1. Introduction Global stability theory Roughness-induced transition Conclusion
Dynamics and global stability analysis of
three-dimensional flows
Jean-Christophe Loiseau1,2
supervisor: Jean-Christophe Robinet1
co-supervisor: Emmanuel Leriche2
(1): DynFluid Laboratory - Arts & M´etiers-ParisTech - 75013 Paris, France
(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France
PhD Defence, May 26th 2014
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2. Introduction Global stability theory Roughness-induced transition Conclusion
What are hydrodynamic instabilities?
• Let us consider the flow of water (ν = 15.10−6 m2.s−1) past a
two-dimensional cylinder of diameter D = 1.5 cm.
• If water flows from left to right at U = 4.5 cm.s−1 (Re = 45),
nothing really fancy takes place: the flow is steady and stable.
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3. Introduction Global stability theory Roughness-induced transition Conclusion
What are hydrodynamic instabilities?
• If you increase the velocity to U = 5 cm.s−1 (Re = 50), the flow
looks very different.
• The steady flow became (globally) unstable and has experienced a
(supercritical) bifurcation.
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4. Introduction Global stability theory Roughness-induced transition Conclusion
How do we study these instabilities?
• Let us consider a non-linear dynamical system
B
∂Q
∂t
= F(Q) (1)
1. Compute a fixed point (or base flow): F(Qb) = 0
2. Linearise the dynamics of an infinitesimal perturbation q in the vicinity
of this solution:
B
∂q
∂t
= Jq with J =
∂F
∂q
(2)
3. Investigate the stability properties of this linear dynamical system.
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5. Introduction Global stability theory Roughness-induced transition Conclusion
How do we study these instabilities?
• In the context of fluid dynamics, this includes several different
approaches depending on the nature of the base flow:
• Local stability analysis for parallel flows:
֒→ Temporal stability, Spatial stability, Absolute/Convective stability,
Response to harmoning forcing, Transient growth
• Global stability analysis for two-dimensional and three-dimensional
flows:
֒→ Temporal stability, Response to harmoning forcing (Resolvent),
Transient growth
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6. Introduction Global stability theory Roughness-induced transition Conclusion
Local stability analysis
• The base flow depends on a single space coordinate:
Ub = (Ub(y), 0, 0)T
• Linear dynamical system (2) is now autonomous in time and in the x
and z coordinates of space.
֒→ The perturbation q can be decomposed into normal modes:
q(x, y, z, t) = ˆq(y) exp(iαx + iβz + λt) + c.c with λ = σ + iω
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7. Introduction Global stability theory Roughness-induced transition Conclusion
Local stability analysis
• Introducing such decomposition into the system (2) yields to a
generalised eigenvalue problem:
λBˆq = J(y, α, β)ˆq (3)
• The stability of the base flow Ub is governed by the growth rate σ:
֒→ If σ < 0, the base flow is said to be locally stable.
֒→ If σ > 0, the base flow is said to be locally unstable.
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8. Introduction Global stability theory Roughness-induced transition Conclusion
Local stability analysis
Theoretical point of view
• Relies on the parallel flow assumption.
• Provides insights into the local stability properties of the flow.
֒→ Requires a good theoretical and mathematical background.
Practical point of view
• The generalised eigenproblem involves small matrices (∼ 100 × 100)
• Can be solved using direct eigenvalue solvers in a matter of seconds
even on a 10 years old laptop.
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9. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability theory
• The base flow has two components both depending on the x and y
space coordinates:
Ub = (Ub(x, y), Vb(x, y), 0)T
• Linear dynamical system (2) is now only autonomous in time and in
z.
֒→ The perturbation q can be decomposed into normal modes:
q(x, y, z, t) = ˆq(x, y) exp(iβz + λt) + c.c with λ = σ + iω
Base flow of the 2D separated boundary layer at Re = 600 as in Ehrenstein & Gallaire (2008).
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10. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability theory
• Introducing such decomposition into the system (2) yields to a
generalised eigenvalue problem once again:
λBˆq = J(x, y, β)ˆq (4)
• The stability of the base flow Ub is governed by the growth rate σ:
֒→ If σ < 0, the base flow is said to be globally stable.
֒→ If σ > 0, the base flow is said to be globally unstable.
Streamwise velocity component of the leading unstable global mode for the 2D separated boundary layer.
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11. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability analysis
Theoretical point of view
• Got rid of the parallel flow assumption.
• Allows to investigate more realistic configurations as separated flows
very common in Nature and industries.
Practical point of view
• The generalised eigenproblem involves relatively large matrices
(∼ 105 × 105)
• Mostly solved using iterative eigenvalue solvers on large workstations.
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12. Introduction Global stability theory Roughness-induced transition Conclusion
Objectives
• Bagheri et al. (2008) and Ilak et al. (2012) performed the first global
stability analysis ever on a 3D flow (jet in crossflow).
• Extension of the global stability tools to a fully three-dimensional
framework.
֒→ Mostly a numerical problem due to the (extremely) large matrices
involved.
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13. Introduction Global stability theory Roughness-induced transition Conclusion
Objectives
λ2 visualisation of the hairpin vortices shed behind a hemispherical roughness element. Courtesy of P. Fischer.
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14. Introduction Global stability theory Roughness-induced transition Conclusion
Context
• PhD thesis part of a larger project: Simulation and Control of
Geometrically Induced Flows (SICOGIF)
֒→ Funded by the French National Agency for Research (ANR)
֒→ Involves several different parties (IRPHE, EPFL, Arts et M´etiers
ParisTech and Universit´e Lille-1)
֒→ Aims at improving our understanding of instability and transition in
complex 2D and 3D separated flows both from an experimental and
numerical point of view.
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15. Introduction Global stability theory Roughness-induced transition Conclusion
Context
• Three flow configurations have been investigated:
֒→ The lid-driven cavity flow
֒→ The asymmetric stenotic pipe flow
֒→ The roughness-induced boundary layer flow
Vertical velocity component of the leading global mode for a LDC having a spanwise extent Λ = 6 at Re = 900.
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16. Introduction Global stability theory Roughness-induced transition Conclusion
Context
• Three flow configurations have been investigated:
֒→ The lid-driven cavity flow
֒→ The asymmetric stenotic pipe flow
֒→ The roughness-induced boundary layer flow
Streamwise velocity component for the two existing steady states of an asymmetric stenotic pipe flow at Re = 400.
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17. Introduction Global stability theory Roughness-induced transition Conclusion
Context
• Three flow configurations have been investigated:
֒→ The lid-driven cavity flow
֒→ The asymmetric stenotic pipe flow
֒→ The roughness-induced boundary layer flow
Identification of the vortical structures by the λ2 criterion (Jeong & Hussain 1995).
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18. Introduction Global stability theory Roughness-induced transition Conclusion
Introduction
Global stability theory and algorithm
Base flows
Global stability theory
How to solve the eigenvalue problem?
Roughness-induced transition
Motivations
Fransson 2005 experiment
Parametric investigation
Physical analysis
Non-linear evolution
Conclusions & Perspectives
Conclusions
LDC & Stenosis
Perspectives
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19. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability analysis of
three-dimensional flows
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20. Introduction Global stability theory Roughness-induced transition Conclusion
How to compute base flows?
• Base flow are given by:
F(Qb) = 0 (5)
• Various techniques can be employed to compute these peculiar
solutions:
֒→ Analytical solutions, impose appropriate symmetries, Newton and
quasi-Newton methods, ...
• In the present work, we use the Selective frequency damping
approach (see Akervik et al. 2006).
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21. Introduction Global stability theory Roughness-induced transition Conclusion
Selective frequency damping
• Enables the stabilisation of the solution by applying a low-pass filter
to the Navier-Stokes equations.
֒→ A forcing term is added to the r.h.s of the equations.
֒→ The system is extended with an equation for the filtered state.



∂Q
∂t
= F(Q) + χ(Q − ¯Q)
∂ ¯Q
∂t
= ωc(Q − ¯Q)
(6)
• The cutoff frequency ωc is connected to the frequency of the most
dominant instabilities and should be smaller than this frequency
(ωc < ω).
• The gain χ needs to be large enough to stabilise the system (χ > σ).
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22. Introduction Global stability theory Roughness-induced transition Conclusion
Selective frequency damping
Pros
→ Really easy to implement within
an existing DNS code.
→ Memory footprint similar to that
of a simple direct numerical
simulation.
→ Easy to use/tune the low-pass
filter.
Cons
→ As time-consuming as a direct
numerical simulation.
→ Requires a priori information
regarding the instability of the
flow.
→ Unable to stabilise the system if
the instability is non-oscillating.
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23. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability theory
• Dynamics of a three-dimensional infinitesimal perturbation
q = (u, p)T evolving onto the base flow Qb = (Ub, Pb)T are
governed by:



∂u
∂t
= −(u · ∇)Ub − (Ub · ∇)u − ∇p +
1
Re
∆u
∇ · u = 0
(7)
• If projected onto a divergence-free vector space, this set of equations
can be recast into:
∂u
∂t
= Au (8)
with A the (projected) Jacobian matrix of the Navier-Stokes
equations.
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24. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability theory
• Using a normal mode decomposition
u(x, y, z, t) = ˆu(x, y, z)e(σ+iω)t
+ c.c
• System (8) can be formulated as an eigenvalue problem
(σ + iω)ˆu = Aˆu (9)
• The sign of σ determines the stability of the base flow Ub:
֒→ If σ < 0, the base flow is said to be asymptoticaly linearly stable.
֒→ If σ > 0, the base flow is said to be asymptoticaly linearly unstable.
• ω determines whether the instability is oscillatory (ω = 0) or not
(ω = 0).
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25. Introduction Global stability theory Roughness-induced transition Conclusion
How to solve the eigenvalue problem?
• Depends on the dimension of the discretised problem.
Base Flow Inhomogeneous Dimension Storage
direction(s) of ˆu of A
Poiseuille U(y) 1D 102 ∼ 1 Mb
2D bump U(x, y) 2D 105 ∼ 1-50 Gb
3D bump U(x, y, z) 3D 107 ∼ 1-100 Tb
• For 3D global stability problem, A is so large that it cannot be
explicitely constructed.
Matrix-free approach is mandatory!
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26. Introduction Global stability theory Roughness-induced transition Conclusion
Time-stepping approach
• Time-stepping approach (Edwards et al. 1994, Bagheri et al. 2008) is
based on the formal solution to system (8):
u(∆t) = eA∆t
u0
• The operator M(∆t) = eA∆t is nothing but a matrix. Its application
on u0 can be computed by time-marching the linearised Navier-Stokes
equations.
֒→ Its stability properties can be investigated by eigenvalue analysis.
MˆU = ˆUΣ (10)
with ˆU the matrix of eigenvectors and Σ the eigenvalue matrix of
M = eA∆t.
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27. Introduction Global stability theory Roughness-induced transition Conclusion
Arnoldi algorithm
• The eigenvalue problem (10) is solved using an Arnoldi algorithm.
1. Given M and u0, construct a small Krylov subspace (compared to the
size of the initial problem),
Km(M, u0) = span u0, Mu0, M2
u0, · · · , M(m−1)
u0
2. Orthonormalize: U = [U1, · · · , Um]
3. Project operator M ≈ UHUT
−→ MUk = Uk Hk + rk eT
k
with Hk : upper Hessenberg matrix.
4. Solve small eigenvalue problem (ΣH , X): HX = XΣH, (m × m),
m < 1000
5. Link with the initial eigenproblem (ΛA, ˆu):
ΛA =
log(ΣH)
∆t
, ˆu = UX
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28. Introduction Global stability theory Roughness-induced transition Conclusion
Arnoldi algorithm
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29. Introduction Global stability theory Roughness-induced transition Conclusion
Summary
• All calculations have been performed with the code Nek 5000
֒→ Legendre spectral elements code developed by P. Fischer at Argonne
National Laboratory.
֒→ Semi-implicit temporal scheme.
֒→ Massively parallel code based on an MPI strategy.
• Base flow computation
֒→ Selective frequency damping approach : application of a low-pass filter
to the fully non-linear Navier-Stokes equations (Akervik et al. 2006).
• Global stability analysis
֒→ Arnoldi algorithm similar to the one published by Barkley et al. (2008).
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30. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
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31. Introduction Global stability theory Roughness-induced transition Conclusion
Motivations
• Roughness elements have numerous applications in aerospace
engineering:
֒→ Stabilisation of the Tollmien-Schlichting waves,
֒→ Shift and/or control of the transition location, ...
• Their influence on the flow has been extensively investigated since the
early 1950’s.
Experimental visualisation of the flow induced by a roughness element. Gregory & Walker, 1956.
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32. Introduction Global stability theory Roughness-induced transition Conclusion
Delay of the natural transition
• Cossu & Brandt (2004): Theoretical prediction of the stabilisation of
TS waves by streamwise streaks.
• Fransson et al. (2004-2006): Experimental demonstration using a
periodic array of roughness elements.
Schematic setup
Experimental observations
Figures from Fransson et al. (2006).
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33. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
• Problem: If the Reynolds number is too high, transition occurs right
downstream the roughness elements!
Illustration of the early roughness-induced transition. λ2 visualisation of the vortical structures.
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34. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
• Since the early 1950’s: Numerous experimental investigations.
֒→ Transition diagram by von Doenhoff & Braslow (1961).
• Despite the large body of literature, the underlying mechanisms are
not yet fully understood.
Transition diagram from von Doenhoff & Braslow (1961).
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35. Introduction Global stability theory Roughness-induced transition Conclusion
Motivations
• Methods used until now rely on a parallel flow assumption:
֒→ Local stability theory (Brandt 2006, Denissen & White 2013, ...),
֒→ Local transient growth theory (Vermeersch 2010, ...)
• Objective:
֒→ Might a 3D global instability of the flow explain the roughness-induced
transition?
֒→ If so, what are the underlying physical mechanisms?
• Methods:
֒→ Fully three-dimensional global stability analyses,
֒→ Direct numerical simulations,
֒→ Comparison with available experimental data.
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36. Introduction Global stability theory Roughness-induced transition Conclusion
Problem formulation
z X
y
d h
Lz
l
Lx
Ly
δ
0
Sketch of the computational arrangement and various scales used for DNS and stability analysis.
- (Lx , Ly , Lz ) = (105, 50, 8η)
- η = d/h = 1, 2, 3
- Re = Ueh/ν
- Reδ∗
= Ueδ∗/ν
- Inflow: Blasius profile,
- Outflow: ∇U · x = 0,
- Top: U = 1, ∂y V = ∂y W = 0,
- Wall: no-slip B.C.
- Lateral: periodic B.C.
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37. Introduction Global stability theory Roughness-induced transition Conclusion
Numerical informations
η Number of SEM Gridpoints (N = 6-12) Number of cores used
1 10 000 2-17.106 256
2 17 500 3.5-30.106 512
3 20 000 4.5-35.106 512
Typical size of the numerical problem investigated. N is the order of the Legendre polynomials used in the three directions
within each element.
Typical SEM distribution in a given horizontal plane. Full mesh with Legendre polynomials of order N = 8.
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38. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
The Fransson 2005 experiment
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39. Introduction Global stability theory Roughness-induced transition Conclusion
Experimental setup
• Experimental demonstration of the ability for finite amplitude streaks
to stabilise TS waves.
• Unfortunately, transition takes place right downstream the array of
roughness elements if the Reynolds number is too high.
h D η Lz/h xk/h Rec
δ∗
1.4mm 4.2mm 3 10 57.14 ≃ 290
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40. Introduction Global stability theory Roughness-induced transition Conclusion
Base flow
(a)
(b)
• Upstream and downstream
reversed flow regions:
֒→ Induces a central low-speed
region.
• Vortical system stemming:
֒→ Investigated by Baker (1978)
֒→ Horseshoe vortices whose legs
are streamwise oriented
counter-rotating vortices.
֒→ Creation of streamwise
velocity streaks (lift-up effect)
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41. Introduction Global stability theory Roughness-induced transition Conclusion
Base flow
(a) X=20 (b) X=40 (c) X=60 (d) X=80
Visualisation of the base flow deviation from the Blasius boundary layer flow in various streamwise planes for Re = 466. High
speed streaks are in red while low-speed ones are in blue.
• Low-speed region generated by the roughness element’s blockage.
֒→ Fades away quite rapidly in the streamwise direction.
• High- and low-speed streaks on each side of the roughness element
due to the horshoe vortex.
֒→ Sustains over quite a long streamwise distance.
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42. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
Eigenspectrum of the linearised Navier-Stokes operator.
• Hopf bifurcation taking place in-between 550 < Rec < 575.
֒→ Linear interpolation: Rec = 564, i.e. Rec
δ∗
= 309.
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43. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
(a) Top view of u = ±10% iso-surfaces
(b) X = 23 (c) X = 40
Visualisation of streamwise velocity component of the leading unstable mode for Re = 575.
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44. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
• Leading unstable mode exhibits a varicose symmetry:
֒→ Streamwise alternated patches of positive and negative velocity mostly
localised along the central low-speed region.
֒→ Non-linear DNS have revealed that it gives birth to hairpin vortices.
• Rec predicted by global stability analysis only 6% larger than the
experimental one from Fransson et al. (2005):
֒→ Global instability of the flow appears as one of the possible
explanations to roughness-induced transition.
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45. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
Parametric investigation
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46. Introduction Global stability theory Roughness-induced transition Conclusion
Parametric investigation
• Aims of the parametric investigation:
֒→ How do the Reynolds number and the aspect ratio of the roughness
elements impact the base flow and its stability properties?
֒→ Does the leading unstable mode always exhibit a varicose symmetry?
• To do so:
֒→ The spanwise extent of the domain is taken large enough so that the
roughness element behaves as being isolated.
֒→ δ99/h is set to 2 to isolate the influence of the Reynolds number only.
֒→ The roughness element’s aspect ratio varies from η = 1 up to η = 3.
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47. Introduction Global stability theory Roughness-induced transition Conclusion
Base flows
(a) (Re, η) = (600, 1) (b) (Re, η) = (1250, 1)
Influence of the Reynolds number on the flow within the z = 0 plane. Streamwise velocity iso-contours ranging from U = 0.1
up to U = 0.99.
• Influence of the Reynolds number:
֒→ Does not qualitatively change the shape of the downstream reversed
flow region.
֒→ Strengthen the gradients and reduces the thickness of the shear layer.
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48. Introduction Global stability theory Roughness-induced transition Conclusion
Base flows
(a) (Re, η) = (600, 1)
(b) (Re, η) = (1250, 1)
Top view of the low-speed (white) and high-speed (black) streaks induced by the roughness element. Streak have been
identified using the deviation of the base flow streamwise velocity from the Blasius boundary layer flow.
• Influence of the Reynolds number:
֒→ Strongly increases the amplitude and the streamwise extent of the
central low-speed region.
֒→ Slightly increases the amplitude of the outer velocity streaks.
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49. Introduction Global stability theory Roughness-induced transition Conclusion
Base flows
(a) (Re, η) = (600, 2) (b) (Re, η) = (600, 3)
Influence of the aspect ratio on the flow within the z = 0 plane. Streamwise velocity iso-contours ranging from U = 0.1 up to
U = 0.99.
• Influence of the aspect ratio:
֒→ Strengthen the gradients and reduces the thickness of the shear layer.
֒→ Strongly increases the amplitude and the streamwise extent of the
central low-speed region.
֒→ Strongly increases the amplitude of the outer velocity streaks.
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50. Introduction Global stability theory Roughness-induced transition Conclusion
Base flows
(a) (Re, η) = (600, 2)
(b) (Re, η) = (600, 3)
Top view of the low-speed (white) and high-speed (black) streaks induced by the roughness element. Streak have been
identified using the deviation of the base flow streamwise velocity from the Blasius boundary layer flow.
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51. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
ω
σ
0 0.5 1 1.5 2
-0.04
-0.02
0
0.02
0.04
(a) (Re, η) = (1200, 1)
ω
σ
0 0.5 1 1.5
-0.04
-0.02
0
0.02
0.04
(b) (Re, η) = (900, 2)
ω
σ
0 0.5 1 1.5
-0.04
-0.02
0
0.02
0.04
(c) (Re, η) = (700, 3)
Eigenspectra of the linearised Navier-Stokes operator for different roughness elements.
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52. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
(a) (Re, η) = (1200, 1)
(b) (Re, η) = (900, 2)
Evolution of the leading unstable mode when the roughness element’s aspect ratio is changed from η = 1 to η = 2. Isosurfaces
u = ± 10% of the modes streamwise velocity component.
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53. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
(a) (Re, η) = (1200, 1) (b) (Re, η) = (900, 2)
Evolution of the leading unstable mode when the roughness element’s aspect ratio is changed from η = 1 to η = 2 in the
X = 25 plane.
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54. Introduction Global stability theory Roughness-induced transition Conclusion
Global stability
• Increasing the roughness element’s aspect ratio decreases the critical
Reynolds number.
η 1 2 3 Fransson (η = 3)
Rec 1040 850 656 564
Rec
h 813 630 513 519
Symmetry S V V V
Summary of the global stability analyses. V: varicose, S: sinuous. Reh is the roughness Reynolds number.
• Exchange of symmetry in qualitative agreements with Sakamoto &
Arie (1983) and Beaudoin (2004).
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55. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
Physical analysis
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56. Introduction Global stability theory Roughness-induced transition Conclusion
Physical analysis
• Aims of the analysis:
֒→ Unravel the underlying physical mechanisms for each mode.
֒→ How and where do they extract their energy?
֒→ Where do they originate?
• Type of analysis:
֒→ Kinetic energy transfer between the base flow and the perturbation
(Brandt 2006).
֒→ Computation of the wavemaker region (Giannetti & Luchini 2007).
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57. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget
• The evolution of the perturbation’s kinetic energy is governed by the
Reynolds-Orr equation:
∂E
∂t
= −D +
9
i=1 V
Ii dV (11)
• with the total kinetic energy E and dissipation D given by:
E =
1
2 V
u · u dV , and D =
1
Re V
∇u : ∇u dV (12)
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58. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget
• The integrands Ii representing the different production terms are
given by:
I1 = −u2 ∂Ub
∂x
, I2 = −uv
∂Ub
∂y
, I3 = −uw
∂Ub
∂z
I4 = −uv
∂Vb
∂x
, I5 = −v2 ∂Vb
∂y
, I6 = −vw
∂Vb
∂z
I7 = −wu
∂Wb
∂x
, I8 = −wv
∂Wb
∂y
, I9 = −w2 ∂Wb
∂z
(13)
• Their sign indicates whether the associated local transfer of kinetic
energy acts as stabilising (negative) or destabilising (positive).
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59. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Sinuous mode
0
0.5
1
I1 I2 I3 I4 I5 I6 I7 I8 I9 D
(a) (Re, η) = (1125, 1)
0
0.5
1
I1 I2 I3 I4 I5 I6 I7 I8 I9 D
(b) (Re, η) = (1250, 1)
X0 30 60 90
2.0x10
-03
4.0x10
-03
6.0x10
-03
∫I2dydz
∫I3dydz
(c) (Re, η) = (1125, 1)
X0 30 60 90
2.0x10
-03
4.0x10
-03
6.0x10
-03
∫I2dydz
∫I3dydz
(d) (Re, η) = (1250, 1)
Top: Sinuous unstable mode’s kinetic energy budget integrated over the whole domain. Bottom: Streamwise evolution of the
production terms y,z I2 dydz (red dashed line) and y,z I3 dydz (blue solid line).
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60. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Sinuous mode
(a) I2 = −uv∂U/∂y (b) I3 = −uw∂U/∂z
Spatial distribution of the I2 = −uv∂Ub/∂y (a) and I3 = −uw∂Ub/∂z (b) production terms in the plane x = 25 for
(Re, η) = (1125, 1). Solid lines depict the base flows streamwise velocity isocontours from Ub = 0.1 to 0.99, whereas the red
dashed lines stand for the location of the shear layer.
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61. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Sinuous mode
(a) I2 = −uv∂U/∂y
(b) I3 = −uw∂U/∂z
Spatial distribution of I2 (c) and I3 (d) in the y = 0.75 horizontal plan.
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62. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Varicose mode
0
1
2
I1 I2 I3 I4 I5 I6 I7 I8 I9 D
(a) (Re, η) = (850, 2)
0
1
2
I1 I2 I3 I4 I5 I6 I7 I8 I9 D
(b) (Re, η) = (1000, 2)
X0 30 60 90
.0x10
+00
4.0x10
-03
8.0x10
-03
∫I2dydz
∫I3dydz
(c) (Re, η) = (850, 2)
X0 30 60 90
.0x10
+00
4.0x10
-03
8.0x10
-03
∫I2dydz
∫I3dydz
(d) (Re, η) = (1000, 2)
Top: Varicose unstable mode’s kinetic energy budget integrated over the whole domain. Bottom: Streamwise evolution of the
production terms y,z I2 dydz (red dashed line) and y,z I3 dydz (blue solid line).
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63. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Varicose mode
(a) I2 = −uv∂U/∂y (b) I3 = −uw∂U/∂z
Spatial distribution of the I2 = −uv∂Ub/∂y (a) and I3 = −uw∂Ub/∂z (b) production terms in the plane x = 25 for
(Re, η) = (850, 2). Solid lines depict the base flows streamwise velocity isocontours from Ub = 0.1 to 0.99, whereas the red
dashed lines stand for the location of the shear layer.
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64. Introduction Global stability theory Roughness-induced transition Conclusion
Kinetic energy budget: Varicose mode
Spatial distribution of the I3 = −uw∂Ub/∂z production term in the plane y = 0.5 for (Re, η) = (850, 2).
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65. Introduction Global stability theory Roughness-induced transition Conclusion
Wavemaker
• Kinetic energy budgets provide valuable insights into the mode’s
dynamics but very limited about its core region, i.e. the wavemaker.
• Defined by Giannetti & Luchini (2007) as the overlap of the direct
global mode u and its adjoint u†:
ζ(x, y, z) =
u† u
u†, u
(14)
• Allows the identification of the most likely region for the inception of
the global instability under consideration.
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66. Introduction Global stability theory Roughness-induced transition Conclusion
Wavemaker
Figure: Sinuous wavemaker in the y = 0.75 plane.
Figure: Varicose wavemaker in the z = 0 plane.
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67. Introduction Global stability theory Roughness-induced transition Conclusion
Wavemaker
Figure: Sinuous wavemaker in the y = 0.75 plane.
Figure: Varicose wavemaker in the z = 0 plane.
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68. Introduction Global stability theory Roughness-induced transition Conclusion
Wavemaker
• Sinuous wavemaker:
֒→ Exclusively localised within the spatial extent of the downstream
reversed flow region.
֒→ Shares close connections with the von K´arm´an global instability in the
2D cylinder flow (Giannetti & Luchini 2007, Marquet et al. 2008).
• Varicose wavemaker:
֒→ Localised on the top of the central low-speed region shear layer.
֒→ Quite extended in the streamwise direction.
֒→ Yet, its amplitude in the reversed flow region is almost ten times larger
than its amplitude in the wake.
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69. Introduction Global stability theory Roughness-induced transition Conclusion
Sinuous instability mechanism
What we know from local stability
approaches?
• Central low-speed region can
sustain local convective
instabilities (Brandt 2006).
• Related to the work of the
Reynolds stresses against the
wall-normal and spanwise
gradients of Ub.
• Not the dominant local
instability though.
What global stability analyses
revealed?
• Existence of a global sinuous
instability.
• Related to the downstream
reversed flow region.
• Similar to the von K´arm´an
instability in the 2D cylinder
flow.
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70. Introduction Global stability theory Roughness-induced transition Conclusion
Varicose instability mechanism
What we know from local stability
approaches?
• Central low-speed region can
sustain local convective
instabilities (Brandt 2006,
Denissen & White 2013).
• Related to the work of the
Reynolds stresses against the
wall-normal gradient of Ub.
• Dominant local instability and
possible large transient growth
(Vermeersch 2010)
What global stability analyses
revealed?
• Existence of a global varicose
instability.
• Find its roots in the reversed
flow region.
• Mechanism might be similar to
the one proposed by Acarlar &
Smith (1987).
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71. Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transition
Non-linear evolution
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72. Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Varicose instability
֒→ Induces a varicose modulation of the central low-speed region and
surrounding streaks.
֒→ Numerous hairpin vortices are shed right downstream the roughness
element and trigger very rapid transition to turbulence.
֒→ Dominant frequency and wavelength of this vortex shedding is well
captured by global stability analyses.
Streamwise velocity distribution in the y = 0.5 plane for (Re, η) = (575, 3).
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73. Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Varicose instability
֒→ Induces a varicose modulation of the central low-speed region and
surrounding streaks.
֒→ Numerous hairpin vortices are shed right downstream the roughness
element and trigger very rapid transition to turbulence.
֒→ Dominant frequency and wavelength of this vortex shedding is well
captured by global stability analyses.
Identification of the vortical structures by the λ2 criterion (Jeong & Hussain 1995).
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74. Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
֒→ Induces a sinuous wiggling of the central low-speed region (Beaudoin
2004, Duriez et al. 2009).
֒→ Frequency of this sinuous wiggling well captured by global stability
analysis.
֒→ Hairpin vortices are nonetheless observed to be shed downstream the
roughness element..
Streamwise velocity distribution in the y = 0.5 plane for (Re, η) = (1125, 1).
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75. Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
֒→ Induces a sinuous wiggling of the central low-speed region (Beaudoin
2004, Duriez et al. 2009).
֒→ Frequency of this sinuous wiggling well captured by global stability
analysis
֒→ Hairpin vortices are nonetheless observed to be shed downstream the
roughness element..
Identification of the vortical structures by the λ2 criterion (Jeong & Hussain 1995).
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76. Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
֒→ Monitoring the amplitude of the spanwise velocity in the central
mid-plane revealed the bifurcation is supercritical.
−20 0 20 40 60 80 100
−0.1
−0.05
0
0.05
0.1
ε=Re−Rec
Amplitude
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77. Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion & Perspectives
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78. Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion
• Sinuous instability
֒→ Dominant instability for low aspect ratio roughness elements.
֒→ von K´arm´an-like global instability of the reversed flow region.
֒→ Vortices shed from this region then experiences weak spatial transient
growth.
֒→ The creation of hairpin vortices by sinuous global instability is not yet
understood.
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79. Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion
• Varicose instability
֒→ Dominant instability for large aspect ratio roughness elements.
֒→ Mechanism similar to the one proposed by Acarlar & Smith (1987).
֒→ Triggers rapid transition to a turbulent-like state by promoting the
creation of hairpin vortices.
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80. Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion
• Critical roughness Reynolds numbers and observations from DNS in
qualitatively good agreements with the transition diagram by von
Doenhoff & Braslow (1961).
֒→ Three-dimensional global instability of the flow appears as one of
the possible explanations to roughness-induced transition.
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81. Introduction Global stability theory Roughness-induced transition Conclusion
Lid-driven cavity flow
• Same instability mechanism as
before:
֒→ Centrifugal instability of the
primary vortex core.
• For large LDC, Rec in good
agreements with predictions
from 2.5D stability analysis.
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82. Introduction Global stability theory Roughness-induced transition Conclusion
Lid-driven cavity flow
• DNS revealed bursts of kinetic energy related to intermittent chaotic
dynamics.
֒→ Koopman modes decomposition suggests it would type-2 intermittent
chaos (Pomeau & Manneville 1980).
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83. Introduction Global stability theory Roughness-induced transition Conclusion
Stenotic pipe flow
• Asymmetry of the stenosis triggers the wall-reattachment at lower Re
compared to the axisymmetric case.
• Existence of a hysteresis cycle related to a subcritical pitchfork
bifurcation.
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84. Introduction Global stability theory Roughness-induced transition Conclusion
Stenotic pipe flow
• Nonetheless, predictions from global stability analyses are
uncorelatted to the experimental observations (Passaggia et al.)
֒→ Transition is dominated by transient growth.
• Preliminary optimal perturbation analysis appears to be more
conclusive.
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85. Introduction Global stability theory Roughness-induced transition Conclusion
Perspectives
• Several questions are still unanswered and require further in-depth
investigations:
֒→ What is the mechanism responsible for the creation of hairpin vortices
in the sinuous case?
֒→ Is the varicose bifurcation super- or subcritical?
֒→ How does global optimal perturbation influence these transition
scenarii? Can they trigger subcritical transition to turbulence?
֒→ How does the shape of the roughness element impact the stability
properties of the flow?
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86. Introduction Global stability theory Roughness-induced transition Conclusion
Perspectives
• How to answer these questions?
֒→ More direct numerical simulations!
֒→ Non-linear analyses of these DNS (Koopman modes decomposition,
POD, statistical analysis, ...).
֒→ Linear and non-linear transient growth analysis.
֒→ Conduct similar investigations for smooth bumps and hemispherical
roughness elements to assess the robustness of these results.
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