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Research Papers

Direct Numerical Simulations of Wake-Perturbed Separated Boundary Layers

[+] Author and Article Information
Ayse G. Gungor

School of Aeronautics,  Universidad Politécnica de Madrid, Madrid 28040, Spaingungor@torroja.dmt.upm.es

Mark P. Simens, Javier Jiménez

School of Aeronautics,  Universidad Politécnica de Madrid, Madrid 28040, Spain

J. Turbomach 134(6), 061024 (Sep 04, 2012) (9 pages) doi:10.1115/1.4004882 History: Received July 13, 2011; Accepted July 25, 2011; Published September 04, 2012; Online September 04, 2012

A wake-perturbed flat plate boundary layer with a streamwise pressure distribution similar to those encountered on the suction side of typical low-pressure turbine blades is investigated by direct numerical simulation. The laminar boundary layer separates due to a strong adverse pressure gradient induced by suction along the upper simulation boundary, transitions, and reattaches while still subject to the adverse pressure gradient. Various simulations are performed with different wake passing frequencies, corresponding to the Strouhal number 0.0043< fθb /ΔU <0.0496 and wake profiles. The wake profile is changed by varying its maximum velocity defect and its symmetry. Results indicate that the separation and reattachment points, as well as the subsequent boundary layer development, are mainly affected by the frequency, but that the wake shape and intensity have little effect, and that the forcing is effective as long as the wake-passing period is shorter than the bubble-regeneration time. Moreover, the effect of the different frequencies can be predicted from a single experiment in which the separation bubble is allowed to reform after having been reduced by wake perturbations. The stability characteristics of the mean flows resulting from the forcing at different frequencies are evaluated in terms of local linear stability analysis based on the Orr-Sommerfeld equation.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Instantaneous streamwise velocity (a) W 0 and (b) W 4. The black solid line indicates the separation bubble.

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Figure 2

(a) Separation (○) and reattachment (□) points (left y axis) and the minimum velocity (▴, right y axis) as a function of Stθb. The vertical thick line indicates the Kelvin-Helmholtz frequency Stθb=0.018. (b) The space-time development of the negative streamwise velocity at y/θ0  = 0.17 for cases W1-W 5 left to right, respectively. (c) The time history of the streamwise velocity at y/θ0  = 0.17 as a function of time at x/θ0  = 426, located inside the separated region, for cases W1-W 5 bottom to top, respectively. Data shifted vertically for visual clarity.

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Figure 3

(a) The development of the wall-normal velocity very close to the wall V, and (b) its second time-derivative d2 V/dt2 at y/θ0  = 0.17 for case W 4 as a function of space and time. The blue line traces the wake center and the red one wave trains generated by the wake passing. The propagation speed of these waves is 0.5U∞ . The vertical lines mark separation and reattachment points of the time averaged bubble.

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Figure 4

The phase-averaged spanwise velocity fluctuations (shaded contours ranging 0 < wrms /wrms,max  < 1) and spanwise vorticity contours (ωzz,max —, 0.1; - -, 0.3; ·····, 0.5) over one forcing period. The vertical thick solid line marks the location of the wake center. Dashed lines trace the development of roll-up vortices. The x axes labels 295 and 490 mark separation and reattachment points of the time-averaged bubble for case W 4.

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Figure 5

The phase-averaged skin-friction coefficient over one forcing period for case W 4. The vertical thick solid line marks the location of the wake center. The x axes labels 295 and 490 mark separation and reattachment points of the time-averaged bubble.

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Figure 6

The phase-averaged streamwise momentum budget over one forcing period for case W 4 at x/θ0  = 295 (marked in Fig. 5). □: Convective term ∂xUU⟩ + ∂yUV⟩, ▿: Reynolds stress term ∂xuu⟩ + ∂yuv⟩, ⋄: pressure gradient term ∂xP⟩, Δ: viscous term 1/Re(∂xx  + ∂yy )⟨U⟩.

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Figure 7

(a) Shape factor; (b) Reynolds number based on momentum thickness; (c) maximum turbulent intensity; (d) skin friction coefficient; and (e) wall pressure coefficient. No symbol: W 0, □: W 1, ▾: W 2, ▵: W 3, ▸: W 4, ◃: W 5, ——-: separated flow, Cf  < 0, and – – –: attached flow, Cf >0.

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Figure 8

Mean streamwise velocity U¯ profiles along the flat plate around the separated region for cases W 0-W 5 from left to right, top to bottom, respectively. □: Inflection points, and ▾: location of the maximum of the streamwise velocity intensity in the velocity profiles. The thick solid line in all figures is the zero contour of U¯, and marks the locations of the separated region.

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Figure 9

(a), (d) Streamwise momentum budget at x/θ0  = 295. □: Convective term, ▿: Reynolds stress term, ⋄: pressure gradient term, ▵: viscous term. Streamwise Reynolds stress balance at (b), (e) x/θ0  = 295 and (c), (f) x/θ0  = 650. □: Production: -2u′·u→′¯∇·U¯, ∇: convection: -∇u′u′¯·U¯→, ⋄: transport: -∇·u′u′u→′¯, ▵: pressure transport: -2u′∂xp′¯, ▹: dissipation: 1/Re∇2u′u′¯, ◃: diffusion: 2/Re∇u′·∇u′¯. (a), (b), (c) The unperturbed case (W 0). (d), (e), (f) The perturbed case (W 2).

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Figure 10

(a) The space-time development of the separated region and for cases R 1-R 5 top to bottom, respectively. The dashed line indicates the mean wake deficit at the inflow x/θ0  = 0. (b) The time history of the streamwise velocity just above the wall as a function of time at x/θ0  = 426, for cases R1-R 5 bottom to top, respectively. Data shifted vertically for visual clarity. (c) Separation (○) and reattachment (□) points.

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Figure 11

(a) The space-time development of the separated region. The dashed line traces the wake. (b) The space-time development of the time-averaged separated region. The thick solid lines mark the location of the numerical experiments. (c) The size of the separated region estimated from the regeneration experiment (——-) and obtained from DNSes (□: W 0-W 5).

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Figure 12

(a) Growth rate of the unstable modes αci  = Im(ω) as a function of (x-xS)/Lb0. (b) Amplification of the unstable modes within the range of λ/Lb0=0.10 (□) and 0.27 (▪). (c) Amplification of the unstable modes for the most amplified wavelengths: ——-: W 0; – – – : W 2; –· – ·–: W 4.

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Figure 13

(a) Maximum of the total amplification of the unstable modes: —–: W 0; – – –: W 2; –· – ·–: W 4. (b) Most amplified wavelengths predicted from LSA (—×—-) and from DNSes (□ and ▿).

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Figure 14

u-v perturbation streamlines at (x-xS)/Lb0=0.034, 0.2, and 0.3. The solid lines correspond to clockwise-rotating rollers.

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