Errors In Large Eddy Simulation Research Articles

Challenges of high-fidelity air quality modeling in urban environments – PALM sensitivity study during stable conditions

Abstract. Urban air quality is an important part of human well-being, and its detailed and precise modeling is important for efficient urban planning. In this study the potential sources of errors in large eddy simulation (LES) runs of the PALM model in stable conditions for a high-traffic residential area in Prague, Czech Republic, with a focus on street canyon ventilation, are investigated. The evaluation of the PALM model simulations against observations obtained during a dedicated campaign revealed unrealistically high concentrations of modeled air pollutants for a short period during a winter inversion episode. To identify potential reasons, the sensitivities of the model to changes in meteorological boundary conditions and adjustments of model parameters were tested. The model adaptations included adding the anthropogenic heat from cars, setting a bottom limit of the subgrid-scale turbulent kinetic energy (TKE), adjusting the profiles of parameters of the synthetic turbulence generator in PALM, and limiting the model time step. The study confirmed the crucial role of the correct meteorological boundary conditions for realistic air quality modeling during stable conditions. Besides this, the studied adjustments of the model parameters proved to have a significant impact in these stable conditions, resulting in a decrease in concentration overestimation in the range 30 %–66 % while exhibiting a negligible influence on model results during the rest of the episode. This suggested that the inclusion or improvement of these processes in PALM is desirable despite their negligible impact in most other conditions. Moreover, the time step limitation test revealed numerical inaccuracies caused by discretization errors which occurred during such extremely stable conditions.

Evaluating large eddy simulation results based on error analysis

The quantification of the prediction accuracy in large eddy simulations (LES) is very challenging due to various interacting errors associated with this approach. When dealing with errors in LES using implicit filtering, numerical and modeling errors have drawn the interest of many researchers. Little attention has been paid to other sources of discrepancies between LES results and reference data, namely sampling errors, influence of the initial conditions, improper boundary conditions or uncertainties issuing from reference data. A framework of metrics that includes all these issues is addressed in the present paper to study subgrid-scale (SGS) models for LES and to quantify their prediction accuracy and computational costs. The method is applied to a simple wall-bounded turbulent flow at moderate Reynolds number. It turns out from the results obtained with six commonly used SGS models that wall-adapting models (WALE and SIGMA) and localized dynamic models reproduce the physics of the flow field more faithfully, reveal a superior prediction accuracy and have a similar computational cost than models using van Driest wall damping. Especially at the viscous wall region (r^+<50), wall-adapting and localized dynamic models are more accurate, reflecting the proper near wall behavior of such models. Relying on the analysis of sources of various errors, uncertainties in LES are estimated and systematically assessed, and their influence on simulation results is quantified. Finally, engineering estimations of the required averaging time to obtain basic estimates of statistical quantities with a predetermined degree of accuracy are suggested.

Quantification of errors in large-eddy simulations of a spatially evolving mixing layer using polynomial chaos

A stochastic approach based on generalized polynomial chaos (gPC) is used to quantify the error in large-eddy simulation (LES) of a spatially evolving mixing layer flow and its sensitivity to different simulation parameters, viz., the grid stretching in the streamwise and lateral directions and the subgrid-scale (SGS) Smagorinsky model constant (CS). The error is evaluated with respect to the results of a highly resolved LES and for different quantities of interest, namely, the mean streamwise velocity, the momentum thickness, and the shear stress. A typical feature of the considered spatially evolving flow is the progressive transition from a laminar regime, highly dependent on the inlet conditions, to a fully developed turbulent one. Therefore, the computational domain is divided in two different zones (inlet dependent and fully turbulent) and the gPC error analysis is carried out for these two zones separately. An optimization of the parameters is also carried out for both these zones. For all the considered quantities, the results point out that the error is mainly governed by the value of the CS constant. At the end of the inlet-dependent zone, a strong coupling between the normal stretching ratio and the CS value is observed. The error sensitivity to the parameter values is significantly larger in the inlet-dependent upstream region; however, low-error values can be obtained in this region for all the considered physical quantities by an ad hoc tuning of the parameters. Conversely, in the turbulent regime the error is globally lower and less sensitive to the parameter variations, but it is more difficult to find a set of parameter values leading to optimal results for all the analyzed physical quantities. A similar analysis is also carried out for the dynamic Smagorinsky model, by varying the grid stretching ratios. Comparing the databases generated with the different subgrid-scale models, it is possible to observe that the error cost function computed for the streamwise velocity and for the momentum thickness is not significantly sensitive to the used SGS closure. Conversely, the prediction of the shear stress is much more accurate when using a dynamic subgrid-scale model and the variance of the error is lower in magnitude.

Modeling and discretization errors in large eddy simulations of hydrodynamic and magnetohydrodynamic channel flows

We assess the performances of three different subgrid scale models in large eddy simulations (LES) of turbulent channel flows. Two regimes are considered: hydrodynamic and magnetohydrodynamic (i.e. in the presence of a uniform wall-normal magnetic field). The simulations are performed using a second-order finite volume (FV) and a pseudo-spectral (PS) method. The LES results are compared with under-resolved results (obtained without model) and direct numerical simulations (DNS). We show that discretization errors affect the FV results in two ways: (1) the flow statistics differ from the spectral estimates in the absence of subgrid model; and (2) the eddy viscosity systematically underestimates the spectral value in the presence of a subgrid model. This is mainly because numerical errors affect the computation of the derivatives, and in particular, they lower the discrete strain rate appearing in the viscous term and the subgrid model. The magnitude of the numerical errors further varies with the mesh resolution and the intensity of the turbulent fluctuations. In this manuscript, a novel formulation of the discrete strain, which was proven successful in homogeneous isotropic turbulence, is used to compute the FV eddy viscosities. Although the average norm of the discrete strain is largely increased using this formulation, the effect on the flow dynamics is marginal. This is explained by analysing the contribution of each term of the discrete kinetic energy balance. It is shown how the underestimation of the discrete viscous dissipation inhibits the effect of the improved discrete strain.

Growth of Rounding Errors and Repetitivity of Large Eddy Simulations

This paper studies the propagation of rounding errors in large-eddy simulation and shows that instantaneous flowfields produced by large-eddy simulation are partially controlled by these rounding errors and depend on multiple parameters: number of processors used for parallel simulation (even in an explicit code), changes in initial conditions (even of the order of machine accuracy), machine precision (simple, double, or quadruple), etc. Using a laminar Poiseuille pipe flow, a fully developed turbulent channel flow, and a complex burner geometry as test cases, results show that only turbulent flows exhibit a high sensitivity to these parameters. These results confirm that large-eddy simulation reflects the true nature of turbulence insofar as it may exponentially amplify infinitely small perturbations on initial conditions in time. However, they highlight an often overlooked limitation of large-eddy simulation in terms of validation and prediction of unsteady phenomena.

A study of differentiation errors in large-eddy simulations based on the EDQNM theory

This paper is concerned with the investigation of numerical errors in large-eddy simulations by means of two-point turbulence modeling. Based on the eddy-damped quasi-normal Markovian (EDQNM) theory, a stochastic model is developed in order to predict the time evolution of the kinetic energy spectrum obtained by a large-eddy simulation (LES), including the effects of the numerics. Using this framework, the influence of the accuracy of the approximate space differencing schemes on LES quality is studied, for decaying homogeneous isotropic incompressible turbulence, with Reynolds numbers Re λ based on the transverse Taylor scale equal to 780, 2500 and 8000. The results show that the discretization of the filtered Navier–Stokes equations leads to differentiation and aliasing errors. Error spectra are also presented, and indicate that the numerical errors are mainly originating from the approximate differentiation. In addition, increasing the order of accuracy of the differencing schemes or using algorithms optimized in the Fourier space is found to widen the range of well-resolved scales. Unfortunately, for all the schemes, the smaller scales with wavenumbers close to the grid cut-off wavenumber, are badly calculated and generate differentiation errors over the whole energy spectrum. The eventual use of explicit filtering to remove spurious motions with short wavelength is finally shown to significantly improve LES accuracy.

Analysis of numerical errors in large eddy simulation using statistical closure theory

This paper develops a dynamic error analysis procedure for the numerical errors arising from spatial discretization in large-eddy simulation. The analysis is based on EDQNM closure theory, and is applied to the LES of decaying isotropic turbulence. First, the effects of finite-differencing truncation error, aliasing error and the dynamic Smagorinsky model are independently considered. The time-evolution of kinetic energy and spectra predicted by the analysis are compared to actual LES using the Navier–Stokes equations, and good agreement is obtained. The analysis is then extended to simultaneously consider all sources of error in a second-order discretely energy conserving, central-difference LES solver. Good agreement between the analysis and actual LES is obtained. The analysis is used to compare the contribution of the subgrid model to that of numerical errors, and it is shown that the contribution of the subgrid scale model is much higher than the numerical errors. The proposed one-dimensional EDQNM-LES model shows potential as a more general tool for the analysis of numerical error, and SGS model in simulations of turbulent flow.

A priori tests on numerical errors in large eddy simulation using finite differences and explicit filtering

AbstractWhen low‐order finite‐difference methods are applied in large eddy simulation (LES), the magnitude of the numerical error may be larger than that of the subgrid‐scale (SGS) term. In this paper, the effect of explicit filtering on the numerical error related to the spatial discretization of the convection term and the exact SGS term is studied a priori in the turbulent fully developed channel flow. As the filter width is increased the grid resolution is kept constant. Also filtering in the inhomogeneous wall‐normal direction is discussed. The main conclusions are related to two approaches to explicit filtering. In the traditional approach, the whole velocity field is filtered explicitly while in the alternative approach, only the non‐linear convection term of the Navier–Stokes equations is filtered explicitly. Based on the results presented in the paper it seems that the first approach leads to an unphysical situation. However, the later approach works in the desired way, and the numerical error becomes clearly smaller than the SGS term. The main difference between the two approaches seems to be the interpretation of the resolved non‐linear term in the filtered Navier–Stokes equations. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Commutator errors in large-eddy simulation

Commutator errors arise in large-eddy simulation of incompressible turbulent flow from the application of non-uniform filters to the continuity -- and Navier-Stokes equations. For non-uniform, high order filters with bounded moments the magnitude of the commutator errors is shown to be of the same order as that of the turbulent stress fluxes. Consequently, one cannot reduce the size of the commutator errors independently of the turbulent stress terms by any judicious construction of such filter operators. Independent control over the commutator errors compared to the turbulent stress fluxes can, instead, be obtained by appropriately restricting the spatial variations of the filter-width and filter-skewness. For situations in which the dynamical consequences of the commutator errors are significant, e.g., near solid boundaries, explicit similarity modelling for the commutator errors is proposed, including the application of Leray regularization. The performance of this commutator error parametrization is illustrated for the one-dimensional Burgers equation. The Leray approach is found to capture the filtered flow with higher accuracy than conventional similarity modelling, which is particularly relevant for large filter-width variations.

Assessment measures for URANS/DES/LES: an overview with applications

A survey of quality assessment methods is presented covering a range of possible errors in applications of CFD. Beside modeling and numerical errors further uncertainties in unsteady flow simulations are related to inlet conditions and statistical averaging. The main focus is on measures that can be applied to unsteady Reynolds averaged Navier–Stokes (URANS) solvers, hybrid large eddy simulations (HLES), detached eddy simulations (DES), and mostly LES applications. The quantification of uncertainty in LES (large eddy simulations) is difficult for two reasons: (i) substantial grid refinement is too expensive and time consuming; (ii) the modeling errors and discretization errors are convoluted in such a nonlinear manner that it is difficult to segregate one from the other. In hybrid URANS/LES this task is even more difficult because the transitional regions cannot be identified a priori. A brief review of attempts to quantify the accuracy of LES focusing on Richardson extrapolation (RE) is presented. Selected measures are applied to URANS, DES and LES cases from the literature. An attempt is made to separate the modeling contribution from the numerical errors in LES. The methods proposed for the quantification of LES and DES accuracy are still in their infancy stage. Early results are however, promising.

Interacting errors in large-eddy simulation: a review of recent developments

The accuracy of large-eddy simulations is limited, among others, by the quality of the subgrid parameterisation and the numerical contamination of the smaller retained flow structures. We review the effects of discretisation and modelling errors from two different perspectives. We first show that spatial discretisation induces its own filter and compare the dynamic importance of this numerical filter to the basic large-eddy filter. The spatial discretisation modifies the large-eddy closure problem as is expressed by the difference between the discrete ‘numerical stress tensor’ and the continuous ‘turbulent stress tensor’. This difference consists of a high-pass contribution associated with the specific numerical filter. Several central differencing methods are analysed and the importance of the subgrid resolution is established. Second, we review a database approach to assess the total simulation error and its numerical and modelling contributions. The interaction between the different sources of error is shown to lead to their partial cancellation. From this analysis one may identify an ‘optimal refinement strategy’ for a given subgrid model, discretisation method and flow conditions, leading to minimal total simulation error at a given computational cost. We provide full detail for homogeneous decaying turbulence in a ‘Smagorinsky fluid’. The optimal refinement strategy is compared with the error reduction that arises from grid refinement of the dynamic eddy-viscosity model. The main trends of the optimal refinement strategy as a function of resolution and Reynolds number are found to be adequately followed by the dynamic model. This yields significant error reduction upon grid refinement although at coarse resolutions significant error levels remain. To address this deficiency, a new successive inverse polynomial interpolation procedure is proposed with which the optimal Smagorinsky constant may be efficiently approximated at a given resolution. The computational overhead of this optimisation procedure is shown to be well justified in view of the achieved reduction of the error level relative to the ‘no-model’ and dynamic model predictions.

Lagrangian dynamics of commutator errors in large-eddy simulation

In large-eddy simulations of turbulent flow only the flow structures with length scales larger than the local filter width Δ are explicitly resolved. We analyze the dynamic effect associated with spatial variations in the filter width. With the introduction of such a nonuniform filter width a number of additional closure terms emerges, generally referred to as commutator errors. The dynamic effect of the commutator errors is shown to correspond to the apparent local creation or destruction of turbulent flow scales, depending on, respectively, a decrease or an increase in Δ along the flow path. This Lagrangian context suggests significant correlation between the material derivative of the filter width and the production or dissipation of kinetic energy due to the commutator error. This is confirmed by novel a priori analysis of turbulent mixing. An explicit Lagrangian model for the commutator error in the momentum equations is proposed. Additionally, the dynamic effect of a skewed filter on the commutator error is investigated. It is shown that skewed filters induce both dissipative and dispersive behaviors, which are explicitly retained in the new Lagrangian model.

Discretization errors in large eddy simulation: on the suitability of centered and upwind-biased compact difference schemes

The suitability of high-order accurate, centered and upwind-biased compact difference schemes for large eddy simulation (LES) is evaluated through the static and dynamic analyses. For the static error analysis, the power spectra of the finite-differencing and aliasing errors are evaluated in the discrete Fourier space, and for the dynamic error analysis LES of isotropic turbulence is performed with various dissipative and non-dissipative schemes. Results from the static analysis give a misleading conclusion that both the aliasing and finite-differencing errors increase as the numerical dissipation increases. The dynamic analysis, however, shows that the aliasing error decreases as the dissipation increases and the finite-differencing error overweighs the aliasing error. It is also shown that there exists an optimal upwind scheme of minimizing the total discretization error because the dissipative schemes decrease the aliasing error but increase the finite-differencing error. In addition, a classical issue on the treatment of nonlinear term in the Navier–Stokes equation is revisited to show that the skew-symmetric form minimizes both the finite-differencing and aliasing errors. The findings from the dynamic analysis are confirmed by the physical space simulations of turbulent channel flow at Re=23000 and flow over a circular cylinder at Re=3900.

Database analysis of errors in large-eddy simulation

A database of decaying homogeneous, isotropic turbulence is constructed including reference direct numerical simulations at two different Reynolds numbers and a large number of corresponding large-eddy simulations at various subgrid resolutions. Errors in large-eddy simulation as a function of physical and numerical parameters are investigated. In particular, employing the Smagorinsky subgrid parametrization, the dependence of modeling and numerical errors on simulation parameters is quantified. The interaction between these two basic sources of error is shown to lead to their partial cancellation for several flow properties. This leads to a central paradox in large-eddy simulation related to possible strategies that can be followed to improve the accuracy of predictions. Moreover, a framework is presented in which the global parameter dependence of the errors can be classified in terms of the “subgrid activity” which measures the ratio of the turbulent to the total dissipation rate. Such an analysis allows one to quantify refinement strategies and associated model parameters which provide optimal total simulation error at given computational cost.

A further study of numerical errors in large-eddy simulations

Numerical errors in large-eddy simulations (LES) arise from aliasing and discretization errors, and errors in the subfilter-scale (SFS) turbulence model. Using a direct numerical simulation (DNS) dataset of stably stratified shear flow to perform a priori tests, we compare the numerical error from several finite difference schemes to the magnitude of the SFS force. This is an extension of Ghosal’s analysis [J. Comput. Phys. 125 (1996) 187] to realistic flow fields. By evaluating different grid resolutions as well as different filter–grid ratios, we provide guidelines for LES: for a second-order finite difference scheme, a filter–grid ratio of at least four is desired; for a sixth-order Padé scheme, a filter–grid ratio of two is sufficient.

Analysis of Numerical Errors in Large Eddy Simulation

We consider the question of "numerical errors" in large eddy simulation. It is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the Reynolds number and dependence only on the resolution ``$\delta$' of the eddies sought. This report considers this question analytically: Is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on $\delta$ but not Re? We consider the most common, simplest, and most mathematically tractable model and the most mathematically clear discretization. In two cases, we prove such an error estimate and try to explain why our technique of proof fails in the most general case. Our analysis aims to assume as little time regularity on the true solution as possible.

On the Effect of Numerical Errors in Large Eddy Simulations of Turbulent Flows

Aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods. Analytical and numerical studies show that aliasing errors are more destructive for spectral and high-order finite-difference calculations than for low-order finite-difference simulations. Numerical errors have different effects for different forms of the nonlinear terms in the Navier–Stokes equations. For divergence and convective forms, spectral methods are energy-conserving only if dealiasing is performed. For skew-symmetric and rotational forms, both spectral and finite-difference methods are energy-conserving even in the presence of aliasing errors. It is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite-difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences. The relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed. For low-order finite-difference simulations, truncation errors can exceed the magnitude of the subgrid scale term.

An Analysis of Numerical Errors in Large-Eddy Simulations of Turbulence

The reliability of numerical simulations of turbulence depend on our ability to quantify and control discretization errors. In the classical literature on error analysis, typically, simple linear equations are studied. Estimates of errors derived from such analyses depend on the assumption that each dependent variable can be characterized by a unique amplitude and scale of spatial variation that can be normalized to unity. This assumption is not valid for strongly nonlinear problems, such as turbulence, where nonlinear interactions rapidly redistribute energy resulting in the appearance of a broad continuous spectrum of amplitudes. In such situations, the numerical error as well as the subgrid model can change with grid spacing in a complicated manner that cannot be inferred from the results of classical error analysis. In this paper, a formalism for analyzing errors in such nonlinear problems is developed in the context of finite difference approximations for the Navier–Stokes equations when the flow is fully turbulent. Analytical expressions for the power spectra of these errors are derived by exploiting the joint-normal approximation for turbulent velocity fields. These results are applied to large-eddy simulation of turbulence to obtain quantitative bounds on the magnitude of numerical errors. An assessment of the significance of these errors in made by comparing their magnitudes with that of the nonlinear and subgrid terms. One method of controlling the errors is suggested and its effectiveness evaluated through quantitative measures. Although explicit evaluations are presented only for large-eddy simulation, the expressions derived for the power spectra of errors are applicable to direct numerical simulation as well.

Joint effects of subgrid-scale diffusion and truncation errors in large-eddy simulations of the convective boundary layer

Two runs of a large-eddy simulation model with Deardorff's and Schumann's subgrid parametrizations have been compared in order to analyze spurious effects at the top of the mixed layer due to a presence of a strong temperature “jump”. Simulations performed showed that Schumann's subgrid eddy viscosity was sufficient to spreads out sharp ripples which appeared in the numerical solution due to numerical dispersion. Deardorff's subgrid eddy viscosity was found too small near the top of the mixed layer, and as a result truncation dispersion errors caused unphysical solutions in this region.