Numericalmethodo_省略_orlargeairliners_.pdf

SCIENCE CHINA Technological Sciences Science China Press and Springer-Verlag Berlin Heidelberg 2012 *Corresponding author (email: luzl) RESEARCH PAPER September 2012 Vol.55 No.9: 24472452 doi: 10.1007/s11431-012-4936-0 Numerical method of static aeroelastic correction and jig-shape design for large airliners HUANG Wei1, 2, LU ZhiLiang1*, GUO TongQing1, XUE Fei2 2 Shanghai Aircraft Design and Research Institute, Shanghai 200232, China Received March 11, 2012; accepted May 7, 2012; published online July 20, 2012 In this paper, a coupled CFD-CSD method based on N-S equations is described for static aeroelastic correction and jig-shape design of large airliners. The wing structural flexibility matrix is analyzed by a finite element method with a double-beam model. The viscous multi-block structured grid is used in aerodynamic calculations. Flexibility matrix interpolation is fulfilled by use of a surface spline method. The load distributions on wing surface are evaluated by solving N-S equations with a paral- lel algorithm. A flexibility approach is employed to calculate the structural deformations. By successive iterations between steady aerodynamic forces and structural deformations, a coupled CFD-CSD method is achieved for the static aeroelastic cor- rection and jig-shape design of a large airliner. The present method is applied to the static aeroelastic analysis and jig-shape design for a typical large airliner with engine nacelle and winglet. The numerical results indicate that calculations of static aeroelastic correction should employ tightly coupled CFD-CSD iterations, and that on a given cruise shape only one round of iterative design is needed to obtain the jig-shape meeting design requirements. N-S equations, large airliner, static aeroelasticity, flexibility matrix, jig-shape Citation: Huang W, Lu Z L, Guo T Q, et al. Numerical method of static aeroelastic correction and jig-shape design for large airliners. Sci China Tech Sci, 2012, 55: 24472452, doi: 10.1007/s11431-012-4936-0 1 Introduction Aeroelastic research on aircraft mainly includes static aero- elasticity and flutter 1. Conventional static aeroelastic analysis generally employs a panel method 2 to calculate the linearized aerodynamic forces or uses the experimental aerodynamic forces 3, and then the divergence speed and static deformation on specific flight condition are analyzed according to the variation relationship of aerodynamic force with torsional deformation. The aeroelastic problem of large civil airliners is mainly focused on the elastic wing. A large civil airliner usually adopts supercritical wings with high aspect ratio, and its cruise Mach number is commonly in the transonic region. Conven- tional aeroelastic analysis method uses linearized potential flow theory, which cannot handle the viscous effects or simu- late the shock waves on wing surface. Hence, the linearized potential flow theory is unreliable for the aeroelastic analysis of large civil airliners. Compressible full-potential equations can well deal with a weak shock wave. During the past, full-potential equations were mostly employed for the aeroe- lastic correction calculations of civil airliners. However, the method of full-potential equations is a finite difference method, which is not suitable for the complex configurations, such as a large civil airliner with engine nacelle and winglet. Modern CFD techniques based on Euler/N-S equations trend to mature in aspects of the shock wave capture and the viscous boundary layer simulation, and have been widely used in many fields. In order to accurately analyze the static aeroelastic problems of 2448 Huang W, et al. Sci China Tech Sci September (2012) Vol.55 No.9 large civil airliners, it is necessary to take advantage of the nonlinear aerodynamic equations to develop a coupled CFD- CSD method. In order to simplify the load calculations and interpolations during the coupled CFD-CSD iterations, the multi-block grid technique is employed to generate the viscous structured grids for a typical large civil airliner. Deforming grids are generated by use of a rapid elastic deforming technique. The structural flexibility matrix is interpolated from the structural model to the aerodynamic model with a surface spline method. The finite volume scheme and the 5-stage Runge-Kutta explicit time -stepping approach are used to solve the N-S equations, and a parallel algorithm is utilized to improve the computa- tional efficiency. The pressure distributions on the wing sur- face are evaluated with N-S equations, and the structural de- formations are calculated with a flexibility approach. Finally, a coupled CFD-CSD method for the static aeroelastic correc- tion and jig-shape design of large airliners is developed by means of successive iterations between steady aerodynamic forces and structural deformations. The present method is successfully used for the static aeroelastic analysis and jig-shape design of a typical large airliner, and the effects of engine nacelle and winglet are reasonably considered. 2 Structural model of wing The double-beam model is applied to a high aspect ratio wing shown in Figure 1. Stiffness and flexibility matrices are obtained using structural analysis, which will be used in the later static aeroelastic calculations. For a more accurate structural analysis, the flexibility matrix can also be directly extracted from the structural finite element model. 3 Parallelized CFD method for large airliners 3.1 Static and deforming multi-block grid generation Multi-block structured grid is very suitable for numerical simulations of unsteady flow and viscous flow around com- plex assemblies, such as a large civil airliner. For the wing-body-nacelle-winglet combination of a typical large civil airliner, its multi-block grid topology is designed as Figure 1 Structural model of a high aspect ratio wing. follows: The physical space is decomposed on the wing surface along the longitudinal direction, and decomposed at the nacelle and wingtip along the spanwise direction, and the inner space of nacelle is decomposed into two blocks. Total eight blocks are generated. In order to simplify the moving or deforming grid generation during aeroelastic calculations, the H-H type grid is adopted in each block, where the algebraic technique is used to generate the initial grids and the elliptic partial differential equations are used for the grid optimization. Figures 2 and 3 show the surface grid and volume grid of a typical large airliner, respectively. As for static aeroelastic calculations, deforming a grid is needed to model the structural deformations. Here, a rapid elastic deforming technique 4 is used to generate the de- forming grid of the multi-block grid system, in which the wall grids move synchronously with the structural motion and the Figure 2 Surface grid of a typical large airliner. Figure 3 Spatial grid of a typical large airliner. Huang W, et al. Sci China Tech Sci September (2012) Vol.55 No.9 2449 far-field grids are fixed. 3.2 N-S simulations of steady aerodynamic forces In the Cartesian coordinate system, the three-dimensional N-S equations in a conservative differential form can be written as , RST txyzxyz Wfgq (1) where W is the vector of conservative variables, and defined as , u v w E W (2) f, g and q are the convective flux vectors defined as 2 , u up uv uw uEup f 2 , v vu vp vw vEvp g 2 . w wu wv wp wEwp q In the above formulas, p, and E are the pressure, the density and the total energy per unit mass, respectively; u, v and w are the x-, y- and z- velocity components of the fluid velocity; R, S and T are the viscous flux vectors. Pressure p is obtained from 222 1 1. 2 pEuvw (3) The finite volume scheme and the 5-stage Runge-Kutta explicit time-stepping approach are used to solve the N-S equations 5. The Spalart-Allmaras one-equation turbu- lence model is employed to simulate the turbulent flows 6. The local time step, residual smoothing and enthalpy damping are utilized to accelerate the convergence 7. 3.3 Parallelization The N-S solver is parallelized for more efficient calcula- tions. Based on the basic block of a multi-block system, the flow in each block is calculated on a single processor 8. The data exchange between adjacent blocks is done on the block interfaces using functions of MPICH2. The mul- ti-block decomposition of the structured grid is used for the load balancing. Meanwhile, the decomposition should make the data exchanges as few as possible 9. For each block, the interface mapping relationships with adjacent blocks and the orders of data sending and data receiving should be established prior to the iterative calculations, and then they can directly be used to accomplish the data exchange during the iterative calculations. 4 Coupled CFD-CSD method for static aeroe- lastic analysis 4.1 Interpolation of flexibility matrix The flexibility matrix on structural nodes is directly ex- tracted by a structural analysis. However, the flexibility matrix on aerodynamic nodes is needed in the present method. The interpolation of flexibility matrix is fulfilled using a surface spline method 1, formulated as T , ArSr CT CT (4) where Tr is the interpolating matrix, CS the given flexibility matrix on structural nodes, and CA the flexibility matrix on aerodynamic nodes. According to the flexibility approach, the longitudinal deformation of aircraft surface is calculated by , A ZC B (5) where B is the aerodynamic force vector, which is defined as the force difference calculated with the gird cell on the wing lower surface and the corresponding cell on the upper surface. This kind of simplification is usually employed in engineering. Although this will change the real wingspan to some extent, the error can be accepted in the small flexibil- ity and small deformation cases. 4.2 Coupled iterative procedure For a given airliner, the elastic correction calculations on certain flight conditions are performed by means of succes- sive iterations between steady aerodynamic forces and structural deformations. In this way, the deforming process, the final equilibrium position of wing and the corrected aerodynamic forces are obtained. The solution procedure of the coupled CFD-CSD method is described in detail as fol- lows. 1) Evaluate the pressure distributions on the original shape of the given aircraft with N-S equations. 2) Calculate the structural deformation of wing using the flexibility method, and obtain a deformed aerodynamic shape. 3) Generate the deforming grids for the deformed aero- dynamic shape. 4) Once again, evaluate the new load distributions on the deformed aerodynamic shape. 5) Repeat steps 24 until the numerical results converge to an equilibrium state; then the static aeroelastic correc- 2450 Huang W, et al. Sci China Tech Sci September (2012) Vol.55 No.9 tions under the given angle of attack , Mach number Ma, Reynolds number Re and dynamic pressure q are analyzed. Because the aerodynamic nonlinearity has been consid- ered, the whole solution procedure should be done again for another angle of attack. The iterative procedure described above is loosely coupled, and about 58 times of steady computation are needed to obtain the convergent solution. Therefore, it is quite time-consuming. By taking advantage of the rapid deforming elastic de- forming technique for deforming grid generation, the con- vergence of aerodynamic forces is kept synchronous with that of the coupled CFD-CSD iterations to enable tightly coupled CFD-CSD iterations. Since the tightly coupled CFD-CSD method does not need to march the aerodynamic equations to a steady state strictly at each iteration, its total computing time is about 1.11.2 times of that of the corre- sponding steady problem. 4.3 Effects of components Once the distributed loads on airliner surfaces are evaluated with N-S equations, the effects of other components, such as engine nacelle and winglet, can be taken into account by means of concentrated forces. According to the equivalence theory, the distributed loads on nacelle surfaces are ap- proximately equivalent to a concentrated force acting on the main hanging point of engine and a y-moment. Then the y-moment is decomposed into two concentrated forces, Fqy and Fhy, acting on the front and rear hanging points, respec- tively. The concentrated forces of nacelle are illustrated in Figure 4, where the circle symbol represents the action point of force. All concentrated forces are directly exerted on the corresponding positions of wing during numerical simula- tions. The effect of the winglet is considered in the same man- ner. The coordinate system is the same as shown in Figure 4, Figure 4 Concentrated forces of engine nacelle. where its distributed loads are approximately equivalent to a concentrated force Facting on the main beam tip and two moments, Mx and My. Mx is decomposed into two concen- trated forces, acting on the tip and a spanwise point close to the tip, respectively. My is decomposed into two concen- trated forces, acting on the tip and a chordwise point close to the tip, respectively. In a simplified way, the winglet moves synchronously with the wing tip. 5 Jig-shape design of wing During the practical wing design of a large civil airliner, the static aeroelastic effect is considered with the coupled CFD-CSD method to design a jig-shape. The design proce- dure is described in detail in this section. Firstly, steady aerodynamic forces on the given cruise shape in cruise state are calculated with N-S equations, and the static bending and torsional deformations are calculated with the flexibility approach. Secondly, if the bending deformation does not have obvious effect on the aerodynamic forces, an initial jig-shape can be obtained by deducting the torsional defor- mation from the given cruise shape. Thirdly, the static de- formations and aerodynamic characteristics of the initial jig-shape at the cruise condition are solved with the tightly coupled CFD-CSD method described in section 4.2. Finally, the static deformations of the initial jig-shape are compared with those of the given cruise shape. If the relevant indexes meet the design requirements, a jig-shape is determined. Otherwise, based on the jig-shape of last round design, a next round of iterative design should be done by deducting the deformation difference, until the jig-shape meets all design requirements. 6 Results and discussions 6.1 Validation calculations of CFD solver The present CFD solver is employed to calculate the tran- sonic steady aerodynamic forces on the rigid wind tunnel model for the wing-body combination of a typical large civil airliner, which was tested in a 2 m transonic wind tun- nel. As shown in Figure 5, there are some differences in stall characteristics, but the calculated results at small and moderate angles of attack agree well with the test. Hence, it is reliable to carry out the elastic correction calculations with the present CFD solver. 6.2 Static aeroelastic calculations Static aeroelastic calculations are performed on the wing- body-nacelle-winglet combination of a typical large airliner with Mach number 0.785, flight altitude 12000 m and Reynolds number 6106. The lift and drag comparisons Huang W, et al. Sci China Tech Sci September (2012) Vol.55 No.9 2451 Figure 5 Steady aerodynamic force on a rigid wind tunnel model for a wing-body combination of a typical large airliner. between the rigid model and the elastic model are shown in Figures 6 and 7, respectively. When the structural elasticity is taken into account, the lift and drag will decrease a lot at the same angles of attack. The static deformations of the wing are described in terms of deflection and twist angle. The deflection is de- fined as the vertical deformation displacement, and is posi- tive when the wing deforms upwards. The twist angle is defined as the trailing edge angle deformation relative to the leading edge, and is positive when the leading edge deforms Figure 6 Lift coefficient curve. Figure 7 Drag coefficient curve. upwards relatively to the trailing edge. Figures 8 and 9 show the spanwise deflection distribution and twist angle distribution, respectively. Figure 10 illustrates the static deformation at angle attack of 2. 6.3 Calculations of wing jig-shape design The jig-shape design of a wing is performed on the wing-body-nacelle-winglet combination of a typical large airliner with an aspect ratio of 11.5. Its cruise Mach number Ma = 0.785, cruise altitude H = 12000 m and cruise angle of attack = 2.5. Table 1 gives the lift and drag comparisons Figure 8 Spanwise deflection distribution. Figure 9 Spanwis twist angle distribution. Figure 10 Static deformation at angle attack of 2. 2452 Huang W, et al. Sci China Tech Sci September (2012) Vol.55 No.9 Table 1 Lift and drag comparisons between the given cruise shape and designed jig-shape () Cl Cd Cruise shape Jig-shape Cruise shape Jig-shape 2.5 0.605 0.608 0.0458 0.0460 between the given cruise shape and the jig-shape with one round of iterative design. Figures 11 and 12 show the com- parison of the corresponding deflection and twist angle, respectively. Due to the great difference between basic aerodynamics and cruise aerodynamics, multiple rounds of the coupled CFD-CSD iterations are needed for the aeroelastic correc- tion calculations based on a given jig-shape. For the jig- shape design of the wing, however, the aerodynamic forces of the basic shape and the target shape both correspond to the cruise state, and the error of reverse deformation deduc- tion is small,