## 1 Introduction
They concluded that [PERSON] correlation is the most reliable one among existing local correlations for RANS computations. The key point in many of these approaches is that the correlation has to be defined and validated for each geometry. Another different approach is to define the turbulent heat diffusivity with specific time scales resolved by the introduction of the dissipation rate of both temperature and velocity fluctuations. These models have been studied by some authors for forced convection and buoyant flows in different geometrical configurations [7; 8; 14; 15]. The effects of geometries with injection and suction at the wall are simulated in [14; 15] with a two-equation turbulent thermal model. In [PERSON] et al. [7] an overview of different realistic geometries for liquid-metal applications in the field of nuclear engineering is summarized with the help of several Computational Fluid Dynamic (CFD) models and tools. A four-parameter turbulence model \(k\)-\(\epsilon\)-\(k_{\theta}\)-\(\epsilon_{\theta}\), based on [PERSON] and [PERSON]'s model [15; 16; 17; 18], was introduced by [PERSON] and [PERSON] [8]. The model has been validated against DNS and experimental correlation data, with simulations of \(Pr=0.025\) fluids for several geometries, involving plane channels, cylindrical ducts and triangle and square bundle lattices [8; 19; 20]. With the purpose of improving the numerical stability of the model, a change of state variables has been considered in [21], where specific dissipation rates \(\omega\) and \(\omega_{\theta}\) have been used in place of \(\varepsilon\) and \(\varepsilon_{\theta}\), and in [22], where, by following the work of [23], logarithmic values of \(\omega\) and \(\omega_{\theta}\), i.e., \(\Omega\) and \(\Omega_{\theta}\), have been used. As a consequence of the change of state variables, some model coefficients had to be changed in order to improve the numerical stability, and the resulting models have been validated with simulations of plane channels and cylindrical ducts for \(Pr=0.025\). The model, in a fully logarithmic version and with additional terms accounting for buoyancy, has also been used to simulate a sodium turbulent flow over a backward facing step geometry, with \(Pr=0.0088\), in the cases of forced and mixed convection [24]. Many studies have been performed to both examine the range of validity of the model and to improve its numerical stability. The purpose of the present paper is to test fully developed turbulent flows in plane channels and cylindrical ducts, with \(Pr=0.01\), in order to further extend the validation tests of the model presented in [8; 19; 20]. These works presented in previous papers deal with a single \(Pr\) number and therefore these heat transfer turbulence models may not be suitable for different liquid metals. Now we present numerical results obtained with two Prandtl numbers, namely, \(Pr=0.01\) and \(0.025\), in order to test the possibility of studying, with a unique set of parameters, different \(Pr\) numbersand different operational temperature ranges. Furthermore, we present the model in a logarithmic \(k\)-\(\Omega\)-\(k_{\theta}\)-\(\Omega_{\theta}\) formulation with the purpose of improving its numerical robustness. In two-equation turbulence models, when state variable limiters are not used, numerical instabilities usually appear due to negative values of the state turbulence variables, especially near boundaries where temperature and velocity field profiles show steep variations. A logarithmic formulation of the problem avoids these numerical undershooting or overshooting errors and improves the smoothness of dynamical and thermal field profiles at the walls for a thinner boundary refinement [23]. Since a four-equation turbulence model \(k\)-\(\Omega\)-\(k_{\theta}\)-\(\Omega_{\theta}\) could be computationally expensive, with respect to a SED model, we report also the results obtained by using Kays correlation.