## 1 Introduction  
A pressure turbine of the jet engine is the part where the thermal energy is transformed into mechanical work [1]. The energy transformation is ensured by increasing the hot gas's speed between the turbine blades and by their impact on the airfoils of the turbine blades. Hot gases emit significantly high-temperature, which has an impact on the turbine blades. In order to predict this impact, a number of analyses for particular engine partsmust be carried out [2]. Numerical simulations of the temperature fields of the turbine blades are required for every element of a gas turbine during the design process of the engine. However, as it is reviewed below, there are also many other cases that require the determination of the temperature fields of the turbine blades. Increasing the turbine inlet temperature leads to the improvement of the efficiency, but the temperature cannot exceed the melting point of the material. Therefore, blade metal temperature distribution and temperature gradients are the most important parameters when determining the blade life. The temperature of the blade has a significant impact on the jet engine lifetime. It also must be considered that during the jet engine operation, there are multiple phases, such, for example, nominal regime, idle run, maximal regime, etc. with are characterized by completely different temperatures. Therefore, it is essential to know the temperature fields on the surfaces of the turbine blade also for the specific regimes of the jet engine operation. Mentioned facts are clearly proving the importance of the turbine temperature field prediction. There are multiple reasons why it is necessary to know the temperature fields of the turbine blades. For example, in [1], the study of the temperature of the cooling turbine blade is presented. In addition, for the Campbell diagram creation, the temperature fields of the turbine blade are needed because, with the increase of the temperature, natural frequencies are decreasing [3]. Another application area where the knowledge of temperatures of the turbine blade is necessary [4] is the fatigue life analysis, particularly considering the cyclic loading, as presented in [5; 6; 7]. The next example is the sensitivity study, where the knowledge of the temperature fields must be included when the simulations of the turbine blade application are performed [8]. For the sensitivity study presented in [4], the temperature fields of the turbine blade must be available for the multiple regimes [9]. When simulating the models of internally cooled gas turbine blades [10], the temperature characteristics also must be used. An example of the simulation of the cooled gas turbine blades is shown in [10], where the impact of the internal cooling system is studied [11; 12]. Another example, where the temperature fields are needed, is presented in [13], where the effect of internal cracks on the surface profile change due to low loading of a turbine blade using the finite element method is studied. From this brief overview, it is obvious that the need for the temperature field prediction of the turbine blade is indisputable because it serves as the input of many simulations and analyses and can help to significantly improve the results of many types of analyses. Nowadays the numerical methods based on the finite element method (FEM) or computational fluid dynamics (CFD) methods are used for this purpose [14; 15; 16]. The application of the CFD [17] and FEM [18] methods for the surface temperature field of the turbine blades is nowadays widespread; however, the calculation of the turbine blade surface temperature during the multiple regimes can be time-consuming.