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Simulation and Optimization of Flow Control Strategies for Novel High-L ift Configurations

M. Meunier1 ONERA, Châtillon, 92322, France

A Kriging-based optimization algor ithm is presented and applied to the design of a novel high-lift air foil having a simply hinged, slotless flap and equipped with fluidic flow control. The position, or ientation and continuous blowing character istics of the actuator are set as design var iables for the maximization of lift in landing conditions. Compar isons with a reference, fully slotted configuration are presented and conclusions drawn. In par ticular , it was found that a careful parameter ization of the jet is capable of an efficient control strategy and leads to a complete reattachment of the flap flow, with considerable gains in lift and aerodynamic efficiency.

Nomenclature CFD = computational fluid dynamics GA = genetic algorithm H3(.) = Hartmann’s three-variable function LHS = latin hypercube sampling LU-SSOR = lower-upper symmetric successive over

relaxation (U)RANS = (unsteady) Reynolds averaged Navier-Stokes RMSE = root mean square error SA = Spalart-Allmaras Cl = lift coefficient Cp = pressure coefficient

Cµ = jet momentum coefficient, Cµ= 2

2

..

..

∞∞ Vrefl

Vd injinjinj

ρ

ρ

E[ I(.)] = expected improvement criterion lref = reference length N = number of samples P = number of variables Rel = lref-based Reynolds number Sth = h-based Strouhal number Vinj = injection velocity Visentropic = isentropic boundary velocity Vx = streamwise velocity V� = freestream velocity VR = jet to local, isentropic velocity ratio xinj = injection abscissa y+ = normalized first cell height α = angle of attack δ = boundary-layer thickness δinj = injection angle δflap = flap deflection angle

1Ph.D. Student, Applied Aerodynamics Department, mickael.meunier@onera.fr.

Introduction HE requirement, for aircraft manufacturers, to meet more and more stringent regulations has pushed

industrials and applied research towards the in-depth evaluation of innovative design strategies for boundary layer separation delay or prevention. In this context, active flow control1-3 is unanimously regarded as a key evolution, offering new solutions for the performance maximization of existing designs, and, to an even greater extent, the apprehension of new concepts, where it could be used in a multi-disciplinary approach, fully integrated to the initial screening and conception phases. Existing Studies on High-L ift Geometr ies In 1999, McLean et al.4 examined some of the benefits to be gained from the application of unsteady excitation to civil transport aircraft, ranging from exhaust mixing to engine inlet or landing-gear scenarios. They identified high-lift trailing-edge separation management as the most promising and feasible application, through the enhancement of conventional designs5 or the replacement of slotted flaps. Applying flow control to such cases could not only improve take-off lift-to-drag ratio and landing maximum lift, but could also result in drastic reductions in radiated noise, mechanical complexity, weight and manufacturing costs, from simplifications and/or size reductions relative to current systems. High-lift devices consisting of simply hinged moveable slats and/or flaps have had the interest of a growing number of studies in fairly recent years. Seifert et al.6 conducted wind-tunnel tests on a NACA 0015 airfoil equipped with a 25% chord-length flap deflected up to 40 deg. Actuation in the form of steady/unsteady blowing was applied from a two-dimensional slot located on the suction side of the profile, above the flap pivot. A significant lift augmentation was observed together with a

T

25th AIAA Applied Aerodynamics Conference25 - 28 June 2007, Miami, FL

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cancellation of form drag and no adverse effects from Reynolds number increase. The superposition of an unsteady component was also seen to reduce massflow requirements by one order of magnitude compared to steady-only blowing. Nishri and Wygnansky7 and Darabi and Wygnanski8,9 experimentally studied flow reattachment over an inclined plane surface simulating a simple flap, with similar conclusions. They reported however on the presence of a strong hysteresis effect and as such the existence of a dual management scheme (where it is actually “cheaper” to maintain an attached flow for increasing flap deflection angles than it is to force reattachment at a given inclination), which ultimately pushes for closed loop separation control10. With the aim of reducing blade-vortex interaction noise in helicopter applications, Hassan11 performed unsteady, thin-layer Navier-Stokes computations over a NACA 0012 airfoil having a 20% chord-length flap, deflected at 40 deg., and equipped with an oscillatory air jet at the upper flap shoulder. Partial flap reattachment was achieved. The author indicates that while the importance of transition and turbulence modeling remains of primary concern for RANS-based simulations of controlled flows, the latter are capable of capturing global flow features with enough fidelity for general aerodynamic trends to be extracted. A parametric study also demonstrated that the aerodynamic benefits to be hoped for were mostly dependant on the jet peak Mach number and oscillating frequency (which are intrinsic characteristics of the actuator itself) rather than its injection angle. It is to be noted however that these computations do not account for the presence of a cavity below the orifice, which has since been shown to be of fundamental importance for trustworthy comparisons to be made12. In a first attempt to combine leading and trailing-edge separation control, Greenblatt13 successfully performed experiments on a rectangular planform wing equipped with three equal-span flaps hinged at 70% of a NACA 0015 airfoil. Unsteady control is provided by two independently operated two-dimensional slots located at the leading edge (behaving, to some extent, as a proper leading-edge device) and at the upper flap-shoulder (equivalent to an additional flap deflection) of the wing. An important factor for maximized control efficiency was found in the operating phase between each actuator, where aft-suction should be applied when the upcoming shear layer (generated by the forebody excitation) sits closest to the flap surface. Galbraith14 performed URANS simulations of the flow around a novel high-lift airfoil having a conventional slotted slat and a simply hinged flap deflected at 40 deg. Separation control was performed via an unsteady surface boundary condition applied near the flap shoulder. A parametric study, interested in the influences of injection angle, frequency, jet magnitude and amplitude, is presented and discussed for different blowing

characteristics (pulsed blowing, pulsed suction, and zero net mass-flux). In general, the computations were found to be in good agreement with already available experiments. Lift improvements are reported but flowfield analysis suggest a single momentum source might not be capable of fully reattaching the flap flow for reasonable magnitude inputs. In the frame of the ADVINT program, Kiedaisch et al.15 conducted two- and three-dimensional (swept wing) experiments on a series of high-lift designs (conventional slotted slat, cruise and drooped leading edges, slotted or simply hinged flaps), actuated at several locations and backed-up by a CFD survey by Shmilovich and Yadlin16 (see also the work by Nagib et al.17 for a discussion on similarity parameters). To meet their performance goals, the authors suggest that a jet to local velocity ratio beyond two would be required, which could not be achieved for the leading-edge part. On the other hand, the use of active flow control near the flap shoulder, although not capable of complete reattachment, was seen to improve the complete airfoil performance, from increased circulation. Eventually, the introduction of a spanwise flow component did not alter flow control efficiency, as also demonstrated by Seifert et al.18. Computational results yielded the same conclusions with favorable comparisons against the experiments. Tests with distributed actuation were performed and showed that the combination of five, equally spaced slots operating in-phase could fully reattach the flow over the flap. In a series of publications, Pack Melton et al.19-21 performed wind-tunnel measurements on a supercritical airfoil fitted with a 15% and a 25% chord-length simply hinged slat and flap. Synthetic control was applied at several leading- and trailing-edge positions and tests with combined actuated locations performed. Overall, it was found that larger momentum inputs are required for an effective management of the flap separation (when compared to the slat counterpart), with emphasis put on the importance of surface curvature. Encouraging results were also reported from synchronous fore- and aft-excitations. Objectives

Although appealing, the suppression of the slat and flap slots is not as easy as it would originally appear. They are important elements in which the entire performance of conventional high-lift configurations is to be found, from limitations to local pressure rises (the primary cause of flow separation) and a more homogeneous repartition of the complete airfoil loading over its three characteristic elements. Adding the presence of rapid and rather severe geometry changes to the loss of these previously mentioned elliptical effects, it is easy to understand that the aerodynamic behavior of uncontrolled, slotless geometries is relatively poor in its original basis, being subject to massive separations and early stall.

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While the above investigations demonstrate various degrees of improvements from the introduction of flow control concepts, the sensitivity of aerodynamic performance to design choices makes it a non trivial and certainly expensive problem, the number of actuation parameters and combinations thereof being almost infinite. As a possible answer22,23, a completely automated, surrogate-based optimization24,25 algorithm was developed and applied to the design of a two-dimensional, high-lift system having a conventional, slotted slat and a simply hinged, slotless flap. The process will now be described in details and results for a continuously blowing slot presented.

Overview of the Optimization Process A global flow chart for any surrogate-based optimization loop is presented in Fig. 1. Details for each individual procedure are given in the following sections. The framework of this algorithm is a collection of Fortran/Shell routines encompassed in a Python interface.

Fig. 1 Outline of surrogate-based optimization algor ithms

Sampling The construction of a surrogate model of any black-box function requires the creation of a database of precisely-evaluated individuals. By database, we mean a collection of samples, spread all around or slightly outside the design space of interest, which should be as representative as possible of the interactions existing between the design variables, on the one side, and the objective functions to be optimized on the other side. It is

ideal for the sampled points to be distributed homogeneously across the search space so that no region is left without a single representative. The apparition of Design of Experiments (DoE) concepts has introduced a large number of sampling methods, including : fully random, Monte-Carlo, Tagushi, Halton/Hammersley, central Voronoï tessellation, or Latin hypercube each with its advantages and drawbacks26,27. Out of these, the constrained space-filling Latin hypercube sampling (LHS) was chosen, for its simplicity and yet good overall performance compared to other, more complex formulations. In the case where we wish to gather N samples of P variables (where N ≥ 3P is a first rule of thumb for a correct initial representation), the LHS strategy performs as follow: • divide the range of each variable into N non-

overlapping, equal probability (hence equal size) intervals,

• from a user-specified (usually uniform) probability density, arbitrarily select one value from each interval in every direction and randomly pair (equally likely combinations) the N values of each dimension.

Design Analyzer : RANS CFD Flow Solver The aerodynamic performances of each design were evaluated using ONERA’s structured, multiblock, cell-centred, elsA CFD software28. A second order centred scheme with added artificial viscosity was chosen for spatial discretisation, the system of equations being resolved with an implicit LU-SSOR relaxation method and a multigrid algorithm. Turbulence is modelled using the one transport-equation formulation of Spalart and Allmaras29 (SA). Precise, fully turbulent, boundary-layer resolutions gave average y+ values below unity for all the meshes considered in this work. Surrogate Model

The major disadvantage of classical optimization techniques is the important number of calls (at least once per iteration) to the high fidelity analyzer for objective function evaluations, which remains very expensive indeed in computational time and costs. An alternative is the construction of a response model that links the design variables with the inputs for the optimizer (i.e. objective functions) through a simple and inexpensive, yet accurate, relation. The most widely used substitute models include neural network, polynomial-based formulations and, gaining more and more popularity, the Kriging linear interpolation method, as formulated by Sacks et al.30.

Consider a globally unknown function, Y, expressed as a stochastic process:

)()( xZxY += β (1)

where x is the P-dimensional vector of design variables, � a global constant of the model (its average) and Z(x) an

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error, or deviation, term assumed to have mean zero and covariance: )),((.)(,(( 2

)) jiji xxfRxZxZCoV σ= (2) between Z(xi) and Z(xj), where σ2 is the process variance and R a symmetric, N× N correlation matrix with ones along the diagonal. f is a user-specified correlation function. To estimate the random components of Y, the N sample points are interpolated using a Gaussian correlation function. This makes them closely dependant on the relative distance between each design sites, d(xi,xj). Instead of an Euclidian distance, that would weight all the variables equally, the following norm is employed:

�=

−=P

k

kj

ki

kji xxxxd

1

2.),( θ (3)

where kθ ≥ 0 is a correlation parameter vector, or, if deemed sufficient, constant to be determined. Based on that, the correlation function between the errors at xi and xj becomes:

),(

),( jiji

xxdexxf

−= (4)

Now, let r denote the N-vector of correlations between the estimation error at x*, an untried design set, and the error terms at the already known points.

The derivation of the Kriging predictor, not discussed here, yields the following estimation of Y at x*:

)ˆ.1.(.ˆ*)(ˆ 1 ββ −′+= − yRrxy (5)

where β is an estimation of � and y the true value of Y at each sampling point. The determination of the P+2 unknown parameters (�, σ2, θ) is such that they maximize the likelihood of the samples, i.e. the function:

��

�

�

��

�

� − −′−−2

1

2/12/22/

.2).1.(.).1(

..).().2(

1 σββ

σπ

yRy

eRNN

(6)

The corresponding estimates of µ and σ2 being:

1..1

..1ˆ1

1

−

−

′

′=

R

yRβ and

N

yRy )ˆ.1.(.)ˆ.1(ˆ

12 ββ

σ−′−

=−

(7)

The above problem is solved using the same unconstrained non-linear optimization technique as employed for the global problem and described below. The quality of the prediction is obviously affected by the correlation of the errors, that is the distance between x* and the reference points (the closer, the better). This is

expressed in the following expression for root mean square prediction error (RMSE),

2/1

1

211

1..1)..11(

..1.ˆ*)( ��

���

�−

−−

′

′−+′−=

R

rRrRrxs σ (8)

Thus, a Kriging interpolation is able to provide both an estimation of the objective function and some sort of uncertainty quantification at untested design points, which is a clear advantage of the method compared to most of the other meta-models. The Optimizer Amongst the many optimization strategies available (finite-difference or adjoint-based gradients and descent, simulated-annealing31, simplex and multi-directional search22, Nash game theory32…), a derivative-free, genetic algorithm (see Fig. 2 for an overview), GADO33, was preferred for: • its ease of interfacing with readily existing black-box

solvers (the Kriging model in this case), • its supposedly better robustness and independence

towards the choice of an objective function, especially in aerodynamic problems facing possible non-linearity and/or discontinuities,

• its searching pattern based on a population of individuals, making it less amenable to local minima entrapments.

Fig. 2 Genetic-algor ithm optimization pr inciples

A Validation Example Several validation studies were conducted to assess, and if necessary correct, or improve, the quality, accuracy and robustness of each of the above mentioned tools, on the one hand, and of the entire optimization process on the other hand. The prediction capabilities of the elsA CFD software for high-lift34 and flow-control5 problems have been reported elsewhere and will not be further discussed here. Nor will the individual validations of the Kriging module or genetic algorithm. An academic test function for unconstrained optimization, having the same dimensionality as the

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aerospace application to be discussed later on, was chosen. The aim is to find the design vector, x, which minimizes the following mathematical expression, the so-called Hartmann 3-variable problem:

��

=

=−−

−=4

1

3

1

2

3

).(.)(

i

jijjij

i

PxAexH α (9)

where3,1 [1,0]

=∈

iix is the design space. Constants are defined as follow:

����

����

�

=

2.332.1

1

iα ,

����

����

�

=

35101.03010335101.030103

ijA ,

and

����

����

�

=

8828.05743.00381.05547.08732.01091.07470.04387.04699.02673.01170.06890.0

ijP .

This function is known to have 4 local minima and a global minima for x* = (0.122990, 0.458121, 0.852447) and H3(x* ) = -3.934085.

To initiate the minimization loop, 27 samples are distributed in the design hypercube using LHS, and their associated objective function evaluated using (9). The selection of parents for crossover and mutation in GADO uses a pool of 50 individuals and up to 2000 Kriging-based inner optimization iterations are allowed for each of the 100 outer function evaluations.

A convergence history plot is depicted in Fig. 3 and shows that 45 to 50 function evaluations were required to reach the true global minimum, with very little discrepancy when compared to analytical results.

Fig. 3 Convergence history for Hartmann’s test function

minimization problem

The resulting Kriging interpolation (on a 513 Cartesian grid) iso-surface plot is presented in Fig. 4 together with the initial (LHS) and GA-proposed design sites. It can be seen that the choice of H3 as the objective function does not call for sampling diversity and enhancement, the newly added points being gathered in a single region close to the optimum.

As we have seen in the theoretical developments of the Kriging method, the quality of its estimation is closely linked to the distance between each sample. As such, a non-homogenous repartition will inevitably result in a non-uniformly representative response surface. This is less tricky for problems where the aim is to find a unique, best optimum, regardless of the behaviour of the objective function over the entire design space. However, for applications where the meta-model is to be used as a complete replacement for an expensive design analyser, this lack of accuracy could become somewhat dangerous and certainly questionable.

Fig. 4 Kriging interpolation for Hartmann’ s test function

minimisation problem

Fig. 5 Kriging cross-validation results for Hartmann’ s test

function minimisation problem

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This might be better illustrated by the plot of cross-validation results35, Fig. 5. The aim is to validate the resulting response surface as an accurate representative of the problem to be solved, by leaving out one of the original observations and predicting it back using only the N-1 remaining points.

If the model was perfectly exact, the representation of the true objective function value against its cross-predicted counterpart should lie on a line with a unity slope. In our case, the relation is relatively good but far from being uniform, demonstrating that the model is more accurate near the global optimum (where, comparatively, the number of samples is higher), than it is elsewhere. The other problem that could arise from a lack of proper sample distribution is the complete miss of the real optimum. This is especially true when the number of initial evaluations is kept voluntary small because of the computational cost involved in the creation of a population (a typical CFD issue).

Design of Novel High-L ift Devices Equipped with Flow-Control

This study is interested in the mono-objective optimization of flow-control input for a two-dimensional section equipped with a single, constant blowing nozzle (to be seen as a spanwise tangential slot in three dimensions), having a length to diameter aspect ratio of four. All the computations were performed at a chord-based Reynolds number of Rel=2.33×106 and the optimization conducted at a single design point, for an angle of attack of α=25 deg., which corresponds to the stall incidence of a reference geometry (Fig. 10). Air foil Geometry The latter is the GARTEUR high-lift landing configuration, for which extended experimental36 and elsA computational results34 have been obtained. This design was modified in its aft-part to remove the trailing-edge slot and thus create a simply hinged flap, with a deflection angle of δflap=25 deg. (to be compared with δflap=32.4 deg. for the original flap) and articulated at 75% of its chord, as seen in Fig. 6.

Fig. 6 GARTEUR Reference (up) and simplified (down)

high-lift landing geometr ies

Problem Parameter isation, Meshing Strategy and Objective Function

Variations in blowing characteristics are allowed through modifications to the injection abscissa, xinj, the injection angle relative to the local normal, δinj, and the injection speed, Vinj, as diplayed in Fig. 7.

Fig. 7 Problem parameter ization

Design variables can fluctuate in the following ranges:

64 ≤ xinj/lref ≤ 96% (10)

0 ≤ δinj ≤ 70 deg. (11)

0 ≤ Vinj/V� ≤ 3.6 or 0 ≤ Cµ ≤ 2.55% (12)

The latter were chosen based on manufacturing constraints and to remain achievable in realistic experimental conditions, given the current advances in actuator’s development and integration.

Fig. 8 Mesh topology showing the slat and slot over lapping

gr ids and patched gr id approach

The mesh is generated on a partially-automated component-build-up basis, using overset grids for the slat and injection slot, Fig. 8. The latter is linked to the background mesh through a body-conforming interfacing domain, which ensures a correct flux-continuity at the slot exit. The actuator sits in a much refined region, resulting in a two-dimensional grid having approximately 170,000 nodes. A mass-flow type fluid injection is imposed at the entrance of nozzle while its lateral boundaries are treated

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under a no-slip hypothesis, which allows for some boundary-layer developments inside the cavity. Since we are only interested in approach configurations, the primary objective of the optimization is to maximize lift coefficient, at least up to the performances of the baseline design. However, to overcome the difficulties underlined before regarding the importance of response surface representation and improvement, the above problem was transformed to balance the search for a global optimum with the need for extra-sampling at the locations of highest estimation uncertainty. To illustrate that, let us introduce a conceptual mono-dimensional problem, Fig. 9.

Fig. 9 I llustration of Expected Improvement Pr inciples

Let us suppose we want to find the minimum of an original, globally unknown function, based on the existing samples shown above.

As discussed earlier, the resulting Kriging interpolation is accurate around the initial population but becomes fundamentally false outside certain intervals, around x=8 in this case, as proved by the shape of RMSE. Should one run on optimization on this Kriging model, there is a great probability that the optimizer will find the local minimum between x=2 and x=3, and call for a function evaluation at this position. As a result, the updated response surface will have the exact same shape as the original one and the global minimum be certainly missed. A high RMSE, on the other hand, suggests a good place to search from a global representation point of view, but sampling there would mean only improving the quality of the surface without actually looking for an optimum.

Jones et al.37 proposed a figure of merit that balances this need for a combined local and global search, the expected improvement (EI) criterion. If we model our uncertainty at some untried design point by a normal density function with mean and standard deviation suggested by the Kriging model, the expected improvement represents the probability for the objective

function at that point to be better (lower in that case) than the currently known minimum, Fmin. Mathematically, this can be expressed as:

�

��

�

��

−+

−Φ−=

s

yFs

s

yFyFxIE

ˆ.

ˆ).ˆ()]([ minmin

min φ (13)

where φ and Φ are the standard normal density and distribution functions, respectively. In this work, we define y as the ratio of uncontrolled to controlled lift, which we are looking at minimizing:

Controlledl

edUncontrolll

C

Cy

,

,ˆ = (14)

Computational Results for the Reference and Uncontrolled Configurations Before applying any form of separation control, the performance of the newly-defined, uncontrolled geometry was evaluated and compared with the reference, three-element case. By doing so, we obtained a measure of the lift deficit engendered by the suppression of the flap slot as well as preliminary guidance regarding possible actuator positioning, local boundary layer thicknesses, recirculation topologies and equivalent isentropic boundary velocities. The evolution of lift coefficient with angle of attack is represented in Fig. 10 for the baseline (with reference to higher Reynolds number experiments) and uncontrolled cases, showing a 25 to 30% lift loss over the range of practical incidences

Fig. 10 Lift coefficient vs. angle of attack for the reference

and uncontrolled cases

Turbulence modelling dependency was assessed through a comparison between Spalart-Allmaras29 and k-ω Wilcox38 results. The influence of the closure formulation is seen to be fairly negligible throughout the linear part of the polar and up to the maximum lift area, where the two-equation model predicts a higher stall incidence. As a consequence, and in the absence of back-

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up experiments, the SA model was chosen for all the remaining simulations.

Fig. 11 depicts the evolution of surface pressure coefficient with angle of attack. It is interesting to note that the pressure levels are almost independent on incidence over the aft-part of the airfoil (downstream x/lref=50%). For that particular case, the onset of separation is not driven by high pressure gradients (which is usually the case for conventional high-lift devices), but by invariant geometrical constraints, where the flow is unsuccessfully forced to take a rather sharp turn around the upper flap shoulder. When looking at typical flowfield developments, it can be seen that the general topology observed in Fig. 12 remains basically unchanged with increasing angles of attack, which should guarantee that a one-design-point (α=25 deg. here) optimized control scheme will also be effective at off-design conditions.

Fig. 11 Evolution of sur face pressure distr ibution with

incidence for the uncontrolled case

Fig. 12 Contours of Mach number and streamlines for the

uncontrolled case at αααα=25 deg.

Fig. 13 illustrates the evolution of boundary layer thickness above the main-element suction side, together with isentropic surface velocities. Over the attached aft-part of the airfoil, and within the range of interest for possible flow-control input, the height of the boundary layer varies from approximately 10 to 20 mm, giving us a

ratio of ]1.0,05.0[∈δslotl

, in the average of what is

commonly observed in most existing studies. With our

allowed maximum injection speed, a ratio of jet to local velocity of up to VR=3 will be permitted inside the controllable area, which is also similar to already available experimental setups and results.

Fig. 13 Suction-side boundary layer thickness and isentropic

sur face velocities for the uncontrolled case at αααα=25 deg.

Steady/Unsteady Compar isons for the Uncontrolled Case

It is a well-known fact that flow separation is an unsteady, usually three-dimensional, process. Hence, decision was made to perform a time-accurate simulation at the chosen design point, to try and assess the quality and veracity of our steady optimization. The resulting convergence history is plotted in Fig. 14.

Fig. 14 Convergence of steady/unsteady lift and drag

coefficients for the uncontrolled case at αααα=25 deg.

The URANS computation is initiated using a steady RANS solution (from an average value point of view). A transient departure is observed for lift and drag coefficients, corresponding to the capture of a well established vortex-shedding phenomenon, at a frequency of approximately 150 Hz (or a circulation height-based

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Strouhal number of Sth=0.124, Fig. 15). When averaged in time, this corresponds to a reduction in the extent of the separated region which is translated into a substantial lift increment, as seen in Fig. 10. Galbraith14 obtained similar conclusions.

Fig. 15 Instantaneous contours of crossflow vor ticity and

associated par ticle paths for the unsteady, uncontrolled case at αααα=25 deg.

Yet, because of the computational times involved, it would have been very expensive to perform fully URANS-based objective function evaluations. Moreover, since the application of flow control is supposed to delay or alienate flow separation, the sources of possible unsteadiness will be equally absorbed as the optimization progresses, which allows for steady computations to be realistically retained. As such, it was decided that the steady-result uncontrolled lift coefficient be kept as a reference in the evaluation of EI.

Optimization: Results and Discussion Sampling and modifications to the algorithm (hybridising) The construction of the initial Kriging model is based on a LHS-collection of 27 samples. Compared to Hartmann’s test problem, the optimization procedure has been slightly modified to account for the introduction of EI as the objective function. To begin with, 100 EI-designs are added to the database using GADO. The latter is set-up to use an initial population of 100 individuals and up to 2000 mutation/crossover steps are permitted. The best result is then selected as a starting point for a small number (typically 5 to 10 iterations to reach convergence) of finite difference gradient-based descents using CFSQP39 and equation (14) as the function to minimize. This whole process ensures a correct and global surface enrichment while further seeking improved quality in a more local fashion40. Assessment of convergence and Kriging model validation The convergence history in the course of the optimization is shown in Fig. 16, with the corresponding lift coefficient interpolation (on a 513 Cartesian grid) in Fig. 17. In all, 130 RANS simulations were conducted throughout the procedure, and increasing the number of process iterations did not yield any performance improvements.

Fig. 16 L ift coefficient convergence history dur ing the

optimization process

Fig. 17 Sampling evolution and resulting Kriging

interpolation for lift coefficient at the end of the optimization

When compared to the results obtained during our validation study, it can be seen that the use of EI produces a much more scattered final population (which explains the hectic evolution of lift with the growing number of function evaluations), while still putting emphasis on the search for a best individual, a behaviour that was definitely aimed at.

Similar conclusions can be made by looking at cross-validation results, Fig. 18. The samples are almost uniformly distributed across the achievable performance range and well gathered around a linear regression line (except for a few weak exceptions), indicative of a quite strong relationship. Moreover, very little regularization was needed to interpolate the data, suggesting that the Kriging surrogate we obtained is reasonably accurate and well representative of the problem we are modelling.

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Fig. 18 Kriging cross-validation results for the high-lift

optimization application

Review of the best performing candidate Flow-control parameters for the highest-lift design were found as follow:

xinj,opt./lref = 64.5% (15)

δinj,opt. = 66.5 deg. (16)

Vinj,opt./V� = 3.6 or Cµ = 2.55 % (17)

Fig. 19 depicts Mach number contours as well as streamlines for the optimal, controlled scenario, to be compared with the baseline configuration displayed in Fig. 12.

Fig. 19 Contours of Mach number and streamlines for the

optimally controlled case at αααα=25 deg.

The result is quite remarkable indeed, with complete reattachment over the flap and a large modification of the overall flow pattern following the substantial downwash increase at the flap trailing edge. The main-body leading-edge velocity peak effect is now transported towards the flap shoulder, with important benefits in terms of pressure distribution, as seen in Fig. 20. Also noticeable is the complete change of size and shape of the recirculation bubble sitting in the slat cove area. The consequence of momentum input is perceptible from the plot of boundary layer profiles, Fig. 21. The velocity peak corresponding to the high speed injection is propagated all the way downstream, despite the strong pressure gradient to be sustained. Its normal extension, however, is quite limited, which is characteristic of a

Coandà-effect circulation control, that is to say a balance between centrifugal forces and low static-pressure levels from the high momentum introduction of the wall-bounded jet over the convex upper surface41.

Fig. 20 Compar isons of sur face pressure distr ibutions for the

uncontrolled and optimally controlled cases at αααα=25 deg.

Fig. 21 Streamwise velocity evolutions for the uncontrolled

and optimally controlled cases at αααα=25 deg.

Fig. 22 Lift coefficient vs. angle of attack for the reference

uncontrolled and optimally controlled cases

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The extent of control authority was assessed through an incidence sweep and comparisons of lift levels against the reference configuration, Fig. 22. The original goal and motivation behind the application of fluidic momentum input on that particular configuration (the recovery of the reference, fully slotted performances at minimal cost) have been fully achieved over the entire incidence range. It should be noted as well that the controlled geometry has a 7.5 deg., unfavourable flap deflection difference with the baseline geometry, and, contrary to the latter, does not benefit from chord extension as the flap is un-stowed. In fact, there is little doubt that in exactly similar operating conditions, the fittest sample proposed by the optimization procedure would perform even better than the reference does, which demonstrates the validity of our design framework, on the one hand, and of separation-management concepts on the other hand.

Conclusions and Perspectives A surrogate-based, automated framework for the

optimization of flow control-equipped configurations using overset grids is proposed and discussed. The tool was successfully applied to the challenging case of a two-dimensional airfoil having a simply hinged flap and equipped with constant blowing actuation, with the aim of maximizing lift in landing conditions. A complete reattachment of the flap flow was achieved, with an almost total recovery of the aerodynamic efficiency when compared to a conventional, fully slotted reference geometry.

A Coandà-like circulation augmentation was identified as the main source of performance increase and calls for further investigations, to try and understand the influence of each design parameter on the behaviour of the resulting individual as well as minimize massflow requirements (the most important parameter for future in-flight applications). Consequently, future studies will also include the unsteady, multi-objective (taking instantaneous variations as well as average lift levels into account) optimization of periodically blowing actuators (simulating pulsed or synthetic jets, or slots) to evaluate the resulting lift-loss penalty against power or air consumption benefits. Furthermore, the possible application of flow control strategies will be assessed on more realistic three dimensional cases, based on the general results obtained during simpler and cheaper two-dimensional parametric surveys, as the one proposed and validated here.

Acknowledgments Parts of this study were conducted in the frame of a

national program supported by French government agencies (DPAC).

The author would like to thank D. Bailly for the Kriging model, as well as G. Carrier and I. Salah El Din

for their help with GADO. The comments and reviewing efforts of P. Guillen, J. Reneaux and P. Sagaut are also sincerely acknowledged.

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