Division of Mathematical and Physics Science, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
Faculty of Mathematical and Physics, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
Japan Science and Technology Agency, PRESTO, Kawaguchi 332-0012, Japan
* Corresponding author: Imam Wijaya
* Corresponding author: Imam WijayaThe purposes of this work are to study the $ L^{2} $-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically. The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.
Figure 1. Representative elementary volume (REV)
Figure 2. The order of convergence for scheme (22)
Figure 3. The boundary conditions and the finite element mesh
Figure 4. Time evolution of velocity magnitude
Figure 5. Computation domain and porosity value distribution
Figure 6. Time evolution of magnitude velocity
Table 1. The unit of important symbols
| No | Symbol | Unit | Name of the symbol |
| [0.5ex] 1 | $ u $ | $ {\rm m} \cdot {\rm s}^{-1} $ | Darcy velocity |
| 2 | $ p $ | $ {\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-2} $ | Pressure |
| 3 | $ \phi $ | - | porosity |
| 4 | $ k_{D} $ | $ {\rm kg}^{-1} \cdot {\rm m}^3 \cdot {\rm s} $ | Hydraulic conductivity |
| 5 | $ K $ | $ {\rm m}^2 $ | Permeability |
| 6 | $ \mu $ | $ {\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-1} $ | Dynamic viscosity |
| 7 | $ \rho $ | $ {\rm kg} \cdot {\rm m}^{-3} $ | Density |
| 8 | $ d_{p} $ | $ {\rm m} $ | Particle diameter |
| 9 | $ F $ | - | Forchheimer constant |
| 10 | $ B $ | $ {\rm kg} \cdot {\rm m}^{-2} \cdot {\rm s}^{-2} $ | Drag force per unit volume |
Table 2.
Values of
| $ N $ | $ Er1 $ | $ Er2 $ | Slope of $ Er1 $ | Slope of $ Er2 $ | CPU time [s] |
| 4 | $ 3.4\times10^{-1} $ | $ 1.6\times10^{-1} $ | $ - $ | $ - $ | 1.9 |
| 8 | $ 7.1\times10^{-2} $ | $ 5.8\times10^{-3} $ | 2.26 | 4.76 | 16.4 |
| 16 | $ 1.4\times10^{-2} $ | $ 1.2\times10^{-3} $ | 2.34 | 2.30 | 174.8 |
| 32 | $ 3.5\times10^{-3} $ | $ 2.9\times10^{-4} $ | 2.00 | 2.05 | 577.4 |
| 64 | $ 1.0\times10^{-3} $ | $ 6.3\times10^{-5} $ | 1.81 | 2.20 | 5,953.9 |
| 128 | $ 2.8\times10^{-4} $ | $ 1.5\times10^{-5} $ | 1.84 | 2.07 | 58,150.9 |
Figures(6)
Tables(2)
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Representative elementary volume (REV)
The order of convergence for scheme (22)
The boundary conditions and the finite element mesh
Time evolution of velocity magnitude
Computation domain and porosity value distribution
Time evolution of magnitude velocity
