some observations on the structure and modeling of 3-d turbulent boundary layers...

Some Observations on the Structure and Modeling of 3-D Turbulent

Boundary Layers and Separated Flow

Roger L. SimpsonTurbulence and Shear Flow Phenomena - 4

Williamsburg, VAJune 27, 2005

Outline

• Some features of experimental results from low-speed flows

• Some computational insights• Some future research needs and tools• Cited references 1995-2005 plus earlier

review references

Some 3-D Flows Examined• Wing/body junction

– Effects of Reynolds number– Effect of roughness

• Swept bump (Webster et al.)• Pressure-driven 3DTBL (Compton and Eaton)• ‘S-duct’ (Bruns et al.)• Channel flows with 3-D obstacles and separation• Axisymmetric bump with vortical separation• Prolate spheroid with crossflow separation• Compressor tip gap flow with leakage vortex• Current and future experiments

Surface Vorticity Flux and Vorticity Equations

( )( ) ( ) ( ))/(/)/(2Re

4 LtUdUVdeCeLnUL

refrefnpnref

L

×+∇×=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂ω

DDt

Ur

r r r rωω ν ω= ⋅ ∇ + ∇ 2

Vorticity flux Pressure gradient Accelerating wall

Material derivative Vortex stretching Viscous diffusion

New vorticity from wall

Redistribution of vorticity within flow

Precise measurements needed from experiments

X

Zω upstream

Chaotic horseshoevortices

See Simpson (2001), “JunctureFlows,” Ann. Rev. Fluid Mech.

3:2 elliptic noseNACA 0020 tail form “Rood” wing

zp

yW

w ∂∂

=∂∂

ρν 12

2

refUY

Streamwise vorticity flux

3-D smooth wall TBLoutside horseshoe vortex structure

Approach Reθ = 7300and 23,200.

Webster, et al. 1996:Un/swept Bump

Internal layers begin near leading and trailing edges due to discontinuities

Compton & Eaton, 1997:Spanwise ∇PStreamwise Favorable ∇P

Strong skewing of flow with reductionin shear stress and TKE correlation

Coherent Structures in a Skewed 3DTBL.

Large-scaled structuresthat entrain free-stream fluidmove in different directionsthan near wall vorticalstructures.

Defines the statistically ergodiclength scale for every frequency and wavelength eddy within the flow (Ha and Simpson, 1993).

For example, the coherency γ2from the cross-spectra from sensor signals at span-wise locations ∆Z apart can be modeled by the equation:

γ2 = exp (-kz |∆Z|/Lγz(f))

where kz is a flow dependent decay constant and Lγz(f) is a length scale for each frequency eddy passing the 2 sensors.

Spatial Coherency and PhaseStation 5

Station 5. Flow angle, flow gradient angle, shear stress angle calculated using two separate data sets in wall stress co-ordinates.

Tan-1(FA)= W/U

yUuv

yWvw

Nx

z

∂∂

−

∂∂

−==

/

/

εε

Ha and Simpson, 1993

Confirmed Earlier Insights on 3-D TBL • Outer layer flow and inner layer flow from

different directions with different histories– lower coherencies than for 2-D flows– lower correlation between shearing stress and TKE

• Most cases are non-equilibrium flows• No universal law-of-the-wall mean velocity

profile exists – but may have semi-log U and flow angle variations with y for inner/outer overlap region for moderate skewing

• Shearing stress direction lags velocity gradient direction –non-isotropic eddy viscosity – no preferred co-ordinate system

• Less skewed flows easier to calculate

No data for 3-D flowsat higher Reynoldsnumbers to developor test models

Momentum, continuity, &transport equations neededbecause of non-equilibriumnature of most 3-D flows

Wing/Body Junction FlowU+ vs. y+ mean velocity profiles in local wall shear stress co-ordinates.

Stations 1 – 9, from top to bottom

Solid, approach Reθ = 5940;

Open, Reθ = 23,200.(Őlçmen et al., 1999)

Solid – wall stress; Open - FS

Wing/Body Junction

W/Uτ in tunnel co-ordinateswith same legend.

Top, approach Reθ = 5940;

Bottom, Reθ = 23,200.

For 2 Reynolds numbers:

∂2(W/Uτ)/∂(y+)2 about the same for Stations 1- 6

∂2(W/Uτ)/∂(y+)2 for high Reθabout 2/3 of low Reθ forStations 7-9

( ) ( )2

221

vvwuv

S−+−

= vs. y+ profiles near the wall throughout the flow for 890 < Reθ < 23200, Station 5.

Water tunnel, Fleming andSimpson (1997)

Wind tunnel, Őlçmen and Simpson (1995)Őlçmen et al. (1999)

v2 scales active stress producing motions

Emerging Overlap Structure for High Reynolds Number 3-D Flows, Reθ = 23,200, Őlçmen et al. 1999

Semi-logarithmic variation of correlation coefficient for some triple products.Trend supported by DNW data for 2-D flows. Needs further data at higher Reθ.

Emerging Overlap Structure for High Reynolds Number 3-D Flows, Reθ = 23,200, Őlçmen et al. 1999

Semi-logarithmic variation of correlation coefficient for some tripleproducts. Needs further data at higher Reynolds numbers.

• No Law of the Wall for 3-D Flows.• Profiles show no effects of orientation for y>3k

yUτo/ν

U/U

τo

100 101 102 103 104

0

5

10

15

20

25

30

k=0.38mm, straight patternk=0.38mm, staggered patternk=0.76mm, straight patternk=0.76mm, staggered patternk=1.52mm, straight patternk=1.52mm, staggered patternsmooth wall

Station 1

k=1.52mm

k=0.38mm

k=0.76mm

yUτo/ν

U/U

τo

100 101 102 103 104

0

5

10

15

20

25

30

Station 3k=1.52mm

k=0.38mm

k=0.76mm

yUτo/ν

U/U

τo

100 101 102 103 104

0

5

10

15

20

25

30

Station 5

k=0.38mm

k=0.76mm

k=1.52mm

yUτo/ν

U/U

τo

100 101 102 103 104

0

5

10

15

20

25

30

Station 7k=1.52mm

k=0.76mmk=0.38mm

Rough Wall Wing/Body Junction (J. George and Simpson, AIAA 2002-0580)

W Profiles Collapse at y> 3k

y/δ

W/U

e

10-3 10-2 10-1 100-0.3

-0.2

-0.1

0

0.1

0.2 k=0.38mm, straight patternk=0.38mm, staggered patternk=0.76mm, straight patternk=0.76mm, staggered patternk=1.52mm, straight patternk=1.52mm, staggered patternsmooth wall

Station 1

k=1.52mmk=0.38mm

k=0.76mm

y/δ

W/U

e

10-3 10-2 10-1 100-0.3

-0.2

-0.1

0

0.1

0.2

Station 3

k=1.52mm

k=0.38mm

k=0.76mm

y/δ

W/U

e

10-3 10-2 10-1 100-0.3

-0.2

-0.1

0

0.1

0.2

Station 5

k=0.38mm

k=0.76mm k=1.52mm

y/δ

W/U

e

10-3 10-2 10-1 100-0.3

-0.2

-0.1

0

0.1

0.2

Station 7k=0.76mm

k=0.38mm k=1.52mmMax W propagatesaway from the wallfaster for rougher

surface

BL thicknesses are all different

Rough Wall Wing/Body Junction (J. George and Simpson, AIAA 2002-0580)Same surface pressure gradient as smooth wall cases

Transport of streamwise vorticity andGrowth of boundary layer are similar

y/δ

1/S

10-3 10-2 10-1 1000

0.25

0.5

0.75

1

1.25

1.5

1.75

2k=0.38mm, straight patternk=0.38mm, staggered patternk=0.76mm, straight patternk=0.76mm, staggered patternk=1.52mm, straight patternk=1.52mm, staggered patternsmooth wall

Station 1

k=1.52mmk=0.38mm

k=0.76mm

y/δ

1/S

10-3 10-2 10-1 1000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Station 3

k=1.52mm

k=0.38mm

k=0.76mm

y/δ

1/S

10-3 10-2 10-1 1000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Station 5k=0.38mm

k=0.76mm

k=1.52mm

y/δ

1/S

10-3 10-2 10-1 1000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Station 7

k=1.52mm

k=0.76mmk=0.38mm

( ) ( )2

221

v

vwuvS

−+−=

Axisymmetric 3-D Bump (Byun and Simpson, 2004, 2005)

x,U

y,V

R=2H

z,W

ψ=0°

ψ=90°

ψ=180°

δ=H/2

refU

6.41δ Wind tunnel ceiling

H

j

i

k

zL,WL L

xL,UL L

yL,VL Lθ

ψ

r

i

j

k

The right-handed tunnel coordinates, (x,y,z)

The right-handed local coordinates, (xL,yL,zL) r : local radiusψ : yaw angle about y-axisθ : pitch angle about zL-axisyL : locally perpendicular to the surface

Measurements 0° ≤ ψ ≤ 90°

= 27.5m/s, Reθ ≈ 7300

At least 2 papers atTSFP- 4 discusscomputations of this case.Ma and Simpson discusswake flow experiments atTSFP- 4.

Results - Normalized Mean Velocities , and Intermittency of UL in the Center PlaneU V

1

x/H

y/H

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

0=U

x/H ≈ 0.96 Mean backflow zone

downstream

downstream

Downstream from separationUpstream from separation

r/H=0.644

r/H=0.573

yL0+

Inte

rmitt

ency

,

101 102 103 1040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0.5730.6440.7060.7810.8410.91

r/H

γ pu

L

yL0+

Inte

rmitt

ency

,

101 102 103 1040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0.9761.0441.1121.1791.2441.3131.3861.461.5321.6151.6921.769

r/H

γ pu

L

Results – Nearest Wall Flow Field

RANS with quadratic eddy-viscosity model

Focus separation

00.5

11.5

2

0

0.5

1

1.5

-2

-1.5

-1

-0.5

0

1.781.651.531.401.271.141.020.890.760.640.510.380.250.130.00

→∇CP

Saddle separation

Pressure gradient contour, its vectors.line connecting vectors locally tangent to surface at yL0+ = 11

LLLL kWiU +

x/Hz/H

y/H

Uref

at x/H ≈ 1.2, z/H ≈ -0.7

at x/H ≈ 0.96 on the centerline

Mean backflow region

Saddle-focus separated flow extends up toyL0+ ≈ 340.

Separation line

→∇CP

1

Z

X

Y

Results - TKE Contour and its Normalized Transport Vectors in the Center Plane

0.0460.0430.0400.0360.0330.0300.0260.0230.0200.0170.0130.0100.0070.0030.000

0.2TKE/Uref

2

x/H

y/H

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

≡2/2q )2/( 222 wvu ++TKE ≡

refrefqvqurefq UqjvqiuqUjViVUV222 /)(/)(/ +≡+≡

, normalized by Uref

TKE transport vector, qV

Their magnitudes show some unsteadiness and jitter of the instantaneous flow

Results - TKE Contour and its Normalized Transport Vectors

0.2

x/H

z/H

0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.2

0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

yL0+ = 200yL0+ = 11

yL0+ = 578

locally tangent to the surface

qLV

Lines for visual aid of locally tangential mean velocity direction

4.50E-024.21E-023.91E-023.62E-023.33E-023.04E-022.74E-022.45E-022.16E-021.86E-021.57E-021.28E-029.86E-036.93E-034.00E-03

0.2

TKE/Uref2

x/H0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

Results – 1/S Parameter at r/H = 1.386

yL0+

1/S

100 101 102 103 1040

0.5

1

1.50102030405060708090

Yaw(deg)

very low correlation for 0° ≤ ψ ≤ 40°mean backflow region

2

22 )()(1

L

LL

v

vwuvS

−+−≡ Low A1 and Ruv for 0° ≤ ψ ≤ 40° due to the mean flow

angle, tan-1( ) variation with distance from the wall.The turbulent flow at different yL0+ from differentdirections is not correlated with turbulence at other yL0+.

LL UW /

2-D TBL level

yL0+=90 yL0+=200 yL0+=261

yL0+=456

yL0+=261

yL0+=261

Results – Bimodal Velocity Probability Histograms in UL and WL

Bimodal regionyL0+ < 5000.841 ≤ r/H ≤ 1.7690° ≤ ψ ≤ 30°

yL0+=261

x/H

z/H

0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

10

80

70

60

50

40

30

20

yaw angle

Results – Flow Structures on Leeside of the Bump

x,U

H

z,W

ψ = 0°

ψ = 90°

i

k

ψ = 30°

y,V j

Uref

High TKE levelnear wall

Low Reynolds shearing stresses productionin mean backflow region, yL0+ ≤ 100

Mean backflow supplied by large eddies

Mean backflow supplied by flow from sides

Spanwise ejection(uL’ < 0, vL’ > 0, wL’ < 0)

Sweeping toward the center(uL’ > 0, vL’ < 0, wL’ > 0)

Mean backflow zone, yL0+ ≤ 350

Bimodal histogram zone, yL0+ < 500

Saddle separation

Focus separation

Large eddies

ψ = -30°

Focus separation

Reattachment downstreamNodal or Saddle point ?

Conclusions

• A saddle type 3-D separation at x/H ≈ 0.96 on the centerline of bump.

• One focus separation on each side of bump at x/H ≈ 1.2 and z/H ≈ ±0.7.

• The large eddies and the flow from the bump side supply the mean backflow.

• The large eddies supply the intermittently forward flow in this mean backflow region.

• In the separation zones, the turbulent flow from different directions is decorrelated,

resulting in lower shearing stresses.

• Bimodal probability distributions of UL and WL appear in this region due to the unsteady

and low frequency meandering of the flow field.

• To get complete flow features over this bump, more measurements are needed for

the reattachment and separation regions downstream.

Flowfield for 6:1 Prolate Spheroid at α = 20o

U∞

Solid lines are oilflowseparation lines. Dashed linesshow local minima in skin friction.(Wetzel et al., 1998).

Mean secondary flowstreamlines on leesideof crossflow separatedflow from 3DLDV data(Chesnakas and Simpson,1997)

Computations by othersUsing RANS,LES, DESAt least 2 papersat TSFP-4

ReL = 4.2 X 106

Linear Cascade Model – Devenport et al., Va Tech

Fixed Blades

Moving endwall models casing

Vortex-generator pairs attached to moving endwallmodel IGV necklace vortices

• Large scale flow suited to rotor frame measurements.• Models relative motion of tip and casing and relative motion of vortical wakes and blades • Does not model axisymmetric effects and full IGV wake not important• Allows flow to be manipulated in ways not possible in actual turbomachine

Fixed bladesTip gap←region

Inflowregion

Tip-gap data by Tang, Tian,& Simpson (2003, 2004)

Suction side data locations (Tian et al., 2004)

LINEAR COMPRESSOR CASCADE RESEARCH

-2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.50.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24Y

/Ca

-2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5

Z/Ca

blade tip

7U7P7B7C7SX1X4X2X5X3

Pressure Side

Blade Tip SeparationVortex

Tip LeakageVortex Suction Side

SP

The secondary mean velocity flow field in the tip gap and tip leakage vortex (x/Ca=0.42) in the separation-line coordinates (t/c=3.3%)

:UrefUref=24.5 m/s

FA, FGA, and SSA plot for tip gap flow

y+

1/S

101 102 1030

0.2

0.4

0.6

0.8

1

1.2uu1d2s3s4s5s6s7s8s9s

Suction side t/c=3.3%

1/S vs y+ from endwall along compressor blade suction side.(Tang and Simpson, 2004). Large reduction in 1/S due to large variation of flow angle with y and uncorrelated structures.

( ) ( )2

221

vvwuv

S−+−

=

Some Results from Calculations• Full Reynolds stress model superior to algebraic stress model. (Iavocides)

• Nonlinear turbulent viscosity model superior to linear model. (Suga)

• Models with anisotropic dissipation perform better than those with isotropic dissipation. (Jongen)

• Near-wall region modeling critical to accurate description of 3-D flows.

• Reynolds stress models with wall functions are unsatisfactory. (Sotiropoulos)– near-wall turbulence closures must account for the anisotropy of the

Reynolds stresses.

• V2F model is the ‘best’ linear turbulent viscosity model. (Parneix)– An elliptic operator is used to model non-local effects of near-wall

turbulence.– Makes intrinsic and linear specific assumptions.– Good results by Wu and Squires (1998) for Swept Bump

More Results from Calculations

• Unsteady RANS and DES can simulate 3-D flow features dominated by geometry with relatively simple turbulence models (Paik).

• LES important for separated flows with low frequency large-scale motions (Davidson)

• Stress transport and algebraic stress models preferable over eddy viscosity models (Cummings; Morrison)

• From DNS the directional ‘lag’ between the mean shear stress and the velocity gradient directions depends strongly on the velocity-pressure-gradient correlation. (Coleman et al.)

Wavy Wall 3D TBL – Sadek & Simpson, 2005Waves of stream-wise vorticity produced by spanwise pressure gradients.

Floor BL data

‘Comprehensive’ LDV Technologies

Linearly varying fringe patterns for resolving particle positions

Particle acceleration measurements

12

12 )()(tt

tftfddt

dfda DDD−−

==

2000 )(2

)()( ttattVxtx −+−+=vvvv

Second-order particle trajectory resolved to Kolmogorov length and time scales:

)( 2tfD)( 1tfD

'x

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Blue

Green

ffgx'

Lowe and Simpson, TSFP-4

RST Measurements• Navier-Stokes equations:

Multiply by u’, time-average:

Reynolds stress transport equations:

– Obtain measurements of dissipation rate, acceleration/velocity fluctuation correlations to obtain velocity/pressure gradient correlation within flow field in laboratory flows

– Accomplished by measuring instantaneous position, velocity, and acceleration vectors within Kolmogorov scale (100-200 microns) measurement volume with single seed particles, current data rate 100KHz.

– Rate-of-strain tensor has not been measured on scales smaller than 5-7 times the Kolmogorov length scale.

Some Capabilities of Comprehensive LDV for High Reynolds Number 3-D Flow Research

iji

jji uuxpuua 2∇+∂∂

−= µρ

Recommendations• Data for high Reynolds numbers 3-D flows, Reθ

> 105• Direct measurement of terms in transport

equations• Velocity pressure gradient correlation data to

develop models • Auto & cross-spectral data for coherency &

phase for large-scaled motions• Rate-of-strain tensor for dissipation rate and

enstrophy

• LES, URANS, DES calculations necessary to capture low frequency motions of separated flows that affect diffusion

AcknowledgmentsThe completed work at Virginia Tech reported here has beensupported by the U.S. Office of Naval Research, Drs. L.P. Purtell,E. Rood, R. Joslin, and Ki-Han Kim, Program Managers.

K. Granlund, D. Hunter, and N. Varano contributed to the literature search and review.

some observations on the structure and modeling of 3-d turbulent boundary layers...

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