Mechanics of Wall Turbulence

Parviz Moin

Center for Turbulence Research

Stanford University

Classical View of Wall Turbulence

• Mean Velocity Gradients Turbulent Fluctuations• Predicting Skin Friction was Primary Goal

Classical View of Wall Turbulence

• Eddy Motions Cover a Wide Range of Scales– Energy Transfer from Large to Smaller Scales– Turbulent Energy Dissipated at Small Scales

Major Stepping Stones• Visualization & Discovery of Coherent Motions– Low-Speed Streaks in “Laminar Sub-Layer”• Kline, Reynolds, Schraub and Runstadler (1967)• Kim, Kline and Reynolds (1970)

– Streaks Lift-Up and Form Hairpin Vortices• Head and Bandyopadhyay (1980)

Large Eddies in a Turbulent Boundary Layer with Polished Wall, M. Gad-el-Hak

Low-Speed Streaks in “Laminar Sub-Layer”Kline, Reynolds, Schraub and Runstadler (1967)

• Three-Dimensional, Unsteady Streaky Motions– “Streaks Waver and Oscillate Much Like a Flag”– Seem to “Leap Outwards” into Outer Regions

y+ ≈ 4

Bursts

Ki

m,

Kline

and

Reynolds (1970)

• Streaks “Lift-Up” Forming a Streamwise Vortex

• Near-Wall Reynolds Shear Stress Amplified• Vortex + Shear New Streaks/Turbulence

Major obstacle for LES

• Streaks and wall layer vortices are important to the dynamics of wall turbulence

• Prediction of skin friction depends on proper resolution of these structures• Number of grid points required to capture the streaks is almost like DNS,

N=cRe2

• SGS models not adequate to capture the effects of missing structures (e.g., shear stress).

Early Hairpin Vortex ModelsTheodorsen (1952)

• Spanwise Vortex Filament Perturbed Upward (Unstable)- Vortex Stretches, Strengthens, and Head Lifts Up More (45o)

• Modern View = Theodorsen + Quasi-Streamwise Vortex

Streaks Lift-Up

and

Form

Hairpin

Vortices

Head

and

Bandyopadhyay (1980)

• Hairpins Inclined at 45 deg. (Principal Axis)• First Evidence of Theodorsen’s Hairpins

Reθ = 1700

Streaks Lift-Up

and

Form

Hairpin

Vortices

Head

and

Bandyopadhyay (1980)

• For Increasing Re, Hairpin Elongates and Thins• Streamwise Vortex Forms the Hairpin “Legs”

Forests of HairpinsPerry and Chong (1982)

• Theodorsen’s Hairpin Modeled by Rods of Vorticity - Hairpins Scattered Randomly in a Hierarchy of Sizes

• Reproduces Mean Velocity, Reynolds Stress, Spectra- Has Difficulty at Low-Wavenumbers

Packets of HairpinsKim and Adrian (1982)

• VLSM Arise From Spatial Coherence of Hairpin Packets• Hairpin Packets Align & Form Long Low-Speed Streaks (>2δ)

Packets of HairpinsKim and Adrian (1982)

• Extends Perry and Chong’s Model to Account for Correlations Between Hairpins in a Packet; this Enhanced Reynolds Stress Leads to Large-Scale Low-Speed Flow

Major Stepping Stones• Computer Simulation of Turbulence (DNS/LES)– A Simulation Milestone and Hairpin Confirmation• Moin & Kim (1981,1985), Channel Flow• Rogers & Moin (1985), Homogeneous Shear

– Zero Pressure Gradient Flat Plate Boundary Layer (ZPGFPBL)• Spalart (1988), Rescaling & Periodic BCs

– Spatially Developing ZPGFPBL• Wu and Moin (2009)

A Simulation MilestoneMoin and Kim (1981,1985)

ILLIAC-IV

A Simulation MilestoneMoin and Kim (1981,1985)

LES

Experiment

A Simulation MilestoneMoin and Kim (1981,1985)

Hairpins Found in LESMoin and Kim (1981,1985)

• “The Flow Contains an Appreciable Number of Hairpins”

• Vorticity Inclination Peaks at 45o

• But, No Forest!?!

Shear Drives Hairpin GenerationRogers and Moin (1987)

• Homogeneous Turbulent Shear Flow Studies Showed that Mean Shear is Required for Hairpin Generation

• Hairpins Characteristic of All Turbulent Shear Flows– Free Shear Layers, Wall Jets, Turbulent Boundary Layers, etc.

Spalart’s ZPGFPBL and PeriodicitySpalart (1988)

• TBL is Spatially-Developing, Periodic BCs Used to Reduce CPU Cost• Inflow Generation Imposes a Bias on the Simulation Results• Bias Stops the Forest from Growing!

Analysis of Spalart’s DataRobinson (1991)

• “No single form of vortical structure may be considered representative of the wide variety of shapes taken by vortices in the boundary layer.”

• Identification Criteria and Isocontour Subjectivity• Periodic Boundary Conditions Contaminate Solution

Computing Power

5 Orders of Magnitude Since 1985!

Advanced Computing has Advanced CFD

(and vice versa)

DNS of Turbulent Pipe FlowWu and Moin (2008)

Re_D = 5300 Re_D = 44000

300(r) x 1024(θ) x 2048(z) 256(r) x 512(θ) x 512(z)

DNS at ReD = 24580, Pipe Length is 30R

Very Large-Scale Motions in PipesWu and Moin (2008)

Log Region (1-r)+ = 80

Buffer Region (1-r)+ = 20

Core Region (1-r)+ = 270

(Black) -0.2 < u’ < 0.2 (White)

Experimental energy spectrum

Wavelength

Energy

Experiment, using T.H.Perry & Abell (1975)

Energy Spectrum from Simulations

Wavelength

Energy

Experiment, using T.H.Perry & Abell (1975)

Simulation, true spectrumdel Álamo & Jiménez (2009)

Artifact of Taylor's Hypothesis

Wavelength

Energy

Experiment, using T.H.Perry & Abell (1975)

Simulation, true spectrumdel Álamo & Jiménez (2009)

Simulation, using T.H.del Álamo & Jiménez (2009)

Artifact of Taylor's Hypothesis

Wavelength

Energy

Experiment, using T.H.Perry & Abell (1975)

Simulation, true spectrumdel Álamo & Jiménez (2009)

Simulation, using T.H.del Álamo & Jiménez (2009)

Aliasing

Simulation of spatially evolving BLWu and Moin (2009)

• Simulation Takes a Blasius Boundary Layer from Reθ = 80 Through Transition to a Turbulent ZPGFPBL in a Controlled Manner

• Simulation Database Publically Available:

http://ctr.stanford.edu

Blasius Boundary Layer + Freestream Turbulence

4096 (x), 400 (y), 128 (z)

t = 100.1T

t = 100.2T

t = 100.55T

Isotropic Inflow Condition

Validation of Boundary Layer Growth

Blasius

Blasius

Monkewitz et al

Validation of Skin Friction

Blasius

Validation of Mean Velocity

Murlis et al Spalart

Reθ = 900

Validation Mean Flow Through Transition

Reθ = 200

Reθ = 800

Circle: Spalart

Validation of Velocity Gradient

Circles: Spalart (Exp.)Triangles: Smith (Exp.)Dotted Line: Nagib et al. (POD)Solid Line: Wu & Moin (2009)

Validation of RMS Through Transition

Circle: Spalart

Reθ = 800

Reθ = 200

Validation of RMS fluctuations

circle: Purtell et al other symbols: Erm & Joubert

Reθ = 900

Validation of RMS Fluctuations

Circle: Purtell et alPlus: SpalartLines: Wu & Moin

Total stress through transition

Plus: Reθ = 200Solid Line: Reθ = 800

Near-Wall Stresses

Circle: Spalart

Viscous Stress

Total Stress

Shear Stresses

Circle: Honkan & AndreopoulosDiamond: DeGraaff & EatonPlus: Spalart

Immediately before breakdown

t = 100.55T

u/U∞ = 0.99

Hairpin Packet at t = 100.55 TImmediately Before Breakdown

Winner of 2008 APS Gallery of Fluid Motion

Summary• Preponderance of Hairpin-Like Structures is Striking!• A Number of Investigators Had Postulated The

Existence of Hairpins• But, Direct Evidence For Their Dominance Has Not

Been Reported in Any Numerical or Experimental Investigation of Turbulent Boundary Layers

• First Direct Evidence (2009) in the Form of a Solution of NS Equations Obeying Statistical Measurements

Summary-II• Forests of Hairpins is a Credible Conceptual Reduced Order

Model of Turbulent Boundary Layer Dynamics • The Use of Streamwise Periodicity in channel flows and

Spalart’s Simulations probably led to the distortion of the structures

• In Simulations of Wu & Moin (JFM, 630, 2009), Instabilities on the Wall were Triggered from the Free-stream and Not by Trips and Other Artificial Numerical Boundary Conditions

• Smoke Visualizations of Head & Bandyopadhyay Led to Striking but Indirect Demonstration of Hairpins Large Trips May Have Artificially Generated Hairpins

Conclusion• A renewed study of the time-dependent

dynamics of turbulent boundary layer is warranted. Helpful links to transition and well studied dynamics of of isolated hairpins.

• Calculations should be extended to Re>4000 would require 3B mesh points. • Potential application to “wall modeling” for LES