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# A Family of Orthogonalised Nonlinear LES Models Based on the Velocity Gradient: Discretisation and Analysis

Remmerswaal, R. A. (2016) A Family of Orthogonalised Nonlinear LES Models Based on the Velocity Gradient: Discretisation and Analysis. Master's Thesis / Essay, Applied Mathematics.  Preview
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## Abstract

The Navier-Stokes equations (NSE) are a model for fluid flow. When simulating a turbulent flow at high Reynolds number the required resolution to capture all scales of motion is too high. Therefore we want to find equations that govern the temporal evolution of the local spatial average of the velocity field instead. Due to the nonlinearity of the NSE, expressing the equations merely in terms of the local spatial average of the velocity field requires a model which describes the interaction of the small scales of motion (not represented in the simulation) with the large scales of motion. Such a model is called a Large Eddy Simulation (LES) model. There is a large variety of such LES models, perhaps the most common ones are given by eddy-viscosity models. In an eddy-viscosity model the aim is to model the dissipative behaviour of the interaction between the small and large scales of motion. Eddy-viscosity models are generally functions of the velocity gradient. An example of a non-eddy-viscosity model is the Gradient model, which depends nonlinearly on the velocity gradient. This model is a mathematically consistent approximation to the exact interactions between the small and large scales of motion. The Gradient model is not purely dissipative (like an eddy-viscosity model) and therefore also models some transport mechanism. The Gradient model, however, turns out to be inherently unstable. One way of stabilising this model is by using only it's dissipative part. Thereby decomposing the Gradient model in it's dissipative and transport part. We then propose a generalisation of the previously mentioned LES models: we assume the LES model to be a function of the velocity gradient. Moreover, we generalise the separation of dissipation and transport mechanisms by means of orthogonalisation. This yields a framework of LES models expressed in terms of a linear combination of eleven model tems. Only one of these terms is dissipative and is included in the previously mentioned eddy-viscosity models. The Gradient model can also be represented in this framework. The research done in this report is aimed at finding out how to use the transport terms as a part of a LES model, and therefore also modelling the non-dissipative interactions between the small and large scales of motion. This research will be done by means of numerical experiments which are verified by comparison to other LES models as well as experimental data. Before such simulations can be done, we need a suitable discretisation method. We choose to use a symmetry-preserving Finite Volume (FV) discretisation method. Here, symmetry-preserving indicates the preservation of symmetry properties of certain differential operators present in the NSE in the discretisation. This results in a discrete equivalent to the analytical energy equality. Such preservation of symmetry properties is desirable since it results in stability properties in the simulation. We extend this symmetry-preserving discretisation by discretising the contribution of the LES model to the momentum equation. We consider several methods for discretising the LES model. Such methods are second-order accurate and may be extrapolated. Moreover we emphasise the importance of preserving the energy equality in the discretisation and therefore desire local orthogonality properties as well as zero contribution to the discrete energy equality whenever the model is non-dissipative. This results in choosing a single discretisation method which we use in the research that follows. Simulations using a non-dissipative term confirm the importance of preserving the energy equality in the discretisation: a discretisation method which does not preserve this results in an unstable simulation. Provided with this discretisation method we can now start characterising the terms present in the general framework of LES models. We choose a single transport term. By comparing the simulation of merely this term to a known transport term, given by the convection operator, we are able to characterise this transport term in terms of how it distributes kinetic energy over the scales of motion. We find that the net result is transport from large (medium) to small scales of motion. Therefore we expect that in combination with a dissipative term we can obtain a similar decay in kinetic energy as a purely dissipative model (eddy-viscosity model), while using less eddy-viscosity. To test this hypothesis we use the test case of decaying Homogeneous and Isotropic Turbulence (HIT) in a periodic box. We propose a two-parameter LES model which is a linear combination of a dissipative term and a non-dissipative term given by the previously characterised transport term. We perform numerous simulations and quantify the agreement to the kinetic energy decay from the experimental data. This results in a single set of parameters for which this agreement is optimised. Such a comparison only assesses the dissipative behaviour of the LES model. To obtain good dissipative behaviour however, a purely dissipative model is sufficient. Therefore we also consider other quantities like the energy spectrum function, which, when considered as a function of time, provides insight on how the energy is distributed among the different scales of motion. We find that the introduction of a transport term results in a small pile-up of energy at the smallest scales. This is to be expected, since we simultaneously increase the transport of energy to small scales of motion, as well as decrease the amount of dissipation (which is most active at small scales of motion). Besides kinetic energy (per wavenumber) we also consider several statistical quantities. For instance, we compute the two-point correlation functions which are also available from the experimental data. Here we find that using a transport term as part of a LES model does not worsen the agreement to the experimental data in terms of the statistical measures. This may not sound optimistic at first, however such results are promising in the sense that we have shown that using a transport term to model the non-dissipative part of the interaction between the small and large scales of motion results in good dissipative properties while maintaining similarly good agreement in terms of statistical measures. Hence allowing the modelling of non-dissipative mechanisms in a LES while maintaining the previously mentioned good agreement. Such modelling is of course far from fully explored in this work.

Item Type: Thesis (Master's Thesis / Essay) Applied Mathematics Master's Thesis / Essay English 15 Feb 2018 08:30 15 Feb 2018 08:30 http://fse.studenttheses.ub.rug.nl/id/eprint/15547

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