Dynamics of the boundary layer flow over a

cylindrical roughness element

J.-C. Loiseau(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

European Turbulence Conference 14, Lyon, France, September 1-42013

ANR – SICOGIF

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Backgroud & generalities

◮ Roughness-induced transition has numerous applications in aerospaceengineering :

→ Stabilisation of the Tolmien-Schlichting waves,→ Shift and/or control of the transition location, ...

◮ Despite the large body of literature, the underlying mechanisms arenot yet fully understood.

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

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Motivations

◮ Objectives :

→ Have a better insight of the roughness element’s impact on the flow→ Better understanding of the physical mechanisms responsible for

roughness-induced transition.

◮ Methods :

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

→ Comparison with experimental data whenever possible.

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Geometry & Notations

Geometry under consideration

◮ Box’s dimensions :

→ l = 15, Lx = 105, Ly = 50, Lz = 8.

◮ Roughness element’s characteristics :

→ Diameter d = 1, height h = 1, aspect ratio η = d/h = 1.

◮ Incoming boundary layer characteristics :

→ Ratio δ99/h = 2.

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Methodology : generalities

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

→ order of the polynomials N = 8 to 12,→ Temporal scheme of order 3 (BDF3/EXT3),→ Between 106 and 17.106 gridpoints.

◮ Base flows :

→ Selective frequency damping approach : application of a low-pass filterto the fully non-linear Navier-Stokes equations, see Akervik et al(2006).

◮ Global stability analysis :

→ Home made time-stepper Arnoldi algorithm build-up around Nek 5000temporal loop.

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Base flow

◮ Main features of the base flows :

→ Upstream and downstream reversed flow regions,→ Vortical system stemming from the upstream recirculation bubble and

extending downstream the roughness element.

U = 0 isosurface and some streamlines for the base flow (η, δ99/h,Re) = (2, 2, 600)

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Base flow

◮ Upstream vortical system investigated by Baker in the late 70’s,

◮ Vortical system composed of 4 vortices in all the cases investigated,

◮ Upstream spanwise vorticity wraps around the roughness element andtransforms into streamwise vorticity downstream :

→ Creation of downstream quasi-aligned streamwise vortices,→ Transfer of momentum through the lift-up effect giving birth to

streamwise streaks.

Solutions diagram from Baker (1979) Upstream vortical system’s topology for (η, δ99/h, Re) = (2, 2, 600)

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Base flow

◮ Horsheshoe vortical system :

→ Creation of the two outer pairs of low/high-speed streaks.

◮ Roughness element blockage :

→ Central low-speed streak due to streamwise velocity deficit.

Isosurfaces of the streamwise velocity deviation u = ±0.2 from the Blasius boundary layer flow for

(η, δ99/h, Re) = (1, 2, 1125).

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Linear stability

◮ Base flow and stability computed for (Re, η, δ99/h) = (1250, 1, 2) :

→ Only a sinuous unstable mode (0.0326± i0.68) lies in the upper-halfcomplex plane

→ Existence of a branch of varicose modes in the low-half part of theplane.

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

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Linear stability

◮ Base flow and stability computed for (Re, η, δ99/h) = (1250, 1, 2) :

→ Only a sinuous unstable mode (0.0326± i0.68) lies in the upper-halfcomplex plane

→ Existence of a branch of varicose modes in the low-half part of theplane.

Real part of the unstable eigenmode for (Re, η, δ99/h) = (1250, 1, 2).From left to right : spanwise, streamwise, wall-normal components.

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Linear stability

◮ Base flow and stability computed for (Re, η, δ99/h) = (1250, 1, 2) :

→ Only a sinuous unstable mode (0.0326± i0.68) lies in the upper-halfcomplex plane

→ Existence of a branch of varicose modes in the low-half part of theplane.

Real part of the leading varicose eigenmode for (Re, η, δ99/h) = (1250, 1, 2).From left to right : spanwise, streamwise, wall-normal components.

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Linear stability

Isocontours of uv∂U/∂y (red) and uw∂U/∂z (blue).

Perturbation’s kinetic energy budget analysis.

◮ Sinuous ReC = 1040.

Isocontours of uv∂U/∂y (red) and uw∂U/∂z (blue).

Perturbation’s kinetic energy budget analysis.

◮ Varicose ReC = 1225.

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Direct numerical simulation

◮ DNS at (Re, η, δ99/h) = (1125, 1, 2) :

→ Initialized with the base flow plus a small component flow made fromthe unstable global mode,

→ 9888 spectral elements, order 12 polynomial reconstruction → almost17 millions gridpoints.

→ Computation performed on 256 processors.

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Direct numerical simulation

Instantaneous streamwise velocity component evaluated at z = 0.5 for (η,Re, δ99/h) = (1, 1125, 2).

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Direct numerical simulation

Spanwise velocity signal from probes located at (x, y, z) = (10, 0.5, 0) and (x, y, z) = (80, 0.5, 0).

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Direct numerical simulation

Fourier transforms of the probes’signals.

Linear stability Near-wake region Far-wake region0.68 0.687 0.736

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Conclusion & Outlooks

◮ Major impact of the roughness element on the Blasius boundary layerflow :

→ Creation of streaks : two outer pairs and a central low-speed one.

◮ First instability of the streaks at ReC = 1040 due to a sinuousinstability.

◮ Non-linear evolution investigated by direct numerical simulation :

→ Sinuous eigenmode’signature clearly visible in the near-wake region.→ Enrichment of the Fourier spectrum and transition to turbulence

further downstream.→ Even in the far-wake region, the eigenmode’signature is still present.

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Conclusion & Outlooks

◮ What next ?

→ Further investigation of the instability mechanisms,→ Super/sub-criticality of the different bifurcations,→ Can optimal perturbations yield transition for Re ≤ ReC ?→ Influence of the roughness element’ shape,→ ...

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