Influence of the shape on the roughness-inducedtransition

J.-C. Loiseau(1), S. Cherubini(1) J.-C. Robinet(1) and E. Leriche(2)

(1): DynFluid Laboratory - Arts & Metiers-ParisTech - 75013 Paris, France(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France

International Conference on Instability and Control of MassivelySeparated Flows, Prato, Italy, Sept. 4-6, 2013

ANR – SICOGIF

1/20

Background - generalities

• Three-dimensional wall roughness has numerous applications inaerospace engineering :

↪→ Upstream shift of the transition location↪→ Transition delay↪→ Increase/Decrease of the skin friction ...

• Despite the large body of literature, physical mechanismsinducing transition are still poorly understood :↪→ Empirical transition criterion by von Doenhoff and Braslow,

experimental investigation by Asai et al, ...↪→ Investigations usually focus on one kind of roughness, without

considering the effect of its shape

Experimental visualisation of the flow induced by a roughness element. Gregory & Walker, 1956.

2/20

Motivations

• Objectives :

↪→ Have a better insight of the roughness element’s shape impacton the flow instability

↪→ Understanding the physical mechanisms responsible forroughness-induced transition.

• Methods :

↪→ Joint application of direct numerical simulations and linear globalstability analyses

↪→ Comparison of the instability mechanisms for two chosen shapes of theroughness element

• Cases under consideration :

↪→ Sharp-edged case → CYLINDER (Fransson et al. (2006)),↪→ Smooth case → BUMP (Piot et al. (2008))

3/20

Geometry & Notations

0

z

x

Lz

l Lx

Ly

δy d

h

Geometry under consideration

• Roughness elements’s

characteristics :

↪→ Cubic-cosine bump shape :

h(d) = h0 cos3(π

√(x2+y2)

d)

↪→ Diameter : d = 2↪→ Height : h0 = 1↪→ Aspect ratio : η = d/h0 = 2.

• Incoming boundary layer

characteristics :

↪→ Ratio : δ99/h0 = 2.

↪→ Re = U∞h0ν

= [700, 1000].

• Box’s dimensions :

↪→ Lx = 105↪→ Ly = 50↪→ Lz = 8.

4/20

Methodology : generalities

• All calculations are performed with the spectral elements code Nek5000 :

↪→ order of the polynomials N = 8,↪→ Temporal scheme of order 3 (BDF3/EXT3),↪→ Between 106 and 7.106 gridpoints.

• Base flows :

↪→ Selective frequency damping approach : application of a low-passfilter to the fully non-linear Navier-Stokes equations, see Akervik etal(2006).

• Global stability analysis :

↪→ Home made time-stepper Arnoldi algorithm build-up around Nek5000 temporal loop.

5/20

Numerical method : iterative eigenvalue methods

INPUTS Krylov basis LNS-Solver ORTHOGONALISE

OUTPUTS RESIDUAL

LAPACK LINEAR STABILITY

w = eAtukU = [U uk] h = scal.prod(w,U)

H = [H h]

H = [bek H]

if(k=kmax)

exit loop

f = w - Uh,

b = ||f||,

uk = f/b

U, H

[X,D] = eig(H) [UX,log(D)/t] ~ eig(A)

U = [], H = []

k = 0, uk

Arnoldi algorithm build-up

around Nek5000 temporal loop

6/20

ResultsBase Flows

7/20

Three-dimensional Base Flows

Main features of the base flows with (η, δ99/h,Re) = (2, 2, 1000) :

↪→ Upstream and downstreamreversed flow regions (blue forU = 0),

↪→ Vortical system stemmingfrom the upstream recirculationbubble (green for Q criterion).

↪→ Uptream spanwise vorticitywraps around the roughnesselement and transforms intostreamwise vorticity downstream

8/20

Three-dimensional Base Flows - (2)↪→ Creation of downstream quasi-aligned streamwise vortices↪→ Transfer of momentum through the lift-up effect giving birth to

streamwise streaks• For the bump, the streaks are weaker and more streamwise-localized

than for the cylinder

Streamwise velocity deviation from the Blasius profile, u = ±0.1 (top) u = ±0.05 (bottom)

9/20

Stability

10/20

Eigenspectra

� Cylinder → branch of eigenvalues, unstable mode at Rec = 803 (Reh = 593)

� Bump → isolated mode becoming unstable at Rec = 891 (Reh = 659), followedby a very stable branch

The bump becomes unstable at larger Re than the cylinder 11/20

Cylinder’s leading mode

Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)

Eigenspectrum for(η, δ99/h, Re) = (2, 2, 1000)

• The spatial support of the mode is located on thestreaks, well downstream of the cylinder

• It is composed by streamwise-alternated patches ofpositive/negative velocity perturbation

• It is symmetric w.r.t. the z = 0 axis (varicose mode)

12/20

Bump’s leading mode

Spatial support of the most unstable mode (u ± 0.05, v = ±0.02, and w = ±0.05)

Eigenspectrum for(η, δ99/h, Re) = (2, 2, 1000)

• The spatial support of the mode is located on theseparation zone, close to the bump

• It is composed by streamwise-alternated patches ofpositive/negative velocity perturbation

• It is symmetric w.r.t. the z = 0 axis (varicose mode)

13/20

Varicose eigenmodes at Re = 1000

CYLINDER

↪→ Strong deformation of the base flowstreamwise velocity (solid contours)

BUMP

↪→ Weaker deformation of the base flowstreamwise velocity (solid contours)

• Largest values of the perturbation in the zones of maximum shear (shaded)

• Instability linked with the base flow shear like for optimal streaks ?

• To verify it, we analyze the production terms of the Reynolds-Orr equation :

dE

dt= −

∫V

uiuj∂Ui

∂xjdV − 1

Re

∫V

∂ui

∂xj

∂ui

∂xjdV (1)

14/20

Production terms - CYLINDER case

­0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

uuU uvU uwU uvV vvV vwV uwW vwWx y z x y z x y

↪→ The dominant production termsare TUy = uv ∂U

∂yand TUz = uw ∂U

∂z

↪→ TUz is the largest term, even if themode is varicose (unlike theoptimal streaks case)

Production term TUy

and streamwise perturbation u

� Streamwise displacement of thewall-normal shear

Production term TUz

and spanwise perturbation w

� Spanwise displacement of thespanwise shear

15/20

Production terms - BUMP case

­0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

uuU uvU uwU uvV vvV vwV uwW vwWx y z x y z x y

↪→ The dominant production termsare TUy = uv ∂U

∂yand TUz = uw ∂U

∂z

↪→ TUz is the largest term, as for thecylinder

Production term TUy

and streamwise perturbation u

� Streamwise displacement of thewall-normal shear

Production term TUz

and spanwise perturbation w

� Spanwise displacement of thespanwise shear

16/20

Branch vs. isolated mode

The instability mechanism appears the same for the two roughness elements

↪→ But why for the bump the branch is very far from the most unstable mode ?

CYLINDER

• The two most unstable modes arevery similar, except for a shift in thestreamwise direction

• They are located on the low-speedstreak downstream of the roughnesselement

• Probably related to thequasi-parallelism of the streaks

BUMP

• The two most unstable modes arevery different

• The isolated mode is located on thelow-speed streak close to theroughness ; the modes on the stablebranch at the outlet

• Probably related to thestreamwise-localization of the streaks

17/20

Conclusions• CYLINDER :

↪→ Very strong quasi-parallel streaks downstream of the roughness element↪→ Unstable mode at Re = 803, closely followed by an eigenvalue branch↪→ Spatially localized along the central low-speed streak↪→ Varicose symmetry, but it extracts its energy mostly from the

spanwise shear

• BUMP :↪→ Rather strong streaks which fade away far from the roughness element↪→ An isolated mode is destabilized at Re = 891, followed by a very

stable branch↪→ Spatially localized along the central low-speed streak closer to the

separation zone↪→ Varicose symmetry, but it extracts its energy mostly from the

spanwise shear

⇒ Global counterpart of the local streak’s instability observed by Asaiet al(2002,2007) and Brandt (2006).

18/20

Outlook and future works

• Several questions remain unanswered :

↪→ Why the critical Reynolds number is higher in the bump’s case ?Maybe because of the lower amount of fluid displaced by the roughnesselement ?

↪→ What would happens considering a cylinder having the samesurface area of the bump, instead of the same aspect ratio ? Wouldthe critical Reynolds number be the same ?

↪→ For thin cylinders (η ≤ 1), a sinuous unstable mode has beenrecovered. Does a sinuous mode exists also for the bump ?

↪→ What about non-normal and non-linear effects in the transitionprocess ? (Arnal et al., Cherubini et al., ...)

↪→ Because of the spatial localization of the mode, would a localstability analysis give similar results ?

19/20

Thanks for listening !

20/20

Sinuous eigenmode

ω

σ

0 0.5 1 1.5 2 2.5­0.1

­0.08

­0.06

­0.04

­0.02

0

0.02

0.04A­S mode

Eigenspectrum for (η, δ99/h, Re) = (1, 2, 1250)

Real part of the unstable sinuous mode streamwise component

20/20

Sinuous eigenmode - (2)

Slice in the plane y = 1

Slice in the plane x = 30 Production terms τuv∂yU and τuw∂zU in the planex = 30

20/20

Varicose eigenmode - (2)

Slice in the symmetry plane z = 0

Slice in the plane x = 30 Production terms τuv∂yU and τuw∂zU in the planex = 30

20/20

Discussion

• Convenient definition of the Reynolds number is the roughnessReynolds number Reh :

Reh = U(h)hν

δ99/h 1.75 2 2.25Rec 1175 1225 1310Rech 960 903 899

Table: Evolution of the critical Reynolds numbers Rec and Rech with respect to

δ99/h for the varicose instability and aspect ratio η = 1.

• Rech tends to a value of approximately 900 :

↪→ Good agreements with experimental observations : transition withinthe range 600 ≤ Reh ≤ 900.

20/20

Comparison with von Doenhoff-Braslow transitiondiagram

Reproduction of the von Doenhoff-Braslow transition diagram along with the critical roughness Reynolds numbers for varicoseinstability obtained by global stability analyses.

20/20