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At least 1,153 records · Page 64

The design and implementation of a parallel unstructured Euler solver using software primitives

This paper is concerned with the implementation of a 3D unstructured-grid Euler-solver on massively parallel distributed-memory computer architectures. The goal is to minimize solution time by achieving high computational rates with a numerically efficient algorithm. An unstructured multigrid algorithm with an edge-based data-structure has been adopted, and a number of optimizations have been devised and implemented in order to accelerate the parallel computational rates. The implementation is carried out by creating a set of software tools, which ease the implementation of computational problems on parallel architecture machines by relieving the user of the low-level machine specific issues. The quantitative effect of the various optimizations are demonstrated, and we show that the combined effect of these optimizations leads to roughly a factor of three performance improvement. The overall solution efficiency is compared with that obtained on the CRAY-YMP vector supercomputer.

Das, R.↗

Juncture flow improvement for wing/pylon configurations by using CFD methodology

Transonic flow field around a fighter wing/pylon configuration was simulated by using an implicit upwinding Navier-Stokes flow solver (F3D) and overset grid technology (Chimera). Flow separation and local shocks near the wing/pylon junction were observed in flight and predicted by numerical calculations. A new pylon/fairing shape was proposed to improve the flow quality. Based on numerical results, the size of separation area is significantly reduced and the onset of separation is delayed farther downstream. A smoother pressure gradient is also obtained near the junction area. This paper demonstrates that computational fluid dynamics (CFD) methodology can be used as a practical tool for aircraft design.

Gea, Lie-Mine↗

Nonreflecting boundary conditions for linearized unsteady aerodynamic calculations

The present method for the implementation of nonreflecting boundary conditions in 2D and 3D linearized unsteady flow computations is applied to cases of unsteady flows in turbomachine blade rows. The eigenmodes of a discrete representation of the governing equations are computed and used to construct nonreflecting boundary conditions. In 3D, a mixed numerical method is used; in 2D, the discrete representation of the governing equations is obtained from the discretized equations used by the flow solver itself. Wavenumbers and radial mode shapes are computed.

Hall, Kenneth C.↗

Design and implementation of a parallel unstructured Euler solver using software primitives

This paper is concerned with the implementation of a three-dimensional unstructured-grid Euler solver on massively parallel distributed-memory computer architectures. The goal is to minimize solution time by achieving high computational rates with a numerically efficient algorithm. An unstructured multigrid algorithm with an edge-based data structure has been adopted, and a number of optimizations have been devised and implemented to accelerate the parallel computational rates. The implementation is carried out by creating a set of software tools, which provide an interface between the parallelization issues and the sequential code, while providing a basis for future automatic run-time compilation support. Large practical unstructured grid problems are solved on the Intel iPSC/860 hypercube and Intel Touchstone Delta machine. The quantitative effects of the various optimizations are demonstrated, and we show that the combined effect of these optimizations leads to roughly a factor of 3 performance improvement. The overall solution efficiency is compared with that obtained on the Cray Y-MP vector supercomputer.

Das, R.↗

Development of Multiobjective Optimization Techniques for Sonic Boom Minimization

A discrete, semi-analytical sensitivity analysis procedure has been developed for calculating aerodynamic design sensitivities. The sensitivities of the flow variables and the grid coordinates are numerically calculated using direct differentiation of the respective discretized governing equations. The sensitivity analysis techniques are adapted within a parabolized Navier Stokes equations solver. Aerodynamic design sensitivities for high speed wing-body configurations are calculated using the semi-analytical sensitivity analysis procedures. Representative results obtained compare well with those obtained using the finite difference approach and establish the computational efficiency and accuracy of the semi-analytical procedures. Multidisciplinary design optimization procedures have been developed for aerospace applications namely, gas turbine blades and high speed wing-body configurations. In complex applications, the coupled optimization problems are decomposed into sublevels using multilevel decomposition techniques. In cases with multiple objective functions, formal multiobjective formulation such as the Kreisselmeier-Steinhauser function approach and the modified global criteria approach have been used. Nonlinear programming techniques for continuous design variables and a hybrid optimization technique, based on a simulated annealing algorithm, for discrete design variables have been used for solving the optimization problems. The optimization procedure for gas turbine blades improves the aerodynamic and heat transfer characteristics of the blades. The two-dimensional, blade-to-blade aerodynamic analysis is performed using a panel code. The blade heat transfer analysis is performed using an in-house developed finite element procedure. The optimization procedure yields blade shapes with significantly improved velocity and temperature distributions. The multidisciplinary design optimization procedures for high speed wing-body configurations simultaneously improve the aerodynamic, the sonic boom and the structural characteristics of the aircraft. The flow solution is obtained using a comprehensive parabolized Navier Stokes solver. Sonic boom analysis is performed using an extrapolation procedure. The aircraft wing load carrying member is modeled as either an isotropic or a composite box beam. The isotropic box beam is analyzed using thin wall theory. The composite box beam is analyzed using a finite element procedure. The developed optimization procedures yield significant improvements in all the performance criteria and provide interesting design trade-offs. The semi-analytical sensitivity analysis techniques offer significant computational savings and allow the use of comprehensive analysis procedures within design optimization studies.

Chattopadhyay, Aditi↗

Re-evaluation of an Optimized Second Order Backward Difference (BDF2OPT) Scheme for Unsteady Flow Applications

Recent experience in the application of an optimized, second-order, backward-difference (BDF2OPT) temporal scheme is reported. The primary focus of the work is on obtaining accurate solutions of the unsteady Reynolds-averaged Navier-Stokes equations over long periods of time for aerodynamic problems of interest. The baseline flow solver under consideration uses a particular BDF2OPT temporal scheme with a dual-time-stepping algorithm for advancing the flow solutions in time. Numerical difficulties are encountered with this scheme when the flow code is run for a large number of time steps, a behavior not seen with the standard second-order, backward-difference, temporal scheme. Based on a stability analysis, slight modifications to the BDF2OPT scheme are suggested. The performance and accuracy of this modified scheme is assessed by comparing the computational results with other numerical schemes and experimental data.

Vatsa, Veer N.↗

Microwave Cavity Simulation Using Ansys HFSS

The design of microwave cavity detectors for axion dark matter research is often accomplished using advanced full-wave electromagnetic simulation software tools. These tools provide a cost-effective approach to evaluate a wide variety of cavity configurations, frequency tuning mechanisms, and conductive or dielectric materials and coatings. One simulation software package used for this application is Ansys High Frequency Structure Simulator (HFSS), which is based upon the well-established finite element method. HFSS includes numerous features useful for microwave cavity design such as parametric geometry modeling, adaptive meshing algorithm, curvilinear mesh elements, driven modal and eigenmode matrix solvers, and optimization algorithms. This paper describes the use of the HFSS software to simulate microwave cavities for axion haloscope detectors, with an example tutorial for a cylindrical cavity. Excellent agreement between the simulated and analytical results is shown for the resonant frequency, quality factor, and form factor.

Ansys HFSS, Cavity simulation, finite element mode↗

A numerical investigation of turbulent baseflow

The turbulent base flow of a cone with a sting in hypersonic flow is investigated using a time-dependent Navier-Stokes code called UWIN. A PNS solver provides the upstream flow conditions which drive the development of the base flow. The computational method is described along with its application to the base flow problem. The grid generation and boundary conditions are described. The results and the summary conclude this note.

Chaussee, Denny S.↗

Numerical study to assess sulfur hexafluoride as a medium for testing multielement airfoils

A methodology is described for computing viscous flows of air and sulfur hexafluoride (SF6). The basis is an existing flow solver that calculates turbulent flows in two dimensions on unstructured triangular meshes. The solver has been modified to incorporate the thermodynamic model for SF6 and used to calculate the viscous flow over two multielement airfoils that have been tested in a wind tunnel with air as the test medium. Flows of both air and SF6 at a free-stream Mach number of 0.2 and a Reynolds number of 9 x 10(exp 6) are computed for a range of angles of attack corresponding to the wind-tunnel test. The computations are used to investigate the suitability of SF6 as a test medium in wind tunnels and are a follow-on to previous computations for single-element airfoils. Surface-pressure, lift, and drag coefficients are compared with experimental data. The effects of heavy gas on the details of the flow are investigated based on computed boundary-layer and skin-friction data. In general, the predictions in SF6 vary little from those in air. Within the limitations of the computational method, the results presented are sufficiently encouraging to warrant further experiments.

Bonhaus, Daryl L.↗

Numerical Prediction of Radiation Measurements Taken in the X2 Facility for Mars and Titan Gas Mixtures

Thermochemical relaxation behind a normal shock in Mars and Titan gas mixtures is simulated using a CFD solver, DPLR, for a hemisphere of 1 m radius; the thermochemical relaxation along the stagnation streamline is considered equivalent to the flow behind a normal shock. Flow simulations are performed for a Titan gas mixture (98% N2, 2% CH4 by volume) for shock speeds of 5.7 and 7.6 km/s and pressures ranging from 20 to 1000 Pa, and a Mars gas mixture (96% CO2, and 4% N2 by volume) for a shock speed of 8.6 km/s and freestream pressure of 13 Pa. For each case, the temperatures and number densities of chemical species obtained from the CFD flow predictions are used as an input to a line-by-line radiation code, NEQAIR. The NEQAIR code is then used to compute the spatial distribution of volumetric radiance starting from the shock front to the point where thermochemical equilibrium is nominally established. Computations of volumetric spectral radiance assume Boltzmann distributions over radiatively linked electronic states of atoms and molecules. The results of these simulations are compared against experimental data acquired in the X2 facility at the University of Queensland, Australia. The experimental measurements were taken over a spectral range of 310-450 nm where the dominant contributor to radiation is the CN violet band system. In almost all cases, the present approach of computing the spatial variation of post-shock volumetric radiance by applying NEQAIR along a stagnation line computed using a high-fidelity flow solver with good spatial resolution of the relaxation zone is shown to replicate trends in measured relaxation of radiance for both Mars and Titan gas mixtures.

Palmer, Grant↗

Aeroacoustic Simulations of a Nose Landing Gear Using FUN3D on Pointwise Unstructured Grids

Numerical simulations have been performed for a partially-dressed, cavity-closed (PDCC) nose landing gear configuration that was tested in the University of Florida's open-jet acoustic facility known as the UFAFF. The unstructured-grid flow solver FUN3D is used to compute the unsteady flow field for this configuration. Mixed-element grids generated using the Pointwise(TradeMark) grid generation software are used for these simulations. Particular care is taken to ensure quality cells and proper resolution in critical areas of interest in an effort to minimize errors introduced by numerical artifacts. A hybrid Reynolds-averaged Navier-Stokes/large eddy simulation (RANS/LES) turbulence model is used for these simulations. Solutions are also presented for a wall function model coupled to the standard turbulence model. Time-averaged and instantaneous solutions obtained on these Pointwise grids are compared with the measured data and previous numerical solutions. The resulting CFD solutions are used as input to a Ffowcs Williams-Hawkings noise propagation code to compute the farfield noise levels in the flyover and sideline directions. The computed noise levels compare well with previous CFD solutions and experimental data.

Vatsa, Veer N.↗

A comparative study of diffraction of shallow-water waves by high-level IGN and GN equations

This work is on the nonlinear diffraction analysis of shallow-water waves, impinging on submerged obstacles, by two related theories, namely the classical Green–Naghdi (GN) equations and the Irrotational Green–Naghdi (IGN) equations, both sets of equations being at high levels and derived for incompressible and inviscid flows. Recently, the high-level Green–Naghdi equations have been applied to some wave transformation problems. The high-level IGN equations have also been used in the last decade to study certain wave propagation problems. However, past works on these theories used different numerical methods to solve these nonlinear and unsteady sets of differential equations and at different levels. Moreover, different physical problems have been solved in the past. Therefore, it has not been possible to understand the differences produced by these two sets of theories and their range of applicability so far. We are thus motivated to make a direct comparison of the results produced by these theories by use of the same numerical method to solve physically the same wave diffraction problems. We focus on comparing these two theories by using similar codes; only the equations used are different but other parts of the codes, such as the wave-maker, damping zone, discretion method, matrix solver, etc., are exactly the same. This way, we eliminate many potential sources of differences that could be produced by the solution of different equations. The physical problems include the presence of various submerged obstacles that can be used for example as breakwaters or to represent the continental shelf. A numerical wave tank is created by placing a wavemaker on one end and a wave absorbing beach on the other. The nonlinear and unsteady sets of differential equations are solved by the finite-difference method. The results are compared with different equations as well as with the available experimental data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

OpenMP and MPI implementations of an elasto-viscoplastic fast Fourier transform-based micromechanical solver for fast crystal plasticity modeling

Here, we explore several parallel implementations of an elasto-viscoplastic fast Fourier transform (EVPFFT) model using Message Passing Interface (MPI), OpenMP, and a hybrid of MPI and OpenMP to efficiently predict micromechanical response of polycrystals. Performance studies using EVPFFT are performed based on domain decomposition over voxels of a periodic cell, which is a representative volume element (RVE) of polycrystalline copper. We begin by parallelizing the computationally intensive Newton–Raphson (NR) single crystal solver within EVPFFT. Next, we compare the performance of the serial and parallel FFTW (Fastest Fourier Transform in the West) using OpenMP (OpenMP-FFTW) and MPI (MPI-FFTW) with the original Numerical Recipes-based FOURN routine within EVPFFT. In the parallel environment, we find that the FFT calculations are best performed using the MPI version of FFTW. Finally, the remainder of the code, except read/write subroutines, is parallelized. Significant speedups of the original EVPFFT model are achieved using MPI on shared memory multicore workstations. Furthermore, results achieved on a distributed memory Cray supercomputer show promising strong and weak scalability and in some cases even super scalability for the single crystal NR solver in EVPFFT. MPI-FFTW also scales perfectly for microstructure RVEs larger than 64 3 FFT voxels. For example, the MPI-EVPFFT parallel version of the code accelerates the simulations for approximately two orders of magnitude using 64 cores over the old serial code for an RVE size of 128 3 . Finally, the parallel EVPFFT code developed in this work can run massive voxel-based microstructural RVEs taking the advantages of thousands of logical cores provided by more advanced clusters.

42 ENGINEERING↗

Comparison of High-Order and Low-Order Methods for Large-Eddy Simulation of a Compressible Shear Layer

The objective of this work is to compare a high-order solver with a low-order solver for performing Large-Eddy Simulations (LES) of a compressible mixing layer. The high-order method is the Wave-Resolving LES (WRLES) solver employing a Dispersion Relation Preserving (DRP) scheme. The low-order solver is the Wind-US code, which employs the second-order Roe Physical scheme. Both solvers are used to perform LES of the turbulent mixing between two supersonic streams at a convective Mach number of 0.46. The high-order and low-order methods are evaluated at two different levels of grid resolution. For a fine grid resolution, the low-order method produces a very similar solution to the highorder method. At this fine resolution the effects of numerical scheme, subgrid scale modeling, and filtering were found to be negligible. Both methods predict turbulent stresses that are in reasonable agreement with experimental data. However, when the grid resolution is coarsened, the difference between the two solvers becomes apparent. The low-order method deviates from experimental results when the resolution is no longer adequate. The high-order DRP solution shows minimal grid dependence. The effects of subgrid scale modeling and spatial filtering were found to be negligible at both resolutions. For the high-order solver on the fine mesh, a parametric study of the spanwise width was conducted to determine its effect on solution accuracy. An insufficient spanwise width was found to impose an artificial spanwise mode and limit the resolved spanwise modes. We estimate that the spanwise depth needs to be 2.5 times larger than the largest coherent structures to capture the largest spanwise mode and accurately predict turbulent mixing.

large eddy simulations↗

Single-Bubble Flow Boiling Phenomena - Interface Tracking Simulation

Boiling, as one of the most efficient heat transfer mechanisms, is widely used in various engineering systems. Better understanding and modeling of this process remains a major challenge in multiphase flow research. The distribution of vapor in boiling system affects the heat transfer rate and may cause unfavorable conditions, such as heater burn-out. A number of flow regimes have been observed experimentally. In general, there are three regimes under boiling conditions: nucleate boiling, transition boiling, and film boiling. The nucleate boiling regime is categorized into partial boiling regime and fully developed nucleate boiling, according to behavior of bubble dynamics and heat transfer characteristics. In the first region, as the wall temperature increases, the fraction of the wall surface subject to nucleate boiling increases until bubble formation occupies the entire heated surface. Heat transfer in this region is a complex mixture of single-phase forced convection and nucleate boiling. As bubble density increases, bubbles turn to coalesce and form insulating vapor patches, which impedes liquid flow back to the surface. The heat transfer rate under this condition reduces dramatically. This point is called Critical Heat Flux (CHF), which leads to boiling crisis. The behavior of boiling crisis is dependent on local fluid conditions due to bubble growth and departure. Experiments are the ultimate tool for boiling model development and verification, but they still lack some capabilities needed for better understanding of the local mechanism of this phenomena. It's difficult to capture many important details at the pressure/temperature conditions of interest using experimental techniques, such as full three dimensional measurements of velocity and temperature distribution and estimates of heat flux partitioning during the boiling process. The interface tracking simulation (ITS) is one of the promising approaches to describe heat transfer of boiling phenomena and their underlying mechanisms. The advances of high performance computing (HPC) in recent years made it possible to apply DNS to a wide variety of adiabatic flows. Some work has been done to implement evaporation / condensation models as well. Lee and Nydahl performed a numerical simulation of bubble growth and departure assuming that the bubble has the shape of a truncated sphere and a micro-layer exists during bubble growth. A 2-D cylindrical bubbles simulation has been done by Son with a simplified model to predict the heat flux in a thin liquid micro-layer. A direct simulation of 2D film boiling is conducted by Juric and Tryggvason. The film boiling, below critical pressure with a modified Level Set Method, is studied by Son and Dhir. Ose and Kunugi have developed a boiling and condensation model on subcooled boiling phenomena. Fuchs performed numerical simulations of bubble growth and departure considering wall conduction effects. Fully transient heat and fluid flow is modeled with a free surface of a rising bubble and a periodic calculation of the whole cycling of a growing, detaching, and rising bubble. Mukherjee and Kandikar have done a numerical simulation of an evaporating meniscus on a moving heated surface with level set method. The heat transfer associated with the advancing and receding contact angle has been studied. To develop the capability of physics-based boiling process, in the presented work the energy and momentum equations are coupled to simulate single-bubble growth scenarios. For a single bubble simulation in infinite liquid, the analytical solution exists and can be compared with the numerical results for verification purposes. The presented capabilities will be utilized in the large scale DNS of boiling flows in the future. The finite-element based code, PHASTA is used for the simulation. It is a parallel, hierarchic, higher-order accurate, adaptive, stabilized, transient analysis flow solver, which has been shown to be an effective tool for a wide range of single and two-phase flow problems. It has been shown to be highly scalable on top supercomputers ({approx}85% scaling on up to 3 million mesh partitions using a 92 billion element mesh . PHASTA has been shown to reliably predict the details of adiabatic bubbly flows as well as single-phase flows with heat transfer. To develop the capability of physics-based boiling process, in the presented work the energy and momentum equations are coupled to simulate a single-bubble growth scenario. For a single bubble growth in superheated water, the analytical solution exists and can be compared with the numerical results for verification purposes. The second-order accurate spatial and time discretization is used for the flow solver in the presented work. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Efficient Parallel Kernel Solvers for Computational Fluid Dynamics Applications

Distributed-memory parallel computers dominate today's parallel computing arena. These machines, such as Intel Paragon, IBM SP2, and Cray Origin2OO, have successfully delivered high performance computing power for solving some of the so-called "grand-challenge" problems. Despite initial success, parallel machines have not been widely accepted in production engineering environments due to the complexity of parallel programming. On a parallel computing system, a task has to be partitioned and distributed appropriately among processors to reduce communication cost and to attain load balance. More importantly, even with careful partitioning and mapping, the performance of an algorithm may still be unsatisfactory, since conventional sequential algorithms may be serial in nature and may not be implemented efficiently on parallel machines. In many cases, new algorithms have to be introduced to increase parallel performance. In order to achieve optimal performance, in addition to partitioning and mapping, a careful performance study should be conducted for a given application to find a good algorithm-machine combination. This process, however, is usually painful and elusive. The goal of this project is to design and develop efficient parallel algorithms for highly accurate Computational Fluid Dynamics (CFD) simulations and other engineering applications. The work plan is 1) developing highly accurate parallel numerical algorithms, 2) conduct preliminary testing to verify the effectiveness and potential of these algorithms, 3) incorporate newly developed algorithms into actual simulation packages. The work plan has well achieved. Two highly accurate, efficient Poisson solvers have been developed and tested based on two different approaches: (1) Adopting a mathematical geometry which has a better capacity to describe the fluid, (2) Using compact scheme to gain high order accuracy in numerical discretization. The previously developed Parallel Diagonal Dominant (PDD) algorithm and Reduced Parallel Diagonal Dominant (RPDD) algorithm have been carefully studied on different parallel platforms for different applications, and a NASA simulation code developed by Man M. Rai and his colleagues has been parallelized and implemented based on data dependency analysis. These achievements are addressed in detail in the paper.

Sun, Xian-He↗

Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems

Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.

Preconditioner, Nonlinear, Porous media, Multiphas↗

An Eulerian/Lagrangian coupling procedure for three-dimensional vortical flows

A coupled Eulerian/Lagrangian method is presented for the reduction of numerical diffusion observed in solutions of 3D vortical flows using standard Eulerian finite-volume time-marching procedures. A Lagrangian particle tracking method, added to the Eulerian time-marching procedure, provides a correction of the Eulerian solution. In turn, the Eulerian solution is used to integrate the Lagrangian state-vector along the particles trajectories. While the Eulerian solution ensures the conservation of mass and sets the pressure field, the particle markers describe accurately the convection properties and enhance the vorticity and entropy capturing capabilities of the Eulerian solver. The Eulerian/Lagrangian coupling strategies are discussed and the combined scheme is tested on a constant stagnation pressure flow in a 90 deg bend and on a swirling pipe flow. As the numerical diffusion is reduced when using the Lagrangian correction, a vorticity gradient augmentation is identified as a basic problem of this inviscid calculation.

Felici, Helene M.↗