Volume Source Terms Research Articles

A pore-scale lattice Boltzmann model for solute transport coupled with heterogeneous surface reactions and mineral dissolution

In this paper, we propose a pore-scale lattice Boltzmann model to simulate fluid flow coupled with heterogeneous surface reactions and mineral dissolution. The primary innovation lies in the transformation of surface reactions, originally treated as a boundary condition, into a volume-source term through dimensional augmentation within the framework of sharp liquid–solid interfaces. This significantly simplifies the implementation, particularly for reactions occurring in porous media with intricate geometric structures. Several benchmark tests were performed to validate the accuracy of this model, including a reaction–diffusion problem in a rectangular domain, a two-dimensional reaction and dissolution of a circular grain, as well as a three-dimensional calcite crystal dissolution in a micro-channel. All the obtained simulation results agree well with the reference solutions. In addition, a dissolution problem in a three-dimensional porous medium built with the sand pack is then investigated. Cases with different Péclet numbers (Pe) and Damköhler numbers (Da) were simulated, and five dissolution modes were obtained, which were finally summarized in a diagram of Pe and Da.

Pore-Scaled investigation on dynamic carbonation mechanism of calcium oxide particles

Carbonation of high-temperature calcium oxide is one of the main means of calcium looping thermochemical heat storage and it has unique advantages in carbon capture and energy conversion and utilization. This work is the first to construct the carbonation process of a single calcium oxide particle from pore size by using the lattice Boltzmann method. At the same time, considering the diffusion of gas in the solid, the reaction source term is optimized to a volume source term to be closer to the real situation. The results show that the establishment of the method can better reflect the micro-flow diffusion mass transfer and heat release process inside the pores. The multi-field coupling method can be extended to many areas of the porous microstructure of gas–solid reaction, which guides significance for macro-reactor design and optimization of the particle structure, and produces far-reaching value in carbon neutrality and new energy exploitation.

A Modified Reynolds-Averaged Navier–Stokes-Based Wind Turbine Wake Model Considering Correction Modules

Increasing wind power generation has been introduced into power systems to meet the renewable energy targets in power generation. The output efficiency and output power stability are of great importance for wind turbines to be integrated into power systems. The wake effect influences the power generation efficiency and stability of wind turbines. However, few studies consider comprehensive corrections in an aerodynamic model and a turbulence model, which challenges the calculation accuracy of the velocity field and turbulence field in the wind turbine wake model, thus affecting wind power integration into power systems. To tackle this challenge, this paper proposes a modified Reynolds-averaged Navier–Stokes (MRANS)-based wind turbine wake model to simulate the wake effects. Our main aim is to add correction modules in a 3D aerodynamic model and a shear-stress transport (SST) k-ω turbulence model, which are converted into a volume source term and a Reynolds stress term for the MRANS-based wake model, respectively. A correction module including blade tip loss, hub loss, and attack angle deviation is considered in the 3D aerodynamic model, which is established by blade element momentum aerodynamic theory and an improved Cauchy fuzzy distribution. Meanwhile, another correction module, including a hold source term, regulating parameters and reducing the dissipation term, is added into the SST k-ω turbulence model. Furthermore, a structured hexahedron mesh with variable size is developed to significantly improve computational efficiency and make results smoother. Simulation results of the velocity field and turbulent field with the proposed approach are consistent with the data of real wind turbines, which verifies the effectiveness of the proposed approach. The variation law of the expansion effect and the double-hump effect are also given.

Effect mechanism of exit-water temperature distribution characteristics on the anti-freezing of natural draft dry cooling tower

For thermal power plants with natural draft dry cooling towers (NDDCTs) in arid and severely cold regions, the anti-freezing of finned-tube bundles is critical for the safe and economic running of a thermodynamic system. Using volume source term models representing the water flow in each pass of a cooling column, a 3D numerical model for NDDCT was established and validated. The exit water temperature and its relevant distribution characteristics could then be analyzed from cooling columns, deltas to the whole tower. Based on the analyses, the non-equilibrium temperature difference and relevant anti-freezing parameters were defined. It could be observed that the ambient crosswind, ambient temperature, heat load, circulating water flow rate, and convection pattern have different impact mechanisms on the exit water temperature distribution characteristics, among which, the ambient crosswind plays a major role but the ambient temperature has a negligible effect. Finally, the critical anti-freezing average exit-water temperatures were presented based on real work conditions, which would facilitate the safe and economic operation of NDDCTs in severely cold weather.

Vaporization of liquid droplet with large deformation and high mass transfer rate, I: Constant-density, constant-property case

A numerical model based on decomposed vaporization mechanisms has been developed for the study of vaporization of liquid droplets with large deformation and high mass transfer rate. In this model, the vaporization-induced volume, energy, and species source terms in the gas phase are modeled using a body-fitted layer placed around the interface on the gas side. In this layer, continuous distributions of volume, energy, and species sources are specified by the conservation laws. A corresponding body-fitted sink layer is placed on the liquid side of the interface to account for mass, energy and species loss in the liquid phase due to vaporization. The formulation retains the advantages of interface localization methods and mitigates the error and complexity caused by the combined treatment of mass/heat transfer and interface localization. The new model is versatile and does not depend on the specific interface localization method, and can thus be implemented in any of the existing methods. As a specific example, the present approach is implemented in a volume-of-fluid (VOF)-based code “Gerris”. A number of test cases concerning liquid vaporization in both quiescent and convective environments are presented, with the constant-density assumption, to verify and validate this model in terms of interface localization. In particular, the vaporization of a single n-decane droplet in a uniform air flow of 1000 K at 1 atm is investigated. The predicted liquid volume evolution demonstrates the robustness and accuracy of interface localization for cases with large density ratio and high vaporization rate. Vaporization of n-decane droplets with large deformation and high mass transfer is also investigated systematically.

Accuracy and efficiency improvements in synthetic eddy methods.

With performance in mind, we propose two general improvements to the popular class of turbulent inlet boundary conditions known as the ‘synthetic eddy method’, originally proposed 10 years ago by Jarrin 2006. Our updated version offers improvement in both statistical accuracy and computational efficiency. We first demonstrate that the original approach led to inaccuracies where eddy length-scale prescription was inhomogeneous. We then describe a correction to the normalisation procedure to ensure that prescribed statistics can be correctly recovered. A second improvement focusses on the computational efficiency of the method; by generalising the method to allow for arbitrary eddy placement, the required number of eddies whilst conserving the target ‘eddy density’ is reduced. The former enhancement is observed to deliver a consistent and measurable improvement over the standard formulation, whilst the latter provides efficiency savings of around 1–2 orders of magnitude. The original SEM has spawned a number of derivatives over the past decade, many of which would be expected to benefit from the improvements reported herein (whether they are used as boundary conditions, volume source terms or as part of a dynamic forcing algorithm). We apply the improved formulation to the case of a turbulent channel flow at two Reynolds numbers as well as to the case of an asymmetric planar diffuser, which is set up to exhibit a pressure-induced separation expected to be highly sensitive to upstream flow conditions. It is demonstrated that even apparently small errors in the imposed flow field can persist in such cases, adversely affecting the downstream flow prediction.

Phase-field study of electrochemical reactions at exterior and interior interfaces in Li-ion battery electrode particles

To study the electrochemical reaction on surfaces, phase interfaces, and crack surfaces in lithium ion battery electrode particles, a phase-field model is developed which describes fracture in large strains and anisotropic Cahn–Hilliard-Reaction. Thereby the concentration-dependency of the elastic properties and the anisotropy of diffusivity are also considered. The implementation in 3D is carried out by the isogeometric finite element method in order to treat the higher order terms in a straightforward manner. The electrochemical reaction is modeled through a modified Butler–Volmer equation to account for the influence of the phase change on the reaction on exterior surfaces. The reaction on the crack surfaces is considered through a volume source term weighted by a term related to the fracture order parameter. Based on the model, three characteristic examples are considered to reveal the electrochemical reactions on particle surfaces, phase interfaces, and crack surfaces, as well as their influence on the particle material behavior. The results show that both the anisotropy and the ratio between the timescales of reaction and diffusion can have a significant influence on the phase segregation behavior. In turn, the distribution of the lithium concentration strongly influences the reaction on the surface, especially when the phase interfaces appear on exterior surfaces or crack surfaces. The reaction rate increases considerably at phase interfaces due to the large lithium concentration gradient. Moreover, the simulations demonstrate that the segregation of a Li-rich and a Li-poor phase during delithiation can drive the cracks to propagate. The results indicate that the model can capture the electrochemical reaction on the freshly cracked surfaces.

Modeling and validation of interfacial area transport equation in subcooled boiling flow

The first comprehensive validation of the interfacial area transport equation in subcooled boiling is presented and shown to perform exceptionally when compared with experimental data. The formulation and closure of the bubble layer averaged interfacial area transport equation is reviewed along with the treatment of the two-fluid model in subcooled boiling. Interfacial area concentration source and sink terms in subcooled boiling are presented including the bubble interaction mechanisms (random collision and turbulent impact), as well as phase change terms (wall nucleation and condensation). Additionally, the volume source terms from phase change are described and discussed in terms of their significance to the interfacial area transport equation. The validation of the interfacial area transport equation with a recently proposed wall nucleation source term is shown to have excellent prediction at low and elevated pressure, as well as a wide range of mass flux. With new confidence in the wall nucleation source term, the interfacial area concentration in subcooled boiling can be accurately predicted. Due to its strong dependence in the modeling of active nucleation site density, bubble departure frequency, and departure diameter, the calculation is shown to be very sensitive to wall temperature.

A path-conservative finite volume scheme for compressible multi-phase flows with surface tension

The accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). The basic difficulties are the capturing of the correct interface dynamics between the two fluids as well as the computation of the interface curvature. In this paper, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer–Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. In the CSF method, the surface tension effect is replaced by a volume force, which is usually integrated as a classical volume source term. However, since this source term contains the gradient of a color function that is convected with the flow velocity, we propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path-conservative finite volume schemes put forward by Castro and Parés.For that purpose, we use the new generalized Osher-type Riemann solver (DOT), recently proposed by Dumbser and Toro and compare it with a path-conservative Roe and Rusanov scheme. Via numerical evidence we can show that if the curvature computation is exact, our new scheme is well-balanced for a steady circular bubble in equilibrium according to the Young–Laplace law. This means that the pressure jump term across the interface in the momentum equation is exactly balanced with the surface tension force.We apply our scheme to several one- and two-dimensional test problems, including 1D Riemann problems, an oscillating droplet, as well as a deforming droplet initially attached to a wall and which is subject to gravity and surface tension forces. Finally, we also check the rising of a gas bubble due to buoyancy forces. We compare our numerical results with those published previously in the literature.

On the Extrapolation of Acoustic Waves from Flow Simulations with Vortical out Flow

Feri Farassat was one of the pioneers of the use of the Ffowcs Williams & Hawkings formulation of Lighthill's acoustic analogy as a way to extrapolate radiated waves from simulations of unsteady flows. Current computational limits mean that volume source terms are often neglected, causing inaccurate acoustical predictions when entropy fluctuations or vorticity pass across the extrapolation surface. The derivation of the Ffowcs Williams–Hawkings equation is modified to allow the equivalent surface sources to be distributed over a transition layer of finite thickness rather than being confined to a single layer, in order to reduce the effect of vorticity exiting the computational domain.

Implementation of a roughness element to trip transition in large-eddy simulation

In aerodynamics, the laminar or turbulent regime of a boundary layer has a strong influence on friction or heat transfer. In practical applications, it is sometimes necessary to trip the transition to turbulent, and a common way is by use of a roughness element (e.g. a step) on the wall. The present paper is concerned with the numerical implementation of such a trip in large-eddy simulations. The study is carried out on a flat-plate boundary layer configuration, with Reynolds number Rex=1.3×106. First, this work brings the opportunity to introduce a practical methodology to assess convergence in large-eddy simulations. Second, concerning the trip implementation, a volume source term is proposed and is shown to yield a smoother and faster transition than a grid step. Moreover, it is easier to implement and more adaptable. Finally, two subgrid-scale models are tested: the WALE model of Nicoud and Ducros (Flow Turbul. Combust., vol. 62, 1999) and the shear-improved Smagorinsky model of Leveque et al. (J. Fluid Mech., vol. 570, 2007). Both models allow transition, but the former appears to yield a faster transition and a better prediction of friction in the turbulent regime.

Homogenization of heat diffusion in a cracked medium

We develop in this Note a homogenization method to tackle the problem of a diffusion process through a cracked medium. We assume that the cracks are orthogonal to the surface of the material, where an incoming heat flux is applied. The cracks are supposed to be of depth 1, of small width, and periodically arranged. We show that the cracked surface of the domain induces a volume source term in the homogenized equation.

The acoustic analogy in an annular duct with swirling mean flow

AbstractThis paper is concerned with modelling the effects of swirling flow on turbomachinery noise. We develop an acoustic analogy to predict sound generation in a swirling and sheared base flow in an annular duct, including the presence of moving solid surfaces to account for blade rows. In so doing we have extended a number of classical earlier results, including Ffowcs Williams & Hawkings’ equation in a medium at rest with moving surfaces, and Lilley’s equation for a sheared but non-swirling jet. By rearranging the Navier–Stokes equations we find a single equation, in the form of a sixth-order differential operator acting on the fluctuating pressure field on the left-hand side and a series of volume and surface source terms on the right-hand side; the form of these source terms depends strongly on the presence of swirl and radial shear. The integral form of this equation is then derived, using the Green’s function tailored to the base flow in the (rigid) duct. As is often the case in duct acoustics, it is then convenient to move into temporal, axial and azimuthal Fourier space, where the Green’s function is computed numerically. This formulation can then be applied to a number of turbomachinery noise sources. For definiteness here we consider the noise produced downstream when a steady distortion flow is incident on the fan from upstream, and compare our results with those obtained using a simplistic but commonly used Doppler correction method. We show that in all but the simplest case the full inclusion of swirl within an acoustic analogy, as described in this paper, is required.

Extensions of Lighthill's acoustic analogy with application to computational aeroacoustics

Lighthill's aeroacoustic analogy is formulated for bounded domains in a general way that allows pressure-based alternatives to the fluid density as wave variable. The advantage relative to the standard version (Ffowcs Williams & Hawkings 1969 Phil. Trans. R. Soc. A 264 , 321–342) is that the equivalent surface source terms needed for boundary value problems do not involve the local density. Difficulties encountered in computational aeroacoustics with standard wave extrapolation procedures, due to advection of density inhomogeneities across the control surface, are thereby avoided. Likewise, in initial-value problems, the equivalent volume source terms that represent initial conditions do not involve the density either. The paper ends with an extension to parallel shear flows, in which a modified aeroacoustic analogy due to Goldstein (Goldstein 2001 J. Fluid Mech . 443 , 231–236) is formulated for bounded domains using a similar windowed-variable approach. The results provide a basis for acoustic wave extrapolation from jets and boundary layers, where the control surface cuts through a sheared mean flow.

Mass– and volume–specific views on thermodynamics for open systems

A general framework for the thermodynamics of open systems is presented. The theory is fundamentally based on the generalized balance of mass, which is enhanced by additional surface–flux and volume–source terms. The presentation highlights the influence of the enhanced mass balance on the remaining balance equations. To clarify the impact of the variable reference density, we introduce the notions of volume–specific and mass–specific formats. Particular attention is drawn to the fact that the mass–specific balance equations are free from explicit open–system contributions. They resemble the classical balance equations and can thus be evaluated in complete analogy to the closed–system case. Restrictions for the constitutive equations follow from the second law of thermodynamics, as illustrated for the particular model problem of thermo–hyperelasticity.

On spatial and material settings of thermo-hyperelastodynamics for open systems

The present treatise aims at deriving a general framework for the thermodynamics of open systems typically encountered in chemo- or biomechanical applications. Due to the fact that an open system is allowed to constantly gain or lose mass, the classical conservation law of mass has to be recast into a balance equation balancing the rate of change of the current mass with a possible in- or outflux of matter and an additional volume source term. The influence of the generalized mass balance on the balance of momentum, kinetic energy, total energy and entropy is highlighted. Thereby, special focus is dedicated to the strict distinction between a volume specific and a mass specific format of the balance equations which is of no particular relevance in classical thermodynamics of closed systems. The change in density furnishes a typical example of a local rearrangement of material inhomogeneities which can be characterized most elegantly in the material setting. The set of balance equations for open systems will thus be derived for both, the spatial and the material motion problem. Thereby, we focus on the one hand on highlighting the remarkable duality between both approaches. On the other hand, special emphasis is placed on deriving appropriate relations between the spatial and the material motion quantities.

Global estimates and asymptotics for electro reaction diffusion systems in heterostructures

We treat a wide class of electro reaction diffusion systems with nonsmooth data in two dimensional domains. Forced by applications in semiconductor technology a nonlinear and nonlocal Poisson equation is involved. We state global existence, uniqueness and asymptotic properties of solutions to the evolution problem. Essential tools in our investigations are energetic estimates, Moser iteration, regularization techniques and results for electro diffusion systems with weakly nonlinear volume and boundary source terms. Especially, we discuss the connection between the existence of global lower bounds for the chemical potentials and the property that the energy functional decays exponentially to its equilibrium value as time tends to infinity

Stationary Boltzmann's equation with Maxwell's boundary conditions in a bounded domain

AbstractThe paper deals with the stationary Boltzmann equation in a bounded convex domain Ω. The boundary ∂Ω is assumed to be a piecewise algebraic variety of the C2‐class that fulfils Liapunov's conditions. On the boundary we impose the so‐called Maxwell boundary conditions, that is a convex combination of specular and diffusive reflections. The non‐linear Boltzmann equation is considered with additional volume and boundary source terms and it has been proved that for sufficiently small sources the problem possesses a unique solution in a properly chosen subspace of C(Ω × ℝ3). The proof is a refined version of the proof delivered by Guiraud for purely diffusive reflection.

Linear growth of a liquid droplet divided from its vapour by a “soap bubble”-like fluid interface

The theory proposed in [1,3] is particularized to describe the spherically symmetric growth, in the neighbourhood of an equilibrium state far from the critical temperature, of a liquid incompressible droplet surrounded by its vapour when the interface between the phases behaves as a fluid membrane which resembles a soap bubble. A free moving boundary problem for an integro-differential equation of parabolic type is deduced in which the second-order time derivative of the radius R(t) of the droplet appears, together with a volume source term which depends on the history of R.