## 2 Materials and Methods
### Numerical Method
#### 2.2.7 Mesh Motion  
Due to the motion of the specimen made of Inconel 600 nickel alloy, one needs to ensure the appropriate technique for the establishment of accurate mesh motion. Within the present research, it is done by the application of the arbitrary Lagrangian-Eulerian (ALE) method, in which the two domains (solid and fluid) are distinguished by a solid boundary that is explicitly tracked (Lagrangian approach), while within each domain, the equations are being solved in a Eulerian, control volume approach. The mathematical outline of the ALE method is available in [PERSON] [41]. Since the motion of a specimen is by far more than just an oscillation of, for example, a droplet in a zero-gravity area, it is necessary to use the remeshing method to overcome the huge displacements that are to appear when a body is moving from its initial point toward the quenchant liquid. The applied method herein is briefly described, together with other numerical simulation features, in [PERSON] et al. [42]. The remeshing problem has been tackled by [PERSON] [43], while its application within the ALE solver is shown by [PERSON] [44] and [PERSON] [45].  
2.2.8. Integral Quantities Calculation: Heat Fluxes and Heat Transfer Coefficients  
The wall heat flux and the wall temperature may stem from the following reasoning. Let the temperatures of fluid, \(T_{\text{fluid}}\), and solid, \(T_{\text{solid}}\), in a system shown in Figure 3 be known and, in addition, the local thermal equilibrium may be assumed.  
Therefore, we can write the heat fluxes at each side of the solid wall as follows:  
\[q_{s}=\lambda_{s}\frac{(T_{\text{wall}}-T_{\text{solid}})}{\Delta x_{\text{ solid}}} \tag{6}\]  
\[q_{s}=\lambda_{f}\frac{\left(T_{\text{fluid}}-T_{\text{wall}}\right)}{\Delta x _{\text{fluid}}} \tag{7}\]  
From these two equations, Equations (6) and (7), one may obey the wall heat flux, \(q_{s}\), and the wall temperature, \(T_{\text{wall}}\).  
By further exploration of Equation (7), one may note the necessity for the definition of effective thermal conductivity in the case of turbulent flow, that is, that the thermal conductivity at the liquid side is actually a sum of molecular and turbulent thermal conductivities, \(\lambda_{\text{f}}=\lambda_{\text{m}}+\lambda_{\text{t}}\). In doing so, the molecular thermal conductivity refers to the physical property of the fluid, and it is the only one considered in the laminar flow, while the turbulent thermal conductivity (present if the flow is turbulent) is the flow dependent variable and could be estimated using either the turbulent viscosity via the turbulent Prandtl number or the wall function.  
The heat transfer coefficient, on the other hand, is defined as the ratio between the wall heat flux and the temperature difference between the solid surface and the bulk fluid, as follows:  
\[h=\frac{q_{s}}{T_{wall}-T_{\infty}} \tag{8}\]  
where \(q_{s}\) is the wall heat flux, \(T_{\text{wall}}\) is the wall temperature, and \(T_{\infty}\) is the bulk fluid temperature. Within the scope of the present work, an area weighted average heat transfer coefficient has been taken into consideration within the discussion of the results, as follows:  
\[\underline{u}=\int_{A}hdA \tag{9}\]  
The average heat transfer data has been computed within postprocessing of the results and presented herein for the overall heat transfer surface in the case of silver specimens and partitioned only for a vertical surface of the cylinder in the case of the Inconel 600 superalloy since the dominant heat transfer is to appear on this heat transfer surface. The overbar in Equation (9) will be omitted in further discussions of average heat transfer coefficients in this paper.  
Figure 3: The wall adjacent system in a conjugate heat transfer case.