Designed porous media: Optimally nonuniform flow structures connecting one point with more points

This paper shows analytically and numerically how an originally uniform flow structure transforms itself into a nonuniform one when the objective is to minimize global flow losses. The flow connects one point (source, sink) to a number of points (sinks, sources) distributed uniformly over a two-dimensional domain. In the first part of the paper, the flow between neighboring points is modeled as fully developed through round tubes. It is shown that flow ‘maldistribution’ and the abandonment of symmetry are necessary for the development of flow structures with minimal resistance. The search for better flow structures can be accelerated: tubes that show a tendency of shrinking during the search can be assumed absent in future steps of structure optimization. In the second part of the paper, the flow medium is continuous and permeated by Darcy flow. The development of flow structures (channels) is modeled as a mechanism of erosion, where elements of the original medium are removed one by one, and are replaced with a more permeable medium. The elements selected for removal are identified based on two criteria: maximum pressure integrated over the element boundary, and maximum pressure gradient. The flow structures generated based on the pressure gradient criterion have consistently smaller flow resistances. As flow systems become smaller and more compact, the flow systems themselves become “designed porous media”. These design optimization trends revealed are generally applicable in constructal design, i.e., where miniaturization, global performance, compactness and complexity rule the design.