Direct Numerical Simulations of Incompressible Flow through Porous Packs over a wide Range of Reynolds Numbers
Jost, Antoine Michael Diego
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Direct numerical simulations of incompressible flow through periodic square/rectangular (two-dimensional; 2-D) and cubic (three-dimensional; 3-D) arrays of particles and periodic random packs of polydisperse spheres over a wide range of Reynolds numbers and at various porosities are performed. The unsteady Navier- Stokes equations are solved with either a 3-D parallel finite-difference (FD) in-house research code on a collocated Cartesian grid and with an immersed boundary method (IBM) to treat the internal boundaries, or with the finite-volume (FV) solver OpenFOAM R⃝ on an unstructured grid and with an explicit treatment of the boundaries. The FD explicit solver is used for 2-D flows and 3-D laminar flows while the FV implicit solver is preferred for 3-D transitional and turbulent flows for the consideration of computational cost. Macroscopic friction factors for periodic arrays of cylinders and ellipses and periodic cubic arrays of spheres with a porosity range of 0.30≤σ≤0.70 and 0.55≤σ≤0.85, respectively, and varying Reynolds numbers ranging from the creeping to (pseudo-)turbulent flow regimes are calculated; we reserve “turbulent flow” for 3-D cases and refer to 2-D turbulent flow as “pseudo-turbulent flow”. The effects of Reynolds number and porosity on friction factor and permeability are investigated; the former is in good agreement with the modified Forchheimer friction factor and a modified curve fit based on the Gebart relation is proposed for the latter. A similar study was conducted for periodic random packs of spheres in the creeping and inertial flow regimes with varying distributions of particle diameters, and modified curve fits are proposed. The onset of unsteady flow, identified through the critical Reynolds number and defined as the Hopf bifurcation, is found to be particle shape and porosity dependent. It is also found that transition to turbulence occurs when significant vertical cross flow starts to emerge, causing an imprint on statistics such as vorticity budget and two-point double correlation field. The transitional and pseudo-turbulent flow features are then analyzed for square array of particles in 2-D with peculiar and normal transitional behaviors. Peculiar transitional behavior refers to mixed steady-unsteady flows at different Reynolds numbers in the transitional regime and was discovered by our numerical simulations for certain porosities. Time histories and statistics of probe data and flow fields are used to analyze the time and length scales, respectively. For transitional flows, the length scales of instantaneous vorticities decrease with increasing Reynolds number while the temporal signals experience wider power distributions as the Reynolds number increases. For pseudo-turbulent flows, the instantaneous length scales and the length scales identified by two-point double correlations are similar for the range of pseudo-turbulent flow Reynolds numbers considered, specific to 2-D pseudo-turbulence. High frequency in probe data is identified with periodic flows as the flow pass-through resonance of the “tail”, high u-velocity stream originated from the left boundary. For transitional and pseudo-turbulent flows, this high frequency resonance is superimposed on distributed “tail” flapping. Interesting intermittent behavior of the “tail” tilting is observed and discussed for transitional and pseudo-turbulent flows. Pseudo-turbulent flows also display similar Gaussian PDFs of probe data as a result of a variety of vortex and “tail” motions. The effect of the size of the representative elementary volume (REV) on the instantaneous, mean, and rms flow fields for packs with peculiar transitional behavior is also studied. Finally, initial results of 3-D DNS of turbulent flow are presented, and similar to the 2-D cases, the association of transition with flow perpendicular to the stream-wise direction is observed.