12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

EXPLORATIVE GRADIENT METHOD FOR HIGH-DIMENSIONALACTUATION PARAMETER SPACES

Yiqing Li
Zhigang Yang
School of Mechanical Engineering and Automation Shanghai Automotive Wind Tunnel Center
Harbin Institute of Technology, Shenzhen
Tongji University
Shenzhen 518055, China
Shanghai 201804, Chinaanne.yiqing.li@outlook.com
zhigangyang@tongji.edu.cn

Marek Morzyński
Bernd R. Noack
Department of Virtual Engineering School of Mechanical Engineering and Automation
Poznań University of Technology
Harbin Institute of Technology, Shenzhen
PL 60-965 Poznań, Poland
Shenzhen 518055, China
morzynski@virtual.edu.pl
bernd.noack@hit.edu.cn

ABSTRACTIn this paper, this challenge is addressed by a new optimizer, called explorative gradient method (EGM) (Li et al.,2022). EGM alternatively performs one exploitive downhill
simplex step and an explorative Latin hypercube sampling iteration. Thus, the convergence rate of a gradient based method
is guaranteed while, at the same time, better minima are explored. EGM is found to be two times faster than the Genetic
algorithm (GA) on an analytical function. When applied to
the fuel distribution optimization across the fuel lines, EGM
only spends a quarter of the cost by the Evolutionary algorithm(EA) (Reumschüssel et al., 2022).
The fluidic pinball (Ishar et al., 2019; Deng et al., 2020)
includes three equal, parallel, equidistantly placed cylinders.
The a two-dimensional flow can be changed by the three rotation velocities of the cylinders. The dynamics is rich innonlinear behaviour, yet geometrically simple and physicallyinterpretable. With suitable rotation of the cylinders many
known wake stabilizing and drag-reducing mechanisms canbe realized: (1) Coanda actuation (Geropp, 1995), (2) circulation control (Magnus effect), (3) base bleed (Wood, 1964), (4)high-frequency forcing (Thiria et al., 2006), (5) low-frequency
forcing (Glezer et al., 2005) and (6) phasor control (Protas,
2004). In this study, constant rotations are optimized for net
drag power reduction accounting for the actuation energy. Thissearch space implies the first three mechanisms.
This study also targets the drag reduction of the low-dragAhmed body with rear slant angle of 35 degrees. This Ahmed
body simplifies the shape of many cars. Bideaux et al. (2011),
Gilliéron & Kourta (2013) have achieved 20% drag reduction
for this configuration in an experiment. High-frequency blowing was applied orthogonal to the upper corner of the slanted
rear surface. Intriguingly, the maximum drag reduction wasachieved in a narrow range of frequencies and actuation velocities and its effect rapidly deteriorated for slightly changed
parameters. In addition, the actuation is neither Coanda blowing nor an ideal candidate for shear-layer energization as theauthors noted.
The manuscript is organized as follows. The employed
optimization algorithm is first introduced. Then, the configuration and the numerical methods of the three benchmarks fol-This paper presents the application of explorative gradientmethod (EGM) (Li et al., 2022) to three open-loop flow control
benchmarks with multiple actuators. The fluidic pinball withthree rotating cylinders, the advanced fluidic pinball with six
rotating cylinders, and the Ahmed body with seven distributed
blowing slots.
The first trial is to minimize the net drag power of the fluidic pinball by three rotating cylinders. The net drag powerof the 3-cylinder configuration is reduced by 29%, 52% dragreduction at a price of 23% actuation energy. The configuration is extended further with 3 more cylinders, which ends
with 14% net drag reduction. The 35◦ slanted Ahmed body
employing distributed steady blowing is included as a morepractical example. 17% drag reduction is achieved in this 10dimenional design space.
The boat tailing as an effective strategy is verified in design spaces with the dimension ranging from 3 to 10. EGM
shows an advantage of tackling high-dimensional optimization
for active flow control by reducing the simulation cost fromO(1000) to O(100).

INTRODUCTION
Actuators and sensors become increasingly cheaper, powerful and reliable. This trend makes active flow control ofincreasing interest to industry. In addition, distributed actuation can give rise to performance benefits over single actuator solutions. Here, we focus on the simple case of openloop control with steady or periodic operation of multiple actuators. Even for this simple case, the optimization of actuation constitutes an algorithmic challenge. Often the budget
for optimization is limited to O(100) high fidelity simulations,like direct numerical simulations (DNS) or large-eddy simulations (LES) or O(100) water tunnel experiments, or O(1000)
Reynolds Averaged Navier-Stokes (RANS) simulations, or asimilar amount of wind-tunnel experiments. Moreover, the optimization may need to be performed for multiple operating
conditions. The efficient optimizers are thus of large practicalimportance (Blanchard & Sapsis, 2021).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Following Cornejo Maceda et al. (2021), we aim to minimize of the averaged parasitic drag power J¯a penalizing the
averaged actuation power J¯b . The resulting cost function reads

low. The optimized control commands and the resulted flow
fields are analyzed hereafter. Finally, The outlook is given in
the conclusion.

J¯ = J¯a + J¯b .

(2)

METHODOLOGYThis study employs the explorative gradient method(EGM) (Li et al., 2022) as the optimizer for the active flow
control with multiple parameters. This approach alternates between downhill simplex method (DSM) as a robust gradientmethod and Latin hypercube sampling (LHS) as the most explorative step. The pusedocode is described as algorithm 1.
First, ND + 1 vertices are initialized as the set S for the DSM
operation, after which the S is updated by the best ND + 1 parameters discovered so far. Here, ND is the dimension of theparameter vector b . Then, one sample x L from the samples
by LHS is selected to test, which is the furthest away fromall hitherto studied data points. If b L is better than any of thesimplex vertices, it will replace the worst vertex to formulated
the new simplex set S. The best vertex b∗ is selected from thelatest simplex vertices when the maximum iteration is met.

The first contribution J¯a = cD corresponds to drag coefficient.
Here, F̄D denotes total averaged drag force on all cylinders
per unit spanwise length. The second contribution arises fromthe necessary actuation torque to overcome the skin-friction
resistance. In the subsequent study, the actuation commands
are bounded by 5, i.e, the search space readsΩ := { b ∈ R N : bi ∈ [−5, 5] for i = 1, . . . , N } .

(3)where N is equal to 3 in figure 1a and equal to 6 in figure 1b.
An in-house implicit finite-element method solver’UNS3’ is employed. It is of third-order accuracy in space andtime. The unstructured grids in figure contain 4225 triangles(8633 vertices) and 4901 triangles (11063 vertices) separately.
This resolution is sufficient for up to 2 percent error in drag,lift and Strouhal number.

Algorithm 1: Explorative gradient method
Data: f , Ω, ND , Ni
Result: b∗ = arg minb∈Ω f (b)D, S,YD ,YS ← InitSamples( f , Ω, ND + 1)
while iter ≤ Ni do
S,YS = DSM( f , S)
D,YD ← S,YS
L = LHS(D)b L = arg max minbi − b j .

Ahmed body
The drag reduction of a 1:3-scaled Ahmed body is alsostudied. The model is characterized by a slanted edge angleof α = 35◦ . Five groups of steady blowing slot actuators are
deployed on all edges of the rear window and the vertical base,see figure 11 in Li et al. (2022). The drag coefficient, J = cD , iscomputed by RANS (Reynolds-Averaged Navier-Stokes simulations) with the varying steady blowing as the control input.
The 10-dimensional designed actuation space b includes amplitudes Ui and directions θi , i = 1, . . . , 5. The former five parameters are capped by the incoming velocity, and the latter by
π/2.

b i ∈L b i ∈L,bb j ∈D

D,YD ← b L , f (bbL )S = sort(YS , f (bbL ), ND + 1)
endb∗ = arg minb∈S YSbi ∈ [0, 2]
for i = 1, . . . , 5 10Ω := b ∈ R : bi ∈ [−35/90, 1] for i = 6
. (4)


bi ∈ [−1, 1]
for i = 7, . . . , 10
The RANS simulations are based on the realizable k − εmodel. The spatial discretization is based on a second-order
upwind scheme in the form of SIMPLE scheme based on apressure-velocity coupling method. The simulations are performed with the commercial flow solver Ansys Fluent. The
prediction of the uncontrolled and controlled cases is validated
by the experiment and LES (Large eddy simulation).

CONFIGURATIONS & NUMERICAL SETUPSThe optimization goal in this study is to find the global
minimum of the aerodynamic performance target J defined for
different configurations in the design parameter space Ω of thecontrol input bb ? = arg min J(bb).
b ∈Ω

(1)

RESULTSThe best control of the fluidic pinball is found after 78
evaluations:

Fluidic pinball
The fluidic pinball (Deng et al., 2020) is a benchmark
configuration for wake control. It is geometrically simple yetrich in nonlinear dynamics behaviours. This configurationconsists of a cluster of three equal, parallel and equidistantly
spaced cylinders pointing in opposite to uniform flow (figure1a). The wake can be controlled by the cylinder rotation. This
study advance the complexity by adding one more columns
with another three cylinders (figure 1b). The actuation commands bi corresponds to the rotation velocities Ui , where thepositive value denotes the anti-clockwise direction.

b ? = (−0.08, 1.13, −1.15).

(5)

The cost function J ? = 1.3 reveals a net drag power saving of
29% with respect to the unforced value Ju = 1.8235. This nearoptimal actuation corresponds to 52% drag reduction, which
requires 23% investment in actuation energy.
The optimization cost increases exponentially with thecomplexity added to the 6 cylinders configuration. It takes2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(b)

(a)

Figure 1: Fluidic pinball. (a) 3-cylinder configuration and (b) 6-cylinder configuration.

more than 400 evaluations to find the optimum J = 2.457(Ju = 2.8597), 14% drag reduction, by the control commandb ? = (0.39, −0.15, 1.68, 0.05, −0.74, 0.09).

for the square-back Ahmed body (Barros et al., 2016). Improved drag reduction with inward as opposed to tangential
blowing was also observed for the 25◦ high-drag Ahmed body
(Zhang et al., 2018) and the square back version (Schmidt
et al., 2015). However, this is the first time the additive effects by the actuation direction is found on the slanted lowdrag Ahmed body.
Figure 3 shows the streamwise velocity component andstreamlines of the transversal velocity in the same plane forthe unforced and actuated cases. The subfigures colorcode
the pressure coefficient (left) as well as streamwise (middle)
and vertical velocity components (right). The solid circle
marks the furthest downstream extend of the dead-water region (x, z)DW = arg maxx {u(x, 0, z) ≤ 0}. The squares denote
in-plane velocity equilibrium associated with the vortices. The
optimized flow is associated with an increase of the recirculation bubble and a more symmetric wake.
The more the wake is elongated the smaller the pressure
gradient. A larger pressure in the near wake is related to thelower drag of the bluff body. This is confirmed by the pressure coefficient contours of figure 3 (left). Such correlation
between length of the recirculation bubble and drag reduction has also been reported in actuated cylinder wakes (Gerhard et al., 2003; Thiria et al., 2006). Moreover, the wake becomes more slender and symmetric as featured by the velocity
equilibrium points marking the vortex centers (solid squares).
The increased up-down symmetry is facilitated by the upward
blowing of the bottom jet. This peripheral inward blowing
enables aerodynamic boat-tailing (Geropp, 1995) as new evidently more effective drag reduction mechanism. Similar observation is also reported on the square-back Ahmed body
(Barros et al., 2016; Haffner et al., 2020), and a 25-degree
Ahmed body Rossitto et al. (2016).

(6)

Both the best actuation commands mimic nearly symmetric
Coanda forcing with a circumferential velocity of nearly 1 (figure 2). This actuation deflects the flow towards the positive
x-axis and effectively removes the dead-water region with reversal flow. The slight asymmetry of the actuation is not a
bug but a feature of the optimal actuation after the pitchfork
bifurcation at Re2 ≈ 68. This achieved performance and actuation is similar to the optimization feedback control achieved
by machine learning control (Cornejo Maceda et al., 2019),
comprising a slightly asymmetric Coanda actuation with small
phasor control from the front cylinder. Also, the optimized experimental stabilization of the high-Reynolds number regime
lead to asymmetric steady actuation (Raibaudo et al., 2020).
The asymmetric forcing may be linked to the fact that the unstable asymmetric steady Navier-Stokes solutions have a lower
drag than the unstable symmetric solution.

Topb1 = 0.8611
θ1 = −27◦

Upper sideb2 = 0.9856
θ2 = −42◦

Middleb3 = 0.0726
θ3 = 67◦

Lower sideb4 = 1.0089
θ4 = −44◦

Bottomb5 = 0.8981
θ5 = 22◦

Table 1: Optimized actuation of the Ahmed body

The optimal actuation command for the Ahmed bodyfound by EGM (see table 1) leads to 17% drag reduction (cD =0.2586) compared with the unforced flow (cD = 0.3134). It
takes only 354 RANS evaluations by the subspace-aided strategy. All peripheral actuators are directed inward. The top
and bottom jets have inclinations of 27◦ and 22◦ , respectively,
while side jets feature stronger inward angles of 42◦ and 44◦ ,respectively. Intriguingly, the additional drag reduction by inward deflection of the jet-slot actuators has also been observed
(a)

(b)

(c)

(d)

CONCLUSION
This paper reports an application study of the newly proposed optimizer — explorative gradient method (EGM). Three
configurations with increasingly complex design space are included. Owing to the advantage of EGM over non-convex optimization problems, all the optimal solutions are targeted onlyat O(100) simulations.
The net drag reduction of fluidic pinball with a 3- and6-dimensional control space study foreshadows the benefit of
the boat tailing solution by Coanda forcing. Finally, this strategy turns out to be the optimal drag reduction solution in a
10-dimensional design space composed of the steady blowing
actuation for 35◦ Ahmed body.
EGM is not only an efficient but also a versatile optimizer
framework with various future applications. In addition to parameter optimization, EGM can also be applied to model-freecontrol law optimization, hitherto performed by genetic programming (Gautier et al., 2015; Ren et al., 2020) and deep reinforcement learning (Rabault et al., 2019; Bucci et al., 2019).

Figure 2: The unforced and optimally controlled wakesof 3-cylinder (a, b) and 6-cylinder (c, d) fluidic pinball.
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 3: Flow visualization of the unforced and optimally controlled Ahmed body wakes in the symmetry plane y = 0.

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