Explorative gradient method for active drag reduction of the fluidic pinball and slanted Ahmed body
Yiqing Li, Wenshi Cui, Qing Jia, Qiliang Li, Zhigang Yang, Marek Morzyński, Bernd R. Noack
UUnder consideration for publication in J. Fluid Mech. Optimization of active drag reduction for aslanted Ahmed body in a high-dimensionalparameter space
Yiqing Li , , Wenshi Cui , , Qing Jia , , Qiliang Li , , ZhigangYang , , and Bernd R. Noack , † Shanghai Automotive Wind Tunnel Center, Tongji University, 4800 CaoAn Road, JiadingDistrict, Shanghai 201804 Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems,Shanghai 201804, China Beijing Aeronautical Science & Technology Research Institute, Beijing, 102211, China LIMSI, CNRS, Universit´e Paris-Saclay, Bˆat 507, rue du Belv´ed`ere, Campus Universitaire,F-91403 Orsay, France Hermann-F¨ottinger-Institut, Technische Universit¨at Berlin, M¨uller-Breslau-Straße 8, D-10623Berlin, Germany(Received ?; revised ?; accepted ?. - To be entered by editorial office)
We numerically optimize drag of the 35 ◦ slanted Ahmed body with active flow controlusing Reynolds-Averaged Navier-Stokes simulations (RANS). The Reynolds number is Re H = 1 . × based on the height of the Ahmed body. The wake is controlled withseven local jet slot actuators at all trailing edges. Symmetric operation corresponds to fiveindependent actuator groups at top, midle, bottom, top sides and bottom sides. Each slotactuator produces a uniform jet with the velocity and angle as free parameters. The dragis reduced by 17% optimizing all 10 actuation parameters as free input. The optimalactuation emulates boat tailing by inward-directed blowing with velocities which arecomparable to the oncoming velocity. The results are well aligned with an experimentaldrag reduction of a square Ahmed body by boat tailing with using Coanda actuation(Barros et al.
1. Introduction
In this study, we focus on active drag reduction behind a generic car model usingReynolds-averaged Navier-Stokes (RANS) simulations. Aerodynamic drag is a majorcontribution of traffic-related costs, from airborne to ground and marine traffic. A smalldrag reduction would have a dramatic economic effect considering that transportationaccounts for approximately 20% of global energy consumption, Gad-el Hak (2006); Kim(2011). While the drag of airplanes and ships is largely caused by skin-friction, the † Email address for correspondence: [email protected], [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] M a y Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack resistance of cars and trucks is mainly caused by pressure or bluff-body drag. (Hucho2002) defines bodies with a pressure drag exceeding the skin-friction contribution as bluffand as streamlined otherwise.The pressure drag of cars and trucks originates from the excess pressure at the frontscaling with the dynamic pressure and a low-pressure region at the rear side of lowerbut negative magnitude. The reduction of the pressure contribution from the front sideoften requires significant changes of the aerodynamic design. Few active control solutionsfor the front drag reduction have been suggested Minelli et al. (2016). In contrast, thecontribution at the rearward side can significantly be changed with passive or activemeans. Drag reductions of 10% to 20% are common, (Pfeiffer & King 2014) have evenachieved 25% drag reduction with active blowing. For a car at a speed of 120km/h, thiswould reduce consumption by about 1.8 liter per 100 km. The economic impact of dragreduction is significant for trucking fleets with a profit margin of only 2-3%. Two thirdsof the operating costs are from fuel consumption. Hence, a 5% reduction of fuel costsfrom aerodynamic drag corresponds to over 100% increase of the profit margin.The car and truck design is largely determined by practical and aesthetic considera-tions. In this study, we focus on drag reduction by passive or active means at the rearwardside. Intriguingly most drag reductions of bluff body fall in the categories of Kirchhoffsolution and aerodynamic boat tailing. The first strategy may be idealized by the Kirch-hoff solution, i.e. potential flow around the car with infinitely thin shear-layers from therearward separation lines, separating the oncoming flow and a dead-water region. Thelow-pressure region due to curved shear-layers is replaced by an elongated, ideally in-finitely long wake with small, ideally vanishing curvature of the shear-layer. This wakeelongation is achieved by reducing entrainment through the shear-layer, e.g. by phasor-control control mitigating vortex shedding (Pastoor et al. et al. et al. et al. et al. (2016) has achieved 20% drag reductionof a square-back Ahmed body with high-frequency Coanda blowing in a high-Reynolds-number experiment. A similar drag reduction was achieved with steady blowing but athigher C µ values.This study focuses on drag reduction of the low-drag Ahmed body with rear slantangle of 35 degrees. This Ahmed body idealizes the shape of many cars. (Bideaux et al. et al. rag reduction of an Ahmed body Figure 1.
Dimensions of the investigated 1:3-scaled Ahmed body. a ) Side view. b ) Back view.The length unit is mm and the angle is specified in degrees. angles and amplitudes of steady blowing of five actuator groups: one the top, middleand bottom edge and two symmetric actuators at the corners of the slanted and verti-cal surface. High-frequency forcing is not considered, as the RANS tends to be overlydissipative to the actuation response.The choice of optimization algorithm is critical for an acceptable computational load. Asimple gradient search, like the downhill simplex method, may provide a drag minimum inmany dozens of simulations, but there is no guarantee for a global minimum. On the otherhand, exploratory strategies, like Monte Carlo methods or Latin hypercube sampling, willeventually come close to the global minimum, but tend to be prohibitively expensive. Inthis study, we combine exploitation and exploration in a single novel explorative gradientsearch strategy.The goal of the current study is a simulation-based optimization of actuation for thelow-drag Ahmed body. The manuscript is organized as follows. § § §
4, 5, and 6 describe the optimization for the one-, five- and ten-dimensional actua-tion space, respectively. The first one-dimensional space contains the streamwise velocityof the top jet actuator. The five-dimensional space comprises the streamwise velocities ofthe 5 symmetric actuations. And the ten-dimensional space includes in addition to thesevelocities also the orientation angle. Our results are summarized in §
2. Configuration and RANS simulation
Starting point of the computational fluid dynamics plant is an experimental study of alow-drag 35 ◦ Ahmed body (Li et al. § § § Configuration
Point of departure is an experimentally investigated 1:3-scaled Ahmed body characterizedby slanted edge angle of α = 35 ◦ with length L , width W and height H of 348mm, 130 mm and 96mm, respectively. The front edges are rounded with a radius of 0 . H . The modelis placed on four cylindrical supports with a diameter equal to 10 mm and the groundclearance is 0 . H . The origin of the Cartesian coordinate system ( x, y, z ), is located inthe symmetry plane on the lower edge of the model’s vertical base (see figure 1). Here, x , y and z denote the streamwise, spanwise and wall-normal coordinate, respectively. Thevelocity components in the x , y and z directions are denoted by u , v and w , respectively.The free-stream velocity is chosen to be U ∞ = 30 m / s. Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack
Figure 2.
Deployment and blowing direction of actuators on the rear window and the verticalbase. The angles θ , θ , θ , θ and θ are all defined to be positive when pointing outward orupward. Figure 3.
Computational domain of the RANS simulation.
Five groups of steadily blowing slot actuators (figure 2) are deployed on all edges of therear window and the vertical base. All slot widths are 2 mm. The horizontal actuators atthe top, middle and bottom side have lengths of 109 mm. The upper and lower sidewiseactuators on the upper and vertical rear window have a length of 71 mm and 48 mm,respectively. The actuation velocities U , . . . , U are independent parameters. U refersto the upper edge of the rear window, U to the middle edge and U to the lower edgeof the vertical base. U and U correspond to the velocities at the right and left sides ofthe upper and lower window, respectively.Following the experiment by Zhang et al. (2018), all blowing angles can be varied asindicated in figure 2. This study aims at minimizing drag as represented by the dragcoefficient, J = c D , by varying the actuation control parameters. The actuation velocityamplitudes U i , i = 1 , . . . , §
4. The actuation angles θ i , i = 1 , . . . , ◦ , i.e. streamwise direction, in a 5-dimensional optimization. The actuation angles are later added into the input parametersin 10-dimensional optimization, with variable angles θ ∈ [ − ◦ , ◦ ], θ , θ , θ , θ ∈ [ − ◦ , ◦ ]. 2.2. Reynolds-Average Navier-Stokes (RANS) simulations
A numerical wind tunnel (figure 3) is constructed using the commercial grid genera-tion software Ansys ICEM CFD. The rectangular computational domain is bounded by X (cid:54) x (cid:54) X , 0 (cid:54) z (cid:54) H T , | y | (cid:54) W T /
2. Here, X = − . H, X = 20 . H , H T = 4 H ,and W T = 9 . H . A coarse, medium and fine mesh using unstructured hexahedral com-putational grid are employed in order to evaluate the performance of RANS method for rag reduction of an Ahmed body Mesh grid points 2.5M 5M 10MDrag Coefficient 0.294 0.313 0.318
Table 1.
Drag coefficient based on different mesh resolutions.
Figure 4.
Side view of the computational grid used for RANS. the current problem with different mesh resolutions. The statistics in Table 1 show thatusing a finer mesh can be expected to have negligible improvement on the accuracy of thedrag coefficient. Hence, the more economical medium mesh 4 is used. This mesh consists of5 million elements and features dimensionless wall values ∆ x + = 20 , ∆ y + = 3 , ∆ z + = 30.In addition to resolving the boundary layer, the shear layers and the near-wake region,the mesh near the actuation slots is also refined.Reynolds-Averaged Navier-Stokes (RANS) simulations using the realizable k − (cid:15) modelwith the constant parameters σ k = 1 . , σ ε = 1 . , C = 1 . C = max (cid:18) . , ηη + 5 (cid:19) η = (cid:88) i,j =1 E ij E ji / kεE ij = 12 (cid:18) ∂u i ∂x j + ∂u j ∂x i (cid:19) σ k = 1 . σ (cid:15) = 1 . C = 1 . et al. et al. et al. et al. et al. Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack
Figure 5.
Experimental setup with the Ahmed body in the model wind tunnel (a). Photo (b)shows the open test section with the Ahmed body (B) linked to a six component balance. A isthe wind-tunnel jet; C is its collector.
Companion experiment
The experiment was implemented in a scaled wind-tunnel with 3 : 4 open jet in theShanghai Automotive Wind Tunnel Center (SAWTC) with an geometric blockage ofapproximately 10% (see figure 5). The Ahmed body was placed at the correspondingposition in the wind tunnel as figure 3. The force measurement system consists of anexternal six-component balance, a balance platform, an Ahmed body model supportsystem, a signal amplifier, a data collecting system and a data processing system. At anincoming velocity of 30 m/s , the drag coefficient obtained from the drag measurementis 0 . et al. ◦ slanted Ahmed body is numerically and experimentallyfound to be statistically symmetric without forcing and under symmetric forcing. Thisobservations is consistent with other control studies of this slanted version (Krentel et al. et al. et al. et al. et al.
3. Optimization algorithms
In this section, the employed optimization algorithms for the actuation parametersare described. Let J ( b ) be the cost function—here the drag coefficient—depending on N actuation parameters b = ( b , . . . , b N ) in the domain Ω, b = ( b , . . . , b N ) ∈ Ω ⊂ R N . The optimization goal is to find the global minimum of J in Ω, b (cid:63) = arg min b ∈ Ω J ( b ) . (3.1)Starting point is the downhill simplex algorithm (see, e.g. Press et al. § et al. explorative gradient search algorithm explained in § rag reduction of an Ahmed body J ( b , b ) = 1 − . e − b − . − b − . − e − b − . − b − . (3.2)with the argument in square domain Ω := { ( b , b ) : 0 (cid:54) b , b (cid:54) } . The cost J has twominima, a local one J ≈ . . , .
5) and a global one J ≈ . , . § Downhill simplex search
The downhill simplex method by Nelder & Mead (1965) is a very simple, robust andwidely used gradient search algorithm. This method does not require any gradient infor-mation and is well suited for expensive function evaluations, like the considered RANSsimulation for the drag coefficients, and for experimental optimizations with inevitablenoise. A price is a slow convergence for the minimization of smooth functions as comparedto algorithms which can exploit gradient and curvature information.We briefly outline the employed downhill simplex algorithm, as there are many variants.First, N +1 vertices b m , m = 1 , . . . , N +1 in Ω are initialized as detailed in the respectivesections. Commonly, b is placed somewhere in the middle of the domain and the othervertices explore steps in all directions, b m = b + h e m − , m = 2 , . . . , N + 1. Here, e i = ( δ i , . . . , δ iN ) is a unit vector in i -th direction and h is a step size which is smallcompared to the domain. Evidently, all vertices must remain in the domain b m ∈ Ω.The goal of the simplex transformation iteration is to replace the worst argument b h of the considered simplex by a new better one b N +2 . This is archived in following steps:
1) Ordering:
Without loss of generality, we assume that the vertices are sorted interms of the cost values J m = J ( b m ): J (cid:54) J (cid:54) . . . (cid:54) J N +1 .
2) Centroid:
In the second step, the centroid of the best side opposite to the worstvertex b N +1 is computed: c = 1 N N (cid:88) m =1 b m .
3) Reflection:
Reflect the worst simplex b N +1 at the best side, b r = c + ( c − b N +1 )and compute the new cost J r = J ( b r ). Take b r as new vertex, if J (cid:54) J r (cid:54) J N . b m , m = 1 . . . , N and b r define the new simplex for the next iteration. Renumber the indicesto the 1 . . . N + 1 range. Now, the cost is better than the second worst value J N , but notas good as the best one J . Start a new iteration with step 1.
4) Expansion: If J r < J , expand in this direction further by a factor 2, b e = c + 2 ( b N +1 − c ) . Take the best vertex of b r and b e as b N +1 replacement and start a new iteration.
5) Single contraction:
At this stage, J r (cid:62) J N . Contract the worst vertex half-waytowards centroid, b c = c + (1 /
2) ( c − b N +1 ) . Take b c as new vertex ( b N +1 replacement), if it is better than the worst one, i.e. J c (cid:54) J N +1 . In this case, start the next iteration.
6) Shrink / multiple contraction:
At this stage, none of the above operations wassuccessful. Shrink the whole simplex by a factor 1 / Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m J m b m b (a)(b)(c) Figure 6.
Downhill simplex algorithm applied to the test function (3.2). The figure shows theiteration of (a) b , (b) b and (c) the associated cost J . A red circle marks a newly discoveredminimum, while blue circles indicate the best unchanged value of the simplex. m counts theevaluations of the cost function J . all vertices by b m (cid:55)→ b + 12 ( b m − b ) , m = 2 , . . . , N + 1 . This shrinked simplex represents the one for the next iteration. It should be noted thatthis shrinking operation is the last resort as it is very expensive with N function eval-uations. The rational behind this shrinking is that a smaller simplex may better followlocal gradients.Figure 6 illustrates the simplex algorithm for the analytical function (3.2) with theinitial simplex b = (0 , b = (0 , . b = (0 . , . , . Latin hypercube sampling
The downhill simplex method exploits neighbourhood information to slide down to alocal minimum. Latin hypercube sampling (LHS) (McKay et al. maximin ’ criterion in Mathematica) minimizes the maximum minimal distance rag reduction of an Ahmed body J Figure 7.
Same downhill simplex iteration as in figure 6, but displayed in the two-dimensionalparameter space. Red solid circles mark new optima while blue open circles denote investigatedsuboptimal vertices. The numbers in the circles correspond to evaluation counter m . between the points: { b m } Mm =1 = arg max b m ∈ Ω min i =1 ,...,M − j = i +1 ,...,M (cid:107) b i − b j (cid:107) . In other words, there is no other sampling of M parameters with a larger minimumdistance. M can be any positive integral number.For better comparison with the simplex algorithm, we employ an iterative variant.Note that once M sample points are created and they cannot be augmented anymore,for instance when learning of LHS was not satisfactory. We create a large number ofLHS parameters b (cid:63)j , j = 1 , . . . , M (cid:63) for a dense coverage of the parameter space Ω at thebeginning, typically M (cid:63) = 10 . As first sample b , the center of the initial simplex istaken. The second parameter is taken from b (cid:63)j , j = 1 , . . . , M (cid:63) maximizing the distanceto b , b = argmax b (cid:63)j ,j =1 ,...,M (cid:63) (cid:107) b (cid:63)j − b (cid:107) . The third parameter b is taken from the same set so that the minimal distance to b and b is maximized and so on. This procedure allows to recursively refine sample pointsand to start with an initial set of parameters.Figure 8 illustrates the LHS for the analytical test function. In contrast to the downhillsimplex algorithm, LHS arrives near the global optimum at m = 15. From figure 9, thedomain Ω is nearly homogeneously covered with samples. The next significant improve-ment requires and much finer ‘resolution’ and orders of magnitudes more samples.0 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m b m b m J (a)(b)(c) Figure 8.
Same as figure 6, but for Latin Hypercube Sampling.
Explorative gradient search
In this section, we combine the advantages of the exploitive downhill simplex methodand the explorative LHS in a single algorithm.
Step 0—Initialize.
First, b m , m = 1 , . . . , M + 1 are initialized for the downhill sim-plex algorithm. Step 1—LHS.
Compute the cost J of a new LHS parameter b . As described above,we take a parameter from a precomputed list which is furthest away from all hithertoemployed parameters. Step 2—Downhill Simplex.
Perform one simplex iteration ( § M +1 parameters discovered so far. Step 3—Loop.
Continue with Step 1 until a convergence criterion is met.The algorithm is intuitively appealing. If the LHS discovers a parameter with a cost J inthe top M + 1 values, this parameter is included in the new simplex and correspondingiteration may slide down in another better minimum.The explorative gradient search is applied to the test function starting with the samevertices b , b , b as in § . , . rag reduction of an Ahmed body J Figure 9.
Same as figure 7, but for Latin Hypercube Sampling of figure 8.
A more common approach to avoid descending into a suboptimal minimum is ‘randomrestart or shotgun hill climbing’ (Russell & Norvig 2016), where ‘climbing’ refers to amaximization problem. Here, the gradient search method (here: downhill simplex) isrepeatedly started with random initial values. Note that for O (10) local minima at least O (10) simplex searches with ideally placed initial conditions need to be performed toassess the terrain. The proposed explorative downhill simplex search requires a numberof evaluations corresponding to one additional simplex search at worst: The number ofsimplex and LHS iterations are the same but the simplex algorithm may require (many)more than one function evaluation. Hence, the explorative gradient search increases thecomputation cost by at maximum a factor 2 while profiting the effective exploitation of agradient search. The explorative gradient search has a stochastic explorative componentbut is distinctly different from stochastic gradient methods , like Rechenberg’s evolutionarystrategy (Rechenberg 1973) or particle-swarm optimization (Kennedy 2017). The purposeof stochastic gradient methods is a downhill motion in a high-dimensional parameterspace in which the computation of gradients constitutes a challenge. We refer to theexquisite optimization textbook by Wahde (2008) for other related methods.The no-free-lunch theorem reminds us that there exists no optimal algorithm for un-specified data. For any set of algorithms one can find data favourable to any of thealgorithms. We assume that the landscape is reasonably smooth with with a couple ofminima. 3.4. Computational accelerators
The choice of the initial condition for RANS affects the convergence time for the steadysolution. The first simulation of an optimization starts with the unforced flow as initialcondition. The next iterations exploit the existence of a deterministic mapping fromactuation parameters b to the corresponding averaged velocity field u ( x ). The initial2 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m b m b m J (a)(b)(c) Figure 10.
Same as figure 6, but for the explorative gradient search. condition of the m th simulation is obtained with the 1-nearest-neighbour approach: Thevelocity field associated with the closest hitherto computed actuation vector is taken asinitial condition for the RANS simulation. This simple choice of initial condition savesabout 60% CPU time in reduced convergence time.Another 30% reduction of the CPU time is achieved by avoiding RANS computationswith very similar actuations. This is achieved by a quantization of the b vector: Theactuation velocities are quantized with respect to integral m / s values. This corresponds toincrements of U ∞ /
30 with U ∞ = 30m / s. All actuation vectors are rounded with respect tothis quantization. If the optimization algorithm yields a rounded actuation vector whichhas already been investigated, the drag is taken from the corresponding simulation andno new RANS simulation is performed. Similarly, the angles are discretized into integraldegrees.
4. Formulation of optimization problem based on streamwise blowingat the top edge
The formulation and constraints of the optimization problem is motivated by the dragreduction results from the top actuator blowing in streamwise direction. Figure 12 showsthe drag coefficient in dependency of streamwise blowing velocity, all other actuatorsbeing off. The blowing velocity varies in increments of 5 m / s from 0 m / s to 60 m / s, i.e.reaches twice the oncoming velocity.The drag coefficient is quickly reduced by modest blowing, has a shallow minimum rag reduction of an Ahmed body J Figure 11.
Same as figure 7, but for the explorative gradient search displayed in figure 6. U (m/s) C d M CO
Figure 12.
Drag coefficient as a function of the blowing velocity U of the streamwise-orientedtop actuator. Here, ‘ O ’ marks the drag without forcing, ‘ M ’ the best actuation, and ‘ C ’ thesmallest actuation with worse drag than for unforced flow. Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack near the actuation velocity U b = 25 m / s before quickly increasing with more intenseblowing. This optimal value corresponds to 5 / C d = 0 . U = 45 m / s, thedrag rapidly rises beyond the unforced value.This behaviour motivates the choice of actuation parameters. The first five actuationparameters are normalized jet velocities b i = U i /U b , i = 1 , . . . , § b = 1 corresponds to minimal drag with a single streamwise-oriented top actuator.All b i are capped by 2: b i ∈ [0 , i = 1 , . . . ,
5. At b = 1 .
8, point ‘ C ’ in figure 12,actuation yields already drag increase. The first vertex of the amoeba of the downhillsimplex search is put at b = b = b = b = b = 1 .
8. From figure 12, we expecta drag minimum at lower values, hence the next five vertices test the value 1 .
6, e.g.( b , b , . . . , b ) = (1 . − . δ ,m − , . − . δ ,m − , . . . , . − . δ ,m − ) for m = 2 , . . . , b i (cid:62) − oraround 3% of uniformly distributed sampling points are in the original domain and 97%of the samples are outside.The next five parameters characterize the deflection of the actuator velocity withrespect to the streamwise direction (see § b i +5 = θ i / ( π/ i = 1 . . . ,
5, and arenormalized with 90 ◦ . Now all b i , i = 1 , . . . ,
10 span an interval of width 2, except for themore limited deflection b of the top actuator. Summarizing, the domain for the mostgeneral actuation readsΩ := b ∈ R : b i ∈ [0 ,
2] for i = 1 , . . . , b i ∈ [ − / ,
1] for i = 6 b i ∈ [ − ,
1] for i = 7 , . . . , . (4.1)The choice of b as symbol shall remind about the control B -matrix in control theoryand is consistent with many earlier publications of the authors, e.g. the review article byBrunton & Noack (2015).
5. Optimization of the streamwise trailing edge actuation
The drag of the Ahmed body is optimized with streamwise blowing from the fiveslot actuators. We apply a simplex downhill search, Latin hypercube sampling and theexplorative gradient search of § § § Downhill simplex algorithm
Following §
4, the downhill simplex algorithm is centered around b i = 1 . i = 1 , . . . , b m − = 1 . m = 2 , . . . ,
6. Table 2 shows the values of the individuals and corresponding cost. Allvertices have a larger drag than for the unforced benchmark C d = 0 . b i = 1 . § §
3, solid red circles mark newly found optima while open bluecircles record the best actuation so far. The drag quickly descends after staying shortlyon a plateau at m ≈
20. From there on, the descend becomes gradual. The optimal rag reduction of an Ahmed body m b m b m b m b m b m J (a)(b)(c)(d)(e)(f) Figure 13.
Optimization of the streamwise trailing edge actuation during a downhill simplexsearch. The actuation parameters and cost is visualized like in figure 6. m counts the RANSsumulation calls for drag computation. Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m b b b b b J1 1.8 1.8 1.8 1.8 1.8 0.41532 1.6 1.8 1.8 1.8 1.8 0.40483 1.8 1.6 1.8 1.8 1.8 0.41094 1.8 1.8 1.6 1.8 1.8 0.39965 1.8 1.8 1.8 1.6 1.8 0.40756 1.8 1.8 1.8 1.8 1.6 0.4040
Table 2.
Initial simplex ( m = 1 , . . . ,
6) for the five-dimensional downhill simplex optimization. b i are the normalized actuation velocities and J corresponds to the drag coefficient. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2-1.5-1-0.500.511.522.5 J Figure 14.
Proximity map of the streamwise trailing edge actuation during downhill simplexsearch. The contour plot corresponds to the interpolated cost function (drag coefficient) fromall RANS simulations of this section. As in figure 13, solid red circles mark newly find optimawhile open blue circles mark unsuccessful explorations of cost functions. drag J = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . J ( γ , γ ) describedin §
3. Here ( γ , γ ) feature vectors defining a proximity map of the five-dimensionalactuation parameters ( b , . . . , b ). This landscape indicates a complex topology of thefive-dimensional actuation space by many local maxima and minima in the feature plane.This complexity may explain why most simplex steps did not yield a better cost. Thefeature coordinate γ , γ arise from a kinematic optimization process and have no inherent rag reduction of an Ahmed body γ ≈ (2 ,
0) to the assuminglyglobal minimum at γ ≈ ( − . ,
0) through an elongated curved valley.5.2.
Latin hypercube sampling
Figure 15 shows the slow learning process associated with Latin hypercube sampling(LHS) starting with the simplex reference point b = . . . = b = 1 .
8. Apparently theoptimization is ineffective. Only 4 new optima are successively obtained in 200 RANSsimulations. The remaining simulations yield worse drags than the best discovered before.At the 162th RANS simulation, the best drag coefficient of C d = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 1, b = b = b = b = 0. The correspondingLHS actuation parameters read b = 0 . b = 0 . b = 0 . b = 0 . b = 1 . b i amplitudesnear unity while remaining parameters are less than 13% of the one-dimensional optimum.These results show that near optimal drag reductions can be achieved with quite differentactuations. Moreover, individual actuation effects are far from additive. Otherwise, thealmost complimentary LHS optimum for actuators 2–5 and the one-dimensional optimumof § b ≈ b ≈ b ≈ . b ≈ b ≈ m = 162 is still far from the assumingly globalminimum at γ = ( − . ,
0) (see figure 14). The exploratory steps uniformly cover thewhole range of feature vectors.5.3.
Explorative gradient search
From figure 17, the explorative gradient search is seen to converge much faster than thedownhill simplex algorithm. The best actuation is found at the 65th RANS simulationyielding the same drag coefficient C d = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 .
6. Optimization of the directed trailing edge actuation
In this section, the actuation space is enlarged by the jet directions of all slot actuators.The jets may now be directed inwards or outwards as discussed in § § § Explorative gradient search
We employ the explorative gradient search as best performing method of § (cid:104) lower than the previous8 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m J m b m b m b m b m b (a)(b)(c)(d)(e)(f) Figure 15.
Same as figure 13 but for Latin hypercube sampling (streamwise trailing edgeactuation). rag reduction of an Ahmed body -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2-1.5-1-0.500.511.522.5 J Figure 16.
Same as figure 14 but for Latin hypercube sampling (streamwise trailing edgeactuation). m b b b b b b b b b b J Table 3.
Initial individuals in the optimization of the directed trailing edge actuation. b i , i = 1 , , , , U i of the i th actuator. b i , i = 6 , , , , θ i of the ( i − J is the drag coefficient. section as the RANS integration for the first flow is not fully converged. The next fivevertices represent isolated actuations at the optimal value but directed 45 ◦ outwards forthe side edges and upwards for the middle horizontal actuator. The corresponding dragvalues are larger. The next five vertices deflect the jets in opposite direction by 45 ◦ or themaximum 35 ◦ of the top actuator, giving rise smaller drag than the previous deflection.The drag of middle horizontal actuator remains close to the unforced benchmark becausethe jet velocity is small.Figure 19 illustrates the convergence of the explorative gradient search. After 2890 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m J m b m b m b m b m b (a)(b)(c)(d)(e)(f) Figure 17.
Same as figure 13 but for the explorative gradient search (streamwise trailing edgeactuation). rag reduction of an Ahmed body -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2-1.5-1-0.500.511.522.5 J Figure 18.
Same as figure 14 but for the explorative gradient search (streamwise trailing edgeactuation).
RANS simulations, a drag coefficient of 0 . b = 0 . b = 0 . b = 0 . b = 1 . b = 0 . b = − . θ = − ◦ , b = − . θ = − ◦ ), b = 0 . θ = 67 ◦ ), b = − . θ = − ◦ ), and b = 0 . θ = − ◦ ). All outer actuators have velocity amplitudes near unity and are directedinwards, i.e. emulate Coanda blowing. The third middle actuator blows upward with lowamplitude. The strong inward blowing seems to be related to the additional 10% dragreduction as compared to the 7% of streamwise actuation.Figure 20 shows the search process in a proximity map. It should be noted that thiscontrol landscape is based on data in a ten-dimensional actuation space and hence dif-ferent from the 5-dimensional space in §
5. The algorithm quickly descends in the valleywhile many exploration steps probe suboptimal terrain. One reason for this quick land-ing in good terrain is the chosen initial simplex around the optimized actuation in thefive-dimensional subspace.6.2.
Discussion of streamwise and directed jet actuators
In the following, the physical structures associated with the optimized one-, five- and ten-dimensional actuation are discussed. Evidently, more degrees of freedom are associatedwith more opportunities for drag reduction. Expectedly, the drag reduces by 5% to 7% to17% as the dimension of the actuation parameters increase from 1 to 5 to 10, respectively.Intriguingly, the increase of drag reduction from the optimized top actuator to the best5 streamwise actuators is only 2%. For the square-back Ahmed body, Barros (2015)experimentally observed that the individual drag reductions from the streamwise blowingactuators on the four trailing edges roughly add up to the total drag reduction of 10%with all actuators on. This additivity of actuation effects is not corroborated for the2
Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack m -1-0.500.51 b m -1-0.500.51 b m -1-0.500.51 b m -1-0.500.51 b m -1-0.500.51 b m J m b m b m b m b m b (f)(g)(h)(i)(j)(a)(b)(c)(d)(e)(k) Figure 19.
Optimization of the directed trailing edge actuation during explorative gradientsearch. J : cost function (drag coefficient); m the number of RANS calls by the algorithm. Redsolid circles and blue open circles have the same meaning as in previous convergence plots. Theactuation includes the amplitudes: ( a ) b , ( b ) b , ( c ) b , ( d ) b , ( e ) b , and the angles: ( f ) b ,( g ) b , ( h ) b , ( i ) b , ( j ) b . rag reduction of an Ahmed body -1 -0.5 0 0.5 1 1.5 2 2.5 3 -2.5-2-1.5-1-0.500.511.522.5 J Figure 20.
Proximity map of the directed trailing edge actuation during explorative gradientsearch. The contour indicates the interpolated cost function (drag coefficient). Red solid circlesand blue open circles have the same meaning as in previous control landscapes.Case Drag Actuation parametersreduction Top Upper side Middle Lower side BottomA) 0% — — — — —B) 5% b = 1 b = 0 b = 0 b = 0 b = 0C) 7% b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 0 . b = 1 . b = 0 . θ = − ◦ θ = − ◦ θ = 67 ◦ θ = − ◦ θ = − ◦ Table 4.
Investigated optimized actuations in comparison to the unforced benchmark. The tableshows the achieved drag reduction and corresponding actuation parameters for A) the unforcedbenchmark, and for the optimized B) top streamwise actuator, C) all streamwise actuators, D)all deflected actuators. slanted low-drag Ahmed body. Intriguingly, the inward deflection of the jet-slot actuatorssubstantially decreases drag by 10%. This additional drag reduction of 10% has also beenobserved for the square-back Ahmed body when the horizontal jets were deflected inwardwith Coanda surfaces on all four edges (Barros et al. ◦ high-drag Ahmedbody (Zhang et al. et al. (2015).Table 4 summarizes the discussed flows, associated drag reduction and actuation pa-rameters. For brevity, we refer to flows with no, one-dimensional, five-dimensional and4 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack (a) (b)(c) (d)
Figure 21.
Okubo-Weiss parameter Q of flow. a ) without control and under b ) 1D, c ) 5D and d ) 10D control respectively, where Q = 15000 /s . ten-dimensional actuation spaces as case A, B, C and D, respectively. The actuationenergy may be conservatively estimated by the energy flux through all jet actuators: (cid:80) i =1 (cid:82) dA i ρU i /
2. Here, the actuation jet fluid is assumed to be accelerated from 0 tothe actuation jet velocity U i and then deflected after the outlet, e.g. via a Coanda sur-face. In this case, the actuation energy of cases B , C and D would correspond to 3 . .
0% and 7 .
9% of the parasitic drag power, respectively. This expenditure is significantlyless than the saved drag power. The ratio from saved drag power to actuation energy iscomparable for a truck model where steady Coanda blowing with 7% energy expenditureyields a 25% drag reduction (Pfeiffer & King 2014). This estimate should not be taken tooliterally as actuation energy strongly depends on the realization of the actuator. It wouldbe less, more precisely (cid:80) i =1 (cid:82) dA i ρ cos( θ i ) U i /
2. when the actuation jet fluid leaves theAhmed body through a slot directed with the jet velocity and can be expected much lesswhen this fluid is taken from the oncoming flow, e.g. from the front of the Ahmed body.Figure 21 displays iso-surfaces for the same Okubo-Weiss parameter value Q for all fourcases. The unforced case A (figure 21 a ) shows a pronounced C-pillar vortices extending farinto the wake. Under streamwise top actuation (case B, figure 21 b ), the C-pillar vorticessignificantly shorten. The next change with all streamwise actuators optimized (case C)is modest consistent with the small additional drag decrease. The C-pillar vortices areslightly more shortened (see figure 21 c ). The inward deflection of the actuation (case D)is associated with aerodynamic boat tailing as displayed in figure 21 d . The separationfrom the slanted window is significantly delayed and the sidewise separation is vectoredinward.This actuation effect on the C-pillar vortices is corroborated by the streamwise vorticitycontours in a transverse plane on body height downstream ( x/H = 1). Figure 22 showsthis averaged vorticity component for case A–D in subfigure a – d , respectively. The extendof the C-pillar vortices clearly shrink with increasing drag reduction. rag reduction of an Ahmed body (a) (b)(c) (d) Figure 22.
Streamwise vorticity component in near-wake plane x/H = 1. a ) without forcingand under b ) 1D, c ) 5D and d ) 10D control respectively. (a) (b)(c) (d) Figure 23.
Streamwise velocity component in the near-wake tranversal plane x/H = 1 andstreamlines from the in-plane velocity components. a ) without forcing and under b ) 1D, c ) 5Dand d ) 10D control respectively. Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack (a) (b)(c) (d)
Figure 24.
Streamwise velocity component in the symmetry plane y = 0 and streamlines fromthe in-plane velocity components. a ) without forcing and under b ) 1D, c ) 5D and d ) 10D controlrespectively. Figure 23 shows the streamwise velocity component and streamlines of the transversalvelocity in the same plane for the same cases. Cases B and C feature a larger region ofupstream flow while case D has a narrowed regions of backflow. From these visualizations,one may speculate that the drag reduction from streamwise actuation (cases B and C)is due to a wake elongation towards the Kirchhoff solution while the inward directedactuation (case D) is associated with drag reduction from aerodynamic boat-tailing.This hypothesis about different mechanisms of drag reduction is corroborated fromthe streamlines in the symmetry plane y = 0 in figure 24. The tangential blowing (seesubfigures b , c leads to an elongated fuller wake as compared to the unforced bench-mark (subfigure a ). The top shear-layer is oriented more horizontal under streamwiseactuation—consistent with the Kirchhoff wake solution. The inward-directed actuation(see subfigure d ) also elongates the wake but gives rise to a more streamlined shape. Thetop and bottom shear-layers are vectored inward.The drag reduction can more directly be inferred from the C p distribution of the rear-ward windows in figure 25. The 5% drag reduction in subfigure b ) for case B is associatedwith a pressure increase of the vertical surface. The additional 2% drag decrease for caseC in subfigure c is accompanied by an increase over vertical and slanted surface. Theaerodynamic boat-tailing of case D with 17% drag reduction alleviates significantly thepressures on both surfaces.
7. Conclusions
This numerical study proposes a novel optimization approach for active bluff-body con-trol exploiting local gradients with a downhill simplex algorithm and exploring new betterminima with Latin hypercube sampling. The computational load for the exploration isless than for the simplex iterations. This compares favourably with a shotgun downhilloptimization typically requiring dozens of converged downhill simplex applications.This approach named explorative gradient search (EGS) minimizes the drag of an 35 ◦ slanted Ahmed body at Reynolds number Re H = 1 . × with independent steady rag reduction of an Ahmed body (a) (b)(c) (d) Figure 25.
Pressure coefficient on the slant and vertical base of flow. a ) without forcing andunder b ) 1D, c ) 5D and d ) 10D control respectively. blowing at all trailing edges. The 10-dimensional actuation space includes 5 symmetricjet slot actuators or corresponding actuator groups with variable velocity and variableblowing angle. The resulting drag is computed with a Reynolds-Averaged Navier-Stokes(RANS) simulation.The approach is augmented by auxiliary methods for initial conditions, for acceleratedlearning and for a control landscape visualization. The initial condition for a RANSsimulation with a new actuation is computed by the 1-nearest neighbour method. Inother words, the RANS simulation starts with the converged RANS flow of the closesthitherto examined actuation. This cuts the computational cost by 60% as it acceleratesRANS convergence. The actuation velocities are quantized to prevent testing of toosimilar control laws. This optional element reduces the CPU time by roughly 30%. Thelearning process is illustrated in a control landscape. This landscape depicts the drag ina proximity map—a two-dimensional feature space from the high-dimensional actuationresponse. Thus, the complexity of the optimization problem can be assessed.In a analytical example, the explorative gradient search is found to outperform thedownhill simplex method converging to suboptimal minimum and a Latin hypercubesampling needing too many iterations. The slanted Ahmed body with 1, 5 and 10 actu-ation parameters constitutes a more realistic plant for an optimization algorithm. First,only the upper streamwise jet actuator is optimized. This yields drag reduction of 5%with pronounced global minimum for the jet velocity. Second, the drag can be furtherreduced to 7% with 5 independent streamwise symmetric actuation jets. Intriguingly,the actuation effects of the actuator are far from additive—contrary to the experimental8 Y. Li, Z. Yang, W. Cui, Q. Jia, Q. Li, and B. R. Noack observation for the square-back Ahmed body Barros (2015). The optimal parameters of asingle actuator are not closely indicative for the optimal values of the combined actuatorgroups. The control landscape depicts a long curved valley with small gradient leading toa single global minimum. Interestingly, the explorative step is not only a security policyfor the right minimum. It also helps to accelerate the optimization algorithm by jumpingout of the valley to a point closer to the minimum.A significant further drag reduction of 17% is achieved when, in addition to the jetvelocities, also the jet angles are included in the optimization. Intriguingly, all trailingedge jets are deflected inward mimicking the effect of Coanda blowing and leading tofluidic boat tailing. The C-pillar vortices are increasingly weakened with one-, five- andten-dimensional actuation. Compared with the pressure increase at C-pillar in one- andfive-dimensional control, the ten-dimensional control brings a substantial pressure re-covery over the entire base. The achieved 17% drag decrease with constant blowing iscomparable with the experimental 20% reduction with high-frequency forcing by Bideaux et al. (2011); Gilli´eron & Kourta (2013).For the 25 ◦ high-drag Ahmed body, (Zhang et al. .
361 thanthe low-drag version and is hence not fully comparable. Their reduced drag coefficient of0 .
256 is almost identical with the one of this study.We expect that our RANS-based active control optimization is widely applicable forvirtually all multi-input steady actuations or combinations of passive and active controlBruneau et al. (2010). The explorative gradient search mitigates the chances of slidingdown a suboptimal minimum at a acceptable cost. The 1-nearest neighbour method forinitial condition and the actuation quantization accelerate the simulations and learningprocesses. And the control landscape provides the topology of the actuation performance,e.g. the number of local minima, nature and shape of valleys, etc.
Acknowledgements
This work is supported by Shanghai Key Lab of Vehicle Aerodynamics and VehicleThermal Management Systems (Grant No.18DZ2273300), by public grants overseen bythe French National Research Agency (ANR) ANR-11-IDEX-0003-02 (iCODE Instituteproject, IDEX Paris-Saclay), and ANR-17-ASTR-0022 (FlowCon), ’ACTIV ROAD’.We have profited from stimulating discussions with Steven Brunton, Siniˇsa Krajnovi´c,Francois Lusseyran, Navid Nayeri, Oliver Paschereit, Luc Pastur, Richard Semaan, Wolf-gang Schr¨oder, Bingfu Zhang and Yu Zhou.
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