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Int. J. Aerodynamics, Vol. 1, Nos. 3/4, 2011

Transient growth of coherent streaks for control of turbulent flow separation G. Pujals* LadHyX, CNRS-École Polytechnique, F-91128 Palaiseau, France and PSA Peugeot Citroën, Centre Technique de Velizy, 2 Route de Gisy, 78943 Vélizy-Villacoublay Cedex, France E-mail: [email protected] *Corresponding author

S. Depardon PSA Peugeot Citroën, Centre Technique de Velizy, 2 Route de Gisy, 78943 Vélizy-Villacoublay Cedex, France E-mail: [email protected]

C. Cossu IMFT-CNRS, Allée du Pr. Camille Soula, 31400 Toulouse, France E-mail: [email protected] Abstract: In this paper, we summarise our recent results on turbulent flow separation control using transient growth of large-scale coherent streaks. According to linear stability analysis, the optimal perturbations (i.e., disturbances experiencing the largest transient energy growth) sustained by a turbulent boundary layer are large-scale streamwise uniform coherent vortices leading to streaks, the lift-up effect being responsible for their growth. A first experimental study confirms that using arrays of suitably shaped cylindrical roughness elements, streaks can be artificially forced in a flat plate turbulent boundary layer at a Reynolds number based on the displacement thickness of 1; 000. Interacting with the mean flow at leading order, these streaks induce a strong controlled spanwise modification and that their amplitude transiently grows in space. Eventually, streaks are forced on the roof of a generic car model (Ahmed body, see Ahmed et al., 1984) to test their ability to suppress the separation around the rear-end. Keywords: turbulent boundary layer; streaks; transient growth; turbulent separation control; Ahmed body. Reference to this paper should be made as follows: Pujals, G., Depardon, S. and Cossu, C. (2011) ‘Transient growth of coherent streaks for control of turbulent flow separation’, Int. J. Aerodynamics, Vol. 1, Nos. 3/4, pp.318–336. Biographical notes: G. Pujals received his PhD in Fluid Mechanics in 2009. He now holds a Research Engineer Position in PSA Peugeot Citroën in France. He works on flow control topics. Copyright © 2011 Inderscience Enterprises Ltd.

Transient growth of coherent streaks for control of turbulent flow separation

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C. Cossu is the Directeur de Recherches CNRS at the IMFT, France. S. Depardon is holding an engineer position in PSA Peugeot Citroën.

1

Introduction

In a large majority of industrial applications, boundary layer separation is associated with a large loss of performance that makes separation control of great importance: increased fuel consumption and pollutants emissions and decreased stability for ground vehicles, decrease of the lift-force on airplanes’ wing at high angle of attack. This phenomenon occurs when low momentum flow (i.e., in the near-wall region of a boundary layer) faces an adverse pressure gradient; due to either geometrical constraints (rear-end of a ground vehicle) or operating conditions (airfoil at high angle of attack during take-off or landing manoeuvres); which tends to lower its motion. One way to delay or suppress separation is to enhance momentum in the near-wall region of the boundary layer. While, recent experiments involving active closed-loop flow control device have proved to be effective in reducing the drag of bluff-bodies (see, e.g., Pastoor et al., 2008); passive (open-loop) separation control remains a very attractive approach because, as it is easier to implement and to handle, it has very low production and maintainance costs. The most widely used passive devices are vortex generators (VGs). Introduced in the late forties by Taylor (1947), the early VGs consist in arrays of small vanes which height k is comparable to the undisturbed boundary layer thickness δ0 at the same position. In the late 1980s, experiments of Rao and Kariya (1988) showed that submerged VGs (i.e., k / δ0 ≤ 0.6) can be more efficient than the classical VGs of size ≈δ0. This breakthrough opened the way to many experimental studies aiming at optimising the shape and dimensions of such ‘low-profile’ VGs (see Betterton et al., 2000; Lin, 2002; Angele, 2003; Godard and Stanislas, 2006; Lögdberg, 2006). One of the common results of all these studies is that counter-rotating (CtR) VGs are more efficient than co-rotating VGs. To design efficient VGs, the relevant parameters are their height k, their width l, their spanwise spacing λz and their streamwise location from the separation line. Unfortunately, a survey of literature shows a wide range of shapes and values for most of these parameters. The only consensus being about the height of the devices: to limit the induced drag penalty optimisation have been conducted resulting in a reduction of VGs’ height from k ≈ δ0 to only a fraction of it (namely, k ≤ 0.6δ0). In laminar shear flows, perturbations induced by low energy CtR streamwise vortices can experience large transient energy growth through the lift-up effect (see, e.g., Moffatt, 1967; Landahl, 1980). The mechanism is the following: high momentum fluid is pushed towards the wall while low momentum fluid is transported in the outer region of the flow resulting in streamwise-elongated spanwise modulations of the velocity field called ‘streaks’. The energy growth of these streaks is transient and strongly related to the

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non-normal nature of the linearised Navier-Stokes operator (Trefethen et al., 1993). When the shape and the wavelength of the disturbing vortices are the energy ( optimised, ) amplification of the resulting optimal streaks can be of order O Re2 (see, e.g., Butler and Farrell, 1992; Schmid and Henningson, 2001). In recent experiments, steady and stable streaks of moderate amplitude have been forced by nearly-optimal streamwise vortices generated with cylindrical roughness elements. The resulting streaky flow can be used to manipulate laminar shear flows at leading order and applied to stabilise Tollmien-Schlichting waves (Cossu and Brandt, 2002; Fransson et al., 2004) and to delay transition to turbulence (Fransson et al., 2006). Following a similar guideline, Duriez (2009) showed that laminar streaks could also be used to delay separation on a 2D backward-facing ramp in the laminar régime. An interesting extension of such approach would consist in the manipulation of turbulent boundary layers with optimal or nearly optimal vortices and streaks. To do so, some issues had to be addressed first: what are the shape and scales of such disturbances? Can we artificially force them in a flat plate turbulent boundary layer? If one can do so, do they experience any growth and/or contribute to modify the base flow at leading order? Eventually, is it possible to perform flow control (for instance, to delay turbulent flow separation) using these perturbations? The present paper summarises the main results of the studies that were conducted with this goal (Cossu et al., 2009; Pujals et al., 2010a, 2010b). It is organised as follow: in the first part, we focus on the computation of the optimal perturbations supported by a zero pressure gradient turbulent boundary layer. The second section describes the results of an experimental study in which nearly optimal perturbations are artificially forced in a flat plate turbulent boundary layer by means of spanwise organised roughness elements. The third part investigates the ability of these perturbations to delay separation on the rear-end of a generic car model (Ahmed body, see Ahmed et al., 1984). All those results are discussed and related to previous studies in the last section.

2 Optimal perturbations in zero pressure gradient turbulent boundary layer Progress in linear stability analysis of turbulent mean flows, has been made by Reynolds and Hussain (1972) who considered the Reynolds-averaged Navier-Stokes equations linearised around a turbulent mean flow. These equations rule the dynamics of small amplitude statistically coherent perturbations (i.e., non-zero mean value) while the dynamics of random fluctuations (i.e., statistically uncorrelated) is modeled by means of the turbulent eddy-viscosity in equilibrium with the mean flow. This modelling has ´ been used by del Alamo and Jiménez (2006), Pujals et al. (2009) and Hwang and Cossu (2010) to compute the optimal perturbations sustained by turbulent plane Poiseuille and plane Couette flows. Here, we focus on the optimal perturbations supported by a turbulent boundary layer and follow the analyses of Pujals et al. (2009) and Cossu et al. (2009).

2.1 Linearised equations and optimal growth Small amplitude coherent perturbations u = (u, v, w), p to a turbulent mean flow U = (U (y), 0, 0) described by the self-consistent analytic expression recently proposed

Transient growth of coherent streaks for control of turbulent flow separation 321

by Monkewitz et al. (2007) satisfy both the continuity ∇ · u = 0 and the linearised momentum equation: )] [ ( ∂u ∂u +U + (v ∂U /∂y, 0, 0) = −∇p + ∇ · νT (y) ∇u + ∇uT ∂t ∂x

(1)

In equation (1), νT (y) = ν + νt (y) is the total viscosity (i.e., the sum of the molecular viscosity and the eddy-viscosity, see Cossu et al. (2009) for details). In the following, uτ = (νdU /dy|wall )1/2 is the wall friction velocity, y + = yuτ /ν is the wall normal coordinate scaled in inner units, Ue + = Ue /uτ is the free-stream velocity Ue scaled with uτ , Reδ∗ = Ue δ ∗ /ν is the Reynolds number based on the displacement thickness δ ∗ , and η = y/∆ is the wall normal coordinate scaled with the Rotta-Clauser length scale ∆ = δ ∗ Ue + . The mean flow being homogeneous in the streamwise and spanwise directions, we b(α, y, β, t) ei(αx+βz) , where α and consider perturbations of the form u(x, y, z, t) = u β are the streamwise and spanwise wavenumbers, respectively. Standard manipulations (see Schmid and Henningson, 2001), generalised to include a variable eddy-viscosity (White, 2006), allow to rewrite the linearised system into the following generalised Orr-Sommerfeld and Squire equations for the normal velocity vb(y) and vorticity ω cy (y): [

] { } [ ]{ } D2 − k2 0 ∂ vb LOS 0 vb = 0 1 ∂t ω cy −iβU ′ LSQ ω cy

(2)

with [ ( ) ] LOS = −iα U D2 − k 2 − U ′′ ( 2 ) ( ) ( ) 2 +νT D − k 2 + 2νT′ D3 − k 2 D + νT′′ D2 + k 2

(3)

( ) LSQ = −iαU + νT D2 − k 2 + νT′ D

(4)

where D and (′ ) stand for ∂/∂y and k 2 = α2 + β 2 . Even though, the mean flow is linearly stable for all α and β, so that infinitesimal perturbations decay after enough time (Cossu et al., 2009), some disturbances may support large growth before decaying. The ratio ∥b u (t) ∥2 /∥b u0 ∥2 , where ∥∥ stands for the energy norm, quantifies the energy amplification of a perturbation as it evolves in b time. The temporal optimal growth G(α, β, t) = supub0 ∥b u (t) ∥2 /∥b u0 ∥2 is the maximum energy amplification of a disturbance optimised over all possible initial conditions u b0 . b In this study, we focus on the maximum optimal growth Gmax (α, β) = supt G(α, β, t) reached using the optimal initial conditions.

2.2 Optimal perturbations The computations of streamwise uniform optimals (α∆ = 0) are conducted for a wide range of Reynolds numbers Reδ∗ ranging from 103 to 6 104 . Figure 1(a) presents the gains obtained varying β∆ in the range already investigated for the selected Reynolds numbers. The double peak structure observed in turbulent channel flow case ´ by del Alamo and Jiménez (2006) and Pujals et al. (2009) is observed provided that the Reynolds number is large enough. The secondary peak seems independent of the Reynolds number and is shifted towards smaller values of β∆ as Reδ∗ increases.

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When replotted in wall units (not reported here, see Cossu et al., 2009), we find that this secondary peak is obtained for λz = 81.5y + for all the considered Reynolds numbers. The maximum energy growth corresponding to the primary peak increases with the Reynolds number Reδ∗ and is attained for spanwise wavenumbers in the range β∆ ∈ [1, 10] (λz /δ ∈ [3, 20]), the maximum being reached for β∆ ≈ 3 corresponding to λz /δ ≈ 8 (Cossu et al., 2009). Figure 1 (a) Maximum growth Gmax of streamwise uniform (α∆ = 0) optimal perturbations as a function of the spanwise wavenumber for the selected Reynolds numbers Reδ∗ (b) cross-stream view of the v -w component of the optimal initial vortices (arrows) and of the u component of the corresponding maximally amplified streak (lines) associated with the primary peak optimal (α∆ = 0, β∆ = 3) for Reδ∗ = 10, 000 plotted in outer units. Black lines represent high speed streaks while gray lines represent low speed streaks Reδ∗= 103 3 2 10 4 10 4 2 10 4 4 10 4 6 10

(a)

10

0.5

(b)

0.4

y/∆

Gmax

100

0.3 0.2 0.1 0

1 1

10

β∆

100

1000 10000

(a)

−1

0

1

z/∆ (b)

Source: Adapted from Cossu et al. (2009)

We present in Figure 1(b), the optimal initial conditions, along with their optimal responses, corresponding to the primary peak illustrated in Figure 1 for Reδ∗ = 10, 000. The initial disturbances (arrows) consist in counter rotating streamwise vortices which induce at time of maximum amplification streamwise streaks (black and gray contours). These optimal disturbances consist in very large-scale structures spreading the whole boundary layer, the optimal vortices being centered near the boundary layer edge where δ ≈ 0.224∆.

3 Experimental investigation of large-scale coherent streaks’ transient growth Naturally occurring very large-scale structures with λz ≥ 6δ have not been detected yet in turbulent boundary layers where the most energetic structures have a spanwise scale of rather λz ≈ δ. It could be that the larger optimal structures, even if largely potentially amplified by the mean lift-up, are not able to self-sustain in the boundary layer because they are not selected by the other mechanisms involved in the self-sustained process (see, e.g., Hamilton et al., 1995; Schoppa and Hussain, 2002). It could however, also be that, after all, no mean lift-up exists and that other mechanisms are responsible for the existence of very large-scale streaks.


An experimental study has therefore been set up in order to verify if the transient growth of coherent streaks can actually be experimentally observed in a turbulent flow. This is an important step aimed at validating or discarding the theoretical predictions described above. As a second objective, we want to verify to which extent the most amplified measured scales match the predictions of the linear optimal perturbation analyses. In order to have reproducible results we decided to artificially force large scale streaks in the turbulent boundary layer on a flat plate, our approach being similar to the one previously used to force moderate amplitude streaks in the laminar boundary layer (White, 2002; Fransson et al., 2004, 2005, 2006).

3.1 Description of the facility and measurements The measures have been conducted in the wind-tunnel facility of the technical centre of PSA Peugeot Citr¨ oen. The wind-tunnel is of closed-return type. The test section is 0.8m long with a cross sectional area of 0.3 m × 0.3 m. The temperature can be kept constant and uniform within ± 0.5◦ C. The contraction ratio is 8 and the velocity can be controlled from 7 m.s−1 up to 45 m.s−1 . The boundary layer develops on a flat plat fixed to the roof of the test section and that transition to turbulence is tripped using a strip of sandpaper. The velocity is measured using Dantec’s flow manager particle image velocimetry (PIV) system associated with a 120 mJ Nd:Yag double pulsed laser and a 1,024 × 1,280 Hisense Mk2 CCD camera placed above the test section (resulting in (x, z) planes). A 28 mm optical lens is used resulting in a 300 mm × 220 mm field. The laser sheet is 1 mm thick and, in order to ensure the convergence of the mean velocity fields, 600 pairs of images are acquired. All the data presented here are acquired at Y/k = 0.5 (corresponding to Y = 2 mm) from the wall. Figure 2

Shape of the roughness elements and parameters

3.2 Roughness elements Previous investigations have shown that setting the free-stream velocity to Ue = 20 m.s−1 , the boundary layer is turbulent. However, due to the small dimensions

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of the test section, the developing boundary layer is quite thin: the boundary layer is δ0 = 5.4 mm thick at x0 = 110 mm. The resulting Reynolds number Reδ∗ = Ue δ ∗ /ν is Reδ∗ ≈ 1,000. According to the linear stability analysis, this value is large enough to see the outer peak (see Figure 1). To introduce streamwise vorticity in the boundary layer, we use cylindrical PVC roughness elements which dimensions are of great importance if we want to generate stable streaks (see Figure 2). In this study, we keep the height of the roughness elements constant equal to k = 4 mm (k/δ0 = 0.8), as well as the ratio λz /d = 4 (the same ratio was used in Fransson et al. (2005, 2006) and Hollands and Cossu (2009). Several spacings λz spanning the width of the primary peak discussed in Section 2.2 are tested. The selected spacings are nearby λz /δ0 = 3, 5, 6 ,7.5, 10 and 12 (see Table 1). Table 1 Description of the configurations used for the flat plate study Config. A B C D E F

λz (mm)

d (mm)

λz /d

λz /δ0

k/δ0

15.8 26.8 33 40. 50.8 65.6

3.94 6.7 8.25 10. 12.7 16.4

4 4 4 4 4 4

3 5 6 7.5 10 12

0.8 0.8 0.8 0.8 0.8 0.8

Source: From Pujals et al. (2010a)

3.3 Large-scale coherent streaks Figure 3(a) shows a visualisation of the flow measured downstream of the λz = 6δ0 cylinder array. The flow is from left to right and the cylinders (represented as white disks on the left part of the figure) are located at z/λz = ± 0.5, ± 1.5 and ± 2.5. The streamwise velocity scaled on the free-stream velocity Ue is plotted versus the streamwise and spanwise directions both scaled on the roughness’ spacing λz . The upper part of the figure (i.e., z/λz > 0) represents an instantaneous velocity field. Owing to the roughness elements, the flow is clearly spanwise modulated and an alternating pattern of high speed (clear contours) and low speed (dark contours) streaks is observed. The lower part (z/λz ≤ 0) displays the time-averaged velocity field. As expected, the time-averaging permits to filter all the random small-scales and then the artificially forced streaks appear to be statistically coherent (i.e., the velocity modulation induced by the streaks has a non-zero mean value) and steady. In accordance with White (2002) and Fransson et al. (2005), it appears that the high speed streaks are developing straight behind the cylinders. This last result proves that lift-up effect can exist in turbulent shear flows and, in the present case, can be used to modify the mean flow at leading order. Like in the laminar case, in the present experimental framework the transient growth of the streaks is expected to occur in the streamwise direction. A measure of the amplitude of the streaks must be defined in order to quantify this transient growth. Various definitions are available in the literature like the energy amplification used in linear stability analyses or the min-max criterion expressing the peak-to-peak difference between the velocities in the high and low speed streaks (Andersson et al., 2001). Here, following Hollands and Cossu (2009), we estimate the amplitude of the streaks introducing a ’local’ min-max criterion which is defined as:


bst (x, Y /k) = U (x, Y /k, zhsst ) − U (x, Y /k, zlsst ) A 2Ue

(5)

where zhsst denotes the spanwise location of high speed streaks (i.e., z/λz = ±0.5, ±1.5, ±2.5) and zlsst denotes the location of the neighbouring low speed streaks (i.e., z/λz = 0, ±1, ±2). A more accurate value is then obtained performing a sliding averaging over the whole spanwise window. The so-defined amplitude maximises the velocity difference in the spanwise direction only for the given height Y of the plane where the velocity is measured. This measure therefore represents only a lower bound on the amplitude defined by Andersson et al. (2001) where the min-max is found in both z and y. This approximation has however already proven reasonable when compared to other similar definitions (see, e.g., Hollands and Cossu, 2009) and is sufficient to prove the existence of transient growth. (a) Very large-scale coherent streaks forced in a turbulent boundary layer in the plane situated at Y /k = 0.5 from the wall. The spanwise wavelength used here is λz = 6δ0 (configuration C, see Table 1). Flow is from left to right, the white disks on the left part of the figure indicate the position of the roughness elements. Both, the streamwise x and spanwise directions z are scaled with λz . The upper part (i.e., z/λz ≥ 0.) of the figure displays the instantaneous velocity field while the lower part (z/λz < 0) shows the time-averaged velocity U /Ue computed using 600 PIV fields (b) streamwise evolution of bst (x/λz , Y /k) of the streaks estimated with equation (5) the finite amplitude A

(a)

z/λz

0.8

U/Ue

0.9

3 2 1 0 -1 -2 -3

0.7 0

2

4 x/λz

6

8

(a)

Ast(x/λz,Y/k=0.5)

(b)

0.14 0.12 0.1 0.08 0.06 0.04

^

Figure 3

0.02 0 1

2

3

4

5 x/λz

(b)

Source: Adapted from Pujals et al. (2010a)

6

7

8

326 Figure 4

G. Pujals et al. bmax Black triangles: experimentally measured maximum streak amplitude A scaled with st max b the global maximum amplitude max(Ast ) versus the wavenumber β scaled with the boundary layer thickness δ0 ; Solid line: linear optimal growth Gmax data normalised on the global maximum and computed for streamwise uniform perturbations and the same mean base flow

Gmax/max(Gmax)

^ max ^ max Ast /max(Ast )

1 0.8 0.6 0.4 0.2 0 1

10

100

βδ0 Source: Adapted from Pujals et al. (2010a)

bst (x, Y /k = 0.5) of the amplitude In Figure 3(b), we report the streamwise evolution A of the coherent (averaged) streaks obtained with configuration C. This plot confirms what was already discernible on Figure(a): the amplitude of the streaks grows in the downstream direction until a maximum value is reached, near x − x0 ≈ 4λz and then decays. The maximum amplitude is nearby 13% of the free-stream velocity, a value similar to the largest amplitude of stable streaks experimentally forced in the laminar boundary layer (Fransson et al., 2004, 2005, 2006).

3.4 Influence of the spanwise wavelength on the streaks amplitude The influence of the spanwise spacing on the streaks amplitude is analysed by repeating the measures for each configuration described in Table 1. For each spacing from λz = 3δ0 to λz = 12δ0 , large-scale coherent streaks similar to the ones presented in Figure 3 are observed and their growth has been calculated using equation (5). Those bmax results are reported in Figure 4: the maximum amplitude A obtained for each st max b configuration is scaled with the global maximum amplitude max(Ast ) and reported as black triangles versus βδ0 . A maximum of the amplitude is observed for λz ≈ 6δ0 (Case C). This value is well in the range of the theoretical predictions. We therefore report with a solid line in Figure 4 the optimal growth Gmax data displayed on Figure 1 normalised with the global maximum. We remind that these optimal growth data have been obtained for streamwise uniform perturbations for the turbulent mean flow profile of Monkewitz et al. (2007) at the same Reynolds number Reδ∗ ≈ 1,000. The optimal growth data from the linear stability analysis and the experimental data agree reasonably bst is an amplitude while well even though the two compared quantities are different: A Gmax stands for an energy amplification. The implicit assumption made is that the initial amplitude of the vortices generated by the roughness elements does not change much with λz . The important point is however that both the optimal growth analysis and the experimental data indicate that the maximum amplification of the coherent


streaks by the mean lift-up effect is obtained for spanwise wavelengths of λz ≈ 6δ0 at this Reynolds number and that neighbouring very large-scale spacings λz also lead to noticeable growth.

4 Application to separation control: Ahmed body In the laminar Blasius boundary layer, nearly optimal streaks were successfully used to stabilise Tollmien-Schlichting waves and then delay transition to turbulence (Fransson et al., 2005, 2006). Due to their strong amplification, the forced streaks modify the mean velocity profiles at leading order. The spanwise averaged shape factor of the resulting velocity profile is lower than the unforced case implying a fuller, and then more resistant, mean velocity profile. In the present turbulent case, similar modification of the mean velocity profiles is to be expected since we have observed a strong modulation of the mean velocity field in presence of the streaks. Such modification could result in a turbulent mean velocity profile with a lower shape factor being consequently more resistant to adverse pressure gradients. To find out if it is the case or not, a new experimental study has been carried in which large-scale coherent streaks are forced in presence of an adverse pressure gradient due to geometrical constraints. The generic car model used here is the one originally described in Ahmed et al. (1984) or Lienhart et al. (2003). The rear part of this model consists in a slanted surface which slant-angle can be modified resulting in very different flow topologies. The dimensions and the overall shape of the model are given in Figure 5. The axis system origin is taken at mid-width on the separation line between the roof and the slanted-surface of the model. This experimental campaign was carried in a different PSA in-house facility which is an Eiffel type wind tunnel with a rectangular cross section of 2.1 m high, 5.2 m wide and 6 m long. Velocity can be controlled from 5 m.s−1 up to 55 m.s−1 . Models are fixed on a flat plate placed 500 mm above the floor in order to control the boundary layer development without any suction device. This flat plate is 3 m wide and 52 mm thick. The leading edge is covered with sand-paper to avoid separation in low flow-speed conditions and to promote transition to a turbulent boundary layer. In the present study, we focus on the φ = 25o slanted rear-end, the free-stream velocity being set to Ue =20 m.s−1 and Ue = 40 m.s−1 ; this later velocity being the most commonly studied. This choice of geometry and free-stream velocities leads to an unsteady and three-dimensional flow exhibiting a large separation bubble over the slanted surface along with highly energetic streamwise vortices issuing from the slant side edges. The turbulent boundary layer developing on the roof of the model upstream of the separation line is δ020 ≈ 20 mm thick for Ue = 20 m.s−1 and δ040 ≈ 12 mm thick for Ue = 40 m.s−1 . The Reynolds numbers based on the free-stream velocity Ue , the model length L and the molecular viscosity ν are respectively ReL = 1.35 · 106 and ReL = 2.7 · 106 . The roughness elements used to force the streaks are designed according to the set of criterions described in Section 3.2. However, owing to the thicker boundary layer, their dimensions are much larger: the height k is fixed to k =12 mm (≈ 0.6δ020 and 1δ040 ) while the ratio λz /d = 4 is kept. In the present paper, two spanwise spacing are investigated for both Reynolds numbers ReL : λz = 24 mm (i.e., d = 6 mm) and λz = 48 mm (d = 12 mm). The location of the roughness arrays, denoted x0 , is

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between 4λz and 5λz upstream of the separation line. Table 2 summarises the main parameters used in Pujals et al. (2010b). In the following, all the measurements are carried with the equipment described in Section 3.1. Figure 5

Shape and dimensions of the generic car model, the dimensions are given in millimeters

Source: From Ahmed et al. (1984)

Table 2 Description of the configurations used for the Ahmed body study Config. A20 A40 B20 B40

Ue (m.s−1 )

ReL

λz (mm)

d (mm)

x0 (mm)

λz /d

λz /δ0

k/δ0

x0 /λz

20 40 20 40

1.35·106 2.7·106 1.35·106 2.7·106

24. 24. 48. 48.

6. 6. 12. 12.

–120. –120. –192. –192.

4 4 4 4

1.2 2. 2.4 4.

0.6 1. 0.6 1.

–5 –5 –4 –4

4.1 Streaks developing on the roof Figure 6 shows a visualisation of the time-averaged flow over the roof measured downstream of the cylinders array at height Y /k = 0.5 (Y = 6mm). The flow is from left to right and the cylinders (represented by the white circles) are located on the left part of the figures. The velocity is scaled on the free-stream velocity Ue and plotted versus the spanwise and streamwise directions both scaled on the spacing λz . The data reported here are those obtained with configurations B20 [Figure 6(a)] and B40 [Figure 6(b)]. In both cases, the mean flow is spanwise modulated like in the flat plate case [see Figure 3(b)]. The streaks are observable around x = 1λz downstream of the cylinders array up to the separation line at x = 0λz . Additional experiments with configurations A20 and A40 show similar behaviour. bst (˜ The amplitude A x, Y /k = 0.5) of the streaks has been estimated using the definition of equation (5) and is displayed in Figure 7 for each case reported in Table 2. In spite of the adverse pressure gradient and the modification of the Reynolds number, all the streaks forced here experience an algebraic growth period with a maximum observed amplitude between 9% and 15% of Ue depending on the Reynolds number and the spanwise spacing. This maximum amplitude is reached between 2. and 3. times


the spacing downstream of the array. This behaviour is, however, slightly different from the flat plate case [see Figure 3(b)]; this is probably due to the adverse pressure gradient and the vicinity of the separation line. Figure 6

Large coherent velocity streaks forced on the roof of the model at Y /k = 0.5 (Y = 6 mm) from the wall. The flow is from left to right. Time-averaged streamwise velocity U /Ue as a function of the streamwise and spanwise directions x/λz and z/λz . On the lower horizontal axis x stands for the streamwise position in the axis system described in Section 4. On the upper horizontal axis x ˜ denotes the streamwise distance from the roughness array (a) velocity field for configuration B20 of Table 2 (b) velocity field for configuration B40 of Table 2 ~

x /λz

(a) 0

1

2

3

4 1.2

2 1.1 1.0 0 0.9

U/Ue

z/λz

1

-1 0.8 -2 0.7 -4

-3

-2 x/λz

-1

0

3

4

(a) ~

x /λz

(b) 0

1

2

1.2 2 1.1 1.0 0 0.9

U/Ue

z/λz

1

-1 0.8 -2 0.7 -4

-3

-2 x/λz

-1

0

(b)

bmax attained for each configuration are reported in Table 3. The maximum amplitude A st bmax These results highlight some tendencies relative to the dependence of A on ReL st and λz . Hence, the maximum amplitude seems to increase with the spanwise spacing (cases A20/B20 and A40/B40) and the Reynolds number (cases A20/A40 and B20/B40)

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in agreement with both the predictions of the linear stability theory and the experiments conducted on the flat plate boundary layer. This last result proves that this control strategy, even if resulting on a passive open-loop approach based on a fixed spanwise spacing, could be efficient for a wide range of Reynolds numbers provided that streaks could effectively modify the flow downstream of the separation line. In order to investigate such ability, PIV measurements around the rear-end have been carried in (x, y) plane along the symmetry line of the model. Figure 7

bst (x, Y /k = 0.5) as a function of the distance Estimated streaks’ finite amplitude A from the cylinders array scaled with the spanwise spacing x ˜/λz

0.2

Config. A20 A40 B20 B40

^

Ast

0.15

0.1

0.05

0 1

1.5

2

2.5

3

3.5

4

4.5

~

x/λz Table 3 Amplitude of coherent streaks issuing from configurations listed in Table 2 Config. A20 A40 B20 B40

Ue (m.s−1 )

ReL

λz /δ0

bmax A (%Ue ) st

20 40 20 40

1.35·106 2.7·106 1.35·106 2.7·106

1.2 2. 2.4 4.

9 12 11 15

4.2 Mean flow organisation around the rear-end In Figure 8, the time-averaged streamwise velocity U /Ue is plotted for the uncontrolled case [Figure 8(a) and 8(c)] and for the controlled case [Figure 8(b) and 8(d)] for the two Reynolds numbers studied in the present paper. The flow is from left to right. The uncontrolled flows present a thick shear layer originating from the upper side of the slanted surface. The curvature of the whole shear layer and the dark spot between the surface and the layer indicate a re-circulation bubble. When control is applied, the shear layer becomes thinner and its curvature is highly reduced. This behaviour suggests that the re-circulation bubble has been suppressed or at least highly reduced. To confirm this, the near-wall PIV measures on (x, z) planes are carried out at Y /k = 0.08 (corresponding to Y = 1 mm) over the slanted surface. The resulting streamwise velocity fields (coloured contours) as well as the time-averaged streamlines (vectors field) are reported in Figure 9 for the two cases discussed previously. The flow


is from top to bottom. The uncontrolled flow [Figure 9(a) and 9(c)] presents the common features observed in Lienhart et al. (2003). For ReL = 1.35 · 106 , the re-circulation bubble is clearly apparent since a backward flow region is observed for roughly 0.01 m ≤ x ≤ 0.15 m along the symmetry line z =0 m while it is only suggested by the slight curvature of the constant-velocity lines for the ReL = 2.7 · 106 case. The footprint of one streamwise conical vortex is also observed on the right part of Figure 9(a) and 9(c). The controlled flow displayed on both Figure 9(b) and 9(d) confirms the previous suggestion: the re-circulation bubble is suppressed since the streamlines present straight trajectories with no evidence of backward flow. Nevertheless, no noticeable influence on the streamwise vortices can be observed at this distance from the wall. Time-averaged streamwise velocity U /Ue around the rear-end in the symmetry plane using PIV. The flow is from left to right and configurations B20 and B40 are presented [(a) and (b)] ReL = 1.35 · 106 (c,d) ReL =2.7·106 [(a) and (c)] uncontrolled case [(b) and (d)] controlled case. Contour lines of constant streamwise velocity (black solid lines) are added to highlight the curvature of the shear layer

Figure 8

(b)

-0.05

0.8

0 y (m)

0.6

1

0.05

0.8

0

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0

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0

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(c)

0

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0 U/Ue

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(d)

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0 0

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(c)

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U/Ue

1

0.05 y (m)

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G. Pujals et al.

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(b)

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0

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U/Ue

(a)

Time-averaged streamwise velocity Y /k = 0.08 above the slanted surface using near-wall PIV for [(a) and (b)] ReL = 1.35 · 106 and [(c) and (d)] ReL = 2.7·106 . The line x = 0 is the begining of the slanted surface. The flow is from top to bottom [(a) and (c)] uncontrolled case [(b) and (d)] controlled case with respectively configuration B20 and B40

U/Ue

Figure 9

-0.2 0

0.06

0.12 z (m)

0.18

(d)

5 Conclusions and discussion The present paper summarises the main results of the studies aiming at using nearly optimal coherent perturbations supported by turbulent wall flows to perform flow control. Linear optimal growth analysis of turbulent boundary layer flow has lead to the conclusion that the most amplified perturbations are large-scale streamwise uniform vortices evolving into large-scale coherent streamwise streaks through the lift-up effect. A wide range of spanwise wavelengths ranging from 3δ to 20δ can be largely amplified with an optimal wavelength near 8δ. In some recent studies, evidences of very large-scale streaks present in the outer region of turbulent shear flows (see, e.g., Tomkins and Adrian, 2003; Hutchins and ´ Marusic, 2007; del Alamo and Jiménez, 2003) have been reported. These streaks, being of finite streamwise extension, have a typical spanwise wavelength λz ∼ δ. However, to our best knowledge, there is no experimental or numerical observation of large-scale structures with λz ≈ 4 ∼ 8 δ. In the core region of turbulent Couette flow, large-scale streamwise vortices and streaks are known to exist for a long time (Komminaho et al., 1996). In a recent experimental study, Kitoh and Umeki (2008) confirmed that these very large-scale streaks can be artificially forced in turbulent


Couette flow. Consequently, a first experimental study has been conducted to verify if artificially forced turbulent large-scale coherent streaks could sustain and be amplified in a flat plate turbulent boundary layer in the spirit of previous works by Fransson et al. (2005). Doing so, we have found that very large-scale coherent structures with a spanwise spacing 3δ0 < λz < 12δ0 can be forced in a low Reynolds number turbulent boundary layer. Those streaks experience a transient growth in space owing to the lift-up effect, similarly to the growth observed for streaks forced in the same way in a laminar boundary layer and modify the mean flow at leading order. They can reach similar finite moderate amplitudes ≈ 13% Ue . In agreement with results from the linear optimal perturbations theory, for this Reynolds number, the most amplified wavelength is ≈ 6δ0 at Reδ∗ ≈ 1,000. Eventually, the potential of a separation control strategy based on the growth of large-scale coherent streaks has been investigated. Streaks are generated on the roof of a 25o slant angle Ahmed body for two Reynolds numbers: ReL = 1.35·106 and ReL = 2.7·106 . In spite of the adverse pressure gradient, near wall PIV measurements have shown that the streaks can develop and experience growth. The resulting streaky base flow is responsible for the modification of the flow organisation around the rear end of the model: when controlled with large-scale streaks, the re-circulation bubble is completely suppressed. It is important to emphasise the differences between the technique proposed in the present paper and other passive control strategies based on VGs (Betterton et al., 2000; Lögdberg, 2006) or wake disrupters (Park et al., 2007) in which the separation delay is attributed to vortices that work by pairs to create a vertical mixing on the bubble edge (Godard and Stanislas, 2006). In the present study, the separation delay must be attributed only to the streaks (the spanwise modulation of the streamwise velocity) because streamwise vortices have virtually disappeared at the position of maximum streak amplitude, which is a typical feature of the lift-up effect. Previous studies showed that the major part of the drag force applied to Ahmed body is due to pressure drag at the rear-end. Thus, changes in the flow organisation such as those described in Section 4.2 should result in substantial drag reduction. In order to investigate this last point, we measured the drag coefficient Cd of the model when control is applied and compared it to the uncontrolled cases. The estimated drag reductions are reported in Table 4. One salient feature is that the larger drag reductions are obtained for ReL = 1.35 · 106 . Since near-wall PIV measurements over the slanted surface showed that the re-circulation bubble is suppressed with each configuration, this can be explained by the length of the re-circulation bubble which is larger for ReL = 1.35 · 106 than for ReL = 2.7 · 106 [see Figures 9(a) and (c)]. We can also observe that for a fixed ReL , the total drag reduction decreases with λz in spite of the larger amplitude of the resulting streaks. One possible explanation is that the bigger roughness elements used in, for instance, configuration B20 induce a larger drag penalty than those used for configuration A20. For wind-tunnel availability reasons, we could not investigate the minimum streaks amplitude required to suppress the re-circulation bubble. This type of study would probably lead to a improved total drag reduction owing to even better shaped roughness elements.

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Table 4 Drag reduction obtained with configurations listed in Table 2 Config. A20 A40 B20 B40

ReL

λz /δ0

bmax A (%Ue ) st

∆Cd (%)

1.35·106 2.7·106 1.35·106 2.7·106

1.2 2. 2.4 4.

9. 12. 11. 15.

–10. –2.5 –6. –2.

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List of symbols bst A Cd d k L Reδ∗ ReL U Ue Ue + u uτ v w x x0 y y+ z α β δ δ0 δ∗ ∆ λz ν νt νT ωy

finite amplitude of the streaks drag coefficient diameter of the roughness elements height of the roughness elements length of Ahmed body model Reynolds number Reδ∗ = Ue δ ∗ /ν Reynolds number ReL = Ue L/ν streamwise velocity base flow free-stream velocity free-stream velocity scaled in wall-units Ue + = Ue /uτ streamwise velocity of a perturbation friction velocity normal velocity of a perturbation spanwise velocity of a perturbation streamwise direction/position streamwise position of roughness arrays normal direction/position normal position expressed in wall-units y + = yuτ /ν spanwise direction/position streamwise wavenumber of a perturbation spanwise wavenumber of a perturbation boundary layer thickness boundary layer thickness where roughness arrays are located (flat-plate case) displacement thickness Rotta-Clause thickness spanwise wavelength of a perturbation/ spacing of the roughness elements molecular viscosity eddy viscosity total viscosity, νT = ν + νt normal vorticity

m m m m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m m m m m−1 m−1 m m m m m m2 .s−1 m2 .s−1 m2 .s−1 s−1