An overview of adverse pressure gradient turbulent boundary layers Yvan Maciel Mechanical Engineering Department, Laval University, Quebec City, Canada
Acknowledgments Ayse Gungor and Javier Jiménez Many graduate students Financial support from NSERC and CFI of Canada
First Multiflow Summer Workshop, Madrid, 4 July 2013
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Change paradigm From canonical wall flows to complex wall flows
2
Outline • Basic flow properties • Impact of the pressure gradient on the boundary layer • Equilibrium TBLs • Pressure gradient parameter • Scaling • Layer structure • Coherent structures
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Nomenclature TBL = Turbulent Boundary Layer APG = “Adverse” Pressure Gradient
∂p >0 ∂x
ZPG = Zero Pressure Gradient
∂p =0 ∂x
FPG = Favourable Pressure Gradient
∂p 0 ∂x
ZPG = Zero Pressure Gradient
∂p =0 ∂x
FPG = Favourable Pressure Gradient
∂p 0 dx
(x-xmin)/(xmax-xmin) 37
How does the pressure gradient affect the boundary layer dynamically? To Lo dU e β= = − Te U o dx
= local impact of the PG on the boundary layer
dβ d L dU e = − o = local stabilizing/destabilizing effect of the PG dx dx U o dx dβ 0 ⇒ dx
destabilizing effect 38
An adverse pressure gradient can stabilize a boundary layer ! It can be favorable!
destabilizing
stabilizing
dβ >0 dx
dβ 0 ⇒ APG ZPG
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An adverse pressure gradient can stabilize a boundary layer ! It can be favorable!
destabilizing
stabilizing
dβ >0 dx
dβ 0 ⇒ APG ZPG
dβ d Lo dU e Lo d 2U e = − − 0 dx
dβ 0 ⇒ APG ZPG
Defect decreasing!
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Scaling of APG TBLs
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Definition of a scale •
A scale S is a variable whose evolution follows that of another variable, say X, in a physical system.
⇒
The scaled variable X/S retains the same order of magnitude whatever the spatial, temporal or other parametric changes of X itself.
Definition of a scale •
A scale S is a variable whose evolution follows that of another variable, say X, in a physical system.
⇒
The scaled variable X/S retains the same order of magnitude whatever the spatial, temporal or other parametric changes of X itself.
•
O(X/S) is not necessarily one. Scaling is not normalization!
Scale in fluid mechanics O(X/S) constant in a given flow: 1. for the complete Reynolds-number range of validity of the flow regime (including );
Scale in fluid mechanics O(X/S) constant in a given flow: 1. for the complete Reynolds-number range of validity of the flow regime (including ); 2. in the whole flow region where it should apply (ex. entire TBL over an airfoil)
For a given complex flow at finite Re, a valid scale does not have to: •
reveal asymptotic invariance of a flow variable
•
reveal self-similarity of a flow variable
For a given flow at finite Re, a valid scale does not have to: •
reveal asymptotic invariance of a flow variable
•
reveal self-similarity of a flow variable
Scaling is not similarity! Scaling is not “collapsing of curves”
Outer length scale Lo ∞
δ
or = ∆
∫ 0
δ*
θ
Ue − U Ue dy δ* = Uo Uo
δ* →0 δ
and
θ →0 δ
as
Re → ∞
APG: Outer velocity scale U o 1/ 2
Viscous-friction velocity:
Zagarola-Smits velocity:
τw uτ = ρ U ZS
Zagarola and Smits (1998)
δ* = Ue δ 1/ 2
Pressure-gradient velocity: U po Mellor and Gibson (1966)
Other suggestions:
δ * dp = ρ dx
U e , (uτ U e )1/ 2 ,
(
)
1/ 3
U PS = k −u ′v ′max U e δ */ L
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APG: Outer velocity scale U o Uo ∝ Ue −U
(
)
uτ
U ZS
U po
Ue
Scale in the asymptotic limit for all PG conditions
?
Scale at finite Re for all PG conditions
Scale in the asymptotic limit at separation (Cf = 0)
Scale at finite Re at separation (Cf = 0)
Rij1/ 2
∝ ui′u ′j
1/ 2
APG: Inner scales Ui
Li 1/ 2
Viscous-friction scales:
τw uτ = ρ
uτ 1/3
Viscous-pressure scales: Stratford (1959)
“Extended” scales: Manhart et al. (2007)
u pi
ν dpw = ρ dx
u= τp
uτ + u 2pi 2
ν
ν u pi
ν uτ p
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Inner region: near the wall 1/ 2
τ uτ = w ρ
1/3
u pi
Viscous-friction scales
ν dpw = ρ dx
Viscous-pressure scales
ZPG
y
y 1/2
U u pi
Detachment
y2
y u pi Scaling
ν
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Layer structure of APG TBLs Various propositions: Two-layer structure (not ZPG one) Stratford (1959), Townsend (1961), Mellor (1966), McDonald (1969), Kader and Yaglom (1978), Afzal (1996)…
Three-layer structure Yajnik (1970), Mellor (1972), Melnik (1989), Durbin & Belcher (1992).
Transition from two- to three-layers Scheichl and Kluwick (2007)
Usually implies matching with an assumed overlap law: log, squareroot,… 54
Large defect flows
defect 55
Conclusion Knowledge still missing about pressure gradient TBLs: • Layer structure(s) • Does it depend on: • Equilibrium/non-equilibrium state? • Size of momentum defect?
• Corresponding length and velocity scales • Organization of turbulence
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