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

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

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 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

Packets of Hairpins Kim and Adrian (1982) VLSM Arise From Spatial Coherence of Hairpin Packets Hairpin Packets Align & Form Long Low-Speed Streaks (>2δ)

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

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 Milestone Moin and Kim (1981,1985) ILLIAC-IV

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

A Simulation Milestone Moin and Kim (1981,1985)

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!?!

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.

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!

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

Computing Power 5 Orders of Magnitude Since 1985! Advanced Computing has Advanced CFD (and vice versa)

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

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

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

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

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

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

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: http://ctr.stanford.edu

Blasius Boundary Layer + Freestream Turbulence t = 100.1T t = 100.2T 4096 (x), 400 (y), 128 (z) t = 100.55T

Isotropic Inflow Condition

Validation of Boundary Layer Growth Blasius Monkewitz et al Blasius

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 al Plus: Spalart Lines: Wu & Moin

Total stress through transition Plus: Reθ = 200 Solid Line: Reθ = 800

Near-Wall Stresses Total Stress Circle: Spalart Viscous Stress

Shear Stresses Circle: Honkan & Andreopoulos Diamond: DeGraaff & Eaton Plus: Spalart

Immediately before breakdown t = 100.55T u/U∞ = 0.99

Hairpin Packet at t = 100.55 T Immediately 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