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The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

A new method, herein referred to as optimal mode decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the ‘optimal’ low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.

The approach of combining computational fluid dynamics (CFD) for continuum fluid and the discrete element method (DEM) for discrete particles has been increasingly used to study the fundamentals of coupled particle–fluid flows. Different CFD–DEM models have been used. However, the origin and the applicability of these models are not clearly understood. In this paper, the origin of different model formulations is discussed first. It shows that, in connection with the continuum approach, three sets of formulations exist in the CFD–DEM approach: an original format set I, and subsequent derivations of set II and set III, respectively, corresponding to the so-called model A and model B in the literature. A comparison and the applicability of the three models are assessed theoretically and then verified from the study of three representative particle–fluid flow systems: fluidization, pneumatic conveying and hydrocyclones. It is demonstrated that sets I and II are essentially the same, with small differences resulting from different mathematical or numerical treatments of a few terms in the original equation. Set III is however a simplified version of set I. The testing cases show that all the three models are applicable to gas fluidization and, to a large extent, pneumatic conveying. However, the application of set III is conditional, as demonstrated in the case of hydrocyclones. Strictly speaking, set III is only valid when fluid flow is steady and uniform. Set II and, in particular, set I, which is somehow forgotten in the literature, are recommended for the future CFD–DEM modelling of complex particle–fluid flow.

We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A general operator (referred to as the measure operator) is defined in order to mathematically describe an experimental measure. The presented method is applied to the case of mean flow measurements. Such a flow can be described by the Reynolds-averaged Navier–Stokes (RANS) equations, which can be formulated as the classical Navier–Stokes equations driven by a forcing term involving the Reynolds stresses. The stress term is an unknown of the equations and is thus chosen as the control parameter in our study. The data-assimilation algorithm is derived to minimize the error between a mean flow measurement and the measure performed on a numerical solution of the steady, forced Navier–Stokes equations; the optimal forcing is found when this error is minimal. We demonstrate the developed data-assimilation framework on a test case: the two-dimensional flow around an infinite cylinder at a Reynolds number of $\\mathit{Re}=150$ . The mean flow is computed by time-averaging instantaneous flow fields from a direct numerical simulation (DNS). We then perform several ‘measures’ on this mean flow and apply the data-assimilation method to reconstruct the full mean flow field. Spatial interpolation, extrapolation, state vector reconstruction and noise filtering are considered independently. The efficacy of the developed identification algorithm is quantified for each of these cases and compared with more traditional methods when possible. We also analyse the identified forcing in terms of unsteadiness characterization, present a way to recover the second-order statistical moments of the fluctuating velocities and finally explore the possibility of pressure reconstruction from velocity measurements.

Current macroscopic theories of two-phase flow in porous media are based on the extended Darcy's law and an algebraic relationship between capillary pressure and saturation. Both of these equations have been challenged in recent years, primarily based on theoretical works using a thermodynamic approach, which have led to new governing equations for two-phase flow in porous media. In these equations, new terms appear related to the fluid–fluid interfacial area and non-equilibrium capillarity effects. Although there has been a growing number of experimental works aimed at investigating the new equations, a full study of their significance has been difficult as some quantities are hard to measure and experiments are costly and time-consuming. In this regard, pore-scale computational tools can play a valuable role. In this paper, we develop a new dynamic pore-network simulator for two-phase flow in porous media, called DYPOSIT. Using this tool, we investigate macroscopic relationships among average capillary pressure, average phase pressures, saturation and specific interfacial area. We provide evidence that at macroscale, average capillary pressure–saturation–interfacial area points fall on a single surface regardless of flow conditions and fluid properties. We demonstrate that the traditional capillary pressure–saturation relationship is not valid under dynamic conditions, as predicted by the theory. Instead, one has to employ the non-equilibrium capillary theory, according to which the fluids pressure difference is a function of the time rate of saturation change. We study the behaviour of non-equilibrium capillarity coefficient, specific interfacial area, and its production rate versus saturation and viscosity ratio. A major feature of our pore-network model is a new computational algorithm, which considers capillary diffusion. Pressure field is calculated for each fluid separately, and saturation is computed in a semi-implicit way. This provides more numerical stability, compared with previous models, especially for unfavourable viscosity ratios and small capillary number values.

The term 'immersed boundary method' was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (TB) on the flow.

Reactive processes associated with multiphase flows play a significant role in mass transport in unsaturated porous media. For example, the effect of reactions on the solid matrix can affect the formation and stability of fingering instabilities associated with the invasion of a buoyant non-wetting fluid. In this study, we focus on the formation and stability of capillary channels of a buoyant non-wetting fluid (developed because of capillary instabilities) and their impact on the transport and distribution of a reactant in the porous medium. We use a combination of pore-scale numerical calculations based on a multiphase reactive lattice Boltzmann model (LBM) and scaling laws to quantify (i) the effect of dissolution on the preservation of capillary instabilities, (ii) the penetration depth of reaction beyond the dissolution/melting front, and (iii) the temporal and spatial distribution of dissolution/melting under different conditions (concentration of reactant in the non-wetting fluid, injection rate). Our results show that, even for tortuous non-wetting fluid channels, simple scaling laws assuming an axisymmetrical annular flow can explain (i) the exponential decay of reactant along capillary channels, (ii) the dependence of the penetration depth of reactant on a local Péclet number (using the non-wetting fluid velocity in the channel) and more qualitatively (iii) the importance of the melting/reaction efficiency on the stability of non-wetting fluid channels. Our numerical method allows us to study the feedbacks between the immiscible multiphase fluid flow and a dynamically evolving porous matrix (dissolution or melting) which is an essential component of reactive transport in porous media.

The paper presents a model for particle–particle and particle–wall collisions during interface-resolving numerical simulations of particle-laden flows. The accurate modelling of collisions in this framework is challenging due to methodological problems generated by interface approach and contact as well as due to the greatly different time scales involved. To cope with this situation, multiscale modelling approaches are introduced avoiding excessive local grid refinement during surface approach and time step reduction during the surface contact. A new adaptive model for the normal forces in the phase of ‘dry contact’ is proposed, stretching the collision process in time to match the time step of the fluid solver. This yields a physically sound and robust collision model with modified stiffness and damping determined by an optimization scheme. Furthermore, the model is supplemented with a new approach for modelling the tangential force during oblique collisions which is based on two material parameters: a critical impact angle separating rolling from sliding and the friction coefficient for the sliding motion. The resulting new model is termed the adaptive collision model (ACM). All proposed sub-models only contain physical parameters, and virtually no numerical parameters requiring adjustment or tuning. The new model is implemented in the framework of an immersed boundary method but is applicable with any spatial and temporal discretization. Detailed validation against experimental data was performed so that a general and versatile model for arbitrary collisions of spherical particles in viscous fluids is now available.

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator: We pose the problem for and t  > 0 with bounded and compactly supported initial data, and prove the existence of weak and bounded solutions that propagate with finite speed, a property that is not shared by other fractional diffusion models.

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.