Determination of Numerical Errors in Under-Resolved DNS of Turbulent Non-isothermal Flows

Abstract

The methodology to quantify the numerical dissipation in Under-resolved DNS (UDNS) based on the balance of the kinetic energy equation by Schranner et al. (Comput Fluids 114:84–97, 2015), has been examined in this study. Furthermore, this methodology has been extended for considering active scalars, based on both balance of the kinetic energy and thermal variance equations for assessing the quality of UDNS. As a first step towards the analysis of reactive flows, where temperature is actively coupled to the momentum equation, the turbulent Rayleigh–Bénard convection in a cubical enclosure is considered here. The simulations have been carried out for different Prandtl numbers (Pr= 0.1 , 1.0 , 10 ) to analyse different levels of momentum-temperature coupling for a representative value of nominal Rayleigh number (Ra= 10 7) in order to investigate the effects of grid resolution and discretisation schemes on the numerical dissipation. It has been found that the numerical dissipation for both kinetic energy (Rk) and thermal variance (Rt) can range between positive or negative values depending on the combined effects of temporal and spatial discretisation schemes and grid spacing. The positive values of Rk and Rt indicate a loss of the kinetic energy and thermal variance due to numerical errors, whereas the negative values of Rk and Rt result in a moderate attenuation of these quantities. Accordingly, the numerical dissipation results obtained here (i.e. Rk and Rt) have been utilised for evaluating the effective values of the Rayleigh and Prandtl number (i.e. Raeff, Preff) for UDNS. Using the scaling of the mean Nusselt number (Nu∼ RaaPrb), it has been shown that the error relative to exact mean Nusselt number can be estimated based on Rk, Rt and the effective mean Nusselt number of the UDNS (NuUDNS). Overall, this methodology has been found a promising tool for the quality assessment of UDNS for the applications of non-isothermal flows (e.g. Rayleigh–Bénard convection).