Control Volume Discretization Research Articles

Symmetry Preservation by a Compatible Staggered Lagrangian Scheme Using the Control-Volume Discretization Method in $r–z$ Coordinate

Symmetry Preservation by a Compatible Staggered Lagrangian Scheme Using the Control-Volume Discretization Method in $r–z$ Coordinate

Modelling the complex evaporated gas flow and its impact on particle spattering during laser powder bed fusion

The additive manufacturing (AM) of metals is becoming an increasingly important production process with the potential to replace traditional techniques such as casting. Laser Powder Bed Fusion (LPBF) is used in many applications to print metal parts from powder. The metal powder is heated locally with sufficient laser radiation that the liquid melt easily reaches its boiling temperature, which leads to a metallic vapour jet that can entrain both powder bed particles and molten droplets. The small size of laser-matter interaction site makes a detailed experimental analysis of the process challenging. Synchrotron X-ray imaging experiments are one of the few methods which can capture the dynamic melting and solidification processes. Comparing such experiments with computer simulations of the process is an important approach in order to better understand the manufacturing process and to analyse the influence of process parameters on the evaporated gas jet and the subsequent impact on particle ejection, leading to potentially reduced AM component quality. The melting and solidification of the metal powder is simulated using an Eulerian multiphase approach based on a control volume discretization of powder bed and substrate and a volume of liquid separation from melt and gas phase. The gas phase modelled as an ideal gas reaches velocities up to 100 m/s. Lagrangian particle tracking in the simulation demonstrates that the velocity fields calculated by the Eulerian multi-phase approach in combination with a standard drag-force model lead to particle accelerations in good agreement with those measured experimentally. In order to avoid numerical laborious Lagrangian calculations, a direct method to compare an Eulerian multiphase simulation with synchrotron X-ray experiments was introduced and validated . This approach is used to analyse the influence of process parameters including laser power and laser speed on the maximal acceleration of particles from the melt pool area. While the particle acceleration increases linearly with line energy in the conduction mode, a linear decrease of the acceleration with increasing line energy can be found in the transition mode before the acceleration increases again with line energy in the keyhole mode.

Robust control volume finite element methods for numerical wave tanks using extreme adaptive anisotropic meshes

SummaryMultiphase inertia‐dominated flow simulations, and free surface flow models in particular, continue to this day to present many challenges in terms of accuracy and computational cost to industry and research communities. Numerical wave tanks and their use for studying wave‐structure interactions are a good example. Finite element method (FEM) with anisotropic meshes combined with dynamic mesh algorithms has already shown the potential to significantly reduce the number of elements and simulation time with no accuracy loss. However, mesh anisotropy can lead to mesh quality‐related instabilities. This article presents a very robust FEM approach based on a control volume discretization of the pressure field for inertia dominated flows, which can overcome the typically encountered mesh quality limitations associated with extremely anisotropic elements. Highly compressive methods for the water‐air interface are used here. The combination of these methods is validated with multiphase free surface flow benchmark cases, showing very good agreement with experiments even for extremely anisotropic meshes, reducing by up to two orders of magnitude the required number of elements to obtain accurate solutions.

A Hybrid Backward Euler Control Volume Method to Solve the Concentration-Dependent Solid-State Diffusion Problem in Battery Modeling

Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.

Implementation Strategies for Accurate and Efficient Control Volume-Based Two-Phase Hydrothermal Flow Solutions

Numerical models of magmatic hydrothermal systems have become powerful tools for linking surface and seafloor observations to chemical and fluid-dynamic processes at depth. This task requires resolving multi-phase flow over large distances of several kilometers, a wide range of pressure (p) and temperature (T) conditions, and over timescales of several thousands of years. The key numerical challenge is that realistic simulations have to consider the high nonlinearity and strong coupling of the governing conservation equations for mass and energy, while also being numerically efficient so that the required spatial and temporal scales can be resolved. Here we outline possible solutions to this problem by evaluating different implementation strategies and presenting a numerical scheme for fully coupled accurate and efficient flow solutions. The general scheme, based on the Newton–Raphson (NR) method, is presented for the simplified case of 2-D pure water convection and uses a control volume discretization on unstructured meshes. We find that the presented techniques significantly reduce the computational effort with respect to sequential/decoupled schemes. Key to this is a theta-time-differencing method for better accuracy, stability and convergence behavior of the NR-iterations, as well as improvements regarding upwinding. These features make the presented methods useful for coupled simulations of magmatic hydrothermal systems and a potential basis for future 3-D multi-phase codes.

Coupling LaGrit unstructured mesh generation and model setup with TOUGH2 flow and transport: A case study

There are many applications in science and engineering modeling where an accurate representation of a complex model geometry in the form of a mesh is important. In applications of flow and transport in subsurface porous media, this is manifest in models that must capture complex geologic stratigraphy, structure (faults, folds, erosion, deposition) and infrastructure (tunnels, boreholes, excavations). Model setup, defined as the activities of geometry definition, mesh generation (creation, optimization, modification, refine, de-refine, smooth), assigning material properties, initial conditions and boundary conditions requires specialized software tools to automate and streamline the process. In addition, some model setup tools will provide more utility if they are designed to interface with and meet the needs of a particular flow and transport software suite.A control volume discretization that uses a two point flux approximation is for example most accurate when the underlying control volumes are 2D or 3D Voronoi tessellations. In this paper we will present the coupling of LaGriT, a mesh generation and model setup software suite and TOUGH2 (Pruess et al., 1999) to model subsurface flow problems and we show an example of how LaGriT can be used as a model setup tool for the generation of a Voronoi mesh for the simulation program TOUGH2. To generate the MESH file for TOUGH2 from the LaGriT output a standalone module Lagrit2Tough2 was developed, which is presented here and will be included in a future release of LaGriT.In this paper an alternative method to generate a Voronoi mesh for TOUGH2 with LaGriT is presented and thanks to the modular and command based structure of LaGriT this method is well suited to generating a mesh for complex models.

Dynamic modeling of a sensible thermal energy storage tank with an immersed coil heat exchanger under three operation modes

In this paper we consider control-oriented modeling of a sensible thermal energy storage (TES) tank with a helical immersed heat exchanger (IHX) coil. A key focus of the modeling approach is to minimize the number of dynamic states required to adequately describe the system dynamics. The resulting model is well-suited for model-based control design, real-time simulation, and hardware-in-the-loop testing aimed at intelligent operation of TES systems. We use a discretized approach to model the temperature dynamics of the water within the storage tank. We use a quasi-steady approach to model the IHX coil dynamics, thereby limiting computational complexity. In simulation, the model runs up to 1200× faster than real-time. A simulated case study of model-based feedback control demonstrates the utility of the modeling approach. The model contains four tuning parameters that are empirically determined using experimental data collected from a commercially available domestic hot water storage tank. The model is then validated, both temporally and spatially, against data collected during the simultaneous charge/discharge mode. Finally, we quantify the trade-off between model fidelity and increased control volume discretization, showing that a 60 node model yields a RMSE value under 4.5%.

Experimental and Computational Investigation of Natural Convection in a Rectangular Duct with Internal Heat Source

To design passive air-cooled heat exchangers, an experimental study on natural convection in a square or rectangular duct was felt necessary, as literature on this subject is not available. In the present study, experiments were carried out using a square duct of size 500 × 500 × 800 mm height fabricated with 3.0 mm thick aluminium sheet and a brass cylinder of diameter 25 mm and height 200 mm was used as internal heat source. The heated duct was cooled by natural circulation of air. A 3D Computational Fluid Dynamics (CFD) model for natural convection with variable properties of air and no-slip condition was developed. Laminar simulations of cooling of a vertical cylinder in a rectangular duct was obtained by solving the governing equations using Fluent 6.2.16 along with Gambit v4.6.0 as pre-processor for grid generation. Grid generation resulted in 23,800 cells with 24,185 nodes. Segregated solver solution methodology has been used for simulation with implicit linearization of the Equations (10–14). Finite volume or control volume discretization technique has been used with iterative solution techniques. Residual of the order of for momentum equation, laminar model and for energy equation was used for convergence and stability check. The solver used was based on SIMPLE algorithm. Variation of density and viscosity of air with temperature were accounted for in the simulation. The model was validated by using experimental data. Average heat transfer coefficients were estimated. It was found that the theoretically calculated average heat transfer coefficients were in good agreement with the modelled values.

Numerical Study of Reduced Frequency Effect on Longitudinal Stability Derivatives of Airfoil under Pitching and Plunging Oscillations

In this study, incompressible, unsteady and turbulent flow over an airfoil with pitching and plunging oscillations is numerically studied in order to investigate the effect of reduced frequency on stability derivatives of oscillating airfoil. Linear k – e model called Launder-Sharma and Rhie and Chow model are used for turbulence modeling and overcoming pressure checkerboard problem. This means that a co-located approach is proposed in this paper to study a moving grid problem, and the results demonstrate the high accuracy of the method. Control volume and Crank-Nicholson discretization method are also used for the numerical solution. It is shown that the longitudinal stability derivatives of plunging and pitching motions trend change intensively beyond the stall angle of attack while pitching rate has a completely opposite behavior. The results also show that increasing reduced frequency leads to stability reduction in plunging oscillation but it does not have significant effect on pitching oscillation case in pre-stall, stall and post-stall conditions. Grid convergence is examined to assess the accuracy of the numerical method that shows the high accuracy of it and this is a prominent achievement of the present study. The results of the proposed method in forces and moment show a good agreement with the experimental data.

Unstructured computational aerodynamics on many integrated core architecture

Abstract Shared memory parallelization of the flux kernel of PETSc-FUN3D, an unstructured tetrahedral mesh Euler flow code previously studied for distributed memory and multi-core shared memory, is evaluated on up to 61 cores per node and up to 4 threads per core. We explore several thread-level optimizations to improve flux kernel performance on the state-of-the-art many integrated core (MIC) Intel processor Xeon Phi “Knights Corner,” with a focus on strong thread scaling. While the linear algebraic kernel is bottlenecked by memory bandwidth for even modest numbers of cores sharing a common memory, the flux kernel, which arises in the control volume discretization of the conservation law residuals and in the formation of the preconditioner for the Jacobian by finite-differencing the conservation law residuals, is compute-intensive and is known to exploit effectively contemporary multi-core hardware. We extend study of the performance of the flux kernel to the Xeon Phi in three thread affinity modes, namely scatter, compact, and balanced, in both offload and native mode, with and without various code optimizations to improve alignment and reduce cache coherency penalties. Relative to baseline “out-of-the-box” optimized compilation, code restructuring optimizations provide about 3.8x speedup using the offload mode and about 5x speedup using the native mode. Even with these gains for the flux kernel, with respect to execution time the MIC simply achieves par with optimized compilation on a contemporary multi-core Intel CPU, the 16-core Sandy Bridge E5 2670. Nevertheless, the optimizations employed to reduce the data motion and cache coherency protocol penalties of the MIC are expected to be of value for CFD and many other unstructured applications as many-core architecture evolves. We explore large-scale distributed-shared memory performance on the Cray XC40 supercomputer, to demonstrate that optimizations employed on Phi hybridize to this context, where each of thousands of nodes are comprised of two sockets of Intel Xeon Haswell CPUs with 32 cores per node.

Formulation and validation of a two-dimensional steady-state model of desiccant wheels

Desiccant wheels are rotary desiccant dehumidifiers used in air-conditioning and drying applications. The modeling of simultaneous heat and mass transfer in these components is crucial for estimating their performances, as well as for simulating and optimizing their implementation in complete systems. A steady-state two-dimensional model is formulated and implemented, aiming to obtain good accuracy and short computational times with the purpose of inclusion in complete system models. The model includes mass and energy balances and correlations for heat and mass transfer based on empirical relations from the scientific literature. Convective heat and mass transfer coefficients are computed locally accounting for the entrance length effects. Mass diffusion inside the desiccant material is neglected. Comparison with experimental data from the literature shows that the model reproduces the physical behavior of desiccant wheels satisfactorily, as the deviation between the computed results and available data is always within 12%. The simulation time is as low as 3 s for a model with 200 control volumes. It is found that for the applied case, the model provides accurate results for the overall flow using an equiangular control volume discretization with 200 control volumes and no axial discretization. More computationally expensive configurations with axial discretization give more accurate results and information on local flow and desiccant conditions inside the wheel.

Inexact linear solvers for control volume discretizations in porous media

We discuss the construction of multi-level inexact linear solvers for control volume discretizations for porous media. The methodology forms a contrast to standard iterative solvers by utilizing an algebraic hierarchy of approximations which preserve the conservative structure of the underlying control volume. Our main result is the generalization of multiscale control volume methods as multi-level inexact linear solvers for conservative discretizations through the design of a particular class of preconditioners. This construction thereby bridges the gap between multiscale approximation and linear solvers. The resulting approximation sequence is referred to as inexact solvers. We seek a conservative solution, in the sense of control-volume discretizations, within a prescribed accuracy. To this end, we give an abstract guaranteed a posteriori error bound relating the accuracy of the linear solver to the underlying discretization. These error bounds are explicitly computable for the grids considered herein. The afore-mentioned hierarchy of conservative approximations can also be considered in the context of multi-level upscaling, and this perspective is highlighted in the text as appropriate. The new construction is supported by numerical examples highlighting the performance of the inexact linear solver realized in both a multi- and two-level context for two- and three-dimensional heterogeneous problems defined on structured and unstructured grids. The numerical examples assess the performance of the approach both as an inexact solver, as well in comparison to standard algebraic multigrid methods.

The effects of radiative heat transfer during the melting process of a high temperature phase change material confined in a spherical shell

The influence of radiation heat transfer during the phase change process of a storage material has been numerically analyzed in this study. Emphasis has been placed on the thermal characterization of a single constituent storage module rather than an entire storage system, in order to precisely capture the energy exchange contributions of all the fundamental heat transfer mechanisms during the melting of a phase change material (PCM) with tailored optical properties. The equations describing the conservation of mass, momentum and energy have been solved by using the control volume discretization approach, while the radiative transfer equation (RTE) was solved by the discrete ordinate method (DOM). The enthalpy–porosity method was used to track the PCM liquid/solid interface during the process. A parametric analysis has been performed in order to ascertain the effects of the optical thickness and the Planck, Grashof and Stefan numbers on the melting rate, as well as the total and radiative heat transfer rates at the inner surface of the shell. The results show that the presence of thermal radiation enhances the melting process. Correlations for the melt fraction and modified Nusselt number are developed for application in the design process of packed bed heat exchangers for latent heat thermal energy storage.

Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates

In applications such as astrophysics and inertial confinement fusion, there are many three-dimensional cylindrical-symmetric multi-material problems which are usually simulated by Lagrangian schemes in the two-dimensional cylindrical coordinates. For this type of simulation, a critical issue for the schemes is to keep spherical symmetry in the cylindrical coordinate system if the original physical problem has this symmetry. In the past decades, several Lagrangian schemes with such symmetry property have been developed, but all of them are only first order accurate. In this paper, we develop a second order cell-centered Lagrangian scheme for solving compressible Euler equations in cylindrical coordinates, based on the control volume discretizations, which is designed to have uniformly second order accuracy and capability to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. The scheme maintains several good properties such as conservation for mass, momentum and total energy, and the geometric conservation law. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of accuracy, symmetry, non-oscillation and robustness. The advantage of higher order accuracy is demonstrated in these examples.

Phase Separation Dynamics in Isotropic Ion-Intercalation Particles

Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from diffusion and phase transformations coupled to ion-intercalation reactions. Using the recently developed Cahn--Hilliard reaction (CHR) theory, we investigate a simple mathematical model of ion intercalation in a spherical solid nanoparticle, which predicts transitions from solid-solution radial diffusion to two-phase shrinking-core dynamics. This general approach extends previous lithium-ion battery models, which either neglect phase separation or postulate a spherical shrinking-core phase boundary, by predicting phase separation only under appropriate circumstances. The effect of the applied current is captured by generalized Butler--Volmer kinetics, formulated in terms of diffusional chemical potentials, and the model consistently links the evolving concentration profile to the battery voltage. We examine sources of charge/discharge asymmetry, such as asymmetric charge transfer and surface “wetting" by ions within the solid, which can lead to three distinct phase regions. In order to solve the fourth-order nonlinear CHR initial-boundary-value problem, a control-volume discretization is developed in spherical coordinates. The basic physics are illustrated by simulating many representative cases, including a simple model of the popular cathode material, lithium iron phosphate (neglecting crystal anisotropy and coherency strain). Analytical approximations are also derived for the voltage plateau as a function of the applied current.

Analysis of heat transfer and fluid flow during melting inside a spherical container for thermal energy storage

A numerical model of heat transfer and fluid flow during melting of a phase change material (PCM) inside a closed and uniformly heated spherical shell has been developed to investigate the thermal performance of the system. The control volume discretization technique has been used to solve the conservation equations for mass, momentum, and energy along with the enthalpy-porosity method to track the melting front. The study focused on PCMs with a melting temperature in the range of operation of concentrated solar thermal power (CSP) generation. A comprehensive analysis has been performed in order to determine the influence of the Grashof, Stefan and Prandtl numbers on the melting dynamics of capsules with various diameters (20, 30, 40, and 50 mm). Also the effects of the shell material and initial temperature of the system on the thermal performance have been analyzed. Correlations have been proposed for the melt fraction and average Nusselt number as a function of the main dimensionless parameters that characterized this problem. In order to support the validity of the developed model, the predicted results of a validation case are compared with the experimental results reported by Tan [1] and are found to be in good agreement.

Effect of the Tank Design on the Flow Pattern Generated with a Pitched Blade Turbine

The effect of the tank design on the hydrodynamic structure is carried out in three tanks; cylindrical tank, curved tank and spherical tank; equipped with a six-pitched blade turbine (PBT6). The hydrodynamic behaviour is numeri- cally predicted by the resolution of the Navier-Stokes equations in conjunction with the Renormalization Group (RNG) of the k-e turbulence model. These equations are solved by a control volume discretization method. The numerical results from the application of the CFD code Fluent with the Multi Reference Frame (MRF) model are presented in the impeller stream region. Particularly, the velocity components and the turbulent characteristics are presented in different planes con- taining the blade. The power consumption of these stirred tanks was calculated to choose the most effective system. The comparison between the numerical results and the experimental data showed a good agreement.

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture–Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

Numerical estimation of time lags and decrement factors for wall complexes including Multilayer Thermal Insulation, in two different climatic zones

In the current contribution, time lags and decrement factors for a building wall including different Multilayer Thermal Insulation (MTI) configurations have been calculated numerically. For this investigation purpose, a numerical model which can take care of composite walls was developed. The combined conduction, radiation and convection heat transfer equation was solved explicitly via a control volume discretization. In order to investigate the thermal behavior of a wall under certain outdoor climates, meteorological data that served as boundary conditions were applied to the outer surface of this wall. The outdoor air temperatures were averaged over a number of years after been obtained from hourly measurements in both locations under study. Regarding the solar radiation, it was calculated using a clear-sky model during the summer period. The influence of geographic coordinates’ location on the insulating performance of MTI applications was investigated while the wall configurations were supposed to be located in two different climatic zones. The selection of a particular wall configuration involved not only the outdoor climate, but also the whole building characteristics, orientation, etc. The results of the present study are useful for further development in order to conceive and design the optimum MTI configuration, adjusted in specific geographic coordinates and orientations.

A model of the kinetics of the temperature-induced phase transformation in NiTi alloys and its experimental verification

A model, describing the kinetics of the temperature-induced austenite phase transformation processes in shape memory alloys (SMAs) is presented. The model is formulated for a general stress and temperature state in the solid body and allows the computation of the austenite phase fraction evolution in it, that is, the dependence of the amount of the austenite phase fraction on the values of the temperature and stress. It is shown that the predicted amount of the austenite phase fraction is not dependent on the control volume discretization. The model does not take into account microstructural changes in the material and the existence of different phases in the material (R-phase, twinned martensite). The experimental verification of the model was conducted using SMA nitinol wire that was prestrained (detwinned martensite state) and subjected to a heat load. It was shown that the austenite phase fraction predicted by the model agrees very well with the tabulated (reference) and experimental data.