Abstract
Motivated by the sizable increase of available computing resources, large-eddy simulation of complex turbulent flow is becoming increasingly popular. The underlying filtering operation of this approach enables to represent only large-scale motions. However, the small-scale fluctuations and their effects on the resolved flow field require additional modeling. As a consequence, the assumptions made in the closure formulations become potential sources of incertitude that can impact the quantities of interest. The objective of this work is to introduce a framework for the systematic estimation of structural uncertainty in large-eddy simulation closures. In particular, the methodology proposed is independent of the initial model form, computationally efficient, and suitable to general flow solvers. The approach is based on introducing controlled perturbations to the turbulent stress tensor in terms of magnitude, shape and orientation, such that propagation of their effects can be assessed. The framework is rigorously described, and physically plausible bounds for the perturbations are proposed. As a means to test its performance, a comprehensive set of numerical experiments are reported for which physical interpretation of the deviations in the quantities of interest are discussed.
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The summation convention is adopted for Latin, but not for Greek indices.
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Acknowledgments
This work was funded by the United States Department of Energy’s (DoE) National Nuclear Security Administration (NNSA) under the Predictive Science Academic Alliance Program (PSAAP) II at Stanford University.
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to help improve the quality of the paper.
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Appendices
Appendix A: Framework Implementation Overview
The uncertainty quantification framework introduced in this work is developed with the objective of being suitable to LES solvers in complex geometries. A general example would be, for instance, the unstructured and massively parallel Nalu open-source code [48] utilized in the numerical experiments section. For this purpose, an implementation overview of the framework is described below.
Similar to the calculation of the turbulent viscosity in eddy-viscosity-type models, introduction of the perturbations is performed locally at each time step. Therefore, the framework is inherently parallel and easy to implement on 3-D unstructured meshes. For a general combination of perturbations, four main steps are required.
The first step is to construct \(a^{sgs}_{ij}\) from the base-model definition. For example, in the case of eddy-viscosity models, \(\overline {u_{k} u_{k}}\) and \(-2\nu _{sgs}\overline {S}_{ij}\) need to be calculated. The latter is directly accessible in most LES solvers as ν s g s is typically evaluated from expressions involving \(\overline {S}_{ij}\). The former, however, is less commonly available since it requires modeling \(\tau _{kk}^{sgs}\).
Step number two is to perform the spectral decomposition of \(a^{sgs}_{ij}\). Many efficient and robust methods exist for 3 × 3 symmetric matrices. For instance, optimized algorithms can be found in [51]. Once the eigendecomposition is obtained, the eigenvalues and the corresponding eigenvectors need to be sorted such that \(\lambda _{1}^{sgs} \geq \lambda _{2}^{sgs} \geq \lambda _{3}^{sgs}\) is satisfied.
The following step, number three, is to apply perturbations (individual or a combination) to \(a^{sgs}_{ij}\) within the framework described in Section 4. Next, the perturbed decomposition is reassembled to generate \({a^{sgs}_{ij}}^{*} = {{v}^{sgs}_{in}}^{*}{{\Lambda }^{sgs}_{nl}}^{*}{{v}^{sgs}_{jl}}^{*}\).
Finally, in step number four, \({a^{sgs}_{ij}}^{*}\) is multiplied by \({\overline {u_{k} u_{k}}}^{*}\), and the divergence of the resulting tensor, \({\overline {u_{k} u_{k}}}^{*}{a^{sgs}_{ij}}^{*}\), is introduced into the LES equations. Notice that
Therefore, instead of augmenting the molecular viscosity, ν, with the turbulent viscosity, ν s g s , as it is typical in most LES solvers, the SGS term in this framework is treated independently from the viscous stresses since the eigenvalues and eigenvectors of \(\overline {S}_{ij}\) and \({a^{sgs}_{ij}}^{*}\) are different after the perturbations are applied. The isotropic term \({\tau _{kk}^{sgs}}^{*}/3\) should be computed and integrated into the equations for compressible flows, while it can be absorbed into the filtered pressure when considering incompressible flow.
Appendix B: Mesh Convergence Study
The main objective of this work is to introduce a framework to analyze sensitivity to structural uncertainty in LES closures. Of particular interest is the case of reasonably well-resolved LES calculations in which a balance between overall accuracy and computational cost is a critical aspect. In this regard, a mesh convergence study is presented here for the computational case considered in the numerical experiments section.
The problem under consideration is LES of channel flow at R e τ = 395. Details of the computational setup are described in Section 5. Differences between three increasingly finer meshes are discussed and compared against DNS data from [49]. The size of the meshes in the streamwise, vertical and spanwise directions are 64 × 64 × 64, 64 × 128 × 64, and 64 × 128 × 96. The meshes are uniform in the streamwise and spanwise directions, while stretched following a hyperbolic tangent distribution in the wall-normal direction. The latter mesh is chosen for the numerical tests of structural uncertainty.
Results of averaged streamwise velocity profile (a) and rms velocity fluctuations (b, c, d) for the three meshes considered are shown in Fig. 10. The first observation is that the magnitudes and trends of the results are in accordance with equivalent LES calculations reported in the literature; for example in [52]. The (1) averaged streamwise velocity profile tends to be overpredicted starting from y + ≈ 10, (2) there is an overprediction in streamwise velocity fluctuation at y + ≈ 15, and (3) the vertical and spanwise fluctuations are underpredicted for all y +. However, a clear difference in trend is observed between averaged velocity profile and fluctuations. Irrespective of the spatial direction, refining the mesh improves the LES prediction of averaged velocity. Conversely, refining the mesh in the vertical direction by a factor of two does not improve significantly the accuracy in velocity fluctuation prediction, especially in the near-wall region (y + ≈ 50), while a notable improvement is achieved when refining the mesh in the spanwise direction. An interpretation of this behavior in channel flow is that turbulence tends to organize in long streaks along the streamwise direction. Consequently, a relatively small number of gridpoints is sufficient to capture the evolution of the large structures in the streamwise direction. By contrast, the turbulent eddies in the spanwise direction are small, and therefore many more gridpoints are required per spatial length to properly capture them. Typical meshes considered in LES calculations are not fine enough in the spanwise direction, and consequently large scales tend to survive longer due to the incapacity of the grids to break them. In other words, the overprediction in streamwise and underprediction in vertical and spanwise velocity fluctuations is mainly related to the ratio between mesh resolution and large scales in the spanwise direction, and is effectively independent from the subgrid-scale modeling.
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Jofre, L., Domino, S.P. & Iaccarino, G. A Framework for Characterizing Structural Uncertainty in Large-Eddy Simulation Closures. Flow Turbulence Combust 100, 341–363 (2018). https://doi.org/10.1007/s10494-017-9844-8
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DOI: https://doi.org/10.1007/s10494-017-9844-8