基于AUSM分裂的二维通量分裂格式

基于对流迎风分裂思想构造的AUSM类格式具有简单、高效、分辨率高等优点,在计算流体力学中得到了广泛的应用.传统的AUSM类格式在计算界面数值通量时只考虑网格界面法向的波系,忽略了网格界面横向波系的影响.使用Liou-Steffen通量分裂方法将二维Euler方程的通量分裂成对流通量和压力通量,采用AUSM格式来分别计算对流数值通量和压力数值通量.通过求解考虑了横向波系影响的角点数值通量来构造一种真正二维的AUSM通量分裂格式.在计算一维算例时,该格式保留了精确捕捉激波和接触间断的优点.在计算二维算例时,该格式不仅具有更高的分辨率而且表现出更好的鲁棒性,可以消除强激波波后的不稳定现象.此外,在多维问题的数值模拟中,该格式大大地提高了稳定性CFL数,具有更高的计算效率.因此,它是一种精确、高效并且强鲁棒性的数值方法.     

非定常不可压Navier-Stokes方程两重网格方法的稳定性和L2误差估计

讨论了二维非定常不可压Navier-Stokes方程的两重网格方法．此方法包括在粗网格上求解一个非线性问题,在细网格上求解一个Stokes问题．采用一种新的全离散(时间离散用Crank-Nicolson格式,空间离散用混合有限元方法)格式数值求解N-S方程．证明了该全离散格式的稳定性．给出了L2误差估计．对比标准有限元方法,在保持同样精度的前提下,TGM能节省大量的计算量．     

PSE在边界层中预测转捩位置的应用

层流到湍流的转捩是自然界和各项工程实践中广泛存在的现象,层流和湍流的性质大不相同．因此,预测转捩位置是流体力学中的重要理论和实际问题．针对不可压缩边界层,入口加入展向等幅值型和展向波包型两类扰动,展向等幅值型扰动是由一个二维波(2-D)和两个三维波(3-D)组成,使用抛物化稳定性方程(PSE)的方法来研究扰动的演化和预测转捩位置,并且与数值模拟的结果相比较．结果表明,PSE可以研究扰动的演化和预测转捩位置,同时其计算比数值模拟快得多．     

二维平板可压缩边界层的二次稳定性分析

本文在二维可压缩边界层中应用Floquet分析,建立了控制次谐波稳定性的方程组,研究在二维可压缩边界层转捩过程中二维有限振幅的T-S波对三维线性小扰动的作用,并计算了来流马赫数对次谐波的产生和发展情况的影响,从中可以看出二维和三维扰动波相互作用对二维可压缩流动边界层的发展过程所产生的影响.     

求解二维浅水波方程的移动网格旋转通量法

为提高求解二维浅水波方程数值算法的分辨率,拟构造求解该方程的新算法:基于移动网格法,选用熵稳定数值通量函数,利用旋转不变性得到混合数值通量.该算法中,浅水波方程的数值求解和依据解的特性进行自适应疏密分布的网格计算过程交错进行.利用变分原理进行网格重构,新网格上的物理量采用二阶精度的守恒型插值公式计算,最终采用三阶强稳定Runge-Kutta法与满足热力学第二定律的熵稳定格式实现浅水波方程的数值求解.数值结果表明,新算法具有良好的间断捕捉能力,分辨率高.     

非结构网格上求解二维H-J方程的一种WENO格式

基于WENO(Weighted Essentially Non-Oscillatory)的思想,提出了一种在非结构网格上求解二维Hamilton-Jacobi(简称H-J)方程的数值方法.该方法利用Abgrall提出的数值通量,在每个三角形单元上构造三次加权插值多项式,得到了一个求解H-J方程的高阶精度格式.数值实验结果表明,该方法计算速度较快,具有较高的精度,而且对导数间断有较高的分辨率.     

带间断扩散系数热传导方程的新型自适应数值解法

本文研究带间断扩散系数热传导方程在大变形网格上的高精度数值模拟方法.该方法在算每条边上的能流时采用了本文提出的"孪生逼近"方法,提高了扩散系数间断处能流的计算精度,给出了"孪生逼近"的误差分析.应用该方法于二维大变形网格上热传导问题计算,构造r网格边上能流的一种自适应高精度计算方法,其中自适应指的是自适应选取模板和自适应选取权重大小.数值试验表明该方法能适应网格大变形和扩散系数间断的困难情况.     

黏性不可压流体的自适应网格技术和基本特性方程分离算法的联合分析

组合基本特性方程分离算法和自适应网格技术,分析二维黏性不可压流体.该方法使用3节点三角单元,对速度分量和压力等变量分析,使用等阶次的插值函数.组合解法的主要优点在于,在自适应网格技术中,对解梯度变化大的区域,通过耦合误差估计生成小的单元,利于提高解的精度,在其它区域生成大单元,可以节省时间.最后,通过对一个黏性流体圆柱体绕流问题的瞬态和稳态特性分析,给出了组合解法性能的评价.     

可视球面多分辨率网格建模与求解

针对任意视点下的球面多分辨率网格建模与求解困难的情况,提出了屏幕细分自适应算法.该算法将计算限制在与视点相关的可视球面区域,求解得到视点相关的可视球面多分辨率网格,并在PC机上实现了可视球面多分辨率网格的三维表现.     

二维系数不连续Helmholtz方程的高阶紧致差分格式

针对二维系数不连续Helmholtz方程,提出和研究了高阶紧致差分格式,在波数跳跃位置引入局部网格加密技巧进行网格加密.数值实验验证,该高阶紧致差分格式用于求解二维系数不连续Helmholtz方程可以达到四阶精度,局部网格加密技巧能够有效地提高数值解的精度.     

平面不可压缩的NAVIER-STOKES方程新五模类LORENZ方程组的混沌行为

本文研究了平面正方形区域上不可压缩的Navier-Stokes方程五模类Lorenz方程组的混沌行为问题.利用傅立叶展开方法对Navier-Stokes方程进行模式截断,获得了新五模类Lorenz方程组,给出了该方程组定常解及其稳定性的讨论,证明了该方程组吸引子的存在性,并对其全局稳定性进行了分析和讨论.     

Nonlinear dynamics of laminar-turbulent transition in three dimensional Rayleigh–Benard convection

This paper proposes a new approach to analysis of incompressible 3D fluid motion in Rayleigh–Benard convection in transition from laminar to turbulent regimes. Number of test series were conducted. The analysis indicated that in different test series laminar-turbulent transition follows either the subharmonic bifurcation cascade of two-dimensional tori or the subharmonic bifurcation cascade of limit cycles. Cycles up to the third period were found in the system that indicated the end of the Sharkovskii sequence. All bifurcation cascades agree with the Feigenbaum–Sharkovskii–Magnitskii (FSM) scenario.     

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ?, we prove the existence of the uniform global attractor A^?. Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in Chepyzhov and Vishik (2007) [22].     

有界域上具有部分耗散和磁扩散的二维磁流体方程的全局适定性

探究了具有部分耗散和磁扩散的二维不可压缩磁流体(MHD)方程的初边值问题.在有界区域上,当系统的各个方向上的耗散系数和磁扩散系数都非负时,我们得到了该模型的强解是整体存在且唯一的.此外,对周期域而言,其解仍是全局适定的.     

Simulation of Separating Flows over a Backward-Facing Step

Simulation results are reported for plane two-dimensional viscous incompressible flow in a channel with an abrupt expansion. The mathematical model is provided by the quasi-hydrodynamic equations in the incompressible fluid approximation. The computations are carried out in a range of Reynolds numbers including both laminar and turbulent flow. As the Reynolds number increases, the solution bifurcates and the steady laminar flow changes to time-dependent flow. The computation results are consistent with known experimental data. Turbulence models were not used for large Reynolds number computations.     

Navier–Stokes equations with periodic boundary conditions and pressure loss

We present in this note the existence and uniqueness results for the Stokes and Navier–Stokes equations which model the laminar flow of an incompressible fluid inside a two-dimensional channel of periodic sections. The data of the pressure loss coefficient enables us to establish a relation on the pressure and to thus formulate an equivalent problem.     

On solutions to the Goldshtik problem

The Goldshtik model for separated flows in an incompressible fluid is considered. A solution to this two-dimensional problem of mathematical physics for a finite domain is found with a finite element method. Estimates of the differential operator are obtained. A result on the number of solutions to the Goldshtik problem is obtained using a variational method.     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.     

Coherent structure visiometrics: From the soliton to HEC

In this paper I give a brief personal review of the history of the early soliton days and how they led to thevisiometrics and reduced modeling paradigm that has been a part of my approach to nonlinear science in the last three decades. I illustrate it with HEC (Hybrid Eiliptic-Contour): a fast, minimal, asymptotically motivated model for unforced, 2-dimensional incompressible weakly dissipative turbulence (U2DIT).The work on reduced modeling of two-dimensional turbulence is on-going with David G. Dritschel and Hongbing Yao (incompressible); and Jaideep Ray, Thomas Scheidegger and Ravi Samtaney (compressible).