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 & Metiers-ParisTech - 75013 Paris, France(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France
PhD Defence, May 26th 2014
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What are hydrodynamic instabilities?
• Let us consider the flow of water (ν = 15.10−6 m2.s−1) past atwo-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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What are hydrodynamic instabilities?
• If you increase the velocity to U = 5 cm.s−1 (Re = 50), the flowlooks very different.
• The steady flow became (globally) unstable and has experienced a(supercritical) bifurcation.
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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) = 02. 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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How do we study these instabilities?
• In the context of fluid dynamics, this includes several differentapproaches 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-dimensionalflows:
→ Temporal stability, Response to harmoning forcing (Resolvent),Transient growth
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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 xand z coordinates of space.
→ The perturbation q can be decomposed into normal modes:
q(x , y , z , t) = q(y) exp(iαx + iβz + λt) + c .c with λ = σ + iω
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Local stability analysis
• Introducing such decomposition into the system (2) yields to ageneralised eigenvalue problem:
λBq = J(y , α, β)q (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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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 secondseven on a 10 years old laptop.
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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 inz .
→ The perturbation q can be decomposed into normal modes:
q(x , y , z , t) = q(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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Global stability theory
• Introducing such decomposition into the system (2) yields to ageneralised eigenvalue problem once again:
λBq = J(x , y , β)q (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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Global stability analysis
Theoretical point of view
• Got rid of the parallel flow assumption.
• Allows to investigate more realistic configurations as separated flowsvery 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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Objectives
• Bagheri et al. (2008) and Ilak et al. (2012) performed the first globalstability analysis ever on a 3D flow (jet in crossflow).
• Extension of the global stability tools to a fully three-dimensionalframework.
→ Mostly a numerical problem due to the (extremely) large matricesinvolved.
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Objectives
λ2 visualisation of the hairpin vortices shed behind a hemispherical roughness element. Courtesy of P. Fischer.
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Context
• PhD thesis part of a larger project: Simulation and Control ofGeometrically Induced Flows (SICOGIF)
→ Funded by the French National Agency for Research (ANR)→ Involves several different parties (IRPHE, EPFL, Arts et Metiers
ParisTech and Universite Lille-1)→ Aims at improving our understanding of instability and transition in
complex 2D and 3D separated flows both from an experimental andnumerical point of view.
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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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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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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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Introduction
Global stability theory and algorithmBase flowsGlobal stability theoryHow to solve the eigenvalue problem?
Roughness-induced transitionMotivationsFransson 2005 experimentParametric investigationPhysical analysisNon-linear evolution
Conclusions & PerspectivesConclusionsLDC & StenosisPerspectives
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Global stability analysis of
three-dimensional flows
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How to compute base flows?
• Base flow are given by:F(Qb) = 0 (5)
• Various techniques can be employed to compute these peculiarsolutions:
→ Analytical solutions, impose appropriate symmetries, Newton andquasi-Newton methods, ...
• In the present work, we use the Selective frequency damping
approach (see Akervik et al. 2006).
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Selective frequency damping• Enables the stabilisation of the solution by applying a low-pass filterto 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 mostdominant 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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Selective frequency damping
Pros
→ Really easy to implement withinan existing DNS code.
→ Memory footprint similar to thatof a simple direct numericalsimulation.
→ Easy to use/tune the low-passfilter.
Cons
→ As time-consuming as a directnumerical simulation.
→ Requires a priori informationregarding the instability of theflow.
→ Unable to stabilise the system ifthe instability is non-oscillating.
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Global stability theory
• Dynamics of a three-dimensional infinitesimal perturbationq = (u, p)T evolving onto the base flow Qb = (Ub,Pb)
T aregoverned 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 equationscan be recast into:
∂u
∂t= Au (8)
with A the (projected) Jacobian matrix of the Navier-Stokesequations.
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Global stability theory
• Using a normal mode decomposition
u(x , y , z , t) = u(x , y , z)e(σ+iω)t + c .c
• System (8) can be formulated as an eigenvalue problem
(σ + iω)u = Au (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 6= 0) or not(ω = 0).
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How to solve the eigenvalue problem?
• Depends on the dimension of the discretised problem.
Base Flow Inhomogeneous Dimension Storagedirection(s) of u of A
Poiseuille U(y) 1D 102 ∼ 1 Mb2D bump U(x , y) 2D 105 ∼ 1-50 Gb3D bump U(x , y , z) 3D 107 ∼ 1-100 Tb
• For 3D global stability problem, A is so large that it cannot beexplicitely constructed.
Matrix-free approach is mandatory!
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Time-stepping approach
• Time-stepping approach (Edwards et al. 1994, Bagheri et al. 2008) isbased on the formal solution to system (8):
u(∆t) = eA∆tu0
• The operator M(∆t) = eA∆t is nothing but a matrix. Its applicationon u0 can be computed by time-marching the linearised Navier-Stokesequations.
→ Its stability properties can be investigated by eigenvalue analysis.
MU = UΣ (10)
with U the matrix of eigenvectors and Σ the eigenvalue matrix ofM = eA∆t .
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Arnoldi algorithm
• The eigenvalue problem (10) is solved using an Arnoldi algorithm.
1. Given M and u0, construct a small Krylov subspace (compared to thesize of the initial problem),
Km(M, u0) = span[
u0,Mu0,M2u0, · · · ,M
(m−1)u0
]
2. Orthonormalize: U = [U1, · · · ,Um]3. Project operator M ≈ UHUT −→ MUk = UkHk + rke
Tk
with Hk : upper Hessenberg matrix.4. Solve small eigenvalue problem (ΣH ,X): HX = XΣH , (m ×m),
m < 10005. Link with the initial eigenproblem (ΛA, u):
ΛA =log(ΣH)
∆t, u = UX
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Arnoldi algorithm
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Summary
• All calculations have been performed with the code Nek 5000
→ Legendre spectral elements code developed by P. Fischer at ArgonneNational 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 filterto 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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Roughness-induced transition
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Motivations
• Roughness elements have numerous applications in aerospaceengineering:
→ Stabilisation of the Tollmien-Schlichting waves,→ Shift and/or control of the transition location, ...
• Their influence on the flow has been extensively investigated since theearly 1950’s.
Experimental visualisation of the flow induced by a roughness element. Gregory & Walker, 1956.
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Delay of the natural transition
• Cossu & Brandt (2004): Theoretical prediction of the stabilisation ofTS waves by streamwise streaks.
• Fransson et al. (2004-2006): Experimental demonstration using aperiodic array of roughness elements.
Schematic setup
Experimental observations
Figures from Fransson et al. (2006).
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Roughness-induced transition
• Problem: If the Reynolds number is too high, transition occurs rightdownstream the roughness elements!
Illustration of the early roughness-induced transition. λ2 visualisation of the vortical structures.
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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 arenot yet fully understood.
Transition diagram from von Doenhoff & Braslow (1961).
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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-inducedtransition?
→ 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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Problem formulation
z X
y
d h
Lz
l
Lx
Ly
δ0
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, ∂yV = ∂yW = 0,
- Wall: no-slip B.C.
- Lateral: periodic B.C.
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Numerical informations
η Number of SEM Gridpoints (N = 6-12) Number of cores used
1 10 000 2-17.106 2562 17 500 3.5-30.106 5123 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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Roughness-induced transitionThe Fransson 2005 experiment
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Experimental setup
• Experimental demonstration of the ability for finite amplitude streaksto stabilise TS waves.
• Unfortunately, transition takes place right downstream the array ofroughness elements if the Reynolds number is too high.
h D η Lz/h xk/h Recδ∗1.4mm 4.2mm 3 10 57.14 ≃ 290
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Base flow
(a)
(b)
• Upstream and downstreamreversed flow regions:
→ Induces a central low-speedregion.
• Vortical system stemming:
→ Investigated by Baker (1978)→ Horseshoe vortices whose legs
are streamwise orientedcounter-rotating vortices.
→ Creation of streamwisevelocity streaks (lift-up effect)
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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 elementdue to the horshoe vortex.
→ Sustains over quite a long streamwise distance.
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Global stability
Eigenspectrum of the linearised Navier-Stokes operator.
• Hopf bifurcation taking place in-between 550 < Rec < 575.
→ Linear interpolation: Rec = 564, i.e. Recδ∗
= 309.
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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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Global stability
• Leading unstable mode exhibits a varicose symmetry:
→ Streamwise alternated patches of positive and negative velocity mostlylocalised 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 theexperimental one from Fransson et al. (2005):
→ Global instability of the flow appears as one of the possibleexplanations to roughness-induced transition.
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Roughness-induced transitionParametric investigation
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Parametric investigation
• Aims of the parametric investigation:
→ How do the Reynolds number and the aspect ratio of the roughnesselements 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 theroughness 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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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 reversedflow region.
→ Strengthen the gradients and reduces the thickness of the shear layer.
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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 thecentral low-speed region.
→ Slightly increases the amplitude of the outer velocity streaks.
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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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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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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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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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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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Global stability
• Increasing the roughness element’s aspect ratio decreases the criticalReynolds number.
η 1 2 3 Fransson (η = 3)
Rec 1040 850 656 564Rech 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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Roughness-induced transitionPhysical analysis
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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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Kinetic energy budget
• The evolution of the perturbation’s kinetic energy is governed by theReynolds-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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Kinetic energy budget
• The integrands Ii representing the different production terms aregiven 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 kineticenergy acts as stabilising (negative) or destabilising (positive).
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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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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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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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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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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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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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Wavemaker
• Kinetic energy budgets provide valuable insights into the mode’sdynamics but very limited about its core region, i.e. the wavemaker.
• Defined by Giannetti & Luchini (2007) as the overlap of the directglobal 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 ofthe global instability under consideration.
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Wavemaker
Figure: Sinuous wavemaker in the y = 0.75 plane.
Figure: Varicose wavemaker in the z = 0 plane.
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Wavemaker
Figure: Sinuous wavemaker in the y = 0.75 plane.
Figure: Varicose wavemaker in the z = 0 plane.
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Wavemaker
• Sinuous wavemaker:
→ Exclusively localised within the spatial extent of the downstreamreversed flow region.
→ Shares close connections with the von Karman global instability in the2D 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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Introduction Global stability theory Roughness-induced transition Conclusion
Sinuous instability mechanism
What we know from local stability
approaches?
• Central low-speed region cansustain local convectiveinstabilities (Brandt 2006).
• Related to the work of theReynolds stresses against thewall-normal and spanwisegradients of Ub.
• Not the dominant localinstability though.
What global stability analysesrevealed?
• Existence of a global sinuousinstability.
• Related to the downstreamreversed flow region.
• Similar to the von Karmaninstability in the 2D cylinderflow.
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Introduction Global stability theory Roughness-induced transition Conclusion
Varicose instability mechanism
What we know from local stability
approaches?
• Central low-speed region cansustain local convectiveinstabilities (Brandt 2006,Denissen & White 2013).
• Related to the work of theReynolds stresses against thewall-normal gradient of Ub.
• Dominant local instability andpossible large transient growth(Vermeersch 2010)
What global stability analysesrevealed?
• Existence of a global varicoseinstability.
• Find its roots in the reversedflow region.
• Mechanism might be similar tothe one proposed by Acarlar &Smith (1987).
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Introduction Global stability theory Roughness-induced transition Conclusion
Roughness-induced transitionNon-linear evolution
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Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Varicose instability
→ Induces a varicose modulation of the central low-speed region andsurrounding streaks.
→ Numerous hairpin vortices are shed right downstream the roughnesselement and trigger very rapid transition to turbulence.
→ Dominant frequency and wavelength of this vortex shedding is wellcaptured by global stability analyses.
Streamwise velocity distribution in the y = 0.5 plane for (Re, η) = (575, 3).
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Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Varicose instability
→ Induces a varicose modulation of the central low-speed region andsurrounding streaks.
→ Numerous hairpin vortices are shed right downstream the roughnesselement and trigger very rapid transition to turbulence.
→ Dominant frequency and wavelength of this vortex shedding is wellcaptured by global stability analyses.
Identification of the vortical structures by the λ2 criterion (Jeong & Hussain 1995).
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Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
→ Induces a sinuous wiggling of the central low-speed region (Beaudoin2004, Duriez et al. 2009).
→ Frequency of this sinuous wiggling well captured by global stabilityanalysis.
→ Hairpin vortices are nonetheless observed to be shed downstream theroughness element..
Streamwise velocity distribution in the y = 0.5 plane for (Re, η) = (1125, 1).
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Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
→ Induces a sinuous wiggling of the central low-speed region (Beaudoin2004, Duriez et al. 2009).
→ Frequency of this sinuous wiggling well captured by global stabilityanalysis
→ Hairpin vortices are nonetheless observed to be shed downstream theroughness element..
Identification of the vortical structures by the λ2 criterion (Jeong & Hussain 1995).
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Introduction Global stability theory Roughness-induced transition Conclusion
Non-linear evolution
• Sinuous instability
→ Monitoring the amplitude of the spanwise velocity in the centralmid-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
Am
plitu
de
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Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion & Perspectives
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Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion
• Sinuous instability
→ Dominant instability for low aspect ratio roughness elements.→ von Karman-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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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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Introduction Global stability theory Roughness-induced transition Conclusion
Conclusion
• Critical roughness Reynolds numbers and observations from DNS inqualitatively good agreements with the transition diagram by vonDoenhoff & Braslow (1961).
→ Three-dimensional global instability of the flow appears as one of
the possible explanations to roughness-induced transition.
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Introduction Global stability theory Roughness-induced transition Conclusion
Lid-driven cavity flow
• Same instability mechanism asbefore:
→ Centrifugal instability of theprimary vortex core.
• For large LDC, Rec in goodagreements with predictionsfrom 2.5D stability analysis.
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Introduction Global stability theory Roughness-induced transition Conclusion
Lid-driven cavity flow
• DNS revealed bursts of kinetic energy related to intermittent chaoticdynamics.
→ Koopman modes decomposition suggests it would type-2 intermittentchaos (Pomeau & Manneville 1980).
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Introduction Global stability theory Roughness-induced transition Conclusion
Stenotic pipe flow
• Asymmetry of the stenosis triggers the wall-reattachment at lower Recompared to the axisymmetric case.
• Existence of a hysteresis cycle related to a subcritical pitchforkbifurcation.
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Introduction Global stability theory Roughness-induced transition Conclusion
Stenotic pipe flow
• Nonetheless, predictions from global stability analyses areuncorelatted to the experimental observations (Passaggia et al.)
→ Transition is dominated by transient growth.
• Preliminary optimal perturbation analysis appears to be moreconclusive.
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Introduction Global stability theory Roughness-induced transition Conclusion
Perspectives
• Several questions are still unanswered and require further in-depthinvestigations:
→ What is the mechanism responsible for the creation of hairpin vorticesin 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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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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