Amount Of Numerical Dissipation Research Articles

A multi-limiter method for simulating complex flows on large grids

This paper presents a novel method for aiding the convergence of complex flows simulations on large grids. In this work, large grids are considered those that are of the order of 100 × 106 cells or more. These large grids, which, for example, model the flow around a powered airplane, are typically difficult to generate and modify. If a high-fidelity solution is needed, as in the case of predicting sonic boom, the amount of numerical dissipation must be minimized. This reduced dissipation, however, may result in solution divergence, triggered by improperly resolved large gradients due to grid deficiency. Using a highly dissipative limiter for the entire grid could avoid solution divergence, but the resulting solution may be less accurate. To avoid the loss of solution accuracy due to too much numerical dissipation, we present an approach in which the flow solver uses two limiters: a low-dissipation limiter for the vast majority of the computational domain, and a high-dissipation limiter for the regions where the grid is not adequate for capturing the flow variation and would otherwise lead to the divergence of the solution. The paper demonstrates that (1) the overall solution is not modified by the local use of the high-dissipation limiter, and (2) for a case in which the use of low-dissipation limiter (Venkatakrishnan) resulted in solution divergence, a converged solution is obtained by using the low-dissipation limiter on the vast majority of the grid and a high-dissipation limiter (van Albada) on the region where the mesh was problematic. The effect of a local high-dissipation limiter on the overall solution is explored on a double-wedge canonical airfoil. The multi-limiter approach is then used to predict the near field pressure of the X-59 airplane.

High-order composite implicit time integration schemes based on rational approximations for elastodynamics

A new approach for developing implicit composite time integration schemes is established starting with rational approximations of the matrix exponential in the solution of the equations of motion. The rational approximations are designed to have the same effective stiffness matrix in all sub-steps. An efficient algorithm is devised so that the implicit equation for a sub-step is in the same form as that of the trapezoidal method. The proposed M-schemes are Mth order accurate using M sub-steps. The amount of numerical dissipation is controlled by a user-specified parameter ρ∞. The (M+1)-schemes that are (M+1)th order accurate using M sub-steps can also be constructed, but the amount of numerical dissipation is built in the schemes and not adjustable. The order of accuracy of all the schemes is not affected by external forces and physical damping. Numerical examples demonstrate that the proposed high-order composite schemes are effective for introducing numerical dissipation and allow the use of large time step sizes in wave propagation problems. The source code written in MATLAB is available for download at: https://github.com/ChongminSong/CompsiteTimeIntegration.

High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Padé expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius ρ∞ in the high frequency limit. Using this user-specified parameter as a weight factor, a Padé expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient time-stepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.

Stability effect of multidimensional velocity components in numerical flux SLAU

AbstractIn computational fluid dynamics of compressible fluid flow, the simple low‐dissipation advection upstream (SLAU) scheme formulated with multidimensional velocity components (normal and parallel to a cell interface) is a widely employed all‐speed scheme. As a variant of SLAU, the mSLAU scheme, which adopts only a velocity component normal to the cell interface instead of multidimensional velocity components, is used for rotorcraft calculations. However, although mSLAU has been claimed to be empirically stable, it has been pointed out that using only the cell‐interface‐normal velocity component instead of the multidimensional velocity components causes numerical instability. Therefore, to clarify the roles of the multidimensional velocity components for computational stability, we solved some benchmark problems associated with using SLAU or mSLAU. We discovered that the multidimensional velocity components contributed to stability against poor‐quality grids by isotropically producing enough amount of numerical dissipation, especially in low‐subsonic and supersonic flows. Although mSLAU could practically treat moderate Mach number flows (approximately 0.1 < M < 1.0) when coupled with the minmod limiter, using only the cell‐interface‐normal velocity component can deteriorate convergence of calculations and lead to susceptibility in the grid geometry.

Spectral Properties of High-Order Element Types for Implicit Large Eddy Simulation

The use of high-order schemes continues to increase, with current methods becoming more robust and reliable. The resolution of complex turbulent flows using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) can be computed more efficiently with high-order methods such as the Flux Reconstruction approach. We make use of the implicit form of LES, referred to as ILES, in which the numerical dissipation of the spatial scheme passively filters high-frequency modes, and no subgrid-scale turbulence model is explicitly implemented. Therefore, given the inherent three-dimensional behaviour of turbulent flows, it is important to understand the spectral characteristics of spatial discretizations in three dimensions. The dispersion and dissipative properties of hexahedra, prismatic and tetrahedral element types are compared using Von Neumann analysis. This comparison is performed on a per degree of freedom basis to assess their suitability for ILES in terms of computational cost. We observe dispersion relations that display non-smooth behaviour for tetrahedral and prismatic elements. In addition, the periodicity of the dispersion relations in one dimension is generally not observed in three-dimensional configurations. Semilogarithmic plots of the numerical error are presented. We observe that the amount of numerical dissipation and dispersion added by hexahedral elements is the least, followed by prisms and finally tetrahedra. We validate our analysis comparing results obtained on computational domains with comparable computational cost against DNS data. Hexahedral elements have the best agreement with the reference data, followed by prismatic and finally tetrahedral elements, which is consistent with the spectral analysis.

An improved Roe solver for high order reconstruction schemes

Roe solver is one of the most popular upwind schemes in computational fluid dynamics which is first-order accurate in space. However, the amount of numerical dissipation gets larger than necessary for high order numerical schemes and then becomes the dominant term of the numerical error. This paper proposes a modified Roe solver that can use high order schemes with high stability based on splitting of the upwind term. The novel algorithm is proposed to compute the dissipation term of Roe flux function using low and high reconstruction schemes for computing acoustic and entropy waves information, respectively. It changes the original Roe solver dissipation term to improve the level of accuracy. The third-order WENO-Z and MUSCL schemes are adopted as high reconstruction schemes. Finite difference approach is used to simulate Euler equations with the original and modified Roe solvers. The results of four shock tube problems of Sod, Lax, Mach 3, and supersonic test cases are compared with the exact solution of the Riemann problem. It is found that the proposed Roe solver increases the accuracy and robustness of the original Roe solver, especially around the expansion waves, contact discontinuities, and shock waves.

Effects of numerical dissipation on the predictive accuracy of wall-modelled large-eddy simulation

The effect of numerical dissipation on the predictive accuracy of wall-modelled large-eddy simulation is investigated via systematic simulations of fully-developed turbulent channel flow. A total of 16 simulations are conducted using the open-source computational fluid dynamics software OpenFOAM®. Four densities of the computational mesh are considered, with four simulations performed on each, in turn varying in the amount of numerical dissipation introduced by the numerical scheme used for interpolating the convective fluxes. The results are compared to publicly-available data from direct numerical simulation of the same flow. Computed error profiles of all the considered flow quantities are shown to vary monotonically with the amount of dissipation introduced by the numerical schemes. As expected, increased dissipation leads to damping of high-frequency motions, which is clearly observed in the computed energy spectra. But it also results in increased energy of the large-scale motions, and a significant over-prediction of the turbulent kinetic energy in the inner region of the boundary layer. On the other hand, dissipation benefits the accuracy of the mean velocity profile, which in turn improves the prediction of the wall shear stress given by the wall model. Thus, in the current framework, the optimal choice for the dissipation of the numerical schemes may depend on the primary quantity of interest for the conducted simulation. With respect to the resolution of the grid, the change in the accuracy is much less predictable, and the optimal resolution depends on the considered quantity and the amount of dissipation introduced by the numerical schemes.

A low-dissipation DG method for the under-resolved simulation of low Mach number turbulent flows

In recent years the use of high-order Discontinuous Galerkin (DG) methods for the under-resolved direct numerical simulations (uDNS) of turbulent flows has received special attention. The suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme: while the dispersion errors are small over a broad range of frequencies, a relevant dissipation error mainly acts at the smallest under-resolved scales, resembling a high frequency filter. Nevertheless, it was recognized (Flad and Gassner, 2017; Mengaldo et al., 2018) that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the uDNS approach. In this regard, the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided, in particular when dealing with low-speed flows, since this behaviour is exacerbated approaching the incompressibility limit. Fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods, see for example Weiss and Smith (1995). In this work a simple modification of the dissipation term of the low Mach preconditioned Roe scheme proposed by Weiss and Smith is considered. The aim is to reduce further the amount of numerical dissipation with the intent of improving the results of uDNS. A spatial DG discretization coupled with a linearly-implicit Rosenbrock-type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions. Results on canonical turbulent flow problems as the Taylor–Green vortex and the flow in a straight sided channel are presented. The improved accuracy of the proposed flux function is demonstrated. The new low-dissipation flux can be useful also in the context of standard, lower order, finite volume methods.

Convective-pressure flux split algorithm for incompressible flow computation using artificial compressibility formulation

Purpose – The purpose of this paper is to present two low diffusive convective-pressure flux split finite volume algorithms for solving incompressible flows in artificial compressibility framework. Design/methodology/approach – The present method follows the framework similar to advection upwind splitting method of Liou and Steffen for compressible flows which is used by Vierendeels et al. to solve incompressible flow equations. Instead of discretizing the total inviscid flux using upwind scheme, the inviscid flux is first split into convective and pressure parts, and then discretized the two parts differently. The convective part is discretized using upwind method and the pressure part using central differencing. Since the Vierendeels type scheme may not be able to capture the divergence free velocity field due to the presence of artificial dissipation term, a strategy to progressively withdraw the dissipation with time step is proposed here that can ascertain the divergence free velocity condition to the level of residual error. This approach helps in reducing the amount of numerical dissipation due to upwind discretization, which is evident from the numerical test examples. Findings – Upwind treatment of only the convective part of the inviscid flux terms, instead of the whole inviscid flux term, leads to more accurate solutions even at relatively coarse grids, which is substantiated by numerical test examples. Research limitations/implications – The method is presently applicable to Cartesian grid. Originality/value – Although similar formulation is reported by Vierendeels et al., no detailed study of the accuracy is presented. Discretization and solution reconstructions used in the present approach differ from the approach reported by Vierendeels et al. A modification to Vierendeels type scheme is proposed that can help in achieving divergence free velocity condition. Finally the efficacy of the present approach to produce very accurate solutions even on coarse grids is successfully demonstrated using a few benchmark problems.

Numerical dissipation control in high order shock-capturing schemes for LES of low speed flows

The Yee & Sjögreen adaptive numerical dissipation control in high order scheme (High Order Filter Methods for Wide Range of Compressible Flow Speeds, ICOSAHOM 09, 2009) is further improved for DNS and LES of shock-free turbulence and low speed turbulence with shocklets. There are vastly different requirements in the minimization of numerical dissipation for accurate turbulence simulations of different compressible flow types and flow speeds. Traditionally, the method of choice for shock-free turbulence and low speed turbulence are by spectral, high order central or high order compact schemes with high order linear filters. With a proper control of a local flow sensor, appropriate amount of numerical dissipation in high order shock-capturing schemes can have spectral-like accuracy for compressible low speed turbulent flows. The development of the method includes an adaptive flow sensor with automatic selection on the amount of numerical dissipation needed at each flow location for more accurate DNS and LES simulations with less tuning of parameters for flows with a wide range of flow speed regime during the time-accurate evolution, e.g., time varying random forcing. An automatic selection of the different flow sensors catered to the different flow types is constructed. A Mach curve and high-frequency oscillation indicators are used to reduce the tuning of parameters in controlling the amount of shock-capturing numerical dissipation to be employed for shock-free turbulence, low speed turbulence and turbulence with strong shocks. In Kotov et al. (High Order Numerical Methods for LES of Turbulent Flows with Shocks, ICCFD8, Chengdu, Sichuan, China, July 14–18, 2014) the LES of a turbulent flow with a strong shock by the Yee & Sjögreen scheme indicated a good agreement with the filtered DNS data. A work in progress for the application of the adaptive flow sensor for compressible turbulence with time-varying random forcing is forthcoming. The present study examines the versatility of the Yee & Sjögreen scheme for DNS and LES of traditional low speed flows without forcing. Special attention is focused on the accuracy performance of this scheme using the Smagorinsky and the Germano–Lilly SGS models.

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

In this paper, the ability of high-order flux reconstruction numerical schemes to perform accurate and stable computations of compressible turbulent flows on coarse meshes is investigated. Two new flux reconstruction schemes, which are optimized for wave dissipation and dispersion properties, are compared to the nodal discontinuous Galerkin and spectral difference methods recovered via the energy-stable flux reconstruction method. The Taylor–Green vortex benchmark problem at is used as a simple a priori test of the numerics. Dissipation rates computed from kinetic energy, vorticity, and pressure dilatation are plotted against reference solutions. Results show that, at low mesh resolution, the flux reconstruction schemes are highly accurate across a range of orders of accuracy, although oscillations can appear in the solution at orders of six and above. Although the flux reconstruction method has a built-in stabilization mechanism, an additional means of damping these instabilities is required. The schemes vary in the amount of numerical dissipation and resolution of the turbulent spectrum. One of the optimized flux reconstruction schemes (the optimized flux reconstruction scheme) is shown to have greater spectral accuracy than any of the others tested, motivating its future usage for high-order high-fidelity computational fluid dynamics.

Evaluation of Riemann flux solvers for WENO reconstruction schemes: Kelvin–Helmholtz instability

Accurate and computationally efficient simulations of Euler equations are of paramount importance in both fundamental research and engineering applications. In this study, our main objective is to investigate the efficacy and accuracy of several Riemann solvers for high-order accurate weighted essentially non-oscillatory (WENO) reconstruction scheme as a state-of-the-art tool to study shear driven turbulence flows. The Kelvin–Helmholtz instability occurs when a perturbation is introduced to a continuous fluid system with a velocity shear, or where there is a velocity difference across the interface between two fluids. Here, we solve a stratified Kelvin–Helmholtz instability problem to demonstrate the performance of six different Riemann solvers’ ability to evolve a linear perturbation into a transition to nonlinear hydrodynamic two-dimensional turbulence. A single mode perturbation is used for our evaluations. Time evolution process shows that the vortices formed from the turbulence slowly merge together since both energy and enstrophy are simultaneously conserved in two-dimensional turbulence. Third-, fifth- and seventh-order WENO reconstruction schemes are investigated along with the Roe, Rusanov, HLL, FORCE, AUSM, and Marquina Riemann flux solvers at the cell interfaces resulting in 18 joint flow solvers. Based on the numerical assessments of these solvers on various grid resolutions, it is found that the dissipative character of the Riemann solver has significant effect on eddy resolving properties and turbulence statistics. We further show that the order of the reconstruction scheme becomes increasingly important for coarsening the mesh. We illustrate that higher-order schemes become more effective in terms of the tradeoff between the accuracy and efficiency. We also demonstrate that AUSM solver provides the least amount of numerical dissipation, yet resulting in a pile-up phenomenon in energy spectra for underresolved simulations. However, results obtained by the Roe solver agree well with the theoretical energy spectrum scaling providing a marginal dissipation without showing any pile-up at a cost of around 30% increase in computational time.

Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The present study focuses only on solving the reactive system by the fractional step method using the Strang splitting. Studies shows that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.

Numerical investigation on the three-dimensional unsteady flow past a 5:1 rectangular cylinder

This paper deals with the three-dimensional simulation of the unsteady flow around a stationary 5:1 rectangular cylinder at zero-degree angle of attack, low Mach number ( M ∞ =0.1) and high Reynolds number ( Re=26,400, based on the plate thickness). Detached-Eddy Simulation (DES) was adopted as strategy of turbulence modeling. Results obtained with a hybrid mesh show satisfactory agreement when validated against experimental data and other computational results from the literature. Particular attention is devoted to the effects of the spanwise extension of the computational domain. Results show that the common choice of a spanwise period equal to the chord of the cylinder might not be enough to allow the natural loss of correlation of the pressure fluctuations and the free development of large-scale turbulent structures. The key role played by the amount of numerical dissipation, which is introduced by the second-order central difference scheme used to discretize the inviscid fluxes in the governing equations, is highlighted. The promising results obtained with DES for this benchmark test case suggest that this hybrid method is well suited for complex problems of high-Reynolds number bluff body aerodynamics and fluid–structure coupling.

Convergence and stability in numerical relativity

It is often the case in numerical relativity that schemes that are known to be convergent for well posed systems are used in evolutions of weakly hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used through out the years, this procedure leads to non-convergent schemes. That is, arbitrarily small initial errors are amplified without bound when resolution is increased, independently of the amount of numerical dissipation introduced. The lack of convergence introduced by this instability can be particularly subtle, in the sense that it can be missed by several convergence tests, especially in 3+1 dimensional codes. We propose tests and methods to analyze convergence that may help detect these situations.

Accuracy of Parabolized Navier-Stokes Schemes for Stability Analysis of Hypersonic Axisymmetric Flows

The accuracy of the parabolized Navier ‐Stokes (PNS) and globally iterated PNS (IPNS) schemes for use as basic e ows is studied for stability analysis of hypersonic axisymmetric e ow over blunt and sharp slender cones at Mach 8. The numerical solution of the PNS and IPNS models is obtained by using the implicit e nite difference algorithm of Beam and Warming with a shock-e tting procedure. Both the PNS and IPNS solutions are thoroughly verie ed and compared with the thin-layer Navier ‐Stokes (TLNS) solution and experimental data. The stability analysis and transition prediction are performed using theparallel-e owlinearstability theory together with the eN method.Asystematicsensitivitystudyisperformedforbasicstatesolutions,includingproe lesandtheirderivatives, and spatial growth rates obtained from the PNS and IPNS models. The basic e ow and stability results and the predicted transition location based on the PNS model are qualitatively in good agreement with those of other theoretical investigations using various basic e ow models. Improved agreement with the TLNS model for both basic state solution and stability results is achieved when the IPNS model is applied. Attention is concentrated on those features of the numerical methods of the basic e ow computation that need particular care if adequate stability results are to be obtained. The study shows that the number of grid points, the grid distribution, and the amount of numerical dissipation have signie cant effects on the stability results. Moreover, indications are that carefulnumericalsolutionsofthePNSandIPNSschemesaslaminarbasice owcalculationsleadto amorepractical transition prediction tool in hypersonic e ows.

Inviscid and viscous transonic flows in cascades using an implicit upwind algorithm

This article presents a numerical algorithm for calculating inviscid and viscous flows in transonic and super- sonic turbomachinery cascades. Of particular interest is the prediction of shock-induced boundary-layer sep- aration and the comparison with experiments. The numerical procedure uses upwind-biased differencing with an approximate Riemann solver for the inviscid Euler fluxes. The viscous fluxes are discretized with careful treatment of cross derivatives. Convergence to a steady state is accelerated by use of an implicit relaxation strategy. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. Details of the numerical method are presented along with an appropriate grid generation procedure. Computed results show that the method can predict complicated viscous flow phenomena in cascades. RANSONIC and supersonic flows in turbomachinery blade passages are dominated by shock waves and viscous ef- fects. Typically, the inlet flow of fan blades and compressor cascades is supersonic, causing a more or less complicated and strong shock-wave system. Compressors are designed in general to produce a streamwise pressure gradient that causes boundary-layer thickening and often flow separation. Within turbine bladings, viscous effects are more moderate, whereas shock-boundary-layer interactions may play an important role as well. Most of the numerical cascade flow analyses in use today are based on central differencing algorithms with added ar- tificial dissipation. Node-centered as well as cell-centered fi- nite difference and finite volume schemes have been pre- ferred,1-2 though cell-vertex schemes3'5 can yield more accurate results as long as the amount of numerical dissipation is kept at a low level. Taking into account the limited robustness provided by these central schemes, the complexity of the flowfields arising within modern transonic and supersonic high-performance cascades6 seems to favor a physically motivated upwind scheme. Since the latter is obtained by using a central scheme with the most sophisticated form of added numerical dissipation,7 the concept of upwinding is also pursued in the present in- vestigation. The applied high-order upwind scheme is based on the class of schemes satisfying the total-variation-diminishing (TVD) requirement.8 Therefore, the scheme is particularly well suited for complex shocked flow problems. Exploiting the diagonal dominance property inherent with this scheme, implicit re- laxation is used to achieve computational efficiency.