1. Influence of the shape on the roughness-induced
transition
J.-C. Loiseau(1) , S. Cherubini(1) J.-C. Robinet(1) and E. Leriche(2)
(1): DynFluid Laboratory - Arts & M´tiers-ParisTech - 75013 Paris, France
e
(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France
International Conference on Instability and Control of Massively
Separated Flows, Prato, Italy, Sept. 4-6, 2013
ANR – SICOGIF
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2. Background - generalities
• Three-dimensional wall roughness has numerous applications in
aerospace engineering :
→ Upstream shift of the transition location
→ Transition delay
→ Increase/Decrease of the skin friction ...
• Despite the large body of literature, physical mechanisms
inducing 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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3. Motivations
• Objectives :
→ Have a better insight of the roughness element’s shape impact
on the flow instability
→ Understanding the physical mechanisms responsible for
roughness-induced transition.
• Methods :
→ Joint application of direct numerical simulations and linear global
stability analyses
→ Comparison of the instability mechanisms for two chosen shapes of the
roughness element
• Cases under consideration :
→ Sharp-edged case → CYLINDER (Fransson et al. (2006)),
→ Smooth case → BUMP (Piot et al. (2008))
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4. Geometry & Notations
Lz
Ly
l
Geometry under consideration
Lx
• Roughness elements’s
d
y
δ
h
characteristics :
→
z
x
0
Cubic-cosine bump shape :
√
(x 2 +y 2 )
)
h(d) = h0 cos3 (π
d
→ Diameter : d = 2
→ Height : h0 = 1
→ Aspect ratio : η = d/h0 = 2.
• Incoming boundary layer
characteristics :
→ Ratio : δ99 /h0 = 2.
→ Re = U∞ h0 = [700, 1000].
ν
• Box’s dimensions :
→ Lx = 105
→ Ly = 50
→ Lz = 8.
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5. Methodology : generalities
• All calculations are performed with the spectral elements code Nek
5000 :
→ 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-pass
filter to the fully non-linear Navier-Stokes equations, see Akervik et
al(2006).
• Global stability analysis :
→ Home made time-stepper Arnoldi algorithm build-up around Nek
5000 temporal loop.
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6. Numerical method : iterative eigenvalue methods
Arnoldi algorithm build-up
around Nek5000 temporal loop
INPUTS
Krylov basis
LNS-Solver
ORTHOGONALISE
U = [], H = []
k = 0, uk
U = [U uk]
w = eAtuk
h = scal.prod(w,U)
if(k=kmax)
exit loop
H = [H h]
RESIDUAL
OUTPUTS
U, H
H = [bek H]
f = w - Uh,
b = ||f||,
uk = f/b
LAPACK
LINEAR STABILITY
[X,D] = eig(H)
[UX,log(D)/t] ~ eig(A)
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7. Results
Base Flows
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8. Three-dimensional Base Flows
Main features of the base flows with (η, δ99 /h, Re) = (2, 2, 1000) :
→ Upstream and downstream
reversed flow regions (blue for
U = 0),
→ Vortical system stemming
from the upstream recirculation
bubble (green for Q criterion).
→ Uptream spanwise vorticity
wraps around the roughness
element and transforms into
streamwise vorticity downstream
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9. 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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10. Stability
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11. Eigenspectra
Cylinder → branch of eigenvalues, unstable mode at Rec = 803 (Reh = 593)
Bump → isolated mode becoming unstable at Rec = 891 (Reh = 659), followed
by a very stable branch
The bump becomes unstable at larger Re than the cylinder
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12. Cylinder’s leading mode
Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)
• The spatial support of the mode is located on the
streaks, well downstream of the cylinder
• It is composed by streamwise-alternated patches of
positive/negative velocity perturbation
• It is symmetric w.r.t. the z = 0 axis (varicose mode)
Eigenspectrum for
(η, δ99 /h, Re) = (2, 2, 1000)
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13. Bump’s leading mode
Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)
• The spatial support of the mode is located on the
separation zone, close to the bump
• It is composed by streamwise-alternated patches of
positive/negative velocity perturbation
• It is symmetric w.r.t. the z = 0 axis (varicose mode)
Eigenspectrum for
(η, δ99 /h, Re) = (2, 2, 1000)
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14. Varicose eigenmodes at Re = 1000
CYLINDER
BUMP
→ Strong deformation of the base flow
streamwise velocity (solid contours)
→ Weaker deformation of the base flow
streamwise 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
ui uj
V
∂Ui
1
dV −
∂xj
Re
V
∂ui ∂ui
dV
∂xj ∂xj
(1)
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15. Production terms - CYLINDER case
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
→ The dominant production terms
are TUy = uv ∂U and TUz = uw ∂U
∂y
∂z
uuU x
uvU y
uwU z
uvVx
vvV y
vwV z uwW x vwW y
Production term TUy
and streamwise perturbation u
Streamwise displacement of the
wall-normal shear
→ TUz is the largest term, even if the
mode is varicose (unlike the
optimal streaks case)
Production term TUz
and spanwise perturbation w
Spanwise displacement of the
spanwise shear
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16. Production terms - BUMP case
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
→ The dominant production terms
are TUy = uv ∂U and TUz = uw ∂U
∂y
∂z
uuU x
uvU y
uwU z
uvVx
vvV y
vwV z uwW x vwW y
Production term TUy
and streamwise perturbation u
Streamwise displacement of the
wall-normal shear
→ TUz is the largest term, as for the
cylinder
Production term TUz
and spanwise perturbation w
Spanwise displacement of the
spanwise shear
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17. 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
BUMP
• The two most unstable modes are
very similar, except for a shift in the
streamwise direction
• They are located on the low-speed
streak downstream of the roughness
element
• Probably related to the
quasi-parallelism of the streaks
• The two most unstable modes are
very different
• The isolated mode is located on the
low-speed streak close to the
roughness ; the modes on the stable
branch at the outlet
• Probably related to the
streamwise-localization of the streaks
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18. 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 Asai
et al(2002,2007) and Brandt (2006).
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19. 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 roughness
element ?
→ What would happens considering a cylinder having the same
surface area of the bump, instead of the same aspect ratio ? Would
the critical Reynolds number be the same ?
→ For thin cylinders (η ≤ 1), a sinuous unstable mode has been
recovered. Does a sinuous mode exists also for the bump ?
→ What about non-normal and non-linear effects in the transition
process ? (Arnal et al., Cherubini et al., ...)
→ Because of the spatial localization of the mode, would a local
stability analysis give similar results ?
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20. Thanks for listening !
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21. Sinuous eigenmode
0.04
A-S mode
0.02
0
σ
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
2
2.5
Real part of the unstable sinuous mode streamwise component
ω
Eigenspectrum for (η, δ99 /h, Re) = (1, 2, 1250)
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22. Sinuous eigenmode - (2)
Slice in the plane y = 1
Slice in the plane x = 30
Production terms τuv ∂y U and τuw ∂z U in the plane
x = 30
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23. Varicose eigenmode - (2)
Slice in the symmetry plane z = 0
Slice in the plane x = 30
Production terms τuv ∂y U and τuw ∂z U in the plane
x = 30
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24. Discussion
• Convenient definition of the Reynolds number is the roughness
Reynolds number Reh :
Reh =
δ99 /h
Rec
c
Reh
U(h)h
ν
1.75
1175
960
2
1225
903
2.25
1310
899
c
Table: Evolution of the critical Reynolds numbers Rec and Reh with respect to
δ99 /h for the varicose instability and aspect ratio η = 1.
c
• Reh tends to a value of approximately 900 :
→ Good agreements with experimental observations : transition within
the range 600 ≤ Reh ≤ 900.
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25. Comparison with von Doenhoff-Braslow transition
diagram
Reproduction of the von Doenhoff-Braslow transition diagram along with the critical roughness Reynolds numbers for varicose
instability obtained by global stability analyses.
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