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Research Papers: Flows in Complex Systems

Optimal Perturbations in Boundary-Layer Flows Over Rough Surfaces

[+] Author and Article Information
S. Cherubini

DynFluid, Arts et Metiers ParisTech,
Paris 75013, France;
Dipartimento di Meccanica,
Matematica e Management,
DMMM, CEMeC,
Politecnico di Bari,
Bari 70125, Italy
e-mail: s.cherubini@gmail.com

M. D. de Tullio

e-mail: m.detullio@poliba.it

P. De Palma

e-mail: depalma@poliba.it

G. Pascazio

e-mail: pascazio@poliba.it
Dipartimento di Meccanica,
Matematica e Management,
DMMM, CEMeC,
Politecnico di Bari,
Bari 70125, Italy

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 12, 2012; final manuscript received July 15, 2013; published online September 12, 2013. Assoc. Editor: Zhongquan Charlie Zheng.

J. Fluids Eng 135(12), 121102 (Sep 12, 2013) (11 pages) Paper No: FE-12-1569; doi: 10.1115/1.4025028 History: Received November 12, 2012; Revised July 15, 2013

This work provides a three-dimensional energy optimization analysis, looking for perturbations inducing the largest energy growth at a finite time in a boundary-layer flow in the presence of roughness elements. The immersed boundary technique has been coupled with a Lagrangian optimization in a three-dimensional framework. Four roughness elements with different heights have been studied, inducing amplification mechanisms that bypass the asymptotical growth of Tollmien–Schlichting waves. The results show that even very small roughness elements, inducing only a weak deformation of the base flow, can strongly localize the optimal disturbance. Moreover, the highest value of the energy gain is obtained for a varicose perturbation. This result demonstrates the relevance of varicose instabilities for such a flow and shows a different behavior with respect to the secondary instability theory of boundary layer streaks.

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References

Figures

Grahic Jump Location
Fig. 1

Sketch of the computational domain: Γ1 and Γ2 indicate the fluid and solid regions, respectively, separated by the body surface, in gray

Grahic Jump Location
Fig. 8

Isosurfaces of the streamwise component of velocity (light gray and dark gray for negative and positive values, respectively) for the optimal perturbations at target time T = 50 for four base flows with h0 = 0.75 (first frame), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame)

Grahic Jump Location
Fig. 7

Contours of the streamwise component of velocity and streamlines in the plane at x = 77.5 for the initial optimal perturbation obtained at T = 50 for the base flow with h0 = 0.75 (top frame) and h0 = 1.5 (bottom frame). The spanwise and wall-normal axis are not on the same scale.

Grahic Jump Location
Fig. 6

Isosurfaces of the spanwise component of velocity (dark gray and light gray for negative and positive values, respectively) for the initial optimal perturbations (t = 0) obtained at T = 50 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The inset shows the location and the shape of the roughness element.

Grahic Jump Location
Fig. 5

Optimal perturbations for the Blasius base flow. Top: isosurfaces of the spanwise component of the velocity (dark gray and light gray for negative and positive values, respectively) of the initial (t = 0) optimal perturbation. Bottom: isosurfaces of the streamwise component (light gray and dark gray for negative and positive values, respectively) of the optimal perturbation at t = T = 50.

Grahic Jump Location
Fig. 4

Optimal energy gain versus target time for the Blasius base flow (circles) and for the base flows in the presence of the roughness element with height h0 = 0.75 (triangles up), h0 = 1 (triangles down), h0 = 1.5 (squares), and h0 = 2 (diamonds)

Grahic Jump Location
Fig. 3

Surfaces of the positive (light gray) and negative (dark gray) deviation of the streamwise component of velocity with respect to the spanwise mean for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The black circle represents the bump.

Grahic Jump Location
Fig. 2

Contours of positive (dark gray) and negative (light gray) streamwise vorticity in the plane y = h0 + 1 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The black circle represents the bump.

Grahic Jump Location
Fig. 9

Contours of the streamwise velocity component of the optimal perturbation at target time T = 50 in the plane x = 112.5 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axes are not on the same scale.

Grahic Jump Location
Fig. 10

Isosurfaces for T = 200 of the spanwise component of the velocity (dark gray and light gray for negative and positive values, respectively) for the initial (t = 0) optimal perturbations for the bumps with h0 = 0.75 (top) and h0 = 1.5 (bottom)

Grahic Jump Location
Fig. 11

Isosurfaces of the streamwise component of the velocity (light gray and dark gray for negative and positive values, respectively) for the optimal perturbations at target time t = T = 200 for the bumps with h0 = 0.75 (top) and h0 = 1.5 (bottom)

Grahic Jump Location
Fig. 12

Vectors of the spanwise and wall-normal velocity components and contours of the streamwise velocity component in the plane at x = 77.5 for the initial (t = 0) optimal perturbations obtained for T = 100 and T = 150 (from top to bottom) for the case of the bump with h0 = 1.5

Grahic Jump Location
Fig. 13

Shaded contours of the streamwise velocity component in the planes at x = 133.5 and x = 165 (from top to bottom) for the optimal perturbation at target time t = T = 100 and t = T = 150 (from top to bottom) for the case of the bump with h0 = 1.5. The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axes are not on the same scale.

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