Drag Divergence Mach Number Research Articles

Optimization of transonic low-Reynolds number airfoil based on genetic algorithm

A 2-D airfoil shape optimization in transonic low-Reynolds number regime is conducted. A Navier–Stokes flow solver with a transition model (k-ω SST γ-Reθ) is used to evaluate the fitness function. Single-point and multi-point formulations of the optimization results are compared. In addition, the effects of Mach number and angles of attack on aerodynamic characteristics of the optimized airfoils are investigated under low Reynolds number (Re = 17,000) and high-subsonic-flow ([Formula: see text]) conditions. The results show that the corresponding drag divergence Mach number curves of the conventional airfoil present almost a parallel shifting at the entire Mach number range. By contrast, the unconventional airfoil starts showing a significant drag reduction when Mach number is greater than 0.75. Besides, the maximum lift-to-drag ratio is highly influenced by the Mach number because of the formation, movement, type, and strength of a shock wave. In addition, the distinguishing difference in the conclusion between two airfoils is that the lift fluctuation of the conventional airfoil amplifies with the increase of the Mach number. However, the unconventional airfoil shows an opposite trend.

Knowledge discovery with computational fluid dynamics: Supercritical airfoil database and drag divergence prediction

Aerodynamic rules and knowledge are often obtained through theoretical research and experiments, which have contributed greatly to aircraft design. For example, Korn's equation predicts the airfoil drag divergence Mach number using the airfoil maximum thickness and the lift coefficient. It is very helpful in the aircraft initial design. However, it neither reveals the key factors of fluid features on the drag divergence nor contributes to the detailed design. This paper designs a supercritical airfoil database that covers the typical free stream Mach number, angle of attack, lift coefficient, and geometry of modern transonic commercial aircraft. Correlation screening and multivariate regression are carried out to discover knowledge about the airfoil drag divergence Mach number and pressure distribution features. A new linear correlation is discovered and validated by existed airfoil databases. Compared with Korn's equation, the discovered correlation reduces the maximum prediction error by approximately 40%. It indicates that the drag divergence Mach number can be increased by obtaining a shock wave that is further upstream in the detailed design. Furthermore, it enables the cruise performance and drag divergence Mach number to be predicted with only one simulation of the cruise point, which will greatly save the computational cost of optimizations.

A new many-objective aerodynamic optimization method for symmetrical elliptic airfoils by PSO and direct-manipulation-based parametric mesh deformation

A new many-objective aerodynamic optimization method is proposed to design the distinctive symmetrical elliptic airfoil applied on the canard rotor wing aircraft with special requirements on both low drag in different Mach numbers and large drag divergence Mach number. A parameterized direct manipulation of free form deformation (DFFD) method is proposed to realize the one-step process on geometry parameterization and mesh deformation. Then an efficient multi and many-objective particle swarm optimization algorithm (LMPSO) is built to optimize the symmetrical elliptic airfoil to comprehensively improve its aerodynamics. The biobjective optimization of drag reductions at hover stage and at cruise flight shows that the new method can significantly reduce the drag of the elliptic airfoil but at the cost of drag increases at high angles of attack. The four-objective optimization results indicate that the consideration of the high angle of attack flight conditions can improve the aerodynamic performance of the airfoil over a large lift range, which is more practical in engineering application. The five-objective optimization presents that the maximization of the drag divergence Mach number of the elliptic airfoil can be significantly promoted when adding three more design conditions at high Mach number. In general, the optimization results demonstrate that the developed new method can effectively improve the aerodynamic performance of symmetrical elliptic airfoils and provide support on the development of the canard rotor wing aircraft.

Pressure distribution feature-oriented sampling for statistical analysis of supercritical airfoil aerodynamics

In the field of supercritical wing design, various principles and rules have been summarized through theoretical and experimental analyses. Compared with black-box relationships between geometry parameters and performances, quantitative physical laws about pressure distributions and performances are clearer and more beneficial to designers. With the advancement of computational fluid dynamics and computational intelligence, discovering new rules through statistical analysis on computers has become increasingly attractive and affordable. This paper proposes a novel sampling method for the statistical study on pressure distribution features and performances, so that new physical laws can be revealed. It utilizes an adaptive sampling algorithm, of which the criteria are developed based on Kullback–Leibler divergence and Euclidean distance. In this paper, the proposed method is employed to generate airfoil samples to study the relationships between the supercritical pressure distribution features and the drag divergence Mach number as well as the drag creep characteristic. Compared with conventional sampling methods, the proposed method can efficiently distribute samples in the pressure distribution feature space rather than directly sampling airfoil geometry parameters. The corresponding geometry parameters are searched and found under constraints, so that supercritical airfoil samples that are well distributed in the pressure distribution space are obtained. These samples allow statistical studies to obtain more reliable and universal aerodynamic rules that can be applied to supercritical airfoil designs.

Design of a slotted, natural-laminar-flow airfoil for commercial transport applications

Slotted, natural-laminar-flow (SNLF) airfoils are a novel aerodynamic concept that enable significant performance improvements over conventional, single-element NLF airfoils. The S207, SNLF airfoil has been designed using requirements derived from a transonic, truss-braced wing commercial aircraft configuration. The airfoil is designed for a cruise Reynolds number of 13.2×106 and it exhibits a drag divergence Mach number exceeding 0.71. At these conditions, low drag is predicted to occur between lift coefficients of 0.40 and 0.79. The S207 airfoil exhibits a range-based figure of merit nearly triple that of a turbulent airfoil representative of modern commercial transport aircraft. Slotted configurations also have benefits for low-speed, high-lift conditions representative of approach. The S207 airfoil exhibits a low-speed, maximum lift coefficient in excess of 2.1, and is limited by compressibility effects around the leading edge. Incorporation of this airfoil onto a transonic, truss-braced wing configuration shows strong potential for meeting mid- and far-term goals for reducing aircraft fuel/energy consumption.

Robust optimization for a wing at drag divergence Mach number based on an improved PSO algorithm

In this study, an improved PSO (particle swarm optimization) algorithm is proposed and applied to the robust optimization of a wing at drag divergence Mach number. In order to reduce the number of design variables, a six-order CST (class/shape function transformation) method is employed for airfoil parameterization. For the purpose of improving the optimization efficiency, Delaunay graph mapping method is adopted for mesh deformation in each iteration of the airfoil optimization, and NURBS (non-uniform rational B-splines)-FFD (free-form deformation) method is employed for mesh deformation in each iteration of the wing optimization. For improving the standard PSO algorithm, CVTs (centroidal Voronoi tessellations) method is introduced to generate original positions of the particles more dispersedly, a second-order oscillating scheme is used and an FDR (fitness distance ratio) item is added for updating velocities and positions of the particles. By virtue of the improved PSO algorithm, single point optimization and robust optimization are conducted for both airfoil and wing. The results indicate that, comparing with the single point optimizations, the robust optimizations not only reduce drag coefficients of the airfoil and the wing at cruise Mach numbers, but also attenuate the drag increments as the Mach number increases up to drag divergence Mach numbers.

An optimal control framework for the climb and descent economy modes of flight management systems

This paper proposes an optimal control framework for the climb and descent economy modes of a flight management system (FMS) yielding a solution that can be implemented in real-time flights below the drag divergence Mach number. The problem is formulated as the optimization of a functional that trades off the fuel- and time-related costs of a flight as a function of a (crew-supplied) parameter called the cost index. The work builds on previous research of the authors for the cruise phase and extends it to the climb and descent phases of flight. More specifically, for both climb and descent, it is found that suboptimal solutions can be obtained as the positive real roots of a fifth-degree polynomial lying inside the flight envelope, which can be found using fast-converging algorithms such as Newton's method. The main contributions of this work are threefold. First, the proposed method gives physical insight because there is an analytical expression for each coefficient of the polynomial. Second, this approach eliminates the need to have a performance database in the system, thus making its implementation faster in real-time. Third, the solution exhibits the same behavior of airborne FMS units as a function of the cost index, which is justified in this paper based on Bellman's principle of optimality. This justification is an important theoretical contribution of the paper. A validation of the approximate solution is obtained using the shooting method to compute the optimal trajectories and compare them against the proposed suboptimal solution. Simulation results show that, for an Airbus A320 model and for a Gulfstream-IV aircraft model, the relative error of the suboptimal trajectories when compared to the optimal trajectories is small for climb and descent trajectories, respectively.

Optimal Control Framework for Cruise Economy Mode of Flight Management Systems

The main contribution of this paper is to propose optimal and suboptimal control solutions to the cruise economy mode problem in a flight management system for flights below the drag divergence Mach number. The problem is formulated as an optimization of a functional that trades off the fuel and time-related costs of a flight using a (crew-supplied) cost index CI. A novel approach is proposed based on solving the problem analytically using a combination of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation. A suboptimal analytical solution for the true airspeed is obtained in state feedback form, which reduces to the well-known optimal solution for maximum range when the cost index vanishes. An analytical solution for the speed target as a function of the cost index gives physical insight, allows one to analytically compute sensitivities, and eliminates the need to have a performance database to store the optimal speed schedules in the system. An extension shows that the approach is valid when the aircraft is turning with a small bank angle. Overall, this work provides not only a very efficient means of implementing the optimal speed schedules in an onboard flight management system for flights below the divergence Mach number, but also extends the theory of aircraft performance to the more general case based on a nonzero cost index. The new results are compared with flight simulation data for an A320 Airbus aircraft.

Estimation of Wave Drag of Non-TransonicAirfoils Using Korn Equation

Approximation of Drag Divergence Mach number, Critical Mach Number and the Coefficient of Wave Drag of non-transonic airfoils using Korn equation are presented in this research paper. Due to the presence of the CFD factor which influences the aforementioned parameters heavily, transonic airfoils develop an undue advantage over non-transonic airfoils. A preliminary analysis of the behaviour of the airfoils is done using XFLR5 wherein the aerodynamic characteristics such as coefficient of lift, coefficient of drag, pitching moment, stall angle and aerodynamic efficiency of the airfoils are evaluated at low Reynolds Number. In order to observe the effect of camber and thickness on the performance of airfoils, airfoils of varying camber and thickness are chosen. The airfoils chosen for this research paper are NACA 0012, NACA 0018, MH 23, NLF 0115, FX 83 W108, FX 66 S196, UltraSport 1000 and Clark Y (smooth). The main objective of this paper is to computationally obtain an airfoil which can perform well under both transonic and low Reynolds number conditions.

The drag reduction study of the supercritical airfoil with shock control bump considering the robustness

The shock control bump of supercritical airfoil is numerically investigated by Computational Fluid Dynamics in present paper. The influence of the geometry parameters of the bump is studied by considering the drag reduction performance and robustness. The results show that, the bump location strongly affects on the drag reduction. The best performance is obtained when locating the bump crest at the shock position of the clean airfoil. Higher bump is worse for the drag reduction, while longer bump has good effect in a bigger height range. The bump can effectively weaken the shock wave, suppress the boundary layer separation, and extend the buffet onset boundary, but its working lift coefficient range is relatively narrow. The robustness is an important requirement of engineering application. The results of robustness analysis shows that, the longer and lower bump is more engineering acceptable than the shorter and higher bump, while it could not only improve lift-drag ratio in larger lift coefficient range insensitive to the changes of shapes and Reynolds number, but also increase the drag divergence Mach number.

Using of Particle Swarm for Performance Optimization of Helicopter Rotor Blades

As part of a research activity at Politecnico di Torino, aiming to develop multi-disciplinary design procedures implementing nature inspired meta-heuristic algorithms, a performance design optimization procedure for helicopter rotors has been developed and tested. The procedure optimizes the aerodynamic performance of blades by selecting the point of taper initiation, the root chord, the taper ratio, and the maximum twist which minimize horsepower for different flight regimes. Satisfactory aerodynamic performance is defined by the requirements which must hold for any flight condition: the required power must be minimized, both the section drag divergence Mach number on the advancing side of the rotor disc and the maximum section lift coefficient on the retreating side of the rotor disc must be avoided and, even more important, the rotor must be trimmed. The procedure uses a comprehensive mathematical model to estimate the trim states of the helicopter and the optimization algorithm consists of a repulsive particle swarm optimization program. A comparison with an evolutionary micro-genetic algorithm is also presented.

Extension to the Class-Shape-Transformation Method Based on B-Splines

A new parameterization method to enhance shape optimization for aircraft design was developed based on the class-shape-transformation method. This method was implemented in a tool that combined all aspects of the aerodynamic design process: parameterization, aerodynamic analysis, and optimization. The parameterization method used a combination of Bernstein polynomials and B-splines to allow for both local and global control of a shape. Additionally, the use of B-splines made it possible to efficiently handle volume constraints, which are very common in aircraft design. The parameterization method was coupled to two different aerodynamic analysis tools: the commercial panel method code VSAERO and an in-house Euler code. As a first test case, a wing was optimized for maximum lift-to-drag ratio in subsonic conditions using VSAERO and various optimization algorithms. It was shown that the optimizer was capable of arriving at wing shapes that demonstrated laminar flow up to 80%, without violating the implemented volume constraints. As a second test case, an airfoil was optimized for lift-to-drag ratio in transonic conditions using the in-house Euler solver and a gradient-based optimizer. Refining the shape using B-splines proved to be an efficient way of increasing the design freedom, resulting in additional improvements in the drag divergence Mach number in the order of 0.02.

Reduction of Drag Penalty by means of Plain Flaps in the Boomless Busemann Biplane

This study describes a method to overcome the choked-flow and flow-hysteresis problems of the Busemann biplane at off-design conditions, by using plain flaps at the leading and trailing edges. Two-dimensional analyses of four different biplanes were addressed in order to examine the effect of the flaps, using Computational Fluid Dynamics (CFD) in inviscid flow (Euler) mode. Results show that the leading-edge flaps can alleviate the drag increase due to the choked-flow, and reduce the area of flow-hysteresis. In contrast, the trailing-edge flaps are not effective in overcoming these problems. However, the trailing-edge flaps can reduce the wave drag near to the speed of sound, and also shift the drag-divergence Mach number to a higher one. Thus, the combined effect of both flaps allows us to smoothly achieve the design point (M∞ = 1.7) from subsonic regime, and to avoid the severe drag penalty due to the choked-flow and flow-hysteresis problems.

The development of a GUI program for automated generating rotor blade airfoil performance table

A user-friendly, Windows-based Graphic User Interface (GUI) program that automates the generation of the aerodynamic performance tables for comprehensive helicopter load analysis codes was developed. With this program, computational grids are automatically created for the given airfoil surface coordinates, and users can verify the grids immediately. In addition, the program’s post-processing feature can enable the real-time inspection of the interim results of the computation. The aerodynamic performance tables can be generated by performing automated aerodynamic analysis of the airfoil in various angles of attack and Mach numbers by using validated CFD code. The Mixed-Language technique was employed to combine the FORTRAN-based CFD codes to the C++-based GUI program without needing any transformation between these two languages. The MS Access was used to construct the data base of the aerodynamic tables. The real-time inputs of the analysis results of the ODBC Access data base make it easy to retrieve the aerodynamic tables. Of the entire range of the angle of attack (−180° to 180°), the CFD analysis is only performed in the significant angles depending on the Mach number. The aerodynamic data for the other angles were obtained by using an interpolation method; therefore, the tables could be generated quickly. Various aerodynamic information such as drag divergence Mach number and stall angle can be extracted from the computed results stored in the database.

수정 Sonic Arc 익형의 공력성능

TSD 이론으로 구해진 sonic arc 익형의 형상을 새로운 익형 형상으로 수정하고 천음속 유동에서의 공력 성능을 조사하였다. Euler solver를 이용하여 수정 sonic arc 익형에 대한 수치계산을 수행하고 그 결과를 NACA0012 익형의 공력 성능과 초임계 익형의 공력 성능 그리고 NACA64A210 익형의 공력성능과 비교하였다. 천음속 유동의 같은 자유 유동 마하수에서 수정 sonic arc 익형의 압력 항력은 NACA0012 익형의 압력 항력보다 더 작게 나타났으며 수정 sonic arc 익형의 양항비는 NACA0012 익형의 양항비보다 훨씬 크게 나타났다. 천음속 유동에서의 항력 발산 마하수 비교에서는 수정 sonic arc 익형의 항력 발산 마하수가 NACA64A210 익형의 항력 발산 마하수보다는 크게 나타났지만 초임계 익형의 항력 발산 마하수 보다는 작게 나타났다. Sonic arc airfoil derived from the TSD theory is modified to new airfoil shape and its aerodynamic performance in transonic flow is investigated. The numerical simulation using Euler equations for the modified sonic arc airfoil is performed. The numerical results are compared with the aerodynamic performance of NACA0012 airfoil, of supercritical airfoil, and of NACA64A210 airfoil. In the same free stream Mach number of transonic flow, the pressure drag of the modified sonic arc airfoil is smaller than that of NACA0012 airfoil and the lift-drag ratio of the modified sonic arc airfoil is much larger than that of NACA0012 airfoil. In the comparison of the drag-divergence Mach number of transonic flow, the drag-divergence Mach number of the modified sonic arc airfoil is larger than that of NACA64A210 airfoil but is smaller than that of supercritical airfoil.

Fully integrated aerodynamic/dynamic optimization of helicopter rotor blades

This paper describes a fully integrated aerodynamic/dynamic optimization procedure for helicopter rotor blades. The procedure combines performance and dynamics analyses with a general purpose optimizer. The procedure minimizes a linear combination of power required (in hover, forward flight, and maneuver) and vibratory hub shear. The design variables include pretwist, taper initiation, taper ratio, root chord, blade stiffnesses, tuning masses, and tuning mass locations. Performance constraints consist of limits on power required in hover, forward flight, maneuver, airfoil section stall, drag-divergence Mach number, minimum tip chord, and trim. Dynamic constraints are on blade natural frequencies, minimum autorotational inertia, and maximum blade weight. The procedure is demonstrated for two cases. In the first case, the objective function involves power required (in hover, forward flight, and maneuver) and vibratory hub shear. In the second case, the objective function involves power required in hover and vibratory hub shear. For both cases, the designs from the integrated procedure are compared with designs from a sequential optimization approach. In the sequential optimization approach, the blade is optimized first for performance (minimum power) using only aerodynamic design variables and performance constraints. Then these optimized aerodynamic design variables are held fixed, and the blade is optimized for minimum vibratory hub shear, using only dynamic design variables and dynamic constraints. In both cases, the integrated approach is superior to the sequential approach.

Development of an Inlet for a Tilt Nacelle Subsonic V/STOL Aircraft

A fixed geometry inlet capable of efficient operation to very high angles-of-attack has been developed by the Grumman Aerospace Corporation for a tilt nacelle subsonic V/STOL airplane. The inlet, sized to fit a 5.5-in. (13.97 cm) diameter fan, has been extensively tested and demonstrated very high pressure recoveries and low distortions at higher angles-of-attack than any of the published inlet data to date, including an inlet with the very high contraction ratio of 1.78 on the windward lip. This inlet capability has been achieved by judicious selection of the inlet design parameters affecting flow separation. The substantial angle-of-attack capability of the inlet provides the opportunity for trading off some of the margin at low speed to reduce nacelle maximum diameter, hence weight, and to increase the estimated drag divergence Mach number of 0.75 to higher than Mach 0.80. Tests with lower lip contraction ratio show that this may be possible and investigations exploring this possibility are now in progress at NASA/Lewis.