Boussinesq approximation (water waves)

In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours. While the Boussinesq approximation is applicable to fairly long waves

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

Christoph Bosshard

In nuclear safety, most severe accident scenarios lead to the presence of fission products in aerosol form in the closed containment atmosphere. It is important to understand the particle depletion process to estimate the risk of a release of radioactivity to the environment should a containment break occur. As a model for the containment, we use the three-dimensional differentially heated cavity (DHC) problem. DHC is a cubical box with a hot wall and a cold wall on vertical opposite sides. On the other walls of the cube we have adiabatic boundary conditions. For the velocity field the no-slip boundary condition is valid. The flow of the air in the cavity is described by the Boussinesq equations. Complex flow patterns develop and the flow characteristics depend on the non-dimensional Rayleigh and Prandtl numbers. The predominant flow type in the DHC is a turbulent natural convection flow. This work aims at reaching Rayleigh numbers and turbulent levels as high as possible given the available computational resources. The method used to simulate the turbulent flow is the large eddy simulation (LES) where the dynamics of the large eddies is resolved by the computational grid and the small eddies are modelled by the introduction of subgrid scale quantities using a filter function. Numerically, the LES equations are discretized by the spectral element method. Particle trajectories are computed using the Lagrangian particle tracking method, including the relevant forces (drag, gravity, thermophoresis). Four different particle sets with each set containing one million particles and diameters of 10 μm, 15 μm, 25 μm and 35 μm are simulated. The complexity and the size of the large three-dimensional problem requires the use of massively parallel supercomputers. Spectral element methods are naturally suitable for parallelisation by distributing the elements among the processors. For the Lagrangian particle tracking we use a method where equal numbers of particles are assigned to every processor. The flow field is broadcast and every particle processor tracks the assigned particles, a procedure which leads to a perfect load balancing. Simulation results for the flow field and particle sizes from 15 μm to 35 μm at a Rayleigh number of 109 are compared to previous results from a direct numerical simulation. First order statistics of the LES flow fields are in very good agreement with the direct numerical simulation while the agreement of second order moments is fair. Also the turbulent structures associated to the maximum of turbulent kinetic energy production are correctly reproduced. Particle statistics in the LES and the direct numerical simulation were similar and the settling rates practically identical. Contrary to previous particle simulations in LES, it was found that no model was necessary for the influence of the unresolved flow scales on the particle motions. This can be explained, because the important settling mechanism is through gravity and particle deposition at the walls by turbophoresis is negligible.

EPFL2012

Large eddy simulation of the differentially heated cubic cavity flow by the spectral element method

Christoph Bosshard, Abdelouahab Dehbi, Michel Deville, Emmanuel Leriche, Riccardo Puragliesi, Alfredo Soldati

Large eddy simulations of the turbulent natural convection flow in a differentially heated cavity have been carried out at a Rayleigh number of 10(9) using the spectral element method. To obtain the large eddy simulation equations, a low pass filter given by the numerical space discretisation is applied to the Boussinesq equations. The subgrid tensor in the filtered momentum equation is modelled by a subgrid viscosity computed by the dynamic Smagorinsky model. To model the subgrid heat flux vector in the filtered temperature equation, a subgrid diffusivity is used which is related to the subgrid viscosity by a dynamically computed subgrid Prandtl number. All filtering operations are done in an elementwise defined hierarchical polynomial basis. The test filter for the dynamic procedure is chosen so that the grid filter and the combination of the grid with the test filter are self-similar. An important parameter of the simulation namely the choice of the decomposition of the computational domain into spectral elements is fully discussed. Large eddy simulations for three different grid resolutions are validated and compared with a highly accurate direct numerical simulation. At the end, turbulence structures associated with the maximum of the turbulent kinetic energy production are identified by the lambda(2) criterion. (C) 2013 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2013

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

Christoph Bosshard

EPFL2012

Boussinesq approximation (water waves)

An alternative to Boussinesq approximation

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

Large eddy simulation of the differentially heated cubic cavity flow by the spectral element method

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

An alternative to Boussinesq approximation

Large eddy simulation of the differentially heated cubic cavity flow by the spectral element method