Retardation of break-away separation by enhanced · PDF fileenhanced boundary layer...

Retardation of break-away separation byenhanced boundary layer turbulence:

from the triple-deck to a novel Rayleigh stage

Bernhard [email protected]

Institute of Fluid Mechanics and Heat TransferVienna University of Technology, Austria

AC²T research GmbH — Austrian Center of Competence for TribologyWiener Neustadt, Austria

Fluid Dynamics Group Seminar, 26 October 2012

Introduction Existing theory Present approach Summary

Outline

1 IntroductionMotivationBasic problem of bluff-body separationPreceding work

2 Existing theoryTurbulent boundary layerEmbedding of interacting flowComparison with simulation

3 Present approachPotential flow encompassing a small cavityTurbulent boundary layerA novel interaction mechanism: Rayleigh stageVery recent findings

4 Summary


Some prefatory statements on a long-standing problem

V A Sandborn & C Y Liu [JFM, 32(2):293–304, 1968]

“Turbulent boundary-layer separation is normally listed as one of themost important unsolved problems in fluid mechanics . . . ”

U

A

L

?

θ

−3

−2

0 150 18050 100

1

0

−1

θ [◦]

cP

LES

Re = 6.7× 105Re = 1.26× 106

L Prandtl M Wang, P Catalano & G Iaccarino[Roy. Aeron. Soc., XXXI:720–743, 1927] [CTR Annual Research Briefs 2001]


Some prefatory statements on a long-standing problem

R E Melnik & B Grossman [Numer. & Phys. Aspects of Aerodyn. Flows, 1982]

“Progress . . . will go a long way toward the establishment of rationalasymptotic methods for trailing-edge separation.”

Re → ∞ : Kutta condition

cusp / flat plate

wedge


Another long-standing problem: high-Re bluff-body drag

typical: circular cylinder / sphere with radius R, drag D = ?

cD : =D/(R2

π)

ρ∞U2∞/2

, Re : =2RU∞

ν, M =

U∞

c∞

dimensional analysis ⇒ cD(Re,M)

cylinder sphereJ D Anderson[Fundamentals of Aerodynamics, 1991]




cD : =D/(R2

π)

ρ∞U2∞/2

, Re : =2RU∞

ν, M =

U∞

c∞


cylinder spherecD

Re

4π/Re

ln(3.703/Re)+ · · ·

“drag crisis”,laminar–turbulenttransition (LTT)

? 0

2

1

−1−1 0 1 2 3 4 5 7

3lg cD

lgRe

24/ReLTT

J D Anderson[Fundamentals of Aerodynamics, 1991]




cD : =D/(R2

π)

ρ∞U2∞/2

, Re : =2RU∞

ν, M =

U∞

c∞


cylinder spherecD

Re

4π/Re

ln(3.703/Re)+ · · ·

“drag crisis”,laminar–turbulenttransition (LTT)

? 0

2

1

−1−1 0 1 2 3 4 5 7

3lg cD

lgRe

24/ReLTT

J D Anderson cf. FIFA WC 2010:[Fundamentals of Aerodynamics, 1991] relatively smooth ball, lowered ρ∞


Outline




4 Summary


A long-standing question in high-Re asymptoticsIncompressible nominally 2D time-mean flow past rigid bluff body

u ∼ 1d

open

closed

cavity∼ separation

turbulent boundary layer (TBL)

key parameters

Re≫ 1

turbulence-intensity gauge factor T , 0 ≤ T (Re) . 1

potential-flow parameter d

key question

position of separation ?


Outline




4 Summary


Two controversial concepts

[NS’92] potential flow fully attached: d = 0 , Re≫ 1 , T . 1

[SK’08] resulting trailing-edge structure questioned

[SKS’11] potential flow sought in class of Helmholtz–Kirchhoff flows:

d = O(1) , Re≫ 1 , T ≪ 1

distinguished limit d≪ 1 , Re≫ 1 , T ≪ 1 ?

[NS’92] A Neish & F T SmithOn turbulent separation in the flow past a bluff bodyJFM, 241:443–467, 1992

[SK’08] B Scheichl & A KluwickAsymptotic theory of bluff-body separation: a novel shear-layer scaling deduced from aninvestigation of the unsteady motionJ Fluids Struct, 24(8):1326–1338, 2008

[SKS’11] B Scheichl, A Kluwick & F T SmithBreak-away separation for high turbulence intensity and large Reynolds numberJFM, 670:260–300, 2011


Outline




4 Summary


Small-defect TBLs

B Scheichl & A KluwickTheor Comput Fluid Dyn, 2012

ǫ ≪ 1

y

δ

δWL

u, xus

u

uτ

∼1

κln y+ + C+

{

outer predominantly

inviscid region overlap , y+ := yuτRe → ∞

viscous wall layer

u∂u

∂x+ · · · ∼ −∂u

′v′

∂y

hypothesis: ǫ ∼ 1− u/us ∼ u′i

⇒ u′iu′

j = O(ǫ2) , δ ∼ ǫ

classical: ǫ ∼√τwus

∼ κ

lnRe(1)

δWL

δ. . . exponentially small (2)


“Upstream-dilemma” in classical boundary layer theory . . .

inviscid flow detaches by Brillouin–Villat (B-V) singularity

us(x, k)/uD(k) ∼ 2k√−s , s = x− xD(k) → 0− , k > 0

triggers Goldstein singularity upstream

but both singularities to be avoided

local interaction strategypressure induced by BL displacement alters rapid pressure variationsuch that it affects lowest-order BL approximation adjacent to the wall


“Upstream-dilemma” in classical boundary layer theory . . .. . . and how to resolve it

inviscid flow detaches by Brillouin–Villat (B-V) singularity

us(x, k)/uD(k) ∼ 2k√−s , s = x− xD(k) → 0− , k > 0

triggers Goldstein singularity upstream

but both singularities to be avoided

local interaction strategypressure induced by BL displacement alters rapid pressure variationsuch that it affects lowest-order BL approximation adjacent to the wall

δ

δWL

y

underdeveloped turbulence:

δWL

δ. . . algebraically small


A novel triple deck structurefixes scaling of oncoming TBL

y

δ

uus

O(ǫ)

δWL

log law

{

defectlayer

{

viscouswall layer

y

s

δ ≪ ǫδTD

δTD

TD

outer inner

interaction

δWL = 1/(Re δ2) , δTD/δ = O(ǫ) , δTD = Re−4/9


Outline




4 Summary


Outer inviscid/irrotational interaction

local expansion on BL scale

[X,Y ] = [x− xBV(k), y ]/δBV , Z = X + iY

ψ/δBV ∼ Y + δ1/2BV ψBV + · · ·

︸︷︷︸

irrotational

− ǫ[FBV(Y ; k)− δ1/2BV ψv(X,Y ; k) + · · ·︸︷︷︸

vorticity-induced

] +O(ǫ2)

potential of B-V singularity

ψBV = −4k

3ℜ(Z3/2

)

Log law and Cole’s law of the wake

F ′

BV = −κ−1 lnY + F ′

w(Y ; k)

Y → 0 : F ′

w ∼ A + B Y 1/2︸︷︷︸

separation

+O(Y lnY )


Outer inviscid/irrotational interactionWiener–Hopf-type vortex flow problem

(∂2X + ∂2Y

)ψv = −F ′′′

BV(Y ; k)ψBV

Y → 0 : ψv|X<0 → 0 , induced pressure → 0 for X > 0

Y → ∞ : ψv ∼ 4kℑ√Z induces potential flow

solution

ψv =4k

3ℜ[

Z3/2 F ′

BV(Y ; k)− 3i

∫ Y

0

√

Z − 2i(Y − η)F ′

BV(η; k) dη

]

︸︷︷︸

match with upper deck: O(Z3/2 lnZ) , Z → 0 X

cf. flow past inclined flat plate

R E Melnik & B GrossmanNumerical & Physical Aspects of Aerodynamical Flows, 1982

R E Melnik & R ChowAsymptotic Theory of Two-Dimensional Trailing-Edge Flows, NASA TR, 1975


Outer inviscid/irrotational interactionWiener–Hopf-type vortex flow problem

(∂2X + ∂2Y

)ψv = −F ′′′

BV(Y ; k)ψBV

Y → 0 : ψv|X<0 → 0 ,[∂Y ψv + F ′′

BV ψBV

]

X>0→ 0

Y → ∞ : ψv ∼ 4kℑ√Z induces potential flow

solution

ψv =4k

3ℜ[

Z3/2 F ′

BV(Y ; k)− 3i

∫ Y

0

√

Z − 2i(Y − η)F ′

BV(η; k) dη

]

︸︷︷︸

match with upper deck: O(Z3/2 lnZ) , Z → 0 X

cf. flow past inclined flat plate

R E Melnik & B GrossmanNumerical & Physical Aspects of Aerodynamical Flows, 1982

R E Melnik & R ChowAsymptotic Theory of Two-Dimensional Trailing-Edge Flows, NASA TR, 1975


Unravelling the separation riddle Re : = RU/ν → ∞

A∼ D

∼ D

classical triple deck

outer / inner interaction

underdevelopedturbulent

Re

2R

laminar

transcritical turbulent

transition

u∞ = 1

V V SychevLaminar separationFluid Dyn, 7(3):407–417, 1972

B Scheichl, A Kluwick & M Alletto“How turbulent” is the boundary . . . ?Acta Mech, 201(1–4):131–151, 2008

[SKS’11]


A self-consistent yet incomplete theory

This intricate interaction mechanism is

not mastered by any exisiting RANS closure

holds for any value of xD(k) indicating Helmholtz–Kirchhoff flow

But effective value of k

must be fixed by a rational separation criterion

singled out by splitting of wall layer into lower / main deck ?

θsep − θD(k) = O(Re−4/9

)


A self-consistent yet incomplete theory

This intricate interaction mechanism is

not mastered by any exisiting RANS closureholds for any value of xD(k) indicating Helmholtz–Kirchhoff flow

But effective value of k

must be fixed by a rational separation criterionsingled out by splitting of wall layer into lower / main deck ?

θsep − θD(k) = O(Re−4/9

)

U

A

L

θsep


Outline




4 Summary


Detached-Eddy Simulation Jon Paton (TotalSim Ltd, Brackley, UK, 2010)

Re = RU/ν = 106 (cf. experiments), fully 3D (aspect ratio = 1:10)

viscous sublayer resolved by wall functions

∼ xDu∞ = 1

2

colours distinguish isotachs

Re→ ∞ : xD → ? , k → ?


Outline




4 Summary


Potential flow with small cuspidal stagnation zone

u = us(x)

u = uD

d

D

R

x, u

y, vx = xD

x = xT

infinite / finite cusp

J-M Vanden-BroeckGravity-Capillary Free-Surface FlowsOxford University Press, 2010

outer limit d = xT − xD → 0+ , σ = xT − x→ 0+ , d/σ → 0

[u, v] ∼ [u0, v0](x, y) +O(d4) , [u0, v0] ∼ λ[σ, y] → [0, 0]

circular cylinder: us = 2 sin(x) ⇒ λ = 2

inner limit d/σ = O(1)


Potential flow with small cuspidal stagnation zoneInner limit

u0 − iv0 ∼ dλW ′(Z) , Z = X + iY , (X,Y ) = (−σ, y)/dZ-plane 7→ auxiliary χ-plane

XYO 1

1

−1

−1 D

R

S

D′

R′

S′

ℑ(χ)

ℜ(χ)Z = ∞

W (Z) = (U2D/2)(χ+ 1/χ)2

W ′(Z) = UD i/χ

}

⇒{−Z = (−W )3/2 + (1− W )3/2

UD = 3/4 , W = 9W/8

separatrix S : astroid X2/3+Y 2/3 = 1 , ℑ(W ) = 0 , |W ′| = UD

wall slip US(X) =W ′(X) , X ≤ −1:

(US/UD − UD/US)(X) . . . is root of 4th-order polynomial


Potential flow with small cuspidal stagnation zoneInner limit

u0 − iv0 ∼ dλW ′(Z) , Z = X + iY , (X,Y ) = (−σ, y)/dZ-plane 7→ auxiliary χ-plane

XYO 1

1

−1

−1 D

R

S

D′

R′

S′

ℑ(χ)

ℜ(χ)Z = ∞

TBL

W (Z) = (U2D/2)(χ+ 1/χ)2

W ′(Z) = UD i/χ

}

⇒{−Z = (−W )3/2 + (1− W )3/2

UD = 3/4 , W = 9W/8

separatrix S : astroid X2/3+Y 2/3 = 1 , ℑ(W ) = 0 , |W ′| = UD

wall slip US(X) =W ′(X) , X ≤ −1:

(US/UD − UD/US)(X) . . . is root of 4th-order polynomial


Potential flow with small cuspidal stagnation zoneInner limit extracted from splitter-plate flows

d d

dd

[C’99] [K’40]

let d→ 0

[C’99] S A ChaplyginOn the problems of jets in an incompressible fluidProc Phys Sect Soc Natural Scientists, 10(1):35–40, 1899

[K’40] M KolscherUnstetige Strömungen mit endlichem Totwasser (in German)Luftfahrtforsch, 17(5):154–160, 1940


Potential flow with small cuspidal stagnation zoneB-V singularity recovered

u = us

u = uD

d

D

Rx

x = xD

x = xT

B-V parameter k(d)

outer variables

s = xD(k)− x→ 0+ :

us(x, k)/uD(k) ∼ g(k2s) +O(s3/2) , g(t) = 1 + 2√t+ 10 t/3

d→ 0: k ∼ 1/√6d

inner variables

S = −1−X → 0+ : 4US(X)/3 ∼ g(S/6) +O(S3/2) X


Potential flow with small cuspidal stagnation zoneB-V singularity recovered

u = us

u = uD

d

D

Rx

x = xD

x = xT

3

2

1.5

0.75−3 −2.5 −2 −1.5 −1

2.5

1

US

X

X

X → −∞ :US +X = O(X−3)

B-V parameter k(d)

outer variables

s = xD(k)− x→ 0+ :

us(x, k)/uD(k) ∼ g(k2s) +O(s3/2) , g(t) = 1 + 2√t+ 10 t/3

d→ 0: k ∼ 1/√6d

inner variables

S = −1−X → 0+ : 4US(X)/3 ∼ g(S/6) +O(S3/2) X


BTW: approach recovers classical flow patterns

Z = −(−W )3/2 − (1− W )3/2 , W = 8W/9

Z 7→ [Z2α +H(1/2− α)]1/2 , α > 0 , H . . . Heaviside function

α > 1/2

X

Y

O 1

1

D

R

S

α < 1/2 α = 1/2

X

Y

O

1

D

R

S

X

Y

O1

DR

S

ω = 0

Chaplygin-/Sadovskii-type potential-vortex flow

K KrämerPotential flow with dead water past a kinked plane wall (in German)Arch Appl Mech (Ing-Arch), 33(1):36–50, 1963


Outline




4 Summary


Incident TBL with small velocity deficitRapid growth of deficit layer

2-tiered slightly underdeveloped TBL[

1− u

us,−〈u′v′〉u2s

]

∼[

ǫ∂F

∂η, ǫ2TR

](x, η =

y

δ; d), δ ∼ ǫ T∆(x; d)

T ≪ 1 , ǫ = κ/ lnRe

η → 0:[∂ηF, R

]∼

[−κ−1 ln η + C(x; d), R ∼ 1

︸︷︷︸

wall shear stress

]

3-tiered, closure-, history-independent as [d, σ] → [0+, 0+] , d/σ → 0

[F, R, ∆

]∼

[f Ft(η)

σ2√−lnσ

,f2Rt(η)

σ4(−lnσ),4f

√−lnσ

σFt(1)

]

, f =

1

2

√∫ xT

0

(us(x)

λ

)3

dx

2ηF ′

t = Ft(1)Rt , F ′

t (1) = Rt(1) = Rt(0) = 0




1− u


]

∼[

ǫ∂F

∂η, ǫ2TR

](x, η =

y

δ; d), δ ∼ ǫ T∆(x; d)

T ≪ 1 , ǫ = κ/ lnRe

η → 0:[∂ηF, R

]∼

[−κ−1 ln η + C(x; d), R ∼ 1

︸︷︷︸

wall shear stress

]


[F, R, ∆

]∼

[f Ft(η)

σ2√−lnσ

,f2Rt(η)

σ4(−lnσ),4f

√−lnσ

σFt(1)

]

, f =

1

2

√∫ xT

0

(us(x)

λ

)3

dx

2ηF ′

t = Ft(1)Rt , F ′

t (1) = Rt(1) = Rt(0) = 0

also governs moderately-large-defect equilibrium TBL




1− u


]

∼[

ǫ∂F

∂η, ǫ2TR

](x, η =

y

δ; d), δ ∼ ǫ T∆(x; d)

T ≪ 1 , ǫ = κ/ lnRe

η → 0:[∂ηF, R

]∼

[−κ−1 ln η + C(x; d), R ∼ 1

︸︷︷︸

wall shear stress

]


[F, R, ∆

]∼

[f Ft(η)

σ2√−lnσ

,f2Rt(η)

σ4(−lnσ),4f

√−lnσ

σFt(1)

]

, f =

1

2

√∫ xT

0

(us(x)

λ

)3

dx

2ηF ′

t = Ft(1)Rt , F ′

t (1) = Rt(1) = Rt(0) = 0

η → 0: F ′

t ∼ F ′

t (0) − 2√

2F ′

t (0) η [SK’08] vs. [NS’92]


A nearly inviscid boundary layerLeast-degenerate distinguished limit

D = d/τ!= O(1) , τ = (−2/ ln ǫ)1/4 ǫ1/2

3-tiered universal large-defect TBL

u ∼ τλU(X, Y ;D) , −〈u′v′〉 = O(Tτ2) ⇒ inviscid rotational

δ/τ = O(−T ln ǫ) ⇒ T ≪ −1/ ln ǫ vs. T ∼ ǫ−2Re−4/9

[SKS’11]

U = ∂Y Ψ = ±√

D2U2S (X)− 3F ′

t (Ψ)/2

Rt = ℓ2(η)F ′′

t2

η → 0: ℓ ∼ κη X


A nearly inviscid boundary layerLeast-degenerate distinguished limit

D = d/τ!= O(1) , τ = (−2/ ln ǫ)1/4 ǫ1/2

3-tiered universal large-defect TBL

u ∼ τλU(X, Y ;D) , −〈u′v′〉 = O(Tτ2) ⇒ inviscid rotational

δ/τ = O(−T ln ǫ) ⇒ T ≪ −1/ ln ǫ vs. T ∼ ǫ−2Re−4/9

[SKS’11]

U = ∂Y Ψ = ±√

D2U2S (X)− 3F ′

t (Ψ)/2

Rt = ℓ2(η)F ′′

t2

η → 0: ℓ ∼ κη X

12 1410862 40

1

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

η

F ′

F ′

t

Rt

Rt


Separation criterion: non-interactive limit

US(X;D) =(D2U2

S (X)− 3F ′

t (0)/2)1/2 ≥ 0

B-V singularity ∼ separation

US = 0 ⇔ X = −1 , US = UD = 3/4 ⇔ D =(8F ′

t (0)/3)1/2

d ∼ 27/4(ǫF ′

t (0)/3)1/2

/(− ln ǫ)1/4 , F ′

t (0).= 13.868

[Z’97] M M ZdravkovichFlow Around Circular Cylinders. . . . , Experiments, . . . Vol. 1: FundamentalsOxford University Press, 1997


Separation criterion: non-interactive limit

US(X;D) =(D2U2

S (X)− 3F ′

t (0)/2)1/2 ≥ 0

B-V singularity ∼ separation

US = 0 ⇔ X = −1 , US = UD = 3/4 ⇔ D =(8F ′

t (0)/3)1/2

d ∼ 27/4(ǫF ′

t (0)/3)1/2

/(− ln ǫ)1/4 , F ′

t (0).= 13.868

6

4

2

0−8 −4 −2 0−10 −6.08...

10

8DUS

DX

D = 9, 7, 5, 3, 1, 0

−3DX/4

[NS’92]

165

160

155

150

145

140

135

13012 8 7 6

170

1/ lgRe

xD[◦]

cf. [Z’97]

[Z’97] M M ZdravkovichFlow Around Circular Cylinders. . . . , Experiments, . . . Vol. 1: FundamentalsOxford University Press, 1997


Outline




4 Summary


A novel interaction mechanismInviscid vortex flow

S → 0+ : U ∼ U0(Y ) + S1/2 U1(Y ) , US ∼(6F ′

t (0)2S

)1/4

U0 =∂Ψ

∂Y=

√

3

2

[F ′

t (0)− F ′

t (Ψ)], Y → 0: U0 ∼ 3

κ2/3

[F ′

t (0)

Ft(1)

Y

2

]1/3

2-tiered Euler stage for x− xD = O(δ)

pressure P (X, Y ) by Rayleigh problem

∂2P

∂X2+∂2P

∂Y 2= 2

U ′

0

U0−c∂P

∂Y, X < 0 , Y → ∞ :

P

2∼ −ℑ

√

X + iY

wall slip by Bernoulli’s law ∼ ±√

2[P (XD, 0)− P (X, 0)

]

X → −∞ : P ∼ −2√

−X , X → ∞ : P → 0 (how?)

wall slip drives near-wall tiers

log-region eradicated as wall shear stress changes sign


Outline




4 Summary


Rayleigh stage and new 1/3-power velocity law

BL edge at Y = Ye ⇒ (X, Y ) = (X, Y )/Ye

model Λ = 2H(1− Y )U ′

0(Y )

U0(Y )⇒ U0 =

√

3F ′

t (0)

2exp

∫ Y

1

Λ(ζ)

2dζ

vortex-flow problem(∂2X + ∂2Y )P = Λ(Y ) ∂Y P

Y → 0: P . . . regular , Y > 1 , Z = X + iY → ∞ : P ∼ −2ℑ√Z

Y → 0: Λ ∼ −2/(3Y ) ⇒ P ∼ P0(X) +O(Y 2)


Rayleigh stage and new 1/3-power velocity law

BL edge at Y = Ye ⇒ (X, Y ) = (X, Y )/Ye

model Λ = 2H(1− Y )U ′

0(Y )

U0(Y )⇒ U0 =

√

3F ′

t (0)

2exp

∫ Y

1

Λ(ζ)

2dζ

vortex-flow problem has no solution(∂2X + ∂2Y )P = Λ(Y ) ∂Y P

Y → 0: P . . . regular , Y > 1 , Z = X + iY → ∞ : P ∼ −2ℑ√Z

Y → 0: Λ ∼ −2/(3Y ) ⇒ P ∼ P0(X) +O(Y 2)


Summary

Main achievements

asymptotic theory of bluff-body flow for Re→ ∞turbulence intensity level is highest possible

novel distinguished limit bridges theories for k = O(1) and k = ∞

Further outlook

regularisation of B-V singularity

merging of separated shear layers to wake

⇒ turbulent–turbulent transition

comprehensive flow description on/beyond body scale

⇒ recirculating-flow regions slender

⇒ bluff-body drag of O(1) !


Thank you for your attention!

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