Effects Of Numerical Dissipation Research Articles

Fluid–structure interaction simulation of pulse propagation in arteries: Numerical pitfalls and hemodynamic impact of a local stiffening

When simulating the propagation of a pressure pulse in arteries, the discretization parameters (i.e. the time step size Δt and the grid size Δx) need to be chosen carefully in order to avoid a decrease in amplitude of the traveling wave due to numerical dissipation. In this paper the effect of numerical dissipation is examined using a numerical fluid–structure interaction (FSI) model of the pulse propagation in an artery. More insight in the influence of the temporal and spatial resolution of the wave on the results of these simulations is gained using an analytical study in which the scalar linear one-dimensional transport equation is considered. Although this model does not take into account the full complexity of the problem under consideration, the results can be used as a guidance for the selection of the numerical parameters. Furthermore, this analysis illustrates the difference in accuracy that can be obtained using a second-order implicit time integration scheme instead of a first-order scheme.The results from the analytical and numerical studies are subsequently used to determine the settings necessary to obtain a grid and time step converged simulation of the wave propagation and reflection in a simplified model of an aorta with repaired aortic coarctation. This FSI model allows to study the hemodynamic impact of a stiff segment and demonstrates that the presence of a stiff segment has an important impact on a short pressure pulse, but has almost no influence on a physiological pressure pulse. This phenomenon is explained by analyzing the reflections induced by the stiff segment.

Energy transfer in the Richtmyer-Meshkov instability

The variable-density spectral kinetic energy budget for the Richtmyer-Meshkov-induced turbulent mixing layer is presented using results from a 512{3} implicit large eddy simulation. The budget is presented at several time instants and as a function of the inhomogeneous direction as the layer transitions from the initial impulse through to self-similarity. There are clear parallels in the development of the mixing layer with a previous analysis for the Rayleigh-Taylor instability. In the core of the layer, the quadratic terms are largely negative in the energy-containing scales. The transfer spectra are clearly asymmetric, where the majority of the activity occurrs on the spike side. The quadratic and pressure components are of opposite sign and almost cancel each other out in the spikes. The dilatational terms are negligible in comparison to the difference between the quadratic and pressure transfer. A notable result is that vortex rings are identified as the key source of alternating fields of negative and positive energy transfer within the mixing layer. This helps explain similar observations noted in direct numerical simulations of Rayleigh-Taylor instability. Finally, the spectral numerical dissipation for this scheme is computed for the self-similar layer. This demonstrated that the effects of numerical dissipation are small compared to the other terms at low wave numbers, whereas at higher wave numbers where modes become significantly underresolved the numerical dissipation is approximately twice the nonlinear transfer term and behaves in approximate analogy to an effective spectral eddy viscosity.

Numerical Dissipation Effects on Massive Separation Around Tandem Cylinders

Three spatial schemes, the original Roe scheme and two high-order symmetric total variation diminishing schemes,whose dissipations aremultiplied by a constant parameter or a function (called ), are coupledwith delayed detached eddy simulation to investigate the numerical dissipation effects on the massive separation flow around tandem cylinders. From the comparisons between the computations and the available measurements, the numerical dissipation has a significant influence on the mean and instantaneous flowfields. The original Roe scheme is too dissipative to predict the small scale turbulent structures, and it strongly suppresses the growth of resolved turbulence. The S6WENO5 schemes with constantand adaptivetimes of dissipation have similar performances and they well match the measurements. However, the S6WENO5 with constanttimes (0.12 here) of dissipation is too empirical, and the small constanttimes of dissipation cannot generally suppress the numerical oscillations near the wall and in the far fields. The S6WENO5 with adaptive dissipation provides the best performance.

COMPARING NUMERICAL METHODS FOR ISOTHERMAL MAGNETIZED SUPERSONIC TURBULENCE

We employ simulations of supersonic super-Alfvenic turbulence decay as a benchmark test problem to assess and compare the performance of nine astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss convergence of various characteristics for the turbulence decay test and impacts of various components of numerical schemes on the accuracy of solutions. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the magnetic field using the constrained transport method, and using little to no explicit artificial viscosity. Codes which fall short in one or more of these areas are still useful, but they must compensate higher numerical dissipation with higher numerical resolution. This paper is the largest, most comprehensive MHD code comparison on an application-like test problem to date. We hope this work will help developers improve their numerical algorithms while helping users to make informed choices in picking optimal applications for their specific astrophysical problems.

Numerical simulation of shock tube generated vortex: effect of numerics

Vortices generated at the open end of a planar shock tube are numerically simulated using the AUSM+ scheme. This scheme is known to have low numerical dissipation and therefore is suitable for capturing unsteady vortex motion. However, this low numerical dissipation can also cause oscillations in the vorticity field. Numerical experiments presented here highlight the effect of numerical dissipation on the simulated vortex, as well as the role played by turbulence models. Two turbulence models – the shear-stress-transport (SST) and its modified version for unsteady flows (SST-SAS) – are employed to observe the effect of including turbulence models in such complex flows where the vortex has an embedded shock.

Large Eddy Simulation of Stable Supersonic Jet Impinging on Flat Plate

This paper describes a numerical study based on large eddy simulation of the flow induced by a supersonic jet impinging on a flat plate in a stable regime. This flow involves very high velocities, shocks, and intense shear layers. Performing large eddy simulation on such flows remains a challenge because of the shock discontinuities. Here, large eddy simulation is performed with an explicit third-order compressible solver using a unstructured mesh, a centered scheme, and the Smagorinsky model. Three levels of mesh refinement (from 7 to 22 million cells) are compared in terms of instantaneous and averaged flowfields (shock and recirculation zone positions), averaged flow velocity and pressure fields, wall pressure, root mean square pressure fields, and spectral content using one and two-point analyses. The effects of numerical dissipation and turbulent viscosity are compared on the three grids and shown to be well controlled. The comparison of large eddy simulation with experimental data shows that the finest grid (a 22 million cell mesh) ensures grid-independent results not only for the mean and rms fields but also for higher statistics such as single and two-point correlation functions.

Shock-turbulence interaction: What we know and what we can learn from peta-scale simulations

Many applications in engineering and physical sciences involve turbulent flows interacting with shock waves. High-speed flows around aerodynamic bodies and through propulsion systems for high-speed flight abound with interactions of shear driven turbulence with complex shock waves. Supernova explosions and implosion of a cryogenic fuel pellet for inertial confinement fusion also involve the interaction of shockwaves with turbulence and strong density variations. Numerical simulations of such physical phenomena impose conflicting demands on the numerical algorithms. Capturing broadband spatial and temporal variations in a turbulent flow suggests the use of high-bandwidth schemes with minimal dissipation and dispersion, while capturing the flow discontinuity at a shock wave requires numerical dissipation.Results from three promising shock-capturing schemes a) high order WENO, b) nonlinear artificial diffusivity with compact finite differences, and c) a hybrid approach combining high-order central differencing with WENO near the shocks are compared using the Taylor-Green problem and compressible isotropic turbulence with eddy-shocklets. The performance of each scheme is characterized in terms of an effective bandwidth. The comparison highlights the damaging effect of numerical dissipation when the WENO scheme is applied everywhere. The hybrid approach is found to be best suited for studying shock-turbulence interactions.Results from previous DNS and LES studies of the canonical shock-turbulence interaction problem, i.e. the interaction of isotropic turbulence with a (nominally) normal shock and comparison with available theory and experimental data are recalled. The principal physical effects include the amplification of turbulent kinetic energy across the shock and its anisotropy, change in turbulence length scales across the shock, departure from the common assumption of strong Reynolds analogy (used in modeling turbulence in high-speed flows), and the distortion of the shock due to its interaction with the incident turbulence. In this context new results from our SciDAC sponsored project are shown and open issues are mentioned. New DNS results achieve a significantly larger turbulence Reynolds number and allow an exploration of the nonlinear effects. While the linear interaction theory of Ribner provides useful estimates of the amplification of vorticity fluctuations across the shock, it misses the strong nonlinear dynamics of the energized and highly anisotropic vorticity downstream of the shock. It is found that previous DNS studies also underestimated this effect. The simulations show that turbulent self-stretching and tilting mechanisms bring about a relatively rapid return to isotropy in the turbulent vorticity field. The turbulent velocity field, however, does not show any appreciable tendency towards isotropy. It is further observed that when the turbulence interacting with the shock is sufficiently energetic the instantaneous shock structure is significantly modified; local regions of significant over-compression are found as well as regions where the mean shock compression is nearly isentropic. Estimates for the computational resources necessary for studying this fundamental shock-turbulence interaction problem at higher Reynolds number on peta-scale computing systems are given.

Simulating supersonic turbulence in magnetized molecular clouds

We present results of large-scale three-dimensional weakly magnetized supersonic turbulence simulations with an isothermal equation of state at grid resolutions up to 10243 cells with the Piecewise Parabolic Method on a Local Stencil. The turbulence is driven by a large-scale isotropic solenoidal force in a periodic computational domain and fully develops in a few flow crossing times. We then evolve the flow for a number of flow crossing times and analyze various statistical properties of the saturated turbulent state. We show that the energy transfer rate in the inertial range of scales is surprisingly close to a constant, indicating that Kolmogorov's phenomenology for incompressible turbulence can be extended to magnetized supersonic flows. We also discuss numerical dissipation effects and convergence of different turbulence diagnostics as grid resolution refines from 2563 to 10243 cells.

Nonstationary Westward Translation of Nonlinear Frontal Warm-Core Eddies

Abstract For the first time, an analytical theory and a very high-resolution, frontal numerical model, both based on the unsteady, nonlinear, reduced-gravity shallow water equations on a β plane, have been used to investigate aspects of the migration of homogeneous surface, frontal warm-core eddies on a β plane. Under the assumption that, initially, such vortices are surface circular anticyclones of paraboloidal shape and having both radial and azimuthal velocities that are linearly dependent on the radial coordinate (i.e., circular pulsons of the first order), approximate analytical expressions are found that describe the nonstationary trajectories of their centers of mass for an initial stage as well as for a mature stage of their westward migration. In particular, near-inertial oscillations are evident in the initial migration stage, whose amplitude linearly increases with time, as a result of the unbalanced vortex initial state on a β plane. Such an initial amplification of the vortex oscillations is actually found in the first stage of the evolution of warm-core frontal eddies simulated numerically by means of a frontal numerical model initialized using the shape and velocity fields of circular pulsons of the first order. In the numerical simulations, this stage is followed by an adjusted, complex nonstationary state characterized by a noticeable asymmetry in the meridional component of the vortex’s horizontal pressure gradient, which develops to compensate for the variations of the Coriolis parameter with latitude. Accordingly, the location of the simulated vortex’s maximum depth is always found poleward of the location of the simulated vortex’s center of mass. Moreover, during the adjusted stage, near-inertial oscillations emerge that largely deviate from the exactly inertial ones characterizing analytical circular pulsons: a superinertial and a subinertial oscillation in fact appear, and their frequency difference is found to be an increasing function of latitude. A comparison between vortex westward drifts simulated numerically at different latitudes for different vortex radii and pulsation strengths and the corresponding drifts obtained using existing formulas shows that, initially, the simulated vortex drifts correspond to the fastest predicted ones in many realistic cases. As time elapses, however, the development of a β-adjusted vortex structure, together with the effects of numerical dissipation, tend to slow down the simulated vortex drift.

On the Use of High-Order Accurate Solutions of PNS Schemes as Basic Flows for Stability Analysis of Hypersonic Axisymmetric Flows

Abstract High-order accurate solutions of parabolized Navier–Stokes (PNS) schemes are used as basic flow models for stability analysis of hypersonic axisymmetric flows over blunt and sharp cones at Mach 8. Both the PNS and the globally iterated PNS (IPNS) schemes are utilized. The IPNS scheme can provide the basic flow field and stability results comparable with those of the thin-layer Navier–Stokes (TLNS) scheme. As a result, using the fourth-order compact IPNS scheme, a high-order accurate basic flow model suitable for stability analysis and transition prediction can be efficiently provided. The numerical solution of the PNS equations is based on an implicit algorithm with a shock fitting procedure in which the basic flow variables and their first and second derivatives required for the stability calculations are automatically obtained with the fourth-order accuracy. In addition, consistent with the solution of the basic flow, a fourth-order compact finite-difference scheme, which does not need higher derivatives of the basic flow, is efficiently implemented to solve the parallel-flow linear stability equations in intrinsic orthogonal coordinates. A sensitivity analysis is also conducted to evaluate the effects of numerical dissipation and grid size and also accuracy of computing the basic flow derivatives on the stability results. The present results demonstrate the efficiency and accuracy of using high-order compact solutions of the PNS schemes as basic flow models for stability and transition prediction in hypersonic flows. Moreover, indications are that high-order compact methods used for basic-flow computations are sensitive to the grid size and especially the numerical dissipation terms, and therefore, more careful attention must be kept to obtain an accurate solution of the stability and transition results.

A fast and accurate semi-Lagrangian particle level set method

In this paper, we present an efficient semi-Lagrangian based particle level set method for the accurate capturing of interfaces. This method retains the robust topological properties of the level set method with- out the adverse effects of numerical dissipation. Both the level set method and the particle level set method typically use high order accurate numerical discretizations in time and space, e.g. TVD Runge–Kutta and HJ-WENO schemes. We demonstrate that these computationally expensive schemes are not required. Instead, fast, low order accurate numerical schemes suffice. That is, the addition of particles to the level set method not only removes the difficulties associated with numerical diffusion, but also alleviates the need for computationally expensive high order accurate schemes. We use an efficient, first order accurate semi-Lagrangian advection scheme coupled with a first order accurate fast marching method to evolve the level set function. To accurately track the underlying flow characteristics, the particles are evolved with a second order accurate method. Since we avoid complex high order accurate numerical methods, extending the algorithm to arbitrary data structures becomes more feasible, and we show preliminary results obtained with an octree-based adaptive mesh.

Real-time synthesis of bleeding for virtual hysteroscopy

In this paper we present a method for simulating bleeding in a virtual reality hysteroscopic simulator for surgical training. The simulated bleeding is required to be visually appealing while at the same time instantaneously responsive to any feedback that the surgeon may be conducing to the virtual environment. In order to meet these real-time requirements, we have based the simulation on graphical fluid solvers. These solvers primarily work best over a 2D domain. For correct visualization in the hysteroscopic simulator, it is, however, necessary to perform the simulation fully in 3D. Therefore in this paper we also present the design modifications for 3D graphical fluid solving and show how to use parallelization to maintain real-time behavior. We also discuss how the incorporation of massless particles into the simulation can improve the visual quality of the results by limiting numerical dissipation effects.

Linear and non‐linear stability analysis for finite difference discretizations of high‐order Boussinesq equations

AbstractThis paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non‐linear and extremely dispersive water waves. The analysis demonstrates the near‐equivalence of classical linear Fourier (von Neumann) techniques with matrix‐based methods for formulations in both one and two horizontal dimensions. The matrix‐based method is also extended to show the local de‐stabilizing effects of the non‐linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep‐water non‐linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non‐normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non‐linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non‐linear PDEs. Copyright © 2004 John Wiley & Sons, Ltd.

LES에서 중심 및 상류 컴팩트 차분기법의 적합성에 관하여 (I) - 수치 실험 -

The suitability of high-order accurate, centered and upwind-biased compact difference schemes is evaluated for large eddy simulation of turbulent flow. Two turbulent flows are considered: turbulent channel flow at Re = 23000 and flow over a circular cylinder at Re = 3900. The effects of numerical dissipation on the finite differencing and aliasing errors and the subgrid-scale stress are investigated. It is shown through the simulations that compact upwind schemes are not suitable for LES, whereas the fourth order-compact centered scheme is a good candidate for LES provided that proper dealiasing of nonlinear terms is performed. The classical issue on the aliasing error and the treatment of nonlinear terms is revisited with compact difference schemes.br/

LES에서 중심 및 상류 컴팩트 차분기법의 적합성에 관하여 (II) - 정적 오차 해석 -

The suitability of high-order accurate, centered and upwind-biased compact difference schemes for large eddy simulation is evaluated by a dynamic analysis. Large eddy simulation of isotropic turbulence is performed with various dissipative and non-dissipative schemes to investigate the effect of numerical dissipation on the resolved solutions. It is shown by the present dynamic analysis that upwind schemes reduce the aliasing error and increase the finite differencing error. The existence of optimal upwind scheme that minimizes total numerical error is verified. It is also shown that the finite differencing error from numerical dissipation is the leading source of numerical errors by upwind schemes. Simulations of a turbulent channel flow are conducted to show the existence of the optimal upwind scheme.

LES에서 중심 및 상류 컴팩트 차분기법의 적합성에 관하여 (III) -동적 오차 해석 -

The suitability of high-order accurate, centered and upwind-biased compact difference schemes for large eddy simulation is evaluated by a dynamic analysis. Large eddy simulation of isotropic turbulence is performed with various dissipative and non-dissipative schemes to investigate the effect of numerical dissipation on the resolved solutions. It is shown by the present dynamic analysis that upwind schemes reduce the aliasing error and increase the finite differencing error. The existence of optimal upwind scheme that minimizes total numerical error is verified. It is also shown that the finite differencing error from numerical dissipation is the leading source of numerical errors by upwind schemes. Simulations of a turbulent channel flow are conducted to show the existence of the optimal upwind scheme.<br/>

Large eddy simulation using a curvilinear coordinate system for the flow around a square cylinder

The application of Large Eddy Simulation (LES) in a curvilinear coordinate system to thernflow around a square cylinder is presented. In order to obtain sufficient resolution near the side of therncylinder, we use an O-type grid. Even with a curvilinear coordinate system, it is difficult to avoid thernnumerical oscillation arising in high-Reynolds-number flows past a bluff body, without using an extremelyrnfine grid used. An upwind scheme has the effect of removing the numerical oscillations, but, it isrnaccompanied by numerical dissipation that is a kind of an additional sub-grid scale effect. Firstly, werninvestigate the effect of numerical dissipation on the computational results in a case where turbulentrndissipation is removed in order to clarify the differences between the effect of numerical dissipation. Next,rnthe applicability and the limitations of the present method, which combine the dynamic SGS model withrnacceptable numerical dissipation, are discussed.

A Critical Analysis of Rayleigh–Taylor Growth Rates

Recent simulations of Rayleigh–Taylor instability growth rates display considerable spread. We provide evidence that differences in numerical dissipation effects (mass diffusion and viscosity) due to algorithmic differences and differences in simulation duration are the dominant factors that produce such different results. Within the simulation size and durations explored here, we provide evidence that the principal discrepancies are due to numerical dispersion through comparison of simulations using different algorithms. We present new 3D front tracking simulations that show tentative agreement with the range of reported experimental values. We begin an exploration of new physical length scales that may characterize a transition to a new Rayleigh–Taylor mixing regime.

The Effects of Numerical Dissipation in Large Eddy Simulations

Abstract Large eddy simulations sometimes use monotone advection schemes. Such schemes are dissipative, and the effective subgrid model then becomes the combined effect of the intended model and of the numerical dissipation. The impacts on simulation reliability are examined for the cases of dry convective and neutral planetary boundary layers. In general it is found that the results in the well-resolved flow interior are insensitive to the details of the advection scheme. However, unsatisfactory results may be obtained if numerical dissipation dominates where the flow becomes less well resolved as the surface is approached.

Hydrodynamics of Turbid Underflows. II: Aggradation, Avulsion, and Channelization

A two-dimensional, depth-averaged finite-volume model is used to study the hydrodynamics of turbid underflows. A sensitivity analysis is performed with regard to model input parameters and the initial bed properties. Simulations are then performed to illustrate aggradation and avulsion in an existing submarine channel. The conditions under which channelization of deposits occur are also examined. The effect of numerical dissipation on flow ignition is examined and the high resolution properties of the model are found to produce satisfactory results. The model is applied to historical field-scale events, including the Reserve Fan in Lake Superior and Rupert Inlet in British Columbia.