The tortuosity is a dimensionless property of porous material. It describes an elongation of the flow paths (particle trajectories). It is defined as: $$T = {L_e \over L},$$ where $T$ is tortuosity, $L_e$ is an effective length of paths (the actual length of single path from start to end) and $L$ is the medium length. Thus, tortuosity is 1 for complete straight trajectories (empty duct) and increases with decreasing porosity. Caluclation of tortuosity is important from practical point of view in calculation of permeability where it is used to correct basic empirical laws 1.

I use the Lattice Boltzmann Method (LBM) for fluid flow simulation. In contrast to standard standard CFD solvers, where velocity and pressure are used, LBM uses the velocity distribution function. I am using the D2Q9 variant (two dimensional grid with 9 velocities) in the BGK (single relaxation time) approximation. Time evolution is described by the discrete Boltzmann transport equation 2.

The porous medium is built of overlapping random rectangles. Here we can adjust porosity of the media and change number of overlapping figures.

This directly influences the flow and the value of tortuosity. Below the visualization of the fluid flow is given.

We perform the fluid flow simulation through porous medium. Results are given below.

Fig. Plot of Tortuosity in the function of simulation time step.

I measure tortuosity at various conditions by integrating velocity field (see eg. 5 for details). Additionally I calulate permeability and effective porosity based on the local velocity. These results are stored in the table below.

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Based on the results above we plot $T({\phi})$ relation.

Fig. Plot of the flow tortuosity in the function of porosity of the media.

Fig. Reference results for $T({\phi})$ relation published before in 1. You may download the full paper here .

Description

Summary

The original article about tortuosity in porous media was published in 2008 1. There a systematic study on tortuosity of the flow through a model of an overlapping 2D system was performed. A similar analysis is given here with one difference - the simulation code is exposed to the end-user as a web-based application. I used Bootstrap elements and responsive-design techniques to make this application look good on typical PC computer as well as on mobile devices.

The plan for the future is cover all results given in 1 as well as to extend the range of input parameters that may be changed by the user. The long-term goal is to add another level of complexity of the paragraph synthesized based on the simulation results.

Such web-based interactive paper may repeat some of the results presented in 3 and 4 as well.

Support

The presentation has been prepared as a part of the Support Programme of the Partnership between Higher Education and Science and Business Activity Sector financed by City of Wroclaw.

I am gratefull to Jakub Kossek and Tomasz Bonus for providing hardware necessary for testing of the application. The help in designing the final poster for the presentation goes to Tomasz Bonus and Jakub Jernajczyk from ASP.

Last but not least I would like to acknowledge prof. Andrzej Pękalski for his useful comments at the last few days before the deadline of this project.

References