Grid Convergence Study Research Articles

Time-accurate calculation of two-phase granular flows exhibiting compaction, dilatancy and nonlinear rheology

In this paper we propose a time-accurate algorithm for the calculation of two-phase granular flows via two-velocity, two-pressure Eulerian models. Such flows exhibit a number of important characteristics that make their computation particularly challenging. For example, even at constant-density flows the suspended granular phase can compact or expand so that the velocity fields of the two phases are not solenoidal. Also, the granular material typically exhibits a complex and highly nonlinear rheological behaviour. Moreover, the governing equation for the granular volume fraction (often referred to in the literature as “compaction equation” or “particle migration model”) involves stresses as source terms and, therefore it is strongly coupled to the momentum balance laws. Herein we show that this coupling can be a strong destabilising factor in the computations if it is not treated properly. Our algorithm is based on a predictor–corrector time-integration scheme. Further, it employs a generalized double projection method for non-solenoidal velocity fields that results in second-order elliptic equations for the phasial pressures. The efficiency of the proposed algorithm is illustrated via a series of simulations of transient flows with strong concentration gradients, namely, sedimentation of a granular suspension, compaction of a dense sediment bed, and collapse of a submerged granular column. Numerical grid-convergence studies further show that the algorithm exhibits satisfactory convergence rates. Overall, these numerical results indicate that the proposed algorithm is well suited for the simulation of flows of dense suspensions, flows with high volume-fraction variations or material interfaces, as well as for unsteady fluid–solid interaction problems.

A computational design method for bio-mimicked horizontal axis tidal turbines

PurposeThe purpose of this paper is to conduct a comparative analysis between a straight blade (SB) and a curved caudal-fin tidal turbine blade (CB) and to examine the aspects relating to geometry, turbulence modelling, non-dimensional forces lift and power coefficients.Design/methodology/approachThe comparison utilises results obtained from a default horizontal axis tidal turbine with turbine models available from the literature. A computational design method was then developed and implemented for “horizontal axis tidal turbine blade”. Computational fluid dynamics (CFD) results for the blade design are presented in terms of lift coefficient distribution at mid-height blades, power coefficients and blade surface pressure distributions. Moving the CB back towards the SB ensures that the total blade height stays constant for all geometries. A 3D mesh independency study of a “straight blade horizontal axis tidal turbine blade” modelled using CFD was carried out. The grid convergence study was produced by employing two turbulence models, the standard k-ε model and shear stress transport (SST) in ANSYS CFX. Three parameters were investigated: mesh resolution, turbulence model, and power coefficient in the initial CFD, analysis.FindingsIt was found that the mesh resolution and the turbulence model affect the power coefficient results. The power coefficients obtained from the standard k-ε model are 15 to 20 per cent lower than the accuracy of the SST model. Further analysis was performed on both the designed blades using ANSYS CFX and SST turbulence model. The variation in pressure distributions yields to the varying lift coefficient distribution across blade spans. The lift coefficient reached its peak between 0.75 and 0.8 of the blade span where the total lift accelerates with increasing pressure before drastically dropping down at 0.9 onwards due to the escalating rotational velocity of the blades.Originality/valueThe work presents a computational design methodological approach that is entirely original. While this numerical method has proven to be accurate and robust for many traditional tidal turbines, it has now been verified further for CB tidal turbines.

Harmonic Balance method for nonlinear and viscous free surface flows

The Harmonic Balance method applied to fully nonlinear, viscous, temporally periodic free surface flows is presented in this paper. The Harmonic Balance approach is used to project a transient periodic two phase flow into multiple coupled steady–state problems discretising the periodic time interval. The numerical framework is based on Finite Volume method for Computational Fluid Dynamics in the open–source software foamextend. The disability of the Harmonic Balance method to simulate flows with zero or close to zero mean velocity is overcome by coupling the Harmonic Balance steady–state equations implicitly, presenting a novelty in the field. Von Neumann stability analysis is performed for the governing Harmonic Balance equations to mathematically prove that zero mean velocity can be simulated using the proposed implicit coupling. The method is validated and verified on a periodic free surface flow over a ramp and regular surface wave propagation with current, both including grid convergence studies and spectral resolution sensitivity studies. All results are compared to fully transient computations. A detailed study is performed for wave propagation without current, confirming that convergence of free surface elevation can be achieved without mean velocity. The approach proved accurate and applicable for twophase flows, opening a new research field.

Numerical analysis of wave-induced poro-elastic seabed response around a hexagonal gravity-based offshore foundation

In order to prevent the future risk of soil and structural failures for the offshore foundations, it is essential to evaluate the seabed soil behaviors in the vicinity of the foundation under dynamic wave loadings. The objective of this paper is to investigate the wave-induced soil response and liquefaction risk around a hexagonal gravity-based offshore foundation. Three-dimensional (3D) numerical analysis is performed by applying an integrated multiphysics model developed in the finite volume method (FVM) based OpenFOAM framework. The integrated model incorporates solvers of the nonlinear waves, the linear elastic structure and the anisotropic poro-elastic seabed soil. The free surface model and soil model are verified by grid convergence studies. The wave-induced soil response model is validated by reproducing a laboratory experiment and a good agreement is obtained.Distributions of wave-induced shear stress, pore pressure, vertical displacement and seepage flow structure in the seabed are investigated. It is found that the presence of the foundation significantly amplifies the wave-induced shearing effect and vertical displacement in the underlying seabed soil. Seabed consolidation state in the presence of the structure is evaluated. Since the foundation is embedded in the seabed at a depth, the vertices of the hexagonal foundation cause the stress concentration in the nearby soil during the consolidation process. Therefore, the momentary liquefaction at the vertices is not as significant as that at the edges due to the high initial effective stress. A parametric study with different wave heights is conducted to examine the changes of soil response and momentary liquefaction depth around the hexagonal foundation. Effects of isotropic and anisotropic soil permeability on the pore pressure distribution are investigated. It shows that the effect of anisotropic permeability should be considered for the medium sand that is modelled in the present study.

Exa PowerFLOW Simulations for the Sixth AIAA Drag Prediction Workshop

Unsteady simulations of the NASA Common Research Model using a new transonic lattice Boltzmann method are conducted for the Sixth AIAA CFD Drag Prediction Workshop. The context for the development of this unique new method is briefly described. The results of a grid convergence study are presented and discussed, and it is shown that the numerical method is capable of accurately predicting incremental drag changes. The occurrence of unsteady flow at higher angles of attack during an alpha sweep for the model including static aeroelastic wing deformations is discussed. Buffet is already shown to start locally on the wing for angles of attack considered for the workshop. Additional simulations for higher angles, deep into the buffet range and close to the high-speed stall of the model, are also shown and highlight the method’s capability to handle highly unsteady flows with massive turbulent separations.

RANS Simulations of NASA-CRM by the Finite-volume Method with the SST Turbulence Model

The paper reports numerical simulations of aerodynamic characteristics of a commercial aircraft model known as the NASA-CRM. Specific test cases and detailed experimental results have been provided by JAXA for the CFD workshop “Aerodynamic Prediction Challenge (APC).” We employed the cell-vertex finite-volume method on unstructured grids and carried out the RANS simulations with the k-ω SST turbulence model to present results in the APC-I. After the verification of the model in a benchmark problem, both alpha-sweep simulations from low- to high-angles of attack and the grid-convergence study at a cruise conditions are carried out. The simulations well reproduced the experimental results at moderate angles of attack by both types of computational grids used in the study but the results showed the dependency on the grid type and grid resolutions at the highest angle of attack of 5.72 degrees due to different predictions of flow separation near the root chord of the main wing.

Grid-converged solution and analysis of the unsteady viscous flow in a two-dimensional shock tube

The flow in a shock tube is extremely complex with dynamic multi-scale structures of sharp fronts, flow separation, and vortices due to the interaction of the shock wave, the contact surface, and the boundary layer over the side wall of the tube. Prediction and understanding of the complex fluid dynamics are of theoretical and practical importance. It is also an extremely challenging problem for numerical simulation, especially at relatively high Reynolds numbers. Daru and Tenaud [“Evaluation of TVD high resolution schemes for unsteady viscous shocked flows,” Comput. Fluids 30, 89–113 (2001)] proposed a two-dimensional model problem as a numerical test case for high-resolution schemes to simulate the flow field in a square closed shock tube. Though many researchers attempted this problem using a variety of computational methods, there is not yet an agreed-upon grid-converged solution of the problem at the Reynolds number of 1000. This paper presents a rigorous grid-convergence study and the resulting grid-converged solutions for this problem by using a newly developed, efficient, and high-order gas-kinetic scheme. Critical data extracted from the converged solutions are documented as benchmark data. The complex fluid dynamics of the flow at Re = 1000 are discussed and analyzed in detail. Major phenomena revealed by the numerical computations include the downward concentration of the fluid through the curved shock, the formation of the vortices, the mechanism of the shock wave bifurcation, the structure of the jet along the bottom wall, and the Kelvin-Helmholtz instability near the contact surface. Presentation and analysis of those flow processes provide important physical insight into the complex flow physics occurring in a shock tube.

Numerical comparison of bubbling in a waste glass melter

Radioactive tank waste is scheduled for vitrification at the Waste Treatment and Immobilization Plant (WTP) being constructed at the Hanford Site. Testing of the pilot-scale DuraMelter 1200 at the Vitreous State Laboratory at the Catholic University of America has demonstrated that bubbling increases the melt rate of the batch material, and as a result, melter throughput. Computational fluid dynamics (CFD) models of this pilot-scale waste glass melter are being developed to improve our understanding of the processes that occur within the melter to aid in process optimization and troubleshooting of the WTP melters. Unfortunately, model validation is complicated by the difficulty of obtaining suitable experimental data for operational melters due to the inaccessibility for direct observation and measurements of the high-temperature, opaque fluid through the water-jacketed, refractory-lined steel vessel. This study focuses on assessing the fidelity of the CFD models to accurately predict the bubbling behavior. Because of the paucity of experimental data at the resolution required for CFD validation, a code comparison was used to evaluate two common approaches for simulating flows of two immiscible Newtonian fluids on numerical grids and resolving multiphase interfaces. Here, the volume of fluid and level set methods are used to resolve the dynamically evolving interfaces between the molten glass and the air bubbles. To aid in the validation of the results of these codes, a comparison of the bubble behavior, growth, and frequency of bubble generation are presented and a grid convergence study is performed for the two approaches. The predictions from the two codes are within 6% for the average bubble radius of curvature within the bubble channels, within 2% for average terminal rise velocity of the bubbles, and within 4% for the area mean of the local maxima at the free surface. These parameters are of interest since they affect the convection within the melter and at the interface between the glass and batch layer. Ultimately, the results of this work can assist in confirming the predictive ability of waste glass melter models and provide a better understanding of the flow patterns within the WTP melters.

Numerical algorithms for infinite swept wing ice accretion

A method to simulate ice accretion on swept wings using a two-dimensional Navier-Stokes airflow solver, an Eulerian droplets field formulation and a PDE based thermodynamic model is presented. The current approach, called 2.5D in this paper, is developed by applying a rotation of the system of coordinates to allow simulations on a cut normal to the leading edge. The crossflow velocity component in the simulation plane is computed with the third momentum equation simplified by assuming null derivatives in the spanwise direction. The consistency of the ice accretion software, NSCODE-ICE, is first verified by a grid convergence study on 2D icing cases and an icing layers convergence study on 2D and 2.5D icing simulations. The validation of the approach is performed on an unswept two-dimensional glaze ice case and on a rime ice case. The current 2.5D approach is then validated on an infinite MS-317 swept wing and compared to results from usual semi-empirical infinite swept wing corrections. The new approach succeeds to model the stagnation line where other correction methods fail, leading to a better representation of the convective heat transfer in this area.

Prediction of capillary pressure-saturation relationship for primary drainage in a 3D fibrous porous medium using volume-of-fluid method

The unsaturated flow through fibrous porous media at the macroscale is typically described using the Richards equation which requires constitutive relations for capillary pressure and relative permeability as a function of liquid saturation. In literature, these constitutive relations are typically estimated using reduced-order modeling approaches such as pore-network (PN) models, full-morphology (FM) method etc. In this paper, we determine quasi-static capillary pressure-saturation relationship (Pc-S) for primary drainage in a 3D isotropic fibrous medium by performing direct numerical simulations (at microscale) with the volume-of-fluid (VOF) method, which takes into account the detailed description of the pore structure and interface dynamics. As a first step, the accuracy of the VOF method was verified by simulating a test case of quasi-static drainage between two cylinders. Next, a grid-convergence study was performed to determine the optimal grid resolution required for the 3D simulations of quasi-static drainage. Using this grid, the Pc-S relationship for primary drainage in an isotropic fibrous structure was calculated from the direct simulations, and the results were compared with the Pc-S estimated using full-morphology method, which is a quasi-static reduced-order geometric approach. In the VOF method, no modeling assumptions are made on the geometry of the pore structure and the liquid propagation, and thus, the results from direct simulations can be used to gain insights into the limitations of the reduced-order models.

Contributions to the Sixth Drag Prediction Workshop Using Structured, Overset Grid Methods

Structured-grid solutions obtained for the NASA Common Research Model for the Sixth AIAA Computational Fluid Dynamics Drag Prediction Workshop are detailed. Three different flow solvers were used among the contributors, and the numerical methodologies and turbulence modeling strategies employed by each code are described. Key results for all authors include grid convergence studies for the drag increment of a nacelle and pylon added to a wing–body configuration and a buffet study accounting for static aeroelastic deformation. Additional studies performed include feature-based adaptive mesh refinement and higher-order convective flux discretization, among others.

An improved projection-based embedded discrete fracture model (pEDFM) for multiphase flow in fractured reservoirs

Abstract The discrete fracture-matrix (DFM) approaches based on conforming grids become popular for modeling fractured reservoirs in the last decade. However, the application of conforming DFMs at field scale is limited due to its prohibitive computational cost. In recent years, embedded discrete fracture model (EDFM) has received considerable attention as a promising alternative. EDFM incorporates the effect of each fracture explicitly without requiring the simulation grid to conform to the fracture geometry. A compromise between accuracy and efficiency could be achieved in EDFM by enabling the use of standard corner-point grids for the background matrix domain. Although many works confirm the high accuracy of EDFM for the solutions of pressure and velocity field, very few results have been presented to examine its accuracy for the saturation solutions from multiphase flow problems. This paper shows that EDFM can induce large errors for multiphase displacement processes, due to its incapability to capture the proper flux split through a fracture. For the first time in the literature we present a systematic evaluation on the performances of EDFM for multiphase flow and provide a detailed analysis to illuminate when and why the method fails. The analysis motivates us to exploit the projection-based extension of EDFM (pEDFM) as an effective method to resolve the limitations associated with EDFM. pEDFM is recently developed by Tene et al. (2017) to address the issue of flow barriers, and is based on the introduction of extended fracture-matrix fluxes. Moreover, we make several improvements upon the original pEDFM method. A physical constraint on the preprocessing stage is proposed to overcome the limitation in a ‘naive implementation’ of pEDFM. A number of test cases with different fracture geometries are presented to benchmark the performances of the improved pEDFM method for multiphase flow. Grid convergence studies are conducted for different numerical schemes. The results show that improved pEDFM significantly outperforms the original EDFM method.

해머헤드 형상 주위 압력 섭동 해석에 영향을 미치는 수치적 요인 분석

The pressure perturbation analysis was performed using RANS(Reynolds-Averaged Navier-Stokes) and DDES(Delayed Detached Eddy Simulation) turbulence models. Only the DDES model could capture the pressure perturbation caused by the instability of the shear layer at the boattail of the hammerhead launch vehicle. In order to improve the resolution of the vortex shedding, the sensitivity analysis was performed according to the spatial discretization schemes using Roe-MUSCL, WENO5, and C-WENO5 scheme with lower numerical viscosity. Better resolution of the vortex shedding around the boattail can be obtained by applying higher-order spatial schemes. The grid convergence study also showed the same tendency.

Influence of spanwise no-slip boundary conditions on the flow around a cylinder

First we present high resolution large eddy simulations (LES) for the flow past a circular cylinder at Reynolds number 3900 to prove the quality of the simulations. A thorough grid convergence study is presented, and the agreement in the mean flow statistics between our simulations and the references is excellent. Then we apply a no-slip boundary condition at the spanwise boundaries of the cylinder, with aspect ratios of 6, 12 and 24. This results in large changes in the shear layers and wake topology, even for an aspect ratio of 24. Even though the boundary layers along the side walls are only about 0.4 diameters thick, they nevertheless manage to stabilize the shear layers all the way through the channel, thus effectively delaying the roll-up and transition to a turbulent wake.

Can adaptive grid refinement produce grid-independent solutions for incompressible flows?

This paper studies if adaptive grid refinement combined with finite-volume simulation of the incompressible RANS equations can be used to obtain grid-independent solutions of realistic flow problems. It is shown that grid adaptation based on metric tensors can generate series of meshes for grid convergence studies in a straightforward way. For a two-dimensional airfoil and the flow around a tanker ship, the grid convergence of the observed forces is sufficiently smooth for numerical uncertainty estimation. Grid refinement captures the details of the local flow in the wake, which is shown to be grid converged on reasonably-sized meshes. Thus, grid convergence studies using automatic refinement are suitable for high-Reynolds incompressible flows.

Linearity-preserving monotone local projection stabilization schemes for continuous finite elements

This paper presents some new tools for enforcing discrete maximum principles and/or positivity preservation in continuous piecewise-linear finite element approximations to convection-dominated transport problems. Using a linear first-order advection equation as a model problem, we construct element-level bilinear forms associated with first-order artificial diffusion operators and their two-scale counterparts. The underlying design philosophy is similar to that behind local projection stabilization (LPS) techniques and variational multiscale (VMS) methods. The difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected. The proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation. The value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots. The result is a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The use of variational gradient recovery makes it possible to add high-order background dissipation leading to improved approximation properties in smooth regions. The numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time-dependent test problems in two space dimensions.

CFD investigation to quantify the effect of layered multiple miniature blades on the performance of Savonius rotor

Present study is focussed on improvement of the coefficient of performance (COP) of a Savonius rotor using numerical simulation software. Quantification of the improvement is based on the comparison of the coefficient of performance (COP) of a basic configuration constituting a conventional Savonius rotor to that of a modified configuration developed by adding concentric multiple miniature blades inside the rotor blades of the basic configuration. Validation and grid convergence studies are carried out using k-ε and Shear Stress Transport (SST) turbulence models. Validation study suggested Shear Stress Transport (SST) model as more accurate and better option in present study. Shear Stress Transport (SST) turbulence model is used in the numerical simulations of the modified configuration. Optimum level of grid refinement is achieved through grid convergence study. Boundary layer mesh is created on the rotor blades, by estimating distance of first mesh node from the wall using desired values of y+ for both k-ε and Shear Stress Transport (SST) turbulence models. An improvement in COP spanning between 8.1% and 11.34% is achieved with the modified configuration.

A Validation Study of the Compressible Rayleigh–Taylor Instability Comparing the Ares and Miranda Codes

The compressible Rayleigh–Taylor (RT) instability is studied by performing a suite of large eddy simulations (LES) using the Miranda and Ares codes. A grid convergence study is carried out for each of these computational methods, and the convergence properties of integral mixing diagnostics and late-time spectra are established. A comparison between the methods is made using the data from the highest resolution simulations in order to validate the Ares hydro scheme. We find that the integral mixing measures, which capture the global properties of the RT instability, show good agreement between the two codes at this resolution. The late-time turbulent kinetic energy and mass fraction spectra roughly follow a Kolmogorov spectrum, and drop off as k approaches the Nyquist wave number of each simulation. The spectra from the highest resolution Miranda simulation follow a Kolmogorov spectrum for longer than the corresponding spectra from the Ares simulation, and have a more abrupt drop off at high wave numbers. The growth rate is determined to be between around 0.03 and 0.05 at late times; however, it has not fully converged by the end of the simulation. Finally, we study the transition from direct numerical simulation (DNS) to LES. The highest resolution simulations become LES at around t/τ ≃ 1.5. To have a fully resolved DNS through the end of our simulations, the grid spacing must be 3.6 (3.1) times finer than our highest resolution mesh when using Miranda (Ares).

Application of CFD on the optimization by response surface methodology of a micromixing unit and its use as a chemical microreactor

The optimization of a mixing unit by means of CFD (computational fluid dynamics) and RSM (response surface methodology) is presented in this work. The starting geometry is the one studied by Fang and co-workers [14] consisting in a T-type microchannel with the mixing unit inserted in the straight main channel. The mixing takes place at a very low Reynolds number and is promoted by means of two bars at 69° each with respect to the mixing unit walls. These angles are the input parameters of the optimization process while the output ones are the mixing efficiency, the needed pumping power to run the channel and the mixing energy cost. Fang and co-workers found numerically an efficiency of around 22% when one mixing unit was employed, while an almost perfect mixing could be reached by using 28 of them. However, they did not carry out a grid convergence study and their results were got for just one mesh. Due to this, before the optimization tasks and thanks to the Grid Convergence Index, it is shown that Fang and co-workers’ mixing efficiency has an uncertainty of around 40%. This is due to the fact that, with the optimal grid, the mixing efficiency is around 12.5% which is quite far from what Fang and co-workers said. Additionally, with the RSM and by using the optimal mesh, it is found that the optimal angles α1 and α2 are: ∼76° and ∼62° to get the highest efficiency which is around 14%; and ∼72° and ∼74° to get both the lowest pumping power to run the channel and the lowest mixing energy cost. The performing of the micromixer as a microreactor, where a basic reaction must take place, is finally assessed.

Performance evaluation of RANS-based turbulence models in simulating a honeycomb heat sink

As well-known, there is not a universal turbulence model that can be used to model all engineering problems. There are specific applications for each turbulence model that make it appropriate to use, and it is vital to select an appropriate model and wall function combination that matches the physics of the problem considered. Therefore, in this study, performance of six well-known Reynolds-Averaged Navier–Stokes (RANS) based turbulence models which are the Standard $$k {\text{-}} \varepsilon$$ , the Renormalized Group $$k {\text{-}} \varepsilon$$ , the Realizable $$k {\text{-}} \varepsilon$$ , the Reynolds Stress Model, the $$k {\text{-}} \omega$$ and the Shear Stress Transport $$k {\text{-}} \omega$$ and accompanying wall functions which are the standard, the non-equilibrium and the enhanced are evaluated via 3D simulation of a honeycomb heat sink. The CutCell method is used to generate grid for the part including heat sink called test section while a hexahedral mesh is employed to discretize to inlet and outlet sections. A grid convergence study is conducted for verification process while experimental data and well-known correlations are used to validate the numerical results. Prediction of pressure drop along the test section, mean base plate temperature of the heat sink and temperature at the test section outlet are regarded as a measure of the performance of employed models and wall functions. The results indicate that selection of turbulence models and wall functions has a great influence on the results and, therefore, need to be selected carefully. Hydraulic and thermal characteristics of the honeycomb heat sink can be determined in a reasonable accuracy using RANS-based turbulence models provided that a suitable turbulence model and wall function combination is selected.