Design Optimization of a Subsonic Diffuser. for a Supersonic Aircraft

Transcription

1 Chapter 5 Design Optimization of a Subsonic Diffuser for a Supersonic Aircraft 5. Introduction The subsonic diffuser is part of the engine nacelle leading the subsonic flow from the intake to the turbo-fan engine. It plays an important role in stable operation of the propulsion system. To realize stable operation of the turbo-fan engine, the subsonic diffuser should lead a uniform flow with maximization of pressure recovery. However, leading uniform flow and maximizing pressure recovery are generally considered to have trade-offs. In this chapter, the design of a subsonic diffuser for a supersonic aircraft (Fig. 5.) is discussed. The cross-sectional shape of the subsonic diffuser for the supersonic aircraft is gradually deformed from a rectangle to a circle, because the intake shape is rectangular to generate the oblique shock at the intake ramp and the turbo-fan engine located at the end of the subsonic diffuser is circular. In addition to the diffuser center line, the area and aspect ratio of the cross-sectional shape are determined by biquadratic distributions. Cross-sectional shapes along the diffuser center line are determined by an n-dimensional periodic function. Therefore, not only the center line design but also the rates of change of the area, aspect ratio, and cross-sectional shape are important design specifications []. This study provides the multiobjective design optimization of a subsonic diffuser assuming general supersonic aircraft in which the cross-sectional shape is changed from a rectangle to a circle. The objective functions considered here are maximization of the average total pressure recovery and minimization of the distribution of the total pressure at the end of the subsonic diffuser. In this study, the DRMOGA [, ] discussed in the previous chapter was used. This chapter is organized as follows. In Section 5., the design optimization problem of the subsonic diffuser performance is formulated. In Section 5., the geometry definition is discussed. The biquadratic equation is used for the center line design and definition of the

2 distribution of the area and the aspect ratio. For shape variety, the n-dimensional function is defined. Section 5.4 presents the optimization results. In this study, two cases were handled: straight duct and curved duct. The conclusion of this chapter is described in Section 5.5. bottom top intake diffuser Fig. 5. The intake and subsonic diffuser for a supersonic aircraft. 5. Formulation and Optimization Problem 5.. Objective Function The Objective functions considered here are to maximize the average value of the pressure recovery and to minimize the distortion index at the end of the diffuser. The maximum pressure recovery achieves highly efficient engine operation and the minimum distortion index achieves stable engine operation with a uniform pressure distribution. The objective functions are defined as: (Pressure recovery) (%) = p out /p (5.) (Distortion index) (%) = (p outmax - p outmin )/p (5.) where, p is the total pressure of the mainstream, p out is the average value of the total pressure at the end of the diffuser. p outmax and p outmin are the maximum pressure and the minimum pressure at the end of the diffuser, respectively. Solutions are obtained by DRMOGA [, ]. 5.. Geometry Definition In this study, the geometry of the diffuser was defined by determining the center line, the distribution of the cross-sectional area, and the aspect ratio along the center line based on the

3 biquadratic equation [4] written as: f(x) = a x 4 + b x + c x + d x + e (5.) When coordinates (x, f(x )), (x, f(x )) and gradients f (x ), f (x ) at each boundary and the coefficient c are given, coefficients a, b, d, and e of Eq. (5.) are determined as: ζ δγ / α b = ε δβ / α γ βb a = α d = f '( x ) 4ax e = f ( x ) 4ax 4 bx bx cx cx dx (5.4) where α = x 4 - x 4-4x (x - x ) β = x - x - x (x - x ) γ = f(x ) - f(x ) c(x - x - x (x - x )) (5.5) δ = 4(x - x ) ε = (x - x ) ζ = f (x ) - f (x ) -c(x - x ) Distributions of the area and aspect ratio p/q (Fig. 5.) along the centerline are defined in the same way. Then, the center line and distribution functions of the area and aspect ratio are determined as: f center (x) = a center x 4 + b center x + c center x + d center x + e center f area (x) = a area x 4 + b area x + c area x + d area x + e area (5.6) f ar (x) = a ar x 4 + b ar x + c ar x + d ar x + e ar Figure5. shows the center line of the diffuser defined by Eq. (5.6). y offset is f center (x ). Cross-sectional shapes are defined by n-dimensional function (Fig. 5.) written as:

4 (y/p) n + (z/q) n = (5.6) In Eq. (5.6), n = represent an ellipse and n = represents a rectangle. In this study, n = approximately represents a rectangle. To define cross-sectional shapes along the center line, the exponent number n is defined as: n = A(-t) m + B (5.7) where t is the normalized number of the coordinate x by the length of the diffuser. Coefficients A and B are determined by values of n at t= and t=. In this study, n= represents a rectangle at the entrance and n= represents a circle at the end of the diffuser. Substituting these values, A is determined as 98 and B is determined as. Thus, the value of n is determined as follows: n = 98(-t) m + (5.8) In this study, the geometry of the diffuser was designed by determining c center, c area, c ar, m, f area (x ) and f ar (x ), so the number of design variables was six. The coordinates of the center line were fixed at the entrance (,, ) and the exit (, y offset,). The coordinates of the area and the aspect ratio were also fixed at the entrance f area (x ) = and f ar (x ) = and at the exit f area (x ) =.4 and f ar (x ) =. Every gradient at the entrance and the exit was set to. Cross-sectional shapes were rectangular at the entrance and circular at the exit assuming the diffuser for a supersonic intake. 5.. Evaluation The flowfield through the diffuser was evaluated by solving the structured Navier-Stokes equation with the k-ε turbulent model [5, 6] introduced in chapter (Fig. 5.4). In this study, the freestream Mach number was.7 and the Reynolds number normalized by the height of the entrance was 6. At the boundary of the entrance, the uniformed state variable distributions were given. Evaluations of each individual were parallelized on Numerical Simulator (NS) [7] and a calculation time per individual of about 4 hours. 4

5 Intake Turbo-fun engine y y offset x Fig. 5. Center line definition using biquadratic equation. y p q n=5 p n= q z Fig. 5. Cross-sectional shape definition. 5

6 Fig 5.4 Computational structured grid and total pressure contour obtained by present evaluation method. 5. Optimization Results In this study, two design cases were designed. The first case was the design of a straight duct, i.e., y offset =, while the second case was the design of a curved duct, i.e., y offset =. Although the straight duct is used in many commercial aircraft, the curved duct provides more design opportunities. In each case, the population number was set to, and generations were calculated. These optimizations were performed using DRMOGA the sub-population number of which was 4 and individuals were divided every 4 generations. 5.. Optimization Results of the Straight Duct (Case ) First, design optimization of the straight duct was performed. In this case, non-dominated solutions were found as shown in Fig The geometry of many subsonic diffusers achieves high pressure recovery. Moreover, some solutions achieve not only higher pressure recovery but also lower distortion index. This means that the maximization of pressure recovery is easy 6

7 in the straight duct and the trade-off is not so strong. Figure 5.6 shows the geometry and the flowfield of solution A taken from the non-dominated solutions. The lowest distortion index of 4.4% was achieved in both cases. While separation of the boundary layer was found at the middle part of the diffuser, the streamline was put in order at the end of the diffuser. Figure 5.7 shows the distributions of the center line, the area, and the aspect ratio of solution A along the free stream direction. From this figure, the area became largest at the middle part of the diffuser and was reduced gradually to the end of the diffuser. This suggests that such an area distribution works as a throat, which can achieve uniform flow. Figure 5.8 shows the geometry and the flowfield of solution B. It achieves the highest pressure recovery of 99.% in this case. The boundary layer separation was not so large, and thus the flow can be led to the end of the diffuser, while maintaining high pressure. Figure 5.9 shows the distributions of the center line, the area, and the aspect ratio of solution B along the free stream direction. The distributions of the area and aspect ratio are similar to those of solution B. However, the cross-sectional shape was changed from a rectangle to a circle at the beginning of the diffuser (Fig. 5.8). This geometry can lead the flow with minimal separation. 5.. Optimization Results of the Curved Duct (Case ) Second, design optimization of the curved duct was performed. In this case, non-dominated solutions were found as shown in Fig. 5.. The geometry of many subsonic diffusers achieves higher pressure recovery and lower distortion index. Figure 5. shows the geometry and the flowfield of solution C taken from the non-dominated solutions. It achieved the lowest distortion index of 4.7% in this case. While the separation of the boundary layer was found at the middle part of the diffuser, the streamline was put in order at the end of the diffuser. Its tendency was quite similar to that of solution A. Figure 5. shows the distributions of the center line, the area, and the aspect ratio of solution C along the free stream direction. From this figure, the area became largest at the middle part of the diffuser and was gradually reduced toward the end of the diffuser, while the 7

8 aspect ratio decreased gradually along the center line. Solution A showed the same tendency, i.e., this area distribution is appropriate to achieve uniform flow at the end of the diffuser. Figure 5. shows the geometry and the flowfield of solution D. It achieves the highest pressure recovery of 99.% in both cases. The boundary layer separation was very small, and thus the stable flow could be led. Figure 5.4 shows the distributions of the center line, the area, and the aspect ratio of solution D along the free stream direction. The cross-sectional shape variation was similar to that of solution C. However, the peak of the area distribution of solution D was lower than that of solution C. These results suggest that the distribution should be near flat to obtain higher pressure recovery, while it should be highly convex to obtain lower distortion index. These results also show the trade-off between maximization of pressure recovery and minimization of the distortion index. 5.. Comparison of Non-dominated Solutions in the Two Cases In this study, two cases were designed as discussed above. In each case, many solutions achieved good improvements in both objective functions. Figure 5.5 shows the comparison of non-dominated solutions obtained in the two cases. This figure indicates that a straight duct is good to minimize the distortion index, which indicates stability of a propulsion system. In contrast, a curved duct is good to maximize pressure recovery, which indicates efficiency of the turbo-fan engine. Moreover, in Case, some solutions achieved good improvements, which were close to the maximum values in each objective function. These observations suggest that the straight duct design does not have such strong trade-offs. 8

9 Non-dominated front B Pressure recovery (%) A 5 5 Distortion index (%) Fig. 5.5 All solutions obtained in Case. 9

10 Fig. 5.6 Subsonic diffuser shape A and its total pressure distributions in Case... y/xlength. Area.5.5 Aspect Ratio n distribution Fig. 5.7 Distributions of the center line, area, and aspect ratio of solution A along the x-direction.

11 Fig. 5.8 Subsonic diffuser shape B and its total pressure distributions in Case... y/xlength. Area.5.5 Aspect Ratio n distribution Fig. 5.9 Distributions of centerline, area, aspect ratio and n of solution B along x-direction.

12 Non-dominated front D C Pressure recovery (%) Distortion index (%) Fig. 5. All solutions obtained in Case.

13 Fig. 5. Subsonic diffuser shape C and its total pressure distributions in Case.. y/xlength. Area..5.5 Aspect Ratio n distribution Fig. 5. Distributions of the center line, area, and aspect ratio of solution C along the x-direction.

14 Fig. 5. Subsonic diffuser shape D and its total pressure distributions in Case.. y/xlength. Area..5.5 Aspect Ratio n distribution Fig. 5.4 Distributions of centerline, area and aspect ratio of solution D along x-direction. 4

15 Case Case Pressure recovery (%) Distortion index (%) Fig. 5.5 Comparison of non-dominated solutions obtained by Case and Case. 5

16 5.4 Conclusions The design optimization of a subsonic diffuser assuming a supersonic intake was performed using DRMOGA. In this study, the algebraic geometry definition was used and the Navier-Stokes equation with a low-reynolds number k-ε turbulent model was solved for each individual. Objective functions were used to maximize the total pressure recovery and to minimize the distortion index. Two design problems were examined: a straight duct diffuser and a curved duct diffuser. In Case, the straight duct design, each objective function showed good improvement. Through this design, shape variations along the center line were shown to be effective for minimization of the distortion index. It was demonstrated that the area distribution along the center line should be convex and form a throat at the end of the diffuser. Each objective function was also improved in Case, the curved duct design. Optimization results showed that a convex area distribution along the center line improved the distortion index. On the other hand, the flat area distribution improved pressure recovery. Non-dominated solutions obtained in each case were compared. This comparison suggested that the straight duct design improved the distortion index well, while the curved duct design improved the pressure recovery. The curved duct design showed stronger trade-offs than the straight duct design. In this study, an effective design system of a subsonic diffuser for a supersonic aircraft was developed. However, the flow simulation was simplified giving uniform flow at the entrance boundary. Therefore, high-pressure recovery, close to %, was obtained in each case. In future studies, distributed state variables obtained by intake simulation as the diffuser s entrance boundary condition should be considered and gradual increments in the area distributions should also be handled as a constraint for good productivity. 6

17 5.5 References [] Knight, D., Application of Genetic Algorithms to High Speed Air Intake Design, Lecture Series Programme 999-, Genetic Algorithms for Optimisation in Aeronautics and Turbo Machinery, Von Karman Institute for Fluid Dynamics, pp.-4,. [] Hiroyasu, T., Miki, M. and Watanabe, S., The New Model of Parallel Genetic Algorithm in Multi-Objective Optimization Problems (Divided Range Multi-Objective Genetic Algorithm), IEEE Proceedings of the Congress on Evolutionary Computation, Vol., pp. -4,. [] Kanazaki, M., Morikawa, M., Obayashi, S. and Nakahashi, K., Multiobjective Design Optimization of Merging Configuration for an Exhaust Manifold of a Car Engine, Proceedings of the 7th international conference on parallel problem solving from nature, Springer, pp. 8-87,. [4] McCormic, J. M. and Salvadori, M. G.., Numerical Methods in Fortran, Prentic-Hall, Inc., 964. [5] Fujiwara, H., Murakami, A. and Watanabe, Y., Numerical Analysis on Shock Oscillation of Two-Dimensional External Compression Intakes, AIAA Paper -74,. [6] Myong, H.K. and Kasagi, N., A new approach to the improvement of k-epsilon turbulence model for wall-bounded shear flow, JSME International Journal of Fluid Engineering, Vo. 9, pp.56-6, 99. [7] Matuo, Y., Overview of the Next NAL Numerical Simulator; NS, Special Publication of National Aerospace Laboratory, SP-57, pp. 5-,. [8] Seddon, J. and Goldsmith, E. L. Intake Aerodynamics, AIAA Education Series

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