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