Spurious Vorticity Research Articles

Spurious vorticity in Eulerian and Lagrangian methods

Lagrangian methods for computational continuum mechanics, since their inception, traditionally relied on staggered meshes. This feature, while facilitating their robustness and reliability, presented some difficulties. The latter motivated the search for collocated Lagrangian schemes. One of the attempts to develop such a scheme was the CAVEAT method/code. Numerical solutions produced by this method suffered sometimes from large vorticity errors, which could lead to mesh entanglement and premature run termination. The efforts to devise a more robust collocated scheme began to bear fruit a couple of decades later starting from the groundbreaking method GLACE, closely followed by EUCCLHYD and later on by CCH and others.One of the aims of this paper is to present a novel Lagrangian collocated factorizable scheme. The notion of a factorizable method was introduced more than two decades ago within the Eulerian approach. It designates a numerical scheme that reflects/preserves the mixed character of the Euler equations, i.e. does not introduce non-physical coupling between the different factors of the system of equations - advection and acoustics operators.Another aim of this paper is to explore the connection between the factorizability property of a Lagrangian method and whether or not it suffers from spurious vorticity. Several existing schemes are surveyed for this purpose. A conjecture summarizing our findings is formulated.

Some effects of domain size and boundary conditions on the accuracy of airfoil simulations

This paper investigates a specific case of one of the most popular fluid dynamic simulations, the incompressible flow around an airfoil (NACA 0012 here) at a high Reynolds number (6×106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6 \ imes 10^6$$\\end{document}). OpenFOAM software was used to study the effect of domain size and four common choices of boundary conditions on airfoil lift, drag, surface friction, and pressure. We also examine the relation between boundary conditions and the velocity, pressure, and vorticity distributions throughout the domain. In addition to the common boundary conditions, we implement the “point vortex” boundary condition that was introduced many years ago but is now rarely used. We also applied the point vortex condition for the outlet pressure instead of using the traditional Neumann condition. With the airfoil generating significant lift at incidence angles of 5∘,10∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5^\\circ , 10^\\circ$$\\end{document}, and 14∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$14^\\circ$$\\end{document}, we confirm a previous finding that the boundary conditions combine with domain size to produce an induced (pressure) drag. The change in the pressure drag with domain size is significant for the commonly-used boundary conditions but is much smaller for the point vortex alternative. The point vortex boundary condition increases the execution time, but this is more than offset by the reduction in domain size needed to achieve a specified accuracy in the lift and drag. This study also estimates the error in total drag and lift due to domain size and shows it can be almost eliminated using the point vortex boundary condition. We also used the impulse form of the momentum equations to study the relation between drag and lift and spurious vorticity, which is generated as a result of using non-exact boundary conditions. These equations reveal that the spurious vorticity throughout the domain is associated with cancelling circulation around the domain boundaries.

Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method

The present study focuses on the unphysical effects induced by the use of non-uniform grids in the lattice Boltzmann method. In particular, the convection of vortical structures across a grid refinement interface is likely to generate spurious noise that may impact the whole computation domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. The purpose of this article is to identify the issues occurring at the interface and to propose possible solutions yielding significant improvements for aeroacoustic simulations. More specifically, this study highlights the critical involvement of non-physical modes in the generation of spurious vorticity and acoustics. The identification of these modes is made possible thanks to linear stability analyses performed in the fluid core, and non-hydrodynamic sensors specifically developed to systematically emphasize them during a simulation. Investigations seeking pure acoustic waves and sheared flows allow for isolating the contribution of each mode. An important result is that spurious wave generation is intrinsically due to the change in the grid resolution (i.e. aliasing) independently of the details of the grid transition algorithm. Finally, the solution proposed to minimize spurious wave amplitude consists of choosing an appropriate collision model in the fluid core so as to cancel the non-hydrodynamic mode contribution regardless the grid coupling algorithm. Results are validated on a convected vortex and on a turbulent flow around a cylinder where a huge reduction of both spurious noise and vorticity are obtained.

Toward a reduction of mesh imprinting

SUMMARYThis paper is the initial investigation into a new Lagrangian cell‐centered hydrodynamic scheme that is motivated by the desire for an algorithm that resists mesh imprinting and has reduced complexity. Key attributes of the new approach include multidimensional construction, the use of flux‐corrected transport (FCT) to achieve second order accuracy, automatic determination of the mesh motion through vertex fluxes, and vorticity control. Toward this end, vorticity preserving Lax–Wendroff (VPLW) type schemes for the two‐dimensional acoustic equations were analyzed and then implemented in a FCT algorithm. Here, mesh imprinting takes the form of anisotropic dispersion relationships. If the stencil for the LW methods is limited to nine points, four free parameters exist. Two parameters were fixed by insisting that no spurious vorticity be created. Dispersion analysis was used to understand how the remaining two parameters could be chosen to increase isotropy. This led to new VPLW schemes that suffer less mesh imprinting than the rotated Richtmyer method. A multidimensional, vorticity preserving FCT implementation was then sought using the most promising VPLW scheme to address the problem of spurious extrema. A well‐behaved first order scheme and a new flux limiter were devised in the process. The flux limiter is unique in that it acts on temporal changes and does not place a priori bounds on the solution. Numerical results have demonstrated that the vorticity preserving FCT scheme has comparable performance to an unsplit MUSCL‐H algorithm at high Courant numbers but with reduced mesh imprinting and superior symmetry preservation. Copyright © 2014 John Wiley & Sons, Ltd.

A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere

A well-balanced discontinuous Galerkin (DG) flux-form shallow-water (SW) model on the sphere is developed and compared with a nodal DG SW model cast in the vector-invariant form for accuracy and conservation properties. A second-order diffusion scheme based on the local discontinuous Galerkin (LDG) method is added to the viscous version of the SW model and tested for conservation behaviors. The inviscid flux-form SW model is found to have better conservation of total energy and zonal angular momentum while the vector-invariant form provides better ability of conserving potential enstrophy. The inviscid flux-form tends to generate spurious vorticity but the LDG scheme combined with a well-balanced treatment can effectively eliminate the small-scale noise and generate smooth and accurate results.

Smoothed-particle-hydrodynamics modeling of dissipation mechanisms in gravity waves

The smoothed-particle-hydrodynamics (SPH) method has been used to study the evolution of free-surface Newtonian viscous flows specifically focusing on dissipation mechanisms in gravity waves. The numerical results have been compared with an analytical solution of the linearized Navier-Stokes equations for Reynolds numbers in the range 50-5000. We found that a correct choice of the number of neighboring particles is of fundamental importance in order to obtain convergence towards the analytical solution. This number has to increase with higher Reynolds numbers in order to prevent the onset of spurious vorticity inside the bulk of the fluid, leading to an unphysical overdamping of the wave amplitude. This generation of spurious vorticity strongly depends on the specific kernel function used in the SPH model.

Moving Mesh Finite Element Methods for the Incompressible Navier--Stokes Equations

This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier--Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergence-free for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. Li, T. Tang, and P. W. Zhang, J. Comput. Phys., 170 (2001), pp. 562--588], which decouples the PDE solver and the mesh-moving algorithm. This approach requires interpolating the solution on the newly generated mesh. Designing a divergence-free-preserving interpolation algorithm is the first goal of this work. Selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations, which is the second goal of this study. The performance of the moving mesh scheme is tested on the standard periodic double shear layer problem. No spurious vorticity patterns appear when even fairly coarse grids are used.

Numerical investigation of transonic limit cycle oscillations of a two-dimensional supercritical wing

CFD-based aeroelastic computations are performed to investigate the effect of nonlinear aerodynamics on transonic limit cycle oscillation (LCO) characteristics of the NLR7301 airfoil section. It is found that the LCO solutions from Navier–Stokes computations deviate less from the experiment than an Euler solution but strongly depend on the employed turbulence model. The Degani–Schiff modification to the Baldwin–Lomax turbulence model provokes spurious vorticity spots causing multiple shocks which might be unphysical, while the Spalart–Allmaras turbulence model yields physically reasonable unsteady shocks. In the cases examined, smaller initial perturbations lead to larger LCO amplitudes and vice versa, in contradiction to what one might expect. The amplitude of the initial perturbation is also found to have an impact on the mean position of LCO. Also addressed in the paper are aspects of multiblock message passing interface (MPI) parallel computation techniques as related to the present problem.

Vortex Methods for Direct Numerical Simulation of Three-Dimensional Bluff Body Flows: Application to the Sphere at Re=300, 500, and 1000

Recent contributions to the 3-D vortex methods are presented. Following Cottet, the particles strength exchange (PSE) scheme for diffusion is modified in the vicinity of solid boundaries to avoid a spurious vorticity flux and to enforce a zero-normal component of vorticity during the convection/PSE step. The vortex sheet algorithm used to enforce the no-slip boundary condition through a vorticity flux at the boundary and the technique used to perform accurate redistributions in the presence of bodies of general geometry are extended from their 2-D counterpart. To perform simulations with nonuniform resolution, a mapping of the redistribution lattice is used. Computational efficiency is attained through the use of parallel tree codes based on multipole expansions of vortex particles and of vortex panels. The method is validated, by comparisons with other authors' results, on the flow past a sphere at Re=300. It is then applied to compute the flow at Re=500 and 1000.

Vortex Methods for High-Resolution Simulations of Viscous Flow Past Bluff Bodies of General Geometry

Recent contributions to the 2-D vortex method are presented. A technique to accurately redistribute particles in the presence of bodies of general geometry is developed. The particle strength exchange (PSE) scheme for diffusion is modified for particles in the vicinity of the solid boundaries to avoid a spurious vorticity flux during the convection/PSE step. The scheme used to enforce the no-slip condition through the vorticity flux at the boundary is modified in a way that is more accurate than in the previous method. Finally, to perform simulations with nonuniform resolution, a mapping of the redistribution lattice is also used. In that case, the PSE is still done in the physical domain, using a symmetrized, conservative scheme. The quadratic convergence of this scheme is proved mathematically, and numerical tests are shown to support the proof. These elements are all validated on the benchmark problem of the flow past an impulsively started cylinder. High-resolution, long-time simulations of the flow past other bluff bodies are also presented: the case of a square and of a capsule at angle of attack.

Behavior of Flow over Step Orography

A two-dimensional nonhydrostatic version of the NCEP regional Eta Model together with analytic theory are used to examine flow over isolated mountains in numerical simulations using a step-terrain vertical coordinate. Linear theory indicates that a singularity arises in the steady flow over the step corners for hydrostatic waves and that this discontinuity is independent of height. Analytic solutions for both hydrostatic and nonhydrostatic waves reveal a complex behavior that varies with both horizontal and vertical resolution. Witch of Agnessi experiments are performed with a 2D version of the Eta Model over a range of mountain half-widths. The simulations reveal that for inviscid flow over a mountain using the step-terrain coordinate, flow will not properly descend along the lee slope. Rather, the flow separates downstream of the mountain and creates a zone of artificially weak flow along the lee slope. This behavior arises due to artificial vorticity production at the corner of each step and can be remedied by altering the finite differencing adjacent to the step to minimize spurious vorticity production. In numerical simulations with the step-terrain coordinate for narrow mountains where nonhydrostatic effects are important, the disturbances that arise at step corners may be of the same horizontal scale as those produced by the overall mountain, and the superposition of these disturbances may reasonably approximate the structure of the continuous mountain wave. For wider mountains, where perturbations are nearly hydrostatic, the disturbances above the step corners have horizontal scales that are much smaller than the overall scale of the mountain and appear as sharp spikes in the flow field. The deviations from the ‘‘classic’’ Witch of Agnesi solution are significant unless the vertical resolution is very small compared to the height of the mountain. In contrast, simulations with the terrain-following vertical coordinate produce accurate solutions provided the vertical grid interval is small compared to the vertical wavelength of the mountain waves (typically at least an order of magnitude larger than the mountain height).

A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry

This work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions. Cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations. Written in Lagrangian form, these equations are discretized utilizing compatible, control volume differencing with a staggered-grid placement of the spatial variables. The concept of “compatibility” means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form. Multiple pressures are utilized in each zone; they produce forces that resist spurious vorticity generation. This difficulty can severely limit the utility of the Lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions. An edge-centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks. The particular difficulty of exactly preserving one-dimensional spherical symmetry in three-dimensional geometry is solved. This problem has both practical and pedagogical significance. The algorithm is suitable for both structured and unstructured grids. Limitations that symmetry preservation imposes on the latter type of grids are delineated.

Elimination of Artificial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pressures

The bane of Lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a Lagrangian zonal mass has ceased to be valid. At short spatial grid scales this is usually referred to by the terms “hourglass” mode or “keystone” motion associated, in particular, with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions, respectively. At longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as “spurious vorticity,” or the long-thin zone problem. In both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution, as would be the case for turbulent flow. In this work we show how such motions can be eliminated by the proper use of subzonal Lagrangian masses, and associated densities and pressures. These subzonal pressures give rise to forces that resist these spurious motions. The pressure is no longer a constant in a zone; it now accurately reflects the density gradients that can occur within a zone due to its differential distortion. Subzonal Lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation. However, these masses arise in a natural way from the intersection of the Lagrangian contours, through which no mass flows, that are associated with both the Lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation. This is an extension of the usual Lagrangian assumption that is often applied to only the zonal mass. We show that with a proper discretization of the subzonal forces resulting from subzonal pressures, hourglass motion and spurious vorticity can be eliminated for a very large range of problems.

A fast, high resolution, second-order central scheme for incompressible flows.

A high resolution, second-order central difference method for incompressible flows is presented. The method is based on a recent second-order extension of the classic Lax-Friedrichs scheme introduced for hyperbolic conservation laws (Nessyahu H. & Tadmor E. (1990) J. Comp. Physics. 87, 408-463; Jiang G.-S. & Tadmor E. (1996) UCLA CAM Report 96-36, SIAM J. Sci. Comput., in press) and augmented by a new discrete Hodge projection. The projection is exact, yet the discrete Laplacian operator retains a compact stencil. The scheme is fast, easy to implement, and readily generalizable. Its performance was tested on the standard periodic double shear-layer problem; no spurious vorticity patterns appear when the flow is underresolved. A short discussion of numerical boundary conditions is also given, along with a numerical example.

Vorticity errors in multidimensional lagrangian codes

We investigate the apparent paradox, as exemplified by the well-known Saltzman test problem, of multidimensional lagrangian codes experiencing mesh tangling when computing one-dimensional irrotational flows. We demonstrate that the cause is the generation of spurious vorticity, or vorticity error, by a nonuniform mesh. Based on this, we investigate two methods of constructing improved lagrangian vertex velocities by removing, or filtering out, this spurious vorticity, rather than by the more common practice of introducing artificial viscosity. The first method reconstructs the velocity from the known flow divergence and from the true vorticity computed by means of a transport equation. The second method, which is much simpler and more efficient, subtracts a divergence-free correction from the velocity, such that the resulting velocity possesses the correct vorticity. We then successfully apply this method to solve a two-dimensional shock refraction problem, a problem which exhibits nonzero intrinsic vorticity.

A Multi-Sensor Hot-Wire Probe to Measure Vorticity and Velocity in Turbulent Flows

The design and construction of a miniature hot-wire probe capable of simultaneously measuring all components of the velocity and vorticity vectors in turbulent flows are presented. A brief description of the probe resolution and calibration as well as the procedure to solve the cooling equations are given. Preliminary testing in the irrotational region of the flow shows that the spurious vorticity obtained from the probe due to system measurement error does not exceed 8 percent of the RMS vorticity fluctuation levels measured over the region y+ = 15 − 45 in a turbulent boundary layer.

Author index of volume 55

The bane of Lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a Lagrangian zonal mass has ceased to be valid. At short spatial grid scales this is usually referred to by the terms “hourglass” mode or “keystone” motion associated, in particular, with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions, respectively. At longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as “spurious vorticity,” or the long-thin zone problem. In both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution, as would be the case for turbulent flow. In this work we show how such motions can be eliminated by the proper use of subzonal Lagrangian masses, and associated densities and pressures. These subzonal pressures give rise to forces that resist these spurious motions. The pressure is no longer a constant in a zone; it now accurately reflects the density gradients that can occur within a zone due to its differential distortion. Subzonal Lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation. However, these masses arise in a natural way from the intersection of the Lagrangian contours, through which no mass flows, that are associated with both the Lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation. This is an extension of the usual Lagrangian assumption that is often applied to only the zonal mass. We show that with a proper discretization of the subzonal forces resulting from subzonal pressures, hourglass motion and spurious vorticity can be eliminated for a very large range of problems.

Euler and Navier-Stokes solutions for flow over a conical delta wing

The simulation of vortical flow about a conical delta wing (14:1 elliptic cone) by the Euler equations is studied. Navier-Stokes solutions are generated for comparison with the inviscid results. At the flight condition considered, the viscous solutions reveal a large primary leading-edge separation vortex and a smaller secondary vortex. The viscous solutions are shown to be grid-resolved. On a coarse grid, the Euler solutions result in a primary separation vortex but no secondary vortex. A comparison of pressure coefficient with the viscous solution shows surprising agreement. The agreement is, unfortunately, fortuitous. When the Euler equations are solved on the fine, viscous grid, the large separation vortex is eliminated. Instead, a cross-flow shock appears. Shock curvature introduces sufficient vorticity to produce a small vortex downstream of the shock. This inviscid result, fundamentally different from the viscous case, is a valid solution to the Euler equations. The coarse-grid Euler solution is characterized by the production of spurious vorticity and entropy due to numerical error at the leading edge. It is neither a valid solution to the Euler equations nor a reliable approximation to the real viscous flow. The use of a Kutta condition is investigated for both the fineand coarse-grid Euler solutions. It is shown that the Kutta condition fails to produce a grid-independent simulation of the actual viscous flow.