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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