Implicitly filtered large-eddy simulation (LES) is by nature numerically under-resolved. With the sole exception of Fourier-spectral methods, discrete numerical derivative operators cannot accurately represent the dynamics of all of the represented scales. Since the resolution scale in an LES usually lies in the inertial range, these poorly represented scales are dynamically significant, and errors in their dynamics can affect all resolved scales. This Letter is focused on characterizing the effects of numerical dispersion error by studying the energy cascade in LES of convecting homogeneous isotropic turbulence. Numerical energy and transfer spectra reveal that energy is not transferred at the appropriate rate to wavemodes where significant dispersion error is present. This leads to a deficiency of energy in highly dispersive modes and an accompanying pile up of energy in the well-resolved modes, since dissipation by the subgrid model is diminished. An asymptotic analysis indicates that dispersion error causes a phase decoherence between triad interacting wavemodes, leading to a reduction in the mean energy transfer rate for these scales. These findings are relevant to a wide range of LES, since turbulence commonly convects through the grid in practical simulations. Further, these results indicate that the resolved scales should be defined to not include the dispersive modes.
Large Eddy Simulation (LES) of turbulence in complex geometries is often conducted using strongly inhomogeneous resolution. The issues associated with resolution inhomogeneity are related to the noncommutativity of the filtering and differentiation operators, which introduces a commutation term into the governing equations. Neglect of this commutation term gives rise to commutation error. While the commutation error is well recognized, it is often ignored in practice. Moreover, the commutation error (i.e., projection onto the underlying discretization) has not been well investigated. Modeling the commutator between numerical projection and differentiation is crucial for correcting errors induced by resolution inhomogeneity in practical LES settings, which typically rely solely on implicit filtering. Here, we employ a multiscale asymptotic analysis to investigate the characteristics of the commutator. This provides a statistical description of the commutator, which can serve as a target for the statistical characteristics of a commutator model. Further, we investigate how commutation error manifests in simulation and demonstrate its impact on the convection of a packet of homogeneous isotropic turbulence through an inhomogeneous grid. A connection is made between the commutation error and the propagation properties of the underlying numerics. A modeling approach for the commutator is proposed that is applicable to LES with filters that include projections to the discrete solution space and that respects the numerical properties of the LES evolution equation. It may also be useful in addressing other LES modeling issues such as discretization error.
We present a novel coarse scale solver for the parareal computation of dynamical systems. The coarse scale solver can be de ned through interpolation or as the output of a neural network, and accounts for slow scale motion in the system. When pairing this coarse solver with a ne scale solver that corrects for fast scale motion through a parareal scheme, we are able to achieve the accuracy of the ne solver at the ef ciency of the coarse solver. Successful tests for smaller but challenging problems are presented, which cover both highly oscillatory solutions and problems with strong forces localized in time. The results suggest signi cant speed up can be gained for multiscale problems when using a parareal scheme with this new coarse solver as opposed to the traditional parareal setup.
We present a methodology for optimization of chromatogram alignment using a class separability measure calledthe Hotelling trace criterion (HTC). This metric is a multi-class distance measure that accounts for within-classand between-class variation. We chose the correlation optimized warping algorithm as our alignment method andused the HTC to judge the effectiveness of the alignment based on algorithm parameters called segment lengthand max warp.Biodiesel feedstock samples representing classes of soy, canola, tallow, waste grease, and hybrid were used in ourexperiments. Fatty acid methyl esters in each biodiesel were separated using gas chromatography-mass spectroscopy.The entire data set was baseline corrected, aligned, normalized, and mean-centered prior to principal components(PCs) analysis. The aligned, baseline corrected data sets were used to compute a figure of merit called warping effect,while the PC-transformed data sets were used to evaluate the HTC. The segment length and max warp parametersthat maximized the warping effect and/or HTC were then determined. Scores plots of pairs of PCs, along with 95%confidence ellipses, were created and analyzed.The results demonstrated that the parameters derived from maximizing the HTC more effectively aligned the data,as evidenced by better clustering of the biodiesels in the scores plots. This behavior was robust to the number of PCsused in the computation of the HTC. We conclude that the HTC is an objective measure of alignment quality that allowsfor optimal class separability and can be applied to optimize other methods of chromatogram alignment
Numerical Discretization Effects in Large Eddy Simulation.Fluid Mechanics Seminar, The Johns Hopkins University 2022
Discretization induced statistical artifacts in large eddy simulationAmerical Physical Society, Division of Fluid Dyanamics 2021