A new approach to the numerical simulation of incompressible viscoelastic Rayleigh-Bénard convection in a cavity is presented. Due to the fact that the governing equations are of elliptic-hyperbolic type, a quasi-linear treatment of the hyperbolic part of the equations is proposed to overcome the strong instabilities that can be induced and is handled explicitly in time. The elliptic part related to the mass conservation and the diffusion is treated implicitly in time. The time scheme used is semi-implicit and of second order. Second-order central differencing is used throughout except for the quasi-linear part treated by third order space scheme HOUC. Incompressibility is handled by a projection method. The numerical approach is validated first through comparison with a Newtonian benchmark of Rayleigh-Bénard convection and then by comparing the results related to the convection set-up in a 2 : 1 cavity filled with an Oldroyd-B fluid. A preliminary study is also conducted for a PTT fluid and shows that PTT fluid is slightly more unstable than Oldroyd-B fluid in the configuration of Rayleigh-Bénard convection.

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