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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.

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Abbreviations

\(\beta \) :

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

i :

Layer number

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Appendix

Appendix

The correlation presented in this study was evaluated from the numerical solution to the steady incompressible Continuity and Navier–Stokes Equations

$$\begin{aligned} \begin{array}{l} \nabla \cdot \mathbf u =0 \\ \rho \left( \mathbf{u \cdot \nabla } \right) \mathbf u =-\nabla p+\mu \nabla ^{2}{} \mathbf u \\ \end{array} \end{aligned}$$
(19)

The numerical solution of the axial velocity component determined from simulation is presented in Fig. 11 for the flow through a uniform six layer pore structure for which \(\beta _i =0\,\, i=1,...,6\). The total pressure drop is 100 Pa, the density is \(10^{3}\hbox { kg\,m}^{-3}\) and the viscosity is \(0.1\hbox { Pa\,s}^{-1}\).

In the simulations of the 6 layer structure, the boundary conditions at the inlet and outlet are specified pressure conditions and such uniform pressure distributions will not be found at the layer interfaces in the interior of the domain. It is the intent here to show that the local permeability of the inner layers (especially of layer 3 and of layer 4) are domain independent. It is anticipated that at such low Re, any variations in the flow profile resulting from these boundary conditions will dampen out within 1–2 pore lengths. This may be explicitly demonstrated by comparing the flow profiles within the medium at selected periodic regions. Here, the 1 mm \(\times \) 1 mm surface corresponding a plane of symmetry of each layer is chosen as the representative region. Profiles of the contours of the z component of velocity along these planes at different layers are superimposed upon one another in Fig. 12. In each pane of Fig. 12, the black dashed contour lines correspond to the velocity of layer 2. The red dotted line corresponds to the velocity contours of the remaining layers. The velocity profile of the inner layers are indistinguishable from one another. Only the flow fields of the outer layers (layer 1 and layer 6) are noticeably different from layer 2. This shows that at such low Re, the variation to the flow profile introduced by the boundary condition at the inlet of the first layer, only appears in the first layer. Similarly, the variation to the flow profile introduced by the outlet boundary condition of the last layer, only appears in the last layer. To conclude, the local permeability values calculated from the simulation results using Eq. (12) are plotted for each layer in Fig. 12f. The very small deviations in permeability values of the outer layers are not apparent in the interior layers. This helps to establish the domain independence of the calculated permeability values of layer 3 and layer 4 determined from the simulations of flow through 6 layer structures.

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Becker, S.M. Prediction of Local Losses of Low Re Flows in Non-uniform Media Composed of Parrallelpiped Structures. Transp Porous Med 122, 185–201 (2018). https://doi.org/10.1007/s11242-018-0998-1

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