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**Mechanics of Wall Turbulence**

Parviz Moin Center for Turbulence Research Stanford University

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**Classical View of Wall Turbulence**

Mean Velocity Gradients Turbulent Fluctuations Predicting Skin Friction was Primary Goal

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**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

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**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

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**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

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**Bursts Kim, Kline and Reynolds (1970)**

Streaks “Lift-Up” Forming a Streamwise Vortex Near-Wall Reynolds Shear Stress Amplified Vortex + Shear New Streaks/Turbulence

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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).

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**Early Hairpin Vortex Models Theodorsen (1952)**

Spanwise Vortex Filament Perturbed Upward (Unstable) Vortex Stretches, Strengthens, and Head Lifts Up More (45o) Modern View = Theodorsen + Quasi-Streamwise Vortex

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

Spanwise Separation of Hairpins λ+ ≈ 100 y+ Hairpins Inclined at 45 deg. (Principal Axis) First Evidence of Theodorsen’s Hairpins

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

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

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**Forests of Hairpins Perry 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

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**Packets of Hairpins Kim and Adrian (1982)**

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

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**Packets of Hairpins Kim 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

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**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)

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**A Simulation Milestone Moin and Kim (1981,1985)**

ILLIAC-IV

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**A Simulation Milestone Moin and Kim (1981,1985)**

Experiment

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**A Simulation Milestone Moin and Kim (1981,1985)**

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**Hairpins Found in LES Moin and Kim (1981,1985)**

“The Flow Contains an Appreciable Number of Hairpins” Vorticity Inclination Peaks at 45o But, No Forest!?!

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**Shear Drives Hairpin Generation Rogers 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.

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**Spalart’s ZPGFPBL and Periodicity Spalart (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!

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**Analysis of Spalart’s Data Robinson (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

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**Computing Power 5 Orders of Magnitude Since 1985!**

Advanced Computing has Advanced CFD (and vice versa)

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**DNS of Turbulent Pipe Flow Wu and Moin (2008)**

256(r) x 512(θ) x 512(z) 300(r) x 1024(θ) x 2048(z) Re_D = 5300 Re_D = 44000

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**Very Large-Scale Motions in Pipes Wu and Moin (2008)**

DNS at ReD = 24580, Pipe Length is 30R (Black) -0.2 < u’ < 0.2 (White) Log Region (1-r)+ = 80 Buffer Region (1-r)+ = 20 Core Region (1-r)+ = 270

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**Experimental energy spectrum**

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

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**Energy Spectrum from Simulations**

Experiment, using T.H. Perry & Abell (1975) Simulation, true spectrum del Álamo & Jiménez (2009) Energy Wavelength

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**Artifact of Taylor's Hypothesis**

Experiment, using T.H. Perry & Abell (1975) Simulation, true spectrum del Álamo & Jiménez (2009) Energy Simulation, using T.H. del Álamo & Jiménez (2009) Wavelength

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**Artifact of Taylor's Hypothesis**

Aliasing Experiment, using T.H. Perry & Abell (1975) Simulation, true spectrum del Álamo & Jiménez (2009) Energy Simulation, using T.H. del Álamo & Jiménez (2009) Wavelength

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**Simulation of spatially evolving BL Wu 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:

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**Blasius Boundary Layer + Freestream Turbulence**

t = 100.1T t = 100.2T 4096 (x), 400 (y), 128 (z) t = T

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**Isotropic Inflow Condition**

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**Validation of Boundary Layer Growth**

Blasius Monkewitz et al Blasius

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**Validation of Skin Friction**

Blasius

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**Validation of Mean Velocity**

Murlis et al Spalart Reθ = 900

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**Validation Mean Flow Through Transition**

Reθ = 200 Reθ = 800 Circle: Spalart

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**Validation of Velocity Gradient**

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

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**Validation of RMS Through Transition**

Circle: Spalart Reθ = 800 Reθ = 200

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**Validation of RMS fluctuations**

circle: Purtell et al other symbols: Erm & Joubert Reθ = 900

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**Validation of RMS Fluctuations**

Circle: Purtell et al Plus: Spalart Lines: Wu & Moin

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**Total stress through transition**

Plus: Reθ = 200 Solid Line: Reθ = 800

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Near-Wall Stresses Total Stress Circle: Spalart Viscous Stress

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**Shear Stresses Circle: Honkan & Andreopoulos Diamond: DeGraaff & Eaton**

Plus: Spalart

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**Immediately before breakdown**

t = T u/U∞ = 0.99

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Hairpin Packet at t = T Immediately Before Breakdown

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**Winner of 2008 APS Gallery of Fluid Motion**

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**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

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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

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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

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