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Concept# Navier–Stokes equations

Summary

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).
The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Eule

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This course focuses on the physical mechanisms at the origin of the transition of a flow from laminar to turbulent using the hydrodynamic instability theory.

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We present a multi-level algorithm to approximate the inverse of the fluid block in a Navier-Stokes saddle-point matrix where the coarse level is defined as a restriction of the degrees of freedom to those of lower order finite elements. A one-level scheme involving P1 and P2 finite elements is studied in details and several transfer operators are compared by means of two reference problems. Numerical results show that restriction and prolongation operators based on L2 projection lead to faster GMRES convergence of the fluid part, for all the examined combinations of mesh size, time step and Reynolds number.

2013In this project we consider a simplified model of drug delivery in a stenosed artery. The goal is to simulate the diffusion and transport of drug in blood flows, and to evaluate how physical parameters, such as the drug diffusivity or the geometry of the artery, can influence the process. We model the drug delivery with an unsteady diffusion-advection-reaction equation, where the transport term is given by the blood velocity, obtained as the solution of the Navier-Stokes equations. Since the problem is transport dominated we consider suitable stabilization (namely SUPG and GLS stabilization) in order to control the numerical instabilities.

2014Accurate microscale windfields computations over complex topography is crucial to many particle transport models but remains a challenging task. The objective of this work focuses on the numerical simulations of micro-scale windfields over the steep Gaudergrat ridge, located in the Swiss Alps. These windfields are computed with the objective of driving a snowdrift model, consequently the work concentrates on meteorological situations close to snow storms. As snow transport occurs in the first meters above the surface, this implies a very fine resolution of order tens of meters. Airflow simulations are performed using the meteorological model ARPS (Advanced Regional Prediction System), which is based on a Large Eddy Simulation (LES) formulation of the compressible Navier-Stokes equations. The turbulent airflow features play an important role in the transport of particles. Therefore ARPS turbulence models, the Smagorinsky-Lilly and the 1.5 order Turbulent Kinetic Energy (TKE) closures, have been examined in neutral atmosphere conditions over flat terrain. ARPS mechanical turbulence schemes has hence been tested and the parameters of the Subgrid-Scales (SGS) models have been tuned. ARPS has already been proven suitable for reproducing qualitative features of airflow and over complex alpine terrain with a careful choice of the artificial initialisation and periodic boundary conditions. When lateral periodic boundary conditions are applied for airflow computations over real complex topography, instabilities arise quickly. For a quantitative and stable description of airflow presented in this work, the initialisation and boundary conditions have consequently been improved. In this study, the simulations over the Gaudergrat ridge presented are performed a one-way nesting approach. ARPS is first driven by the outputs of the MeteoSwiss model aLMo which produce initial and time dependent lateral boundary conditions. Then the application of the nesting technique permit to bridge spatial resolutions from 7km (aLMo grid resolution) to 25 m (horizontal resolution in the finer ARPS grid). Such a fine resolution is also required for Large-Eddy Simulations (LES) configuration and it is expected that a large part of the energy is resolved explicitly. The nesting technique has been applied to reproduce two selected days during the Gaudergrat Experiment (Gaudex) with stronger wind, to have conditions as close as possible to winter conditions and when thermal winds are weak. The field measurement campaign, Gaudergrat Experiment (Gaudex), in collaboration with the University of Leeds, was held from June to October 2003 at the Gaudergrat ridge, near Davos, Switzerland. The collected data are used to develop a better understanding of the airflow characteristics and turbulence features as well as to check the model results. The comparison with field data show satisfactory results for the mean flow quantities, whereas the lateral boundary condition forcing suppresses the small scales turbulent motion. A simple method is proposed to spin up turbulent motions in the finer resolution domain. This method is based on the introduction of turbulent perturbations from a precursor simulation onto the mean wind profile at the lateral boundaries. This new configuration facilitate the development of turbulence and resolves explicitly smaller scale motions without altering the mean flow. The spectral analysis of the Gaudex data highlights the fact that the turbulence on the lee side of the Gaudergrat ridge is influenced by local features, whereas at the crest, the effect of the surrounding mountains is recognisable. The statistical analysis of wind speed fluctuations shows that the turbulence in complex terrain is highly intermittent, but can be interpreted as a combination of subsets of isotropic turbulence. In complex terrain, the production of turbulence is not continuous, it is hence difficult to apply the traditional scaling and averaging laws developed for homogeneous horizontal surfaces. The heterogeneous surface conditions are likely to create additional length and time scale to generalise the statistical properties.