Slip-flow lattice-Boltzmann simulations in ducts and porous media: A full rehabilitation of spurious velocities

Slip flow in ducts and porous media is simulated using lattice-Boltzmann method incorporated with interfacial force models. The dependence of the results on the viscosity, LBM scheme (D3Q15 and D3Q19) and the relaxation time model (single- or multirelaxation time) is investigated. The severity of spurious velocities (arisen from classic and advanced interfacial force models) is discussed that leads to entirely nonphysical results for whole flow rates ($0.027<\text{Re}<10.7$). A simple method based on superposition of solutions is proposed to fully rehabilitate the simulations. We validate the method by showing the simulations versus analytical solution of slip flow through circular ducts. The validity of the rehabilitated results for porous media applications are also tested through two approaches: First, we show that the rehabilitation method is independent to the force scheme used, i.e., the rehabilitated results are identical in both pore and macroscales for different force schemes with different distributions of spurious velocities. Second, using an analogy based on the Kozeny-Carman model, we show that the permeability variation in porous media resulted from the flow slippage obtained from rehabilitated simulations is reliable. We argue that to obtain correct results, it is necessary to use the rehabilitation method whenever interfaical force models are used in LB simulations. The results reveal that the permeability of porous media may increase or decrease with positive or negative slippage (repulsive and attractive interfaces), respectively. The permeability enhancement rate increases as the system becomes simpler in its interfaces, i.e., for the same positive slippage of flow, $(\frac{\kappa }{{\kappa }_{\text{NS}}}{)}_{\text{parallel}\text{plates}}>(\frac{\kappa }{{\kappa }_{\text{NS}}}{{)}_{\text{square}\text{ducts}}>(\frac{\kappa }{{\kappa }_{\text{NS}}})}_{\text{porous}\text{media}}$ (where $\kappa$ is the permeability $\mathrm{and}\text{NS}$ denotes no-slip).

Figure 1

Viscosity-dependence of the simulations. The normalized mean velocity of the flow through circular ducts (a) and its relative error (b) vs. the viscosity $\nu$ in l.u and the relaxation time $\tau$. The lattice resolution is $100\times {(d+5)}^2$. The analytical flow mean velocity is given as ${\langle u\rangle }_{\text{Ana}}=R^{2}G/\left(8\mu \right)$. The cross-section of the models is shown in (c).

Figure 2

Evaluation of the no-slip assumption. Velocity profiles of flow between parallel walls with spacing of $d=10$ (a) and 30 (b) in l.u. Simulations are conducted using SRT-D3Q15 scheme with standard bounce-back boundary conditions at walls. The resolution of simulations is $(d+2)\times {110}^2$ lattices. The no-slip velocity profile is obtained from simulations with no interfacial forces ($g_s=0$) and bounce-back boundary conditions. The fitting equations on the no-slip velocity profiles indicate the accuracy of the model for the no-slip assumption, leading to erroneous slip velocities of $(3.4\times {10}^{-3})\times u_{\text{max}}$ and $(3.5\times {10}^{-4})\times u_{\text{max}}$ for $d=10$ and 30, respectively.

Figure 3

Slippage of flow between repulsive (a) and attractive (b) walls, leading to positive and negative slip, respectively. The flow velocity profiles are shown in right-hand-side panels. The definition of slip length $\lambda$ is shown in (a). The insets in left-hand-side panels show the mean velocity of flow versus the force factor $g_s$. The repulsive and attractive interactions resulted in enhanced (a) and decreased flow rates (b), respectively.

Figure 4

Validation of slip-flow simulations for Poiseuille flow between parallel walls. Normalized mean flow velocity as a function of normalized slip length. $d$ is the plate separation. Our LBM results are shown along with the analytical line as driven by Eq. (14) (a linear equation shown in semilog plot).

Figure 5

Normalized mean flow velocity as a function of normalized slip length for polygonal ducts with $M$ sides. Tabulated data obtained from Ref. [82] are transformed into the figure.

Figure 6

Validation of superpositions rehabilitation method. Cross-sectional velocity profiles [(a) and (b)] for a circular duct with interfacial forces ($g_s=1.0$). The diameter of the duct is equal to 90 cells where the resolution of the simulations is ${99}^3$. The simulation velocities $u$ are significantly affected by spurious velocities (a) with nonphysical rotational velocities as shown in the inset at (a). The rehabilitation of flow leads to quadratic flow profiles (b). The simulation velocity profiles $u$ are distorted by spurious velocities (inset at c) for $g_s>0$. $d=2R$ is the pipe diameter. The rehabilitated velocities $u^{*}$ demonstrate the slippage of flow by increasing the force factor (c). The maximum velocity increases with flow slippage and the slip velocity at the duct surface leads to positive slip lengths. The LBM rehabilitated simulations for flow through circular duct are validated with the analytical solution as per Eq. (25) (d). The slip lengths ($\lambda$) are calculated using quadratic fittings on rehabilitated velocity profiles.

Figure 7

Velocity vectors in a single pore space from a 3D porous medium showing spurious currents [(a) and (b)] and physical (no-slip) velocities (c). The net flow generated by gravity in $x$-direction has the mean flow velocity in a no-slip condition (without surface forces) equal to ${\langle u\rangle }_{\text{NS}}=6.7\times {10}^{-5}$ ($\text{Re}=0.027$). With an interfacial attractive force ($g_s=-2.0$) implemented in (a) and (b), the spurious velocities are dominant with different magnitudes and distributions for SC and EFS modes. The same medium with no-slip BC ($g_s=0$) would result in a velocity field as shown in (c). Note the magnification scale in showing vectors in (c) for a comparison to spurious currents.

Figure 8

Significance of spurious velocities in porous media simulations. The ratio of the mean of spurious velocities ${\langle u\rangle }_{\text{SP}}$ to mean velocity of no-slip flow ${\langle u\rangle }_{\text{NS}}$ for flow simulations through porous media where different intensities for solid-fluid interfacial forces ($g_s$) are assigned. The viscosity is equal to 0.1667. The pseudopotential SC model is used here. The characteristic length (sphere size) for the porous medium is 68 cells. A stronger interfacial force increases the spurious velocities (a). The velocity range in (a) is equivalent to $0.027<\text{Re}<10.7$. The relationship collapses into a unique behavior where the variation of velocities are shown versus the ratio of $\frac{g}{g_s}$ with $g$ being the gravitational factor utilized to generate the main flow (b).

Figure 9

Identical macroscopic results for different interfacial force schemes after using the rehabilitation method. The mean velocity, $\langle u\rangle$ of flow through porous media for $g_s=-1.0$ normalized by the mean velocity of the flow in the same model but with no-slip solid boundaries, ${\langle u\rangle }_{\text{NS}}$ ($g_s=0$). $\langle u\rangle$ is the mean velocity of the original simulations and $\langle u^{*}\rangle$ denotes the mean velocity of the rehabilitated simulations using superposition method. Although EFS model provides an improvement in reducing the spurious velocities, the model results are mainly nonphysical in the whole range of flow velocities. The inset is the same graph with logarithmic scales, showing that the model velocities could be several orders of magnitude larger than the physical velocities. The superposition rehabilitation method is shown to achieve the same feature (permeability reduction) regardless of the force model employed (SC or EFS) for the whole velocity range.

Figure 10

The minor difference of rehabilitated velocities applied to different force schemes with the major difference in spurious velocities. The variation of the mean velocity of flow through porous media by using different force models (SC and EFS) for $g_s=-1.0$. $\langle u\rangle$ and $\langle u^{*}\rangle$ are the mean velocity of the original simulations and the simulations rehabilitated using superposition method, respectively. Using an improved force model (EFS) has changed the normalized magnitude of the spurious velocities up to more than one order of magnitude compared to the SC model. However, the rehabilitated velocity fields using superposition method ($u^{*}$) proved to be unaffected by the force model used, with a minor variation between different models.

Figure 11

Identical pore-scale results of different force schemes after rehabilitation. Velocity distributions shown across a cross-section perpendicular to the main flow direction through a 3D porous medium simulation. The surface force factor is $g_s=-0.1$ (a and b) and $-2.0$ (c and d) denoting interfacial attractive behaviours. The mean flow velocity in a no-slip condition (without forces) is ${\langle u\rangle }_{\text{NS}}=6.7\times {10}^{-5}$ ($\text{Re}=0.027$). The force models used are SC (a and c) and EFS (b and d). The simulation velocities, i.e., $u$ in i-plots are impaired with spurious velocities which are much larger than the physical velocities. $u$ in SC model is also roughly fivefold larger than that of EFS. The model of stagnant fluid (with only forces on interfaces) also resulted in similarly large velocities (spurious velocities, i.e., $u_{\text{SP}}$ in ii-plots), indicating the dominance of spurious velocities in the simulations. The results of our superposition rehabilitation method ($u^{*}$ in iii-plots) are slightly different from those obtained from the model with no-slip boundaries ($g_s=0$) illustrated as $u_{\text{NS}}$ in iv-plots. The difference between $u^{*}$ and $u_{\text{NS}}$ is shown in v-plots where the pore flow velocities are reduced (less than $\%1$ or $\%20$ for $g_s=-0.1$ or $-2.0$, respectively) due to the attractive force effects. Despite the significant difference of simulation velocities ($u$) in two force models, the rehabilitation method resulted in accurately similar velocities both in quantities and distributions. We conclude that our rehabilitation model is capable to retrieve the small velocity changes due to surface force effects even for such complex interfaces with large spurious velocities. The 3D porous medium simulated is shown in (e) where the cross-section on which velocities are plotted is highlighted at $x=55$ and flow direction is in $x$.

Figure 12

Identical velocity profiles in pores after rehabilitation of spurious velocities from different force models. Velocity profiles (a) and vectors (b) for 3D porous media simulations where ${\langle u\rangle }_{\text{NS}}=6.7\times {10}^{-5}$ ($\text{Re}=0.027$). $u$ in (a-i) is the simulation velocities for $g_s=-2.0$ simulated using SC and EFS models (D3Q15 scheme) as well as the model of no-slip BC ($g_s=0$). $u$ resulted from two models are entirely corrupted by spurious velocities. They are also largely different, both in values and distributions. Despite the differences in $u$, the rehabilitated velocities ($u^{*}$) of both models as shown in (a-ii) match accurately together indicating a flow pore-wide velocity reduction due to the effects of attractive forces on solid surfaces. The profile of the changes in velocities ($u^{*}-u_{\text{NS}}$) for both models is also shown in (a-iii) demonstrating the close agreement of the results of the models after rehabilitation. In (a-iv) are shown the line (in red) and the cross-section (in $y-z$ plane perpendicular to the flow main direction, $x$) over which the velocity profiles are plotted. The cross-section is highlighted in the porous medium. Original simulation velocity vectors ($u$) for different models where spurious velocities dominate the flow field (before rehabilitation) in both SC and EFS models are shown in (b). The no-slip-flow field is much smaller in magnitudes of velocities than the spurious fields (Note the magnification factor of 500).

Figure 13

Permeability variation due to interfacial forces. Flow mean velocity ($\langle u\rangle$) versus driving force ($\rho g$) for the porous medium (a) and analogic systems composed of square ducts (b) and parallel plates (c). The results of porous media are rehabilitated using our proposed method. The general behavior is enhanced and reduced permeability with repulsive (positive) and attractive (negative) surface forces, respectively, as exhibited by all systems. The analytical solutions of the no-slip assumption for ducts and plates demonstrate the agreement of the results.

Figure 14

The ratio of permeability of analogous systems (ducts and parallel plates) to that of the porous medium (${\kappa }_{PM}$) versus surface force factor ($g_s$).

Figure 15

Variation of the normalised permeability with the surface force factor ($g_s$). The permeability of the porous medium and its analogous systems reduces similarly with attractive interfacial forces. The permeability is enhanced with repulsive forces for all systems too, but with different rates.