Numerical simulation of forced and mixed convection turbulent liquid sodium flow over a vertical backward facing step with a four parameter turbulence model R. Da Vià ⇑ , S. Manservisi ⇑ University of Bologna, via dei Colli 16, 40136 Bologna, Italy article info Article history: Received 20 August 2018 Received in revised form 21 December 2018 Accepted 28 January 2019 Keywords: Liquid metals Vertical backward facing step Mixed convection Buoyancy Four parameter turbulence model abstract Liquid metals are promising fluids for engineering applications since they have interesting physical prop- erties such as high thermal conductivity and low kinematic viscosity with respect to the values of com- mon operating fluids, i.e. air and water. This implies low Prandtl number values and therefore the similarity assumption between velocity and temperature fields does not hold. In the present work we solve a four parameter turbulence model coupled with the Reynolds Averaged Navier Stokes system of equations to simulate a turbulent flow of liquid sodium over a vertical backward facing step. On the wall behind the step a uniform heat flux is applied in order to study both forced and mixed convection flow regimes. This model for turbulent heat transfer, taking into account different boundary conditions, is ana- lyzed and compared with the Kays correlation and Direct Numerical Simulation data. Profiles of velocity, turbulent kinetic energy, turbulent heat flux, temperature and its squared fluctuations are provided for two simulated Richardson number values, namely 0 and 0:338. Obtained results show a reasonable agreement with reference Direct Numerical Simulation values. Ó 2019 Elsevier Ltd. All rights reserved. 1. Introduction In the last few years, turbulent flow over a vertical backward facing step has been extensively studied for low Prandtl number fluids in order to investigate the influence of buoyant forces on fluid behavior and heat exchange. Low Prandtl fluids and liquid metals have gained great interest in engineering applications since most of them can flow in liquid phase on a wide temperature range and low pressure [1]. This characteristic, together with low viscos- ity and high thermal diffusivity, allows scientists and engineers to use liquid metals in applications where huge amounts of heat transfer occur without the need of high pressurized systems. As pointed out in numerous works, from the computational and methodological point of view, more sophisticated models are required in order to accurately simulate turbulent heat transfer involving liquid metals [2–4]. Simplified models, such as the four parameter turbulence models, have been developed based on the evaluation of thermal field characteristic time scales [5–8]. Other turbulence models have been developed to obtain increased numerical stability or to propose more sophisticated expressions of turbulent heat flux, in order to take into account flow anisotro- pic behavior [9]. Since experiments and measurements are rather difficult in liquid metal flows, Direct Numerical Simulation are needed in order to provide reference data for evaluating the accu- racy of turbulence models [10]. In this paper we study liquid sodium turbulent flows over a ver- tical backward facing step for different flow regime characterized by different Richardson number values Ri. This type of flow has been extensively studied in literature. For example a DNS simulation in pure forced convection regime (Ri ¼ 0) is provided in [11] where a constant heat flux is applied on the whole wall behind the step. For the same simulation case with different Reynolds number Re, other studies have been performed [12–14]. In [12] a comparison is provided between the results obtained from DNS simulation and from the solution of a Reynolds Averaged Navier Stokes (RANS) sys- tem of equations closed with various turbulence models. The results are obtained only for Ri ¼ 0 case. It is shown that two equa- tion heat transfer turbulence models, coupled with non linear expressions for Reynolds stresses, allow to improve the predictions of heat flux within the re-circulation zone. In [13] a DNS study is performed for the cases Ri ¼ 0 and Ri ¼ 0:338. In particular, for the mixed convection case, i.e. Ri ¼ 0:338, a different domain con- figuration is considered as an adiabatic section is added behind the heated wall in order to minimize the influence of the outlet on the https://doi.org/10.1016/j.ijheatmasstransfer.2019.01.129 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved. ⇑ Corresponding authors. E-mail addresses: roberto.davia2@unibo.it (R. Da Vià), sandro.manservisi@unibo. it (S. Manservisi). International Journal of Heat and Mass Transfer 135 (2019) 591–603 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt