用高分辨差分格式解SNS方程

本文从Sweby的显式TVD格式^[2,3]出发,构造发一个推进方向一阶精度、侧向二阶精度的隐式TVD差分格式,并用于解可压缩流定常简化Naviev-Stokes(SNS)方程。对不出现流向分离的平板边界层和激波—边界层相互干扰两个问题的数值计算表明,方法可以有效捕捉流场中的间断。     

TVD格式在燃料蒸气云爆炸场数值模拟中的应用

本文采用了有限差分方法——TVD格式(Total Variation Diminishing,A.Harten,1983)对燃料蒸气云爆炸产生的二维轴对称爆炸场进行了数值模拟。爆炸波超压和运动轨迹的数值计算值与实验结果吻合得相当好。     

完全守恒型差分格式对大型变截面激波管中流场的数值模拟

本文研究了А.А.Самарский提出的拉格朗日坐标下气体力学方程组的完全守恒型差分格式^[2],并给出了一种改进形式。对激波管问题所做的数值计算表明改进的格式具有激波过渡区窄,过跳与低亏小的优点,而且所需的计算时间远低于文[2]中的格式。     

二维非定常Sine-Gordon方程辛算法及其孤子数值模拟

在矩形域[-a,a]×[-a,a]内对微分算子L=(ə²)/(əx²)+(ə²)/(əy²)用5点差分格式将二维非定常Sine Gordon方程离散化为一个2×799²阶非线性Hamilton系统.对该系统使用Euler中心格式,得到一个非线性方程组.对此方程组建立迭代解法并给出了这个迭代方法的收敛条件和收敛速度.Sine Gordon方程单孤子和双孤子的数值模拟试验显示该辛算法是有效的.     

MUSCL格式TV性质的非线性分析

分析求解非线性双曲型守恒律的MUSCL类格式的TV性质。首先从该类格式的一般形式出发,提出和证明了该类格式实现TVD的需求。所提TVD需求直接表示为对变量变差符号和量值限制,体现了双曲型方程解的依赖域原理,为分析MUSCL格式的TV性质提供了理论工具。同时提出了基于TVD需求再构数值解分布,以降低数值耗散从而提高接触面及膨胀波头/波尾分辨率的基本思路。     

凹型钝头体超音速粘性流场的数值模拟

本文首先从单个方程出发,根据TVD思想,比较自然地给出了二阶带通量限制器的CSCM^[1]差分方法,然后推广到方程组情形。为了验证方法和程序,将球头绕流计算结果同文献[2]作了比较,然后运用此方法模拟了轴对称凹型钝体的粘性超音速分离绕流流场,对一阶与二阶、粘性与无粘性数值结果作了比较。计算网格采用了简单、快速易行的拟正交代数生成方法。     

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。     

求解Euter方程组的一种简单的守恒误差

本文提出了一种解一维不定常Euler方程组的较简单的一阶,二阶守恒差分格式,一阶格式简称为FUDE(First-Order UPwind Difference Scheme for Euler Equations),二阶格式简记为SUDE。本文的格式避免了大的数值振荡及膨胀激波,同时计算量较MacCormack^[1]格式仅略有增加,文中若干数值试验表明本文格式的分辨率是令人满意的。     

生物力学中片流的边界元分析

用边界元方法^[1,2]研究了生物力学中肺毛细血管血液的流动,并对肺毛细血管的SFH模型,SFC模型进行了数值模拟,给出血液流动的速度场及血管壁上的面力分布和压力分布.对肺毛细管内皮细胞受的切应力,数值计算表明,正常情况下,在0~4.64×10^-5N/cm²之间,这与Dewey实验结果^[8](0~5×10^-5N/cm²)相一致,肺胞的形态将按使呼吸膜受切应力值最小为原则构形.同时,本文方法还可为数值分析心血管内的血液流动提供一种有效的数值方法,这对生物力学和医学工程是有价值的.     

求解二维双曲型方程的一种基本守恒差分格式

构造了一种求解二维双曲型方程的基本守恒型差分格式,并证明了该格式的数值解是全变差有界的,在光滑区域具有二阶精度,按L¹范数及L^∞范数稳定,且其几乎处处有界收敛的极限解是微分方程的物理解。     

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.     

Numerical study on combustion of H2—O2 in a supersonic reactive expansion in a nozzle

A study on expansion flow inside a nozzle considering full mechanism chemistry of hydrogen and oxygen was carried out. In this study, a full implicit scheme for turbulent reactive flow was obtained by combining the second order TVD scheme of Yee and Harten (1987, Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates. American Institute of Aeronautics and Astronautics Journal, 25(2), 266–274) with the efficient implicit lower-upper scheme of Shuen and Yoon (1989, Numerical study of chemically reacting flows using a lower-upper symmetric successive overrelaxation scheme. American Institute of Aeronautics and Astronautics Journal, 27(12), 1752–1756). The species equations, Navier–Stokes equations and turbulence model were implemented in the numerical scheme and solved in conjunction with full detailed finite rate chemistry. The numerical scheme is verified by comparison with experimental results of a converging–diverging nozzle. Effects of inlet pressure, temperature and fuel-oxidant mass ratio on nozzle flow field were studied. Variation of chemical species under different conditions was investigated by considering a chemical mechanism. Results show that increasing inlet pressure increases the rate of reactions due to increasing the concentration of reactants. For lower inlet pressure the radical H increases slightly in the diverging part of the nozzle, while for higher pressures it decreases along the nozzle. Inlet fuel–oxidant mass ratio affects the variation of all species with a greater effect for a near stoichiometric ratio. It was also shown that a higher inlet temperature provides a more enhanced reaction zone in the diverging part of the nozzle.     

用TVD格式求解一维滑坡问题

本文用二阶精度显式TVD格式计算了一维滑坡问题,得到了和实际值基本一致的结果。     

具有TVD性质的三阶精度GODUNOV格式在粘性流场计算中的应用

本文发展了一种具有TVD性质的三阶精度的Godunov格式。隐式部分采用迎风对角线形隐式近似因式分解法。并引入了粘性通量的简化算法。显式部分采用三阶精度TVD格式。为进一步增强格式的稳定性及对间断的捕捉能力,在单元边界上构造Riemann问题。应用上述方法对某型涡轮高压级及压气机进行了数值模拟。     

Simulation of Shock Wave Diffraction over 90° Sharp Corner in Gases of Arbitrary Statistics

The unsteady shock wave diffraction over a 90° sharp corner in gases of arbitrary particle statistics is simulated using an accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space. The numerical method is based on the usage of discrete ordinate method for discretizing the velocity space of the distribution function; whereas a second order accurate TVD scheme (Harten in J. Comput. Phys. 49(3):357–393, 1983) with Van Leer’s limiter (J. Comput. Phys. 32(1):101–136, 1979) is used for evolving the solution in physical space and time. The specular reflection surface boundary condition is assumed. The complete diffraction patterns are recorded using various flow property contours. Different range of relaxation times approximately corresponding to continuum, slip and transitional regimes are considered and the equilibrium Euler limit solution is also computed for comparison. The effects of gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics are examined and depicted.     

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.     

三维高超声速无粘定常绕流的数值模拟

本文采用一种简单有效的通量分裂结合一种二阶TVD格式的数值通量的方法,提出一种隐式的迎风有限体积格式,并利用这种格式,从气体动力学非定常Euler方程组出发,数值模拟了三维不对称物体的高超声速无粘定常绕流。数值结果表明此格式具有分辨率较高和收敛速度较快的优点。     

Multi-dimensional limiting for high-order schemes including turbulence and combustion

In the present paper a fourth/fifth order upwind biased limiting strategy is presented for the simulation of turbulent flows and combustion. Because high order numerical schemes usually suffer from stability problems and TVD approaches often prevent convergence to machine accuracy the multi-dimensional limiting process (MLP) [1] is employed. MLP uses information from diagonal volumes of a discretization stencil. It interacts with the TVD limiter in such a way, that local extrema at the corner points of the volume are avoided. This stabilizes the numerical scheme and enables convergence in cases, where standard limiters fail to converge. Up to now MLP has been used for inviscid and laminar flows only. In the present paper this technique is applied to fully turbulent sub- and supersonic flows simulated with a low Reynolds-number turbulence closure. Additionally, combustion based on finite-rate chemistry is investigated. An improved MLP version (MLP^ld, low diffusion) as well as an analysis of its capabilities and limitations are given. It is demonstrated, that the scheme offers high accuracy and robustness while keeping the computational cost low. Both steady and unsteady test cases are investigated.