## 3 Results
### The Effect of Protrusion Height
#### 3.1.1 Enhancement of Heat Transfer
The highest \(\overline{Nu}\) numbers are obtained for the flat target plate case for all jet \(Re\) numbers considered. For jet \(Re\) numbers of 10,000 and 20,000, the \(\overline{Nu}\) number decreases slightly as \(H_{p}/D\) is increased from 4 to 4.5 due to intense impingement cooling enabled by rapid deceleration of the jet in the small gap between the jet orifice and the CP top face. For a jet \(Re\) number of 15,000, the \(\overline{Nu}\) number remains almost constant when \(H_{p}/D\) is increased from 4 to 4.5. Both the rapid increase in the overall \(\dot{Q}\) and the almost constant \(\overline{Nu}\) number of the cooled surfaces for very small impinging immersed jet lengths are due to a rapid increase in the jet stagnation zone \(Nu\) number. However, for a jet \(Re\) number of 5000, as \(H_{p}/D\) is increased from 4 to 4.5, a notable decrease in the \(\overline{Nu}\) number occurs due to the less effective cooling of the top faces of the CPs by the smaller vortices generated at a shorter jet impingement distance by Kelvin-Helmholtz instabilities. Figure 7: For \(D_{p}/D\) of 3.5, \(\dot{Q}/\dot{Q}_{o}\) according to \(H_{p}/D\) at various jet \(Re\) numbers. Area-averaged \(Nu\) number ratios, the ratio of the \(\overline{Nu}\) number of the protruding surface to that of the flat target surface (\(\overline{Nu}/\overline{Nu}_{o}\)) according to \(H_{p}/D\) for various jet \(Re\) numbers, are presented in Figure 9. For jet \(Re\) numbers above 10,000, at any \(H_{p}/D\), the \(\overline{Nu}/\overline{Nu}_{o}\) decreases with increasing jet \(Re\) number, while for a jet \(Re\) number of 5000, its trend, compared with a jet \(Re\) number of 10,000, varies with \(H_{p}/D\). For any jet \(Re\) number considered, the \(\overline{Nu}/\overline{Nu}_{o}\) decreases as \(H_{p}/D\) is increased due to bigger contribution of the CP lateral surface to cooling. Therefore, for any jet \(Re\) number, the highest \(\overline{Nu}/\overline{Nu}_{o}\) is obtained for \(H_{p}/D\) of 1, which is the smallest among all CP heights considered in this study. For \(H_{p}/D\) of 1, the biggest and smallest \(\overline{Nu}/\overline{Nu}_{o}\)s of 91.7% and 88.3% are obtained for jet \(Re\) numbers of 10,000 and 20,000, respectively. For \(H_{p}/D\) of 4.5, for jet \(Re\) numbers of 5000, 10,000, 15,000, and 20,000, the \(\overline{Nu}/\overline{Nu}_{o}\)s are 0.645, 0.653, 0.642, and 0.629, respectively. For various jet \(Re\) numbers, \(Nu\) number contour plots on the target plate and CP top faces as well as on unrolled lateral faces of CPs are presented in Figure 10. As the \(H_{p}/D\)  
Figure 8: For \(D_{p}/D\) of 3.5, \(\overline{Nu}\) number according to \(H_{p}/D\) at various jet \(Re\) numbers. is increased, the \(\overline{Nu}\) number increases on the top faces of the CPs due to the flatter core velocity profiles and steeper shear layer velocity gradients of the impinging jets and their reduced deflections and deformations due to weakened crossflow interactions. On the other hand, the \(\overline{Nu}\) number decreases on both the lateral faces of CPs and the target plate due to increased thermal boundary layer thickness. For instance, for a jet \(Re\) number of 20,000, while the \(\overline{Nu}\) number on the top faces of the CPs is 94.49 for \(H_{p}/D\) of 2, it increases to 98.23 for \(H_{p}/D\) of 4. On the other hand, the \(\overline{Nu}\) number on the lateral faces of CPs and on the target plate, for stated CP heights, decreases from 36.66 to 28.47 and from 36.12 to 29.74, respectively. As \(H_{p}/D\) is increased by keeping the top face area of a CP constant, its lateral face area increases. While the overall \(\hat{Q}\) is significantly improved by increasing \(H_{p}/D\), the surface heat flux distribution is nonuniform and the \(\overline{Nu}\) number of all cooled surfaces decreases, although that of the top faces of the CPs increases.