icomasef 2013: influence of the shape on the roughness-induced transition
TRANSCRIPT
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Influence of the shape on the roughness-inducedtransition
J.-C. Loiseau(1), S. Cherubini(1) J.-C. Robinet(1) and E. Leriche(2)
(1): DynFluid Laboratory - Arts & Metiers-ParisTech - 75013 Paris, France(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France
International Conference on Instability and Control of MassivelySeparated Flows, Prato, Italy, Sept. 4-6, 2013
ANR – SICOGIF
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Background - generalities
• Three-dimensional wall roughness has numerous applications inaerospace engineering :
↪→ Upstream shift of the transition location↪→ Transition delay↪→ Increase/Decrease of the skin friction ...
• Despite the large body of literature, physical mechanismsinducing transition are still poorly understood :↪→ Empirical transition criterion by von Doenhoff and Braslow,
experimental investigation by Asai et al, ...↪→ Investigations usually focus on one kind of roughness, without
considering the effect of its shape
Experimental visualisation of the flow induced by a roughness element. Gregory & Walker, 1956.
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Motivations
• Objectives :
↪→ Have a better insight of the roughness element’s shape impacton the flow instability
↪→ Understanding the physical mechanisms responsible forroughness-induced transition.
• Methods :
↪→ Joint application of direct numerical simulations and linear globalstability analyses
↪→ Comparison of the instability mechanisms for two chosen shapes of theroughness element
• Cases under consideration :
↪→ Sharp-edged case → CYLINDER (Fransson et al. (2006)),↪→ Smooth case → BUMP (Piot et al. (2008))
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Geometry & Notations
0
z
x
Lz
l Lx
Ly
δy d
h
Geometry under consideration
• Roughness elements’s
characteristics :
↪→ Cubic-cosine bump shape :
h(d) = h0 cos3(π
√(x2+y2)
d)
↪→ Diameter : d = 2↪→ Height : h0 = 1↪→ Aspect ratio : η = d/h0 = 2.
• Incoming boundary layer
characteristics :
↪→ Ratio : δ99/h0 = 2.
↪→ Re = U∞h0ν
= [700, 1000].
• Box’s dimensions :
↪→ Lx = 105↪→ Ly = 50↪→ Lz = 8.
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Methodology : generalities
• All calculations are performed with the spectral elements code Nek5000 :
↪→ order of the polynomials N = 8,↪→ Temporal scheme of order 3 (BDF3/EXT3),↪→ Between 106 and 7.106 gridpoints.
• Base flows :
↪→ Selective frequency damping approach : application of a low-passfilter to the fully non-linear Navier-Stokes equations, see Akervik etal(2006).
• Global stability analysis :
↪→ Home made time-stepper Arnoldi algorithm build-up around Nek5000 temporal loop.
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Numerical method : iterative eigenvalue methods
INPUTS Krylov basis LNS-Solver ORTHOGONALISE
OUTPUTS RESIDUAL
LAPACK LINEAR STABILITY
w = eAtukU = [U uk] h = scal.prod(w,U)
H = [H h]
H = [bek H]
if(k=kmax)
exit loop
f = w - Uh,
b = ||f||,
uk = f/b
U, H
[X,D] = eig(H) [UX,log(D)/t] ~ eig(A)
U = [], H = []
k = 0, uk
Arnoldi algorithm build-up
around Nek5000 temporal loop
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ResultsBase Flows
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Three-dimensional Base Flows
Main features of the base flows with (η, δ99/h,Re) = (2, 2, 1000) :
↪→ Upstream and downstreamreversed flow regions (blue forU = 0),
↪→ Vortical system stemmingfrom the upstream recirculationbubble (green for Q criterion).
↪→ Uptream spanwise vorticitywraps around the roughnesselement and transforms intostreamwise vorticity downstream
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Three-dimensional Base Flows - (2)↪→ Creation of downstream quasi-aligned streamwise vortices↪→ Transfer of momentum through the lift-up effect giving birth to
streamwise streaks• For the bump, the streaks are weaker and more streamwise-localized
than for the cylinder
Streamwise velocity deviation from the Blasius profile, u = ±0.1 (top) u = ±0.05 (bottom)
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Stability
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Eigenspectra
� Cylinder → branch of eigenvalues, unstable mode at Rec = 803 (Reh = 593)
� Bump → isolated mode becoming unstable at Rec = 891 (Reh = 659), followedby a very stable branch
The bump becomes unstable at larger Re than the cylinder 11/20
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Cylinder’s leading mode
Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)
Eigenspectrum for(η, δ99/h, Re) = (2, 2, 1000)
• The spatial support of the mode is located on thestreaks, well downstream of the cylinder
• It is composed by streamwise-alternated patches ofpositive/negative velocity perturbation
• It is symmetric w.r.t. the z = 0 axis (varicose mode)
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Bump’s leading mode
Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)
Eigenspectrum for(η, δ99/h, Re) = (2, 2, 1000)
• The spatial support of the mode is located on theseparation zone, close to the bump
• It is composed by streamwise-alternated patches ofpositive/negative velocity perturbation
• It is symmetric w.r.t. the z = 0 axis (varicose mode)
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Varicose eigenmodes at Re = 1000
CYLINDER
↪→ Strong deformation of the base flowstreamwise velocity (solid contours)
BUMP
↪→ Weaker deformation of the base flowstreamwise velocity (solid contours)
• Largest values of the perturbation in the zones of maximum shear (shaded)
• Instability linked with the base flow shear like for optimal streaks ?
• To verify it, we analyze the production terms of the Reynolds-Orr equation :
dE
dt= −
∫V
uiuj∂Ui
∂xjdV − 1
Re
∫V
∂ui
∂xj
∂ui
∂xjdV (1)
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Production terms - CYLINDER case
0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
uuU uvU uwU uvV vvV vwV uwW vwWx y z x y z x y
↪→ The dominant production termsare TUy = uv ∂U
∂yand TUz = uw ∂U
∂z
↪→ TUz is the largest term, even if themode is varicose (unlike theoptimal streaks case)
Production term TUy
and streamwise perturbation u
� Streamwise displacement of thewall-normal shear
Production term TUz
and spanwise perturbation w
� Spanwise displacement of thespanwise shear
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Production terms - BUMP case
0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
uuU uvU uwU uvV vvV vwV uwW vwWx y z x y z x y
↪→ The dominant production termsare TUy = uv ∂U
∂yand TUz = uw ∂U
∂z
↪→ TUz is the largest term, as for thecylinder
Production term TUy
and streamwise perturbation u
� Streamwise displacement of thewall-normal shear
Production term TUz
and spanwise perturbation w
� Spanwise displacement of thespanwise shear
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Branch vs. isolated mode
The instability mechanism appears the same for the two roughness elements
↪→ But why for the bump the branch is very far from the most unstable mode ?
CYLINDER
• The two most unstable modes arevery similar, except for a shift in thestreamwise direction
• They are located on the low-speedstreak downstream of the roughnesselement
• Probably related to thequasi-parallelism of the streaks
BUMP
• The two most unstable modes arevery different
• The isolated mode is located on thelow-speed streak close to theroughness ; the modes on the stablebranch at the outlet
• Probably related to thestreamwise-localization of the streaks
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Conclusions• CYLINDER :
↪→ Very strong quasi-parallel streaks downstream of the roughness element↪→ Unstable mode at Re = 803, closely followed by an eigenvalue branch↪→ Spatially localized along the central low-speed streak↪→ Varicose symmetry, but it extracts its energy mostly from the
spanwise shear
• BUMP :↪→ Rather strong streaks which fade away far from the roughness element↪→ An isolated mode is destabilized at Re = 891, followed by a very
stable branch↪→ Spatially localized along the central low-speed streak closer to the
separation zone↪→ Varicose symmetry, but it extracts its energy mostly from the
spanwise shear
⇒ Global counterpart of the local streak’s instability observed by Asaiet al(2002,2007) and Brandt (2006).
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Outlook and future works
• Several questions remain unanswered :
↪→ Why the critical Reynolds number is higher in the bump’s case ?Maybe because of the lower amount of fluid displaced by the roughnesselement ?
↪→ What would happens considering a cylinder having the samesurface area of the bump, instead of the same aspect ratio ? Wouldthe critical Reynolds number be the same ?
↪→ For thin cylinders (η ≤ 1), a sinuous unstable mode has beenrecovered. Does a sinuous mode exists also for the bump ?
↪→ What about non-normal and non-linear effects in the transitionprocess ? (Arnal et al., Cherubini et al., ...)
↪→ Because of the spatial localization of the mode, would a localstability analysis give similar results ?
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Thanks for listening !
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Sinuous eigenmode
ω
σ
0 0.5 1 1.5 2 2.50.1
0.08
0.06
0.04
0.02
0
0.02
0.04AS mode
Eigenspectrum for (η, δ99/h, Re) = (1, 2, 1250)
Real part of the unstable sinuous mode streamwise component
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Sinuous eigenmode - (2)
Slice in the plane y = 1
Slice in the plane x = 30 Production terms τuv∂yU and τuw∂zU in the planex = 30
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Varicose eigenmode - (2)
Slice in the symmetry plane z = 0
Slice in the plane x = 30 Production terms τuv∂yU and τuw∂zU in the planex = 30
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Discussion
• Convenient definition of the Reynolds number is the roughnessReynolds number Reh :
Reh = U(h)hν
δ99/h 1.75 2 2.25Rec 1175 1225 1310Rech 960 903 899
Table: Evolution of the critical Reynolds numbers Rec and Rech with respect to
δ99/h for the varicose instability and aspect ratio η = 1.
• Rech tends to a value of approximately 900 :
↪→ Good agreements with experimental observations : transition withinthe range 600 ≤ Reh ≤ 900.
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Comparison with von Doenhoff-Braslow transitiondiagram
Reproduction of the von Doenhoff-Braslow transition diagram along with the critical roughness Reynolds numbers for varicoseinstability obtained by global stability analyses.
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