in Re) for inner & outer region
U+ = f yuτ
ν⎛ ⎝ ⎜ ⎞
⎠ ⎟ = f y+( )
UCL − U
uo
= gyR
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = g η( )
Inner scaling
Outer scaling
Inner
Outer(overlap region)
Superpipe experiment(compressed air up to 200 atm as the working fluid)
Fully-developed pipe flow ReD = 31 x 103 to 35 x 106
Primary test section with test pipe shown
Diffuser section
Pumping section
To motor
Heat exchanger Return leg
34 m
Flow conditioning section
Flow
Test leg
Flow
1.5 m
(Zagarola & Smits; McKeon & Smits)
Superpipe results
R+ = 0.85 x 103
R+ = 2.35 x 103
R+ = 6.59 x 103
R+ = 19.7 x 103
R+ = 54.7 x 103
R+ = 166 x 103
R+ = 528 x 103
(inner variables)
increasing R+
Pipe flow inner scaling
0
5
10
15
20
25
30
100 101 102 103 104 105
U+
y+
U+ = y+
U+ =
10.436
ln y+ + 6.15 U
+ = 8.70 y +( )0.137
(inner variables)
Pipe flow outer scaling
0
5
10
15
0.01 0.1 1
udef
udef
udef
udef
udef
udef
udef
y/R
Re = 31 × 103
Re = 98 × 103
Re = 310 × 103
Re = 1.0 × 106
Re = 3.1 × 106
Re = 10 × 106
Re = 35 × 106
UCL − Uuτ
(inner variables)
R+ = 0.85 x 103
R+ = 2.35 x 103
R+ = 6.59 x 103
R+ = 19.7 x 103
R+ = 54.7 x 103
R+ = 166 x 103
R+ = 528 x 103
Turbulent flow over a “rough” surface
• Reynolds number R+ = Ruτ/ν may be interpreted as ratio of largest turbulent eddies (~R) to smallest turbulent eddies (~ ν/uτ)
• As Reynolds number increases, for a fixed R, the smallest eddies become comparable in size to the surface roughness (~k)
• When k+ = kuτ/ν = O(1), the roughness changes the surface stress τw (= ρuτ
2)• What is k?
– rms roughness height: krms– equivalent sandgrain roughness: ks
• Nikuradse’s rough pipe experiments (sandgrain roughness)– ks
+ < 5, smooth– 5 < ks
+ < 70, transitionally rough– ks
+ > 70, fully rough
Nikuradse's sandgrain experiments
“quadratic resistance" in fully rough regime:Reynolds number independence
fully rough
transitionalsmooth
Recent experiments on roughness
Commercial steel rough pipe, 195μinHoned rough pipe, 98μin
5.0μm3.82
Langelandsvik & SmitsShockling & Smits
Velocity profiles: low Reynolds number
Commercial steel pipe
Velocity profiles: medium Reynolds number
Velocity profiles: high Reynolds number
Velocity profiles: outer scaling
Townsend’s hypothesis: roughness only changes boundary condition
Schultz & Flack show this also holds on ROUGH surfaces for y > 3ks
Turbulence scaling (smooth wall)
• Inner and outer scaling works well for mean flow
• Inner and outer scaling does not work well for turbulent stresses
• Striking example: inner layer peak varies with R+
• Outer layer shows viscous dependence at all R+
• Inner and outer layers interact at all Reynolds number
10000 < R+ < 539
DeGraaf & Eaton
Marusic & Kunkel
Inner and outer scaling applied to spectra
Perry, Henbest & Chong;Perry and Marusic
Model spectra: streeamwise component
inner scaling
outer scaling
start of -5/3
Reynolds number dependence of the outer layer behavior:lines are model results due to Kunkel and Marusic
Turbulence scaling
• Spectra insensitive to roughness (at least for y > 3ks)
• Current work suggests that spectral scaling is more complicated than assumed in Perry et al. analysis
• Very large scale motions of O(10R) exist that seem to show mixed scaling
• Attached eddies (that were assumed to scale with y) also show mixed scaling
• Scaling different for boundary layers, pipe, and channel flows
Hutchins & Marusic
Where we are today
• Mean flow scales well with inner and outer scaling (with minor adjustments)
• Mean flow scaling insensitive to wall roughness for y > 3ks, although inner layer is no longer present in fully rough flow
• Log-law constants different for boundary layers, pipe, and channel flows
• Turbulence displays pronounced inner and outer layer interactions in stress behavior and in spectral content
• Turbulence insensitive to wall roughness for y > 3ks
• No predictive theory for roughness exists• Wall-bounded turbulence continues to surprise, almost 100
years after the boundary layer was first identified by Prandtl
Osborne Reynolds Lewis Ferry Moody
Johann Nikuradse
Ludwig Prandtl
Theodor von KármánCyril F. Colebrook
Where do we go from here?
• More experiments, more data analysis?– Schultz and Flack
• A predictive theory?– Gioia and Chakraborty
• Petascale computing?– Moser, Jimenez, Yeung
Goia and Chakraborty’s (2006) model
• Model the energy spectrum in the inertial and dissipative ranges• Use the energy spectrum to estimate the speed of eddies of size s• Model the shear stress on roughness element of size s as • Hence , then integrate across all scales to find λ
Standard velocity profile
y+ = yuτ /ν
U+ = U/uτ
Inner
Outer
Overlap region
Inner variables
U+ = U/uτ
Reynolds number R+ = Ruτ /ν
Hama roughness function
Colebrook transitional roughness
commercial steel pipe honed
surfaceroughness
Commercial steel pipe friction factor
ks = 1.5krms