Spontaneous imbibition in a microchannel: analytical solution and assessment of volume of fluid formulations
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Abstract
The simulation of multi-phase flow at low capillary numbers (Ca) remains a challenge. Approximate computations of the capillary forces tend to induce parasitic currents (PC) around the interface. These PC induce additional viscous dissipation and shear forces that potentially lead to wrong calculations of the general flow dynamics. Here, we focus on the case of spontaneous imbibition in a microchannel of Hele-Shaw cell symmetry with capillarity being the only driving force. We extend the Lucas–Washburn equation to account for arbitrary viscosity ratios and assess four volume-of-fluid (VOF) formulations against the analytical solution. More specifically, we evaluate the continuum surface force (CSF) formulation, the sharp surface force (SSF) formulation, the filtered surface force (FSF) formulation and the piecewise linear interface calculation (PLIC) formulation extended by a higher order discretisation of the interface curvature through a height function with respect to accuracy, performance and heuristic parameters. We quantify PC for each formulation and investigate their impact on flow with \(\mathrm{Ca} < 10^{-2}\). The magnitude of PC are largest for CSF and are reduced two fold by SSF. FSF reduces PC considerably more but shows periodic short bursts in the velocity field. PLIC shows no PC for the studied \(\mathrm{Ca}\) and viscosity ratios. However, it fails when a denser fluid displaces a lighter fluid. Despite PC, all formulations are accurate within 10%. PLIC is suited to serve as a reference but relies on a structured mesh and is computationally expensive. FSF requires more heuristic parameters. Together with periodic bursts, this prevents a conclusive statement on the best choice between SSF and FSF.
Keywords
Spontaneous imbibition Extended Lucas–Washburn equation Capillary flows Direct numerical simulation Volume of fluid Parasitic currentsList of symbols
- \(\ell\)
Length of the channel (m)
- \(\mathbf F\)
Force (kg m/s\(^2\))
- \(\mathbf n\)
Unit normal
- \(\mathbf S\)
Surface area (m\(^2\))
- \(\mathbf t\)
Unit tangent
- \(\mathbf U\)
Velocity (m/s)
- c
Characteristics of PLIC line segment
- D
Density ratio, \(\rho _n/\rho _w\)
- f
Height function (m)
- h
Height of the channel (m)
- k
Interface curvature (1/m)
- M
Viscosity ratio, \(\mu _n/\mu _w\)
- m
Mass of fluid (kg)
- p
Pressure (kg/ms\(^2\))
- r
Interface normal (1/m)
- t
Time (s)
- V
Volume (m\(^3\))
- w
Width of the channel (m)
- x
Spatial co-ordinates
Coefficients
- \(C_\text {f11}\)
Capillary forces filter
- \(C_\text {fi2}\)
Capillary momentum flux filter
- \(C_\text {shp}\)
Sharp VOF’s colour function
Greek symbols
- \(\alpha\)
Volume-of-Fluid’s colour function
- \(\Delta\)
Spatial/ temporal discretization
- \(\delta\)
Dirac delta function
- \(\lambda\)
Slip length (m)
- \(\mu\)
Dynamic viscosity of fluid (kg/ms)
- \(\phi\)
Flux (m\(^3\)/s)
- \(\rho\)
Density of fluid (kg/m\(^3\))
- \(\sigma\)
Surface tension between fluids (kg/s\(^2\))
- \(\theta\)
Equilibrium contact angle
Subscript
- \(\alpha\)
Fluid phase, w = wetting; n = non-wetting
- BK
Brackbill
- c
Cell centre
- f
Cell face centre
- s
Smooth
- x, y
Co-ordinate plane
- avg
Average
- b
Body
- cap
Capillary term
- fi
Filter
- I
Interface
- m
Meniscus
- max
Maximum
- ref
Reference variable
- shp
Sharp
- th
Threshold
- vis
Viscous term
- w
Wall
1 Introduction
The understanding of multi-phase flow at low flow rates in small confined geometries is crucial for many engineering applications ranging from hydrocarbon recovery, carbon sequestration, microfluidic devices and remediation of contaminated soil. The fluid dynamics depend on the complex interplay of inertial, capillary, viscous and gravitational forces (Méheust et al. 2002; Ferrari and Lunati 2013, 2014). Thus, marginal changes in fluid and rock properties as well as flow conditions can have a huge impact on displacement patterns (Avraam and Payatakes 1995).
In recent years, advancements in imaging techniques have improved the understanding of capillary-dominated flow substantially (Pak et al. 2015; Blois et al. 2015; Duxenneuner et al. 2014). However, numerical simulation tools are still indispensable for progress in theoretical understanding of the dynamic processes as well as for parameter optimisation (Zhou et al. 2017).
A variety of numerical methods exist that differ in their discretisation strategy and come with different challenges. We only provide a brief overview and refer to (Wörner 2012; Teschner et al. 2016) for a comprehensive discussion.
Smooth particle hydrodynamics (SPH) is a mesh-free method that discretizes partial differential equations considering particles that represent a spatial region (Fatehi et al. 2008). Properties of a particle (density, velocity, pressure) are computed using a smoothing kernel considering neighbouring particle properties as well that are within a specific length scale (smoothing length). Further, Tartakovsky and Meakin (2005) used SPH to mimic fluid dynamics by coarse-grained particles with specific interaction terms. SPH is mass conservative and inherently captures a sharp interface. Morris (2000) introduced the concept of modelling two-phase flows by including capillary effects in SPH. Kunz et al. (2016) have validated two phase SPH considering open boundaries for the formation of bubbles during gas injection. Lattice Boltzmann methods solve the Boltzmann equation for particle density in phase space on a lattice and reproduce fluid dynamics by specific collision terms (Zhang 2011) . The local nature of their formulation renders them very well suited for parallel computing. However, they generally require structured meshes and the link between spatio-temporal scales, physical parameters as well as as boundary conditions make extensions to include other physical phenomena cumbersome. In addition, density and viscosity contrasts pose a challenge.
Continuum methods directly solve the Navier–Stokes equations (NSE) through a conventional spatio-temporal discretisation. This renders them flexible with respect to mesh structures and inclusion of additional physics. In addition, density and viscosity contrasts generally do not pose a challenge. Continuum methods differ in the representation of the interface between the fluids. Lagrangian interface tracking methods such as moving mesh (MM) (Quan and Schmidt 2007), marker and cell (MAC) (Harlow et al. 1965) represent a sharp interface but struggle with topological changes such as break up and coalescence. Topological changes are well captured by Eulerian methods. The volume-of-fluid (VOF) method (Hirt and Nichols 1981) represents the two-phase system as a mixture and determines the two-phase properties based on the mixture parameter. The VOF method is simple and conserves mass on any type of mesh. However, the advection of the mixture parameter by conventional schemes lead to smearing of the interface. The level-set (LS) method (Sussman et al. 1994) tracks the distance to the interface by a smooth-signed distance function. The LS method represents a sharp interface but struggles to conserve mass. The phase-field (PF) method (Jacqmin 1999; Lim and Lam 2014) propagates the auxiliary phase-field function using the fourth-order Cahn–Hilliard’s equation. This diffuse interface method captures the contact line dynamics on the wall without developing stress singularity. Higher-order discretisation schemes are required to model the fourth-order term of Cahn–Hilliard’s equation which poses a challenge.
Though at different levels, all methods have in common that they are challenged by strong capillary forces. This stems from the multi-scale nature of the problem. The interface only spreads over a few molecules in distance and an explicit representation of the interface is out of reach. Further, capillary forces depend on the curvature of the interface and are, therefore, highly sensitive to even small errors. These inaccuracies lead to non-physical parasitic currents (PC) around the interface (Lafaurie et al. 1994) that potentially change the interface dynamics. In this paper, we investigate the VOF method due to its simplicity, mass conservative nature and ability to capture topological changes. In addition, the VOF method is available in many open-source and commercial software packages such as OpenFOAM (www.openfoam.com), Gerris (http://gfs.sourceforge.net), Ansys-Fluent (https://www.ansys.com) and others.
The VOF method represents the two-phase system by a mixture. Flow parameters are determined based on the volume fraction function, \(\alpha\) referred to as the colour function. The interface location is determined by steep gradients of the colour function. The capillary forces term is approximated as a body force.
Four implementations of open-source finite-volume VOF formulations are assessed in this manuscript: the continuum surface force (CSF) formulation of the interFoam solver of OpenFoam (version foam-extend 1.6, http://wikki.gridcore.se), the sharp surface force (SSF) and the filtered surface force (FSF) formulations of poreFoam (http://www.imperial.ac.uk/earth-science/research/research-groups/perm/research/pore-scale-modelling/software/direct-two-phase-flow-solver), a separately available solver for OpenFoam (version foam-extend 1.6) and the piecewise linear interface calculation (PLIC) formulation of Gerris (http://gfs.sourceforge.net) with a higher-order discretisation of the interface curvature through a height function. All the formulations differ in approximation of the capillary body force. SSF, FSF and PLIC have previously been benchmarked for a droplet and bubble relaxing in an equilibrium field (Raeini et al. 2012; Popinet 2009). PLIC was further validated for capillary waves and shape oscillations of an inviscid droplet during relaxation (Popinet 2009) and for droplet pinch-off in a T-shaped microchannel (Sivasamy et al. 2011). CSF was validated with experiments for capillary flows in microchannels consisting of integrated pillars (Saha et al. 2009). SSF and its close relatives (Lafaurie et al. 1994; Brackbill et al. 1992) have been used to study complicated flow phenomena in porous materials (Raeini et al. 2014; Ferrari and Lunati 2014) and microfluidic devices (Hoang et al. 2013).
Our contribution in this paper is to bridge the gap between static test cases and applications by introducing a dynamic test case with an analytic solution for spontaneous imbibition in a microchannel of Hele-Shaw cell symmetry considering partial slip on the wall. The imbibition occurs solely due to capillarity. We extend the Lucas–Washburn equation to arbitrary viscosity ratios and assess the performance of the different VOF formulations against this analytical solution. The numerical methods are validated for flows at different capillary numbers as well as different viscosity and density ratios. We quantify the magnitude of PC around the interface and discuss their impact on the flow dynamics.
The paper is structured as follows. Section 2 provides a detailed description of the four VOF formulations highlighting heuristic parameters. In Sect. 3 we develop the extension of the Lucas–Washburn equation to account for arbitrary viscosity ratios. Section 4 discusses the numerical set-up of the initial and boundary value problem. In Sect. 5, we present the assessment of the fourk VOF formulations against the test cases. Conclusions are given in Sect. 6.
2 Physical and numerical description of two-phase flows
The isothermal dynamics of two immiscible and incompressible fluids are governed by the Navier–Stokes Equations for each fluid phase. The NSE of the two fluid phases are coupled through a boundary condition at the interface between them. The interface dynamics are part of the solution and hence, a moving boundary problem has to be solved (Batchelor 2000). The VOF method (Hirt and Nichols 1981) simplifies the moving boundary problem by treating the two-phase fluid system as a mixture. The mixture parameter \(\alpha \in [0,1]\) referred to as a colour function represents the volume fraction of one of the phases in a control volume. The capillary forces at the interface are modelled through a body force.
where the indices w and n denote the two fluids.
2.1 Variants of VOF formulations
The paper assesses the four VOF formulations CSF, SSF, FSF and PLIC. They differ in the discretisation of the interface curvature, the interface normal vector and the Dirac delta function for the capillary body forces of Eq. (4). PLIC further uses a different scheme for the advection equation of the colour function Eq. (5). This section discusses these formulations in detail and highlights any heuristic parameters. The explicit numbers of any heuristic parameters are given in the supplementary material.
2.1.1 Continuum surface force (CSF)
2.1.2 Sharp surface force (SSF)
The SSF has been developed based on CSF to reduce PC. The reduction is achieved by smoothing the colour function for calculating the curvature and sharpening the colour function for calculating the Dirac delta function in Eq. (4) (Raeini et al. 2012; Francois et al. 2006). Here, we use poreFoam solver (Raeini et al. 2012) and switch off the filtering coefficients (discussed in Sect. 2.1.3) to obtain SSF.
2.1.3 Filtered surface force (FSF)
The FSF has been developed based on SSF to reduce PC under dynamic conditions. The reduction is achieved by dampening capillary-induced flows parallel to interface (Raeini et al. 2012). Here, we use the poreFoam solver.
2.1.4 Piecewise linear interface calculation (PLIC)
PLIC constructs a sharp interface using line segments in the cells marked by the gradient of the colour function (cut cells) to geometrically advect the colour function with minimal diffusion (Popinet 2009; Youngs 1982). We use the PLIC formulation available for structured meshes in Gerris (Popinet 2003, 2009).
Advantages, disadvantages and heuristic parameters for different VOF method formulations
VOF formulation | Advantages | Disadvantages | Heuristic parameters |
---|---|---|---|
CSF, solver: interFoam | Simple expression for interface curvature and capillary forces | Inaccurate computation of interface curvature; PCs around the interface | None |
SSF, solver: poreFoam | Reduced PC for static cases | PC for dynamic cases | Number of smoothing operations for colour function to compute interface curvature; sharpening coefficient |
FSF, solver: poreFoam | Reduced PC for static and dynamics cases | Potential removal of physical fluxes; a-priori heuristic filtering coefficients; periodic bursts in PC | In addition to SSF, coefficients for filtering the capillary forces and the capillary flux |
PLIC, solver: Gerris | Negligible PC and negligible interface smearing, adpative meshing | Computationally expensive; code available only for structured meshes and height functions for 2D | Number of smoothing operations for colour function to compute fluid parameters |
2.2 Boundary conditions on the wall
2.3 Discretization and numerical schemes
Foam-extend and Gerris use the finite-volume method for discretization. A collocated Eulerian grid arrangement is used with the field variables stored at the cell centres of a control volume.
In foam-extend, an implicit first-order Euler scheme is used for time discretization. The gradient terms are discretized by the second-order Gauss linear scheme. The advection term in the Navier–Stokes momentum equation is solved using limited linear differencing. A bounded solution for the colour function is ensured by the vanLeer advection scheme. To control the numerical diffusion of the interface while solving the advection equation of the colour function Eq. (5), an artificial compression velocity is added only at the interfacial region in a direction normal to the interface (Rusche 2003). The pressure-velocity coupling of NSE is achieved through a predictor–corrector step of Pressure Implicit with Splitting of Operators (PISO by Issa 1986).
Gerris (Popinet 2003, 2009) uses a second-order staggered time discretization. The advection term of the momentum equation is solved by a Godunov scheme (Bell et al. 1989). For the advection term of the mixture transport equation, a direction-split advection method with geometrical flux estimates is used (Gerlach et al. 2006). The pressure–velocity coupling of the NSE uses the time split projection method based on Hodge decomposition of the velocity field.
3 Analytical solution for spontaneous imbibition in a rectangular channel
3.1 Derivation of extended Lucas–Washburn equation
In this study, we vary the capillary number by changing the channel length \(\ell\) while keeping the other length scales of the channel (\(h, \lambda\)) constant. Thus, small capillary numbers are achieved by long channels.
3.2 Model approximations
4 Numerical set-up
4.1 Initial and boundary conditions
4.2 Mesh dependence
Parameters used for mesh resolution study
\(\mu _w\) | \(\mu _n\) | \(\rho _w\) | \(\rho _n\) | \(\sigma\) | \(\theta _w\) | h (in \(\upmu\)m) |
---|---|---|---|---|---|---|
0.001 | 0.001 | 1000 | 1000 | 0.01 | 45 | 10 |
Figure 5 shows that CSF struggles to converge due to an inaccurate computation of the capillary forces and large magnitude of PC (see Sect. 5). An improvement in computing the capillary forces by SSF compared to CSF is seen to show better convergence and move the numerical solution towards the analytical solution. Compared to CSF and SSF, FSF and PLIC approximate the capillary forces better. At lower Ca \((<4\times 10^{-3})\), the solutions of SSF, FSF and PLIC converge (\(\lesssim 2\%\) with respect to the finest resolution) at \(h/\Delta x = 16\). All formulations at large Ca \((\approx 0.01)\) show a slower convergence rate, which indicate the requirement to either increase the slip length or resolve the mesh with a similar length scale as that of the slip length (Afkhami et al. 2009). Moreover, we see the numerical solution converging to values below the analytical reference solution which is potentially due to the parabolic flow assumption considered in the analytic solution where we expect around 8.5% discrepancy (\(\ell ^{'}/\ell \approx 0.085\), see Fig. 2). At \(Ca \le 3.12 \times 10^{-4}\), the impact of the parabolic flow assumption and stress singularities on the wall boundaries reduce. Hence, the numerical results converge towards the analytic solution.
Time step size (\(\Delta t\)) in seconds and computation time in minutes for all formulations at different mesh resolutions
\(\Delta x\,(\upmu \text {m})\) | \(\Delta\, t (\text {s})\) | Simulation time \((\text {min})\) | |||
---|---|---|---|---|---|
CSF | SSF | FSF | PLIC | ||
1.25 | \(2.5 \times 10^{-7}\) | 6 | 12 | 11 | 58 |
0.625 | \(8.8 \times 10^{-8}\) | 46 | 98 | 83 | 758 |
0.3125 | \(3.1 \times 10^{-8}\) | 490 | 1046 | 976 | 8100 |
Computation time of CSF is two times faster than SSF and FSF but produces the largest discrepancy in terms of the flow dynamics. Note that SSF and FSF are run with the poreFoam solver and for SSF, the filtering coefficients discussed in Sect. 2.1.3 are set to zero. The Gerris implementation of PLIC takes over 5 times longer compared to FSF but produces more accurate solutions.
Emphasising that the scope of this paper is to quantify PC at low Ca \(< 4 \times 10^{-3}\), observing convergence at a resolution of \(h/\Delta x = 16\) and taking into account the computation cost (Table 3) we choose to perform the rest of our analysis with a mesh resolution \(h/\Delta x = 16\) (\(\Delta x = 0.625\, \upmu\)m).
5 Benchmark cases
In this section, we assess the precision of the VOF formulations, CSF, SSF, FSF and PLIC (Sect. 2) for the capillary forces by comparing the numerical solution of the meniscus dynamics during spontaneous imbibition into a capillary with the analytic solution. We focus on flows at different \({\mathrm{Ca}}\) and quantify PC. We also study the resilience of the numerical solvers with respect to viscosity and density contrasts.
5.1 Flow at different capillary numbers
The focus of this study are flows at the brink where PC become dominant. The capillary numbers investigated are \({\mathrm{Ca}} = 3.12 \times 10^{-3}\) and \({\mathrm{Ca}} = 3.12 \times 10^{-4}\). We vary \({\mathrm{Ca}}\) by changing the length of the channel according to Eq. (34). Table 2 lists the parameters used for this case. Equation (29) provides the analytical solution. We terminate the simulations at 10 times the time taken to reach quasi-steady state for \({\mathrm{Ca}} = 3.12 \times 10^{-3}\) and at 15 times the time taken to reach quasi-steady state for \({\mathrm{Ca}} = 3.12 \times 10^{-4}\).
5.1.1 Case 1: \({\mathrm{ Ca}} = 3.12 \times 10^{-3}\)
At quasi-steady state, the numerical solutions of the time-averaged meniscus velocity differ by \(8.25\%\) for CSF, \(5.4\%\) for SSF, \(3.4\%\) for FSF and \(2\%\) for PLIC from the analytic solution.
CSF shows the largest discrepancy due to inaccurate computation of the interface curvature and suffers from larger magnitude of PC at all times. The coarse representation of the capillary forces implies that more cells are affected by PC (marked by a red circle in Fig. 6). The PC increase viscous dissipation and decelerate the flow.
The solution of SSF is closer to the analytical reference (Fig. 7). Though PC occur, their magnitude has reduced compared to CSF. PC are of similar magnitude as the physical flow. They are continuously generated parallel to the interface (marked by a red circle in Fig. 6). The sharper representation of the capillary forces Eq. (10) restricts PC to fewer cells compared to CSF. The additional viscous dissipation from PC that occur close to the sharper interface explain the marginal rise in the amplitude of fluctuations of the meniscus velocity compared to CSF.
FSF and PLIC do not show any significant PC and the meniscus velocity is free from oscillations. Note that, based on the assumption of Hagen–Poiseuille flow for the analytical solution, we expect an error less than 2.5% as discussed in Sect. 3.2. The amount of filtering for FSF explain the remaining difference between the numerical and analytical solutions.
5.1.2 Case 2: \({\mathrm{ Ca}} = 3.12 \times 10^{-4}\)
At quasi-steady state, the numerical solutions of the time-averaged meniscus velocity differ by 6.35% for CSF, 2.27% for SSF, 0.4% for FSF and 0.37% for PLIC from the analytical solution. The better match is due to the reduced impact of the Hagen–Poiseuille flow assumption on the analytic solution (Sect. 3.2), expecting an error less than 0.25%.
The magnitude of PC, however, has increased approximately three times for CSF, showcasing the importance of an accurate representation of the capillary forces to capture the flow dynamics at low \({\mathrm{Ca}}\). The number of interfacial cells affected by PC are also greater compared to the previous case and the amplitude of fluctuations of the meniscus velocity are within 1%.
For SSF, the magnitude of PC reduce approximately by a factor of two compared to CSF. However, PC occur at all times resulting in a fluctuating meniscus velocity with an amplitude of 3.3%.
For FSF, we see minimal PC generated at the interface at all times resulting in minor fluctuations of the meniscus velocity. Additionally, periodic bursts of PC temporarily decelerate the meniscus (represented by a dashed red circle in the inset of Fig. 9) having an amplitude of 2.6%. As the filtering removes these periodic bursts, the meniscus velocity returns back to the correct solution shortly after. For SSF, the bursts corresponding to maximum PC, are observed when the sharp colour function, \(\alpha _{\text {shp}}\) enters a new wall boundary cell. As we see in Fig. 9, the rise to and fall from maximum PC is gradual compared to FSF. Bursts corresponding to maximum PC for FSF are observed when the sharp colour function fills a wall boundary cell and commences to enter a new boundary cell. The bursts are sudden and filtering dampens these bursts much faster compared to SSF. These periodic bursts of PC for SSF and FSF are observed for the cases discussed in the next sections as well dealing with low capillary numbers.
PLIC shows a consistent velocity field with no significant PC around the interface at all times.
5.2 Effect of viscosity ratio
The displacement of the meniscus decelerate or accelerate during the imbibition process if the fluids viscosities differ. The length of the channel is \(400~\upmu\)m that corresponds to reference \({\mathrm{Ca}} = 3.12 \times 10^{-3}\) (Eq. 34). We obtain different viscosity ratios by fixing the viscosity of the non-wetting phase, \(\mu _{n} = 0.001\) kg/ms and by varying the viscosity of the wetting phase, \(\mu _{w}\) accordingly. Table 2 provides the other parameters for this case.
5.2.1 Case 1: \(M = 0.1\)
The flow decelerates over time as the more viscous wetting phase \(\mu _w = 0.01\, \mathrm{kg/ms}\) invades the channel. All formulations qualitatively show a similar behaviour of the meniscus velocity. The numerical solutions of the meniscus velocity at the end of the inertial acceleration stage differ within 10% (for reference: 6.9% for PLIC) from the analytical solution which could potentially be due to the approximations used in the analytical solution, inaccurate computation of the interfacial curvature and due to PC for some formulations (see below). With time, as the flow decelerates, the discrepancy with respect to the analytical solution is seen to reduce.
Note, that for a later time when the meniscus velocity \({U}_{\text {avg}}^*\) reaches approximately 0.1 (not shown here, that is when the imbibition is about to end), the meniscus velocity is approximately equal to that discussed in Sect. 5.1.2.
As the flow decelerates, CSF and SSF show a gradual rise in the amplitude of fluctuations of the meniscus velocity towards the observations made in Sect. 5.1.2. This indicates that CSF and SSF are not sensitive to viscosity contrasts. Again, CSF shows the largest magnitude of PC at all times. For SSF, the magnitude of PC reduce approximately 1.5 times compared to CSF at \(t^* \approx 1.5\).
Periodic generation of PC for FSF results in a widely fluctuating meniscus velocity. The amplitude of fluctuations at \(t^* \approx 1.5\) is 6.3%. At a meniscus velocity 1.9 times (\({U}^* = 0.19\)) that discussed in Sect. 5.1.2, the amplitude of fluctuations of the meniscus velocity increase 2 times. This indicates that FSF is sensitive to viscosity contrasts.
PLIC shows only a minimal amount of PC that lead to marginal fluctuations of the meniscus velocity (at \(t^* = 1.5\), discrepancy with the analytical solution is within 1%)
5.2.2 Case 2: \(M = 10\)
As the less viscous wetting phase \(\mu _w = 0.0001~\mathrm{kg/ms}\) invades the channel, the flow accelerates over time. At the end of the inertial acceleration stage, the numerical solutions of the meniscus velocity differ by 7.1% for CSF, 3.6% for SSF and 1.07% for FSF and PLIC from the analytic solution. This is primarily due to the coarser representation of the capillary forces for CSF and due to the expected discrepancy from the assumptions made for the analytical solution.
Again, CSF generates maximal PC at all times, whereas SSF shows periodic bursts of PC. The magnitude of PC for SSF at \(t^* = 0.15\) is approximately 1.12 times lower compared to CSF. These PC decelerate the meniscus due to additional viscous dissipation. At any given time, CSF and SSF predict more viscous non-wetting fluid in the channel than the analytical solution due to the retardation of the meniscus by PC. This results in a discrepancy of the viscous forces and hence a deterioration of the solutions of CSF and SSF over time. As the flow accelerates, the fluctuations of the meniscus velocity dampen.
The meniscus velocity for FSF and PLIC match closely with the analytical solution at initial stages. As the flow accelerates, the numerical solutions of FSF and PLIC lag behind the analytical solution. This could potentially be due to error we expect from the analytical assumption which leads to predict slightly larger volume of the more viscous non-wetting fluid in the channel compared to the analytical solution. FSF generates periodic PC though their magnitude is not as large compared to the previous case (\(M = 0.1\)). Hence, the impact of PC on the meniscus velocity is small. The periodic bursts of PC dampen as the flow accelerates. PLIC captures the flow dynamics accurately with no large PC observed around the interface.
5.3 Effect of density ratio
In this section, we investigate the sensitivity to density ratios of \(D=1000\) and \(D=10^{-3}\). As density neither occurs in Eq. (24) for the capillary forces nor in Eq. (26) for the viscous forces, we expect a quasi-steady displacement of the meniscus. However, based on the total mass of the fluids to be displaced in the capillary, the inertial acceleration time scales vary.
The channel length is \(4000~\upmu\)m corresponding to \({\mathrm{Ca}} = 3.12 \times 10^{-4}\) and the wetting phase initially occupies 0.2% of the channels length. Apart from the density of fluids, Table 2 provides the parameters for this case. The corresponding case for \(D = 1\) has been discussed in Sect. 5.1.2.
For PLIC, extremely small time steps (\(\Delta t = 0.08~t_{\mathrm{BK}}\), corresponding to a CFL \(\approx\) \(8\times 10^{-5}\) compared to a CFL \(\approx 9\times 10^{-4}\) at \(t_{\mathrm{BK}}\) for FSF) are required to validate \(D = 10^{-3}\) else resulted in an inconsistent flow field which made the validation infeasible. Hence, PLIC for \(D = 10^{-3}\) is omitted in the discussion.
5.3.1 Case 1: \(D = 1000\)
The specific fluid densities here are \(\rho _{n} = 1000\) kg/m\(^3\) and \(\rho _w = 1\) kg/m\(^3\). We terminate the simulation at 15 times the time it takes to reach quasi-steady state. Figure 12 shows the evolution of the meniscus velocity and the maximum velocity for all formulations (solid lines) and the analytical solution. The inset picture shows the difference in the inertial acceleration time scales for the two cases discussed. As the channel is initially filled by the denser non-wetting phase (99.8% of channel length), the total mass of fluids to be displaced is greater (compared to Sect. 5.3.2), and therefore, the inertial acceleration time scales last longer.
All the formulations show a similar behaviour during acceleration stage before PC for CSF, SSF and FSF induce fluctuations of the meniscus velocity at the end of the inertial flow regime. The amplitude of fluctuations of the meniscus velocity and the magnitude of PC for CSF and SSF are similar to the observations made for \(D = 1\).
PC of similar magnitude as the physical flow for FSF induce minimal fluctuations of the meniscus velocity. Compared to \(D = 1\), PC during periodic bursts have increased 1.22 times thereby increasing the amplitude of fluctuation of the meniscus velocity (amplitude = 3% compared to 2.6% for \(D = 1\)). Similar to Sect. 5.1.2, the maximum PC for SSF and FSF occur related to the sharp colour function entering a new cell along the wall boundary (similar observation for \(D=0.001\)).
PLIC captures the flow dynamics accurately at all sections of the channel.
5.3.2 Case 2: \(D = 0.001\)
6 Conclusion
We have presented spontaneous imbibition in a Hele-Shaw cell symmetry considering a partial slip wall boundary to validate the numerical precision of open-source finite-volume VOF formulations namely, CSF (solver: interFoam), Sharp Surface Force (solver: poreFoam), FSF (solver: poreFoam) and PLIC (solver: Gerris) for capillary numbers between \({\mathrm{Ca}} = 3.12 \times 10^{-4}\) to \({\mathrm{Ca}} = 3.12 \times 10^{-3}\). We extended the Lucas–Washburn equation for the meniscus velocity in a horizontal micro-channel for arbitrary viscosity ratios. We considered flows solely driven by capillarity to focus on quantifying parasitic currents at low \({\mathrm{Ca}}\). We also analysed the resilience of the numerical solvers at different density ratios.
Two important steps of the VOF method that impact the capillary forces are, advecting the colour function and computing the interface curvature.
Solving the colour functions advection equation numerically (CSF, SSF, FSF) results in numerical diffusion and a smeared interface. Geometric split advection technique (PLIC) avoids numerical diffusion and preserves the sharpness of the interface. Extending the concept of accurate geometric advection on unstructured meshes is an ongoing research work (Roenby et al. 2016; Maric et al. 2013).
The height function methodology (PLIC) to compute the interface curvature provides an accurate capillary forces with minimal PC compared to other formulations using Brackbill’s expression (CSF, SSF, FSF).
Along with an inaccurate computation of the interface curvature, PC were seen with CSF at all times increasing viscous dissipation thereby slowing the meniscus velocity. Though PC occur at all times, SSF reduced their magnitude twofold compared to CSF. FSF filtered PC around the interface. However, periodic bursts in the velocity field were seen that temporarily slowed down the meniscus. We observe that when the sharp interface (\(\alpha _\text {shp}\)) enters a new cell along the wall boundary resulted in maximum PC for SSF and FSF at low capillary flows. PLIC provided a consistent flow profile with no PC when the viscosity and density ratios were around one. At high viscosity ratios, minimal PC were seen with PLIC too. We observed that the PLIC solver struggled to model few cases with large density contrasts.
In conclusion, the PLIC formulation of Gerris in 2D is a viable numerical tool to model two-phase flows at low \(\mathrm {Ca}\) on structured grids. Even though SSF does not entirely eliminate PC, the smoothing and sharpening of the interface reduces PC substantially.
FSF reduces PC further but shows periodic bursts and requires additional heuristic parameters. The large magnitude of the periodic bursts of PC, raises concern that they can potentially alter the solution.
Notes
Supplementary material
References
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