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To develop a complete numerical model for predicting fluid flow and mass transfer in domains with porous walls such as those found in cross flow filtration is not an easy task to undertake. Many gaps in theory exist that undermine any attempt to build a reliable representation of this process and a sensible effort to solving this problem must be built on sound and logical assumptions. The difficulties with modelling interacting particles, with simulating multiphase flow and with prescribing accurate boundary conditions are very much the essence of the problem. From a comprehensive literature search into areas of combined free and porous flow, mass transfer in porous domains and into the fields of rheology, and mathematical modelling of crossflow filtration it was discovered that present research although great in quantity, is overall limited by the difficulties described above. As well, the present research found in the literature is also limited for use in industrial applications as it generally considers dilute suspensions, it is often found to look at simple flow profiles for Newtonian fluids, the research scarcely looks into the dependency of flow profiles and mass transfer profiles on each other via rheology and many researchers who study crossflow filtration concentrate solely on the porous wall to solve the flux paradox situation and do not generally consider the whole domain. The purpose of the present thesis is to describe the concept, procedure and results behind the integration of a solids transport model into a previously developed flow algorithm and the explanation of ideas for solving the problem of prescribing appropriate concentration boundary conditions at the porous wall. The aim of the research is to develop a fast and cost effective tool for solving the given problem based on rational assumptions.

The work presented concerns the way small particles attached to a surface are resuspended when exposed to a turbulent flow. Of particular concern to this work is the remobilization of radioactive particles as a consequence of potential nuclear accidents. In this particular case the focus is on small particles, < 5 microns in diameter, where the principal force holding such particles onto a surface arises from van der Waals inter-molecular forces. Given its suitable treatment of the microphysics of small particles, it was decided here to aim to develop improved versions of the Rock’n’Roll (R’n’R) model; the R’n’R model is based on a statistical approach to resuspension involving the rocking and rolling of a particle about surface asperities induced by the moments of the fluctuating drag forces acting on the particle close to the surface. Firstly, a force (moment) balance model has been modified by including the distribution of the aerodynamic force instead of considering only its mean value. It was also possible to improve the representation of the adhesive-force distribution where it is customary to include a substantial reduction factor to take account of surface roughness. The R’n’R model is significantly improved by using realistic statistical fluctuations of both the stream-wise fluid velocity and acceleration close to the wall obtained from Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) of turbulent channel flow; in the standard model a major assumption is that these obey a Gaussian distribution. The flow conditions are translated into the moments of the drag force acting on the particle attached to the surface (using O’Neill’s formula for the aerodynamic drag forces in terms of the local flow velocities). In so doing the influence of highly non-Gaussian forces (associated with the sweeping and ejection events in a turbulent boundary layer) on the resuspension rate has been examined along with the sensitivity of the fluctuation statistics to LES and DNS. We have found most importantly that the statistics of both fluctuating forces and its derivative (normalized on their rms values) are noticeably independent of the normalized distance from the wall, y+ within the viscous sublayer (y+ < 6) – if this were not the case then modelling fluctuations with different particle sizes would be far more complex. In particular as a result of the analysis of our DNS/LES data 3 distinct features of the modified R’n’R model have emerged as playing an important part in the resuspension. The first is the typical forcing frequency ω due to the turbulent (fluctuating) aerodynamic drag forces acting on the particle attached to a surface (in the modified R’n’R model based on the DNS results (y+ = 0.1) it is a factor of 4 > the value in the original model based on Hall’s measurements of the lift force). This naturally has a significant effect of increasing the fraction resuspended for very short times (ωt ~< 1) iv and is the controlling influence over the entire range of times from short to long term resuspension. The second is the value of the ratio of the root-mean-square (rms) drag force to its mean value which in the modified model is nearly twice (1.8) than that in the original. This feature of the model is largely responsible for the greater fraction resuspended after times ~ 1s (times which are sufficient to include the transition period from short term resuspension to long term resuspension rates (~t-1). The third feature introduces changes in the resuspension because the distribution of aerodynamic drag forces in the modified model is distinctly non-Gaussian behaving more like a Rayleigh distribution. This means that the distribution of the drag force decays much more slowly in the wings of the distribution than the equivalent Gaussian (with the same rms) so that for very large values of the adhesive force / rms drag force ~ 8 (at the extreme end of the DNS measurements), the resuspension rate constant is a factor of 30 larger than that for an equivalent Gaussian model. Thus although the fraction of particles resuspended is very small in these instances, the differences between the modified and original models can be very large. This is particularly important when we consider resuspension from multilayer deposits. When we consider these influences in the context of a broad range of adhesive forces due to surface roughness, we find that in general, the modified model gives around 10% more for the fraction of particle resuspension fraction than the original R’n’R model (for an initial log normal distribution of adhesive forces), however the difference could become significant (3 to 7 times greater depending on the range of values of the adhesive-force spread factor) when the friction velocity is small (i.e., smaller resuspension fraction). As for the short-term resuspension rate, the difference between the modified and original model becomes significant when this is dominated by the typical forcing frequency (ω+ is 0.0413 for the original model, 0.08553 for LES approach and 0.127143 for DNS for y+ = 6). The sensitivity to the adhesive-force spread factor has also been studied and the results indicate that the modified model removes particles much more easily than the original model in conditions of small friction velocity and a smoother surface (i.e., small spread factor). Finally in this phase of the work, the correlation between the distribution of the fluctuating force and its derivative has been checked for both LES and DNS statistics. The results demonstrate that this correlation has a very slight effect on particle resuspension compared with the result from the uncorrelated curve-fitted model. In view of recent numerical data for lift and drag forces in turbulent boundary layers (Lee & Balachandar), the lift and drag we have considered and the impact of these data on predictions made by the non-Gaussian R’n’R model are compared with those based on O’Neill formula. The results indicate that, in terms of the long-term resuspension fraction, the difference is minor. It is concluded that as the particle size decreases the L&B method will lead to less-and-less long-term resuspension. Finally the ultimate model that has been developed in this work is a hybrid version of the R’n’R model adapted for application to multilayer deposits based on the Friess and Yadigaroglu multilayer v approach. The deposit is modelled in several overlying layers where the coverage effect (masking) of the deposit layers has been studied; in the first instance a monodisperse deposit with a coverage ratio factor was modelled where this was subsequently replaced by the more general case of a polydisperse deposit with a particle size distribution. The results indicate that, in general, as the number of modelled layers increases the resuspension fraction of the whole deposit after a certain time decreases significantly. In other words, it takes a much longer time to resuspend a thicker deposit. Taking account of the particle size distribution slightly increases the short-term resuspension. However, this change decreases the long-term resuspension significantly. The model results have been compared with data from the STORM SR11 test (ISP-40) and the BISE experiments. In general, both comparisons indicate that with smaller spread of the adhesive force distribution (i.e., the range of adhesive force distribution is narrower) the new multilayer model agrees very well with the experimental data. It can be inferred that multilayer deposits lead to much narrower distributions of adhesive force.

For nearly 40 years, engineers, researchers and scientists from the nuclear industry across the World have been trying to understand the behaviors of deposition, bounce and re-suspension of heavy, radioactive particles suspended as a dilute secondary phase in the cooling circuits of primary reactor systems. The aim is to understand the mechanism of transport and deposition of such particles through large, complex geometry systems, so that the risk of dispersal of radioactive particles may be assessed, and confirmed to be acceptably small both in closed containers and in the atmosphere in the case of an accident scenario. The first part of the present work addresses the challenge of robustly and efficiently predicting the behaviors of rigid and spherical particles (referred to as heavy particles) within turbulent boundary layers, the underlying physics of which is the controlling factor on particle deposition in smooth pipes and ducts. In the second component of work we study the deposition and bounce of heavy particles suspended in turbulent flows across heat exchanger tube banks, using Large Eddy Simulation (LES). It was originally proposed to extend the boundary layer work to this application, but it was quickly identified that the deposition mechanisms here are governed by the high core flow turbulence, rather than boundary layer phenomena, so that LES provides the only realistic modelling approach. In both cases the dispersed heavy particles are expressed in a Lagrangian framework solved in an independently developed large-scale parallel code; whilst the fluid phase is described in an Eulerian framework, either based on correlations from published Direct Numerical Simulation (DNS) for the boundary layer models, or from Computational Fluid Dynamics (CFD) simulations for both the boundary layer and tube-bank models, making use of the unstructured-grid based Navier-Stokes solver ANSYS FLUENT. Underpinning this work we implement a complete stochastic Lagrangian particle tracking module, based on a robust and efficient particle localization algorithm which can determine and update the cell containing each particle as the particles move through an unstructured finite volume grid overlying the flow domain. The module can handle correctly the interactions of particles with complex boundaries, and uses a novel numerical scheme for interpolating the carrier-phase velocity field seen by the particles from cell-centred values obtained from CFD computation. It implements a Gear three-level implicit scheme to compute the particle velocity, which is more robust, accurate and efficient than the conventional explicit and implicit schemes. The module has been fully parallelized using MPI (Message Passing Interface) settings on a Linux cluster consisting of 20 single CPU node, and further been successfully integrated with both the steady and unsteady ANSYS FLUENT solvers, complete replacing the built-in Lagrangian particle tracking model provided by ANSYS FLUENT. The algorithm and numerical schemes have been validated against analytical solutions of particle transport in a two-dimensional straining shear flow and other cases. For turbulent boundary layer flows, a simpler but more promising stochastic quadrant model, inspired by the discrete random walk model of Kallio and Reeks and the quadrant analysis of Wu and Willmarth, is developed in order to account for the effects of near wall large-scale coherent structures, e.g. sweeps and ejections, on particle transport. The input parameters for the stochastic quadrant model are educed from the corresponding statistics obtained from a Large Eddy Simulation (LES) of a fully developed channel flow. The model is applied to the prediction of deposition of heavy particles in a turbulent boundary layer; both using a Kallio and Reeks correlation based model of the flow, and also a Reynolds-Averaged Navier-Stokes (RANS) flow solution of using ANSYS FLUENT, the latter flow model having the potential to be extended to complex duct geometries. These solutions are compared to those of by solving an alternative Langevin equation of Dehbi, or continuous random walk model, which satisfies the fully mixed condition and describes the fluid velocity fluctuations seen by heavy particles. Prior to the current work no systematic investigation of the potential errors in particle deposition in turbulent boundary layers due to the modified hydrodynamic forces experienced by particles when very close to the wall has been carried out, possibly because of the complexity of the correlations involved. The effect is explored with the present stochastic quadrant model, using recently published composite correlations of Zeng and Balachandar for the particle drag coefficient CD and lift coefficient CL for near wall particles. This work provides an important first confirmation that for practical cases hydrodynamic effects can reasonably be neglected for particle deposition in turbulent boundary layers. The boundary layer methods developed in the first part of this thesis are applicable to the prediction of heavy particle deposition in fairly complex duct geometries, but are shown to be inappropriate for flow over tube-banks, where the boundary layers are no longer the rate limiting feature. Consequently the parallel Lagrangian stochastic particle tracking model is extended to study the particle impaction efficiency on tube banks in a turbulent flow in the framework of Large Eddy Simulation (LES). The flow field, obtained from Large Eddy Simulation with the dynamic Smagorinsky sub-grid scale model within ANSYS FLUENT, is fully validated against existing experimental data. As far as the dispersed particle phase is concerned, the energy losses when particles impact on and generally, but not always, rebound from cylinders within the tube-bank is taken into account using an empirical critical-impact velocity model. The efficiency of particle impaction is measured for particles of three Stokes number, and the results are compared with existing experimental data.

This thesis describes experiments conducted in 3He-B to investigate the properties of quantum turbulence created by an oscillating grid. The turbulence is detected using an array of vibrating wire resonators. We have measured various steady-state properties of the turbulence includ- ing fluctuations, and investigated how turbulence propagates and evolves by examining the correlations between pairs of detectors. The quantum turbu- lence generated in our experiment is found to decay in a way similar to that found in classical fluids, as described by the Richardson cascade.

The first part is the derivation of one-particle vertical diffusion for stably stratified turbulence with or without rapid rotation. Nicolleau & Vassilicos (2000) have analytically calculated vertical one-particle diffusion in stably stratified turbulence without rotation. One-particle vertical diffusion for turbulence with stable stratification and with or without rapid rotation has been derived here analytically using the solutions of the linearized equations of motions. The second part is an attempt to explain the depletion of horizontal pair diffusion in strongly stratified turbulence. "Recently, Nicolleau et al. (2005) have shown that in their Kinematic Simulations (KS) of vertically stably and strongly stratified homogeneous turbulence (Froude number smaller than 1). horizontal pair diffusion is significantly depleted by comparison to unstratified isotropic and homogeneous two- and three-dimensional turbulence. We have seeked to explain this depletion of horizontal pair diffusion by vertical stratification in terms of the probability density function of the horizontal divergence of the velocity field and the statistics of stagnation points following the recent approach to Richardson pair diffusion by Davila & Vassilicos (2003), Goto & Vassilicos (2004), Goto et al. (2005) and Osborne et al. (2005). We measure the number density of stagnation points in the KS of three-dimensional strongly stratified turbulence and find that it is virtually identical to what it is in KS of three-dimensional isotropic turbulence The third part is a study of the vertical motions of small, spherical inertial particles in strongly stratified turbulence.