Derivation of the Navier–Stokes equations

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids. Basic assumptions The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are at least weakly differentiable. The equations are derived from the basic principles of continuity of mass, conservation of momentum, and conservation of energy. Sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by Ω and its bounding surface ∂Ω. The control volume can remain fixed in space or can move with the fluid. The material derivative material derivat

Microscale airflow simulations over complex alpine terrain

Françoise Faure

Accurate 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.

EPFL2008

Isogeometric Analysis for Navier-Stokes equations

Elena Queirolo

In this project, we present some applications of Isogeometric Analysis in stationary fluid dynamics problems. Specially, we focus on the numerical solution of the Stokes and Navier-Stokes equations and the fulfillment of the inf-sup conditions, specifically by computing the Brezzi and Babuška inf-sup constants. The Isogeometric Taylor-Hood spaces are presented and the stability of the Stokes and Navier-Stokes equations has been numerically tested in a variety of settings, including smooth NURBS basis functions and the multiple patches case.

2014

Numerical study of transitions in Taylor-Couette flow

This thesis is divided into two parts. In the first one the creation of a complete numerical tool, from the mesh generation to the data treatment, is presented. However time consuming this part has been, this is not the most important one. The second and more important part is concerned with the description and physical interpretation of the results obtained with the numerical tool previously developed. The numerical tool, which exists in its quasi-final version since mid-1997, fully satisfies the following requirements. It is specifically designed to study the flow in the annular space between coaxial, differentially-rotating cylinders of finite length. The spectral element method is used for the space discretization. The study of transition requires the high accuracy warranted by this type of method. The numerical scheme has also to be efficient. This requirement is satisfied, due to the fully-explicit time scheme adopted, where both diffusive and non-linear terms of the Navier-Stokes equations are treated explicitly, and the direct inversion of the pseudo-Laplacian matrix applied to the pressure. This inversion is performed in the most efficient way with a fast diagonalization technique. In the Reynolds numbers range we are interested in, the time-step limitation due to the linear viscous term is only slightly more stringent than the one due to the non-linear term. The last requirement that has been fulfilled has been to design as simple a code as possible. The entire code is constructed from a number of well-known algorithms, fitted together to enhance efficiency. However, the way we regularize the boundary conditions is new. It represents the physics more precisely than Tavener et al. [65]. There is a second original feature in our code, which is linked to the time scheme. We derived our time discretization from the scheme of Gavrilakis et al. [31]. Their scheme cannot be applied to cylindrical coordinates as it is. We thus had to modify it. The second part of the thesis consists of the numerical study of the first transitions of the Taylor-Couette flow in a finite-length geometry. The aspect ratio between the length of the cylinders and the gap between them has been chosen equal to twelve; this is small enough for the effects of the upper and lower boundaries of the flow to be significant. It is believed that these end-effects play a non-negligible role in the transition of the flow. On the other hand, the aspect ratio is large enough to make qualitative comparisons with the infinite-length case, which has been studied extensively both theoretically and numerically. The transition process depends on a relatively large number of parameters. Our investigation focuses on the case where the cylinders rotate in opposite directions. The study of counter-rotating Taylor-Couette flow for a large but finite aspect ratio is the main originality of this thesis. Furthermore, the physical mechanism of the appearance of the second bifurcation in classical Taylor-Couette flow, where only the inner cylinder is rotating, is described in the light of the non-linear interaction between velocity and vorticity.

EPFL1999

Numerical study of transitions in Taylor-Couette flow

EPFL1999

Microscale airflow simulations over complex alpine terrain

Françoise Faure

EPFL2008