Implicit discontinuous Galerkin method on agglomerated high-order grids for 3D simulations

High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is inves-tigated in combination with discontinuous Galerkin (DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry represen-tation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomer-ated high-order mesh works well with DG method in dealing with flow problems of curved geome-tries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant effi-ciency improvements compared to its explicit counterparts.     

A multi-aircraft conflict detection and resolution method for 4-dimensional trajectory-based operation

Conflict Detection and Resolution(CDR) is the key to ensure aviation safety based on Trajectory Prediction(TP). Uncertainties that affect aircraft motions cause difficulty in an accurate prediction of the trajectory, especially in the context of four-dimensional(4D) Trajectory-Based Operation(4DTBO), which brings the uncertainty of pilot intent. This study draws on the idea of time geography, and turns the research focus of CDR from TP to an analysis of the aircraft reachable space constrained by 4D waypoint constraints. The concepts of space–time reachability of aircraft and space–time potential conflict space are proposed. A novel pre-CDR scheme for multiple aircraft is established. A key advantage of the scheme is that the uncertainty of pilot intent is accounted for via a Space-Time Prism(STP) for aircraft. Conflict detection is performed by verifying whether the STPs of aircraft intersect or not, and conflict resolution is performed by planning a conflict-free space–time trajectory avoiding intersection. Numerical examples are presented to validate the efficiency of the proposed scheme.     

Robust Navier-Stokes method for predicting unsteady flowfield and aerodynamic characteristics of helicopter rotor

A robust unsteady rotor flowfield solver CLORNS code is established to predict the complex unsteady aerodynamic characteristics of rotor flowfield. In order to handle the difficult problem about grid generation around rotor with complex aerodynamic shape in this CFD code,a parameterized grid generated method is established, and the moving-embedded grids are constructed by several proposed universal methods. In this work, the unsteady Reynolds-Averaged Navier-Stokes(RANS) equations with Spalart-Allmaras are selected as the governing equations to predict the unsteady flowfield of helicopter rotor. The discretization of convective fluxes is accomplished by employing the second-order central difference scheme, third-order MUSCL-Roe scheme, and fifth-order WENO-Roe scheme. Aimed at simulating the unsteady aerodynamic characteristics of helicopter rotor, the dual-time scheme with implicit LU-SGS scheme is employed to accomplish the temporal discretization. In order to improve the computational efficiency of holecells and donor elements searching of the moving-embedded grid technology, the ‘‘disturbance diffraction method" and ‘‘minimum distance scheme of donor elements method" are established in this work. To improve the computational efficiency, Message Passing Interface(MPI) parallel method based on subdivision of grid, local preconditioning method and Full Approximation Storage(FAS) multi-grid method are combined in this code. By comparison of the numerical results simulated by CLORNS code with test data, it is illustrated that the present code could simulate the aerodynamic loads and aerodynamic noise characteristics of helicopter rotor accurately.     

Identification method for helicopter flight dynamics modeling with rotor degrees of freedom

A comprehensive method based on system identification theory for helicopter flight dynamics modeling with rotor degrees of freedom is developed. A fully parameterized rotor flapping equation for identification purpose is derived without using any theoretical model, so the confidence of the identified model is increased, and then the 6 degrees of freedom rigid body model is extended to 9 degrees of freedom high-order model. Bode sensitivity function is derived to increase the accuracy of frequency spectra calculation which influences the accuracy of model parameter identification. Then a frequency domain identification algorithm is established. Acceleration technique is developed furthermore to increase calculation efficiency, and the total identification time is reduced by more than 50% using this technique. A comprehensive two-step method is established for helicopter high-order flight dynamics model identification which increases the numerical stability of model identification compared with single step algorithm. Application of the developed method to identify the flight dynamics model of BO 105 helicopter based on flight test data is implemented. A comparative study between the high-order model and rigid body model is performed at last. The results show that the developed method can be used for helicopter high-order flight dynamics model identification with high accuracy as well as efficiency, and the advantage of identified high-order model is very obvious compared with low-order model.     

An SOR Implicit Time-accurate Scheme in Calculating Unsteady Flows

A new time-accurate marching scheme for unsteady flow calculations is proposed in the present work. This method is the combination of classical Successive Over-Relaxation (SOR) iteration method and Jacobian matrix diagonally dominant splitting method of LUSGS. One advantage of this algorithm is the second-order accuracy because of no factorization error. Another advantage is the low computational cost because the Jacobian matrices and fluxes are only calculated once in each physical time step. And, the SOR algorithm has better convergence property than Gauss-Seidel. To investigate its accuracy and convergency, several unsteady flow computa- tional tests are carried out by using the proposed SOR algorithm. Roe’s FDS scheme is used to discritize the inviscid flux terms. Un- steady computational results of SOR are compared with the experiment results and those of Gauss-Seidel. Results reveal that the numerical results agree well with the experimental data and the second-order accuracy can be obtained as the Gauss-Seidel for unsteady flow computations. The impact of SOR factor is investigated for unsteady computations by using different SOR factors in this algorithm to simulate each computational test. Different numbers of inner iterations are needed to converge to the same criterion for different SOR factors and optimal choice of SOR factor can improve the computational efficiency greatly.     

Numerical Simulation of Rotor Flow Field Based on Overset Grids and Several Spatial and Temporal Discretization Schemes

A numerical method based on solutions of Euler/Navier-Stokes (N-S) equations is developed for calculating the flow field over a rotor in hover. Jameson central scheme, van Leer flux-vector splitting scheme, advection upwind splitting method (AUSM) scheme, upwind AUSM/van Leer scheme, AUSM+ scheme and AUSMDV scheme are implemented for spatial discretization, and van Albada limiter is also applied. For temporal discretization, both explicit Runge-Kutta method and implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method are attempted. Simultaneously, overset grid technique is adopted. In detail, hole-map method is utilized to identify intergrid boundary points (IGBPs). Furthermore, aimed at identification issue of donor elements, inverse-map method is implemented. Eventually, blade surface pressure distributions derived from numerical simulation are validated compared with experimental data, showing that all the schemes mentioned above have the capability to predict the rotor flow field accurately. At the same time, vorticity contours are illustrated for analysis, and other characteristics are also analyzed.     

Calibration of a γ-Re_θ transition model and its validation in low-speed flows with high-order numerical method

Based on the Reynolds-averaged Navier–Stokes(RANS)equations and structured grid technology,the calibration and validation of γ-Re_θ transition model is preformed with fifth-order weighted compact nonlinear scheme(WCNS),and the purpose of the present work is to improve the numerical accuracy for aerodynamic characteristics simulation of low-speed flow with transition model on the basis of high-order numerical method study.Firstly,the empirical correlation functions involved in the γ-Re_θ transition model are modified and calibrated with experimental data of turbulent flat plates.Then,the grid convergence is studied on NLR-7301 two-element airfoil with the modified empirical correlation.At last,the modified empirical correlation is validated with NLR-7301 two-element airfoil and high-lift trapezoidal wing from transition location,velocity profile in boundary layer,surface pressure coefficient and aerodynamic characteristics.The numerical results illustrate that the numerical accuracy of transition length and skin friction behind transition location are improved with modified empirical correlation function,and obviously increases the numerical accuracy of aerodynamic characteristics prediction for typical transport configurations in low-speed range.     

Numerical analysis on noise of rotor with unconventional blade tips based on CFD/Kirchhoff method

A solver is developed aiming at efficiently predicting rotor noise in hover and forward flight. In this solver, the nonlinear near-field solutions are calculated by a hybrid approach which includes the Navier-Stokes and Euler equations based on a moving-embedded grid system and adaptive grid methodology. A combination of the third-order upwind scheme and flux-difference splitting scheme, instead of the second-order center-difference scheme which may cause larger wake dissipation, has been employed in the present computational fluid dynamics (CFD) method. The sound pressure data in the near field can be calculated directly by solving the Navier-Stokes equations, and the sound propagation can be predicted by the Kirchhoff method. A harmonic expansion approach is presented for rotor far-field noise prediction, which gives an analytical expression for the integral function in the Kirchhoff formula. As a result, the interpolation process is simplified and the efficiency and accuracy of the interpolation are improved. Then, the high-speed impulsive (HIS) noise of a helicopter rotor at different tip Mach numbers and on different observers is calculated and analyzed in hover and forward flight, which shows a highly directional characteristic of the rotor HIS noise with a maximum value in the rotor plane, and the HSI noise weakens rapidly with the increasing of the directivity angle. In order to investigate the effects of the rotor blade-tip shape on its aeroacoustic characteristics, four kinds of blade tips are designed and their noise characteristics have been simulated. At last, a new unconventional CLOR-II blade tip has been designed, and the noise characteristics of the presented CLOR-II model rotor have been simulated and measured compared to the reference rotors with a rectangular or swept-back platform blade tip. The results demonstrate that the unconventional CLOR-II blade tip can significantly reduce the HSI noise of a rotor.     

Springback prediction of thick-walled high-strength titanium tube bending

Significant springback occurs after tube rotary-draw-bending （RDB）, especially for a high-strength Ti-3A1-2.5V tube （HSTT） due to its high ratio of yield strength to Young＇s modulus. The combination scheme of explicit and implicit is preferred to predict the springback. This simulation strategy includes several numerical parameters, such as element type, number of elements through thickness （NEL）, element size, etc. However, the influences of these parameters on spring- back prediction accuracy are not fully understood. Thus, taking the geometrical specification 9.525 mm × 0.508 mm ofa HSTT as the objective, the effects of numerical parameters on prediction accuracy and computation efficiency of springback simulation of HSTT RDB are investigated. The simulated springback results are compared with experimental ones. The main results are： （1） solid and continuum-shell elements predict the experimental results well; （2） for C3DSR elements, NEL of at least 3 is required to obtain reliable results and a relative error of 29% can occur as NEL is varied in the range of 1-3; （3） specifying damping factor typically works well in Abaqus/Emplicit simulation of springback and the springback results are sensitive to the magnitude of damping factor. In addition, the explanations of the effect rules are given and a guideline is added.     

Numerical study on flow fields and aerodynamics of tilt rotor aircraft in conversion mode based on embedded grid and actuator model

A method combining rotor actuator disk model and embedded grid technique is presented in this paper, aimed at predicting the flow fields and aerodynamic characteristics of tilt rotor aircraft in conversion mode more efficiently and effectively. In this method, rotor's influence is considered in terms of the momentum it impacts to the fluid around it; transformation matrixes among different coordinate systems are deduced to extend actuator method's utility to conversion mode flow fields' calculation. Meanwhile, an embedded grid system is designed, in which grids generated around fuselage and actuator disk are regarded as background grid and minor grid respectively, and a new method is presented for ‘donor searching' and ‘hole cutting' during grid assembling. Based on the above methods, flow fields of tilt rotor aircraft in conversion mode are simulated, with threedimensional Navier–Stokes equations discretized by a second-order upwind finite-volume scheme and an implicit lower–upper symmetric Gauss–Seidel(LU-SGS) time-stepping scheme. Numerical results demonstrate that the proposed CFD method is very effective in simulating the conversion mode flow fields of tilt rotor aircraft.     

非结构网格紧致高精度气体动理学格式

简要介绍了非结构网格上基于内自由度高效紧致重构方法的几种高精度气体动理学格式(GKS),包括结合三阶GKS通量求解器与子单元有限体积法(SCFV)及通量重构方法(CPR)构造的两种单步时空三阶格式SCFV-GKS和CPR-GKS,以及结合二阶GKS通量和两步四阶方法构造的时空四阶CPR-GKS。进而在CPR-GKS中混合SCFV增强对流场间断的分辨能力,并拓展到三维六面体网格。通过几种典型数值算例,对比分析了这几种格式在可压缩流动问题中具有的高精度、高效率和良好的间断捕捉能力。GKS通量的多维时空演化特性与紧致重构方法的高效性为发展高精度格式提供了有力的支撑。     

跨音速压气机级三维紊流流场数值模拟

在对跨音速压气机级的动／静双叶排中的三维定常紊流流场数值的模拟中，利用三阶高分辨率ＮＮＤ格式和ＬＵ－ＳＧＳ隐式推进迭代法，既保证了流场中激波的模拟质量和粘性流动特征的正确预估，也实现了求解过程的高效率。对某单级跨音速压气机的计算及其与实验数据的对比证实了本文方法的有效性。将计算推广至多级轴流叶轮机械的情况将会是十分简便的     

叶栅绕流计算的高分辨率隐式解法

以一种性能优越的高分辨率格式为基础, 发展了其隐式解法, 提出并实现了与这种格式相适应的隐式边界处理过程, 从而显着提高了高分辨率格式的收敛速度。用不同来流条件下叶栅绕流数组计算结果与实验做了对比, 以说明解的准确程度和本文算法的效果。      

三维可压缩 Navier-Stokes 方程的间断Galerkin 有限元方法研究

拓展了二维间断 Galerkin(DG)有限元方法研究，将该数值方法用于三维可压缩欧拉方程和 Navier-Stokes方程的求解。基于六面体网格单元，采用插值方法将物面的四边形面网格单元构造为弯曲面网格单元，更好地表述了真实物面特征；物面边界相邻体网格单元相应构造为高阶体网格单元，其余体网格单元采用八节点六面体单元，以较小的计算代价使网格满足 DG 方法计算需求。通过对三维带 bump 管道内流、圆球绕流以及旋转流线体绕流进行的数值求解，验证了边界弯曲方法的可行性及 DG 方法的高精度特性。此外，由于采用了隐式计算方法，仅需较少的时间步就能迭代收敛。     

双时间步长加权ENO-强紧致高分辨率格式及在叶轮机械非定常流动中的应用

在非结构网格下,针对非定常三维N-S方程组发展了一种双时间步长高精度快速迭代格式。该格式在时间上具有二阶精度,在空间上将r=3的加权ENO格式与强紧致格式相结合去处理N-S方程中的对流项以及离散方程的右端项,并用四阶精度的紧致格式去计算N-S方程中的粘性项。典型的3个算例从不同侧面对格式进行了考核。计算表明:该算法具有高效率与高分辨率的特征,所得的计算结果与相关实验数据比较吻合,初步表明了该算法可以在非结构网格下模拟非定常流动的物理过程。      

Revisiting the space-time gradient method: A time-clocking perspective,high order difference time discretization and comparison with the harmonic balance method

This paper revisits the Space-Time Gradient (STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric Gauss-Seidel/Gauss-Seidel (LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-count-reduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.     

A novel high-order scheme for numerical simulation of wake flow over helicopter rotors in hover

Accurate prediction of tip vortices is crucial for predicting the hovering performance of a helicopter rotor. A new high-order scheme(we call it WENO-K) proposed by our research group is employed to minimize numerical dissipation and extended to numerical simulation of unsteady compressible viscous flows dominated by tip vortices over hovering rotors. WENO-K is referred to as an adaptively optimized WENO scheme with Gauss-Kriging reconstruction, and its advantage is to reduce dissipation in smoo...     

一种适用于三维混合网格的GMRES加速收敛新方法

为提高流场计算收敛效率,发展了一套适用于三维混合网格Naiver-Stokes方程求解的并行广义最小残差（GMRES）隐式时间推进方法。该方法由科学计算可移植扩展工具包（PETSc）中的Krylov子空间求解器实现,线性方程系统中的系数矩阵直接以显式给出以提高算法的稳定性。为进一步提高GMRES方法的收敛速度,对非结构网格的序号进行了重排序,使得系数矩阵的非零元素尽量向主对角线靠近。利用所发展的GMRES方法,完成了对ONERA-M6机翼、AIAA阻力预测会议通用研究模型（CRM）等算例的计算,计算结果与试验结果吻合良好。通过与其他隐式推进方法进行比较,对算法的收敛特性进行了研究。结果表明,所发展的GMRES方法计算更加稳定,残差下降速度相对LU-SGS（Lower-Upper Symmetric Gauss-Seidel）方法更快,尤其是气动力系数向着收敛解逼近的速度更加明显,提高了计算效率。