## 1 Introduction  
Since the invention of integrated circuit (IC) in 1964, the speed of computers has rapidly increased. As one IC can hold tens of electronic modules, the function of an electronic module is equivalent to a transistor or vacuum tube, so that the computer becomes smaller and more powerful. At present, the IC has evolved from early SSI (Small Scale Integrated Circuit, 10-100 modules/per unit chip) into LSI (Large Scale Integrated Circuit, 1000-10,000 modules/per unit chip), then VLSI (Very Large Scale Integrated Circuit, 100,000-1,000,000 modules/per unit chip), and finally even ULSI (Ultra Large Scale Integrated Circuit, more than a million modules/per unit chip). Such a high density package of semiconductor chip increases the heat density greatly, and the reliability or lifetime of semiconductor chip decays as the temperature rises, usually in the range of below junction temperature 100 \({}^{\circ}\)C. A lower temperature leads to a lower failure rate of the chip. Therefore, a good thermal control cooling design is increasingly important for high-speed computer. Multilayer PCB substrate is mounted inside general superhuge computer. Each PCB substrate is about tens of square centimeter, and carries a hundred LSIs. The cooling technique is that the finned heat sink is mounted on the LSI chip for forced air cooling. In such a configuration, there are gaps on both sides and top of the finned heat sink. A part of forced cold flow enters the internal flow channels of heat sink for cooling, and a part of cold flow leaves passing by the heat sink through the gaps. The bigger by-pass gap results in the smaller amount of the effective cold flow through the heat sink, which is disadvantageous to the overall cooling of the heat sink. Therefore, the by-pass gap and heat sink have optimal relative dimensions, so that there is maximum cooling air entering into the heat sink at fixed pumping power. This is an important topic considering actual forced air cooling. The issue about the heat transfer characteristics of finned heat sinks without [1; 2; 3; 4; 5; 6] or with bypass effect [7; 8; 9; 10; 11; 12] attracts wide attention continuously. This work is to discuss putting the pin-fin heat-sink array into a rectangular channel in in-line arrangement with laminar side-bypass effect. This configuration is similar to the PCB substrate of LSI array with heat sinks inside the general supercomputer. Figure 1 shows the typical flow field of such cooling system with seven rows of heat sinks. Considering the periodical and symmetric configuration of the present system, it only shows the upper half of the single-column flow field. According to Figure 1, when the cold flow enters the heat sink array, a part of fluid exchanges heat with the heat sink, and the other part of fluid passes by the heat sink and passes through the gaps between adjacent heat-sink columns. The cold flow in the heat sink leaks out of the heat sink from side continuously under the flow-resistance effect of pin-fins, so that the downstream heat sink has less incoming cold flow. This phenomenon is especially obvious when the permeability of heat sink is low, even forming recirculating flow between downstream heat sinks. There is also a large recirculation zone behind the heat sink array. Such a change of flow field is mainly related to the flow resistance of pin-fin heat sink and the relative spacing between heat sinks, and affects the heat transfer characteristics of each heat sink significantly. In the numerical simulation of heat transfer and fluid flow characteristics of finned heat sink without bypass effect, some studies regard the finned heat sink as a porous medium, using the assumption of the volume-averaging technique for computations of the porous approach model [13, 14, 15, 16, 17]. Besides, [PERSON] [18] used porous approach model to simulate the fluid flow and heat transfer characteristics of single square pin-fin heat sink with laminar side-bypass effect.