Dynamics and global stability of three-dimensional flows

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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• 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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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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• 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



Streamwise velocity component for the two existing steady states of an asymmetric stenotic pipe flow at Re = 400.

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Context



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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• 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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• 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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• 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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• 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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(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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(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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(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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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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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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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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Roughness-induced transitionNon-linear evolution

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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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→ 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.


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• 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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→ 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..


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→ 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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Conclusion & Perspectives

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Conclusion


→ 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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Conclusion


→ 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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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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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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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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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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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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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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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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Science

Dynamics and global stability of three-dimensional flows