Volume-averaged Navier-Stokes Equations Research Articles

Coupled CFD-DEM modeling of surface erosion in granular soils: Simulation of erosion function apparatus experiments

This study introduces a numerical modeling approach that couples the computational fluid dynamics (CFD) with the discrete element method (DEM) to simulate grain-scale soil erosion processes induced by water flows. In this modeling framework, CFD simulates fluid flows by solving the volume-averaged Navier-Stokes equations, and uses the k-ω turbulent model for turbulent flows. Simultaneously, DEM computes the displacement of solid particles by incorporating the fluid-particle interactions driven by fluid flows while adhering to Newton's laws of motion. These interactions encompass drag force, buoyancy force, pressure-gradient force, and viscous force exerted by fluid flows and acting on the particles. The coupled CFD-DEM modeling adeptly replicates soil erosion processes, demonstrating good alignment with results obtained from laboratory erosion function apparatus (EFA) tests. In particular, the DEM facilitates the estimation of shear stress acting on the soil surface based on fluid-particle interaction forces, which has been roughly approximated by empirical or semi-empirical models. This study underscores the capability of coupled CFD-DEM in providing valuable insights into the grain-scale behavior of soil particles subjected to fluid flows, with the potential for extension to address soil erosion and fines migration.

A pressure-based lattice Boltzmann method for the volume-averaged Navier-Stokes equations

The volume-averaged Navier-Stokes equations (VANSE) play a key role in simulating multiphase flow and fluidization. However, the lack of effective models for VANSE has hindered the integration of miscible methods into the lattice Boltzmann framework. To bridge this gap, a novel method named as the pressure-based lattice Boltzmann method for VANSE (PLBM-VANSE) has been proposed. PLBM-VANSE distinguishes itself by decoupling pressure from density, tackling two significant challenges: ensuring volume conservation under large pressure gradients and maintaining stability across discontinuities in void fractions. The numerical results show that PLBM-VANSE effectively eliminates spurious velocities and maintains second-order accuracy in both spatial and spatiotemporal changes of void fraction. The pattern of flow past fixed particle bed is successfully simulated, demonstrating its feasibility and practicality for accurate simulation of complex multiphase flows using the lattice Boltzmann method.

Direct numerical simulation of compressible turbulent channel flows over porous boundaries

The impact of porous boundaries on skin friction and heat transfer in compressible turbulent channel flows is explored through direct numerical simulations. Utilizing the volume-averaged Navier–Stokes equations for the porous media, we examine flows at Mach numbers Mb=0.3 and Mb=1.5, with friction Reynolds numbers ranging from 176 to 220, across three permeability levels: 0 (impermeable), 1.6×10−5, and 1.0×10−4. Our findings reveal a dual impact of porous boundaries on skin friction: a direct reduction due to averaged slip velocity at the interface and an indirect increase from amplified Reynolds shear stress due to enhanced wall-normal permeability. This permeability weakens the wall-blocking effect, leading to intensified wall-normal velocity fluctuations and consequently higher Reynolds stress. Additionally, the Mach number significantly influences flow structures, particularly at higher permeabilities, where it disrupts near-wall streaks at Mb=0.3 but has a diminished effect at Mb=1.5. Thermal analysis confirms the validity of the generalized Reynolds analogy for the mean temperature in these flows, with the turbulent Prandtl number approximating 1, in line with the strong Reynolds analogy. Moreover, a strong spatial correlation between heat flux and wall shear stress fluctuations underscores the intertwined dynamics of momentum and thermal transport at the porous–fluid interface.

Vegetation effects on radiation-induced natural convection in sloping water bodies

In this work, vegetation effects on the characteristics of radiation-induced natural convection (isotherms, circulation patterns, exchange flow rate) in sloping water bodies are investigated numerically. The water body consists of (a) a sloping vegetated region (with a bottom slope equal to 0.1) and (b) a deep region with a horizontal bottom. The vegetation of porosity 0.85 (typical of aquatic plants found in lakes) has a length equal to the length of the sloping region. It can block (totally or partially) the radiation and as a result a non-uniform (differential) heating is developed along the free surface of the water body. The Volume-Averaged Navier–Stokes equations together with the Volume-Averaged Energy equation are solved numerically in the vegetated region. The radiation-induced natural convection in a water body with only a sloping region (with no vegetation) is also considered for validation purposes since numerical and scaling analysis results are available in literature. The results indicate significant vegetation effects on the thermal and flow patterns especially for vegetation which blocks completely surface heating.

Diurnal heating and cooling of a sloping water body with vegetated shallow regions

In this work, diurnal heating, by solar radiation, and cooling in a sloping water body with vegetated shallow regions are investigated numerically. The sloping water body consists of a vegetated region with a bottom slope equal to 0.1 and a deep region with a horizontal bottom. The Volume-Averaged Navier-Stokes equations together with the Volume-Averaged Energy equation are solved numerically in the vegetated region. The latter has porosity equal to 0.85 (typical of aquatic plants found in lakes) and a length equal to the total length of the sloping region. At the top free surface, a time-dependent thermal forcing is applied, which is reduced in the vegetated region. Shallow water is considered with a maximum water depth being less than the penetration depth of the solar radiation. The non-vegetated sloping water body is also considered for validation purposes. The results (isotherms, streamlines, exchange flow rate), after ten full thermal forcing cycles, indicate significant vegetation effects on the daytime and night circulation and the exchange flow rates.

3D CFD-DEM study on fine particle migration in packed proppant layers

Fine particle migration is a crucial issue in hydraulic fracturing-based production from unconventional reservoirs and can significantly affect the effective conductivity of the packed proppant layers. In this study, to investigate the conveying behavior of fine particles, a Eulerian-Lagrangian method, namely the computational fluid dynamics–discrete element method (CFD-DEM), is adopted for 3D simulations. Two types of particles are involved in this process: larger proppant particles and fine particles. Proppant particles form a background skeleton, and a two-phase flow of fine particles and fluid is constrained in the connected pore networks. We adopt a unified CFD-DEM framework to resolve the non-slip boundary condition on the proppant particle surface and track fine particles using a four-way coupling strategy. In particular, a forcing term based on the fictitious domain method is embedded in a regular volume-averaged Navier-Stokes equation to solve the fluid motion. A perfectly packed skeleton of a face-centered cube is designed for subsequent numerical studies. The results show that the fluid and particle flow velocities are higher near the fracture surfaces than inside the interlayers. In addition, it is found that as the fine particle concentration increases, the average horizontal velocity increases, whereas the settling velocity tends to decrease.

Developed and quasi-developed macro-scale flow in micro- and mini-channels with arrays of offset strip fins

We investigate to what degree the steady laminar flow in typical micro- and mini-channels with offset strip fin arrays can be described as developed on a macro-scale level, in the presence of channel entrance and sidewall effects. Hereto, the extent of the developed and quasi-developed flow regions in such channels is determined through large-scale numerical flow simulations. It is observed that the onset point of developed flow increases linearly with the Reynolds number and channel width but remains small relative to the total channel length. Furthermore, we find that the local macro-scale pressure gradient and closure force for the (double) volume-averaged Navier–Stokes equations are adequately modeled by a developed friction factor correlation, as typical discrepancies are below 15% in both the developed and developing flow region. We show that these findings can be attributed to the eigenvalues and mode amplitudes, which characterize the quasi-developed flow in the entrance region of the channel. Finally, we discuss the influence of the channel side walls on the flow periodicity, the mass flow rate, as well as the macro-scale velocity profile, which we capture by a displacement factor and slip length coefficient. Our findings are supported by extensive numerical data for fin height-to-length ratios up to 1, fin pitch-to-length ratios up to 0.5, and channel aspect ratios between 1/5 and 1/17, covering Reynolds numbers from 28 to 1224.

Consistent lattice Boltzmann methods for the volume averaged Navier–Stokes equations

We derive a novel lattice Boltzmann scheme, which uses a pressure correction forcing term for approximating the volume averaged Navier–Stokes equations (VANSE) in up to three dimensions. With a new definition of the zeroth moment of the Lattice Boltzmann equation, spatially and temporally varying local volume fractions are taken into account. A Chapman–Enskog analysis, respecting the variations in local volume, formally proves the consistency towards the VANSE limit up to higher order terms. The numerical validation of the scheme via steady state and non-stationary examples approves the second order convergence with respect to velocity and pressure. The here proposed lattice Boltzmann method is the first to correctly recover the pressure with second order for space-time varying volume fractions.

Natural convection due to surface cooling in sloping water bodies with vegetation

Natural convection induced by surface cooling in sloping water bodies with vegetation is investigated numerically. The sloping water body consists of a vegetated region with a bottom slope equal to 0.1 and a deep region with a horizontal bottom. The volume-averaged Navier–Stokes equations together with the volume-averaged energy equation are solved numerically in the vegetated region. Submerged vegetation has an equivalent porosity of 0.85. The vegetation length is either equal to the total or half the sloping region. A differential cooling is applied at the top free surface with a varying heat loss rate between the two regions. The non-vegetated sloping water body is also considered for validation purposes. The results indicate that, when cooling is reduced in the vegetated region (a) there is a time delay for the establishment of the quasi-steady regime and (b) the exchange flow rate and the gravity current's velocity decreases with increasing vegetation length. When cooling is completely blocked, different circulation patterns are developed.

Turbulent channel flow of suspensions of neutrally buoyant particles over porous media

This study discusses turbulent suspension flows of non-Brownian, non-colloidal, neutrally buoyant and rigid spherical particles in a Newtonian fluid over porous media with particles too large to penetrate and move through the porous layer. We consider suspension flows with the solid volume fraction ${{\varPhi _b}}$ ranging from 0 to 0.2, and different wall permeabilities, while porosity is constant at 0.6. Direct numerical simulations with an immersed boundary method are employed to resolve the particles and flow phase, with the volume-averaged Navier–Stokes equations modelling the flow within the porous layer. The results show that in the presence of particles in the free-flow region, the mean velocity and the concentration profiles are altered with increasing porous layer permeability because of the variations in the slip velocity and wall-normal fluctuations at the suspension-porous interface. Furthermore, we show that variations in the stress condition at the interface significantly affect the particle near-wall dynamics and migration toward the channel core, thereby inducing large modulations of the overall flow drag. At the highest volume fraction investigated here, ${{\varPhi _b}}= 0.2$ , the velocity fluctuations and the Reynolds shear stress are found to decrease, and the overall drag increases due to the increase in the particle-induced stresses.

How interfacial dynamics controls drainage pore-invasion patterns in porous media

Immiscible two-phase flow through porous media is composed of a series of pore invasions; however, the consequences of pore-scale processes on macroscopic fluid front behavior remain to be clarified. In this work, we perform an analytical and experimental investigation of front behavior and pore invasions dynamics during drainage for various viscosity ratios and capillary numbers. We use a microfluidic setup that includes a fully controlled pore-doublet geometry to isolate and explore pore-invasion mechanisms. We apply a model based on volume-averaged Navier–Stokes equations to capture interface dynamics. Different invasion mechanisms are characterized and correlated with front behavior for various flow conditions. For the viscous flow regime, a succession of continuous pore invasions is observed, leaving a thick layer of wetting phase behind at pore curvatures. Abrupt interfacial jumps, followed by an apparent stagnant condition, are observed for the capillary flow regime. We identified a new regime, called crossover flow regime, for which pore invasion shows a mixed behavior between capillary and viscous dominated regimes. The global front behavior is predicted based on the numerical simulation and experimental results for all flow regimes.

A three-dimensional model of wave interactions with permeable structures using the lattice Boltzmann method

To simulate the three-dimensional interaction of waves and porous structures using the lattice Boltzmann method, a novel multi-relaxation-time lattice Boltzmann scheme corresponding to the volume-averaged Navier-Stokes equations incorporating porous flow is developed. The porosity is introduced into the equilibrium distribution function, and the frictional forces produced by the porous media are added by the discrete force model. Through the Maxwell iteration, the relation between the mesoscopic lattice Boltzmann scheme and the macroscopic governing equations is established. Large eddy simulation and the single-phase volume-of-fluid techniques are modified to take porous media into account. The friction parameters in this model are calibrated using the experimental data of two-dimensional dam-break waves interacting with porous media composed of crushed rocks. Validations are carried out by comparing the simulation result of the proposed model with the laboratory data of three-dimensional dam-break waves impacting a prism and three-dimensional surface gravity waves (solitary waves and cnoidal waves) interacting with a vertical permeable breakwater. Furthermore, the interaction of regular waves and a rubble mound breakwater is used to test the capability of the model to handle spatially varying porous media. The different simulation results strongly agree with the experimental data, which prove that the lattice Boltzmann model has the ability to simulate complex wave motions near porous structures.

Resolvent-based predictions for turbulent flow over anisotropic permeable substrates

Abstract

A Comparative Study of Closure Relations for CFD Modelling of Bubbly Flow in a Vertical Pipe

Commercial code CFX was used to examine the performance of a two-fluid model to predict the details of upward isothermal bubbly flow of air and water in a vertical pipe. The model equations are volume-averaged Navier-Stokes equations that require closure models for interfacial forces and bubble-induced turbulence effects. Two-equation SST and k-epsilon RANS turbulence models were also used. A parametric study of closure models included both standard options in CFX and previously published novel closure models that were implemented with user-defined functions. The CFD simulations were compared with two cases from the MTLoop experiments by Lucas et al. at the Helmholtz-Zentrum Dresden Rossendorf: one with wall-peak void fraction profile (MT039), and another with a core-peak void fraction profile (MT118). The effect of changing the drag force closures was not significant for the set examined. Poor predictions were found when the lift force and wall lubrication models were incompatible in magnitude. There was no significant effect of changing the liquid phase turbulence model. Changing the bubble-induced turbulence models, however, had a significant impact on the radial void fraction profile. The novel wall force from Lubchenko et al. at the Massachusetts Institute of Technology significantly improved the prediction of the near wall void fraction in the wall peak profile.

Velocity measurements of dilute particulate suspension over and through a porous medium model

We experimentally examine pressure-driven flows of 1%, 3%, and 5% dilute suspensions over and through a porous media model. The flow of non-colloidal, non-Brownian suspensions of rigid and spherical particles suspended in a Newtonian fluid is considered at very low Reynolds numbers. The model of porous media consists of square arrays of rods oriented across the flow in a rectangular channel. Systematic experiments using high-spatial-resolution planar particle image velocimetry and index-matching techniques are conducted to accurately measure the velocity measurements of both very dilute and solvent flows inside and on top of the porous media model. We found that for 1%, 3%, and 5% dilute suspensions, the fully developed velocity profile inside the free-flow region is well predicted by the exact solution derived from coupling the Navier–Stokes equation within the free flow-region and the volume-averaged Navier Stokes equation for the porous media. We further analyze the velocity and shear rate at the suspension–porous interface and compare these data with those of pure suspending fluid and the related analytical solutions. The exact solution is used to define parameters necessary to calculate key values to analyze the porous media/fluid interaction, such as Darcy velocity, penetration depth, and fractional ratios of the mass flow rate. These parameters are comparable between the solvent, dilute suspensions, and exact solution. However, we found clear effects between the solvent and the suspensions, which shows different physical phenomenon occurring when particles are introduced into a flow moving over and through a porous media.

A coupled CFD-DEM method with moving mesh for simulating undrained triaxial tests on granular soils

A coupled computational fluid dynamics and distinct element method (CFD-DEM) is proposed to simulate undrained triaxial tests on granular soils. In this method, the compressible volume-averaged Navier–Stokes equations of continuity and momentum with an additional state equation are used for modelling liquid flows. In addition, the volume-averaged Navier–Stokes equations are modified by including the velocity of moving grid in their convective terms to settle the problem of variable boundaries in the process of undrained triaxial compression. Then the capability of the developed method is mainly examined by comparisons with the constant volume method of simulating undrained triaxial tests, which shows that the results of the CFD-DEM method can compare favourably with those obtained from the constant volume method. Finally, the influences of strain rate, initial pore pressure and fluid compressibility on the undrained behavior of granular soils are carefully studied.

Aggregation in mixing tanks – The role of inter-particle forces

We study aggregation of equally-sized spherical particles under mildly turbulent flow conditions (Reynolds numbers in the range 4000–8000) in a mixing tank through numerical simulation. The dynamics of the liquid flow is solved in terms of the volume-averaged Navier-Stokes equations by an extended lattice-Boltzmann method on a fixed uniform cubic grid. The particle dynamics is updated through applying Newton’s second law to each particle. The simulations include a two-parameter model for the attractive force between the particles that causes aggregation. The dynamics of solids and liquid are two-way coupled through a mapping procedure. An overall solids volume fraction of 10% has been investigated. The level of aggregation of particles in the mixing tank mainly depends on the strength of the attractive force and on the impeller-based Reynolds number, not so much on the distance over which the aggregative force is active. A higher Reynolds number leads to less aggregation.

The effect of pressure pulsing on the mechanical dewatering of nanofiber suspensions

Dewatering processes are invariably encountered in the chemical manufacturing and processing of various bioproducts. In this study, Computational Fluid Mechanics (CFD) simulations and theory are utilized to model and optimize the dewatering of commercial nanofiber suspensions. The CFD simulations are based on the volume-averaged Navier-Stokes equations, while the analytical model is deduced from the empirical Darcy’s law for dewatering flows. The results are successfully compared to experimental data on commercial cellulose suspensions obtained with a Dynamic Drainage Analyzer (DDA). Both the CFD simulations and the analytical model capture the dewatering flow profiles of the commercial suspensions in an experiment using a constant pressure profile. However, a temporally varying pressure profile offers a superior dewatering performance, as indicated by both the simulations and the analytical model. Finally, the analytical model also predicts an optimized number of pressure pulses, minimizing the time required to completely dewater the suspension.

A Subgrid-Scale Model for Turbulent Flow in Porous Media

Given the analogy between the filtered equations of large eddy simulation and volume-averaged Navier–Stokes equations in porous media, a subgrid-scale model is presented to account for the residual stresses within the porous medium. The proposed model is based on the kinetic energy balance of the filtered velocity field within a pore; hence, when using the model, numerical simulations of the turbulent flow in the pores are not required. The accuracy of the model is validated with available data in the literature on turbulent flow through packed beds and staggered arrangement of square cylinders. The validation yields that the model successfully captures the effect of the pore-scale turbulent motion. The model is then used to study turbulent flow in a wall-bounded porous media to assess its accuracy.

Numerical simulation of proppant transport in propagating fractures with the multi-phase particle-in-cell method

In this work, the proppant transport process in large-scale propagating fractures is simulated using an Eulerian–Lagrangian method. Fracture propagation is solved using the Perkins–Kern–Nordgren(PKN) model, while the fluid-particle system is solved with the multi-phase particle-in-cell (MP-PIC) method. The fluid motion is governed by volume-averaged Navier–Stokes equations, and solved using the finite volume method, and the particle motion is solved by applying Newton’s second law in a Lagrangian manner. Based on the original MP-PIC method, an extended 2D system of governing equations for fluid-particle flow is derived to solve the moving boundary problems associated with fracture propagation. By means of this method, the fluid-particle interaction is fully coupled, and the propagating fracture is considered as a prior-known boundary for the fluid and particle phases. Several numerical experiments are performed to validate the method for simulating fluid motion and proppant settling behaviors in a fracture through comparison with results in the literature. The simulation results of the 2D framework are also compared with those of 3D framework and show a good agreement. Large-scale problems of proppant transport in propagating fractures for different proppant and fracturing fluid properties, including the leak-off effect, are then simulated using this method. The Lagrangian feature of the MP-PIC method allows for flexible design of proppant injection, such as injection of proppant with multi-densities and/or multi-sizes.