Convective Transport Terms Research Articles

Robust discontinuous Galerkin-based scheme for the fully-coupled nonlinear thermo-hydro-mechanical problem

Abstract We present and analyze a discontinuous Galerkin method for the numerical modeling of the nonlinear fully-coupled thermo-hydro-mechanic problem. We propose an arbitrary-order weighted symmetric interior penalty scheme that supports general polytopal grids and is robust with respect to strong heterogeneities in the model coefficients. We focus on the treatment of the nonlinear convective transport term in the energy conservation equation and we propose suitable stabilization techniques that make the scheme robust for advection-dominated regimes. The stability analysis of the problem and the convergence of the fixed-point linearization strategy are addressed theoretically under mild requirements on the problem data. A complete set of numerical simulations is presented in order to assess the convergence and robustness properties of the proposed method.

A chemistry load balancing model for OpenFOAM

Efficient simulation tools are crucial for studying complex systems such as reacting flows where computational costs of computing chemical reaction rates can vastly exceed the costs for the integration of the convective and diffusive transport terms. Load imbalance in parallel computing poses a significant challenge for massively parallel reacting flow simulations. In response, a novel load balancing library has been developed to enhance OpenFOAM's solver performance in parallel environments. This library seamlessly integrates with OpenFOAM, offering ease of use and applicability to any OpenFOAM reacting solver incorporating finite-rate chemistry. In addition, it supports the standard and the dynamic adaptive chemistry model (TDAC) of OpenFOAM.The newly developed load-balanced standard and TDAC models address significant load imbalances by exchanging information between processes via MPI calls and tracking ODE solution times on a cell level. The TDAC model introduces dual tables on each core and enables immediate addition of computed solutions, enhancing computational efficiency. Validation on various test cases, including simulations on the HLRS Hawk supercomputer with up to 8000 cores, confirms identical results compared to the original unbalanced models, with notable speed-up factors of up to 6 for the standard and 5 for the TDAC model. Despite non-linear scaling at lower cell count per processor, load-balanced models consistently outperform unbalanced counterparts, making them the preferred choice for reacting flow simulations in OpenFOAM.

Upwind reproducing kernel collocation method for convection-dominated problems

Because of the best approximation property, traditional Bubnov–Galerkin numerical methods have proven immensely successful in modeling self-adjoint problems, such as heat conduction, elasticity, and so on. However, a numerical instability arises in these (and central finite difference) methods for problems with strong convection. In this class of problems, the convective transport term can lead to large spurious oscillations but can be handled by the class of Petrov–Galerkin methods. In particular, the upwind-type schemes and their variational and subgrid descendants have been substantially developed over the years for an effective weak-form Galerkin solution that precludes these instabilities. Nevertheless, the scale of development of upwind methods for strong-form collocation is substantially smaller, where numerical oscillations are also observed when they are straightforwardly applied to convection-dominated problems without special treatment. To this end, this paper presents a new upwind collocation method. First, the connection between the upwind finite difference scheme and the gradient smoothing technique in meshfree methods is established. It is then shown that selecting the collocation points as meshfree nodal points is not optimal; selecting the collocation points according to the flow direction and Péclet number is then studied. The upwind effect is achieved without introducing artificial parameters and is trivial to generalize for multi-dimensional cases. Cross-wind diffusion is also not observed in the solution. An error analysis is presented, and the effectiveness of the proposed methodology is well demonstrated by the steady and unsteady numerical examples.

A coupling of Galerkin and mixed finite element methods for the quasi-static thermo-poroelasticity with nonlinear convective transport

In this paper, we propose a combined Galerkin and mixed finite element methods to analyze the fully coupled nonlinear thermo-poroelastic model problems. We design the Galerkin finite element method for the temperature, the mixed finite element method for the pressure, the Galerkin finite element method for the elastic displacement. We linearize the nonlinear convective transport term in the energy balance equation and establish the fully discrete finite element schemes. The stability and convergence of the coupled method are obtained. In particular, all previous works have required certain time step restrictions, but we unconditionally prove the optimal error estimates without certain extra restrictions on both time step and spatial meshes. Finally, some numerical examples are presented to illustrate the accuracy of the method and confirm the unconditional stability of the method.

Discontinuous Galerkin Approximation of the Fully Coupled Thermo-poroelastic Problem

We present and analyze a discontinuous Galerkin method for the numerical modeling of the nonlinear fully coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem. To handle the nonlinear convective transport term in the energy conservation equation we adopt a fixed-point linearization strategy. We perform a robust stability analysis for the linearized semidiscrete problem under mild requirements on the problem’s physical parameters. A priori hp-version error estimates in suitable energy norms are also derived. A complete set of numerical simulations is presented in order to validate the theoretical analysis, to inspect numerically the robustness properties, and to test the capability of the proposed method in a practical scenario inspired by a geothermal problem.

Thermodynamically admissible 13-moment equations

Grad's 13-moment equations describe transport in mildly rarefied gases well, but are not properly embedded into nonequilibrium thermodynamics since they are not accompanied by a formulation of the second law. In this work, the Grad-13 equations are embedded into the framework of GENERIC (general equation for the nonequilibrium reversible–irreversible coupling), which demands additional contributions in the equations to guarantee thermodynamic structure. As GENERIC building blocks, we use a Poisson matrix for the basic convection behavior and antisymmetric friction matrices to correct for additional convective transport terms. The ensuing GENERIC-13 equations completely match the Grad-13 equations up to second-order terms in the Knudsen number and fulfill all thermodynamic requirements.

Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport

This paper concerns monolithic and splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE’s, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare six approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from the mechanics, suitable for varying coupling strength between the three physical processes. The approaches have in common a simultaneous application of the so-called L-scheme, which works both to stabilize iterative splitting as well as to linearize nonlinear problems, and can be seen as a generalization of the Undrained and Fixed-Stress Split algorithms. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes.

Mathematical Modeling of Sucrose Hydrolysis with Product and Substrate Inhibition

In this work, the enzymatic hydrolysis reaction of sucrose through invertase under unsteady-state conditions has been investigated. The aim is to evaluate the inhibition phenomena influence on the reaction rate and, then, on the concentration and temperature profiles by simulating the process in a tubular reactor, varying the enzyme concentration and the reactant mixture velocity. The transport phenomena considered during the enzymatic hydrolysis process have been described by means of unsteady-state momentum, mass and energy balance equations, taking into account molecular and convective transport and generation terms. Interpretation and discussion of the results obtained by FEM resolution of PDEs involved allow to understand the relevance of the operating parameters.

Large Eddy Simulation of a spray jet flame using Doubly Conditional Moment Closure

A spray jet flame is modelled using Large Eddy Simulation (LES) with Doubly Conditional Moment Closure (DCMC). Since turbulent spray flames may include multiple combustion modes, the DCMC model uses both mixture fraction and reaction progress variable as conditioning variables. Conditional spray terms were included in the DCMC model to consider the coupling between evaporation and the flame structure. In the case of spatial homogeneity and in the limit of negligible mixture fraction scalar dissipation rate (SDR), the DCMC equation is shown to reproduce the flame structure of freely propagating laminar flames. For the spray jet flame, a good agreement between the simulation results and the experiments is achieved, in terms of the spray statistics, as well as the instantaneous and mean flame shape. The simulation shows important differences in the flame structure between the turbulent inner and the quasi-laminar outer flame branch. The doubly-conditional parametrisation appears to be advantageous for resolving small scale effects related to droplet evaporation. Analysis of the DCMC equation suggests that the behaviour of the flame at its anchoring point is strongly influenced by non-premixed burning modes. The solution appears to be weakly affected by terms of convective transport in the DCMC equation, but significant spatial variations and temporal fluctuations of the conditional reaction rate, around 10% of the time-based mean, persist.

Well-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport

This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence proof for this algorithm is then inferred by a contraction of successive difference functions of the iterates using suitable norms.

Fully coupled numerical model of actin treadmilling in the lamellipodium of the cell.

Cells rely on an interplay of subcellular elements for motility and migration. Certain regions of motile cells, such as the lamellipodium, are made of a complex mixture of actin monomers and filaments, which polymerize at the front of the cell, close to the cell membrane, and depolymerize at the rear. The dynamic actin turnover induces the so-called intracellular retrograde flow, and it is a fundamental process for cell motility. Apart from some comprehensive mathematical models, the computational modelling of actin treadmilling has been based on simpler biophysical models. Here, we adopt a highly detailed theoretical model of the actin treadmilling process and develop a coupled unsteady finite element formulation. We clearly describe the structure and implementation of the coupled problem within the finite element method. Our numerical results show an excellent correlation with experimental results from literature and with previous models. We include time dependent effects and convective transport terms, which expose puzzling dynamics in the retrograde flow. We propose several biological scenarios to analyze the behavior of the actin treadmilling along space and time. We observed response times of the main density variables in the order of seconds. Compared with previous analytical solutions, which make assumptions related to convective transport, transient dynamics, and actin fluxes, the generic solution can have significant influence on the retrograde flow. All together, our results unveil a promising applicability of classical finite element methods to derive an in silico testing platform for the actin treadmilling processes in motile cells, which could allow for an extension to other biophysical effects.

Numerical study on rarefied unsteady jet flow expanding into vacuum using the Gas-Kinetic Unified Algorithm

The unsteady process of rarefied jet flows expanding into a vacuum is numerically solved by the Gas-Kinetic Unified Algorithm (GKUA) based on the Boltzmann model equations. The discrete velocity ordinate method (DVOM) is adopted to discretize the velocity space of the molecular velocity distribution function, while the corresponding numerical integral technique is developed to evaluate macroscopic flow variables, including number density, flow velocity, temperature and so on. The time-explicit finite difference scheme is constructed to capture the unsteady evolution of the discrete velocity distribution functions at each DVO point by using the unsteady time-splitting method. The multi-block docking grid technique is designed for different regions of flow field in physical space with self-adaptive adjustment. Then, the supersonic planar jet flows with the wide range of Knudsen numbers, from 100 to 0.1, are solved numerically and discussed, including startup to steady state and shutting down. The GKUA results of free-molecule jet flows are analyzed and compared with the Maxwellian analytical solutions for collisionless plume flows, in which good agreement shows the validity and accuracy of the present numerical method. When taking the collision effect into account, the unsteady jet flows with different Knudsen numbers are computed by the present GKUA method. It is shown that the collision term of the Boltzmann model equation plays an important role in this rarefied gas diffusion into the back flow when Kn ≤ 10. However, the colliding relaxation term have a little influence on the kernel region in the startup process for Kn ≥ 0.1. In general, the convective transport term of the Boltzmann model equation dominates the kernel region of the jet flow during the unsteady process of gas expanding into a vacuum. The numerical experience indicates that the present GKUA can provide a vital tool for solving unsteady flow problems covering various flow regimes by directly tracing the time evolution of the Boltzmann-type velocity distribution function.

Self-adjoint integral operator for bounded nonlocal transport.

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

Nodally integrated implicit gradient reproducing kernel particle method for convection dominated problems

Convective transport terms in Eulerian conservation laws lead to numerical instability in the solution of Bubnov–Galerkin methods for these non-self-adjoint PDEs. Stabilized Petrov–Galerkin methods overcome this difficulty, however gradient terms are required to construct the test functions, which are typically expensive for meshfree methods. In this work, the implicit gradient reproducing kernel particle method is introduced which avoids explicit differentiation of test functions. Stabilization is accomplished by including gradient terms in the reproducing condition of the reproducing kernel approximation. The proposed method is computationally efficient and simplifies stabilization procedures. It is also shown that the implicit gradient resembles the diffuse derivative originally introduced in the diffuse element method in Nayroles et al. (1992), and maintains the desirable properties of the full derivative. Since careful attention must be paid to efficiency of domain integration in meshfree methods, nodal integration is examined for this class of problems, and a nodal integration method with enhanced accuracy and stability is introduced. Numerical examples are provided to show the effectiveness of the proposed method for both steady and transient problems.

A Lagrangian–Eulerian formulation for reeling analysis of history-dependent multilayered beams

This paper presents a novel Lagrangian–Eulerian finite element formulation for reeling analysis of multilayered beams with gross interlayer slippage. In contrast to the conventional Lagrangian approach, the mesh becomes practically fixed in space, which yields significant benefits for the performance of the contact algorithms and the overall computational efficiency. The needed Lagrangian–Eulerian kinematic relations are derived, special attention is given to the convective transport term for the constitutive variables and an implicit update scheme for the elasto-plastic bending model is formulated. The proposed formulation is shown to predict responses with the same accuracy as offered by the conventional Lagrangian formulation.

Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

Investigation of membrane degradation in polymer electrolyte fuel cells using local gas permeation analysis

The effect of chemical and mechanical membrane degradation on the gas separation is investigated in the early stage. The aim is to identify the trigger processes which lead to a nonuniform degradation. Chemical and mechanical degradation are investigated in accelerated stress tests at OCV, relative humidity cycling and a combination of both. Using a tracer gas concept, gas permeation in perfluorosulfonic acid membranes (Nafion) is analyzed locally and online in terms of diffusive and convective gas transport. Local phenomena, such as humidity fluctuations, cracks in the micro porous layer and carbon fibers, are found to initiate the degradation of the gas separation. Pinhole formation, induced by mechanical degradation processes, is crucial for the onset of a synergetic effect. Combined chemical and mechanical degradation accelerates pinhole growth, which is the major source of the increasing gas permeation in Nafion membranes with stabilized end-groups. The spatial nonuniformities of the relative humidity and humidity fluctuations are the main source of inhomogeneous membrane degradation.

Parametric dependences of momentum pinch and Prandtl number in JET

Several parametric scans have been performed to study momentum transport on JET. A neutral beam injection modulation technique has been applied to separate the diffusive and convective momentum transport terms. The magnitude of the inward momentum pinch depends strongly on the inverse density gradient length, with an experimental scaling for the pinch number being -Rvpinch/χϕ = 1.2R/Ln + 1.4. There is no dependence of the pinch number on collisionality, whereas the pinch seems to depend weakly on q-profile, the pinch number decreasing with increasing q. The Prandtl number was not found to depend either on R/Ln, collisionality or on q. The gyro-kinetic simulations show qualitatively similar dependence of the pinch number on R/Ln, but the dependence is weaker in the simulations. Gyro-kinetic simulations do not find any clear parametric dependence in the Prandtl number, in agreement with experiments, but the experimental values are larger than the simulated ones, in particular in L-mode plasmas. The extrapolation of these results to ITER illustrates that at large enough R/Ln > 2 the pinch number becomes large enough (>3–4) to make the rotation profile peaked, provided that the edge rotation is non-zero. And this rotation peaking can be achieved with small or even with no core torque source. The absolute value of the core rotation is still very challenging to predict partly due to the lack of the present knowledge of the rotation at the plasma edge, partly due to insufficient understanding of 3D effects like braking and partly due to the uncertainties in the extrapolation of the present momentum transport results to a larger device.

Electrodeposition in highly viscous media: Experiments and simulations

Electrolyte viscosity plays an important role in ion transport. Here we study the effects of high viscosity variations in thin-layer electrochemical deposition (ECD) under constant-current conditions through experimental measurements and theoretical modelling. The viscosity was varied through glycerol and polymer additions and the tracking of convective fronts was performed through the use of optical and particle image velocimetry techniques with micron sized particles. The theoretical model, written in terms of dimensionless quantities, describes diffusive, migratory and convective ion transport in a fluid under constant-current conditions. Experiments reveal that as viscosity increases, convection decreases, while concentration gradients increase. These effects are more pronounced when the current increases. Theory and simulations predict that as viscosity increases, the Poisson and Reynolds numbers decrease whereas the Peclet and electric Grashof numbers increase. Therefore, electroconvection becomes more relevant.

Cosmic‐Ray Diffusion Approximation with Weak Adiabatic Focusing

Large-scale spatial variations of the guide magnetic field of interplanetary and interstellar plasmas give rise to the adiabatic focusing term in the Fokker-Planck transport equation of cosmic rays. The consequences of the adiabatic focusing term for the diffusion approximation to cosmic-ray transport are investigated in the weak focusing limit, where the focusing length L is much larger than the parallel scattering length. Whereas the cosmic-ray anisotropy is unaffected, three new transport terms arise in the diffusion-convection transport equation that represent convective transport terms parallel to the guide field, perpendicular to the guide field, and in momentum space. The convective terms perpendicular to the guide field only occur in nonaxisymmetric electromagnetic turbulence. The new convection term in momentum space represents a continuous momentum loss term or a first-order Fermi-type acceleration term depending on the product a11L of the focusing length L and the rate of adiabatic deceleration a11, which depends sensitively on the cross helicity and magnetic helicity of the magnetic field turbulence.