1. 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 & M´tiers-ParisTech - 75013 Paris, France
e
(2): LML - University of Lille 1 - 59655 Villeneuve d’Ascq, France
European Turbulence Conference 14, Lyon, France, September 1-4
2013
ANR – SICOGIF
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2. Backgroud & generalities
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Roughness-induced transition has numerous applications in aerospace
engineering :
֒→ Stabilisation of the Tolmien-Schlichting waves,
֒→ Shift and/or control of the transition location, ...
◮
Despite the large body of literature, the underlying mechanisms are
not yet fully understood.
Experimental visualisation of the flow induced by a roughness element. Gregory & Walker, 1956.
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3. Motivations
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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 global
stability analyses,
֒→ Comparison with experimental data whenever possible.
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4. Geometry & Notations
Geometry under consideration
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Box’s dimensions :
֒→ l = 15, Lx = 105, Ly = 50, Lz = 8.
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Roughness element’s characteristics :
֒→ Diameter d = 1, height h = 1, aspect ratio η = d/h = 1.
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Incoming boundary layer characteristics :
֒→ Ratio δ99 /h = 2.
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5. Methodology : generalities
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All calculations are performed with the spectral elements code Nek
5000 :
֒→ 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 filter
to the fully non-linear Navier-Stokes equations, see Akervik et al(2006).
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Global stability analysis :
֒→ Home made time-stepper Arnoldi algorithm build-up around Nek 5000
temporal loop.
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6. Base flow
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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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7. Base flow
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Upstream vortical system investigated by Baker in the late 70’s,
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Vortical system composed of 4 vortices in all the cases investigated,
Upstream spanwise vorticity wraps around the roughness element and
transforms 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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8. Base flow
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Horsheshoe vortical system :
◮
Roughness element blockage :
֒→ Creation of the two outer pairs of low/high-speed streaks.
֒→ 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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9. 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-half
complex plane
֒→ Existence of a branch of varicose modes in the low-half part of the
plane.
Eigenspectrum (Re, η, δ99 /h) = (1250, 1, 2).
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10. 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-half
complex plane
֒→ Existence of a branch of varicose modes in the low-half part of the
plane.
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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11. 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-half
complex plane
֒→ Existence of a branch of varicose modes in the low-half part of the
plane.
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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12. Linear stability
Isocontours of uv∂U/∂y (red) and uw∂U/∂z (blue).
Isocontours of uv∂U/∂y (red) and uw∂U/∂z (blue).
Perturbation’s kinetic energy budget analysis.
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Sinuous ReC = 1040.
Perturbation’s kinetic energy budget analysis.
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Varicose ReC = 1225.
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13. Direct numerical simulation
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DNS at (Re, η, δ99 /h) = (1125, 1, 2) :
֒→ Initialized with the base flow plus a small component flow made from
the unstable global mode,
֒→ 9888 spectral elements, order 12 polynomial reconstruction → almost
17 millions gridpoints.
֒→ Computation performed on 256 processors.
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14. Direct numerical simulation
Instantaneous streamwise velocity component evaluated at z = 0.5 for (η, Re, δ99 /h) = (1, 1125, 2).
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15. 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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16. Direct numerical simulation
Fourier transforms of the probes’signals.
Linear stability
0.68
Near-wake region
0.687
Far-wake region
0.736
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17. Conclusion & Outlooks
◮
Major impact of the roughness element on the Blasius boundary layer
flow :
֒→ Creation of streaks : two outer pairs and a central low-speed one.
◮
◮
First instability of the streaks at ReC = 1040 due to a sinuous
instability.
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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18. Conclusion & Outlooks
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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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