Bernard J. Geurts

Large-eddy simulation of the turbulent mixing layer

Six subgrid models for the turbulent stress tensor are tested by conducting large-eddy simulations (LES) of the weakly compressible temporal mixing layer: the Smagorinsky, similarity, gradient, dynamic eddy-viscosity, dynamic mixed and dynamic Clark models. The last three models are variations of the first three models using the dynamic approach. Two sets of simulations are performed in order to assess the quality of the six models. The LES results corresponding to the first set are compared with filtered results obtained from a direct numerical simulation (DNS). It appears that the dynamic models lead to more accurate results than the non-dynamic models tested. An adequate mechanism to dissipate energy from resolved to subgrid scales is essential. The dynamic models have this property, but the Smagorinsky model is too dissipative during transition, whereas the similarity and gradient models are not sufficiently dissipative for the smallest resolved scales. In this set of simulations, at moderate Reynolds number, the dynamic mixed and Clark models are found to be slightly more accurate than the dynamic eddy-viscosity model. The second set of LES concerns the mixing layer at a considerably higher Reynolds number and in a larger computational domain. An accurate DNS for this mixing layer can currently not be performed, thus in this case the LES are tested by investigating whether they resemble a self-similar turbulent flow. It is found that the dynamic models generate better results than the non-dynamic models. The closest approximation to a self-similar state was obtained using the dynamic eddy-viscosity model.

On the formulation of the dynamic mixed subgrid-scale model

The dynamic mixed subgrid‐scale model of Zang et al. [Phys. Fluids A 5, 3186 (1993)] (DMM1) is modified with respect to the incorporation of the similarity model in order to remove a mathematical inconsistency. Compared to DMM1, the magnitude of the dynamic model coefficient of the modified model (DMM2) is increased considerably, while it is still significantly smaller than as occurs in the dynamic subgrid‐scale eddy‐viscosity model of Germano [J. Fluid Mech. 238, 325 (1992)] (DSM). Large eddy simulations(LES) for the weakly compressible mixing layer are conducted using these three models and results are compared with direct numerical simulation (DNS) data. LES based on DMM1 gives a significant improvement over LES using DSM, while even better agreement is achieved with DMM2.

Realizability conditions for the turbulent stress tensor in large-eddy simulation

The turbulent stress tensor in large-eddy simulation is examined from a theoretical point of view. Realizability conditions for the components of this tensor are derived, which hold if and only if the filter function is positive. The spectral cut-off, one of the filters frequently used in large-eddy simulation, is not positive. Consequently, the turbulent stress tensor based on spectrally filtered fields does not satisfy the realizability conditions, which leads to negative values of the generalized turbulent kinetic energy k. Positive filters, e. g. Gaussian or top-hat, always give rise to a positive k. For this reason, subgrid models which require positive values for k should be used in conjunction with e. g. the Gaussian or top-hat filter rather than with the spectral cutoff filter. If the turbulent stress tensor satisfies the realizability conditions, it is natural to require that the subgrid model for this tensor also satisfies these conditions. With respect to this point of view several subgrid models are discussed. For eddy-viscosity models a lower bound for the generalized turbulent kinetic energy follows as a necessary condition. This result provides an inequality for the model constants appearing in a ‘Smagorinsky-type’ subgrid model for compressible flows.

Regularization modeling for large-eddy simulation

A new modeling approach for large-eddy simulation (LES) is obtained by combining a regularization principle with an explicit filter and its inversion. This regularization approach allows a systematic derivation of the implied subgrid model, which resolves the closure problem. The central role of the filter in LES is fully restored, i.e., both the interpretation of LES predictions in terms of direct simulation results as well as the corresponding subgrid closure are specified by the filter. The regularization approach is illustrated with Leray-smoothing of the nonlinear convective terms. In turbulent mixing the new, implied subgrid model performs favorably compared to the dynamic eddy-viscosity procedure. The model is robust at arbitrarily high Reynolds numbers and correctly predicts self-similar turbulent flow development.

A framework for predicting accuracy limitations in large-eddy simulation

The accuracy of large-eddy simulations is limited, among other things, by the quality of the subgrid parametrization and the numerical contamination of the smaller retained flow structures. We characterize the total simulation error in terms of the “subgrid-activity” s, which measures the relative turbulent dissipation rate (0⩽s⩽1) and the “subgrid resolution” r. This analysis is applied to turbulent mixing of a “Smagorinsky fluid” using a finite volume discretization of fourth order accuracy. On fixed coarse grids, i.e., at constant computational cost, the total simulation error decreases monotonically with filter width Δ for large s while for smaller s the total error may even increase with decreasing Δ. The corresponding modeling- and spatial discretization-error contributions are quantified at various resolutions.

A priori tests of large eddy simulation of the compressible plane mixing layer

Three important aspects for the assessment of the possibilities of Large Eddy Simulation (LES) of compressible flow are investigated. In particular the magnitude of all subgrid-terms, the role of the discretization errors and the correlation of the turbulent stress tensor with several subgrid-models are studied. The basis of the investigation is a Direct Numerical Simulation (DNS) of the two- and three-dimensional compressible mixing layer, using a finite volume method on a sufficiently fine grid. With respect to the first aspect, the exact filtered Navier-Stokes equations are derived and all terms are classified according to their order of magnitude. It is found that the pressure dilatation subgrid-term in the filtered energy equation, which is usually neglected in the modelling-practice, is as large as e.g. the pressure velocity subgrid-term, which in general is modelled. The second aspect yields the result that second- and fourth-order accurate spatial discretization methods give rise to discretization errors which are larger than the corresponding subgrid-terms, if the ratio between the filter width and the grid-spacing is close to one. Even if an exact representation for the subgrid-scale contributions is assumed, LES performed on a (considerably) coarser grid than required for a DNS, is accurate only if this ratio is sufficiently larger than one. Finally the well-known turbulent stress tensor is investigated in more detail. A priori tests of subgrid-models for this tensor yield poor correlations for Smagorinskys model, which is purely dissipative, while the non-eddy viscosity models considered here correlate considerably better.

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.

COMPARISON OF NUMERICAL SCHEMES IN LARGE-EDDY SIMULATION OF THE TEMPORAL MIXING LAYER

A posteriori tests of large-eddy simulations for the temporal mixing layer are performed using a variety of numerical methods in conjunction with the dynamic mixed subgrid model for the turbulent stress tensor. The results of the large-eddy simulations are compared with filtered direct numerical simulation (DNS) results. Five numerical methods are considered. The cell vertex scheme (A) is a weighted second-order central difference. The transverse weighting is shown to be necessary, since the standard second-order central difference (A) gives rise to instabilities. By analogy, a new weighted fourth-order central difference (B) is constructed in order to overcome the instability in simulations with the standard fourth-order central method (B). Furthermore, a spectral scheme (C) is tested. Simulations using these schemes have been performed for the case where the filter width equals the grid size (I) and the case where the filter width equals twice the grid size (II). The filtered DNS results are best approximated in case II for each of the numerical methods A, B and C. The deviations from the filtered DNS data are decomposed into modelling error effects and discretization error effects. In case I the absolute modelling error effects are smaller than in case II owing to the smaller filter width, whereas the discretization error effects are larger, since the flow field contains more small-scale contributions. In case I scheme A is preferred over scheme B, whereas in case II the situation is the reverse. In both cases the spectral scheme C provides the most accurate results but at the expense of a considerably increased computational cost. For the prediction of some quantities the discretization errors are observed to eliminate the modelling errors to some extent and give rise to reduced total errors.

Subgrid-modelling in LES of Compressible Flow

Subgrid-models for Large Eddy Simulation (LES) of compressible turbulent flow are tested for the three-dimensional mixing layer. For the turbulent stress tensor the recently developed dynamic mixed model yields reasonable results.A priori estimates of the subgrid-terms in the filtered energy equation show that the usually neglected pressure-dilatation and turbulent dissipation rate are as large as the commonly retained pressure-velocity subgrid-term. Models for all these terms are proposed: a similarity model for the pressure-dilatation, similarity andk-dependent models for the turbulent dissipation rate and a dynamic mixed model for the pressure-velocity subgrid-term. Actual LES demonstrates that for a low Mach number all subgrid-terms in the energy equation can be neglected, while for a moderate Mach number the effect of the modelled turbulent dissipation rate is larger than the combined effect of the other modelled subgrid-terms in the filtered energy equation.

Direct and large-eddy simulation IV

Preface. Experimental measurements of subgrid passive scalar anisotropy and universality H.-S. Kang, C. Meneveau. Simulation of the motion of particles in turbulent flow J.G.M. Kuerten, et al. On the accuracy of symmetry-preserving discretization R.W.C.P. Verstappen. DNS of turbulent supersonic channel flow R. Lechner, et al. LES of wall-bounded turbulence based on a 6th-order compact scheme H.J. Kaltenbach, D. Driller. Large eddy simulations using the subgrid-scale estimation model and truncated Navier-Stokes dynamics J.A. Domaradzki, et al. Assessment of some models for LES without/with explicit filtering G.S. Winckelmans, H. Jeanmart. Alignment of eigenvectors for strain rate and subgrid-scale tress tensors K. Horiuti. A finite-mode spectral model of homogeneous and isotropic Navier-Stokes turbulence E. Leveque, C.R. Koudella. Analysing near-wall behaviour in a separating turbulent boundary layer by DNS M. Manhart. A dynamic subgrid-scale model based on the turbulent kinetic energy O. Debliquy, et al. A model using incremental scales applied to LES of turbulent channel flow F. Bouchon, T. Dubois. Progress in direct simulations of 3d turbulent flames D. Thevenin, et al. Direct numerical simulation of premixed turbulent combustion Th.C. Treurniet, et al. Partially-premixed combustion during autoignition of a turbulent nonpremixed flame R. Hilbert, et al. DNS of turbulent H2/O2 premixed flames using compressible and low-mach number formulations J. de Charentenay, et al. Numerical simulation of turbulent reacting flow A.J. Vrieling, et al. Transition in LES of bluff body flows and airfoils J. Frohlich, C.P. Mellen. Large-eddy simulation of flow around a high lift airfoil I. Mary, P. Sagaut. An LESinvestigation of the separated flow past an airfoil at high angle of attack M. Breuer, N. Jovicic. Direct numerical simulation of three-dimensional transition in the incompressible flow around a wing Y. Hoarau, et al. DNS of turbulent flows in a channel with roughness S. Leonardi, P. Orlandi. Dynamics of transitional noncircular buoyant reactive jets with side-wall effects X. Jiang, K.H. Luo. Direct and large-eddy simulation of a transitional rectangular jet B. Rembold, et al. Coherent structures in excited spatially evolving round jets C.B. Da Silva, O. Metais. Direct and large-eddy simulations of compressible open cavity flows E.J. Avital. Diagonal cartesian method on staggered grids for a DNS in a tube bundle C. Moulinec, et al. A new approach towards subgrid modeling in magnetohydrodynamic turbulence W.-C. Muller, D. Carati. Numerical diffusion based on high-order derivatives in MUSCL schemes for LES on unstructured grids S. Camarri, et al. Large-eddy simulation of variable-density turbulent flows impinging on wall plates and cavity enclosures K.H. Luo, X. Zhou. An adaptive wavelet method for fluid-structure interaction N.K.R. Kevlahan, O.V. Vasilyev. Development of coherent perturbations in a laminar boundary layer H.C. de Lange, R.J.M. Bastiaans. LES of supersonic boundary layers using the approximate deconvolution model S. Stolz, N.A. Adams. Large-eddy simulation of a spatially growing thermal boundary layer in a turbulent square duct M.S. Vazquez, O. Metais. DNS of transition near the leading edge of an aerofoil M. Alam, N.D. Sandham. DNS of turbulent channel flow in the presence of a thin liquid film S. Solbakken, et al. Direct Simulation of transonic flow over a bump A.A. Lawa