Post on 12-Mar-2018
Retardation of break-away separation byenhanced boundary layer turbulence:
from the triple-deck to a novel Rayleigh stage
Bernhard Scheichlbernhard.scheichl@tuwien.ac.at
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
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
Introduction Existing theory Present approach 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]
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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]
Introduction Existing theory Present approach Summary
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]
Introduction Existing theory Present approach Summary
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 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]
Introduction Existing theory Present approach Summary
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 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∞
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
Introduction Existing theory Present approach 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 ?
Introduction Existing theory Present approach 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 ?
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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
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
Introduction Existing theory Present approach 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)
Introduction Existing theory Present approach 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)
Introduction Existing theory Present approach Summary
“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
Introduction Existing theory Present approach Summary
“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
Introduction Existing theory Present approach Summary
“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
Introduction Existing theory Present approach Summary
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
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
Introduction Existing theory Present approach 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 )
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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]
Introduction Existing theory Present approach Summary
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
)
Introduction Existing theory Present approach Summary
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
)
Introduction Existing theory Present approach Summary
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
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
Introduction Existing theory Present approach 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 → ?
Introduction Existing theory Present approach 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 → ?
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
Introduction Existing theory Present approach 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)
Introduction Existing theory Present approach 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)
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
also governs moderately-large-defect equilibrium TBL
Introduction Existing theory Present approach 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
η → 0: F ′
t ∼ F ′
t (0) − 2√
2F ′
t (0) η [SK’08] vs. [NS’92]
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
Introduction Existing theory Present approach Summary
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
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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
Introduction Existing theory Present approach 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
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
Introduction Existing theory Present approach 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)
Introduction Existing theory Present approach 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 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)
Introduction Existing theory Present approach Summary
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) !
Introduction Existing theory Present approach Summary
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) !