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Classical Darcy's equation for multiphase flow assumes that gravity and the gradient in fluid pressure are the only driving forces and resistance to the flow is parameterized by (relative) permeability as a function of saturation. It is conceivable that, in multiphase flow, other driving forces may also exist. This would mean that such nonequilibrium effects are lumped into a permeability coefficient. Indeed, many studies have shown that the relative permeability coefficient generally depends not only on saturation but also on dynamics of the system. Through the application of rational thermodynamics, a theory of two‐phase flow had been developed in which interfacial areas were introduced as separate thermodynamic entities and their macroscale effects were explicitly included. This theory includes new driving forces whose significance needs still to be established. To study new terms in the theory, we employ a dynamic pore network model called DYPOSIT. A long pore network (several representative elementary volumes connected in series) is generated, which represents as a one‐dimensional porous medium column. This model provides pore scale distribution of local phase pressures, capillary pressure, interfacial area, saturation, and flow rate, which are averaged to obtain the macroscale distributions of these variables. Our analysis shows that there are discrepancies between the simulation results and the classical equations to describe the transient behavior, especially for the nonwetting phase transient permeability. The coefficients in the extended equations are quantified and parameterized. Although under the applied Dirichlet boundary conditions, flow varies significantly, there is a clear trend illustrating dependency of coefficients on saturation, independent of dynamic conditions. Furthermore, using the new coefficients, it is possible to explain either transient or steady state flow regimes, which is a new achievement. Key Points The classical Darcy's law for two‐phase flow has flaws The gradient of pressure alone cannot explain the two‐phase flow correctly Interfacial area can be the missing state variable in two‐phase flow

A series of primary drainage experiments was carried out in order to investigate nonequilibrium capillarity effects in two‐phase flow through porous media. Experiments were performed with tetrachloroethylene (PCE) and water as immiscible fluids in a sand column 21 cm long. Four drainage experiments were performed by applying large pressures on the nonwetting phase at the inlet boundary: 20, 30, 35, 38 kPa. Our results showed that the nonequilibrium local fluids pressure difference‐saturation curves are above the capillary pressure saturation curve. Moreover, the nonequilibrium pressure difference showed a nonmonotonic behavior with an overshoot that was more pronounced at higher injection pressures. The dynamic capillarity coefficient was calculated from measured local pressures and saturations (the scale of sensor devices, 0.7 cm). Its value was found to vary between 1.3 × 105 to 2 × 105 Pa s. Within the saturation range of 0.50 > Sw > 0.85, no clear dependency of the dynamic coefficient on the wetting saturation was observed. Also, no dependency of the dynamic capillarity coefficient on the applied boundary pressure was found. Averaged values of at the length scales of 11 and 18 cm were also estimated from averaged pressures and saturations. The upscaled dynamic coefficient was found to vary between 0.5 × 106 and 1.2 × 106 Pa s at the average window size of 11 cm. This is one order of magnitude larger than the local‐scale coefficient. Larger values were found for the length scale of 18 cm: 1.5 × 106 and 2.5 × 106 Pa s. This suggests that the value of dynamic coefficient increases with the scale of observation. Key Points The non‐equilibrium pressure difference shows a non‐monotonic behavior Pressure difference overshoot is more pronounced at higher injection pressures The dynamic capillarity coefficient τ is a length scale dependent

Numerical models have been widely used to simulate multiphase flow in porous media for a variety of applications (e.g., NAPL migration in subsurface aquifers, carbon sequestration, agriculture, paper production, and petroleum reservoir development). The relationship between the difference in phase pressures and saturation is used as one of the important constitutive relationships in numerical models. Theoretical studies have suggested that this relationship should include a damping coefficient or capillarity coefficient () on the basis of thermodynamic considerations. A literature review suggests that the magnitude of this capillarity coefficient varies by over three orders of magnitude. While recent experimental studies have explored the effect of porous medium properties, effect of domain size, hysteresis, and the imposed boundary conditions on the magnitude of , there has been no experimental study investigating the impact of fluid viscosity on . This study reports on a series of primary drainage experiments conducted under both static and dynamic conditions in F70 silica sand. Fluid pairs used included water and silicone oil with two differing viscosities and slightly different densities (used as model nonaqueous phase liquids) in addition to air. Water saturation and both wetting and nonwetting phase pressures were measured in a custom‐built aluminum column using EC‐5 probes and tensiometers at three levels. Results show a strong dependence of the magnitude of the capillarity coefficient on effective fluid viscosity. This implies that consideration should be given for the inclusion of a capillarity coefficient in modeling tools used to simulate multiphase flow when fluids saturations are changing rapidly and when fluids have a large viscosity ratio. Key Points Rate dependence of the phase pressure difference Capillarity coefficient is a function of fluid viscosity Capillarity coefficient is relatively constant when normalized to effective viscosity

In the top submerged lance (TSL) smelting process, flow boiling may occur in the lance’s inner pipe due to the heat coming from the furnace when liquid fuel is adopted. In the current study, a numerical simulation was carried out by coupling the Eulerian two-fluid model with the improved RPI wall boiling model to investigate the subcooled n-heptane flow boiling in the inner pipe. The effects of inlet velocity and pipe wall emissivity on two-phase flow and heat transfer are elucidated. The results show that, for pipes with inlet velocity ranging from 0.3 m·s−1 to 1.0 m·s−1, an increase in inlet velocity leads to a lower void fraction near the outlet, as well as a lower average velocity and a lower average temperature of each phase. Meanwhile, the Onset of Nucleate Boiling (ONB) position approaches to the outlet, and the total pressure drop of the entire pipe reduces when the inlet velocity increases. However, the opposite trends appear when increasing the pipe wall emissivity. The maximum wall temperature corresponding to the critical heat flux (CHF) point is slightly affected by inlet velocity but significantly affected by pipe wall emissivity. The non-equilibrium effect and the specific components of pressure drop are also further investigated.

Phase change energy storage technology has great potential for enhancing the efficient conversion and storage of energy. While triply periodic minimal surface (TPMS) structures have shown promise in improving heat transfer, research on their application in phase change heat transfer remains limited. This paper presents numerical simulations of composite phase change materials (PCMs) featuring TPMS skeletons, specifically gyroid, diamond, primitive, and I-graph and wrapped package-graph (I-WP) utilizing the lattice Boltzmann method (LBM). A comparative analysis of the effects of four TPMS skeletons on enhancing the phase change process reveals that the PCM containing the gyroid skeleton melts the fastest, with a complete melting time of 24.1% shorter than that of the PCM containing the I-WP skeleton. The PCM containing the gyroid skeleton is further simulated to explore the effects of the Rayleigh ( Ra) number, Prandtl ( Pr) number, and Stefan ( Ste) number on the melting characteristics. Notably, the complete melting time is reduced by 60.44% when Ra is increased to 10 6 compared to the case with Ra at 10 4. Increasing the Pr number accelerates the migration of the mushy zone, resulting in fast melting. Conversely, the convective heat transfer effect from the heating surface decreases as the Ste number increases. The temperature differences caused by the local thermal non-equilibrium (LTNE) effect over time are significant and complex, with peaks becoming more pronounced nearer the heating surface. This study intends to provide theoretical support for the further development of TPMS skeletons in enhancing the phase change process.

Hybrid particle–continuum computational frameworks permit the simulation of gas flows by locally adjusting the resolution to the degree of non-equilibrium displayed by the flow in different regions of space and time. In this work, we present a new scheme that couples the direct simulation Monte Carlo (DSMC) with the lattice Boltzmann (LB) method in the limit of isothermal flows. The former handles strong non-equilibrium effects, as they typically occur in the vicinity of solid boundaries, whereas the latter is in charge of the bulk flow, where non-equilibrium can be dealt with perturbatively, i.e. according to Navier–Stokes hydrodynamics. The proposed concurrent multiscale method is applied to the dilute gas Couette flow, showing major computational gains when compared with the full DSMC scenarios. In addition, it is shown that the coupling with LB in the bulk flow can speed up the DSMC treatment of the Knudsen layer with respect to the full DSMC case. In other words, LB acts as a DSMC accelerator. This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

The dynamic non-equilibrium effect (DNE) describes the non-unique character of saturation–capillary pressure relationships observed under static, steady-state, or monotonic hydrodynamic conditions. Macroscopically, the DNE manifests as variations in soil hydraulic characteristic curves arising from varying hydrodynamic testing conditions and is fundamentally governed by soil matrix particle size distribution. Changes in the DNE across porous media with discrete particle size fractions are investigated via stepwise drying experiments. Through quantification of saturation–capillary pressure hysteresis and DNE metrics, three critical signatures are identified: (1) the temporal lag between peak capillary pressure and minimum water saturation; (2) the pressure gap between transient and equilibrium states; and (3) residual water saturation. In the four experimental sets, with the finest material (Test 1), the peak capillary pressure consistently precedes the minimum water saturation by up to 60 s. Conversely, with the coarsest material (Test 4), peak capillary pressure does not consistently precede minimum saturation, with a maximum lag of only 30 s. The pressure gap between transient and equilibrium states reached 14.04 cm H2O in the finest sand, compared to only 2.65 cm H2O in the coarsest sand. Simultaneously, residual water saturation was significantly higher in the finest sand (0.364) than in the coarsest sand (0.086). The results further reveal that the intensity of the DNE scales inversely with particle size and linearly with wetting phase saturation (Sw), exhibiting systematic decay as Sw decreases. Coarse media exhibit negligible hysteresis due to suppressed capillary retention; this is in stark contrast with fine sands, in which the DNE is observed to persist in advanced drying stages. These results establish pore geometry and capillary dominance as fundamental factors controlling non-equilibrium fluid dynamics, providing a mechanistic framework for the refinement of multi-phase flow models in heterogeneous porous systems.

Rayleigh-Taylor (RT) instability widely exists in nature and engineering fields. How to better understand the physical mechanism of RT instability is of great theoretical significance and practical value. At present, abundant results of RT instability have been obtained by traditional macroscopic methods. However, research on the thermodynamic non-equilibrium (TNE) effects in the process of system evolution is relatively scarce. In this paper, the discrete Boltzmann method based on non-equilibrium statistical physics is utilized to study the effects of the specific heat ratio on compressible RT instability. The evolution process of the compressible RT system with different specific heat ratios can be analyzed by the temperature gradient and the proportion of the non-equilibrium region. Firstly, as a result of the competition between the macroscopic magnitude gradient and the non-equilibrium region, the average TNE intensity first increases and then reduces, and it increases with the specific heat ratio decreasing; the specific heat ratio has the same effect on the global strength of the viscous stress tensor. Secondly, the moment when the total temperature gradient in y direction deviates from the fixed value can be regarded as a physical criterion for judging the formation of the vortex structure. Thirdly, under the competition between the temperature gradients and the contact area of the two fluids, the average intensity of the non-equilibrium quantity related to the heat flux shows diversity, and the influence of the specific heat ratio is also quite remarkable.

Based on the framework of our previous work [H.L. Lai et al., Phys. Rev. E, 94, 023106 (2016)], we continue to study the effects of Knudsen number on two-dimensional Rayleigh–Taylor (RT) instability in compressible fluid via the discrete Boltzmann method. It is found that the Knudsen number effects strongly inhibit the RT instability but always enormously strengthen both the global hydrodynamic non-equilibrium (HNE) and thermodynamic non-equilibrium (TNE) effects. Moreover, when Knudsen number increases, the Kelvin–Helmholtz instability induced by the development of the RT instability is difficult to sufficiently develop in the later stage. Different from the traditional computational fluid dynamics, the discrete Boltzmann method further presents a wealth of non-equilibrium information. Specifically, the two-dimensional TNE quantities demonstrate that, far from the disturbance interface, the value of TNE strength is basically zero; the TNE effects are mainly concentrated on both sides of the interface, which is closely related to the gradient of macroscopic quantities. The global TNE first decreases then increases with evolution. The relevant physical mechanisms are analyzed and discussed.

How to solve the hypersonic aerothermodynamics around large-scale uncontrolled spacecraft during falling disintegrated process from outer space to earth, is the key to resolve the problems of the uncontrolled Tiangong-No.1 spacecraft reentry crash. To study aerodynamics of spacecraft reentry covering various flow regimes, a Gas-Kinetic Unified Algorithm (GKUA) has been presented by computable modeling of the collision integral of the Boltzmann equation over tens of years. On this basis, the rotational and vibrational energy modes are considered as the independent variables of the gas molecular velocity distribution function, a kind of Boltzmann model equation involving in internal energy excitation is presented by decomposing the collision term of the Boltzmann equation into elastic and inelastic collision terms. Then, the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions by developing the discrete velocity ordinate method and numerical quadrature technique. The unified algorithm of the Boltzmann model equation involving thermodynamics non-equilibrium effect is presented for the whole range of flow regimes. The gas-kinetic massive parallel computing strategy is developed to solve the hypersonic aerothermodynamics with the processor cores 500~45,000 at least 80% parallel efficiency. To validate the accuracy of the GKUA, the hypersonic flows are simulated including the reentry Tiangong-1 spacecraft shape with the wide range of Knudsen numbers of 220~0.00005 by the comparison of the related results from the DSMC and N-S coupled methods, and the low-density tunnel experiment etc. For un-controlling spacecraft falling problem, the finite-element algorithm for dynamic thermal-force coupling response is presented, and the unified simulation of the thermal structural response and the hypersonic flow field is tested on the Tiangong-1 shape under reentry aerodynamic environment. Then, the forecasting analysis platform of end-of-life large-scale spacecraft flying track is established on the basis of ballistic computation combined with reentry aerothermodynamics and deformation failure/disintegration.