Simulations were performed using Matlab on a PC. The velocity and diffusion/dispersion fields were created using experimental velocity (Uy, Uz) and diffusion/dispersion (Dy, Dz) maps for axial flow at Q = 7.2 cm3 min−1, Taylor vortex flow at ω = 1 Hz and VFR flow at Q = 7.2 cm3 min−1and ω =1 Hz. A more detailed analysis of the properties of these flows is provided in chapters 3 and 4. Molecular displacements in the azimuthal direction were assumed to be symmetric. Also the flow fields present symmetry with respect to the inner cylinder. This allowed considering only the velocity and diffusion/dispersion fields obtained
on one side of the Couette cell gap.
As discussed earlier, diffusion/dispersion maps do not provide with quantitative values of diffusion and dispersion within each voxel (cf. 3.3.1.2). This is due to the fact that the measured values are affected by both velocity shear and the pixel size. An alternative to the use of these maps would have been to use a constant self-diffusion coefficient for these simulations. But for sufficiently long mixing times, the effect of diffusion-dispersion on overall molecular displacements was expected to be negligible compared to the effect of the velocity field. In fact, for all the simulations with mixing times Δ > 100 ms presented in this chapter, replacing the diffusion/dispersion maps with a constant diffusion coefficient gave similar molecular distribution results. Hence, the errors coming from the use of the experimental diffusion/dispersion maps were only expected to play a role for short simulation times. Moreover, using these maps allowed accounting for effects related to spatial inhomogeneity of dispersion.
The simulations used n molecules identified by their two-dimensional coordinates in the velocity field plan. For simulations of stationary and translating vortices, the molecules were evenly distributed throughout several vortex pairs within the velocity maps. The same distribution was used in axial flow simulations. The dimensions of the experimental velocity and diffusion/dispersion maps do not allow simulating displacements over more than a distance corresponding to five vortex pairs. For pipe flow, where velocity is constant along the axial direction, an average velocity profile was considered over longer distances. For the vortical flows, spatial periodicity allowed for longer velocity maps to be constructed by juxtaposition of series of an experimental map region corresponding to one vortex pair. For
the VFR flow, as experimental diffusion/dispersion maps were not available, an estimation of the dispersion field was made. Regions of high and low dispersion could be identified from the diffusive attenuation produced by the PGSE velocity imaging experiments13. The estimated diffusion/dispersion maps were then obtained by scaling the diffusive attenuation field to the TVF diffusion/dispersion map. Also, an additional modification was required in order to take the observed translation of the vortices into account, which allowed the velocity and diffusion/dispersion fields to translate at a velocity UTV.
Time steps, dt, were considered for these simulations. The total number of steps, N, was given by N = Δ / dt, where Δ is the observation time to be simulated. At every time step, dt, the displacement of each molecule ni of coordinates (yi,zi) was defined by a velocity and a diffusion/dispersion step. The velocity step was determined using the experimental velocity maps Uy and Uz. For the diffusive step, the direction was determined by a randomly generated angle θ and the magnitude was calculated using the diffusion/dispersion maps Dy and Dz. Displacements due to velocity and diffusion in each direction where then added to the initial coordinates to define the final coordinates yi(t+dt) and zi(t+dt).
A bounce-back boundary condition was employed for molecules near the reactor walls. If the distance to the wall was found to be smaller than the distance ddt to be traveled during dt, the molecule was simply considered to travel until the wall and perform the rest of ddt in the reverse direction. A minimum of three repetitions for each set of parameters was performed. The Maltab code used for these simulations is included in Appendix B.
After N × dt steps, the overall displacement for each molecule ni along the z-axis was determined by comparing the zi coordinates of the initial and final positions. The displacement
axis was separated in bins and the number of molecules to attain each bin at the end of the simulation was counted. Simulated normalised propagators for a given observation time Δ were obtained by plotting the proportion of molecules at each displacement bin against displacements.
Molecular paths were tracked either by calculations of molecular density per pixel at different observation times, either by calculation of the residence time of molecules per pixel. In this second approach, a matrix counting how many molecules attend each pixel during the simulation was produced to give information on molecular displacements over time. A 2D plot of theses matrices shows high and low circulation pathways within the flow. Because the attendance in some pixels was orders of magnitude higher than in others, a logarithmic scale was used to reveal minor paths taken by molecules.
500 000 molecules were used for all the simulations. The observation times Δ were of 20 ms, 100 ms, 1 s and 1 min. For simulations with Δ < 1 s, dt was of 0.1 ms and 40 bins were used for the displacement axis. For simulations with Δ ≥ 1 s, dt was of 1 ms and 60 bins were used for the displacement axis. For the VFR simulations the velocity and diffusion/dispersion fields were translating at a velocity UTV = 0.81 mm s−1, in accordance with experimental results obtained in chapter 4.