SFB Colloquium Guest Talk Abstract - 2017/2018 - Titi

Edriss S. Titi, Texas A&M University ALSO The Weizmann Institute of Science

On Recent Advances of the 3D Euler Equations by Means of Examples

In this talk we will use a basic example of shear flow to demonstrate some of the recent advances in the three-dimensional Euler equations. Specifically, this example was introduced by DiPerna and Majda to show that weak limit of classical solutions of Euler equations may, in some cases, fail to be a weak solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to  C1,α . Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface (vortex sheet). This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz (Birkhoff-Rott) problem where a minimal regularity implies the real analyticity of the interface. Furthermore, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. Eventually, we will discuss the recent remarkable work of De Lellis and Székelyhidi concerning the wild weak solutions of Euler equations and their non-uniqueness. In particular, we propose the following ruling out criterion for non-physical weak solutions of Euler equations: "Any weak solution which is not a vanishing viscosity limit of weak solutions of the Navier-Stokes equations should be ruled out". We will use this shear flow, and other solutions of Euler equations with certain spatial symmetry, to provide nontrivial examples for the use of this ruling out criterion.

This is a joint work with Claude Bardos.

SFB Colloquium Talk Abstract - 2017/2018 - Boyko

Vyacheslav Boyko, Freie Universität Berlin

"What is .. dynamical system identification with FEM-BV-VARX?"

The system identification (SI) is applied as an alternative to the analytical modeling. The objective is to determine a mathematical model that is able to characterize the output dynamics induced by a given input. Depending on the data and the research task several questions can be answered. Is there a relationship between the variables, how can this relationship be quantified, what amount of output signal energy can be described by a specific input, what type of nonlinearity describes the system, can we predict the output and how accurate it will be, and etc.? In this context one important step is the parameter estimation. To accomplish this the FEM-BV-VARX method will be introduced. One of its strength is to solve the regression and classification task simultaneously, providing a solution as a set of systems operating locally in time.
The aim within this talk is to explain the general idea of SI starting from linear system theory, going to nonlinear-spatio-temporal SI and analysis. In case to be able to identify the letter one the theory, methods and examples are presented in a way, to give an idea of the iterative SI process, its difficulties and capabilities using FEM-BV-VARX.

 

SFB Colloquium Talk Abstract - 2017/2018 - Vercauteren

Nikki Vercauteren, Freie Universität Berlin

"What is… Turbulence closure models based on scale similarity principles?"

In the range of turbulent flow prediction tools, Large Eddy Simulations (LES) stand in the middle, between direct numerical simulations (DNS) where all the scales of motion are resolved, and Reynolds Averaged Navier-Stokes (RANS) methods where all the turbulent scales are modeled. In LES, all the large, energy containing scales that one can computationally afford to capture on a numerical grid (the resolved scales) are simulated, and the dynamics of the small turbulent eddies (subgrid scales) that cannot be captured and their effect on the larger scales are parameterized based on resolved, or filtered, quantities.
With scale similarity in the inertial range of turbulence, a subgrid-scale (SGS) model which respects the scale-similarity should in principle be applicable at different filter scales. This principle is exploited in the dynamic SGS model to determine numerical coefficients in the model. The most widely used version is the dynamic Smagorinsky model, in which the dynamic procedure is applied to determine the appropriate value of the Smagorinsky model coefficient. Usually the appropriate coefficient is determined as the one that most accurately represents energy transfer across scales and calculating it involves averaging over directions of statistical homogeneity of the flow (for example over flow trajectories). With a refined dynamic procedure, scale-dependent coefficients are used to mitigate the assumption of scale invariance; this has proven to be useful in the vicinity of the lower boundary where the subgrid scales account for a large portion of the flow and in stably stratified conditions.
Another, structural approach is to treat the turbulent flow as a set of flow structures moving in a Lagrangian frame and to track their interactions. This is used in coherent vortex simulation methods (CVS), in which a wavelet-filer decomposes the Navier-Stokes equations solutions Wavelets are then dynamically selected to track the flow evolution with a reduced number of modes. Quantifying self-similarity in such coherent vortex approaches would give a way to extrapolate ensemble of coherent flow structures from a coarse grid to generate unresolved fluctuations, thereby defining a new turbulence closure based on self-similarity principles. This is part of the aims of project B07.

SFB Colloquium Guest Talk Abstract - 2017/2018 - Podladchikov

Yuri Podladchikov, University of Lausanne

Direct numerical simulations and parametrization of Thermo-Hydro-Mechanical-Chemical fully coupled flows

Unavoidable upscaling from millimeters to hundreds of kilometers of subsurface flows of porous fluids requires validation by natural observations, laboratory or numerical experiments. We use hints from effective media relationships applied to homogenization of pore-scale models to develop fully coupled continuum models and investigate them at wide range of scales by systematic numerical simulations. The most important features of the solutions are the spontaneous development of spatial and temporal localization into traveling shock or solitary waves or self-similar spreading solutions typical for degenerate parabolic equations. We identify the key features of numerical solutions to collapse the numerical data to simple functional relationships between averaged properties of the flows essential for building of the upscaled models.

 

SFB Colloquium Talk Abstract - 2017/2018 - von Larcher

Thomas von Larcher, Freie Universität Berlin

"What is... Scale similarity and self organisation in turbulent flows?"

Self organisation in turbulent flows leads to the emergence of coherent vortices at different scales. Such coherent structures have been highlighted by flow visualisation methods, for example by defining vortices as areas where the vorticity magnitude is greater than the rate of strain (so-called Q-criterion). To enable self-similar extrapolation of structures to small, unresolved scales, quantitative description of self-similar structures needs to be achieved. One challenging aspect is that the geometry of coherent structures can be variable; with increasing vorticity level, one typically sees an evolution from ribbon-like structures to elongated tubes. In order to be used in this context, pattern recognition techniques need to be able to detect structures despite being stretched or rotated. Furthermore, intense small-scale structures are not randomly distributed in space and time but rather form clusters of inertial-range extent, leading to an intermittent flow organization. With increasing Reynolds number, the intermittency becomes more pronounced and fluctuations in velocity gradients become more extreme, with longer tails in their probability distribution. Non-local scale interactions appear to also impact intermittency, to an extent that scales with the Reynolds number. Studying the organization in turbulent flows using data-driven methodologies will be part of project B07.

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