Traian Iliescu

Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

This paper puts forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.

Three-Dimensional Turbulent Bottom Density Currents from a High-Order Nonhydrostatic Spectral Element Model

Overflows are bottom gravity currents that supply dense water masses generated in high-latitude and marginal seas into the general circulation. Oceanic observations have revealed that mixing of overflows with ambient water masses takes place over small spatial and time scales. Studies with ocean general circulation models indicate that the strength of the thermohaline circulation is strongly sensitive to representation of overflows in these models. In light of these results, overflow-induced mixing emerges as one of the prominent oceanic processes. In this study, as a continuation of an effort to develop appropriate process models for overflows, nonhydrostatic 3D simulations of bottom gravity are carried out that would complement analysis of dedicated observations and large-scale ocean modeling. A parallel high-order spectral-element Navier‐Stokes solver is used as the basis of the simulations. Numerical experiments are conducted in an idealized setting focusing on the startup phase of a dense water mass released at the top of a sloping wedge. Results from 3D experiments are compared with results from 2D experiments and laboratory experiments, based on propagation speed of the density front, growth rate of the characteristic head at the leading edge, turbulent overturning length scales, and entrainment parameters. Results from 3D experiments are found to be in general agreement with those from laboratory tank experiments. In 2D simulations, the propagation speed is approximately 20% slower than that of the 3D experiments and the head growth rate is 3 times as large, Thorpe scales are 1.3‐1.5 times as large, and the entrainment parameter is up to 2 times as large as those in the 3D experiments. The differences between 2D and 3D simulations are entirely due to internal factors associated with the truncation of the Navier‐Stokes equations for 2D approximation.

A Numerical Study of a Class of LES Models

This paper tests if two related types of LES models satisfy some simple necessary conditions for acceptability: replication of laminar flows and boundedness of total kinetic energy. The considered LES models are based on the approximation of the Fourier transform of the Gaussian filter by a simpler function. One uses a Taylor polynomial approximation (Taylor LES model), whereas the other model is obtained by a rational approximation (rational LES model). The numerical experiments at high Reynolds number 2D and 3D driven cavity flows show a blow up of the total kinetic energy of the solutions computed with the Taylor LES model. The details of the calculations and the review of this models derivation point to this blow up being clearly a shortcoming of the model. In contrast, the rational LES model gives solutions with bounded total kinetic energy. In addition, the large eddies are well captured on a coarse grid.

Entrainment in bottom gravity currents over complex topography from three-dimensional nonhydrostatic simulations

[1] By recognizing that oceanic overflows follow the seafloor morphology, which shows a self-similar structure at spatial scales ranging from 100 km to 1 m, the impact of topographic bumps on entrainment in gravity currents is investigated using a 3D nonhydrostatic spectral element model. It is found that a bumpy surface can lead to a significant enhancement of entrainment compared to a smooth surface. The change in entrainment is parameterized as a function of statistical estimates of the amplitude and wavenumber parameters of bumps with respect to the background slope.

Large eddy simulation of turbulent channel flows by the rational large eddy simulation model

The rational large eddy simulation (RLES) model is applied to turbulent channel flows. This approximate deconvolution model is based on a rational (subdiagonal Pade) approximation of the Fourier transform of the Gaussian filter and is proposed as an alternative to the gradient (also known as the nonlinear or tensor-diffusivity) model. We used a spectral element code to perform large eddy simulations of incompressible channel flows at Reynolds numbers based on the friction velocity and the channel half-width Reτ=180 and Reτ=395. We compared the RLES model with the gradient model and the Smagorinsky model with Van Driest damping. The RLES model was much more stable than the gradient model and yielded improved results. Both the RLES model and the gradient model predicted the off-diagonal Reynolds stresses better than the Smagorinsky model with Van Driest damping. The latter, however, yielded better results for the diagonal Reynolds stresses.

Approximate deconvolution large eddy simulation of a barotropic ocean circulation model

Abstract A new large eddy simulation closure modeling strategy is put forth for two-dimensional turbulent geophysical flows. This closure modeling approach utilizes approximate deconvolution, which is based solely on mathematical approximations and does not employ additional phenomenological arguments in the model. The new approximate deconvolution model is tested in the numerical simulation of the wind-driven circulation in a shallow ocean basin, a standard prototype of more realistic ocean dynamics. The model employs the barotropic vorticity equation driven by a symmetric double-gyre wind forcing, which yields a four-gyre circulation in the time mean. The approximate deconvolution model yields the correct four-gyre circulation structure predicted by a direct numerical simulation, but on a coarser mesh and at a fraction of the computational cost. This first step in the numerical assessment of the new model shows that approximate deconvolution could represent a viable tool for under-resolved computations in the large eddy simulation of more realistic turbulent geophysical flows.

A numerical investigation of velocity-pressure reduced order models for incompressible flows

This report has two main goals. First, it numerically investigates three velocity-pressure reduced order models (ROMs) for incompressible flows. The proper orthogonal decomposition (POD) is used to generate the modes. One method computes the ROM pressure solely based on the velocity POD modes, whereas the other two ROMs use pressure modes as well. To the best of the authors? knowledge, one of the latter methods is novel. The second goal is to numerically investigate the impact of the snapshot accuracy on the results of the ROMs. Numerical studies are performed on a two-dimensional laminar flow past a circular obstacle. Comparing the results of the ROMs and of the simulations for computing the snapshots, it turns out that the latter results are generally well reproduced by the ROMs. This observation is made for snapshots of different accuracy. Both in terms of reproducing the results of the underlying simulations for obtaining the snapshots and of efficiency, the two ROMs that utilize pressure modes are superior to the ROM that uses only velocity modes.

Approximate deconvolution large eddy simulation of a stratified two-layer quasigeostrophic ocean model

Abstract We present an approximate deconvolution (AD) large eddy simulation (LES) model for the two-layer quasigeostrophic equations. We applied the AD-LES model to mid-latitude two-layer square oceanic basins, which are standard prototypes of more realistic stratified ocean dynamics models. Two spatial filters were investigated in the AD-LES model: a tridiagonal filter and an elliptic differential filter. A sensitivity analysis of the AD-LES results with respect to changes in modeling parameters was performed. The results demonstrate that the AD-LES model used in conjunction with the tridiagonal or differential filters provides additional dissipation to the system, allowing the use of a smaller eddy viscosity coefficient. Changing the spatial filter makes a significant difference in characterizing the effective dissipation in the model. It was found that the tridiagonal filter introduces the least amount of numerical dissipation into the AD-LES model. The differential filter, however, added a significant amount of numerical dissipation to the AD-LES model for large values of the filter width. All AD-LES models reproduced the DNS results at a fraction of the cost within a reasonable level of accuracy.

A New Closure Strategy for Proper Orthogonal Decomposition Reduced-Order Models

Proper orthogonal decomposition (POD) is one of the most significant reduced-order modeling (ROM) techniques in fluid mechanics. However, the application of POD based reduced-order models (POD-ROMs) is primarily limited to laminar flows due to the decay of physical accuracy. A few nonlinear closure models have been developed for improving the accuracy and stability of the POD-ROMs, which are generally computationally expensive. In this paper we propose a new closure strategy for POD-ROMs that is both accurate and effective. In the new closure model, the Frobenius norm of the Jacobian of the POD-ROM is introduced as the eddy viscosity coefficient. As a first step, the new method has been tested on a one-dimensional Burgers equation with a small dissipation coefficient 1⁄4 10 . Numerical results show that the Jacobian based closure model greatly improves the physical accuracy of the POD-ROM, while maintaining a low computational cost. [DOI: 10.1115/1.4005928]