A generalized k−ϵ model for turbulence modulation in dispersion and suspension flows

Abstract

A large amount of published data show that particles with diameter above 10% of the turbulence integral length scale (D/l>0.1) tend to increase the turbulent kinetic energy of the carrier fluid above the single-phase value, and smaller particles tend to suppress it. We attempted to remove limitations in earlier modelling efforts for solids on the coupling between the particles and turbulence, and better fits to the turbulence modulation amplitude as function of D/l was achieved for a number of data sets. Explicit algebraic forms of the full model were derived using asymptotic analysis, and these are general enough for application to emulsions, bubbles and solids in bulk regions of multiphase turbulent flow. Rigorous particle-kinetic theory was used to derive the work exchanged between the particles and the fluid due to both drag and added mass forces, where the latter is essential for low or moderate particle/fluid density ratios, enabling a well justified model also for emulsions and bubbles. A novel sub-model for turbulence production by vortex shedding due to turbulence-generated slip velocity was incorporated, where earlier models took the slip velocity as an input parameter. The correct asymptotic limit of vanishing turbulence modulation for small tracer particles was also provided, giving better fit to the data for small particles. We found that turbulence augmentation for large diameter solids is due to vortex shedding, and turbulence suppression for small diameters is due to mainly to turbulent drag forces and extra fluid dissipation – a conclusion that agrees with earlier models for solids, despite their possible shortcomings. An important finding is that the mechanisms for turbulence suppression for bubbles and emulsion droplets are similar to those of solids, but with the addition of added mass scaling factors. Another important observation is that augmentation may not occur at all for bubbles or emulsion droplets since the larger diameters require moderate turbulence levels to prevent breakup, so that vortex shedding may be insignificant.