Grid Convergence Index Method Research Articles

Experimental and numerical investigation of rectangular labyrinth weirs in an open channel

Labyrinth weirs are commonly used to increase the discharge efficiency in free-overflow discharges. These weirs provide a higher discharge efficiency than conventional linear weirs at the same headwater. In this study, the hydraulic performance of rectangular labyrinth weirs (RLWs) under different geometries and flow conditions was investigated experimentally and numerically. The numerical model was verified and validated using the grid convergence index method and experimental data. Numerical modelling results showed increased performance of the RLW due to the distribution of lateral velocities in the inlet keys, while nappe interference in the downstream keys led to a decrease in performance at high headwater. Within the limitations of 1.5 ≤ L/B ≤ 2.33 and 0.1 < Ho/P < 0.61 (L is the weir crest length, B is the channel width, Ho is the total water head and P is the weir height), a performance increase of 44% on average (maximum increase of 67%) for unit channel width was found for RLWs compared with linear weirs. For these limitations, two new empirical formulas with a high correlation were derived to estimate the discharge coefficients of RLWs based on B and L for Ho/P > 0.1, which are widely used in practice. When compared with some data in the literature, the empirical formulas produced satisfactory results.

Evaluating uncertainties in CFD simulations of patient-specific aorta models using Grid Convergence Index method

Cardiovascular diseases are among the most important causes of global mortality. Computational Fluid Dynamics (CFD) is a powerful research tool that analyzes the hemodynamics of artery and blood flow patterns. In this study, CFD simulations are performed to assess the patient-specific healthy aorta, fusiform, and saccular aneurysm with various mesh types, including tetrahedral, polyhedral, and poly-hexacore. The aim of this study is to explore how different mesh types and grid densities impact the hemodynamic properties of physiological flows, with the goal of identifying the most cost-effective meshing approach. A mesh independence study is carried out to ensure the precision of the results, considering the wall shear stress distribution. For this, five different mesh resolutions are generated for each geometry. The uncertainties of the simulations associated with the discretization techniques and solutions are evaluated using the Grid Convergence Index (GCI) method. The findings showed that increasing the mesh density provides smaller uncertainty. GCI values for the wall shear stress are in the range of convergence, indicating that the results are reliable and accurate. Mesh type selection affects the accuracy and computational cost of our simulations. The polyhedral and poly-hexacore meshes lead to a good compromise between precision and computational cost, while the tetrahedral mesh style gives the most precise results with fluctuation. This work provides a systematic approach based on the Grid Convergence Index method in order to select the most appropriate mesh type for evaluating uncertainties in CFD simulations of patient-specific healthy aortas and aortas with abdominal aneurysms. According to the findings and GCI analysis, the polyhedral mesh type was chosen for all patient-specific aorta models. The study clearly demonstrated its superiority over other mesh types, considering uncertainties and computational costs associated with different mesh styles.

Prediction of the Hydrodynamic Loads on a Full-Scale Caisson at High Reynolds Number

The paper reports on the computation of the hydrodynamic loads on a full-scale caisson at high Reynolds number in the presence of vortex shedding. The objective was to obtain reliable predictions of the resulting mean and fluctuating forces to guide the design of an actual caisson in the absence of relevant experimental data. A further objective was to investigate the effectiveness of alternative methods for the control of vortex shedding that can be implemented in practice. Two such methods were evaluated: (1) by rounding the corners of the rectangular-sectioned caisson, and (2) by the placement of a splitter plate in the separated wake region. The computations, which were performed using the OpenFOAM open-source software, were for a fixed caisson and hence did not account for motions due to vortex-induced vibrations. The effects of turbulence were accounted for by performing large-eddy simulations, and by using two-equation eddy-viscosity closures, one of which was specifically adapted to account for the interactions between the periodic vortex shedding and the random turbulence. The numerical accuracy was checked using the grid convergence index method, and the computations were extensively validated against data from relevant benchmark flows. The recommendations of this research were implemented in the design of a full-scale caisson that has since been deployed in a bridge construction project.

Effect of cavitation evolution on power characteristics of tidal current turbine

With the development of large-scale tidal current turbines and the increase in tidal current velocity, the possibility of cavitation increases. Furthermore, unsteady cavitation is a complicated multiphase flow that causes power degradation of tidal current turbine blade. There has been no comprehensive investigation of it so far. In this study, the blade captured power is obtained at different cavitation numbers using the Schnerr–Sauer cavitation model. The numerical uncertainty for the mesh and the time step is calculated by the grid convergence index method. It has been shown that, when the cavitation number is 5 and 2, cavitation has no effect on the blade power. With the decrease in the cavitation number, the rise in cavitation intensity occurs when the vapor distribution area stretches from the blade tip to the blade root and from the leading edge to the trailing edge, respectively. With a fall in cavitation number to 1.3, the vapor volume fraction rises, and the viscosity of the mixed phase reduces, resulting in a reduction in viscous power. When the cavitation number is 0.8, there exists a larger region in which an absolute value of minimum pressure coefficient is less than the cavitation number, a smaller blade load is present, and the pressure difference power is substantially decreased. Because of the huge inverse pressure gradient created by cavitation, the negative pressure difference power is generated, resulting in a decline of the blade power coefficient to 14%, when the cavitation number is 0.5.

A Code Verification for the Cavity SGEMP Simulation Code LASER-SGEMP

The stochastic Richardson extrapolation-based numerical error quantification (StREEQ) technique has been applied to accomplish a code verification for the cavity systemgenerated electromagnetic pulse (SGEMP) simulation code LASER-SGEMP. Here, the grid convergence index method estimates the discretization error caused by the finite difference, while the bootstrap sampling demonstrates the sampling error arising from the inherent stochastic noise. The electric field waveforms obtained by the LASER-SGEMP code agree well with the high precision solution from a reference. According to the StREEQ method, the convergence value of the electric field at 1.8 ns is calculated to be 0.955 V/m and the numerical error characterized by the 95% confidence interval is [0.948, 0.964] V/m, while the high precision solution is 0.961 V/m. Therefore, the code verification has supported the accuracy of the LASER-SGEMP code, and the StREEQ technique is proved to be capable of estimating the numerical error of the cavity SGEMP simulation.

An investigation of scale effects on marine propeller under cavitating and non-cavitating conditions

ABSTRACT In this study, scale effects on the hydrodynamic performance of DTMB 4119 propeller have been investigated in uniform flow under non-cavitating and cavitating conditions by a simple practical method based on OpenFOAM. A potential based Lifting Surface Method has also been applied. A verification study of non-cavitating simulations has been carried out by Grid Convergence Index method at one scale ratio. Then the results have been validated with experiments. Schnerr-Sauer cavitation model of OpenFOAM has been used for the phase change between vapor and liquid. Cavitation simulations have been performed as a time-dependent with dynamic mesh while non-cavitating simulations have been carried out as a steady-state with static mesh. Effects of the scale ratio and cavitation simulations have been investigated and a simple fitting procedure on the thrust and torque values based on logarithm of Reynolds number have been applied. Results have also been compared with ITTC corrections.

Grid Convergence Index 방법을 통한 프란시스 수차 모델의 격자계 분석

Grid Convergence Index 방법을 통한 프란시스 수차 모델의 격자계 분석

Application of grid convergence index to shock wave validated with LS-DYNA and ProsAir

The discretization error is not always calculated, even though it is essential for the studies of computational solid mechanics. However, it is well known that an error committed by the mesh used can be as large as the measured variable, which greatly invalidates the results obtained. The grid convergence index (GCI) method makes possible to determine on a solid basis, the order of convergence and the asymptotic solution. This method seems to be a suitable estimator despite further research is needed in the context of blast situations and finite element (FE) calculations. For this purpose, field trials were performed consisting in the detonation of a spherical hanging load of homemade explosive. The pressure generated by the shock wave was measured in different positions at two distances. With these data, a TNT equivalent has been obtained and used to calculate the shock propagation with the solvers LS-DYNA and ProsAir. This work aims to verify the GCI method by comparing its results with field data along with the simulations carried out. The comparison also seeks to validate the methodology used to obtain the TNT equivalent.This research shows that the GCI gives good results for both solvers despite the complexity of the physical problem. Besides, LS-DYNA displays better correlation with the experimental data than the ProsAir results, with an error of less than 10% in all values.

Numerical roll damping prediction of a planing hull

ABSTRACT In this study, the roll damping coefficients of a high-speed planing hull are calculated numerically in order to show the effects of forward speed, roll amplitude and oscillation frequency. Forced roll simulations are carried out to calculate roll damping coefficients. The numerical study is performed at high forward speeds by using a commercial URANS (Unsteady Reynolds-averaged Navier–Stokes) solver. The uncertainty analysis of the numerical results is performed using the GCI (Grid Convergence Index) method in terms of grid spacing and time step size. The obtained numerical results are compared with existing experimental data and empirical methods. Results show that lift contribution is the major component of the roll damping at high speeds since the roll damping does not change considerably at different roll amplitudes and frequencies. It is also found that numerical calculations show better accuracy compared to the implemented empirical method.

NUMERICAL AND EXPERIMENTAL STUDY OF A TWO-STAGE SAVONIUS WIND TURBINE

This article presents a numerical and experimental study of vertical axis wind turbine performance comparison involving a two-stage Savonius rotor with similar parameters. The experimental study is conducted in the aerodynamic tunnel at the Fluid Mechanics Laboratory of the Federal University of Rio Grande do Sul. The aerodynamics rotors are manufactured by 3D prototyping technique. Numerical simulations are performed using the Finite Volumes Method performed by the solution of the Reynolds Averaged Navier-Stokes (RANS) and continuity equations using the SST k-ω turbulence model. The numerical domain is modeled in order to maintain the same characteristics of the experimental model. The mesh quality is evaluated through the GCI (Grid Convergence Index) method. The static and dynamic torque coefficients and the power coefficients are compared. The tests are made without blockage corrections due to the small blockage ratio from 7.5%. Results show that the turbine has a positive static torque coefficient for any rotor angles. The dynamic torque reaches the maximum value for a tip speed ratio (λ) of 0.2 for the experimental and numerical cases. The relative difference between the numerical simulations and the experimental results are between 3.8% and 13.4%.

Simplified CFD model of coolant channels typical of a plate-type fuel element: an exhaustive verification of the simulations

The use of parallel plate-type fuel assemblies is common in nuclear research reactors. One of the main problems of this fuel element configuration is the hydroelastic instability of the plates caused by the high flow velocities. The current work is focused on the hydrodynamic characterization of coolant channels typical of a flat-plate fuel element, using a numerical model developed with the commercial code ANSYS CFX. Numerical results are compared to accurate analytical solutions, considering two turbulence models and three different fluid meshes. For this study, the results demonstrated that the most suitable turbulence model is the k-e model. The discretization error is estimated using the Grid Convergence Index method. Despite its simplicity, this model generates precise flow predictions.

Assessment of CFD for unheated gas-liquid flows with high void fraction

Abstract Dispersed gas-liquid two-phase flows with high void fraction are commonly encountered in the nuclear and process industries. Detailed understanding of the fluid flow behavior under these conditions is imperative to reactor safety. Traditional modelling approaches (e.g. system thermalhydraulic codes) often use geometry specific correlations that limit their applicability. Canadian Nuclear Laboratories is systematically developing leading-edge CFD capabilities for applications to various reactor flows, including two-phase flows. In the present study, the existing turbulent momentum closures and bubble breakage/coalescence models were assessed for modeling the turbulent air-water flow in a horizontal pipe with two 90° bends. CFD predictions were compared against data from a new in-house air-water experiment that adopted an advanced wire-mesh sensor for the measurement of the void fraction distributions. An initial precursor CFD analysis for a simple pipe geometry was performed to determine the suitability and relative importance of interphase closures at the high void fraction range (15% ≤ αavg ≤ 30%); then a selected single set of interphase closures and breakage/coalescence kernels were applied to simulate three horizontal bend tests. The recommended procedure by ASME to determine discretization error was demonstrated through the Grid Convergence Index method. Fair agreement with measurements was obtained at the straight pipe section. Simulations qualitatively captured the kidney-bean shaped void distribution at the bend, but the location of the void peak value was offset by a minimum of 90° compared to the CNL measurements.

Optimization of mesh density for numerical simulations of hydrodynamic lubrication considering textured surfaces

In this work, a parametric study of a lubricated textured contact is carried out using a numerical model. The model solves the Reynolds equation coupled with a mass-conserving cavitation algorithm, and addresses the hydrodynamic lubrication of a mechanical seal with textured surfaces. Three shapes of texture (square, spherical cap, and triangle) are used, and the effect of the aspect ratio is also discussed for the case involving a triangular dimple. In order to estimate quantitatively the discretization error, the grid convergence index method is applied, considering several operating parameters (leakage, cavitation, load-carrying capacity) and using three grid levels: coarse (100 × 100), medium (200 × 200), and fine (400 × 400). As could be expected, grid convergence index analysis showed that there was a reduction in the discretization error when the grid system was refined. However, the study also showed that medium grid (200 × 200) solution has a grid convergence index of <6%. Comparison between fine and medium mesh densities, with regard to accuracy and computational time requirements, indicated that a medium mesh density was appropriate in the case of the present program, since the computational time is reduced by a factor of 9 when compared with the highest mesh density. Moreover, the size of the optimal mesh also depends on the operating conditions. The higher the lubrication number, the higher the number of nodes.

Conjugate Heat Transfer from an Impingement and Film-Cooled Flat Plate

The effect of solid thermal conductivity on conjugate heat transfer from a flat plate with combined impingement and film cooling is studied both experimentally and computationally. One side of the plate is exposed to hot mainstream, whereas the other side is impinged with air jets. A flow configuration with multiple staggered rows of cylindrical film holes and a matrix of impingement holes is considered. A thermochromic liquid crystal technique is used in the experiment to measure the surface temperature of the plate. The physical domain is meshed with fine-sized hybrid: hexahedral and tetrahedral grids suitable for the finite volume computations. Reynolds-averaged Navier–Stokes equations are solved with shear-stress transport -omega turbulence modeling. A grid independence study is carried out using a grid convergence index method. A validation exercise of computational results is carried out against the full coverage film cooling data available in literature (“Total Cooling Effectiveness on A Staggered Full-Coverage Film Cooling Plate with Impinging Jet,” ASME-TurboExpo2010, American Soc. of Mechanical Engineers Paper GT2010-23725, Fairfield, NJ, June 2010). Results from the conjugate heat transfer study are presented for three blade materials: A, B, and C with thermal conductivities of 0.2, 1.5, and , respectively, and for three blowing ratios of 0.6, 1.0, and 1.6. Limited by the experimental facility, the mainstream Reynolds number is varied from 49,050 to 130,800; and the coolant jet Reynolds number is kept constant at 825. The computations for the impingement surface reveal multiple peaks and valleys in the heat flux and temperature plots. Material C has experienced significant changes in the values of impingement surface heat flux and temperature with blowing ratio, whereas these changes are only minor for material A. On the interaction surface, convective heat flux values are the lowest for material A and progressively increase with increasing thermal conductivity. However, the effectiveness values vary significantly for material A in the streamwise direction. A good agreement is found between the effectiveness distributions obtained from the measurements and the computations. The Nusselt number variations on the interaction surface are presented by defining a factor that accounts for the direction of heat flux.

Application of grid convergence index in FE computation

Abstract This paper presents an application of the grid convergence index (GCI) concept based on the Richardson extrapolation to a selected simple problem of a cantilever beam loaded with vertical forces at the tip end. The GCI method, popular in computational fluid dynamics, has been recently recommended for finite element (FE) applications in solid and structural mechanics. Based on the results obtained usually for three meshes, the GCI method enables one to determine, in an objective manner, the order of convergence to estimate the asymptotic solution and the bounds for discretization error. The example shows that the characteristics of the convergence depend on the selection of the quantity of interest, which can be local or a global functional such as the deflection considered here. The results differ for different FE formulations, and the difference is bigger when the nonlinearities (e.g., due to plastic response) are taken into account

Isothermal Air-Ingress Validation Experiments

Idaho National Laboratory has conducted air-ingress experiments as part of a campaign to validate computational fluid dynamics (CFD) calculations for very high-temperature gas-cooled reactor (VHTR) analysis. An isothermal test loop was designed to recreate exchange or stratified flow that occurs in the lower plenum of VHTR after a break in the primary loop allows helium to leak out and reactor building air to enter the reactor core. The experiment was designed to measure stratified flow in the inlet pipe connecting to the lower plenum of the General Atomics gas turbine-modular helium reactor (GT-MHR). Instead of helium and air, brine and sucrose were used as heavy fluids, and water was used as the lighter fluid to create, using scaling laws, the appropriate flow characteristics of the lower plenum immediately after depressurization. These results clearly indicate that stratified flow is established even for very small density differences.Corresponding CFD results were validated with the experimental data. A grid sensitivity study on CFD models was also performed using the Richardson extrapolation and the grid convergence index method for the numerical accuracy of CFD calculations. The calculated current speed showed very good agreement with the experimental data, indicating that current CFD methods are suitable for simulating density gradient stratified flow phenomena in an air-ingress accident.

Discussion: “Factors of Safety for Richardson Extrapolation” (Xing, T., and Stern, F., 2010, ASME J. Fluids Eng., 132, p. 061403)

Discussion: “Factors of Safety for Richardson Extrapolation” (Xing, T., and Stern, F., 2010, ASME J. Fluids Eng., 132, p. 061403)

Uncertainty Analysis in the Numerical Simulation of the Air-Water Droplet Motion Through Drift Eliminators

This work presents an uncertainty study in the numerical simulation of the air-water droplet motion through three types of drift eliminators using the grid convergence index (GCI) method. The analysis of independence of the results with respect to the fineness of mesh is developed with special emphasis on the influence of the nondimensional sub-layer scaled distance. The coefficient of pressure drop and the droplet collection efficiency are numerically calculated. It may be concluded that using the GCI method leads to reliable numerical results, but it is also necessary to establish a sufficiently fine mesh near the walls.

Verification of Calculations of the Potential Flow Around Two-Dimensional Foils

An example of verification of calculations using methods based on grid-refinement studies and Richardson extrapolation is presented. The study is performed for numerical simulations of the potential flow around two-dimensional lifting foils by one low-order and two higher-order panel methods based on Morino's perturbation potential formulation. Flows with known analytical solutions have been selected to assess the reliability of the uncertainty estimations. The grid-convergence index method is the basis for these estimations. Two options are compared: estimating the uncertainty of the solution on a specific grid and of the solution found by extrapolation to zero grid cell size