global optimization of vortex generators … · 2018-08-02 · cfd & optimization 2011 - 075 an...

CFD & OPTIMIZATION 2011 - 075An ECCOMAS Thematic Conference23-25 May 2011, Antalya TURKEY

GLOBAL OPTIMIZATION OF VORTEX GENERATORS PARAMETERSFOR DRAG REDUCTION OF GROUND VEHICLES.

T. ROUILLON ∗,†, F. HARAMBAT∗ , L. MATHELIN† and C. TENAUD†

∗PSA PEUGEOT-CITROEN,Centre Technique de Velizy 2,route de Gisy

78943 VELIZY-VILLACOUBLAY CEDEX, FRANCEe-mail: {thomas.rouillon,fabien.harambat}@mpsa.com

†LIMSI UPR CNRS 3251Campus Universitaire, Bt 508, B.P. 133, F-91403 ORSAY CEDEX, FRANCE

e-mail: {Lionel.Mathelin,Christian.Tenaud}@limsi.fr

Key words: Drag reduction, Vortex Generator, Global Optimization, RANS simulation.

Abstract. Energy consumption in automotive is mainly due to the large boundary layerseparation on the rear-end of the car, main origin of the drag.It hence makes reducing ordelaying the separation a major issue in automotive industry. To manipulate the flow,small flat plates (Vortex Generators or VGs) could be arranged on the car wall. Theylocally create counter-rotating vortex pairs that inject momentum from the high speedouter part of the boundary layer to the low velocity inner region. The boundary layer isthen less prone to detachment.

In this work, an optimization loop is proposed, associating a RANS numerical solverand an optimizer, to determine the best set of VG parameters to reduce the drag of anacademic geometrical configuration. The geometry represents a simplified rear part of acar. To prevent fine-grained meshing in the VG vicinity, and hence a prohibitive CPUtime consumption, source terms (BAY model) are used.

Both a response surface-based optimizer and a geometrical optimizer are compared withevolutionary-based and gradient-based techniques on an analytical function and shown toachieve good results with a fewer number of evaluations. They are then used to find thebest set of VG parameters to control the flow over the geometry. A 25 % reduction of thedrag is achieved.

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T. Rouillon, F. Harambat, L. Mathelin and C. Tenaud

1 INTRODUCTION

Reduction of pollutant emissions is one of the major objective in the industrial world.For instance, in automotive industry, the CAFE (Corporate Averaged Fuel Economy)treaty imposes an average CO2 emission of 120 g/km by 2012. In Europe, the averagewas 171 g/km in 2000 and 140 g/km in 2008: these values indicate the work yet to do toreach CAFE constraints. For an automobile vehicle on highway, 70% of energy waste isdue to the total drag and 80% of it arises from the pressure decrease within the boundarylayer separation1. This explains why reducing the total drag is one of the main issues intransport industry. The adverse pressure gradient at the rear-end of the car (see figure 1)has a major role in flow separation. It hence constitutes the motivation for the presentwork which aims at studying how to manipulate this flow separation. To this end, ageneric configuration mimicking the rear part of a car is considered (see figure 3).

Figure 1: Boundary-layer separations on vehicle. PSA source.

Two main ways can be considered to reduce or suppress the separation. A first oneconsists in improving the whole car geometry to affect the pressure distribution. However,in automotive commercial industry, the body-shape optimization for fuel-efficiency is oftenin competition with other constraints (design, security, ...) and is unfortunately not aleading criterion in the development process. A second way is flow control, introducinglocal perturbations within the boundary layer to manipulate the flow. A feedback fromthe sensors to the actuator defines a closed-loop control. However, this requires energyto operate as well as reliable and inexpensive actuators and sensors. In contrast, passivecontrol relies on inert manipulators (e.g., spoiler) designed for a given operating settingand in this paper, we focus on manipulation using small fins called Vortex Generators(VGs).

The concept of VGs relies on the fact that a boundary layer with a high mixing rate isweakly prone to separation in presence of an adverse pressure gradient2. The VG devicesincrease the momentum transfer from the high speed outer part of the boundary layerto the low velocity inner region (see figure 2). Taylor3 used VGs in 1946 to control theboundary layer in a diffuser and they were since employed in many configurations, see

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Lin4 and Godard et al.5 for reviews. Different configurations were considered (backward-facing ramp, bump, flat plate) and it was shown that small flat plates arranged normalto the surface create counter-rotative vortices and reduce the most the boundary layerseparation. Furthermore, devices with height, denoted Z0, about 60% or less of theboundary-layer thickness δ exhibit the best results. The VG array we use is defined by7 parameters shown in figure 3. The goal sought-after is to reach a global optimal setof the parameters defining the VGs to reduce the drag of a specific configuration. Anoptimization loop for these parameters is then built using a RANS numerical solver tocompute the flow solutions.

(a) co-rotating (b) counter-rotating

Figure 2: Passive device configurations, from5

Figure 3: Vortex generator parameters and dihedron configuration.

The device height is very small w.r.t. the bulk simulation domain and simulationsrequire a fine mesh in the vicinity of the VGs. For simulations to remain tractable inthe context of the optimization procedure, the meshed VGs are substituted with sourceterms that mimic their impact on the flow. Two main approaches are usually considered.Wendt6 developed a model based on the strength and size of the vortex bringing vorticitydownstream of the VGs. Alternatively, Bender et al.7 (the so-called BAY model) substi-tute the meshed VGs with a lift force. Recently, Dudek8 compared results obtained with

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these two models with both meshed VG simulations and experimental measurements inthe case of a circular S-Duct flow. The BAY model exhibits better results than the Wendtmodel and it is retained for the present study.

The paper is organized as follows. Section 2 describes the geometry and the VGsconfigurations used. The experimental and numerical setups are also presented. In sec-tion 3, the BAY model is recalled and its implementation in a RANS numerical solver isexposed. Following the procedure employed by Dudek, we validate this model on an aca-demic dihedron configuration, a proxy of a simplified car geometry. In section 4, we focuson the choice of an appropriate optimization method by evaluating several approacheson an analytical function. The retained method is then applied to the VG parametersoptimization to reduce the aerodynamic drag of the configuration at hand.

2 CASE STUDY

2.1 Dihedron geometry and VG configurations

The geometry used in this study consists in a slanted surface equipped with an upstreamand a downstream horizontal flat plates (see figure 3). The retained slant angle (26.5◦)has been chosen since it corresponds to the angle between the roof and the rear windowof real car. The free-stream velocity is set to Ue = 19 m.s−1. The junction betweenthe upstream flat plate and the slanted surface is sharp so that the turbulent boundarylayer developing on the upstream horizontal plane detaches at the dihedral edge. A largeseparation then occurs that reattaches on the downstream horizontal plane. Dimensionsof the configuration are given on figure 3. It spans over the whole width of the wind tunnel(300 mm) so as to minimize 3D effects in the middle region of the span (see figure 4).The Reynolds number based on the dihedron height is Reh = 3.8 × 104. The boundarylayer thickness is δ = 4.5 × 10−3 m at location X0 = −45 × 10−3 m, with X0 = 0 beingthe upstream edge of the dihedron. Different configurations of tested VGs have beenmachined by fused deposition modeling (FDM) on a strip (figure 5) and mounted on theupstream horizontal plane at a distance X0 upstream of the corner. Note that X0 will beconsidered as a VG parameters in the optimization process.

The configurations tested here are presented in the table 1. Parameters defining thegeometry are presented on figure 3.

2.2 Experimental setup

Experiments are carried out in the PSA closed loop wind tunnel. The test sectionis 800 mm long and has a 300 × 300 mm2 cross-section. The contraction ratio of theconvergent upstream of test-section is 8. Velocity can vary from 7 m.s−1 to 45 m.s−1 andthe temperature can be kept almost constant within ±0.5 K.

In order to measure the velocity, particle image velocimetry (PIV) method is usedwith Dantec’s Flow Manager system. The laser is a 120mJ Nd:Yag double cavity andthe cameras is a 1600×1168 FlowSense CCD. A 105 mm optical lens is added. The laser

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Figure 4:

Figure 5: Experimental VG

Angle of Chord Height Spacing length Spacing length Angle ofConfiguration attack length in pair between pairs the top side

α c/δ Z0/δ d/δ L/δ β1 5 2.5 0.35 1 3 02 10 2.5 0.35 1 3 03 15 1.4 0.35 1 3 0

Table 1: Experimental VG configurations

sheet thickness was 1mm and 500 pairs of images were recorded. We used DEHS dropletslike tracers, with nominally diameter of 0.25 µm.

The influence of the VGs on the wall pressure distribution on the dihedron is studiedusing wall static pressure measurements. A regular grid of 9×14 pressure sensors areplaced on the dihedron equally spaced within X0/δ ∈ [−6.7, 42] and y/δ ∈ [−10, 10]. The

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wall pressure coefficient is defined as:

Cp =P − P∞12ρeU2

e

, (1)

where P is the measured wall static pressure, ρe, Ue and P∞ are respectively the density,streamwise velocity and the static pressure external to the boundary layer.

2.3 Numerical setup

All the simulations presented here have been carried out with the commercial CFDflow solver Fluent to solve the steady state RANS equations. The two equation k − ωSST turbulence model has been used. Allan et al.9 have shown that the k−ω SST modelpresents better results than the one equation Spalart-Allmaras (S-A) turbulence modelas far as the prediction of the location, magnitude et evolution of the created vortex areconcerned. A structured mesh is used, tightened close to solid walls with a first pointover the wall at y+ < 1 wall unit. The mesh size is ≈ 7 × 106 cells. At the inlet, avelocity profile based on an analytic form is prescribed, see section 3.4 for details. No slipconditions are imposed on solid walls, i.e., on the dihedron surface as well as on the roof.Periodic conditions in the spanwise direction are imposed. At the outlet, the pressure isadjusted to ensure mass flow rate conservation.

3 SOURCE TERM MODELING

3.1 Presentation

Bender et al.7 developed a source term model for the Navier-Stokes equations thatmimics the effect of the VG on the flow. The model is implemented by control volume ina finite volume code. The basic idea is to introduce a volume force as a source term inthe momentum equation that imposes a deviation to the flow similar to the one createdby a meshed VG. The strength of the force is adjusted so as to maintain the local flowaligned with the modeled VG. The force imposes in a cell i takes the original form:

~Li = cvgSvg∆ViVtot

γiρU2i~li (2)

where cvg is an empirical calibration constant, Svg is the plan-form surface area of theVG, (∆Vi)/Vtot is the ratio of the cell volume with the total volume of cells where thesource term is applied to, γi is the angle of attack between the local flow and the planeof the VG, ρ is the local flow density, ~Ui is the local flow velocity and ~li is the unit vectordefining the direction of ~Li. The force is oriented normal to the velocity vector and tothe unit vector along the span of the VG. Allan et al.9, Waithe10;11 and Iannelli et al.12

investigated this model on a flat plate with a single modeled VG. The model was alsotested in the S-Duct configuration by Jirasek13 and Dudek8. All these results are in goodagreement with the meshed VG simulations.

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3.2 Implementation

To introduce the force ~Li in the momentum equation, we express the equation (2) in

the frame of the VG (see figure 6). The unit vector ~li is defined as follows:

~li =~Ui

| ~Ui|× ~bi. (3)

The side force imposes to the flow to be aligned with the VG. Although the small angleapproximation is used to express γi,

γi ≈ sinγi =~Ui · ~ni| ~Ui|

, (4)

the BAY model ensures that ~Li → 0 when γi → 90◦ as expected.To recover the right behaviour, namely the decrease of the force at high angle of attack,

the BAY model (2) finally reads

~Li = cvgSV G∆ViVtot

ρ( ~Ui · ~ni)( ~Ui × ~bi)

(~Ui

| ~Ui|· ~ti

). (5)

Figure 6: Local VG orientation.

The source term is implemented in a group of cells around the modeled VG. Benderet al.7 applied the BAY model in a row or group of rows enclosing the VG geometry(figure 7(a)). Jirasek13 presented a new cell selection, considering the VG as an infinitelythin plate. Furthermore, the VGs are usually placed in a boundary layer where the grid isfine enough to accurately represent the VGs. Jirasek suggested to only select cells crossedby the device (figure 7(b)). This allows to accurately set the parameters of the VGs (size,shape, orientation, ...). Wallin et al.14 and Iannelli et al.12 validated this selection bycomparing simulations with fully meshed VGs and results obtained using the VG sourceterm model. Wallin et al.14 demonstrated the accuracy of the model by analyzing a VGarray placed in a sector of S-Duct. Iannelli et al.12 analyzed the effect of a VG mounted

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on a flat plate. Another advantage of this selection is that the source terms are introducedin a small region of the domain, limiting the occurrence of instabilities13. In this work,use is made of the Jirasek’s selection method.

(a) Bender et al. implementa-tion

(b) Jirasek implementation

Figure 7: Selection of cells for implementation of source terms.

3.3 Calibration

cvg is an empirical constant that needs be calibrated to control the strength of the force.When the constant is too small, the force is not strong enough to make flow aligned withthe VG. On the opposite, using a high cvg value, the magnitude of the force is too largeand spurious instabilities could appear. The appropriate value of cvg can be determinedempirically following the approach proposed by Bender et al.7. The evolution of the cross-flow kinetic energy k (6) integrated over a transverse plane ({y, z}) is plotted versus cvgat various locations downstream of the VGs (see figure8). The value of the constant ischosen in the range where

√k plateaus, in other words, where the flow is independent of

cvg:

k =

∫Aρ (v2 + w2) dA∫

Aρu2dA

. (6)

The calibration has been carried out with two counter-rotating pairs (see figure 2) on a flatplate with a free stream velocity Ue = 34m.s−1 and a boundary-layer thickness δ =45mmat the VGs location. From this result, cvg = 40 is used hereafter. Extensive comparisonsbetween the BAY model and meshed VG simulations have also been conducted on a flatplate configuration in Rouillon et al.15.

3.4 Application on the dihedron configuration

First we need to validate the numerical approach on the non-perturbed flow (i.e.,without VG). As the flow is homogeneous in the spanwise direction (y), the followingresults are analyzed on mean spanwise values in the {x, z} plane.

The streamwise velocity inlet condition is obtained by extrapolating the experimen-tal measurements coming from PIV with the Spalding16 profile using the Kendall et al.

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Figure 8:√k vs. cvg — © : x/δ = 5 ; � : x/δ = 10 ; N : x/δ = 15 ; × : x/δ = 20.

method17 (see figure 9). Table 2 presents the boundary layer parameters.

Figure 9: Nondimensionalized inlet velocity profile U+ vs. y+ —© : experimental data ; — : extendedvelocity profile.

Ue Uτ δ δ∗ θ18.88 m.s−1 0.99 m.s−1 59.98 mm 2.63 mm 2.31 mm

Table 2: Experimental VG configurations.

Figure 10 shows the experimental and numerical streamwise velocity field around thedihedron. The white isolines denotes the edge of the separation, i.e., the boundary be-tween the negative and positive streamwise velocity regions. A qualitative agreement isobtained. The main discrepancy is observed for the flow over the horizontal plate upstreamof the detachment. It comes from spurious measurements induced by laser reflections onthe walls. A good agreement is however exhibited on the shape of the separation bubble.The predicted reattachment location is in good agreement with the one from experiments.

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(a) Experimental flow field (b) Numerical flow field

(c) legend

Figure 10: Streamwise velocity U in m.s−1. The flow is from the left to the right. The white arearepresents U=0m.s−1.

To confirm these results, experimental and numerical streamwise velocity profiles areplotted for several downstream locations, figure 11. At the separation location, the numer-ical solution predicts a thinner boundary-layer than in the experiments, inducing a highershear in the boundary layer. Consequently, further downstream, the mixing layer edgingthe separation is thinner. Nevertheless, the growth rate of the mixing layer from the nu-merical simulation is high and the experimental profiles are recovered beyond x = 0.06m.At most, discrepancies of 5% are observed.

Figure 11: Downstream velocity profile U at various locations. The flow is from the left to the right —© : experimental datas ; — : numerical datas.

Figure 12 presents the streamwise distribution of the wall pressure coefficient (Cp). Thetwo peaks at x = 0m and x = 0.06m are due to the sudden evolutions of the geometry,

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leading to abrupt changes in the pressure gradient. As probes cannot be implementedat the corners, these peaks are not visible on the experimental plots. The predicted Cpvalues fit very well the experiments. Simulations correctly predict the location from wherethe pressure raise occurs and the deceleration up to the reattachment location.

The numerical simulations are then accurate and can hence serve as a reliable tool tostudy the effect of the VGs on the flow.

Figure 12: Streamwise distribution of the wall pressure coefficient Cp — © : experimental data; —: numerical data.

Flow with VGs

The different VG configurations (see table 1) are evaluated both experimentally andnumerically. Their influence on the dihedron drag is measured by the value of Cp inte-grated over the slanted surface. Indeed, the pressure seems to be an adequat variable toprobe the influence of the manipulators since the pressure defect directly affects the drag.Figure 13 plots the relative pressure increase for three configurations at three differentlocations upstream of the corner. The ranking between the different configurations frompredicted solutions using the BAY model is roughly similar as the one from experiments.Although the values are not quite the same, the general trend of the experiments andrecovered by using the BAY model. This model demonstrates its ability to predict theflow manipulated by VG arrays.

The numerical simulations based on the BAY model can then serve as a reliable toolfor the optimization procedure.

4 OPTIMIZATION

4.1 Presentation of evaluated optimizers

In order to choose the appropriate optimizer to minimize the drag of the dihedron, weevaluate four different approaches on an analytical function. Two standard optimizationmethods are tested: a gradient-based algorithm (Quasi-Newton BFGS18) and a geomet-rical algorithm based on a Simplex method19. These methods need a small number of

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(a) Experimental Cp (b) Numerical Cp

Figure 13: Difference between the averaged Cp of the slanted surface without VGs and with the differentconfigurations at various downstream locations — � : Config 1 ; � : Config 2 N : Config 3.

evaluations to reach a minimum but are prone to converge to a local minimum. Thesemethods are compared to a recent evolutionary algorithm (CMA-ES20), which guaranteesto reach a global minimum with less evaluations than classical evolutionary-based opti-mizers (e.g., GA, PSO). However this method remains relatively CPU-intensive. In thelast decade, new optimization methods, based on the fit of a response surface, have beenproposed. The knowledge of an approximated solution allows to only evaluate promisingparameter sets and to reduce the computational cost. In this paper, we focus on theEfficient Global Optimizer (EGO), based on the work of Jones21.

An EGO evaluation comprises two steps. A response surface, approximating the func-tion F to be minimized, is first built by kriging with the data-base constituted by allprevious evaluations. Next, an optimization is performed to find the minimum of severalcost functions involving statistical quantities of the assumed underlying process (mean µand variance σ2) identified during kriging. The method used in this study was developedat INRIA22. At each iteration, the minimum of each three cost functions f (f0 = µ,f1 = µ − σ and f2 = µ − 2σ) is searched by PSO. The function F is then evaluated atthese three parameter sets, eventually leading to the best minimum.

4.2 Applications

Analytical function

First, the four optimizers are evaluated on the 7D Griewank23 analytical function (7)that presents several local minima and only one global minimum in (0, 0) with a valueof −1.

F =7∑i=1

x2i50−

7∏i=1

xi√i. (7)

Table 3 and figure 14 exhibit the minimum values reached by the optimization methodsconsidered in this work versus the number of evaluations needed to reach it. The presented

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results are the mean of 100 optimization runs with random initializations.The Quasi-Newton BFGS converges towards a local minimum and fails in reaching the

global minimum although a large number of evaluations is used. The CMA-ES finds theglobal minimum with more than three times the number of evaluations used by EGOand Simplex. CMA-ES and Quasi-Newton methods could hence not be retained for theaerodynamic application at hand.

The Simplex and the EGO both nearly reach the global minimum with a few evaluationswhile the Simplex method prematurely stops due to the allowed mimimal simplex size.While the resulting minimum is similar, the paths of the optimization process are differentas can be seen from figure 14.

Figure 14: Minimization of 7D Griewank function. Mean of 100 procedures — Comparison of Simplex,CMA-ES, EGO and Quasi-Newton BFGS.

Optimizer Minimum found # evaluationsSimplex −0.89 159CMA-ES −1.00 549EGO −0.99 150Quasi-Newton BFGS −0.44 320

Table 3: Results of Griewank optimization for 7 variables. Mean of 100 procedures. The right minimumvalue is −1.

Due to the cost of each evaluation of numerical aerodynamic solutions, obtained bysolving the Navier-Stokes equations, the limiting criterion is the number of evaluations.

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We then focus on the minimum found by the Simplex and the EGO at three prescribednumbers of evaluations, see Table 4. Simplex quickly finds a cost function value (−0.88)close to the global minimum (−1) while EGO is trapped in a local minimum, as we cansee on figure 14. EGO visits every local minimum before heading towards the global one.While Simplex reaches the basin of the global minimum within only 30 evaluations, itssubsequent evolution slows down. Beyond 120 evaluations, EGO outperforms the Simplexmethod.

This test does not allow us to discriminate the methods. Therefore, the EGO andSimplex will both be evaluated in the aerodynamic case.

Optimizer 50 evaluations 100 evaluations 150 evaluationsSimplex −0.77 −0.88 −0.89EGO 0.00 −0.42 −0.99

Table 4: Minimum found vs. number of evaluations for Simplex and EGO. The true minimum is −1.

Aerodynamic case

In this application, the optimization aims at minimizing the total drag of the dihedrondescribed in section 2.1. Seven parameters are considered, see figure 3.

The optimization procedure has been performed in an industrial context, i.e., witha prescribed maximum number of evaluations of 40. Within this maximum evaluationnumber, neither Simplex nor the EGO reaches their stopping criteria. Figure 15 plotsthe evolutions of the drag reduction achieved by both the EGO and Simplex. As alreadyseen previously with the analytical function, the Simplex lowers the cost function in fewerevaluations than the EGO method. With only 30 evaluations, a 25% decrease is achievedby the Simplex in constrast with 5.60% only from the EGO. The EGO is dramaticallyaffected by the small number of evaluations allowed. Better performance could certainlybe achieved by the EGO with a larger allowed number of evaluations, at the cost of aprohibitive CPU time in industrial applications.

5 CONCLUSIONS

An optimization loop is presented to reduce the drag of a generic geometry correspon-ding to a simplified rear part configuration of a car. It associates a RANS numericalsolver with a geometrical optimizer (Simplex).

To reduce the mesh size and CPU time, VGs are substituted with source terms (BAYmodel). The comparison of experimental results with numerical ones for several config-urations shows the ability of the model to predict the VG influence on the flow over thedihedron.

The Simplex is compared to three other optimizers: EGO, CMA-ES and Quasi-NewtonBFGS. Tests on an analytical function and a fluid mechanics problem demonstrate the

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Figure 15: Drag reduction of the dihedron — Comparison between Simplex and the EGO methods.

ability of Simplex to compete with a global meta-model (EGO). It reachs an optimumvalue very close to the global one with a similar number of evaluations as the EGO. TheEGO first visits different local minima while the Simplex quickly converges towards aminimum. In the aerodynamic case, the Simplex finds a minimum value five times lowerthan EGO in the prescribed number of evaluations.

Future works will be devoted to the drag reduction of a simplified car geometry.

Acknowledgments

The EGO used in this study is the one incorporated in Famosa, the optimizationplatform developed at INRIA Sophia Antipolis - Mediterranee by Opale Project-Team.Authors greatly acknowledge Regis Duvigneau for his help and his valuable commentsand fruitful discussions.

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global optimization of vortex generators … · 2018-08-02 · cfd & optimization 2011 - 075 an...

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