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 (R_{k}) and thermal variance (R_{t}) 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 R_{k} and R_{t} indicate a loss of the kinetic energy and thermal variance due to numerical errors, whereas the negative values of R_{k} and R_{t} result in a moderate attenuation of these quantities. Accordingly, the numerical dissipation results obtained here (i.e. R_{k} and R_{t}) have been utilised for evaluating the effective values of the Rayleigh and Prandtl number (i.e. Ra_{eff}, Pr_{eff}) for UDNS. Using the scaling of the mean Nusselt number (Nu sim {Ra^a} {Pr^b}), it has been shown that the error relative to exact mean Nusselt number can be estimated based on R_{k}, R_{t} and the effective mean Nusselt number of the UDNS (Nu_{UDNS}). 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).
Highlights
Turbulent non-isothermal flow is a phenomenon where there is a two-way coupling between fluid flow and heat transfer due to density variations arising from temperature differences
In order to better understand the methodology of evaluating the numerical dissipation on the fly by using a balance equation for kinetic energy, the analysis is started with a simple 1D linear convection-diffusion equation in a periodic domain of length L
Determination of numerical errors in Under-resolved Direct Numerical Simulation (DNS) (UDNS) of Rayleigh–Bénard convection in enclosed spaces has been analysed by extending the method used by Cadieux et al (2017)
Summary
Turbulent non-isothermal flow is a phenomenon where there is a two-way coupling between fluid flow and heat transfer due to density variations arising from temperature differences This phenomenon is prevalent in heat exchangers, chemical reactors, combustion and atmospheric flows. Rayleigh–Bénard convection (i.e. fluid between horizontal walls heated from below and cooled from above) is a canonical type of non-isothermal flow which is significantly important for numerous applications in environment and technology. Industrial processes such as chemical and food processing, heating and ventilation of indoor spaces, cooling of electronics, solar and nuclear power systems are few important examples (Bodenschatz et al 2000). According to the taxonomy for obtaining error estimates, as suggested by Roache (1998), this method belongs to the category of “Auxiliary algebraic evaluations on the same grid”, more precisely “Non-conservation of higher order moments”, and in their particular case: turbulent kinetic energy. Komen et al (2017) proposed a complementary method to Schranner et al (2015) based on the computation of the residuals in the transport equation for the mean turbulent kinetic energy
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