transient growth of coherent streaks for control of ... · reference to this paper should be made...

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

Copyright © 2011 Inderscience Enterprises Ltd.

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.

Transient growth of coherent streaks for control of turbulent flow separation 319

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

320 G. Pujals et al.

non-normal nature of the linearised Navier-Stokes operator (Trefethen et al., 1993).When the shape and the wavelength of the disturbing vortices are optimised, the energyamplification 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 andstable streaks of moderate amplitude have been forced by nearly-optimal streamwisevortices generated with cylindrical roughness elements. The resulting streaky flow canbe used to manipulate laminar shear flows at leading order and applied to stabiliseTollmien-Schlichting waves (Cossu and Brandt, 2002; Fransson et al., 2004) and todelay 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 a2D backward-facing ramp in the laminar régime.

An interesting extension of such approach would consist in the manipulation ofturbulent boundary layers with optimal or nearly optimal vortices and streaks. To doso, some issues had to be addressed first: what are the shape and scales of suchdisturbances? 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 baseflow 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 conductedwith 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 azero pressure gradient turbulent boundary layer. The second section describes the resultsof an experimental study in which nearly optimal perturbations are artificially forced in aflat plate turbulent boundary layer by means of spanwise organised roughness elements.The third part investigates the ability of these perturbations to delay separation on therear-end of a generic car model (Ahmed body, see Ahmed et al., 1984). All those resultsare 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 Reynoldsand Hussain (1972) who considered the Reynolds-averaged Navier-Stokes equationslinearised around a turbulent mean flow. These equations rule the dynamics of smallamplitude statistically coherent perturbations (i.e., non-zero mean value) while thedynamics of random fluctuations (i.e., statistically uncorrelated) is modeled by meansof the turbulent eddy-viscosity in equilibrium with the mean flow. This modelling hasbeen used by del Álamo and Jiménez (2006), Pujals et al. (2009) and Hwang and Cossu(2010) to compute the optimal perturbations sustained by turbulent plane Poiseuilleand plane Couette flows. Here, we focus on the optimal perturbations supported by aturbulent 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 flowU = (U(y), 0, 0) described by the self-consistent analytic expression recently proposed


by Monkewitz et al. (2007) satisfy both the continuity ∇ · u = 0 and the linearisedmomentum equation:

∂u∂t

+ U∂u∂x

+ (v ∂U/∂y, 0, 0) = −∇p+∇ ·[νT (y)

(∇u+∇uT

)](1)

In equation (1), νT (y) = ν + νt(y) is the total viscosity (i.e., the sum of the molecularviscosity 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 normalcoordinate scaled in inner units, Ue+ = Ue/uτ is the free-stream velocity Ue scaled withuτ , Reδ∗ = Ueδ∗/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

consider perturbations of the form u(x, y, z, t) = û(α, y, β, t) ei(αx+βz) , where α andβ 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 generalisedOrr-Sommerfeld and Squire equations for the normal velocity v̂(y) and vorticity ω̂y (y):[

D2 − k2 00 1

]∂

∂t

{v̂ω̂y

}=

[LOS 0−iβU ′ LSQ

]{v̂ω̂y

}(2)

with

LOS = −iα[U(D2 − k2

)− U ′′

]+νT

(D2 − k2

)2+ 2ν′T

(D3 − k2D

)+ ν′′T

(D2 + k2

) (3)LSQ = −iαU + νT

(D2 − k2

)+ ν′TD (4)

where D and (′) stand for ∂/∂y and k2 = α2 + β2.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 maysupport large growth before decaying. The ratio ∥û (t) ∥2/∥û0∥2, where ∥∥ stands forthe energy norm, quantifies the energy amplification of a perturbation as it evolves intime. The temporal optimal growth Ĝ(α, β, t) = supû0 ∥û (t) ∥

2/∥û0∥2 is the maximumenergy amplification of a disturbance optimised over all possible initial conditions û0.In this study, we focus on the maximum optimal growth Gmax(α, β) = supt Ĝ(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 widerange of Reynolds numbers Reδ∗ ranging from 103 to 6 104. Figure 1(a) presentsthe gains obtained varying β∆ in the range already investigated for the selectedReynolds numbers. The double peak structure observed in turbulent channel flow caseby del Álamo and Jiménez (2006) and Pujals et al. (2009) is observed provided thatthe Reynolds number is large enough. The secondary peak seems independent of theReynolds number and is shifted towards smaller values of β∆ as Reδ∗ increases.


When replotted in wall units (not reported here, see Cossu et al., 2009), we findthat this secondary peak is obtained for λz = 81.5y+ for all the considered Reynoldsnumbers. The maximum energy growth corresponding to the primary peak increaseswith the Reynolds number Reδ∗ and is attained for spanwise wavenumbers in the rangeβ∆ ∈ [1, 10] (λz/δ ∈ [3, 20]), the maximum being reached for β∆ ≈ 3 correspondingto λz/δ ≈ 8 (Cossu et al., 2009).

Figure 1 (a) Maximum growth Gmax of streamwise uniform (α∆ = 0) optimal perturbations as afunction 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) andof the u component of the corresponding maximally amplified streak (lines) associatedwith the primary peak optimal (α∆ = 0, β∆ = 3) for Reδ∗ = 10, 000 plotted in outerunits. Black lines represent high speed streaks while gray lines represent low speedstreaks

1

10

100

1 10 100 1000 10000

Gm

ax

β ∆

(a) Reδ∗ = 103

2 103

104

2 104

4 104

6 104

(a)

−1 0 10

0.1

0.2

0.3

0.4

0.5

z/∆

y/∆

(b)

(b)

Source: Adapted from Cossu et al. (2009)

We present in Figure 1(b), the optimal initial conditions, along with their optimalresponses, corresponding to the primary peak illustrated in Figure 1 for Reδ∗ = 10, 000.The initial disturbances (arrows) consist in counter rotating streamwise vortices whichinduce at time of maximum amplification streamwise streaks (black and gray contours).These optimal disturbances consist in very large-scale structures spreading the wholeboundary layer, the optimal vortices being centered near the boundary layer edge whereδ ≈ 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 yetin turbulent boundary layers where the most energetic structures have a spanwise scaleof rather λz ≈ δ. It could be that the larger optimal structures, even if largely potentiallyamplified by the mean lift-up, are not able to self-sustain in the boundary layer becausethey 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 bethat, after all, no mean lift-up exists and that other mechanisms are responsible for theexistence of very large-scale streaks.


An experimental study has therefore been set up in order to verify if the transientgrowth of coherent streaks can actually be experimentally observed in a turbulent flow.This is an important step aimed at validating or discarding the theoretical predictionsdescribed above. As a second objective, we want to verify to which extent the mostamplified measured scales match the predictions of the linear optimal perturbationanalyses. In order to have reproducible results we decided to artificially force large scalestreaks in the turbulent boundary layer on a flat plate, our approach being similar to theone 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 centreof PSA Peugeot Citröen. The wind-tunnel is of closed-return type. The test section is0.8m long with a cross sectional area of 0.3 m × 0.3 m. The temperature can be keptconstant and uniform within ± 0.5◦C. The contraction ratio is 8 and the velocity canbe controlled from 7 m.s−1 up to 45 m.s−1. The boundary layer develops on a flat platfixed to the roof of the test section and that transition to turbulence is tripped using astrip 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 × 1,280Hisense Mk2 CCD camera placed above the test section (resulting in (x, z) planes). A28 mm optical lens is used resulting in a 300 mm × 220 mm field. The laser sheetis 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 toUe = 20 m.s−1, the boundary layer is turbulent. However, due to the small dimensions


of the test section, the developing boundary layer is quite thin: the boundary layer isδ0 = 5.4 mm thick at x0 = 110 mm. The resulting Reynolds number Reδ∗ = Ueδ∗/ν isReδ∗ ≈ 1,000. According to the linear stability analysis, this value is large enough tosee the outer peak (see Figure 1).

To introduce streamwise vorticity in the boundary layer, we use cylindrical PVCroughness elements which dimensions are of great importance if we want to generatestable streaks (see Figure 2). In this study, we keep the height of the roughness elementsconstant equal to k = 4 mm (k/δ0 = 0.8), as well as the ratio λz/d = 4 (the sameratio was used in Fransson et al. (2005, 2006) and Hollands and Cossu (2009). Severalspacings 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. λz (mm) d (mm) λz/d λz/δ0 k/δ0A 15.8 3.94 4 3 0.8B 26.8 6.7 4 5 0.8C 33 8.25 4 6 0.8D 40. 10. 4 7.5 0.8E 50.8 12.7 4 10 0.8F 65.6 16.4 4 12 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 = 6δ0cylinder array. The flow is from left to right and the cylinders (represented as whitedisks 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 thestreamwise and spanwise directions both scaled on the roughness’ spacing λz . The upperpart of the figure (i.e., z/λz > 0) represents an instantaneous velocity field. Owingto the roughness elements, the flow is clearly spanwise modulated and an alternatingpattern 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, thetime-averaging permits to filter all the random small-scales and then the artificiallyforced streaks appear to be statistically coherent (i.e., the velocity modulation inducedby 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 straightbehind the cylinders. This last result proves that lift-up effect can exist in turbulent shearflows 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 growthof the streaks is expected to occur in the streamwise direction. A measure of theamplitude 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 inlinear stability analyses or the min-max criterion expressing the peak-to-peak differencebetween 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 streaksintroducing a ’local’ min-max criterion which is defined as:


Âst (x, Y /k) =U(x, Y /k, zhsst)− U(x, Y /k, zlsst)

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 slidingaveraging over the whole spanwise window. The so-defined amplitude maximises thevelocity difference in the spanwise direction only for the given height Y of the planewhere the velocity is measured. This measure therefore represents only a lower boundon the amplitude defined by Andersson et al. (2001) where the min-max is found in bothz and y. This approximation has however already proven reasonable when compared toother similar definitions (see, e.g., Hollands and Cossu, 2009) and is sufficient to provethe existence of transient growth.

Figure 3 (a) Very large-scale coherent streaks forced in a turbulent boundary layer in the planesituated 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 partof the figure indicate the position of the roughness elements. Both, the streamwise x andspanwise directions z are scaled with λz . The upper part (i.e., z/λz ≥ 0.) of the figuredisplays the instantaneous velocity field while the lower part (z/λz < 0) shows thetime-averaged velocity U/Ue computed using 600 PIV fields (b) streamwise evolution ofthe finite amplitude Âst (x/λz, Y /k) of the streaks estimated with equation (5)

(a)

0 2 4 6 8

x/λz

-3

-2

-1

0

1

2

3

z/λ

z

0.7

0.8

0.9

U/U

e

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

8 7 6 5 4 3 2 1

x/λz

^ Ast

(x/λ

z,Y

/k=

0.5

)

(b)

(b)

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


Figure 4 Black triangles: experimentally measured maximum streak amplitude Âmaxst scaled withthe global maximum amplitude max(Âmaxst ) versus the wavenumber β scaled with theboundary layer thickness δ0; Solid line: linear optimal growth Gmax data normalised onthe global maximum and computed for streamwise uniform perturbations and the samemean base flow

0

0.2

0.4

0.6

0.8

1

1 10 100

^ Ast

max

/max

(^ Ast

max

)

Gm

ax/m

ax(G

max

)

βδ0Source: Adapted from Pujals et al. (2010a)

In Figure 3(b), we report the streamwise evolution Âst (x, Y /k = 0.5) of the amplitudeof the coherent (averaged) streaks obtained with configuration C. This plot confirmswhat was already discernible on Figure(a): the amplitude of the streaks grows in thedownstream direction until a maximum value is reached, near x− x0 ≈ 4λz and thendecays. The maximum amplitude is nearby 13% of the free-stream velocity, a valuesimilar to the largest amplitude of stable streaks experimentally forced in the laminarboundary 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 repeatingthe measures for each configuration described in Table 1. For each spacing fromλz = 3δ0 to λz = 12δ0, large-scale coherent streaks similar to the ones presented inFigure 3 are observed and their growth has been calculated using equation (5). Thoseresults are reported in Figure 4: the maximum amplitude Âmaxst obtained for eachconfiguration is scaled with the global maximum amplitude max(Âmaxst ) and reportedas 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 thereforereport with a solid line in Figure 4 the optimal growth Gmax data displayed on Figure 1normalised with the global maximum. We remind that these optimal growth data havebeen obtained for streamwise uniform perturbations for the turbulent mean flow profileof Monkewitz et al. (2007) at the same Reynolds number Reδ∗ ≈ 1,000. The optimalgrowth data from the linear stability analysis and the experimental data agree reasonablywell even though the two compared quantities are different: Âst is an amplitude whileGmax stands for an energy amplification. The implicit assumption made is that theinitial amplitude of the vortices generated by the roughness elements does not changemuch with λz . The important point is however that both the optimal growth analysisand 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 atthis Reynolds number and that neighbouring very large-scale spacings λz also lead tonoticeable growth.

4 Application to separation control: Ahmed body

In the laminar Blasius boundary layer, nearly optimal streaks were successfully used tostabilise Tollmien-Schlichting waves and then delay transition to turbulence (Franssonet al., 2005, 2006). Due to their strong amplification, the forced streaks modify the meanvelocity profiles at leading order. The spanwise averaged shape factor of the resultingvelocity profile is lower than the unforced case implying a fuller, and then moreresistant, mean velocity profile. In the present turbulent case, similar modification of themean velocity profiles is to be expected since we have observed a strong modulationof the mean velocity field in presence of the streaks. Such modification could result ina turbulent mean velocity profile with a lower shape factor being consequently moreresistant to adverse pressure gradients. To find out if it is the case or not, a newexperimental study has been carried in which large-scale coherent streaks are forced inpresence 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 slantedsurface 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 axissystem origin is taken at mid-width on the separation line between the roof and theslanted-surface of the model. This experimental campaign was carried in a different PSAin-house facility which is an Eiffel type wind tunnel with a rectangular cross sectionof 2.1 m high, 5.2 m wide and 6 m long. Velocity can be controlled from 5 m.s−1up to 55 m.s−1. Models are fixed on a flat plate placed 500 mm above the floor inorder to control the boundary layer development without any suction device. This flatplate is 3 m wide and 52 mm thick. The leading edge is covered with sand-paper toavoid separation in low flow-speed conditions and to promote transition to a turbulentboundary layer.

In the present study, we focus on the φ = 25o slanted rear-end, the free-streamvelocity being set to Ue =20 m.s−1 and Ue = 40 m.s−1; this later velocity being themost commonly studied. This choice of geometry and free-stream velocities leads toan unsteady and three-dimensional flow exhibiting a large separation bubble over theslanted surface along with highly energetic streamwise vortices issuing from the slantside edges. The turbulent boundary layer developing on the roof of the model upstreamof the separation line is δ200 ≈ 20 mm thick for Ue = 20 m.s−1 and δ400 ≈ 12 mmthick 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 andReL = 2.7 · 106.

The roughness elements used to force the streaks are designed according to the setof 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 (≈ 0.6δ200 and1δ400 ) while the ratio λz/d = 4 is kept. In the present paper, two spanwise spacingare 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


between 4λz and 5λz upstream of the separation line. Table 2 summarises the mainparameters used in Pujals et al. (2010b). In the following, all the measurements arecarried 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. Ue (m.s−1) ReL λz (mm) d (mm) x0 (mm) λz/d λz/δ0 k/δ0 x0/λzA20 20 1.35·106 24. 6. –120. 4 1.2 0.6 –5A40 40 2.7·106 24. 6. –120. 4 2. 1. –5B20 20 1.35·106 48. 12. –192. 4 2.4 0.6 –4B40 40 2.7·106 48. 12. –192. 4 4. 1. –4

4.1 Streaks developing on the roof

Figure 6 shows a visualisation of the time-averaged flow over the roof measureddownstream of the cylinders array at height Y /k = 0.5 (Y = 6mm). The flow is fromleft to right and the cylinders (represented by the white circles) are located on the leftpart of the figures. The velocity is scaled on the free-stream velocity Ue and plottedversus the spanwise and streamwise directions both scaled on the spacing λz . Thedata 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 flatplate case [see Figure 3(b)]. The streaks are observable around x = 1λz downstream ofthe cylinders array up to the separation line at x = 0λz . Additional experiments withconfigurations A20 and A40 show similar behaviour.

The amplitude Âst (x̃, Y /k = 0.5) of the streaks has been estimated using thedefinition 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 maximumobserved amplitude between 9% and 15% of Ue depending on the Reynolds numberand 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 fromthe flat plate case [see Figure 3(b)]; this is probably due to the adverse pressure gradientand 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 streamwisevelocity 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 systemdescribed in Section 4. On the upper horizontal axis x̃ denotes the streamwise distancefrom the roughness array (a) velocity field for configuration B20 of Table 2(b) velocity field for configuration B40 of Table 2

x/λz

~x /λz

0 1 2 3 4

(a)

-4 -3 -2 -1 0

-2

-1

0

1

2

z/λ

z

0.7

0.8

0.9

1.0

1.1

1.2

U/U

e

(a)

x/λz

~

x /λz

0 1 2 3 4

(b)

-4 -3 -2 -1 0

-2

-1

0

1

2

z/λ

z

0.7

0.8

0.9

1.0

1.1

1.2

U/U

e

(b)

The maximum amplitude Âmaxst attained for each configuration are reported in Table 3.These results highlight some tendencies relative to the dependence of Âmaxst on ReLand 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)


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

Figure 7 Estimated streaks’ finite amplitude Âst(x, Y /k = 0.5) as a function of the distancefrom the cylinders array scaled with the spanwise spacing x̃/λz

0

0.05

0.1

0.15

0.2

1 1.5 2 2.5 3 3.5 4 4.5

^ Ast

~x/λz

Config. A20A40B20B40

Table 3 Amplitude of coherent streaks issuing from configurations listed in Table 2

Config. Ue (m.s−1) ReL λz/δ0 Âmaxst (%Ue)A20 20 1.35·106 1.2 9A40 40 2.7·106 2. 12B20 20 1.35·106 2.4 11B40 40 2.7·106 4. 15

4.2 Mean flow organisation around the rear-end

In Figure 8, the time-averaged streamwise velocity U/Ue is plotted for the uncontrolledcase [Figure 8(a) and 8(c)] and for the controlled case [Figure 8(b) and 8(d)] for thetwo Reynolds numbers studied in the present paper. The flow is from left to right. Theuncontrolled flows present a thick shear layer originating from the upper side of theslanted surface. The curvature of the whole shear layer and the dark spot between thesurface and the layer indicate a re-circulation bubble. When control is applied, the shearlayer becomes thinner and its curvature is highly reduced. This behaviour suggests thatthe 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 atY /k = 0.08 (corresponding to Y = 1 mm) over the slanted surface. The resultingstreamwise 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 commonfeatures observed in Lienhart et al. (2003). For ReL = 1.35 · 106, the re-circulationbubble is clearly apparent since a backward flow region is observed for roughly0.01 m ≤ x ≤ 0.15 m along the symmetry line z =0 m while it is only suggested by theslight curvature of the constant-velocity lines for the ReL = 2.7 · 106 case. The footprintof 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 theprevious suggestion: the re-circulation bubble is suppressed since the streamlines presentstraight trajectories with no evidence of backward flow. Nevertheless, no noticeableinfluence on the streamwise vortices can be observed at this distance from the wall.

Figure 8 Time-averaged streamwise velocity U/Ue around the rear-end in the symmetry planeusing 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 solidlines) are added to highlight the curvature of the shear layer

(a)

-0.1 0 0.1 0.2

x (m)

-0.15

-0.1

-0.05

0

0.05

0.1

y (m

)

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

(a)

(b)

-0.1 0 0.1 0.2

x (m)

-0.15

-0.1

-0.05

0

0.05

0.1

y (m

)

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e(b)

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

x (m)

y (m

)

(c)

0.2 0.1 0-0.1

-0.15

-0.1

-0.05

0

0.05

0.1

(c)

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

x (m)

y (m

)

(d)

0.2 0.1 0-0.1

-0.15

-0.1

-0.05

0

0.05

0.1

(d)


Figure 9 Time-averaged streamwise velocity Y /k = 0.08 above the slanted surface usingnear-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 respectivelyconfiguration B20 and B40

(a)

0 0.06 0.12 0.18

z (m)

0

0.05

0.1

0.15

0.2

x (m

)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

(a)

(b)

0 0.06 0.12 0.18

z (m)

0

0.05

0.1

0.15

0.2

x (m

)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

(b)(c)

0 0.06 0.12 0.18

z (m)

0

0.05

0.1

0.15

0.2

x (m

)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e

(c)

(d)

0 0.06 0.12 0.18

z (m)

0

0.05

0.1

0.15

0.2

x (m

)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

U/U

e(d)

5 Conclusions and discussion

The present paper summarises the main results of the studies aiming at using nearlyoptimal coherent perturbations supported by turbulent wall flows to perform flowcontrol. Linear optimal growth analysis of turbulent boundary layer flow has lead tothe conclusion that the most amplified perturbations are large-scale streamwise uniformvortices 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 amplifiedwith an optimal wavelength near 8δ.

In some recent studies, evidences of very large-scale streaks present in the outerregion of turbulent shear flows (see, e.g., Tomkins and Adrian, 2003; Hutchins andMarusic, 2007; del Álamo 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 observationof large-scale structures with λz ≈ 4 ∼ 8 δ. In the core region of turbulent Couetteflow, 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 ifartificially forced turbulent large-scale coherent streaks could sustain and be amplifiedin a flat plate turbulent boundary layer in the spirit of previous works by Franssonet al. (2005). Doing so, we have found that very large-scale coherent structures with aspanwise spacing 3δ0 < λz < 12δ0 can be forced in a low Reynolds number turbulentboundary layer. Those streaks experience a transient growth in space owing to the lift-upeffect, similarly to the growth observed for streaks forced in the same way in a laminarboundary layer and modify the mean flow at leading order. They can reach similarfinite moderate amplitudes ≈ 13% Ue. In agreement with results from the linear optimalperturbations theory, for this Reynolds number, the most amplified wavelength is ≈ 6δ0at Reδ∗ ≈ 1,000.

Eventually, the potential of a separation control strategy based on the growth oflarge-scale coherent streaks has been investigated. Streaks are generated on the roofof a 25o slant angle Ahmed body for two Reynolds numbers: ReL = 1.35·106 andReL = 2.7·106. In spite of the adverse pressure gradient, near wall PIV measurementshave shown that the streaks can develop and experience growth. The resulting streakybase flow is responsible for the modification of the flow organisation around the rearend of the model: when controlled with large-scale streaks, the re-circulation bubble iscompletely suppressed.

It is important to emphasise the differences between the technique proposed in thepresent 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 delayis attributed to vortices that work by pairs to create a vertical mixing on the bubbleedge (Godard and Stanislas, 2006). In the present study, the separation delay mustbe attributed only to the streaks (the spanwise modulation of the streamwise velocity)because streamwise vortices have virtually disappeared at the position of maximumstreak amplitude, which is a typical feature of the lift-up effect.

Previous studies showed that the major part of the drag force applied to Ahmedbody is due to pressure drag at the rear-end. Thus, changes in the flow organisationsuch as those described in Section 4.2 should result in substantial drag reduction. Inorder to investigate this last point, we measured the drag coefficient Cd of the modelwhen control is applied and compared it to the uncontrolled cases. The estimated dragreductions are reported in Table 4. One salient feature is that the larger drag reductionsare obtained for ReL = 1.35 · 106. Since near-wall PIV measurements over the slantedsurface 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 forReL = 1.35 · 106 than for ReL = 2.7 · 106 [see Figures 9(a) and (c)]. We can alsoobserve that for a fixed ReL, the total drag reduction decreases with λz in spite ofthe larger amplitude of the resulting streaks. One possible explanation is that the biggerroughness elements used in, for instance, configuration B20 induce a larger drag penaltythan those used for configuration A20. For wind-tunnel availability reasons, we couldnot investigate the minimum streaks amplitude required to suppress the re-circulationbubble. This type of study would probably lead to a improved total drag reduction owingto even better shaped roughness elements.


Table 4 Drag reduction obtained with configurations listed in Table 2

Config. ReL λz/δ0 Âmaxst (%Ue) ∆Cd (%)A20 1.35·106 1.2 9. –10.A40 2.7·106 2. 12. –2.5B20 1.35·106 2.4 11. –6.B40 2.7·106 4. 15. –2.

ReferencesAhmed, S.R., Ramm, R. and Faltin, G. (1984) ‘Some salient features of the time-averaged ground

vehicle wake’, SAE 840300.Andersson, P., Brandt, L., Bottaro, A. and Henningson, D.S. (2001) ‘On the breakdown of boundary

layers streaks’, J. Fluid Mech., Vol. 428, pp.29–60.Angele, K. (2003) ‘Experimental studies of turbulent boundary layer separation and control’, PhD

thesis, Royal Institute of Technology (KTH), Stockholm.Betterton, J.G., Hackett, K.C., Ashill, P.R., Wilson, M.J., Woodcock, I.J., Tilman, C.P. and Langan,

K.J. (2000) ‘Laser doppler anemometry investigation on sub boundary layer vortex generators forflow control’, 10th Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon,pp10–12.

Butler, K.M. and Farrell, B.F. (1992) ‘Three-dimensional optimal perturbations in viscous shear flow’,Phys. Fluids, Vol. A 4, pp.1637–1650.

Cossu, C. and Brandt, L. (2002) ‘Stabilization of Tollmien-Schlichting waves by finite amplitudeoptimal streaks in the Blasius boundary layer’, Phys. Fluids, Vol. 14, pp.L57–L60.

Cossu, C., Pujals, G. and Depardon, S. (2009) ‘Optimal transient growth and very large scalestructures in turbulent boundary layers’, J. Fluid Mech., Vol. 619, pp.79–94.

del Álamo, J.C. and Jiménez, J. (2003) ‘Spectra of the very large anisotropic scales in turbulentchannels’, Phys. Fluids, Vol. 15, L41.

del Álamo, J.C. and Jiménez, J. (2006) ‘Linear energy amplification in turbulent channels’, J. FluidMech., Vol. 559, pp.205–213.

Duriez, T. (2009) ‘Application des générateurs de vortex au contrôle d’écoulements décolĺes’, Phdthesis, Universit́e Paris Diderot.

Fransson, J., Brandt, L., Talamelli, A. and Cossu, C. (2004) ‘Experimental and theoretical investigationof the non-modal growth of steady streaks in a flat plate boundary layer’, Phys. Fluids, Vol. 16,pp.3627–3638.

Fransson, J., Brandt, L., Talamelli, A. and Cossu, C. (2005) ‘Experimental study of the stabilizationof Tollmien-Schlichting waves by finite amplitude streaks’, Phys. Fluids, Vol. 17, 054110.

Fransson, J., Brandt, L., Talamelli, A. and Cossu, C. (2006) ‘Delaying transition to turbulence by apassive mechanism’, Phys. Rev. Lett., Vol. 96, 064501.

Godard, G. and Stanislas, M. (2006) ‘Control of a decelerating boundary layer. Part 1: optimizationof passive vortex generators’, Aero. Sci. Tech., Vol. 10, pp.181–191.

Hamilton, J.M., Kim, J. and Waleffe, F. (1995) ‘Regeneration mechanisms of near-wall turbulencestructures’, J. Fluid Mech., Vol. 287, pp.317–348.

Hollands, M. and Cossu, C. (2009) ‘Adding streaks in the plane Poiseuille flow’, Comptes-Rendusde l’Académie des Sciences, Mécanique, Vol. 337; pp.179–183.

Hutchins, N. and Marusic, I. (2007) ‘Evidence of very long meandering features in the logarithmicregion of turbulent boundary layers’, J. Fluid Mech., Vol. 579, pp.1–28.

Hwang, Y. and Cossu, C. (2010) ‘Amplification of coherent streaks in the turbulent Couette flow: aninput-output analysis at low Reynolds number’, J. Fluid Mech., Vol. 643, pp.333–348.


Kitoh, O. and Umeki, M. (2008) ‘Experimental study on large-scale streak structure in the core regionof turbulent plane Couette flow’, Phys. Fluids, Vol. 20, 025107.

Komminaho, J., Lundbladh, A. and Johansson, A.V. (1996) ‘Very large structures in plane turbulentcouette flow’, J. Fluid Mech., Vol. 320, pp. 259–285.

Landahl, M.T. (1980) ‘A note on an algebraic instability of inviscid parallel shear flows’, J. FluidMech., Vol. 98, pp.243.

Lienhart, H., Stoots, C. and Becker, S. (2003) ‘Flow and turbulence structure in the wake of asimplified car model’, SAE transactions, Vol. 112, pp.785–796.

Lin, J.C. (2002) ‘Review of research on low-profile vortex generators to control boundary layerseparation’, Prog. Aero. Sci., Vol. 38, pp.389–420.

Lögdberg, O. (2006) ‘Vortex generators and turbulent boundary layer separation control’, Master’sthesis, Royal Institute of Technology (KTH), Stockholm.

Moffatt, H.K. (1967) ‘The interaction of turbulence with strong wind shear’, Proc. URSI-IUGG Coloq.on Atoms. Turbulence and Radio Wave Propag., Nauka, Moscow, pp.139–154.

Monkewitz, P.A., Chauhan, K.A. and Nagib, H.M. (2007) ‘Self-consistent high-Reynolds-numberasymptotics for zero-pressure-gradient turbulent boundary layers’, Phys. Fluids, Vol. 19, 115101.

Park, H., Jeon, W.P., Choi, H. and Yoo, J.Y. (2007) ‘Mixing enhancement behind a backward-facingstep using tabs’, Phys. Fluids, Vol. 19, 105103.

Pastoor, M., Henning, L., Noack, B., King, R. and Tadmor, G. (2008) ‘Feedback shear layer controlfor bluff body drag reduction’, J. Fluid Mech., Vol. 608, pp.161–196.

Pujals, G., Cossu, C. and Depardon, S. (2010a) ‘Forcing large-scale coherent streaks in a zero pressuregradient turbulent boundary layer’, J. Turbulence, Vol. 11, 25.

Pujals, G., Depardon, S. and Cossu, C. (2010b) ‘Drag reduction of a 3D bluff body using coherentstreamwise streaks’, Exp. Fluids, Vol. 49, Issue 5, pp.1085–1094.

Pujals, G., Garćıa-Villalba, M., Cossu, C. and Depardon, S. (2009) ‘A note on optimal transientgrowth in turbulent channel flows’, Phys. Fluids, Vol. 21, 015109.

Rao, D.M. and Kariya, T.T. (1988) ‘Boundary-layer submerged vortex generators for separation control– an exploratory study’, AIAA, ASME, SIAM and APS National Fluid Dynamics Congress, 1st,Cincinnati, OH; USA, pp.839–846.

Reynolds, W.C. and Hussain, A.K.M.F. (1972) ‘The mechanics of an organized wave in turbulentshear flow. Part 3. Theoretical models and comparisons with experiments’, J. Fluid Mech.,Vol. 54, pp263–288.

Schmid, P.J. and Henningson, D.S. (2001) ‘Stability and transition in shear flows’, Srpinger, NewYork.

Schoppa, W. and Hussain, F. (2002) ‘Coherent structure generation in near-wall turbulence’, J. FLuidMech., Vol. 453, pp.57–108.

Taylor, H. (1947) ‘The elimination of diffuser separation by vortex generators’, Report No. R-4012-3,United Aircraft Corporation.

Tomkins, C.D. and Adrian, R.J. (2003) ‘Spanwise structure and scale growth in turbulent boundarylayers’, J. Fluid Mech., Vol. 490, pp.37–74.

Trefethen, L.N., Trefethen, A.E., Reddy, S.C. and Driscoll, T.A. (1993) ‘A new direction inhydrodynamic stability: beyond eigenvalues’, Science, Vol. 261, pp.578–584.

White, E.B. (2002) ‘Transient growth of stationary disturbances in a flat plate boundary layer’, Phys.Fluids, Vol. 14, pp.4429–4439.

White, F. (2006) Viscous Fluid Flow Third Edition, McGraw-Hill, New York.


List of symbols

Âst finite amplitude of the streaksCd drag coefficientd diameter of the roughness elements mk height of the roughness elements mL length of Ahmed body model mReδ∗ Reynolds number Reδ∗ = Ueδ∗/νReL Reynolds number ReL = UeL/νU streamwise velocity base flow m.s−1Ue free-stream velocity m.s−1Ue+ free-stream velocity scaled in wall-units Ue+ = Ue/uτu streamwise velocity of a perturbation m.s−1uτ friction velocity m.s−1v normal velocity of a perturbation m.s−1w spanwise velocity of a perturbation m.s−1x streamwise direction/position mx0 streamwise position of roughness arrays my normal direction/position my+ normal position expressed in wall-units y+ = yuτ/νz spanwise direction/position mα streamwise wavenumber of a perturbation m−1β spanwise wavenumber of a perturbation m−1δ boundary layer thickness mδ0 boundary layer thickness where roughness arrays are located (flat-plate case) mδ∗ displacement thickness m∆ Rotta-Clause thickness mλz spanwise wavelength of a perturbation/ spacing of the roughness elements mν molecular viscosity m2.s−1νt eddy viscosity m2.s−1νT total viscosity, νT = ν + νt m2.s−1ωy normal vorticity s−1

transient growth of coherent streaks for control of ... · reference to this paper should be made...

Documents