Numerical simulations of transitional and particle-laden viscoelastic flows

Abstract

The foundation for fully resolved simulations of particle-laden viscoelastic flows is presented, with detailed assessment of its various elements. The predictive capability of the viscoelastic algorithm is demonstrated in a detailed study of natural transition to turbulence in channel flow. The computations start from infinitesimally small Tollmien-Schlichting (TS) waves and track the development of the instability from the early linear stages through nonlinear amplification, secondary instability and full breakdown to turbulence. At low elasticity, the primary TS wave is more unstable than the Newtonian counterpart, and its secondary instability involves the generation of Λ-structures which are narrower in the span. As elasticity increases, the growth rate of the primary TS wave is weaker than the Newtonian value, and the spanwise size of the secondary instability becomes wider, with weaker ejection and without hairpin packets. All viscoelastic flows considered finally reach a drag-reduced turbulent state via different routes to breakdown to turbulence. Notably, at high elasticity, streamwise elongated streaks are formed and break down to turbulence via secondary instability rather than cascading hairpin packets. With the viscoelastic algorithm established, the focus is shifted to particle-laden configurations. An immersed boundary method is adopted, where the particle geometry does not conform to the grid. However, all flow scales related to the motion of particle are resolved. The no-slip condition is enforced via momentum forcing and mass conservation is ensured by introducing a mass source term. An iterative force evaluation is derived for semi-implicit discretisations of the Navier-Stokes equations, which reduces the diffusive errors near the immersed boundary. As for inter-particle interactions, the immersed boundary method is complemented with two mechanical models: a soft-sphere or repulsive potential models for particle collision and, for Newtonian fluids, a lubrication model that captures the unresolved approaching motion prior and after the collision. A number of flow simulations are presented to assess the performance of the numerical algorithm.