Comparison of
Abstract
The focus of this chapter is on the shape optimization of the Busemann-type biplane airfoil for drag reduction under both non-lifting and lifting conditions using genetic algorithms. The concept of biplane airfoil was first introduced by Adolf Busemann in 1935. Under design conditions at a specific supersonic flow speed, the Busemann biplane airfoil eliminates all wave drag due to its symmetrical biplane configuration; however, it produces zero lift. Previous research has shown that the original Busemann biplane airfoil shows a poor performance under off-design conditions. In order to address this problem of zero lift and to improve the off-design-condition performance, shape optimization of an asymmetric biplane airfoil is performed. The commercially available computational fluid dynamics (CFD) solver ANSYS FLUENT is employed for computing the inviscid supersonic flow past the biplane airfoil. A single-objective genetic algorithm (SOGA) is employed for shape optimization under the non-lifting condition to minimize the drag, and a multi-objective genetic algorithm (MOGA) is used for shape optimization under the lifting condition to maximize both the lift and the lift-to-drag ratio. The results obtained from both SOGA and MOGA show a significant improvement in the design and off-design-condition performance of the optimized Busemann biplane airfoil compared to the original airfoil.
Keywords
- genetic algorithm
- shape optimization
- biplane airfoil
- computational fluid dynamics
- supersonic flow
1. Introduction
For decades, the speed of commercial aircrafts has been bounded by the sound barrier. Even the most successful supersonic transport (SST) plane, the Concorde, could only be deployed in very few routes due to government regulation, low efficiency, and excessive noise generation. Since the retirement of Concorde in 2003, the desire for developing a replacement for Concorde space still remains. In order to accomplish that, one of the biggest design challenges is to eliminate, or at least greatly reduce, the strong bow shock wave generated during supersonic flight, which can cause high wave drag and substantial noise. At supersonic speed, a bow shock is generated ahead of the airplane. This shock wave generates a substantially high wave drag which needs to be overcome by the engine by providing a much higher thrust compared to that for a conventional subsonic/transonic airplane, which results in higher fuel consumption and low propulsive efficiency. In order to address the high-wave-drag problem at supersonic speed, a biplane concept was proposed by Adolf Busemann in 1935 [1] which can potentially avoid the formation of bow shock and thus does not create sonic boom.
During the period from 1935 to 1960, substantial research was conducted on the Busemann biplane concept. Moeckel [2] and Licher [3] performed the theoretical analysis of the optimized lifting supersonic biplanes. Tan [4] took a further step and derived analytical expressions for the drag and lift of a three-dimensional supersonic biplane with a finite span of rectangular plan form. Some experimental results were obtained by Ferri [5] using a wind tunnel, and comparisons were made between the experimental and analytical results. During the past 10 years, considerable interest has been shown in supersonic biplane airfoils. Igra and Arad [6] tested and analyzed the effects of different parameters on the drag coefficient of the Busemann airfoil under various flow conditions. Kusunose et al. [7] proposed a concept for the next-generation supersonic transport using the Busemann biplane design. A series of studies using both computational fluid dynamics (CFD) and wind-tunnel experimental methods have been completed by Kusunose’s research group [7–16].
Recently, Hu et al. [17] employed a multi-point adjoint-based aerodynamic design and optimization method to improve the baseline Busemann biplane airfoil’s off-design performance and alleviate the flow-hysteresis problem. They also addressed the problem of minimizing its drag by shape optimization. This chapter addresses the aerodynamic shape optimization of two-dimensional symmetric Busemann biplane airfoil under both design and off-design conditions for reducing its wave drag, and the alleviation of the flow-hysteresis and choked-flow effects using a single-objective genetic algorithm. The symmetric Busemann biplane airfoil generates zero lift; this problem is addressed by introducing asymmetry in the shape of two profiles of the biplane. The asymmetric configuration is shape optimized for maximum lift and minimum drag by employing a multi-objective genetic algorithm. The flow field is computed using the commercial CFD software ANSYS FLUENT. Body-fitted H-grids around the airfoils are generated using the ICEM software. Random airfoil shapes with constraints in a given generation of the genetic algorithm are obtained using the Bezier curves (third-order polynomials).
2. Flow field simulation of the standard diamond-shaped airfoil and the baseline Busemann biplane airfoil under design and off-design conditions
The flow fields of both the standard diamond-shaped airfoil and the Busemann biplane airfoil are computed at zero angle of attack (the zero-lift condition). For meaningful comparison, the total thickness of the two airfoils is set to be equal. For this specific case under consideration, the thickness-to-chord ratio of the diamond-shaped airfoil is
2.1. Mesh generation
The commercial meshing software ANSYS ICEM CFD is used to generate the mesh for computing the flow field. The far-field boundary is set at 21 chord length by a 20.5-chord-length rectangle from the center of the airfoil. Figure 1 shows the H-mesh configuration generated for the simulations. There are 64 nodes around the two components both horizontally and vertically. In between the two components, the grid has a dimension of 32 nodes (vertically) × 64 nodes (horizontally). An ICEM replay script file is created to automatically generate the mesh for the flow past different airfoil shapes once it is called.
2.2. Steady-state flow field simulation at M < 1.7
As has been mentioned above, the baseline Busemann biplane airfoil has a much lower drag under its design condition at Mach number
Figure 3 shows the drag coefficient
As shown in Figure 3, the baseline Busemann biplane airfoil has a higher drag compared to the standard diamond-shaped airfoil when the Mach number is low (0.3 ≤
2.3. Flow field simulation during acceleration and deceleration
Despite the high drag being generated under off-design conditions, there is an even worse problem caused by the flow-hysteresis phenomenon during acceleration and choked-flow phenomenon during deceleration that need to be addressed under off-design conditions. To demonstrate these phenomena, flow field simulations are conducted under both acceleration and deceleration using the previous simulation results shown in Figure 4 as the initial condition. Figure 5 shows two separated
2.4. Flow field of the baseline Busemann biplane airfoil during acceleration
In this section, the flow-hysteresis phenomenon during acceleration is examined for the baseline Busemann biplane airfoil under the non-lifting condition. The pressure coefficient contours for flow past the baseline Busemann biplane airfoil during acceleration are shown in Figure 6. To simulate the non-lifting condition, the angle of attack is set to 0. The resulting non-lifting flow field around the baseline Busemann airfoil is shown in Figure 6 at various supersonic Mach numbers ranging from
2.5. Flow field of the baseline Busemann biplane airfoil during deceleration
In this section, the choked-flow phenomenon of the baseline Busemann biplane airfoil during deceleration is examined. The contours of pressure coefficients for flow past the baseline Busemann airfoil during deceleration are shown in Figure 7. To simulate the non-lifting condition, the angle of attack is set to 0 as before. The resulting non-lifting flow field around the baseline Busemann airfoil is shown in Figure 7 at various supersonic Mach numbers ranging from
In conclusion, the baseline Busemann biplane airfoil produces a substantially higher drag in the low Mach number range (below the design Mach number of
3. Optimization of Busemann-type biplane airfoil
In this chapter, the shape optimization procedure for the baseline Busemann biplane airfoil using the genetic algorithms (GAs) and the results of the optimized Busemann-type airfoil under both non-lifting and lifting conditions are presented. The optimization process is established by coupling a single-objective genetic algorithm (SOGA)- or a multi-objective genetic algorithm (MOGA)-based optimization method with the mesh generation software ANSYS-ICEM and the CFD solver ANSYS FLUENT as shown in Figure 8.
Every individual (airfoil) in each generation of SOGA/MOGA is represented by a set of control points, which randomly generate the airfoil shape by using the Bezier curves. The mesh generation software ICEM is used to generate a two-dimensional structured mesh around the airfoil as an input to the CFD solver FLUENT, which is then used to calculate the supersonic inviscid flow fields for specific flow conditions. Based on the fitness values of all airfoil shapes in a given generation, SOGA/MOGA is applied to create the next generation of airfoils, and this whole process is repeated until the optimal fitness value is obtained. The airfoil shape that corresponds to the optimal fitness value is the final shape of the optimized airfoil [18].
3.1. Overview of genetic algorithm
Genetic algorithms (GAs) are a class of stochastic optimization algorithms inspired by the biological evolution [19]. They efficiently exploit historical information to speculate on new offspring with improved performance [20]. For the Busemann biplane airfoil in particular, genetic algorithms are used to generate new shapes that produce much lower drag by minimizing the flow-hysteresis phenomenon during acceleration and the choked-flow phenomenon during deceleration. Generally, The GA employs the following steps to complete the optimization process:
Initialization: randomly generates a group of individuals.
Evaluation: evaluates the fitness of each individual generated.
Natural selection: individuals who have the lowest fitness are removed.
Reproduction: pairs of individuals are picked to produce the offspring, which is often done by roulette wheel sampling. A crossover function is then used to produce the offspring.
Mutation: randomly modifies some small percentage of the population.
Check for convergence: if the current generation has converged, the individual having the best fitness will be returned. Otherwise, the process will be repeated starting from Step 2 until the predefined tolerance criteria for acceptable change in fitness between generations are met.
3.1.1. Application of single-objective genetic algorithm (SOGA)
For the non-lifting condition, when the lift coefficient
where
As the wave drag of the Busemann biplane airfoil is much lower when the flow is unchoked, it is highly desirable that the strong bow shock wave in front of the airfoil is swallowed into the area between the upper and lower components of the airfoil before the flow speed approaches the design Mach number (
Mach number | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 |
---|---|---|---|---|---|---|---|
Baseline | 1050 | 996 | 957 | 928 | 906 | 889 | 873 |
Optimized (even) | 527 | 473 | 419 | 376 | 332 | 112 | 106 |
Optimized (uneven) | 539 | 486 | 428 | 382 | 336 | 107 | 101 |
Mach number | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 1.2 | 1.1 |
---|---|---|---|---|---|---|---|
Baseline | 873 | 889 | 906 | 928 | 957 | 996 | 1050 |
Optimized (even) | 106 | 112 | 127 | 152 | 419 | 473 | 527 |
Optimized (uneven) | 101 | 107 | 122 | 146 | 428 | 486 | 539 |
GA parameters | Description |
---|---|
Generation size | 8 individuals per generation |
Number of generations | Maximum of 50 generations if convergence not obtained |
Number of design variables | 14 in total, 7 for acceleration and 7 for deceleration |
Selection type | Roulette Wheel Selection |
Crossover rate | 0.7 |
Mutation rate | 0.1 |
Error of mutation constant | 0.8, which determines how much mutation affects the curves as generations proceed |
Here, we have a total of seven design points in the Mach numbers ranging from
A code in the MATLAB package is developed and utilized in the optimization process of the airfoil as well as for ICEM meshing and FLUENT flow field calculations. All SOGA parameters are defined based on the GA methodology, as shown in Table 3.
3.1.2. Airfoil parameterization
The shape of the airfoil is generated by using the Bezier curves (third-order polynomials). Bezier curves are frequently used in computer graphics to obtain curves that appear to be reasonably smooth at all scales. One of the main reasons that Bezier curves are used in computer graphics is that they can be efficiently constructed; each Bezier curve is simply defined by a set of control points [21].
For the non-lifting case, with
In Eqs. (3) and (4), the subscripts “low” and “up” correspond to the lower and upper lines of the airfoil; the origin is at the center of the upper line. Two Bezier curves are used to generate the shape-defining upper line. Each Bezier curve is defined by a set of four control points. Each control point is constrained by a specified range of
3.1.3. Optimization results
After implementing SOGA for 20 generations with eight individuals in each generation, an optimal shape result for symmetric Busemann-type biplane airfoil under the non-lifting condition with a minimum drag is obtained. Figure 10 shows the geometry of the original Busemann biplane airfoil (red) and the optimized Busemann biplane airfoil (blue) under the non-lifting condition.
Mach number | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 |
---|---|---|---|---|---|---|---|
Baseline | 1097 | 1040 | 999 | 967 | 943 | 926 | 923 |
Optimized (GA) | 622 | 506 | 428 | 389 | 132 | 114 | 104 |
Mach number | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 1.2 | 1.1 |
---|---|---|---|---|---|---|---|
Baseline | 32 | 926 | 940 | 967 | 999 | 1040 | 1098 |
Optimized (GA) | 104 | 114 | 132 | 163 | 428 | 506 | 621 |
The drag coefficients for the seven design points are compared in Tables 4 and 5 for both the original and optimized Busemann biplane airfoil under the non-lifting condition during acceleration and deceleration, respectively. As shown in Tables 4 and 5, the baseline Busemann biplane airfoil is choked at all Mach numbers within the optimization range, while the optimized Busemann biplane airfoil un-chokes at
Figures 11 and 12, respectively, show the change in the pressure coefficient
Figure 15 shows the comparison of the drag coefficients for the standard diamond-shaped airfoil, the baseline Busemann biplane airfoil, and the optimized Busemann biplane airfoil under the non-lifting condition. As shown in this figure, the separation between the acceleration and deceleration curves for
Next, we examine the details of the flow field during acceleration and deceleration for the optimized Busemann biplane airfoil and compare them with the original baseline Busemann biplane airfoil and the optimization results obtained by Hu et al. [17] using an adjoint-based optimization technique. Figures 16 and 17 show the pressure coefficient contours of the optimized Busemann biplane airfoil under acceleration and deceleration, respectively. Figures 18 and 19 show the pressure coefficient contours of the optimized Busemann biplane airfoil using the adjoint-based technique [17] under acceleration and deceleration, respectively. During acceleration, the flow-hysteresis effect still exists, and a bow shock wave is formed in front of the airfoil. The swallowing of the bow shock wave happens when the Mach number increases from 1.49 to 1.50 in our GA optimization as shown in Figure 16(f); it happens when the Mach number increases from 1.52 to 1.53 in the adjoint-based optimization as shown in Figure 18(e), and it happens when the Mach number increases from 2.12 to 2.13 for the original Busemann biplane airfoil as shown in Figure 6(d). The drag coefficient decreases from 0.03556 to 0.01316 in the present GA-based optimization and from 0.03336 to 0.01221 in the adjoint-based optimization [17].
In conclusion, as shown in Figure 20, the drag coefficient of the GA-optimized Busemann biplane airfoil is significantly reduced when compared to the original Busemann biplane airfoil, and it matches with the adjoint-based optimization result obtained by Hu et al. [17].
3.2. Shape optimization of Busemann biplane airfoil under lifting conditions
3.2.1. Application of multi-objective genetic algorithm (MOGA)
Under the lifting condition with the lift coefficient
GA parameters | Description |
---|---|
Generation size | 8 individuals per generation |
Number of generations | Maximum of 50 generations if convergence not obtained |
Number of design variables | 28 in total, 14 (7 for |
Selection type | Roulette Wheel Selection |
Crossover rate | 0.7 |
Mutation rate | 0.1 |
Error of mutation constant | 0.8, which determines how much mutation affects the curves as generations go on |
3.2.2. Airfoil parameterization
Similar to that for the non-lifting case, the random shapes of the airfoil are generated by using the Bezier curves with control points. For the lifting case with
3.2.3. Optimization results
After implementing MOGA for 20 generations with eight individuals in each generation, an optimal shape for asymmetric Busemann-type airfoil under lifting conditions with maximum
The drag coefficients for the seven design points are compared in Tables 7 and 8 for both the original and optimized Busemann biplane airfoil under lifting condition. As shown in Tables 7 and 8, the baseline Busemann biplane airfoil is choked at all Mach numbers within the optimization range, while the optimized Busemann biplane airfoil is unchoked at
Mach number | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 |
---|---|---|---|---|---|---|---|
Baseline | 1097 | 1040 | 999 | 967 | 943 | 926 | 923 |
Optimized | 683 | 633 | 529 | 461 | 418 | 148 | 136 |
Mach number | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 1.2 | 1.1 |
---|---|---|---|---|---|---|---|
Baseline | 32 | 926 | 940 | 967 | 999 | 1040 | 1098 |
Optimized | 136 | 148 | 173 | 224 | 529 | 633 | 682 |
Figures 23 and 24 show a change in the pressure coefficient
Figure 25 shows the comparison of the drag coefficients for the standard diamond-shaped airfoil, the baseline Busemann biplane airfoil and the optimized Busemann biplane airfoil under lifting condition. As shown in the figure, the separation between the acceleration and deceleration
Next, we examine the details of the flow field during acceleration and deceleration for the optimized Busemann biplane airfoil under lifting conditions and compare them with the flow field of the original Busemann biplane airfoil. Figures 26 and 27 show the pressure coefficient contours of the optimized Busemann biplane airfoil using GA during acceleration and deceleration, respectively. During acceleration, the flow-hysteresis effect still exists, and a bow shock wave is formed in front of the airfoil. The swallowing of the bow shock wave happens when the Mach number increases from 1.49 to 1.50 as shown in Figure 16(f) and from 2.12 to 2.13 for the original Busemann biplane airfoil as shown in Figure 6(d). The corresponding drag coefficient decreases from 0.04116 to 0.01665 for the optimized Busemann biplane airfoil.
During deceleration, the choked-flow effect still exists; however, it is shifted to a lower Mach number of 1.38 compared to 1.6 for the original Busemann biplane airfoil.
In conclusion, the drag coefficient of the optimized Busemann biplane airfoil under lifting conditions is significantly reduced compared to the original Busemann biplane airfoil as shown in Figures 25–27. Table 9 gives the lift coefficient of the optimized Busemann airfoil under lifting conditions at different design points.
Mach number | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 |
---|---|---|---|---|---|---|---|
0.3319 | 0.3553 | 0.3061 | 0.2842 | 0.2858 | 0.2525 | 0.2713 | |
0.3306 | 0.3551 | 0.3064 | 0.3032 | 0.2678 | 0.2525 | 0.2712 |
4. Conclusions
In this chapter, numerical simulations of the flow past the standard diamond-shaped airfoil and the baseline Busemann-type biplane airfoil have been conducted. An impulsive uniform flow, a flow during acceleration, and a flow during deceleration are simulated. The original Busemann biplane airfoil shows a poor performance under off-design conditions due to the flow-hysteresis phenomenon during acceleration and the choked-flow phenomenon during deceleration. For shape optimization, a single-objective genetic algorithm (SOGA) and a multi-objective genetic algorithm (MOGA) are employed to optimize the shape of the Busemann-type biplane airfoil under non-lifting and lifting conditions, respectively, to improve its performance under off-design conditions. The commercially available CFD solver FLUENT is employed to calculate the flow field on an unstructured mesh generated using the ICEM grid generation software. A second-order accurate steady-density-based solver in FLUENT is employed to compute the supersonic flow field. The optimization results for the non-lifting case show a significant improvement in reducing the drag coefficient under off-design conditions for the optimized Busemann-type biplane airfoil shape compared to the original shape. The flow-hysteresis phenomenon during acceleration and the choked-flow phenomenon during deceleration are both alleviated significantly for the optimized shape. For the lifting case, the optimized Busemann biplane airfoil is able to significantly reduce the drag coefficient under off-design conditions while generating lift at the same time.
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