2阶精度时间后差隐式格式TVD充分条件

非定常流动问题计算中常用到含三个时间层的二阶精度时间后差隐式格式,并且希望构成TVD格式,然而理论上的问题多年一直没有解决。本文找到了解决办法,构造了这种类型的隐式TVD格式,证明了其为TVD的充分条件。理论结果为计算所验证,并表明通常未采取本文对时间差分处理方法的格式尚不具备TVD性质。     

隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI_LU,LGS,LU_SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

非定常隐式TVD格式的基本理论和验证

用于非定常问题的时间二阶精度的隐式TVD格式的理论多年来一直是个空白。本文按新的途径构造了一种时间二阶精度的隐式TVD格式,以发展了的手段证明了其为无条件(任意时间步长)TVD的,且限量因子Φ不受传统上界的限制,具体构造了这样的新的Φ。所得理论结果均为计算所验证,格式显示了良好的稳定性和分辨率。     

高分辨率激波捕捉格式及其CFD应用

在曲线坐标下将高分辨率AUSMPW激波捕捉格式扩展到三维问题。为了提高计算精度,采用三阶MUSCL格式,将AUSMPW格式与隐械时间推进法LU－SGS方法结合,应用于可压缩无粘和有粘流动的数散模拟,计算结果与文献中计算结果和实验数据相符,并将该格式与CD、TVD、FVS和FD格式进行了比较。     

非定常喷流有限差分方法数值试验

以欠膨胀自由喷流初期流动为例,采用Euler方程和Beam-Warming／TVD有限差分格式,对比分析了Beam-Warming AF方法,陷式亚迭代方法和简化Runge-Kutta五步格式的非定常数场描述能力,结果表明：（1）隐式近似因式分解方法基本上可以描述非定常流动现象;（2）隐式亚迭代一阶时间精度格式会导致流场结构的变化,其精度可能是不足的;（3）隐式亚迭代二阶时间精度与简化Runge-Kutta五步格式的计算结果一致,可以认为是计算非定常问题的适当方法。     

汽机末级静叶平衡态湿蒸汽流动数值模拟

本文对存在汽、液平衡态相变的两相流动建立了完全欧拉坐标系统下的数理模型。通过引入考虑真实流体性质的数值算法,并直接从IAPWS水及水蒸汽性质数据库中获取流体工质的性质,使数值计算的精度得到显著提高。采用包括LU-SGS-GE隐式格式和改良型高精度、高分辨率的MUSCL TVD格式的时间推进算法求解平衡态两相流动控制方程组以及低Reynolds数双方程湍流模型,对某汽轮机末级静叶进行了数值模拟,计算结果表明本文采用的模型及方法在某些条件下可以对叶栅主要性能参数进行准确的预测。     

三维不可压N-S方程的多重网格求解

应用全近似存储(Full Approximation Storage,FAS)多重网格法和人工压缩性方法求解了三维不可压Navier-Stokes方程.在解粗网格差分方程时,对Neumann边界条件采用增量形式进行更新,离散方程用对角化形式的近似隐式因子分解格式求解,其中空间无粘项分别用MUSCL格式和对称TVD格式进行离散.对90.弯曲的方截面管道流动和41椭球体层流绕流的数值模拟表明,多重网格的计算时间比单重网格节省一半以上,且无限制函数的MUSCL格式比TVD格式对流动结构有更好的分辨能力.     

平面叶栅气膜冷却流动的数值模拟

为了能够准确地对透平叶栅气膜冷却效率进行数值预测,本文采用了FNM形式的结构化网格,对一个平面叶栅中的气膜冷却流场进行了数值模拟。计算中采用了包括LU-SGS-GE隐式格式和改良型高精度、高分辨率的MUSCL TVD格式的时间推进算法求解三维RANS方程以及低Reynolds数q-ω双方程湍流模型。计算结果表明本文采用的模型及方法在低吹风比的条件下可以较准确地对气膜冷却效率进行数值预测。     

双曲型方程无振荡选取（NOS）有限差分格式

从几何观点解释了双曲型方程差分的TVD条件,导出了常用二阶差分的无振荡条件,发展了一种具有时空三阶精度的无振荡选取NOS差分格式,从单个双曲型方程的一此典型算例,显示了该格式蒿精度、无振荡和逻辑简单的特点,并能有效避免通常使用维数分裂法向二维推广时带来的空间耗散不对称性。     

一种高精度的修正Hermite-ENO格式

设计一种基于三单元具有六阶精度的修正Hermite-ENO格式（CHENO），求解一维双曲守恒律问题.CHENO格式利用有限体积法进行空间离散，在空间层上，使用ENO格式中的Newton差商法自适应选择模板.在重构半节点处的函数值及其一阶导数值时，利用Taylor展开给出修正Hermite插值使其提高到六阶精度，并设计了间断识别法与相应的处理方法以抑制间断处的虚假振荡；在时间层上采用三阶TVD Runge-Kutta法进行函数值及一阶导数值的推进.其主要优点是在达到高阶精度的同时具有紧致性.数值实验表明对一维双曲守恒律问题的求解达到了理论分析结果，是有效可行的.     

脉冲爆震发动机尾焰温度测量与数值模拟

本文采用新型研制的脉冲温度和水蒸气浓度测量仪测量了吸气式脉冲爆震发动机多循环工作过程中的尾焰温度,结果表明该测量仪不仅能够很好地捕捉每一个脉冲温度,与热成像仪测量的结果比较证明其测量温度在误差范围内.并采用了三阶TVD迎风格式和 Strang-splitting 算子分裂法,C3H8/air单步反应机理对混合物爆震燃烧进行了二维数值计算.与测量结果对比,说明采用本文的计算方法能够很好地捕捉到爆震波的点火和传播过程.     

高分辨率TVD格式的改进及应用

详细分析对比了已有的Harten TVD格式在三维任意曲线坐标系下的形式,尤其是其校正流量的构造。通过几何意义及物理意义的分析,指出了其缺陷,即校正流量中包含不必要的几何参数。由此,对校正流量的构造进行了几何改进,使其满足几何与物理要求。这一改进简单、意义明确、而且不增加计算量与存储量,在网格变化剧烈处可以明显改善计算的精度。同时,也考虑了预处理下的Harten TVD格式的几何修正,并通过计算证明了在预处理修正下几何改进仍然有效。     

On large time step TVD scheme for hyperbolic conservation laws and its efficiency evaluation

A large time step (LTS) TVD scheme originally proposed by Harten is modified and further developed in the present paper and applied to Euler equations in multidimensional problems. By firstly revealing the drawbacks of Harten’s original LTS TVD scheme, and reasoning the occurrence of the spurious oscillations, a modified formulation of its characteristic transformation is proposed and a high resolution, strongly robust LTS TVD scheme is formulated. The modified scheme is proven to be capable of taking larger number of time steps than the original one. Following the modified strategy, the LTS TVD schemes for Yee’s upwind TVD scheme and Yee–Roe–Davis’s symmetric TVD scheme are constructed. The family of the LTS schemes is then extended to multidimensional by time splitting procedure, and the associated boundary condition treatment suitable for the LTS scheme is also imposed. The numerical experiments on Sod’s shock tube problem, inviscid flows over NACA0012 airfoil and ONERA M6 wing are performed to validate the developed schemes. Computational efficiencies for the respective schemes under different CFL numbers are also evaluated and compared. The results reveal that the improvement is sizable as compared to the respective single time step schemes, especially for the CFL number ranging from 1.0 to 4.0.     

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework  and  with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK)  and  by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.     

An operator splitting method for the Degasperis–Procesi equation

An operator splitting method is proposed for the Degasperis–Procesi (DP) equation, by which the DP equation is decomposed into the Burgers equation and the Benjamin–Bona–Mahony (BBM) equation. Then, a second-order TVD scheme is applied for the Burgers equation, and a linearized implicit finite difference method is used for the BBM equation. Furthermore, the Strang splitting approach is used to construct the solution in one time step. The numerical solutions of the DP equation agree with exact solutions, e.g. the multipeakon solutions very well. The proposed method also captures the formation and propagation of shockpeakon solutions, and reveals wave breaking phenomena with good accuracy.     

叶片的周向弯曲与弦向弯曲及其数值分析

六十年代提出的静叶片的倾斜或弯曲方法是以透平轴的平行线为转动轴将叶片转动（构成倾斜叶片）或沿叶高弯曲（构成弯曲叶片）。本文提出将转动轴在Ｓ１平面旋转到弦向，然后倾斜或弯曲叶片，称这种叶片为弦向倾斜或弯曲叶片。应用欧拉方程和高精度ＴＶＤ格式对不同弯曲方法的叶片的数值分析表明弦向弯曲叶片控制径向二次流的能力大于周向弯曲叶片。     

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

涡轮叶栅双排孔气膜冷却数值模拟

采用具有三阶精度TVD性质的有限差分格式、自由型曲面技术以及分区网格算法,对某型具有冷气孔形状的涡轮叶栅进行了全三维N-S方程数值求解,描述了相邻两排冷气射流在叶栅吸力面形成的冷却气膜以及壁面附近冷却射流运动的特点,分析了不同吹风比和喷射角度情况下冷却绝热效率的分布规律。结果表明,在较大的吹风比和喷射角下,交错排列的两排冷却射流运动规律非常复杂,在两排孔之间的区域与冷气孔下游区域冷却气膜的形成规律具有明显的区别。