Statistical-learning method for predicting hydrodynamic drag, lift, and pitching torque on spheroidal particles

A statistical learning approach is presented to predict the dependency of steady hydrodynamic interactions of thin oblate spheroidal particles on particle orientation and Reynolds number. The conventional empirical correlations that approximate such dependencies are replaced by a neural-network-based correlation which can provide accurate predictions for high-dimensional input spaces occurring in flows with nonspherical particles. By performing resolved simulations of steady uniform flow at $1\le \mathrm{Re}\le 120$ around a 1:10 spheroidal body, a database consisting of Reynolds number- and orientation-dependent drag, lift, and pitching torque acting on the particle is collected. A multilayer perceptron is trained and validated with the generated database. The performance of the neural network is tested in a point-particle simulation of the buoyancy-driven motion of a 1:10 disk. Our statistical approach outperforms existing empirical correlations in terms of accuracy. The agreement between the numerical results and the experimental observations prove the potential of the method.

Figure 1

Sketch of the computational domain ${\mathrm{\Omega }}_{\mathrm{f}}$ and the particle mounted inside the domain.

Figure 2

Example of the mesh in the region close to the spheroid. To accurately resolve the flow in the vicinity of the particle, a spherical swept region with radius $r_{\mathrm{s}}$ is considered around the particle.

Figure 3

An illustration of the $\mathrm{Re}-\mathrm{\Phi }$ space on which the resolved simulations are performed. Each dot represents one simulation.

Figure 4

Deviations of drag coefficients of a 1:10 spheroid and a thin disk at $\mathrm{\Phi }={90}^{\circ }$ from the drag coefficient of a sphere. The results of the present paper for a 1:10 spheroid [solid bullet ($w=1$)] are compared with the correlation of O'Donnell and Helenbrook [43] for a 1:10 spheroid [(open red bullet); $w=0.1$ and (open small left red triangle); $w=0$] and correlation of Clift et al. [1] for thin disks [(solid small left triangle) ($w\le 0.05$)].

Figure 5

Drag coefficient as a function of incidence angle at (a) ${\mathrm{Re}}_{\mathrm{e}}=0.5({\mathrm{Re}}_1=1)$, (b) ${\mathrm{Re}}_{\mathrm{e}}=27.8({\mathrm{Re}}_1=60)$, and (c) ${\mathrm{Re}}_{\mathrm{e}}=55.7({\mathrm{Re}}_1=120)$ [(red solid bullet); $w_1=1/10$: (green solid square); $w_2=1/5$: and (solid small right triangle); $w_3=2/5$].

Figure 6

Contributions of pressure and viscous stress to the drag coefficient at (a) ${\mathrm{Re}}_{\mathrm{e}}=0.5({\mathrm{Re}}_1=1)$, (b) ${\mathrm{Re}}_{\mathrm{e}}=27.8({\mathrm{Re}}_1=60)$, and (c) ${\mathrm{Re}}_{\mathrm{e}}=55.7({\mathrm{Re}}_1=120)$ [crosses; viscous: and solid bullets; pressure].

Figure 7

Lift coefficient as a function of incidence angle at (a) ${\mathrm{Re}}_{\mathrm{e}}=0.5$, (b) ${\mathrm{Re}}_{\mathrm{e}}=27.8$, and (c) ${\mathrm{Re}}_{\mathrm{e}}=55.7$ [red solid bullet; $w_1=1/10$: green solid square; $w_2=1/5$: (solid small right triangle); $w_3=2/5$].

Figure 8

Pitching torque coefficient as a function of incidence angle at (a) ${\mathrm{Re}}_{\mathrm{e}}=0.5$, (b) ${\mathrm{Re}}_{\mathrm{e}}=27.8$, and (c) ${\mathrm{Re}}_{\mathrm{e}}=55.7$ [red solid bullet; $w_1=1/10$: green solid square; $w_2=1/5$: (solid small right triangle); $w_3=2/5$].

Figure 9

Coefficients of (a) drag, (b) lift, and (c) pitching torque as a functions of Reynolds number at $\varphi ={45}^{\circ }$ (top) and $\varphi ={85}^{\circ }$ (bottom) incidence angles [red solid bullet; $w_1=1/10$: green solid square; $w_2=1/5$: (solid small right triangle); $w_3=2/5$].

Figure 10

Viscous and pressure contributions to the lift force (top) and pitching torque (bottom) on a 1:10 spheroid at (a) $\mathrm{\Phi }=5^{\circ }$, (b) $\mathrm{\Phi }={30}^{\circ }$, and (c) $\mathrm{\Phi }={85}^{\circ }$ crosses; viscous: and solid bullets; pressure].

Figure 11

Pressure field and streamline pattern in the center plane $z=0$ for the flow around a 1:10 spheroid at (a) $({\mathrm{Re}}_{\mathrm{e}},\mathrm{\Phi })=(37.1,{45}^{\circ })$, (b) $({\mathrm{Re}}_{\mathrm{e}},\mathrm{\Phi })=(55.7,{45}^{\circ })$, (c) $({\mathrm{Re}}_{\mathrm{e}},\mathrm{\Phi })=(37.1,{85}^{\circ })$, and (d) $({\mathrm{Re}}_{\mathrm{e}},\mathrm{\Phi })=(55.7,{85}^{\circ })$.

Figure 12

Graphical representation of the architecture of a typical feedforward neural network with $n_{\mathrm{i}}$ inputs, $n_{\mathrm{o}}$ outputs, and $k$ hidden layers each consisting of $n_k$ neurons.

Figure 13

Validation of the neural network prediction for (a) drag, (b) lift, and (c) pitching torque coefficients [(solid line); perfect network: (open bullet); present network].

Figure 14

Reynolds-number dependency of the steady drag coefficient of the 1:10 spheroid at $\mathrm{\Phi }={90}^{\circ }$ predicted by the DNN model is compared to correlation (10) (solid red line). The green dashed line represents the correlation of Clift et al. [1], and $\blacksquare$ indicates the training data points.

Figure 15

Magnetic field strength and effective mass density on a vertical line passing through the center of the tank. A particle with ${\rho }_{\mathrm{p}}/{\rho }_{\mathrm{f}}=1.02$ stably levitates at $y=48.7\mathrm{mm}$ [(solid black line); exponential fit; (solid bullet); measured)].

Figure 16

(a) A sketch of the experimental setup. A tank filled with manganese(II) chloride solution is placed on top of a magnet. The particle is released from the top by a rotating release mechanism. Two cameras record the motion of the released particle. (b) Schematic side view.

Figure 17

Vertical position of a 1:10 disk in a magnetized paramagnetic liquid as a function time. The red dashed line corresponds to the numerical solution of (21) with the DNN-based drag coefficient. To enable the comparison, the numerical results are slightly shifted in time. The experimentally obtained trajectory is indicated by the dotted line.