Grid Refinement Research Articles

A local grid refinement scheme for B‐spline material point method

SummaryA local grid refinement scheme for the material point method with B‐spline basis functions (BSMPM) is developed based on the concept of bridging domain approach. The fine grid is defined in the local large‐deformation regions to accurately capture the complex material responses, whereas other spatial domains are discritized by coarse grids. In the overlapping domain between the fine and coarse grids, the constraint of particle displacements obtained with different grids is enforced using the Lagrange multiplier method to eliminate the spurious stress reflection at the fine/coarse grid interface. Representative numerical examples have shown that the BSMPM simulations with the proposed local grid refinement scheme could provide the solutions in good agreement with those obtained with the uniformly fine grid, and that no significant spurious stress reflection is induced at the fine/coarse grid interface, even for the bridging domain size as small as the cell size of the fine grid. It is also found that the proposed local grid refinement method can significantly reduce the BSMPM computational time compared with the cases for uniformly fine grids. A multitime‐step algorithm is presented and shown to considerably enhance the efficiency of the present local grid refinement scheme with no compromise in accuracy.

A Priori Assessment of Subgrid-Scale Models and Numerical Error in Forced Convective Flow at High Prandtl Numbers

An extensive a priori analysis has been carried out on data from Direct numerical simulation of fully developed heated turbulent pipe flow at high molecular Prandtl numbers Pr=10/20, testing three popular modelling candidates for subgrid-scale closure in Large-Eddy simulation (LES). Aside from assessing the models’ capabilities to describe quantitatively the unresolved turbulent fluxes, a special focus is also put on the role of the numerical error, which arises from the discretization of the filtered advective fluxes on a coarse LES grid. The present analysis extends here previous studies on subgrid-scale momentum transport in a isothermal mixing layer and channel flow carried out by Brandt (J Numer Methods Fluids 51: 635–657, 2006) and Vreman et al. (J Eng Math 29: 299–327, 1995), respectively, to the subgrid-scale transport of heat at high Prandtl numbers. The statistical dependence between the individual contributions (resolved, subgrid-scale, numerical discretization error) constituting the filtered advective flux divergence in the LES formulation is investigated as well, in terms of corresponding cross-correlations. The sensitivity of the tested sgs-models to a grid refinement is further examined performing also a posteriori LES, where the basically more sophisticated candidates turn out to be more demanding in terms of required grid resolution.

Wind resource mapping and energy estimation in complex terrain: A framework based on field observations and computational fluid dynamics

The proposed framework produces wind resource maps and estimates the energy production in complex terrain by combining computational fluid dynamics (CFD) with observations from an arbitrary number of masts. It provides a fast and comprehensive solution to the research gap that is to obtain accurate CFD inflow conditions and to mitigate the modeling error by incorporating observations to produce a wind resource map and estimate energy yield. Wind data are processed by direction bins both to initialize simulations and to combine with the latter. The wind resource map is then obtained by superposing the power density maps from each sector. Ultimately, a database containing different turbine models and hub heights is used to filter the best performing cases. Validation at two complex terrain sites reveals that the wind flow model is sufficiently accurate using the proper parameters, and that reasonable inflow conditions and assimilated wind speed fields are achieved. The framework was tested at a complex site in Brazil and was more sensitive to the number of simulated wind directions than to grid refinement. This research provides a comprehensive contribution toward wind resource mapping in complex terrain because it is computationally fast and flexible in the number of masts.

Performance analysis of two vertical-axis hydrokinetic turbines using variational multiscale method

The performance prediction of two counter-rotating vertical axis hydrokinetic turbines is presented in this paper. The flow field is governed by the 3D time-dependent incompressible Navier–Stokes equations. The system of equations is discretized using the Arbitrary Lagrangian-Eulerian Variational Multi-scale formulation for turbulence modeling on moving domains. Sliding interfaces are used to handle the rotor-stator interactions. Weak enforcement of essential boundary conditions is used to relax the requirement of boundary layers resolution. A grid convergence study based on the evaluation of the grid convergence index for the computed torque and the time-averaged axial wake velocity is performed. The grid convergence study shows good convergence with grid refinement in our formulation. Experimental validation of the averaged torque is performed for two different flow conditions and different turbine rotational velocities with good agreement. The finest mesh resolution from the grid convergence study is used for the multiple turbines simulation. The simulation shows almost 25% deficiency in the averaged torque of the downstream turbine. A multi-domain method is introduced to predict the performance of the turbine array at a lower computational cost than the full array simulation. The results from the multi-domain method show good matching on the averaged torque with the full array simulation. The initial study on the effect of struts on the torque generation is presented. The results assure the robustness of the ALE-VMS formulation and how it can be used to simulate multiple turbines at full-scale and full geometric complexity.

Single-Step Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

A stable and non-dissipative kinetic energy and entropy preserving (KEEP) scheme for non-conforming block boundaries on Cartesian grids

A stable and non-dissipative numerical scheme for Cartesian methods is proposed. The proposed scheme is based on the second-order kinetic energy and entropy preserving scheme, which the authors proposed recently, and conservation is satisfied at non-conforming boundaries where the grid refinement level is different across the computational block boundaries. In the proposed scheme, ghost cells are used at computational block boundaries for efficient parallel computation, and conservation is satisfied by assigning appropriate values to the ghost cells. Also, although the five conservative variables (i.e., mass, momentum, and total energy) are typically transferred from computational cells to corresponding ghost cells at computational block boundaries, the proposed scheme transfers two more conservative variables to satisfy conservation at non-conforming block boundaries. In a vortex convection test, the convergence rates of the L2- and L∞-error norms are examined, and the proposed scheme preserves the second-order of accuracy without inducing destructive errors at non-conforming boundaries. In an inviscid Taylor-Green vortex simulation, the proposed scheme demonstrates superior numerical stability by preserving kinetic energy and entropy. Also, the proposed scheme performs more stable computations on a non-conforming computational grid than a typical kinetic energy preserving scheme calculated on a uniform computational grid.

Numerical Modelling of Round Impinging Jet Heat Transfer Using Scale-Resolving-Simulation Methods: Influence of Local Grid Refinement

An accurate prediction of heat transfer is crucial in many engineering applications and a typical representative is heat transfer of impinging jets. The present study addresses numerical simulations using three Scale-Resolving-Simulation methods, namely the Detached Eddy Simulation, Large Eddy Simulation and the Partially Averaged Navier-Stokes. The study focus on the influence of a local grid refinement on predicted heat transfer. The closest match with experimental data was achieved with DES and PANS (fk = 0.5) models for all computational grids. The grid refinement led to an improvement of predicted results for LES and PANS with a high physical resolution (fk = 0.25).

MESH RESOLUTION EFFECTS ON PRIMARY ATOMIZATION SIMULATIONS

In this work, we use direct numerical simulation through a volume of fluid solver with adaptivemesh refinement to analyze the atomization of a pulsating round liquid jet with a narrow length-scale range configuration. We propose three grid sizes based on characteristic scales we estimate from the deformation and fragmentation processes of this problem. We compute drop statistics and general spray features for each case. We found that mesh resolution affects the atomization rate and the probability density function of droplet size and position, not only for the underresolved drops but for all liquid structures. The two simulations with higher grid refinement presented volume-weighted distribution with minor differences. Therefore, we propose assessing the accuracy of atomization simulations based on the volume fraction of underresolved structures.

Classification-Based One-Bit DOA Estimation for Sparse Arrays

Nowadays, high-resolution DOA estimation techniques have been widely used in many fields such as sonar, communication, radio astronomy and biomedicine etc. Unfortunately, conventional algorithms cannot achieve high precision due to the limitation of the array aperture, and usually suffer from high computational complexity. In this paper, we innovatively model the DOA estimation problem of incoherent signals as a binary classification problem and greatly reduce physical complexity by virtue of sparse arrays. We first propose a classification framework for DOA estimation by benefiting from one-bit quantification. In this framework, any classification algorithm can be exploited to estimate DOA, such as logistic regression used in this paper. And then, sparse array is considered to reduce the exceeding number of antennas. Moreover, an iterative grid refinement procedure is presented to achieve more accurate DOA estimation and further reduces the actual number of physical antennas. Simultaneously, we derive Cramer-Rao bound (CRB) for the proposed algorithm. Finally, simulations are conducted for correctness and validation and the results illustrate the significant performance and reduction of complexity in hardware and computation over the existing methods.

Standard and consistent Detached-Eddy Simulation for turbulent engine flow modeling: an application to the TCC-III engine

Multidimensional modeling of Cycle-to-Cycle Variability (CCV) has become a crucial support for the development and optimization of modern direct-injection turbocharged engines. In that sense, the only viable modeling options is represented by scale-resolving approaches such as Large Eddy Simulation (LES) or hybrid URANS/LES methods.Among other hybrid approaches, Detached-Eddy Simulation (DES) has the longest development story and is therefore commonly regarded as the most reliable choice for engineering-grade simulation. As such, in the last decade DESbased methods have found their way through the engine modeling community, showing a good potential in describing turbulence-related CCV in realistic engine configurations and at reasonable computational costs.In the present work we investigate the in-cylinder modeling capabilites of a standard two-equation DES formulation, compared to a more recent one which we call DESx. The DESx form differs from standard DES in the turbulent viscosity switch from URANS to LES-like behavior, which for DESx is fully consistent with Yoshizawa’s one-equation sub-grid scale model. The two formulations are part of a more general Zonal-DES (ZDES) methodology, developed and validated by the authors in a series of previous publications. Both variants are applied to the multi-cycle simulation of the TCC-III experimental engine setup, using sub-optimal grid refinement levels in order to stress the model limitations in URANS-like numerical resolution scenarios. Outcomes from this study show that, although both alternatives are able to ouperform URANS even in coarse grid arrangements, DESx emerges as sligthly superior and thus it can be recommended as the default option for in-cylinder flow simulation.

Numerical study of additional element transport in wire feed laser beam welding

The transport phenomena in the wire feed laser beam welding are investigated numerically. A three-dimensional transient heat transfer and fluid flow model coupled with free surface tracing and element transport is developed. A ray-tracing method with local grid refinement algorithm is used to calculate the multiple reflections and Fresnel absorption on the keyhole wall. The filler material flows backward along the lateral side of the weld pool, and subsequently flows forward along the longitudinal plane. The occurrence of the bulging phenomenon may further prevent the downward transfer of the additional elements to the root of the weld pool.

A Robust Multi Sample Compressive Sensing Technique for DOA Estimation Using Sparse Antenna Array

In this paper, a multi sample compressive sensing (CS) technique is presented for the direction of arrival (DOA) estimation using sparse antenna array that has applications in several fields including radars and sonars. Two different types of sparse antenna arrays are considered. One is linear sparse array for DOA estimation in one dimension and other is L shaped sparse array for DOA estimation in two dimensions. To make the algorithm robust against impulsive and Gaussian noise, a preprocessing stage is introduced. First, in the preprocessing stage median difference correntropy is used that combines median difference and the generalized correntropy. This suppresses the amplitude of impulsive noise. Second, the strength of weighted moving average filter is exploited before applying the CS technique to make the algorithm more robust. In the CS techniques, the source energy is distributed among the adjacent grid due to grid mismatch. Therefore, a fitness function based on the difference of the source energy among the adjacent grid is introduced. This provides the best discretization value through iterative grid refinement for the grid. The effectiveness and robustness of the proposed method is verified through exhaustive simulations for different number of sources and noise scenarios using one dimensional and two-dimensional sparse array structures.

Discrete attachment to a cellulolytic biofilm modeled by an Itô stochastic differential equation.

We propose a mathematical framework for introducing random attachment of bacterial cells in a deterministic continuum model of cellulosic biofilms. The underlying growth model is a highly nonlinear coupled PDE-ODE system. It is regularised and discretised in space. Attachment is described then via an auxiliary stochastic process that induces impulses in the biomass equation. The resulting system is an Itô stochastic differential equation. Unlike the more direct approach of modeling attachment by additive noise, the proposed model preserves non-negativity of solutions. Our numerical simulations are able to reproduce characteristic features of cellulolytic biofilms with cell attachment from the aqueous phase. Grid refinement studies show convergence for the expected values of spatially integrated biomass density and carbon concentration. We also examine the sensitivity of the model with respect to the parameters that control random attachment.

Efficient coupling of direct forcing immersed boundary‐lattice Boltzmann method and finite element method to simulate fluid‐structure interactions

SummaryFluid‐structure interactions (FSI) of rigid and flexible bodies are simulated in this article. For the fluid flow, multidirect forcing immersed boundary method (IBM) is adopted to capture the moving boundary, and lattice Boltzmann method (LBM) is used to evolve the flow field. Compared with our previous no‐penetration IBM, less iterations are required in this work. In addition, larger velocity in lattice units can be used and the nonphysical force oscillations are suppressed due to the C3 6‐point kernel. Multi‐relaxation‐time collision operator and local grid refinement are also adopted in LBM to enhance the numerical stability and efficiency. To improve the efficiency of the FSI coupling algorithm, the mesh of the deformable structure can be coarser than the Lagrangian mesh using Newton‐Cotes formulas to integrate the traction on the structure surface. A variety of benchmarks, including flow around a circular cylinder with Reynold number ranging from 20 to 200, forced oscillation of a circular cylinder, vortex‐induced vibration (VIV) of an elastically mounted circular cylinder in two degrees of freedom, and VIV of an elastic cantilever beam attached to a circular cylinder, are carried out to evaluate the accuracy and stability of the present coupling algorithm. For the benchmark FSI problem considered in this article, a reduction of 54% of the calculation time is achieved using coarser structure mesh. As an application of the FSI coupling algorithm, the mechanism of an elastic beam in the wake of a circular cylinder is discussed.

Circulating fluidized bed reactors – part 01: analyzing the effect of particle modelling parameters in computational particle fluid dynamic (CPFD) simulation with experimental validation

A CPFD hydrodynamic model was developed for a circulating fluidized bed system and the simulation results were validated against experimental data based on particle circulation rate. Sensitivity of the computational mesh was primarily tested and extended grid refinement was needed at the loopseal to match the particle circulation rate with experimental data. The particle circulation rate was independent of the range of number of computational particles used in this study. A 10% reduction of the particle circulation rate was observed as the particle-wall interaction parameter was changed from 0.85 to 0.55 and 17% increment when the close-packed volume fraction was changed from 0.56 to 0.62. The pressure constant in the particle stress model showed the greatest impact for the circulation rate with 57% increment as the constant was changed from 2.5 to 5. The highest absolute variation in the pressure was observed at the loop seal and pressure values were under predicted in all sections. Highlights CPFD simulations are efficient in analyzing fluidized bed systems. Manipulating of particle circulation rate is important in circulating fluidized bed. Pressure constant in particle stress model is the most influential factor. Uncertainties should be minimized prior to optimization of model parameters.

Investigation of LES models for a stratified shear layer

The performance of three large-eddy simulation (LES) models in predicting the evolution of a shear layer at moderate Reynolds number in a linearly stratified background is investigated. Results of the Dynamic Smagorinsky, Ducros, and Wall-Adapting Local Eddy Viscosity (WALE) models are compared against those of direct numerical simulation (DNS). Two levels of grid refinement are employed to assess the change in the models’ capabilities with varying resolution. Of particular interest is the ability of the models to capture the evolution of instabilities as well as accurately quantify turbulence statistics. Evolution of momentum thickness, local buoyancy flux, shear, and gradient Richardson number show good agreement of the LES models with the DNS. A comparison of the turbulent kinetic energy (TKE) and its budget indicates good capture of turbulence evolution in the LES models. There is a moderate over prediction of spatially integrated TKE during a short period when the integrated TKE is at its maximum. This feature is traced to the underestimate of turbulent dissipation rate in the LES. Coarsening grid resolution significantly increases the discrepancy in dissipation between the WALE model and the DNS while defects due to the coarser grid resolution are mild in the case of the Ducros model. All of the LES models overestimate the peak values of eddy viscosity and eddy diffusivity although the Ducros model produces the closest agreement with the DNS. Furthermore, the Ducros model is found to require fewer CPU hours than the DNS or other LES models.

Thermal performance investigation in a novel corrugated plate heat exchanger

Compact heat exchangers have become an essential necessity for power production and multi other purposes on a daily basis. The corrugated plate heat exchangers (CPHEs) are well-known for their high thermal performance. This study proposes a unique CPHE with a simple modification that can boost its thermal performance significantly. The overall tests have been conducted on four CPHEs for two symmetric chevron angles (β) of 30°/30° and 60°/60° Two CPHEs belong to the newly CPHEs, and the other two belong to the well-known basic CPHE. Data are obtained for steady-state, single-phase (water-water), counter-current arrangements, and for Reynolds number (Re) ranges from 500 to 2500. Sophisticated mesh techniques have been adopted to develop the mesh for the plates and the fluids between the plates. An appropriate grid refinement test has been carried out for the accuracy of the numerical results. The results have been validated with benchmark experimental and numerical data. A realizable k−ε turbulence model with scalable wall treatment found to provide the most consistent and accurate prediction of the thermal performance of CPHE. The numerical results showed that the Nusselt number (Nu) and the effectiveness (ϵ) of the newly developed CPHEs are much higher than that of the basic one, which can be very useful when a heavy heat duty is required. The enhancement for Nu is up to 75% and for ϵ is up to 42%, and generally both exhibit a direct proportional relationship with Re. Based on the numerical result, a new correlation to predict Nu has been developed.

An adaptive WENO algorithm for one-dimensional convection-dominated partial differential equations

In this work, a versatile numerical algorithm was proposed for solving one-dimensional nonlinear convection-dominated partial differential equations as these found in the simulation of diverse chemical engineering applications. The proposed algorithm uses a fully adaptive scheme, in which both the grid spacing and the time discretization are dynamically adjusted. It uses a finite volume discretization with a variable number of grid cells and an explicit time integration with time-step control. The high-order Weighted Essentially Non-Oscillatory (WENO) scheme on non-uniform grid was combined with a grid refinement technique based on the equitable distribution principle and a spatial regularization procedure. The time-stepping procedure as well as some implementation issues are minutely discussed. The capability and efficiency of the new algorithm was demonstrated through the numerical simulation of five chemical engineering applications: the Viscous Burgers Equation, a Chromatographic column, the Buckley-Leverett Equation, and two examples of reaction-convection systems. For all analyzed cases, it was verified that the number of grid cells required to capture steep gradients can be greatly reduced with the proposed grid adaptation scheme.

Assessment of RANS turbulence models and Zwart cavitation model empirical coefficients for the simulation of unsteady cloud cavitation

The numerical simulation of unsteady cavitation flows is sensitive to the selected models and associated parameters. Consequently, three Reynolds Average Navier-Stokes (RANS) turbulence models and the Zwart cavitation model were selected to assess their performance for the simulation of cloud cavitation on 2D hydrofoils. The experimental cavitation tests from a NACA65012 hydrofoil at different hydrodynamic conditions were used as a reference to tune the modeling parameters and the experimental tests from a NACA0015 were finally used to validate them. The effects of near wall grid refinement, time step, iterations and mesh elements were also investigated. The results indicate that the Shear Stress Transport (SST) model is sensitive to near wall grid resolution which should be fine enough. Moreover, the cavitation morphology and dynamic behavior are sensitive to the selection of the Zwart empirical vaporization, Fv , and condensation, Fc , coefficients. Therefore, a multiple linear regression approach with the single objective of predicting the shedding frequency was carried out that permitted to find the range of coefficient values giving the most accurate results. In addition, it was observed that they provided a better prediction of the vapor volume fraction and of the instantaneous pressure pulse generated by the main cloud cavity collapse.

Numerical analysis on the effects of a submerged bottom-mounted barrier in the head-on collision of two solitary waves

The collision of solitary waves is closely related to many physical problems. In this paper, the effects of a submerged bottom-mounted barrier in the head-on collision of two solitary waves are analyzed using a self-developed two-dimensional two-phase flow model. Advanced Adaptive Mesh Refinement (AMR) grid is adopted to reduce the computational cost. The free surface is captured with the Volume of Fraction (VOF) method combined with Piecewise Linear Interface Calculation (PLIC). The Immersed Boundary (IB) method is used to account for the existence of the barrier. Several benchmark problems including solitary wave propagation, head-on collision of two solitary waves, and solitary wave past a submerged barrier are considered to validate the accuracy of the model. Then the model is utilized to simulate the head-on collision of two solitary waves with the existence of a submerged bottom-mounted barrier. Both the symmetric (two waves with equal amplitudes) and asymmetric (two waves with unequal amplitudes) head-on collisions are investigated. The results with and without the barrier are compared and analyzed in detail. Special attention is paid to the variations of the wave amplitudes and the vortex sheddings.