A finite difference method for turbulent thermal convection of complex fluids

Abstract

An efficient and robust finite difference algorithm for three-dimensional direct numerical simulations (DNS) of turbulent thermal convection of complex fluids has been developed. To study the complicated fluid elasticity and plasticity, the simulated non-Newtonian fluids are modelled by either viscoelastic Oldroyd-B or FENE-P, or Saramito elastoviscoplastic constitutive equations based on the conformation tensor. The non-Newtonian solver is built on top of the open-source AFiD (www.afid.eu) code, which uses a pencil distributed parallel strategy to efficiently handle the large-scale wall-bounded turbulence computations. The present algorithm is demonstrated to preserve the properties of symmetry, boundedness and positive definiteness of the conformation tensor up to large Weissenberg numbers $Wi\sim {10}^2$ and high Rayleigh number $Ra\sim {10}^{10}$. To validate and assess the code, both two-dimensional and three-dimensional DNS of viscoelastic Rayleigh–Bénard convection are performed. A comparison with available DNS results in the literature shows a very good agreement. Moreover, the results for the heat transport modification for highly turbulent thermal convection with polymer additives agree not only qualitatively but also quantitatively with previous experiments in a similar parameter range. To validate the elastoviscoplastic model used in the code, the DNS of elastoviscoplastic turbulent channel flows at friction Reynolds number $R{e}_{\tau }=180$ and different Bingham numbers Bi are performed, which also show good agreement with the available results. Single plume dynamics and turbulent Rayleigh–Bénard convection of Newtonian, viscoplastic, viscoelastic and elastoviscoplastic fluids are also studied in the DNS to show the versatility of the code.