Grid Refinement Research Articles

A fourth-order orthogonal spline collocation method to fourth-order boundary value problems

In this article, we use orthogonal spline collocation methods (OSCM) for the fourth-order linear and nonlinear boundary value problems (BVPs). Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution. We establish the existence and uniqueness solution to the discrete problem. We discuss dynamics of the stationary Swift–Hohenberg equation for different values of α. Finally, we perform some numerical experiments and using grid refinement analysis, we compute the order of convergence of the numerical method. Comparative to existing methods, we show that the OSCM gives optimal order of convergence for norms and superconvergent result for -norm at the knots with minimal computational cost.

Numerical simulation of dynamic fracture in functionally graded materials using peridynamic modeling with composite weighted bonds

In this paper, the peridynamic (PD) method with the composite weighted bond for functionally graded materials (FGMs) was proposed. The cracks propagation in FGMs under dynamic loading was analyzed. Both uniform refinement grid (m-convergence) and decreasing radius of PD horizon (δ-convergence) of FGMs with double pre-crack were given. The effects of the micromodulus functions and material gradient form on the crack propagation were discussed, and it was found that the micromodulus function's effect on the crack propagation pattern was limited, while the effect of the gradient form of FGMs is existed. The numerical results show that there is almost no difference between the proposed composite weighted bond model and the previous model for FGMs with single pre-crack. By comparing the numerical results of the proposed model with the results from the experiment, the present PD model for FGMs is valid and effective.

Estimation of Utsira CO2 Storage Capacity Based on Water Producing Modelling and Identification of Proper Water Producer Architecture

Utsira aquifer is well known from the CCS community as Sleipner gas project is storing CO2 in that aquifer since 1996. More than 17 million tons of CO2 have been injected to date. According to previous studies, CO2 storage capacity of the Utsira varies between 0.3 and 50.4 Gt. In this study, assuming specific parameters, the storage capacity is estimated about 1.7 Gt using analytical method. A full scale flow model of the aquifer has been used to model CO2 storage, considering CO2 migration and pressure raise to estimate the CO2 storage capacity, with or without water production. Water-CO2 relative permeabilities are selected based on a vertical equilibrium approach and validated when CO2 migration is compared to high resolution grid refinement around the injector areas. Optimizing injector well locations lead to an estimation of CO2 storage capacity of about 1.6 Gt without water production. This capacity can be increased to 3 Gt with about 1 Gt of formation water being produced during injection. The location and architecture of the water production wells were optimized: the wells should be subsea and should be operated with minimal requirement. Two production methods were compared: eruptive wells, producing water based on the formation overpressure induced by the CO2 storage; and activated wells using electrical submersible pumps (ESP). For each scenario, an optimization is proposed in terms of well count, well locations and well architecture designs. For the reservoir depth ranging between 500–1200 m below sea level and for such a high water production rate, the electrical submersible pumps (ESP) technology is a suitable application.

Spatial scaling in multiscale models: methods for coupling agent-based and finite-element models of wound healing.

Multiscale models that couple agent-based modeling (ABM) and finite-element modeling (FEM) allow the dynamic simulation of tissue remodeling and wound healing, with mechanical environment influencing cellular behaviors even as tissue remodeling modifies mechanics. One of the challenges in coupling ABM to FEM is that these two domains typically employ grid or element sizes that differ by several orders of magnitude. Here, we develop and demonstrate an interpolation-based method for mapping between ABM and FEM domains of different resolutions that is suitable for linear and nonlinear FEM meshes and balances accuracy with computational demands. We then explore the effects of refining the FEM mesh and the ABM grid in the setting of a fully coupled model. ABM grid refinement studies showed unexpected effects of grid size whenever cells were present at a high enough density for crowding to affect proliferation and migration. In contrast to an FE-only model, refining the FE mesh in the coupled model increased strain differences between adjacent finite elements. Allowing strain-dependent feedback on collagen turnover magnified the effects of regional heterogeneity, producing highly nonlinear spatial and temporal responses. Our results suggest that the choice of both ABM grid and FEM mesh density in coupled models must be guided by experimental data and accompanied by careful grid and mesh refinement studies in the individual domains as well as the fully coupled model.

A quantitative comparison of phase-averaged models for bubbly, cavitating flows

We compare the computational performance of two modeling approaches for the flow of dilute cavitation bubbles in a liquid. The first approach is a deterministic model, for which bubbles are represented in a Lagrangian framework as advected features, each sampled from a distribution of equilibrium bubble sizes. The dynamic coupling to the liquid phase is modeled through local volume averaging. The second approach is stochastic; ensemble-phase averaging is used to derive mixture-averaged equations and field equations for the associated bubble properties are evolved in an Eulerian reference frame. For polydisperse mixtures, the probability density function of the equilibrium bubble radii is discretized and bubble properties are solved for each representative bin. In both cases, the equations are closed by solving Rayleigh–Plesset-like equations for the bubble dynamics as forced by the local or mixture-averaged pressure, respectively. An acoustically excited dilute bubble screen is used as a case study for comparisons. We show that observables of ensemble- and volume-averaged simulations match closely and that their convergence is first order under grid refinement. Guidelines are established for phase-averaged simulations by comparing the computational costs of methods. The primary costs are shown to be associated with stochastic closure; polydisperse ensemble-averaging requires many samples of the underlying PDF and volume-averaging requires repeated, randomized simulations to accurately represent a homogeneous bubble population. The relative sensitivities of these costs to spatial resolution and bubble void fraction are presented.

Theoretical Analysis of Computational Fluid Dynamics–Discrete Element Method Mathematical Model Solution Change With Varying Computational Cell Size

Successful verification and validation is crucial to build confidence in the application of coupled computational fluid dynamics–discrete element method (CFD–DEM). Model verification includes ensuring a mesh-independent solution, which poses a major difficulty in CFD–DEM due to the complicated relationship between solution and computational cell size. In this paper, we investigate the production of numerical error in the CFD–DEM coupling procedure with computational grid refinement. The porosity distribution output from simulations of fixed-particle beds is determined to be Gaussian, and the average and standard deviation of the representative distribution are reported against cell size. We find that the standard deviation of bed porosity increases exponentially as the cell size is reduced. The average drag calculated from each drag law is very sensitive to changes in the porosity standard deviation. When combined together, these effects result in an exponential change in expected drag force when the cell size is small relative to the particle diameter. The divided volume fraction method of porosity calculation is shown to be superior to the centered volume fraction (CVF) method. The sensitivity of five popular drag laws to changes in the porosity distribution is presented, and the Ergun and Beetstra drag laws are shown to be the least sensitive to changes in the cell size. A cell size greater than three average particle diameters is recommended to prevent errors in the simulation results. A grid refinement study (GRS) is used to quantify numerical error.

Large-Eddy Simulation of Sandia Flame D with Efficient Explicit Filtering

A uniform Gaussian filter has been applied explicitly to the LES conservation equations for mass, momentum and mixture to simulate a piloted non-premixed methane-air flame (Sandia Flame D). Using a basic property of exponential functions, the three dimensional Gaussian filter is decomposed into the product of three one dimensional filters, greatly reducing the cost of filtering. Seven simulations on three different grids have been performed to investigate the influence of grid refinement with a purely implicit filter, the effects of explicit filtering with increasing filter width and the effect of grid refinement at constant filter-size. Overall, consistent results have been achieved at a cost that is moderate with implicit or explicit filtering, so that explicit filtering can be applied in cases where the numerical error should be independent of the modelling error.

Adaptive finite element simulations of waveguide configurations involving parallel 2D material sheets

We discuss analytically and numerically the propagation and energy transmission of electromagnetic waves caused by the coupling of surface plasmon polaritons (SPPs) between two spatially separated layers of 2D materials, such as graphene, at subwavelength distances. We construct an adaptive finite-element method to compute the ratio of energy transmitted within these waveguide structures reliably and efficiently. At its heart, the method is built upon a goal-oriented a posteriori error estimation with the dual-weighted residual method (DWR).Furthermore, we derive analytic solutions of the two-layer system, compare those to (known) single-layer configurations, and compare and validate our numerical findings by comparing numerical and analytical values for optimal spacing of the two-layer configuration. Additional aspects of our numerical treatment, such as local grid refinement, and the utilization of perfectly matched layers (PMLs) are examined in detail.

Numerical simulation of a silicon-based latent heat thermal energy storage system operating at ultra-high temperatures

This study investigates numerically a silicon-based latent heat storage system operating at ultra-high temperatures (∼1410–2000 °C). Owing to the silicon’s high latent heat (1230 kWh·m−3), storage densities of almost an order of magnitude higher than the state-of-the-art molten salt-based systems can be achieved. Prior to fabricating this system, there is a necessity to scrutinize the complex heat transfer mechanisms occurring during silicon phase change and indicate a vessel design that enables quick charge rates. A validated transient computational fluid dynamics model, combining the multiphase volume of fluid method with the enthalpy porosity approach and an adaptive local grid refinement technique is applied. This model simultaneously takes into account the phase-change material (PCM) volumetric change and the effect of dendritic formations and buoyancy-driven natural convection on the melting process. Numerical results reveal that the system charge rate is highly dependent on the PCM permeability, vessel design and operating conditions. Between five shapes investigated, i.e. cube, truncated cone, sphere, cut-off sphere and cylinder of volume V = 3.75E−03 m3, the truncated cone is the most preferable considering both design flexibility- its height can be easily altered, without increasing its volume- and melting rates. Actually, the PCM melting time achieved with the cone is 21% quicker compared to sphere and 6% slower compared to cube. Concerning the vessel size, the melting time increases by almost 28%, when the vessel volume increases to V′ = 2V. Finally, reduction in the silicon melting time, almost by 31%, is achieved by increasing the Stefan number by 35%.

Coarse-grained modeling of sheared granular beds

In this work, a volume-filtered Eulerian–Lagrangian (VFEL) approach is employed to evaluate the model fidelity required to accurately predict the dynamics of sub-aqueous sedimentary flows. The VFEL approach allows for interphase exchange terms (e.g., volume fraction and drag) to be computed on a length scale independent of the grid resolution, enabling a comprehensive analysis of the errors associated with numerical discretization and physical models on bed height, particle flux, and bed wavelength. Simulations of initially flat granular beds under both laminar and turbulent shear flow are carried out in featureless and pattern forming regimes. Resulting errors are evaluated by comparing VFEL to the direct numerical simulation (DNS) data of [1,2]. Grid refinement demonstrates convergence of the bed height and particle flux in all of the cases considered. However, traditional drag laws based on local volume fraction and particle Reynolds number lead to systematic errors in the featureless laminar regime. A stochastic drag law that accounts for the variance in local particle configuration provides improvement of the steady state statistics. The use of a stochastic drag law was found to be less important when the rise in bed height is minimal. Finally, the VFEL approach with stochastic drag is used in a large-eddy simulation framework to simulate pattern formation in turbulent shear flow at relatively high Galileo numbers.

Compositional Simulation of CO2 Huff ’n’ Puff in Eagle Ford Tight Oil Reservoirs With CO2 Molecular Diffusion, Nanopore Confinement, and Complex Natural Fractures

Summary Although numerous studies proved the potential of carbon dioxide (CO2) huff ’n’ puff, relatively few models exist to comprehensively and efficiently simulate CO2 huff ’n’ puff in a way that considers the effects of molecular diffusion, nanopore confinement, and complex fractures for CO2. The objective of this study was to introduce a numerical compositional model with an embedded-discrete-fracture-model (EDFM) method to simulate this process in an actual Eagle Ford tight oil well. Through nonneighboring connections (NNCs), the EDFM method can properly and efficiently handle any complex fracture geometries. We built a 3D reservoir model with six fluid pseudocomponents. We performed history-matching with measured flow rates and bottomhole pressure (BHP). Good agreements between field data, EDFM, and local grid refinement (LGR) were achieved. However, the EDFM method performed faster than the LGR method. After that, we evaluated the CO2-enhanced-oil-recovery (EOR) effectiveness for molecular diffusion and nanopore confinement effects. The traditional phase equilibrium calculation was modified to calculate the critical fluid properties with nanopore confinement. The simulation results showed that the CO2 EOR with larger diffusion coefficients performed better than the primary production. In addition, both effects were favorable for the CO2 huff ’n’ puff effectiveness. The relative increase of cumulative oil production after 20 years was approximately 12% for this well. Furthermore, when considering complex natural fractures, the relative increase of cumulative oil production was approximately 8%. This study provided critical insights into a better understanding of the impacts of CO2 molecular diffusion, nanopore confinement, and complex natural fractures on well performance during the CO2-EOR process in tight oil reservoirs.

Large Eddy Simulation Requirements for the Flow over Periodic Hills

Large eddy simulations are carried out for flows in a channel with streamwise-periodic constrictions, a well-documented benchmark case to study turbulent flow separation from a curved surface. Resolution criteria such as wall units are restricted to attached flows and enhanced criteria, such as energy spectra or two-point correlations, are used to evaluate the effective scale separation in the present large eddy simulations. A detailed analysis of the separation above the hill crest and of the early shear layer development shows that the delicate flow details in this region may be hardly resolved on coarse grids already at Re = 10595, possibly leading to a non monotonic convergence with mesh refinement. The intricate coupling between numerical and modeling errors is studied by means of various discretization schemes and subgrid models. It is shown that numerical schemes maximizing the resolution capabilities are a key ingredient for obtaining high-quality solutions while using a reduced number of grid points. On this respect, the introduction of a sharp enough filter is an essential condition for separating accurately the resolved scales from the subfilter scales and for removing ill-resolved structures. The high-resolution approach is seen to provide solutions in very good overall agreement with the available experimental data for a range of Reynolds numbers (up to 37000) without need for significant grid refinement.

Investigation of the Effect of Natural Fractures on Multiple Shale-Gas Well Performance Using Non-Intrusive EDFM Technology

The influence of complex natural fractures on multiple shale-gas well performance with varying well spacing is poorly understood. It is difficult to apply the traditional local grid refinement with structured or unstructured gridding techniques to accurately and efficiently handle complex natural fractures. In this study, we introduced a powerful non-intrusive embedded discrete fracture model (EDFM) technology to overcome the limitations of exiting methods. Through this unique technology, complex fracture configurations can be easily and explicitly embedded into structured matrix blocks. We set up a field-scale two-phase reservoir model to history match field production data and predict long-term recovery from Marcellus. The effective fracture properties were determined thorough history matching. In addition, we extended the single-well model to include two horizontal wells with and without including natural fractures. The effects of different numbers of natural fractures on two-well performance with varying well spacing of 200 m, 300 m, and 400 m were examined. The simulation results illustrate that gas productivity almost linearly increases with the number of two-set natural fractures. Furthermore, the difference of well performance between different well spacing increases with an increase in natural fracture density. A larger well spacing is preferred for economically developing the shale-gas reservoirs with a larger natural fracture density. The findings of this study provide key insights into understanding the effect of natural fractures on well performance and well spacing optimization.

A novel spatio-temporally adaptive parallel three-dimensional DSMC solver for unsteady rarefied micro/nano gas flows

An efficient parallel multi-scale direct simulation Monte Carlo algorithm to simulate three-dimensional rarefied gas flows over complex geometries is presented. The proposed algorithm employs a novel spatio-temporal adaptivity scheme. Based on the gradients of flow macro-properties, the spatio-temporal adaptivity scheme computes the cell size distribution and assigns the appropriate number of time sub-steps for each cell. The temporal adaptivity scheme provides local time step adaptation through different temporal levels employed in different cells. Spatial representation is based on a hierarchical octree Cartesian grid with low memory storage requirement. The hierarchical octree grid endows the method with straightforward and efficient data management suitable for particle ray tracing and dynamic grid refinement and coarsening. Solid objects, represented by triangulated surfaces, are incorporated using a cut-cell algorithm. A new parallelization scheme suitable for simulating strongly unsteady, non-equilibrium flows is proposed. The parallelization scheme, implemented for multi-core Central Processing Units (CPUs), significantly reduces the computational cost of modeling these flows. Performance of the method is assessed by comparing with benchmarked test cases for various rarefied gas flows.

Regularized single and double layer integrals in 3D Stokes flow

We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The singular Stokeslet and stresslet kernels are regularized and, for the nearly singular case, corrections are added to reduce the regularization error. These corrections are derived analytically for both the Stokeslet and the stresslet using local asymptotic analysis. For the case of evaluating the integrals on the surface, as needed when solving integral equations, we design high order regularizations for both kernels that do not require corrections. This approach is direct in that it does not require grid refinement or special quadrature near the singularity, and therefore does not increase the computational complexity of the overall algorithm. Numerical tests demonstrate the uniform convergence rates for several surfaces in both the singular and near singular cases, as well as the importance of corrections when two surfaces are close to each other.

Grid Convergence Study for Detached-Eddy Simulation of Flow over Rod-Airfoil Configuration Using OpenFOAM

A rod-airfoil is a benchmark configuration for simulating Airfoil-Turbulence Interaction Noise (ATIN). The numerical simulation is modelled using unsteady Detached Eddy Simulation (DES). The grid refinement involves two stages of assessments. The first compares the sensitivity of two reasonable estimated cells to the flow behaviours of the case. Based on the finding from the first stage, further grid refinement is proceed in stage two. In stage two, a minimum of three different grid resolutions are considered; the fine, medium and coarse grids in order to investigate the grid independency. Richardson extrapolation and Grid Convergence Index (GCI) are introduced to quantitatively evaluate the grid independency. Based on the results between those three different grids, a monotonic convergence criterion has been achieved. The reduction in GCI value indicates that the grid convergence error has been significantly reduced, in which the fine grid has a GCI value is around less than 0.5%.

Effect of dynamical downscaling to cyclone simulation: a study case for Haiyan typhoon

The capacity of a global gridded climate data in describing tropical cyclone (TC) activity might be still less adequate due to its coarse resolution or the TC intensity itself. Grid refinement method such as a dynamical downscaling is one way to add values of the climate attributes in the data. However, the performance of the dynamical downscaling itself still con-tains many uncertainties. This paper aims to evaluate the effect of the dynamical downscaling model on the climate simulation during a strong tropical cyclone Haiyan. The snapshots and the best track data from the historical satellite imagery will be used for validation. In general the downscaled cyclone simulation does not give a significant change to the driven data, but it results in more finer resolution of wind field and improves the time and the position of the eye of the cyclone.

Uncertainty estimation of a CFD-methodology for the performance analysis of a collective and cyclic pitch propeller

Estimation and analysis of the uncertainty introduced by using a numerical model for the investigation and study of any type of flow problem have become common industry practice. Through understanding and evaluation of the uncertainty introduced by a numerical model, the accuracy and applicability of the model itself are evaluated. In this paper, the numerical uncertainty of a CFD-methodology developed to analyse the hydrodynamic performance of a collective and cyclic pitch propeller (CCPP) is estimated and analysed. The CCPP is a novel propulsion and manoeuvring concept for autonomous underwater vehicles, aimed to generate both propulsion and manoeuvring forces through advanced control of the propeller's blade pitch. The numerical uncertainty is established for three performance parameters, the generated propulsive force, the side-force magnitude, and the side-force orientation, by conducting a grid and time-step refinement study over three operational conditions. Additionally, the influence of the oscillatory uncertainty, introduced by the periodic nature of the problem, is investigated although shown to have a minimal effect when properly monitored. Based on a least-squares regression analysis of the refined simulation results, the numerical uncertainty is proven to be dominated by the introduced discretisation errors. In the case of the propulsive and side-force magnitude, the total uncertainty is dictated by the time discretisation uncertainty under bollard pull conditions, while the total uncertainty of the captive cases is mainly a result of the spatial discretisation uncertainty. The total uncertainty in the side-force orientation is observed to be primarily a consequence of the time discretisation uncertainty for all simulated cases. Overall, the total uncertainty for captive cases can be considered satisfactory for all three performance parameters, while further work is needed to reduce the observed uncertainty of the simulations under bollard pull conditions.

Eulerian–Lagrangian flow-vegetation interaction model using immersed boundary method and OpenFOAM

This paper presents a novel coupled flow-vegetation interaction model capable to resolve the flow and motion of flexible vegetation with large deflections simultaneously on a hybrid Eulerian–Lagrangian grid. The hydrodynamics model is based on a Navier–Stokes flow solver with the Volume of Fluid surface capturing method in OpenFOAM. The deforming and moving vegetation are tracked through a series of Lagrangian grids attached to the vegetation and embedded in the computational domain of OpenFOAM. The governing equations for the flexible vegetation motion are based on a slender rod theory and are solved by a Finite Element Method on a vegetation-following Lagrangian grid. The hydrodynamics and vegetation models are coupled through the vegetation-induced hydrodynamic forces using a novel diffused immersed boundary method to avoid excessive fluid grid refinements around the vegetation. The standard two-equation k − ε turbulence model is extended for vegetated flow by incorporating additional closure terms of vegetation effects. The newly developed coupled flow-vegetation model is validated against experiments for both individual single-stem vegetation and a vegetation patch in a large-scale wave flume. Model results of asymmetric displacement of the vegetation motion, wave height decay, and wave kinematics within and outside the vegetation patch are in good agreement with measurements. The drastical change in wave radiation stress by the presence of vegetation patch leads to a strong current jet near the top of vegetation patch, which in turn drives a local circulation pattern around the vegetation patch. The effect of vegetation bending stiffness on the model results and empirical drag and inertia coefficients for calculating the hydrodynamic forces are examined in detail. Maximum turbulence energy is observed close to the bed and the top of vegetation patch just before and after the wave crest/trough, where the relative flow velocity to vegetation is large.

Pore-Scale Level Set Simulations of Capillary-Controlled Displacement with Adaptive Mesh Refinement

Multiphase flow simulations on imaged porous rock structures require numerical methods that are accurate and robust when applied on complex geometries. A key element in this context is to investigate how simulations behave under grid refinement. In this work, we couple an existing software for structured adaptive mesh refinement (AMR) and parallelism to a previously developed level set method for capillary-controlled displacement on the pore scale. The level set method accounts for wettability by using different evolution velocities in pore and solid space rather than implementing it as a boundary condition on the pore walls. We perform simulations with up to three nested refinement levels on idealized pore geometries to validate the AMR technology. Based on simulations, we identify suitable cell refinement criteria for our applications. We demonstrate effects of AMR in simulations relevant for drainage in porous media, such as in the computation of capillary entry pressures in pore throats with different shapes and wetting states, and obtain excellent agreement with analytic results. Finally, we investigate the effects of AMR during simulation of quasi-static drainage on sandstone. The comparison of capillary equilibrium fluid configurations, capillary pressure, and specific fluid/fluid interfacial area–saturation relationships for drainage with and without AMR shows differences that diminish for less water-wet states. The general behavior is that the capillary pressure and interfacial area for a given saturation increase in simulations with AMR. The largest deviation occurs for small water saturations, suggesting AMR can be an important component in simulation tools to describe more accurately capillary behavior in the low water saturation regime where interface curvature is high.