Turbulent bluff-body separation: recent advances of a self ... · Turbulent bluff-body separation: recent advances of a self-consistent ﬂow description for large Reynolds numbers

Turbulent bluff-body separation: recent advances of aself-consistent flow description for large Reynolds numbers

Bernhard [email protected]

Institute of Fluid Mechanics and Heat Transfer

Seminar lecture

Department of Mathematics, UCL, 10 Nov 2009

B. Scheichl (TU Vienna) Turbulent Bluff-Body Separation 1 / 34

The research presented has been carried out in collaboration with

Prof Alfred Kluwick

Institute of Fluid Mechanics and Heat Transfer

Prof Frank T. Smith, FRS

Department of Mathematics

Their contributions are greatly acknowledged.


Sandborn & Liu (JFM, 1968)

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


Overview

1 Turbulent wall-bounded flowsNumerically-based methodsAnalytically-based methods

2 Classical theory of turbulent small-defect BLsGeneral frameworkRoutes to separation

3 Problem formulation – motivationGlobal flow pictureSeparation – experimental findings

4 Asymptotic structure of the flow‘Ideal-fluid limit’ – Kirchhoff-type potential flowBoundary layer

5 Incident boundary layer – numerical results

6 Preliminary conclusions – outlook


Wall-bounded high-Reynolds-number flows : Re := UL/ν ≫ 1

x ,u

y

y ∼ δ

ue(x)

uBL

inviscid, irrotational

Navier–Stokes eqs (ρ = const)

∇ · u = 0 ,∂u∂t

+ u · ∇u = −∇p +1

Re∇2u , u|y=0 = 0

Reynolds-averaging, ergod hypothesis (nominally steady 2D flow)

Q(x , t ; Re) = 〈Q〉(x , y ; Re) + Q′(x , t ; Re)

〈Q〉 := limtav→∞

1tav

∫ tav /2

−tav /2Q(x , t + t ′; Re) dt ′


Wall-bounded high-Reynolds-number flows : Re := UL/ν ≫ 1

x ,u

y

y ∼ δ

ue(x)

uBL

inviscid, irrotational

Navier–Stokes eqs (ρ = const)

∇ · u = 0 ,∂u∂t

+ u · ∇u = −∇p +1

Re∇2u , u|y=0 = 0

Reynolds-averaging, ergod hypothesis (nominally steady 2D flow)

Q(x , t ; Re) = 〈Q〉(x , y ; Re) + Q′(x , t ; Re)

〈Q〉 := limtav→∞

1tav

∫ tav /2

−tav /2Q(x , t + t ′; Re) dt ′


Outline








Numerically-based methods

• DNS : resolution of all scales ⇒ no modelling needed but at presentrestricted to moderately large values of Re

• LES : modelling of short scales ⇒ reduction of computational efforts,which, however, are still too massive to be useful for the solutionof engineering problems

• RANS : modelling of all scales ⇒ engineering problems can be solvedwith an acceptable amount of computational efforts, which,however, increase with increasing values of Re












Outline








Analytically-based methods

• full NS eqs : starting efforts, but still no complete theory existsDeriat & Guiraud (J Mec Theor Appl, 1986), Walker (AIAA J, 1989, CISM, 1998),

Scheichl & Kluwick (JFS, 2008)

• non-dimensional Reynolds-averaged NS eqs(nominally 2D, curvature effects on BL flow of higher order):

∂u∂x

+∂v∂y

= 0−τ

u∂u∂x

+ v∂u∂y

= −∂p∂x

− ∂〈u′2〉∂x

− ∂︷︸︸︷

〈u′v ′〉∂y

+1

Re∇2u

∇2 =∂2

∂x2 +∂2

∂y2

u∂v∂x

+ v∂v∂y

= −∂p∂y

− ∂〈u′v ′〉∂x

− ∂〈v ′2〉∂y

+1

Re∇2v

y = 0 : u = v = u′ = v ′ = 0, y ∼ δ(x ; Re) : u ∼ ue(x), τ ∼ 0

⇒ asymptotic theory faces closure problem for τ


Analytically-based methods

• full NS eqs : starting efforts, but still no complete theory existsDeriat & Guiraud (J Mec Theor Appl, 1986), Walker (AIAA J, 1989, CISM, 1998),

Scheichl & Kluwick (JFS, 2008)

• non-dimensional Reynolds-averaged NS eqs(nominally 2D, curvature effects on BL flow of higher order):

∂u∂x

+∂v∂y

= 0−τ

u∂u∂x

+ v∂u∂y

= −∂p∂x

− ∂〈u′2〉∂x

− ∂︷︸︸︷

〈u′v ′〉∂y

+1

Re∇2u

∇2 =∂2

∂x2 +∂2

∂y2

u∂v∂x

+ v∂v∂y

= −∂p∂y

− ∂〈u′v ′〉∂x

− ∂〈v ′2〉∂y

+1

Re∇2v

y = 0 : u = v = u′ = v ′ = 0, y ∼ δ(x ; Re) : u ∼ ue(x), τ ∼ 0

⇒ asymptotic theory faces closure problem for τ


Outline








Classical theory of turbulent small-defect BLs : uτ:=

√τw → 0

Yajnik (JFM, 1970), Bush & Fendell (JFM, 1972), Fendell (J Astronaut Sci, 1972),

Mellor (Int J Eng Sci, 1972), Gersten & Herwig (1992), Walker (CISM, 1998)

yδ

uue

O(uτ ) uuτ

∼ 1κ

ln y+ + Ci

{

outer predominately

inviscid region overlap region : y+ := yuτ Re → ∞

viscous wall layer

τw = Re−1(∂u/∂y)y=0

assumptions Walker (CISM, 1998), Scheichl, Kluwick & Alletto (Acta Mech, 2008)

(a) locally isotropic turbulence ⇒ 〈u′2〉, 〈u′v ′〉, 〈v ′2〉 of same magnitude

(b) wall layer : total shear stress essentially unaffected by pressure gradient

(c) direct match with outer fully turbulent region ⇒ two-tiered BL



√τw → 0



yδ

uue

O(uτ ) uuτ

∼ 1κ

ln y+ + Ci

{

outer predominately


viscous wall layer

τw = Re−1(∂u/∂y)y=0







√τw → 0



yδ

uue

O(uτ ) uuτ

∼ 1κ

ln y+ + Ci

{

outer predominately


viscous wall layer

τw = Re−1(∂u/∂y)y=0







√τw → 0



yδ

uue

O(uτ ) uuτ

∼ 1κ

ln y+ + Ci

{

outer predominately


viscous wall layer

τw = Re−1(∂u/∂y)y=0






Distinguished limit γ := uτ/ue → 0 , Re → ∞

viscous wall layer : y+ = yuτ Re outer defect layer : η = y/δ

uue

∼ γ(x ,Re) u+(x , y+) + · · ·τ

u2e∼ γ2(x ,Re) t+(x , y+) + · · ·

p ∼ p0(x) + · · ·

uue

∼ 1 − γ∂F1(x , η)

∂η+ O(γ2)

τ

u2e∼ γ2T1(x , η) + O(γ3)

p ∼ pe(x) + O(γ2)

experimental observation / result from first principles (?)Österlund et al. (Phys Fluids, 2000) Walker (AIAA J, 1989, CISM, 1998)

u+(y+) ∼ κ−1 ln y+ + Ci , y+ → ∞ , κ ≈ 0.384 , Ci ≈ 4.1

provides expansion in outer layer and matching with wall layer

η → 0 : ∂F1/∂η ∼ −κ−1 ln η + Co(x) , T1 → 1 ; p0(x) = pe(x)

skin-friction law κ/γ ∼ ln(Re γδue) + κ(Ci + Co) ∼ ln Re


Distinguished limit γ := uτ/ue → 0 , Re → ∞

viscous wall layer : y+ = yuτ Re outer defect layer : η = y/δ

uue

∼ γ(x ,Re) u+(x , y+) + · · ·τ

u2e∼ γ2(x ,Re) t+(x , y+) + · · ·

p ∼ p0(x) + · · ·

uue

∼ 1 − γ∂F1(x , η)

∂η+ O(γ2)

τ

u2e∼ γ2T1(x , η) + O(γ3)

p ∼ pe(x) + O(γ2)

experimental observation / result from first principles (?)Österlund et al. (Phys Fluids, 2000) Walker (AIAA J, 1989, CISM, 1998)

u+(y+) ∼ κ−1 ln y+ + Ci , y+ → ∞ , κ ≈ 0.384 , Ci ≈ 4.1

provides expansion in outer layer and matching with wall layer

η → 0 : ∂F1/∂η ∼ −κ−1 ln η + Co(x) , T1 → 1 ; p0(x) = pe(x)

skin-friction law κ/γ ∼ ln(Re γδue) + κ(Ci + Co) ∼ ln Re


Substitution into Reynolds-averaged NS eqs

viscous wall layer : du+/dy+ + t+ = 1 . . . constant total shear stress

outer defect layer : δ ∼ γ∆1(x) + O(γ2)

leading-order equation

(E + 2β0)ηF ′

1 − E F1 − ∆1F1,e ∂F1/∂x = F1,e (T1 − 1) , F1,e = F1(x ,1)

E := 1 − ∆1 dF1,e/dx , β0 := −(∆1F1,e due/dx)/ue

layer thickness ratio (Kármán number) δ+ exponentially small

δ+ =(uτ Re)−1

δ∼ 1

Re γ2∆1ue∼ 1

∆1ue

exp(−κ/γ)γ2

consequences

classical theory not capable of describing BL separation Sykes (JFM, 1980)

“manipulate” velocity defect 1 − u/ue = O(ǫ), BL thickness δ






(E + 2β0)ηF ′

1 − E F1 − ∆1F1,e ∂F1/∂x = F1,e (T1 − 1) , F1,e = F1(x ,1)



δ+ =(uτ Re)−1

δ∼ 1

Re γ2∆1ue∼ 1

∆1ue

exp(−κ/γ)γ2

consequences








(E + 2β0)ηF ′

1 − E F1 − ∆1F1,e ∂F1/∂x = F1,e (T1 − 1) , F1,e = F1(x ,1)



δ+ =(uτ Re)−1

δ∼ 1

Re γ2∆1ue∼ 1

∆1ue

exp(−κ/γ)γ2

consequences








(E + 2β0)ηF ′

1 − E F1 − ∆1F1,e ∂F1/∂x = F1,e (T1 − 1) , F1,e = F1(x ,1)



δ+ =(uτ Re)−1

δ∼ 1

Re γ2∆1ue∼ 1

∆1ue

exp(−κ/γ)γ2

consequences




Outline








Routes to separation

η

1

uue

O(ǫ)

δ+ =1

Re δ2

log law

defectlayer

viscouswall layer

consider BLs having . . .

• ǫ≫ γ ⇒ ∂p/∂x = O(1) ⇒marginal separation Scheichl & Kluwick (AIAA J, 2007)

• δ+ =(δRe)−1

δ= O(δr ) ⇒ locally ∂p/∂x ≫ 1 ⇒

massive separation Scheichl, Kluwick & Smith



η

1

uue

O(ǫ)

δ+ =1

Re δ2

log law

defectlayer

viscouswall layer



• δ+ =(δRe)−1





η

1

uue

O(ǫ)

δ+ =1

Re δ2

log law

defectlayer

viscouswall layer



• δ+ =(δRe)−1





η

1

uue

O(ǫ)

δ+ =1

Re δ2

log law

defectlayer

viscouswall layer



• δ+ =(δRe)−1




Problem formulation – motivationLocal description of turbulent break-away separation . . .

. . . requires answers to three basic questions:

(1) Global topology of flow for ‘vanishing viscosity’?

(2) Characteristics of incident boundary layer flow?

(3) How do these issues interdepend?

basic assumptions

• flow incompressible,nominally steady, 2D

• free-stream turbulencedisregarded

• Re := UL/ν → ∞

canonical example: circular-cylinder flow

How turbulent are the BL and the separated SL ?Scheichl, Kluwick & Alletto (Acta Mech, 2008, AIAA J)







basic assumptions



• Re := UL/ν → ∞









basic assumptions



• Re := UL/ν → ∞









basic assumptions



• Re := UL/ν → ∞









basic assumptions



• Re := UL/ν → ∞


u∞ = 1L = 1

F R

S

nn ∼ δ

sθ

TBL TSL



Outline








Global flow picture

Re ' 3 × 106 : postcritical regime, Re → ∞ : transition point approximates F

Re−1 = 0 : ultimate or T-state of flow, transition in F

u∞ = 1

F R ∼ S ?

SθS ≈ 115◦ uS . u∞

pS . p∞

transition

TBL TSL

Neish & Smith (JFM, 1992)

unlikely Scheichl & Kluwick (JFS, 2008)

corroborated experimentally up to Re ≈ 4 × 107

Roshko (JFM, 1961), Achenbach (JFM, 1968, Int J Heat Mass Trans, 1975), Zdravkovich (1997),

Schewe (J Wind Eng Ind Aerod, 2001), Gölling (2006)


Global flow picture



u∞ = 1

F R ∼ S ?

SθS ≈ 115◦ uS . u∞

pS . p∞

transition

TBL TSL







Global flow picture



u∞ = 1

F R ∼ S ?

SθS ≈ 115◦ uS . u∞

pS . p∞

transition

TBL TSL







Outline








SeparationIs there really a fully developed TBL as Re → ∞ ?

oil-flow measurements : ReC := UC/ν , C = airfoil chord length

sub- trans- super- post-

by courtesy of G. Schewe (Göttingen, 2001)

-critical


SeparationIs there really a fully developed TBL as Re → ∞ ?

oil-flow measurements : ReC := UC/ν , C = airfoil chord length

C

W ≈ 6 × C

ReC = 7.4 × 105 :

ReC = 7.7 × 106 :

cD ≈ 0.11,

cD ≈ 0.14,

cL ≈ 1.1

cL ≈ 0.65

3.5 2.5 7.5

cD

cL

ReC

sub- trans- super- post-

by courtesy of G. Schewe (Göttingen, 2001)

-critical


U

S

T

T

A

A

Re

laminar

turbulent

Prandtl (Göttingen, 1914)


Outline








‘Ideal-fluid limit’Hierarchy of external flow – shear layer: potential flow approached as Re → ∞Oseen (1927), Lamb (1932), Batchelor (1956), Prandtl (1961), Sychev et al. (1998)

Re−1 = 0 : external potential flows

Imai (J Phys Soc Japan, 1953),

Birkhoff & Zarantonello (1957), Gurevich (1966)

• free streamlines confine (open) dead-water region

• class of flows / θS controlled by

Brillouin–Villat parameter k ≥ 0 , free-stream velocity uS ≤ u∞

ue

∼ b(k)θ + O(θ2) , θ → 0+

∼ uS[1 + 2k(−s)1/2 + O(−s)

], s → 0−

≡ uS , s ≥ 0

κ

{

= κB(s) , s < 0

∼ −k s−1/2 + κB(0) + O(s1/2) , s → 0+





Birkhoff & Zarantonello (1957), Gurevich (1966) 1u∞ = 1 F R

S n

s

ue, κB

uS , κ

θ θS




ue

∼ b(k)θ + O(θ2) , θ → 0+

∼ uS[1 + 2k(−s)1/2 + O(−s)

], s → 0−

≡ uS , s ≥ 0

κ

{

= κB(s) , s < 0

∼ −k s−1/2 + κB(0) + O(s1/2) , s → 0+





Birkhoff & Zarantonello (1957), Gurevich (1966) 1u∞ = 1 F R

S n

s

ue, κB

uS , κ

θ θS




ue

∼ b(k)θ + O(θ2) , θ → 0+

∼ uS[1 + 2k(−s)1/2 + O(−s)

], s → 0−

≡ uS , s ≥ 0

κ

{

= κB(s) , s < 0

∼ −k s−1/2 + κB(0) + O(s1/2) , s → 0+


Kirchhoff-type flows – numerical solutionsOpen cavity: uS ≡ u∞ = 1

surface velocity ue(θ, k)

k = 0, 0.05, 0.1, . . . ,0.5

55◦ 2′ 30′′ ≤ θS / 126◦

k = 0 :

Brillouin–Villat condition

0

2

1.5

1

0.5

030 6055 12012690≈ θ [◦]

k

k = 0

k = 0.5

free streamlines

126◦ / θS ≤ 180◦

0.5 / k < ∞?

1 ≥ uS ≥ 0


Kirchhoff-type flows – numerical solutionsOpen cavity: uS ≡ u∞ = 1

surface velocity ue(θ, k)

k = 0, 0.05, 0.1, . . . ,0.5

55◦ 2′ 30′′ ≤ θS / 126◦

k = 0 :

Brillouin–Villat condition

0

2

1.5

1

0.5

030 6055 12012690≈ θ [◦]

k

k = 0

k = 0.5

free streamlines

126◦ / θS ≤ 180◦

0.5 / k < ∞?

1 ≥ uS ≥ 0

2.5 2 1.5 1 0.5 0

1

2.5 2

1.5

0.5 0−1 2 3 1 4 5 6 7 0

kk = 0

k = 0.5

S

R?


Question: overall flow structure as Re → ∞

125

105

55

95

85

75

65

0 0.45 .05 .1 .15 .2 .25 .3 .35 .4 .5

115

θS

k ?

≈





Question: overall flow structure as Re → ∞

125

105

55

95

85

75

65

0 0.45 .05 .1 .15 .2 .25 .3 .35 .4 .5

115

θS

k ?

≈





Outline








Asymptotic framework – turbulence intensity levelNeish & Smith (1992), Scheichl, Kluwick & Alletto (Acta Mech, 2008, AIAA J)

Prandtl-type BL: k , T , Re ≫ 1

N = Re1/2n{

Re1/2[ψ, −〈u′v ′〉

], p

}

∼{[

Ψ,T r](θ,N; k ,T ), pF −u2

e(θ; k)/2}

+ O(Re−1/2

)

∂NΨ ∂NθΨ − ∂θΨ ∂NNΨ = ue ∂θue + ∂N [T r + ∂NNΨ], 0 ≤ T <∞N → 0 : Ψ → 0, ΨN → 0, r = O(N3), N → ∞ : ΨN → ue, r → 0

solution as s = θ − θS → 0−

T ≫ 1 :

Goldstein singularity at

s = sG = O[k6T−8(ln T )16

]

⇒ interaction length scale δTD




N = Re1/2n{

Re1/2[ψ, −〈u′v ′〉

], p

}

∼{[

Ψ,T r](θ,N; k ,T ), pF −u2

e(θ; k)/2}

+ O(Re−1/2

)



T ≫ 1 :


s = sG = O[k6T−8(ln T )16

]





N = Re1/2n{

Re1/2[ψ, −〈u′v ′〉

], p

}

∼{[

Ψ,T r](θ,N; k ,T ), pF −u2

e(θ; k)/2}

+ O(Re−1/2

)



T ≫ 1 :


s = sG = O[k6T−8(ln T )16

]


−s

τw = ∂NNΨ(θ,0; k ,T )

−sG


Concept of ‘underdeveloped’ TBLBL originating in F parametrised by T ≫ 1

small defect: ue − u = O(ǫ), r = O(ǫ2)

BL eq: −〈u′v ′〉 = Re−1/2T r = O(σǫ), δ = O(σ)

}

⇒ T = Re1/2σ/ǫ

fully developed TBL: −〈u′v ′〉 = O(ǫ2), σ ∝ ǫ ∼ κ/ ln Re

small-defect layer: σ ≪ 1, ǫ = κ/(σ2 ln Re) ≪ 1{[uen − ψ

ue δǫ,−〈u′v ′〉

u2e δǫ

]

,δ

σ

}

∼{

[F ,Σ](θ, η; k), ∆(θ; k)}

+ O(ǫ), η =nδ





}

⇒ T = Re1/2σ/ǫ




u2e δǫ

]

,δ

σ

}

∼{

[F ,Σ](θ, η; k), ∆(θ; k)}

+ O(ǫ), η =nδ





}

⇒ T = Re1/2σ/ǫ




u2e δǫ

]

,δ

σ

}

∼{

[F ,Σ](θ, η; k), ∆(θ; k)}

+ O(ǫ), η =nδ

η

1

uue

O(ǫ)1

Re δ2

log law

defectlayer

viscouswall layer

η

sS

δ ≪ ǫδTD

δTD

TD


Outline








Solutions of outer-defect eqs for 0 ≤ θ < θS.= 113.5◦ k = 0.45

u2e ∂θ(ue∆)ηF ′−∂θ(u3

e∆F ) = u3e(Σ−1), Σ = l2F ′′|F ′′|, F (θ, η; k), ∆(θ; k)

η → 0 : F ′ ∼ −κ−1 ln η + B(θ) ⇔ Σ ∼ 1, η = 1 : F ′ = F ′′ = Σ = 0

θ → 0 : F ∼ F0(η), ∆ ∼ ∆0 θ, ue ∼ b(k) θ stagnant-flow

• algebraic closure for ℓ = δl , Klebanoff’s intermittency factorMichel, Quémard & Durant (1969), Klebanoff (1955)

• Keller–Box scheme / method of lines, adaptive grid remeshing





η → 0 : F ′ ∼ −κ−1 ln η + B(θ) ⇔ Σ ∼ 1, η = 1 : F ′ = F ′′ = Σ = 0








η → 0 : F ′ ∼ −κ−1 ln η + B(θ) ⇔ Σ ∼ 1, η = 1 : F ′ = F ′′ = Σ = 0








η → 0 : F ′ ∼ −κ−1 ln η + B(θ) ⇔ Σ ∼ 1, η = 1 : F ′ = F ′′ = Σ = 0




1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 0 5 10 15 20 25 30

η

F ′

i

θ = θi = i θS/10i = 0, 1, . . . , 10

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 0 0.5 1 1.5 2 2.5 3 3.5

η

Σ

i


Solutions of outer-defect eqs, cont’d k = 0.45Comparison with leading-order theory as s = θ − θS → 0−

1

0

1.5

0.5

−0.5

−1

−1.5

0 90 80 50 40 30 20 10 100 113

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2 70

ue

cp = 1−u2e

∆

∆ ∼ ∆0 θ

θ

[F , ∆

]∼

[FS(η; k), ∆S(k)

]− ∑

∞

i=1

[fi(s, k)Fi(η; k), gi(s, k)∆i(k)

]

[F , ∆

]∼

[FS(1 − 2g1), ∆S(1 − g1)

]+ O(s), g1 = 2k(−s)1/2

F ′

i = O[ln(η)], η → 0, i = 1,2, . . . ⇒ sublayer for η = O(−s)

indicates onset of viscous/inviscid interaction



1

0

1.5

0.5

−0.5

−1

−1.5

0 90 80 50 40 30 20 10 100 113

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2 70

ue

cp = 1−u2e

∆

∆ ∼ ∆0 θ

θ

0.1

0.01

0.001 0.001 0.1 0.01 0.25

∆−∆S

− lg ∆s

∆S.= 0.4464

g1

[F , ∆

]∼

[FS(η; k), ∆S(k)

]−

∑∞

i=1


]

[F , ∆

]∼

[FS(1 − 2g1), ∆S(1 − g1)

]+ O(s), g1 = 2k(−s)1/2

F ′





1

0

1.5

0.5

−0.5

−1

−1.5

0 90 80 50 40 30 20 10 100 113

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2 70

ue

cp = 1−u2e

∆

∆ ∼ ∆0 θ

θ

0.1

0.01

0.001 0.001 0.1 0.01 0.25

∆−∆S

− lg ∆s

∆S.= 0.4464

g1

[F , ∆

]∼

[FS(η; k), ∆S(k)

]−

∑∞

i=1


]

[F , ∆

]∼

[FS(1 − 2g1), ∆S(1 − g1)

]+ O(s), g1 = 2k(−s)1/2

F ′




Outline








status quo of research

• increasing turbulence intensity shifts separation downstream, increases k

• BL never attains fully developed turbulent state

interactive BL ⇒ scaling in terms of k , Re ≫ 1

ǫ ∼ κ/ ln Re, δTD = k8/9Re−4/9, δ = O(δTD/ǫ), −〈u′

i u′

j 〉 = O(ǫδ)

ongoing research

• TD problem (akin to laminar counterpart)

• self-induced separation ⇔ (maximum) values of k , θS(k) ?

• structure of (large-scale) separated flow







i u′

j 〉 = O(ǫδ)

ongoing research










i u′

j 〉 = O(ǫδ)

ongoing research










i u′

j 〉 = O(ǫδ)

ongoing research





Thank you for your attention !


Documents

Turbulent bluff-body separation: recent advances of a self ... · Turbulent bluff-body separation: recent advances of a self-consistent ﬂow description for large Reynolds numbers