We correct a data processing error in the article "Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor-Green vortex" by Dieter Fauconnier, Chris De Langhe and Erik Dick published in the Journal of Computational Physics 228 (2009), pp. 8053-8084.@en

The adaptive local deconvolution method (ALDM) is proposed as a new nonlinear discretization scheme designed for implicit large-eddy simulation (ILES) of turbulent flows. In ILES the truncation error of the discretization of the convective terms functions as a subgrid-scale model. Therefore, the model is implicitly contained within the discretization, and an explicit computation of model terms becomes unnecessary. The discretization is based on a solution-adaptive deconvolution operator which allows to control the truncation error. Deconvolution parameters are determined by an analysis of the spectral numerical viscosity. An automatic optimization based on an evolutionary algorithm is employed to obtain a set of parameters which results in an optimum spectral match for the numerical viscosity with theoretical predictions for isotropic turbulence. Simulations of large-scale forced and decaying three-dimensional homogeneous isotropic turbulence show an excellent agreement with theory and experimental data and demonstrate the good performance of the implicit model. As an example for transitional flows, instability and breakdown of the three-dimensional Taylor-Green vortex are considered. The implicit model correctly predicts instability growth and transition to developed turbulence. It is shown that the implicit model performs at least as well as established explicit models.@en

Flows over airfoils and blades in rotating machinery, for unmanned and micro-aerial vehicles, wind turbines, and propellers consist of a laminar boundary layer near the leading edge that is often followed by a laminar separation bubble and transition to turbulence further downstream. Typical RANS turbulence models are inadequate for such flows. Direct numerical simulation (DNS) is the most reliable, but is also the most computationally expensive alternative. This work assesses the capability of Immersed Boundary (IB) methods and Large Eddy Simulations (LES) to reduce the computational requirements for such flows and still provide high quality results. Two-dimensional and three-dimensional simulations of a laminar separation bubble on a NACA-0012 airfoil at Rec=5×104 at 5° of incidence have been performed with an IB code and a commercial code using body fitted grids. Several Subgrid Scale (SGS) models have been implemented in both codes and their performance evaluated. For the two-dimensional simulations with the IB method the results show good agreement with DNS benchmark data for the pressure coefficient Cp and the friction coefficient Cf but only when using dissipative numerical schemes. There is evidence that this behavior can be attributed to the ability of dissipative schemes to damp numerical noise coming from the IB. For the three-dimensional simulations the results show a good prediction of the separation point, but inaccurate prediction of the reattachment point unless full DNS resolution is used. The commercial code shows good agreement with the DNS benchmark data in both two and three-dimensional simulations, but the presence of significant, unquantified numerical dissipation prevents a conclusive assessment of the actual prediction capabilities of very coarse LES with low order schemes in general case.@en

In this chapter we make a connection between the filtering approach (Leonard 1974) and the averaged-equation approach (Schumann 1975) to large eddy simulation (LES). With the averaged-equation approach, the discrete system for evolving a grid-function approximation of the continuous solution is considered directly as a truncated representation of the continuous system. With the filtering approach, a continuous filtered system is considered as an approximation; the numerical error in solving this continuous system is considered to be negligibly small. The filtering approach provides an analytic framework for deriving LES equations and commonly is employed as a basis for the development of functional and structural models (Sagaut 2005) and Chapter 3 of this book. In practice, models derived on the basis of the filtering approach were plagued by the problem that the numerical error in most cases was nonnegligible. The effect of discretizing the filtered continuous equations on the subgrid-scale (SGS) force was analyzed in detail for the first time by Ghosal (1996). It was revealed that, over a large wave-number range, the truncation error of commonly employed nonspectral discretizations can be as large as the SGS stress, if not larger. During the attempt of improving eddy-viscosity-based models, it was revealed that the correlation of predicted SGS stresses with the exact SGS stresses is much less than unity. This fact is reviewed by Meneveau and Katz (2000) on the basis of experimental data. A much larger correlation is achieved by the scale-similarity model (Bardina, Ferziger, and Reynolds 1983), which does, however, underpredict SGS dissipation.@en

Subgrid-scale models in LES operate on a range of scales which is marginally resolved by the discrete approximation. Accordingly, the discrete approximation method and the subgrid-scale model are linked. One can exploit this link by developing discretization methods from subgrid-scale models, or vice versa. Approaches where SGS models and numerical discretizations are fully linked are called implicit SGS models. Different approaches to SGS modeling can be taken. Mostly, given nonlinearly stable discretizations schemes for the convective fluxes are used as main element of implicit SGS models. Recently we have proposed to design nonlinear discretization schemes in such a way that their truncation error functions as SGS model in regions where the flow is turbulent and as a second-order accurate discretization in regions where the flow is laminar. In this paper we review the current status on this so-called adaptive local deconvolution method (ALDM) and provide some application results.@en

Flows over airfoils and blades in rotating machinery, for unmanned and micro-aerial vehicles, wind turbines, and propellers consist of a laminar boundary layer near the leading edge that is often followed by a laminar separation bubble and transition to turbulence further downstream. Typical RANS turbulence models are inadequate for such flows. Direct numerical simulation (DNS) is the most reliable but is also the most computationally expensive alternative. This work assesses the capability of LES to significantly reduce the resolution requirements for such flows and still provide results of DNS quality. Two flows are considered. A flow over a flat plate with suitable velocity boundary conditions away from the plate to produce a separation bubble and a 2-D flow on a NACA-0012 airfoil. By employing several different codes we conclude that accurate LES are possible using O(1%) of the DNS resolution and that the numerical dissipation plays a significant role in LES of such flows.@en

The Adaptive Local Deconvolution Method (ALDM) is a new nonlinear discretization scheme designed for Implicit Large-Eddy Simulation (ILES). Free parameters of a solution-adaptive deconvolution operator allow to adjust the numerical truncation error which acts as implicit subgrid-scale (SGS) model. To assign physically appropriate values to these parameters the spectral numerical viscosity is analyzed. By an automatic optimization employing an evolutionary algorithm a set of parameters was found which yields an excellent spectral match for the numerical viscosity with theoretical predictions. Computational results for forced and decaying three-dimensional homogeneous isotropic turbulence show an excellent agreement with theory and experimental data.@en