Dieter Fauconnier

On the performance of relaxation filtering for large-eddy simulation

In this work, the performance of large-eddy simulation (LES) based on the relaxation-filtering (RF) technique has been investigated quantitatively. In RF-based LES, the velocity field is filtered each nth time step, using a standard finite-difference filter, characterized by a specific order of accuracy m, and a fixed filtering strength σ. Hence, the procedure dissipates the amount of energy related to the residual stresses, and thus models the dissipative effect of the unresolved scales on the resolved scales. Since the order m and strength σ are related to the spectral distribution and the magnitude of the dissipation, respectively, these predefined parameters are crucial for the success of the method. Here, their influence is systematically investigated for the Taylor–Green vortex flow at a Reynolds number of 3000. First, the effects of m and σ are studied a priori in Fourier space. Further, 36 LESs are performed, each with a different combination of order m=4, 6, 8, 10, 12, 14 and strength σ=0.15, 0.2, 0.4, 0.6, 0.8, 1, and the turbulent statistics are compared with those of a direct numerical simulation, filtered at identical resolutions. The a priori, as well as the a posteriori results indicate that, for low filter orders m⩽4, the LES accuracy is rather poor and depends strongly on the filtering strength σ. However, for higher order filters, i.e. m⩾8, the accuracy is quite good and the results, including the resolved and subgrid dissipation rates, are nearly independent of the strength σ for σ⩾0.4. In this case, the spectral dissipation-distribution, determined by m, turns out to be the dominant parameter, whereas the dissipation strength, determined by σ, is of minor importance.

Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor-Green vortex

A general class of explicit and implicit dynamic finite difference schemes for large-eddy simulation is constructed, by combining Taylor series expansions on two different grid resolutions. After calibration for Re->~, the dynamic finite difference schemes allow to minimize the dispersion errors during the calculation through the real-time adaption of a dynamic coefficient. In case of DNS resolution, these dynamic schemes reduce to Taylor-based finite difference schemes with formal asymptotic order of accuracy, whereas for LES resolution, the schemes adapt to Dispersion-Relation Preserving schemes. Both the explicit and implicit dynamic finite difference schemes are tested for the large-eddy simulation of the Taylor-Green vortex flow and numerical errors are investigated as well as their interaction with the dynamic Smagorinsky model and the multiscale Smagorinsky model. Very good results are obtained.

On the spectral and conservation properties of nonlinear discretization operators

Following the study of Pirozzoli [1], the objective of the present work is to provide a detailed theoretical analysis of the spectral properties and the conservation properties of nonlinear finite difference discretizations. First, a Nonlinear Spectral Analysis (NSA) is proposed in order to study the statistical behavior of the modified wavenumber of a nonlinear finite difference operator, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. Second, the necessary conditions for local and global conservation of momentum and kinetic energy are derived and verified for nonlinear discretizations. Because the nonlinear mechanisms result in a violation of the energy conservation conditions, the NSA is used to quantify the energy imbalance. Third, the effect of aliasing errors due to the nonlinearity is analyzed. Finally, the theoretical observations are verified for two simple, thought relevant, numerical simulations.

The dynamic procedure for accuracy improvement of numerical discretizations in fluid mechanics

In CFD computations, discretization or truncation errors should be small providing an acceptable level of accuracy. In this paper, an extension is made of the recently proposed LES formalism based on sampling operators. It is shown that the sampling-based dynamic procedure, in combination with an appropriate truncation error model, can be used as a technique to increase the numerical accuracy of a discretization. The technique is resemblant to the well-known Richardson extrapolation. The procedure is tested on a 1D convection-diffusion equation and a 2D lid-driven cavity at Re=400, using a finite difference method. Promising results are found.

Large eddy simulations of differential molecular diffusion in non-reacting turbulent jets of H2/CO2 mixing with air

Large eddy simulations of non-reacting H2/CO2 jets mixing with air are performed and the calculations are compared with the experiments reported by Smith et al. [“Laser Raman scattering measurements of differential molecular diffusion in non-reacting turbulent jets of H2/CO2 mixing with air,” Phys. Fluids 7, 1455–1466 (1995)]. The influence of differential diffusion effects for Reynolds numbers Re = 1000–8000 is analyzed and a differential diffusion parameter, ξ, is defined on the basis of normalized H2 and CO2 concentrations in order to quantify the effects of differential diffusion with increasing Reynolds number. The analysis is made not only in physical space but also with scatter plots and histograms. The simulation results reveal that differential diffusion effects are significant at downstream locations (more than 15 nozzle diameters away from the inlet) only for the lower Reynolds numbers (Re = 1000–2000). However, differential diffusion effects are present for all Reynolds numbers examined close to the inlet (closer than 10 nozzle diameters). This is not only confirmed by the mean results of the differential diffusion parameter, ξ, but also by looking at the histograms of ξ. This is an important indication that differential diffusion can be important in turbulent reacting flows if laminarization of the flow or weakening of turbulent diffusion occurs. Including differential diffusion effects in turbulent reactive flows involving mixtures with vastly different mass diffusivities can, therefore, improve the accuracy of numerical simulations. Results obtained assuming equal species mass diffusivities revealed that differential diffusion effects do not have any significant influence in the velocity field.Large eddy simulations of non-reacting H2/CO2 jets mixing with air are performed and the calculations are compared with the experiments reported by Smith et al. [“Laser Raman scattering measurements of differential molecular diffusion in non-reacting turbulent jets of H2/CO2 mixing with air,” Phys. Fluids 7, 1455–1466 (1995)]. The influence of differential diffusion effects for Reynolds numbers Re = 1000–8000 is analyzed and a differential diffusion parameter, ξ, is defined on the basis of normalized H2 and CO2 concentrations in order to quantify the effects of differential diffusion with increasing Reynolds number. The analysis is made not only in physical space but also with scatter plots and histograms. The simulation results reveal that differential diffusion effects are significant at downstream locations (more than 15 nozzle diameters away from the inlet) only for the lower Reynolds numbers (Re = 1000–2000). However, differential diffusion effects are present for all Reynolds numbers examined close ...

Spectral analysis of nonlinear finite difference discretizations

The objective of the present work is to provide a theoretical analysis of the spectral properties of a selection of nonlinear finite difference schemes. The Nonlinear Spectral Analysis (NSA) of Fauconnier and Dick (2011) [6] is applied in order to study the statistical behavior of the modified wavenumber of TVD schemes in combination with five common limiter functions and also the 5th-order WENO scheme, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. The theoretical observations are verified numerically for the linear propagation of a broadband wave.

Analytical and numerical study of resolution criteria in large-eddy simulation

The present work investigates the influence of the primary filter resolution on various turbulence statistics and the representation of vortical structures in Large-Eddy Simulation (LES) of homogeneous isotropic turbulent flow. The resolution effects are investigated both analytically and numerically for an ideal LES solution with negligible modeling and numerical errors, and as such equivalent to filtered direct numerical simulation data. The Taylor-Green vortex is considered for the numerical investigation. Several resolution criteria, found in the literature, which prescribe the filter width requirements for LES, are investigated and their effect on various turbulent statistics is evaluated analytically. Further, the resolution effect on vortical structures is evaluated numerically using the Taylor-Green vortex. Finally, an optimal resolution for LES is derived via a multi-objective optimization, maximizing the resolved fractions of specifically chosen turbulent quantities while minimizing the computat...

Quality assessment of dynamic finite difference schemes on the taylor-green vortex

The performance of a class of explicit and implicit dynamic finite difference schemes (Fauconnier et al. J. Comput. Phys., 228(6):1830–1861 (2009), J. Comput. Phys., Accepted (2009)) is investigated for the Large-Eddy Simulation of the three-dimensional Taylor-Green Vortex flow (Brachet et al. (1983)), in which the dynamic Smagorinsky model and the small-small multiscale Smagorinsky model are used. The numerical errors and the modeling errors and their interactions are investigated.

On the Performance of Optimized Finite Difference Schemes in Large-Eddy Simulation of the Taylor–Green Vortex

We assess the quality of three types of finite difference schemes for LES of Taylor–Green vortex flow: the standard Taylor-based schemes, Dispersion-Relation Preserving (DRP) schemes (Tam and Webb, J. Comput. Phys. 107:262–281, 1993) and the dynamic finite difference (DFD) schemes of Fauconnier et al. (J. Comput. Phys. 228(21):8053–8084, 2009). DRP schemes and DFD schemes reduce substantially the numerical errors in comparison with standard schemes of similar stencil width.

A dynamically optimized finite difference scheme for Large-Eddy Simulation

A low-dispersive dynamic finite difference scheme for Large-Eddy Simulation is developed. The dynamic scheme is constructed by combining Taylor series expansions on two different grid resolutions. The scheme is optimized dynamically through the real-time adaption of a dynamic coefficient according to the spectral content of the flow, such that the global dispersion error is minimal. In the case of DNS-resolution, the dynamic scheme reduces to the standard Taylor-based finite difference scheme with formal asymptotic order of accuracy. When going to LES-resolution, the dynamic scheme seamlessly adapts to a dispersion-relation preserving scheme. The scheme is tested for Large-Eddy Simulation of Burgers equation. Very good results are obtained.