# Prediction of Local Losses of Low *Re* Flows in Non-uniform Media Composed of Parrallelpiped Structures

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## Abstract

A method is presented to predict the local losses of low *Re* flow through a porous matrix composed of layers of orthogonally oriented parallelepipeds for which the local geometry varies discreetly in the direction of bulk flow. In each layer, the variations in the pore lengths perpendicular to and parallel to the direction of bulk flow are restricted to be proportional to one another so that the variation in the geometry of each layer may be characterized by a single parameter, \(\beta \). The solutions to the Navier–Stokes equations are determined for flows through geometries that vary in a forward expansion about this parameter. These provide the data used in the development of a correlation that is able to directly relate local hydraulic permeability to the variation in local pore geometry. In this way, the local pressure losses (as well as the relationship between the volumetric flow rate and the total pressure drop) may be determined without requiring the explicit solution of the entire flow field. Test cases are presented showing that the correlation predicts the local pressure losses to be within 0.5% of the losses determined from the numerical solution to the Navier–Stokes equations. When the magnitude of the variation to the geometry is such that the change in the parameter \(\beta \) between layers is constant throughout the medium, a reduced form of the correlation (requiring the evaluation of only three constants) is able to provide predictions of flow rate and interface pressures that agree to within about 1% with the results of the numerical solutions to the Navier–Stokes equations.

### Keywords

Permeability prediction Correlation Darcy flow Parallelpiped Deterministic model### List of symbols

- \(\beta \)
Variation parameter

- \(\Delta \beta ^{+}\)
Downstream change in variation parameter

- \(\Delta \beta ^{+}\)
Upstream change in variation parameter

- \(\mu \)
Fluid viscosity

- \(\nu \)
Ratio of lateral variation to longitudinal variation

- \(A_\mathrm{T} \)
Total cross-sectional area (void plus solid)

- \(\ell \)
Lateral side length scale associated with the solid matrix

*L*Side length corresponding to layer height

- \(L_0 \)
Unperturbed side length corresponding to layer height

- \(\Delta L\)
Difference between the unperturbed layer height and the perturbed layer height

*K*Permeability

*N*Number of layers

- \(\Delta P\)
Difference in average pressure

- \(\Delta P_\mathrm{T} \)
Difference in average total pressure

*Q*Volumetric flow rate (in the direction of bulk flow)

*U*Seepage velocity

### Subscript

*i*Layer number

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