# The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization
## 2 CFD Solver and Its Validation
### Result Analysis  
The optimization process took about 120 h with each CFD simulation occupying approximately 45 min, considering the fact that the evolution has to be processed in non-parallel way for each candidate of each generation. When the iteration with improved design requirement comes to generation no. 79, there has been no updates of candidates in objective space for the last 10 generations. Therefore, a 'convergence' of Differential Evolution is considered to be obtained. The result in Figure 9 shows that the improvement in design requirement formulation takes effect in enhancing the optimization by applying the intrinsic nature of the problem via objective space analysis. With the improved design requirement formulation, the Pareto front can be obtained faster than conventional configuration. The cost for extracting information from objective space brings rewards in the final optimization process. The Pareto front represented as thick, purple dashed line shows that due to the various constraint in the problem description, most of the candidates are in the region relatively lower than the corner of the line. Therefore, most of the optimization is done mainly in the aspect of pressure drag reduction(e.g., candidate no. 2), while some in the aspect of laminar length increasing (e.g., candidate no. 12). The candidate no. 5 is selected as the optimized airfoil to be compared with original airfoil RAE2822 because it is located in the corner of the obtained Pareto front (therefore possessing good performance in two aspects). The comparison of aerodynamic performance is displayed in Figure 10 in terms of pressure coefficient contour and Figure 11 in terms of two-dimensional comparison of pressure distribution and laminar length in form of skin friction drag coefficient, respectively. From Figure 10, a shock with less strength and more rearward position is shown in optimized case that gives explanation of the better pressure drag performance. The quantitative improvement in laminar-related performance is also shown in Figure 11.  
Figure 11: The two-dimensional pressure distribution and skin friction drag coefficient plot of the initial and optimized airfoil (i.e., candidate no. 5 in Figure 9). Figure 10: The pressure coefficient contour of the initial and optimized airfoil (i.e., candidate no. 5 in Figure 9). Figure 9: The comparison of the two optimization processes when the case of improved design requirement formulation has converged to the Pareto front. Proof-of-Concept Study 2: Improvement of Accessory Aerodynamic Performance via Formulation of Design Requirement with Information of Objective Space  
NLF Optimization Adopting Information of Objective Space of Laminar-Related Performance, Pressure Drag, and Pressure Loss  
In this case of aero engine blade design, the effect of NLF optimization can not only reduce skin friction drag by laminar design, but can also enhance comprehensive performance in case of aero engine blade design with a preview from three-aerodynamic-performance objective space information. The idea of natural laminar flow blade is feasible by modifying the shape even in the coming flow with high temperature and high pressure [62]. The shape modification in the leading-edge region is responsible for the alleviation of separation of peak pressure profile [63], and the curve tendency in the mid-chord also needs reconsideration for maintaining laminar flow. The CFD simulation in this case is to analyze the flow through the tunnel established between blades, where the aerodynamic performance and its optimization is considered (cf. Figure 12). On the boundary of pressure inlet (total pressure of 122,800 Pa) and outlet (static pressure of 101,325 Pa), the Mach number of coming flow is kept as 0.89. The total temperature is 293 K, and the turbulent intensity is 0.2%. The solver is Fluent by using its SST four-equation model. For each blade, a mesh is created where the farfield is approximately 4.5 chord length away.