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FIGURE 11. Evolution in time of the linear optimal perturbation obtained for Re = 300, k = 1.5, T = 50, with initial energy Eo = 10~°: streamwise component of the base flow velocity (flat light blue surface for U = 0.2); streamwise vorticity perturbation (white and black surfaces for we = +0.001) at t = 0 (a) and t = 25 (b), streamwise velocity perturbation (light grey, green online, and dark grey, red online, for u?™’"* = +0.00002) at t = 50 (c) and t = 100 (d). Transient growth in the flow past a three-dimensional smooth roughness element 17

Figure 11 Evolution in time of the linear optimal perturbation obtained for Re = 300, k = 1.5, T = 50, with initial energy Eo = 10~°: streamwise component of the base flow velocity (flat light blue surface for U = 0.2); streamwise vorticity perturbation (white and black surfaces for we = +0.001) at t = 0 (a) and t = 25 (b), streamwise velocity perturbation (light grey, green online, and dark grey, red online, for u?™’"* = +0.00002) at t = 50 (c) and t = 100 (d). Transient growth in the flow past a three-dimensional smooth roughness element 17

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FIGURE 1. Sketch of the computational domain: I; and [2 indicate the fluid and solid regions respectively, separated by the body surface, in gray.
FIGURE 1. Sketch of the computational domain: I; and [2 indicate the fluid and solid regions respectively, separated by the body surface, in gray.
where V is the computational volume. The objective function of the procedure is the energy gain at the target time t = T, i.e. E(T)/E(0). The Lagrange multiplier tech- nique consists in seeking extrema of the augmented functional £ with respect to every independent variable. Such a functional is written as:
where V is the computational volume. The objective function of the procedure is the energy gain at the target time t = T, i.e. E(T)/E(0). The Lagrange multiplier tech- nique consists in seeking extrema of the augmented functional £ with respect to every independent variable. Such a functional is written as:
FIGURE 2. Contours of the streamwise vorticity in the plane y = k + 1, for the two base flows with k = 0.75 (a), and k = 1.5 (b) at Re = 235. The black circle represents the bump. TABLE 1. Values of Re; for all of the base flows considered.
FIGURE 2. Contours of the streamwise vorticity in the plane y = k + 1, for the two base flows with k = 0.75 (a), and k = 1.5 (b) at Re = 235. The black circle represents the bump. TABLE 1. Values of Re; for all of the base flows considered.
Transient growth in the flow past a three-dimensional smooth roughness element FIGURE 3. Surfaces of the positive (light grey, yellow online) and negative (dark grey, blue online) deviation of the streamwise component of velocity with respect to the spanwise-average value for two base flows at Re = 170 (top frames), Re = 235 (middle frames), and Re = 300 (bottom frames) with k = 0.75 (left frames, @ = £0.01), and k = 1.5 (right frame, U = +0.018). The black circle represents the bump.
Transient growth in the flow past a three-dimensional smooth roughness element FIGURE 3. Surfaces of the positive (light grey, yellow online) and negative (dark grey, blue online) deviation of the streamwise component of velocity with respect to the spanwise-average value for two base flows at Re = 170 (top frames), Re = 235 (middle frames), and Re = 300 (bottom frames) with k = 0.75 (left frames, @ = £0.01), and k = 1.5 (right frame, U = +0.018). The black circle represents the bump.
FIGURE 4. Optimal energy gain versus target time for the Blasius base flow (dotted line) and in the presence of the bumps of different heights (increasing bump height following the arrow), for Re = 170 (a), Re = 235 (b), and Re = 300 (c). The symbols correspond to: k = 0.5 (squares), k, = 0.75 (gradients), k = 1 (deltas), k = 1.25 (diamonds), and k = 1.5 (circles).
FIGURE 4. Optimal energy gain versus target time for the Blasius base flow (dotted line) and in the presence of the bumps of different heights (increasing bump height following the arrow), for Re = 170 (a), Re = 235 (b), and Re = 300 (c). The symbols correspond to: k = 0.5 (squares), k, = 0.75 (gradients), k = 1 (deltas), k = 1.25 (diamonds), and k = 1.5 (circles).
FIGURE 5. Iso-surfaces of the spanwise component of velocity (dark grey, blue online, and light grey, yellow online, for negative and positive values, respectively) for the initial optimal pertur- bations obtained at T = 50 with Re = 235 and k = 0 (a), k = 0.5 (b), k= 1 (c), and k = 1.5 (d). Transient growth in the flow past a three-dimensional smooth roughness element 11
FIGURE 5. Iso-surfaces of the spanwise component of velocity (dark grey, blue online, and light grey, yellow online, for negative and positive values, respectively) for the initial optimal pertur- bations obtained at T = 50 with Re = 235 and k = 0 (a), k = 0.5 (b), k= 1 (c), and k = 1.5 (d). Transient growth in the flow past a three-dimensional smooth roughness element 11
FIGURE 6. Iso-surfaces of the streamwise component of velocity (light grey, green online, and dark grey, red online, for negative and positive values, respectively) for the optimal perturbations at target time T = 50 with Re = 235, k = 0 (a), k = 0.5 (b), k = 1 (c), and k = 1.5 (d).
FIGURE 6. Iso-surfaces of the streamwise component of velocity (light grey, green online, and dark grey, red online, for negative and positive values, respectively) for the optimal perturbations at target time T = 50 with Re = 235, k = 0 (a), k = 0.5 (b), k = 1 (c), and k = 1.5 (d).
FIGURE 7. Optimal energy gain versus target time obtained with: (a) Re = 170 and k = 0.75, for three values of the streamwise domain length, namely, Lz = 108, Lz = 216 (the reference one), and Lz = 324; (b) Re = 300, & = 0.75 and k = 1.5, for two values of the streamwise domain length, namely, Lz = 108,216; (c) Re = 300, k = 0.75 and k = 1.5, for two values of the spanwise domain length, namely, L, = 48, L, = 72 (the reference one).
FIGURE 7. Optimal energy gain versus target time obtained with: (a) Re = 170 and k = 0.75, for three values of the streamwise domain length, namely, Lz = 108, Lz = 216 (the reference one), and Lz = 324; (b) Re = 300, & = 0.75 and k = 1.5, for two values of the streamwise domain length, namely, Lz = 108,216; (c) Re = 300, k = 0.75 and k = 1.5, for two values of the spanwise domain length, namely, L, = 48, L, = 72 (the reference one).
FIGURE 8. Iso-surfaces of the spanwise component of velocity (dark grey, blue online, and light grey, yellow online, for negative and positive values, respectively), and of the streamwise com- ponent of velocity (light grey, green online, and dark grey, red online, for negative and positive values, respectively) for the optimal perturbation at t = 0 (a) and t = T (b) obtained at Re = 235, T = 200 and k = 1.5. Transient growth in the flow past a three-dimensional smooth roughness element 13
FIGURE 8. Iso-surfaces of the spanwise component of velocity (dark grey, blue online, and light grey, yellow online, for negative and positive values, respectively), and of the streamwise com- ponent of velocity (light grey, green online, and dark grey, red online, for negative and positive values, respectively) for the optimal perturbation at t = 0 (a) and t = T (b) obtained at Re = 235, T = 200 and k = 1.5. Transient growth in the flow past a three-dimensional smooth roughness element 13
FIGURE 9. Cross-stream vectors and streamwise velocity component contours in the plane at x = 77.5 for the initial optimal perturbation obtained at T = 100 (a) and T = 150 (b) for Re = 235 and k = 1.5. The spanwise and wall-normal axis are not on the same scale.
FIGURE 9. Cross-stream vectors and streamwise velocity component contours in the plane at x = 77.5 for the initial optimal perturbation obtained at T = 100 (a) and T = 150 (b) for Re = 235 and k = 1.5. The spanwise and wall-normal axis are not on the same scale.
FIGURE 10. Shaded contours of the streamwise velocity component of the optimal perturbation at target time T = 100 (a) and T = 150 (b) in the planes « = 135 (a) and x = 165 (b) with Re = 235 and k = 1.5. The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axis are not on the same scale.
FIGURE 10. Shaded contours of the streamwise velocity component of the optimal perturbation at target time T = 100 (a) and T = 150 (b) in the planes « = 135 (a) and x = 165 (b) with Re = 235 and k = 1.5. The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axis are not on the same scale.
FIGURE 12. Evolution in time of the linear optimal perturbation obtained for Re = 300, k = 1.5, T = 200, with initial energy Zo = 107°: streamwise component of the base flow velocity (flat light blue surface for U = 0.2); streamwise vorticity perturbation (white and black surfaces for wz = +0.00001) at t = 0 (a) and t = 100 (b), streamwise velocity perturbation (light grey, green online, and dark grey, red online, for u’ = 4 +0.00001) at t = 200 (c) and t = 300 (d).
FIGURE 12. Evolution in time of the linear optimal perturbation obtained for Re = 300, k = 1.5, T = 200, with initial energy Zo = 107°: streamwise component of the base flow velocity (flat light blue surface for U = 0.2); streamwise vorticity perturbation (white and black surfaces for wz = +0.00001) at t = 0 (a) and t = 100 (b), streamwise velocity perturbation (light grey, green online, and dark grey, red online, for u’ = 4 +0.00001) at t = 200 (c) and t = 300 (d).
FIGURE 13. Perturbation kinetic energy time derivative, and production and dissipation terms versus time extracted by a linearized DNS initialized by the VP* (a) and the SP* (b).
FIGURE 13. Perturbation kinetic energy time derivative, and production and dissipation terms versus time extracted by a linearized DNS initialized by the VP* (a) and the SP* (b).
FIGURE 14. Base flow shear (grey, b ue online), streamwise vorticity perturbation (white and black for w; = +0.001) at t = 0 (a) and streamwise velocity perturbation (dark grey, red online, and light grey, green online, for u’ = 4 for Re = 300, k= 1.5, T = 50. +10~°) at t = 25 (b) for the optimal perturbation obtained
FIGURE 14. Base flow shear (grey, b ue online), streamwise vorticity perturbation (white and black for w; = +0.001) at t = 0 (a) and streamwise velocity perturbation (dark grey, red online, and light grey, green online, for u’ = 4 for Re = 300, k= 1.5, T = 50. +10~°) at t = 25 (b) for the optimal perturbation obtained
FIGURE 15. Base flow shear (grey, b ue online), streamwise vorticity perturbation (white and black for wz = +0.001) at t = 0 (a) and streamwise velocity perturbation (dark grey, red online, and light grey, green online, for u’ = 4 for Re = 300, k = 1.5, T = 200. t10~°) at ¢ = 25 (b) for the optimal perturbation obtained
FIGURE 15. Base flow shear (grey, b ue online), streamwise vorticity perturbation (white and black for wz = +0.001) at t = 0 (a) and streamwise velocity perturbation (dark grey, red online, and light grey, green online, for u’ = 4 for Re = 300, k = 1.5, T = 200. t10~°) at ¢ = 25 (b) for the optimal perturbation obtained
FIGURE 16. Energy gain versus time extracted by a DNS initialized by the optimal perturbation obtained for Re = 300, k = 1.5, T = 50 (VP, left frame) and T = 200 (SP*, right frame) with the initial energies indicated within the figure. The black dots indicate the optimal energy gain obtained at target time by the linear optimization.
FIGURE 16. Energy gain versus time extracted by a DNS initialized by the optimal perturbation obtained for Re = 300, k = 1.5, T = 50 (VP, left frame) and T = 200 (SP*, right frame) with the initial energies indicated within the figure. The black dots indicate the optimal energy gain obtained at target time by the linear optimization.
FIGURE 17. Contours of the instantaneous streamwise velocity at y = 2 and surfaces of the streamwise vorticity perturbation (white and black) extracted by a DNS initialized by the op- 300, k = 1.5, T = 50 with initial energy Eo = 0.001 at timal perturbation obtained for Re t = 0 (surfaces for w. = 4 +0.005, first rame), t = 25 (surfaces for wz t = 75 and t = 150 (surfaces for wz = 4 +0.03, second frame), +0.08, third and fourth frames, respectively).
FIGURE 17. Contours of the instantaneous streamwise velocity at y = 2 and surfaces of the streamwise vorticity perturbation (white and black) extracted by a DNS initialized by the op- 300, k = 1.5, T = 50 with initial energy Eo = 0.001 at timal perturbation obtained for Re t = 0 (surfaces for w. = 4 +0.005, first rame), t = 25 (surfaces for wz t = 75 and t = 150 (surfaces for wz = 4 +0.03, second frame), +0.08, third and fourth frames, respectively).
FIGURE 18. Contours of the instantaneous streamwise velocity at y = 2 and surfaces of the streamwise vorticity perturbation (white and black) extracted by a DNS initialized by the op- timal perturbation obtained for Re = 300, k = 1.5, T = 200 with initial energy Eo = 0.01 at t = 0, t = 25 (surfaces for w, = +0.015, first and second frame, respectively), t = 75 and t = 15( +0.1, third and fourth frame, respectively). (surfaces for wz = 4
FIGURE 18. Contours of the instantaneous streamwise velocity at y = 2 and surfaces of the streamwise vorticity perturbation (white and black) extracted by a DNS initialized by the op- timal perturbation obtained for Re = 300, k = 1.5, T = 200 with initial energy Eo = 0.01 at t = 0, t = 25 (surfaces for w, = +0.015, first and second frame, respectively), t = 75 and t = 15( +0.1, third and fourth frame, respectively). (surfaces for wz = 4
Transient growth in the flow past a three-dimensional smooth roughness element 23 FIGURE 19. Surface of the instantaneous streamwise velocity u = —0.2 and of the Q-criterion (light gray for Q = 100) extracted by a DNS initialized by the optimal perturbation obtained for Re = 300, k = 1.5, T = 50 with initial energy Eo = 0.001 at t = 50 (a), t = 100 (b), t = 150 (c.)
Transient growth in the flow past a three-dimensional smooth roughness element 23 FIGURE 19. Surface of the instantaneous streamwise velocity u = —0.2 and of the Q-criterion (light gray for Q = 100) extracted by a DNS initialized by the optimal perturbation obtained for Re = 300, k = 1.5, T = 50 with initial energy Eo = 0.001 at t = 50 (a), t = 100 (b), t = 150 (c.)
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Boundary LayersAerodynamicsTurbulent FlowsBoundary layer transitionLES and DNSWall JetsFree Shear FlowsAtmospheric Boundary Layer Flows
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