Engineering of polydisperse porous media for enhanced fluid flows through systematic topology tuning via differentiable direct numerical simulation

While predicting the flow behavior for a well-defined particulate microstructure is fairly tractable, systematic tailoring of the particle properties to yield desired flows remains illusive. For example, we still do not know how to precisely control rheological response by tuning particle types and properties or how to tailor porous media for filtration applications. Here we show that the recent advances in automatic differentiation provide a platform for methodologically addressing such questions. We optimize the packing of a polydisperse system of periodically arranged circular rods to maximize flow rate in the direction of applied pressure gradient. We show that the optimum topology of the structured porous medium depends on the solid volume fraction and approaches a staggered square with the particles' diameter ratio of $\sqrt{2}-1$ as the solid volume fraction reaches $\pi /4[1+{(\sqrt{2}-1)}^2]$. Regardless of the porous medium topology, the divergence exponent of the pressure gradient is universal and is equal to $5/2$, as one finds from lubrication theory for a simple square lattice. These results are independent of the flow's Reynolds number and can provide a foundation to study inverse problems for suspension systems.

Figure 1

Problem setup. (a) Porous medium configuration. (b) Flowchart of the optimization algorithm using a differentiable Navier-Stokes equation solver. Here $F$, $g$, and $h$ are the loss function and the equality and inequality constraints, respectively.

Figure 2

Permeability of a periodic square lattice, i.e., $\xi =1$, for different solid packing fractions. The squares represent results obtained by Edwards et al. [5] for $\text{Re}=0$ while the circles are results obtained from the differentiable solver for ${\text{Re}}_p/\sqrt{\varphi }=6.4\times {10}^{-2}$, where ${\text{Re}}_p=\frac{\rho L_x^3\mathrm{\Delta }P}{{\mu }^2}$. The orange line is an empirical formula generated by fitting the data of Edwards et al. To generate the simulation results, $\beta$ was set to $2\times {10}^5$ and $w=1.5\times {10}^{-5}$.

Figure 3

Loss function for different particle diameter ratio $\lambda$ for the case where the periodic porous medium forms a staggered square. The blue line represents the case of the tuning parameters of the logistic function and the final time at which the loss function is evaluated. The orange line was generated using the tunable parameters used to reproduce the permeability of a square lattice that was compared with the data of Edwards et al. in Fig. 2.

Figure 4

Optimum structured periodic porous media. (a) Minimum pressure drop for different porous media. The red dash-dotted curve corresponds to a square lattice, i.e., $\{\lambda =0,\xi =1\}$; the golden dashed curve corresponds to a centered square lattice, i.e., $\{\lambda ={\lambda }^{*}\left(\varphi \right),\xi =1,r=\sqrt{2}{L}_y,\theta ={45}^{\circ }\}$; the dark blue dashed curve corresponds to an off-centered staggered square lattice, i.e., $\{\lambda ={\lambda }^{*}\left(\varphi \right),\xi =1,r=r^{*}\left(\varphi \right),\theta ={\theta }^{*}\left(\varphi \right)\}$, the cyan dash–double-dotted curve represents a rectangular lattice, i.e., $\{\lambda =0,\xi ={\xi }^{*}\left(\varphi \right)\}$; the purple solid curve corresponds to a centered rectangular lattice, i.e., $\{\lambda ={\lambda }^{*}\left(\varphi \right),\xi ={\xi }^{*}\left(\varphi \right),r=\sqrt{1+{{\xi }^{*}\left(\varphi \right)}^2}{L}_y,\theta ={sin}^{-1}[1/{\xi }^{*}\left(\varphi \right)]\}$; and the green solid line with triangles corresponds to the off-centered rectangular lattice where $\{\lambda ={\lambda }^{*}\left(\varphi \right),\xi ={\xi }^{*}\left(\varphi \right),r=r^{*}\left(\varphi \right),\theta ={\theta }^{*}\left(\varphi \right)\}$. (b) Optimized lattice parameters for the off-centered rectangular case. The optimum porous medium structure is visualized for $\varphi =0.08$ (regime I), $\varphi =0.18$ (regime II), $\varphi =0.59$ (regime III), and $\varphi =0.88$ (regime IV). (c) Visualization of the different optimization cases: (i) square, (ii) rectangular, (iii) centered square, (iv) off-centered square, (v) centered rectangular, and (vi) off-centered rectangular. In all cases, the solid volume fraction is equal to 0.72.

Figure 5

Minimum pressure drop of optimized bidisperse porous media for different number density ratios. The green up-pointing triangles, red down-pointing triangles, black diamonds, and blue stars correspond to $n_1/n_2=1$, 2, 3, and 4, respectively. The optimum porous medium is visualized at $\varphi =0.84$ for (i) $n_1/n_2=1$, (ii) $n_1/n_2=2$, (iii) $n_2/n_1=3$, and (iv) $n_2/n_1=4$.

Figure 6

Minimum pressure drop of the optimized polydisperse porous medium for different total numbers of distinct cylinders. The green up-pointing triangles, black squares, blue left-pointing triangles, and purple stars correspond to the cases where $N=2$, 3, 4, and 5, respectively. The optimum porous medium is visualized at $\varphi =0.72$ for (i) $N=2$, (ii) $N=3$, and (iii) $N=4$.

Figure 7

Universal divergence of the pressure drop curve of the different optimized topologies. The slope of the black line is equal to $5/2$.