Turbulence in Pipes:Turbulence in Pipes:The Moody Diagram and MooreThe Moody Diagram and Moores Laws Law

Alexander SmitsPrinceton University

ASME Fluids Engineering ConferenceSan Diego, July 30-August 2, 2007

Thank you

Mark Zagarola, Beverley McKeon, RongrongZhao, Michael Shockling, Richard Pepe, Leif Langelandsvik, Marcus Hultmark, Juan Jimenez

James Allen, Gary Kunkel, Sean Bailey Tony Perry, Peter Joubert, Peter Bradshaw,

Steve Orszag, Jonathan Morrison, Mike Schultz

Lewis Ferry Moody 1880-1953

Professor of Hydraulic Engineering at Princeton University, 1930-1948

Friction factors for pipe flow, Trans. of the ASME, 66, 671-684, 1944

(from Glenn Brown, OSU)

The Moody Diagram

Smooth pipe(Prandtl)

Smooth pipe(Blasius)

Laminar

Increasing roughness k/D

Outline

Smooth pipe experiments ReD = 31 x 103 to 35 x 106

Rough pipe experiments Smooth to fully rough in one pipe Honed surface roughness Commercial steel pipe roughness

A new Moody diagram(s)? Predicting arbitrary roughness behavior

Theory Petascale computing

National Transonic Facility, NASA-LaRC

80 x 120NASA-Ames

High Reynolds number facilities

High Reynolds Number in the lab:Compressed air up to 200 atm as the working fluid

Princeton/DARPA/ONR Superpipe: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

Standard velocity profile

y+ = yu /

U+ = U/u

Inner

Outer

Overlap region

Inner variables

Similarity analysis for pipe flow

Incomplete similarity (in Re) for inner & outer region

Uu

= U+ = f yu , Ru

= f y

+ , R+( )

UCL Uuo

= g yR

, Ru

= g , R

+( )Complete similarity (in Re) for inner & outer region

U+ = f yu

= f y

+( )

UCL Uuo

= g yR

= g ( )

Inner scaling

Outer scaling

Inner

Outer

Overlap analysis: two velocity scales

At low Re:

uou

= h R+( ) > Match velocities and velocity gradients power law

At high Re:

uou

= constant > Match velocities and velocity gradients power law > Match velocity gradients log law

Superpipe results

Pipe flow inner scaling

0

5

10

15

20

25

30

100 101 102 103 104 105

U+

y+

U+ = y+

U+ = 1

0.436ln y+ + 6.15

U+ = 8.70 y +( )0.137

Smooth pipe summary

Log law only appears at sufficiently high Reynolds number

New log law constants: =0.421, B=5.60 (cf. 0.41 and 5.0) Spalart: = 0.01, gives CD=1% at flight Reynolds

numbers

New outer layer scaling velocity for low Reynolds number

What about the friction factor? Need to integrate velocity profile.

Johann Nikuradse, 1933(from Glenn Brown)

Ludwig Prandtl

Gottingen, Germany

Theodor von Krmn

Prandtls Universal Friction Law

Prandtl:

Nikuradses data

Prandtl

Superpipe results

Prandtl (1935)

Blasius (1911)

Prandtl (1935)

McKeon et al. (2004)

McKeon et. al. (2004)

Two complementary experiments

101 102 103 104 105 106 107 108

0.01

0.1

1

10

Princeton SuperpipeOregon

Re

Roughness

How do we know the smooth pipe was really smooth at all Reynolds numbers?

Were the higher friction factors at high Reynolds numbers evidence of roughness?

What is k? rms roughness height: krms equivalent sandgrain roughness: ks

Nikuradses rough pipe experiments (sandgrain roughness) ks+ < 5, smooth 5 < ks+ < 70, transitionally rough ks+ > 70, fully rough

Nikuradse's sandgrain experiments

Transition from smooth to fully rough included inflection

"Quadratic Resistance" in fully rough regime - Reynolds number independence

fully rough

transitionalsmooth

Colebrook and the Moody Diagram

Data from Colebrook & White (1938), Colebrook (1939)

Tested various roughness types Large and small elements

Sparse and dense distributions

Studied many different pipes with commercial roughness

Nikuradse sand-grain trend with inflection deemed irregular

Focus on the behavior in the transitional roughness regime

Cyril F. Colebrook Lewis Moody(from Glenn Brown)

The Moody Diagram

Colebrook transitional roughness function

Where did Colebrooks function come from?

Colebrook (1939)

Colebrook & White boundary layer results

New experiments on roughness

Use Superpipe apparatus to study different roughness types by installing different rough pipes

Advantage: able to cover regime from smooth to fully rough with one pipe

Honed surface roughness

To help establish where Superpipe data becomes rough (10 x 106, or 28 x 106, or 35 x 106, or what?)

To help characterize an important roughness type (honed and polished finish) (ks = 6krms, ks = 3krms?)

Commercial steel pipe roughness

Most important surface for industrial applications

"Smooth pipe, 6in Honed rough pipe, 98in

Honed surface finish

Results in smooth regime

Results: transitional/rough regime

Inner scaling - all profiles

Hama roughness function

Rough pipe: Inner scaling

Moody

rough pipe

ks+

Therefore smooth Superpipe was smooth for ReD

Friction factor results for rough pipe

Inflectional

Monotonic (Moody)

Nikuradse

Revised resistance diagram for honed surfaces

"Smooth"

"Rough"

Commercial steel surface roughness

Commercial steel rough pipe, 195inHoned rough pipe, 98in

5.0m3.82

Sample 2: non-rust spot

Sample 2: rust spot

Velocity profiles: inner scaling

Velocity profiles: inner scaling

Velocity profiles: inner scaling

Hama roughness function

Colebrook transitonalroughness

commercial steel pipe honed

surfaceroughness

Commercial steel pipe friction factor

ks = 1.5krms

Moody diagram for commercial steel pipe

Commercial steel pipe roughness

Smooth transitional fully rough

krms/D = 38 x 10-6

Pipe L/D = 200

ReD = 93 x 103 to 20 x 106

Smooth for ks+ < 3.1

ks = 1.5krms (instead of 3.5krms)!

Friction factor monotonic (but not Colebrook)

Honed surface roughness

Smooth transitional fully rough

krms/D = 19 x 10-6

Pipe L/D = 200

ReD = 57 x 103 to 21 x 106

Smooth for ks+ < 3.5

ks = 3.0krms Inflectional friction factor not

monotonic (Nikuradse not Colebrook)

Rough pipe summary

Why the Moody Diagram needs updating

Prandtls universal friction factor relation is not universal (breaks down at higher Reynolds numbers: >3 x 106)

Transitional roughness regime is represented by Colebrooks transitional roughness function using an equivalent sandgrain roughness, which takes no account of individual roughness types

Honed surfaces are inflectional not monotonic Commercial steel pipe monotonic but not Colebrook

The limitations of the Moody Diagram were well-known (e.g., Hama), but no match for text book orthodoxy

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 Chakrabortys (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

Prospects for Computation: Moores Law

Intel co-founder Gordon Moore

April, 19652005

Petascale computing

Earth Simulator (2004): 36 x 1012 flops peak DNS of 40963 isotropic turbulence

Petascale computing (2007): Blue Gene/P 3 x 1015

flops peak Remarkable resource, but what questions can it answer?

Example: DNS of channel flow Bob Moser, UT Austin

Re = uR/ (approx = ReD/40)

Channel flow simulations

Domain L2000

Resources for L2000 (ReD approx 80,000)

How much higher can we go? How much higher need we go?

Extended log-law (Moser)

Comparison with DNS channel data

From Re = 2000 to 5000

Resource requirements

Are we done with channels at Re = 5000?

Will give about an octave of log-law

Will display true inner and outer regions

Inadequate for high Reynolds number scaling (need Re > 50,000)

What about roughness?

With a 10 Petaflop machine Re = 5000 is cheap enough to do experiments Roughness studies?

Maybe we can do roughness with a teraflop machine (if we are clever)

Conclusions

Moody diagram should be revised, or used with caution

Colebrook is pessimistic (makes us look good)

Transitional roughness behavior not universal: depends on roughness

Gioia model combined with better surface characterization may lead to predictive theory

Petascale computing will provide powerful resource for fluids engineering, but maybe well solve roughness without it

A Golden Age in the study of wall-bounded turbulence?

Questions??

Osborne Reynolds

Turbulence in Pipes:The Moody Diagram and Moores LawThank youLewis Ferry Moody 1880-1953The Moody DiagramOutlineHigh Reynolds Number in the lab:Compressed air up to 200 atm as the working fluidStandard velocity profileSuperpipe resultsSmooth pipe summaryGottingen, GermanyPrandtls Universal Friction LawNikuradses dataSuperpipe resultsTwo complementary experimentsRoughnessNikuradse's sandgrain experimentsColebrook and the Moody DiagramThe Moody DiagramWhere did Colebrooks function come from?Colebrook & White boundary layer resultsNew experiments on roughnessHoned surface finishResults in smooth regimeResults: transitional/rough regimeInner scaling - all profilesRough pipe: Inner scalingFriction factor results for rough pipeRevised resistance diagram for honed surfacesCommercial steel surface roughnessVelocity profiles: inner scalingVelocity profiles: inner scalingVelocity profiles: inner scalingHama roughness functionCommercial steel pipe friction factorMoody diagram for commercial steel pipeRough pipe summaryWhy the Moody Diagram needs updatingWhere do we go from here?Goia and Chakrabortys (2006) modelProspects for Computation: Moores LawPetascale computingAre we done with channels at Ret = 5000?ConclusionsQuestions??