Aeroacoustic shape optimization using adjoint sensitivity analysis based on lattice Boltzmann method for bluff bodies Kazuya Kusano 1 Kyushu University 744 Motooka, Nishi-ku, Fukuoka, Japan

ABSTRACT The present study developed an adjoint-based shape optimization method aimed at suppressing flow- induced sounds. In this method, the direct aeroacoustic simulation was conducted using the lattice Boltzmann equation (LBE) under low-Mach-number conditions. In the simulation, complex shapes of objects were considered using the interpolated bounce-back (IBB) scheme. Furthermore, the sen- sitivities of far-field sounds to object shapes were evaluated by solving the adjoint equation, which was derived from the LBE with IBB scheme. The present method was applied to the cylinder Aeolian tone, and the rear part of the cylinder surface was optimized. The optimized shape suppressed the sound generation by delaying the vortex formation in the wake.

1. INTRODUCTION

In the last few decades, computational aeroacoustics (CAA) have been advanced with the rapid increase in computing power. In CAA, there are two typical approaches for low-Mach-number flows, which are important in various engineering applications. One is the hybrid approach based on Lighthill’s acoustic analogy [1] with the incompressible Navier-Stokes simulation. The other is the lattice Boltzmann method (LBM), which can directly simulate flow-induced sounds under low-Mach- number conditions [2, 3]. In the engineering, these approaches have enabled to predict the sound level of the newly designed products before manufacturing. In addition, the sound generation mechanism can be elucidated by analyzing the simulation results. However, it is still difficult to control flow- induced sounds by modifying object shapes in the aerodynamic design.

A possible approach to overcome this difficulty is the shape optimization based on CAAs. How- ever, the shape optimization is not yet practical for unsteady flow problems including flow induced sounds due to large computational cost. To reduce the computational cost in the aerodynamic shape optimization, the adjoint method has been proposed [4] and widely used in the Navier-Stokes frame- work. The adjoint method can efficiently evaluate the gradients of the objective function to numerous design variables by solving the adjoint equation. Recently, the adjoint method based on the LBM has been developed for several steady flow problems [5, 6], but flow-induced sound problems have not been considered yet.

The present study developed the adjoint sensitivity analysis using the LBM for aeroacoustic shape optimization of bluff bodies. This approach was tested for the Aeolian tone generated from the flow past a circular cylinder, and the optimization of the cross-sectional shape was attempted.

1 kusano@mech.kyushu-u.ac.jp

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2. NUMERICAL METHOD

2.1. Lattice Boltzmann method

Flow-induced sounds were directly simulated using the LBM. In this subsection, the outline of the LBM was described. Details are given in a previous report [2].

The lattice Boltzmann equation (LBE) with the BGK model is written as follows:

𝑓 𝑖 ሺ𝐱+ ∆𝑡𝐜 𝑖 , 𝑡+ ∆𝑡ሻ= 𝑓 𝑖 ሺ𝐱, 𝑡ሻ− 1

eq ሺ𝐱, 𝑡ሻ൧ , (1)

𝜏 ൣ𝑓 𝑖 ሺ𝐱, 𝑡ሻ−𝑓 𝑖

where 𝑓 𝑖 is the distribution function of a particle with velocity 𝐜 𝑖 . As a particle velocity model, D2Q9 model was used. 𝑓 𝑖

eq is the local equilibrium distribution function, and 𝜏 is the relaxation time. Fluid density 𝜌 and velocity 𝐮 are calculated from the distribution functions as follows:

. (2)

𝜌= ෍𝑓 𝑖

, 𝜌𝐮= ෍𝑓 𝑖 𝐜 𝑖

𝑖

𝑖

2 , where 𝑐 𝑠 = 𝑐ξ3 Τ is the speed of the sound. To consider objects with complex geometries, the interpolated bounce-back (IBB) condition [7] was applied for the wall boundary condition.

The fluid pressure is related to the density as 𝑝= 𝜌𝑐 𝑠

2.2. Adjoint method

The shape optimization was conducted using the adjoint method. In this optimization, the objec- tive function I is defined as the square mean of the sound pressure in an arbitrary observation region and at an arbitrary observation time.

The minimization problem of the objective function was formulated based on the Lagrange mul- tiplier method under a constraint condition 𝑅 𝑗 ሺ𝐱 𝑘 , 𝑡 𝑛 ሻ= 0 , which was the LBE for each computa- tional element. Consequently, the adjoint equation was derived as follows:

∗ ൫𝐱 𝑘 0 −∆𝑡𝐜 𝑖 , 𝑡 𝑛 0 ൯

𝑓 𝑖

∗ ൫𝐱 𝑘 0 , 𝑡 𝑛 0 +1 ൯+ 1

∗,eq ൫𝐱 𝑘 0 , 𝑡 𝑛 0 +1 ൯൧− 𝜕𝐼

𝜕𝑓 𝑖 ൫𝐱 𝑘 0 , 𝑡 𝑛 0 ൯ , (3)

∗ ൫𝐱 𝑘 0 , 𝑡 𝑛 0 +1 ൯−𝑓 𝑖

= 𝑓 𝑖

𝜏 ൣ𝑓 𝑖

∗,eq is the local equilibrium distribution function for the adjoint equation [8]. This adjoint equation has a form similar to that of the LBE (1). Therefore, the adjoint equation can be computed as easily procedure as the LBM.

∗ are the adjoint states (Lagrange multiplier), and 𝑓 𝑖

where 𝑓 𝑗

∗ , the sensitivity of the objective function I to design variables 𝛼 𝑚 is given as follows:

Using the adjoint state 𝑓 𝑗

𝜕𝑅 𝑗 ሺ𝐱 𝑘 , 𝑡 𝑛 ሻ

𝑑𝐼 𝑑𝛼 𝑚

= 𝜕𝐼

∗ ሺ𝐱 𝑘 , 𝑡 𝑛+1 ሻ

𝜕𝛼 𝑚 . (4)

+ ෍෍෍𝑓 𝑗

𝜕𝛼 𝑚

𝑛 𝑘 𝑗

In the optimization process, the design variables were updated based on the steepest descent method using the sensitivity data.

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3. TEST PROBLEM

The present optimization method was tested for the Aeolian tone of a circular cylinder. The Mach number of the uniform flow was 0.2, and the Reynolds number based on the circular cylinder diameter D was 150.

The cylinder shape was defined using an interpolation curve that was controlled with 11 points, as shown in Fig. 1. The shape was symmetrical to the center line in the streamwise direction. The rear part of the cylinder was optimized in this test, where the design variables were x and y coordinates of the 5 control points in the rear side. The other control points were fixed.

The size of the computational domain was 2048 D × 2048 D . The objective function was evaluated at the location 20D away from the cylinder center, as shown in Fig. 2. In the simulation, a hierarchi- cally refined Cartesian grid was used. The total number of grid points was 6.86 million.

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Fixed Variable Flow

Figure 1 Cylinder shape defined by an interpolation curve.

32

Observation line

16

20 D

0

40 D

20 D

-16

4.0E-5

40 D

-4.0E-5

-32

0 16 32 -16 -32

Figure 2 Evaluation position of the objective function.

4. RESULTS AND DISCUSSION

Figure 3 shows the history of the objective function in the optimization. In the first few iterations, the sound pressure level (SPL) was dropped rapidly. With more iterations, the objective function tends to converge, and the optimization was finished in 46 iterations. Finally, the SPL was decreased by 3.3 dB compared with the initial shape. Figure 4 shows the history of the mean drag coefficient and the RMS lift coefficient in the optimization. The lift fluctuation, which is known to correlates with the Aeolian tone, decreased by the optimization. In addition, the mean drag also decreased along with the sound reduction.

The shape change is described in Fig. 5. In the first iteration, the surface near x = 0.15 was shrunken. Subsequently, the surface near y = 0.4 was expanded to the downstream direction. Finally, the raised surface reached close to x = 0.8.

Figure 6 shows vorticity distributions at the moment when the lift coefficient is maximum for the initial and optimized shapes. These results show that the vortex formation from the optimized shape was delayed compared with the initial shape. Consequently, the lift fluctuation and the Aeolian tone are significantly suppressed.

0.0

-0.5

-1.0

ΔSPL [dB]

-1.5

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-2.5

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-3.5

0 10 20 30 40 50

Interations

Figure 3 History of the objective function in the optimization.

0.40

1.35

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Cd

Cl

0.20

1.20

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0.00

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Figure 4 Histories of the drag and lift coefficients in the optimization.

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Iteration 1 Iteration 3 Iteration 5

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Iteration 10 Iteration 20 Iteration 46

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x

x

x

Figure 5 Shape changes in the optimization.

2

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-4.0 2 4 6

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(a) Initial shape (b) Optimized shape Figure 6 Instantaneous vorticity distributions. 5. CONCLUSIONS

The present study developed an adjoint-based shape optimization method aimed at suppressing flow-induced sounds. In this method, the direct aeroacoustic simulation was conducted using the LBE under low-Mach-number conditions. In the simulation, complex shapes of objects were considered using the IBB scheme. Furthermore, the sensitivities of far-field sounds to object shapes were evalu- ated by solving the adjoint equation, which was derived from the LBE with IBB scheme.

The present method was applied to the cylinder Aeolian tone, and the rear part of the cylinder surface was optimized. Consequently, the SPL of the Aeolian tone was decreased by 3.3 dB compared with the initial shape.

6. ACKNOWLEDGEMENTS

The present study work was supported by JSPS KAKENHI (Grant Number JP22K03929). Com- putational resources were provided by the Research Institute for Information Technology, Kyushu University. 7. REFERENCES

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put. Fluids , 224, 104970 (2021). 4. Jameson A., “Aerodynamic design via control theory”, J. Sci. Comput. , 3, pp.233–260 (1988). 5. Pingen G., Evgrafov A., Maute K., “Adjoint parameter sensitivity analysis for the hydrodynamic

lattice Boltzmann method with applications to design optimization”, J. Comput. Fluids , 38, 4, pp. 910–923 (2009). 6. Dugast F., Favennec Y., Josset C., Fan Y., Luo L., “Topology optimization of thermal fluid flows

with an adjoint Lattice Boltzmann Method”, J. Comput. Phys. , 365, pp. 376–404 (2018). 7. Yu D., Mei R., Shyy W., “A unified boundary treatment in lattice Boltzmann method”, 41st Aero-

apace Sciences Meeting and Exhibit (2003). 8. Vergnault E., Sagaut P., “An adjoint-based lattice Boltzmann method for noise control problems”,

J. Comput. Phys. , 276, pp. 39–61 (2014).

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