## 4 Results and Discussion  
### Flow Behaviors  
Figure 6 shows the flow visualizations of 3 \(\times\) 3 pin-fin heat-sink array in a rectangular channel in the A-A and B-B sectional views of laser sheet (top-view surface tangential to inside of heat-sink array). From the visualizations of initial, interim and later phases, it can be found that the smoke flows mainly through each longitudinal passage between in-line heat sinks; minority plume goes through the heat sinks. The flow visualizations of various top views give the similar flow behaviors. Of course, the real flow pattern in the present rectangular channel with 3 \(\times\) 3 pin-fin heat-sink array cannot be two-dimensional completely due to the end-wall effect. However, by comparing with the variations of fluid flow in the \(x\) and \(y\) directions, the change of fluid flow in the \(z\) direction is small and can be ignored. In other words, the present 2D simulations actually express the mean results of different cross sections along the \(z\)-axis. Figure 7 depicts the measured values of air velocity at the inlet and outlet of the center column of 3 \(\times\) 3 heat-sink array. It generally agrees to the numerical simulation. Figure 8 shows the simulation result of streamline and velocity vector of heat-sink array in different models with Reynolds number \(Re\) = 500 and relative distances between heat sinks \(X_{T}^{\star}(=S_{T}^{\star}/L)=X_{L}^{\star}(=S_{L}^{\star}/L)=1.5\). It is observed that when the uniform air flows into the heat sink array, a part of air flows through the side-bypass gap between transverse adjacent heat sinks, the other part of air flows into the heat sink. In addition, due to the flow resistance, the air flow through the heat sink leaks from the side of heat sink to the side-bypass gap continuously. Therefore, the by-pass gap in more downstream location contains faster air flow; this phenomenon is more obvious in the heat sink array with bigger flow resistance. For example, the heat sink with smaller porosity or larger number of pin fins has this feature. For the heat sink array with bigger flow resistance, there are vortices formed earlier behind the last row of heat-sink array. Generally speaking, the heat-sink array with bigger flow resistance has more bypass air flow, which is adverse to heat transfer. The heat transfer of the downstream heat sink is thus worsened. Figure 5: Configuration and dimensions of test section and positions of measured points. Figure 6: Flow visualization images. (\(Re=52\); 15 liter/min). Figure 7: Measured values of air velocity at the center column of 3 \(\times\) 3 heat-sink array. (\(Re=500\)). Figure 9 shows the influence of variation in the \({X_{T}}^{*}\) and \({X_{L}}^{*}\) values on the streamline and velocity vector taking Sample 1 heat sink array as an example. According to Figure 9a-c, when the \({X_{L}}^{*}\) value is fixed at 2, a larger \({X_{T}}^{*}\) represents larger relative transverse distance between heat sinks, _i.e.,_ larger lateral by-pass gap. In other words, a majority of air flow will pass through this gap, so that the air flow entering the heat sink decreases greatly (see Figure 9c). A recirculating flow is even formed between adjacent front and back rows of heat sinks in the downstream region, which is very adverse to heat transfer. According to Figure 9c-e, in the arrangement of \({X_{T}}^{*}\) value fixed at 2 but different \({X_{L}}^{*}\) values, the air velocity in the lateral by-pass gap is almost the same. This suggests that although the relative longitudinal distance between adjacent front and back rows of heat sinks is changed, the bypass air flow passing by the heat sink is not influenced (_i.e._, total air flow amount through heat sinks has not changed). Figure 8: Streamlines and velocity vectors for various test samples with \({X_{T}}^{*}={X_{L}}^{*}=1.5\) (the scales of \(X\)-axis and \(Y\)-axis are different): (**a**) Sample 1; (**b**) Sample 2; (**c**) Sample 3; (**d**) Sample 4; and (**e**) Sample 5.