Dirac方程的数值解法及其在原子总能量计算中的应用

中心势近似下径向Dirac方程的求解是相对论性原子(离子)结构计算的基础.本文通过相对论性方程中径向波函数大分量与非相对论方程径向波函数的类比,提出了径向Dirac方程的一种数值解法.为了验证数值解法的精度和可靠性,首先将数值结果与类氢势作用下的解析解进行比较.然后,将这种算法扩展到基于解析势的相对论性原子结构计算中,并将计算出的总能量与实验结果和其他方法得到的结果进行对比.     

An unconditionally gradient stable numerical method for solving the Allen-Cahn equation

We consider an unconditionally gradient stable scheme for solving the Allen-Cahn equation representing a model for anti-phase domain coarsening in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. We also show the pointwise boundedness of the numerical solution for the Allen-Cahn equation. We describe various numerical experiments we performed to study properties of the Allen-Cahn equation.     

三维湍流Rayleigh-Bénard热对流的高效并行直接求解方法

高Ra数Rayleigh-Bénard热对流的湍流特性研究是当前国际上的一个热门研究课题, DNS模拟计算是研究该课题的重要手段之一. 当计算规模增大而网格数巨大时计算工作难以实现, 高Ra湍流热对流的数值模拟研究面临重大挑战. 本文创建了大规模高效并行计算的三维湍流热对流直接求解方法. 采用FFT变换解耦压力泊松方程, 将其变换成沿z方向上的块三对角方程组, 并利用块三对角方程的MPI与OpenMP联立的大规模高效并行近似解求解方案, 创建了可以高效并行计算的热对流直接求解方法. 通过对该方法并行效率的验证计算, 证明新的直接求解并行计算方法具有很好的并行效率和计算时效. 三维窄方腔热对流的计算结果表明, 本文方法计算的三维热对流特性是合理的. 本文创建的可大规模高效并行计算的三维湍流热对流直接求解方法, 也很可能是关于计算流体力学不可压NS方程大规模高效并行计算在特殊情况中计算技术上的一个突破.     

非线性波动方程的新数值迭代方法

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.     

A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation

Bird's direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird's collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.     

热对流条件下双颗粒沉降的直接数值模拟

在任意拉格朗日-欧拉(ALE)算法模拟等温惰性颗粒两相流的基础上,增加对能量方程的联立求解,对热对流条件下双颗粒的沉降进行了直接数值模拟.结果表明:双颗粒在等温流体、热流体和冷流体中运动形式是不同的,造成不同的主要因素是热对流引起了颗粒不同的动态尾迹,在热流体中颗粒尾部形成了涡的脱落,在冷流体中颗粒尾部形成了羽流.     

Mechanism of controlling turbulent channel flow with the effect of spanwise Lorentz force distribution

A direct numerical simulation(DNS) is performed to investigate the control effect and mechanism of turbulent channel flow with the distribution of spanwise Lorentz force. A sinusoidal distribution of constant spanwise Lorentz force is selected, of which the control effects, such as flow characters, mean Reynolds stress, and drag reductions, at different parameters of amplitude A and wave number k_x are discussed. The results indicate that the control effects vary with the parameter A and k_x. With the increase of A, the drag reduction rate D_r first increases and then decreases rapidly at low k_x,and slowly at high k_x. The low drag reduction(or even drag increase) is due to a weak suppression or even the enhancements of the random velocity fluctuation and mean Reynolds stress. The efficient drag reduction is due to the quasi-streamwise vortex structure induced by Lorentz force, which contributes to suppressing the random velocity fluctuation and mean Reynolds stress, and the negative vorticity improves the distribution of streamwise velocity. Therefore, the optimal control effect with a drag reduction of up to 58% can be obtained.     

Turbulence structures in high-speed air flow along a thin cylinder

Turbulent flow in the axial direction along a cylinder representing a monofilament yarn was reproduced for a relatively wide range of radius Reynolds numbers using direct numerical simulations. In the simulation of the thinnest cylinder, the friction coefficient agreed with a previously published formula. A pair of high- and low-speed streaks was detected around even the thinnest cylinder, whereby it was confirmed that turbulence was sustained according to the analysed turbulence statistics. Even when only a single pair of streaks around the cylinder was detected, the characteristics of the turbulence structures, such as the mean streak spacing based on the viscous length scale, were the same as those in flows over a flat plate. It was found that the friction coefficient changes in a way that maintains the structural characteristics of the flow, consistent with the view that universal characteristics of turbulence structures exist.     

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.     

Boundary value problems for the steady Boltzmann equation

We discuss steady boundary value problems for the Boltzmann equation with inflow and diffusive boundary conditions in one, two, and three dimensions, with suitable truncations of the collision kernel. General existence and uniqueness results are obtained if the domain is sufficiently small. In one dimension, the existence of solutions on general intervals is obtained by abstract fixed-point theory.     

Experimental and numerical study on instability structure of the supersonic mixing layer (Mc = 0.5)

The parameters of instability wave of supersonic mixing layer(Mc=0.5,M1=3.5/M2=1.4) are measured by flow visualization and calculated by means of two-dimensional direct numerical simulaitons of the compressible Navier-Stokes equations.In both cases of the mixing layer with harmonic disturbance or not,the comparative results indicate that the wavelength of the two-dimensional wave is equal to the vortex spacing in the streamwise direction because the difference between them is less than 1%.However,the measur...     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

Direct perturbation method for perturbed complex Burgers equation

So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear Schr?dinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.     

Direct numerical simulation of transitional flow in a finite length curved pipe

Transition to turbulent flow in a curved pipe has been well studied through experiments and numerical simulations. Numerical simulations often use a helical pipe with an infinite length such that the inlet and outlet boundary conditions can be modelled as periodic which greatly reduces computational time. In this study, we examined a finite length curved pipe with Poiseuille flow imposed at the inlet and a stress-free boundary condition at the outlet. Direct numerical simulation of the Navier-Stokes equations for rigid walls and a Newtonian fluid was performed using nek5000. Straight extensions were added to the inlet and outlet such to diminish the impact of boundary conditions on the flow field in the region with curvature. The examined model has a pipe radius of curvature that is three times the pipe radius. The model has ～355 million nodes and required an order of magnitude greater computational time when compared with an infinite length curved pipe. Results show that the critical Reynolds number, the lowest value with instabilities present in the flow, is much greater than that of a straight pipe and occurs near Re=5000–5200. This is larger than the critical Reynolds number typically reported for an infinite length curved pipe (Re=4200–4300).     

Coherent structure extraction in turbulent channel flow using boundary adapted wavelets

We introduce boundary adapted wavelets, which are orthogonal and have the same scale in the three spatial directions. The construction thus yields a multiresolution analysis. We analyse direct numerical simulation data of turbulent channel flow computed at a friction Reynolds number of 395, and investigate the role of coherent vorticity. Thresholding of the vorticity wavelet coefficients allows us to split the flow into two parts, coherent and incoherent flows. The coherent vorticity is reconstructed from its few intense wavelet coefficients and the coherent velocity is reconstructed using Biot–Savart's law. The statistics of the coherent flow, i.e. energy and enstrophy spectra, are close to the statistics of the total flow, and moreover, the nonlinear energy budgets of the total flow are very well preserved. The remaining incoherent part, represented by the large majority of the weak wavelet coefficients, corresponds to a structureless, i.e. noise-like, background flow whose energy is equidistributed.     

用松弛法解薛定谔方程

求解定态薛定谔方程常常会涉及到常微分方程的本征值问题.目前解常微分方程本征值用的比较多的方法是以龙格-库塔方法为基础的打靶方法.打靶方法常用,但是计算时间长.当边界条件比较复杂或比较敏感的时候,用松弛法会有更好的效果.本文用松弛法解薛定谔方程,并和理论解进行比较.发现松弛法得到的数值解和理论解符合度很高,而且使用松弛法...     

一种激光驱动高压状态方程绝对测量方法的探索

数值模拟了激光直接驱动铝飞片空腔靶的物理过程,模拟结果表明,如果靶结构参数与激光条件匹配,飞片与靶可以近似实现对称碰撞,而且冲击波在靶中存在稳定传播区,采用结构参数合理的飞片双面台阶靶,可能实现状态方程的绝对测量.同时进行了实验探索,实验结果与数值模拟结果基本吻合. 关键词： 绝对测量 飞片双面台阶靶 数值模拟 状态方程     

Direct numerical simulation of the flow around a wall-mounted square cylinder under various inflow conditions

The flow around a wall-mounted square cylinder of side d is investigated by means of direct numerical simulation (DNS). The effect of inflow conditions is assessed by considering two different cases with matching momentum-thickness Reynolds numbers Re_θ ? 1000 at the obstacle: the first case is a fullyturbulent zero pressure gradient boundary layer, and the second one is a laminar boundary layer with prescribed Blasius inflow profile further upstream. An auxiliary simulation carried out with the pseudo-spectral Fourier–Chebyshev code SIMSON is used to obtain the turbulent time-dependent inflow conditions which are then fed into the main simulation where the actual flow around the cylinder is computed. This main simulation is performed, for both laminar and turbulent-inflows, with the spectral-element method code Nek5000. In both cases the wake is completely turbulent, and we find the same Strouhal number St ? 0.1, although the two wakes exhibit structural differences for x > 3d downstream of the cylinder. Transition to turbulence is observed in the laminar-inflow case, induced by the recirculation bubble produced upstream of the obstacle, and in the turbulent-inflow simulation the streamwise fluctuations modulate the horseshoe vortex. The wake obtained in our laminar-inflow case is in closer agreement with reference particle image velocimetry measurements of the same geometry, revealing that the experimental boundary layer was not fully turbulent in that dataset, and highlighting the usefulness of DNS to assess the quality of experimental inflow conditions.     

模式匹配与积分方程在HPM模拟中的应用

针对具有电大尺寸、厚介质罩且馈源采用特定模式激励的高功率微波(HMP)传输及辐射结构研究了一种新型的电磁建模技术。将模式匹配方法与积分方程方法进行混合,构建了电磁模型的方程组,采用多层快速多极子技术、预条件器等进行求解加速,最终形成了可对电大尺寸HPM传输、辐射系统进行高效全波电磁仿真的技术。以电大变张角喇叭馈源、功率合成天线、波束波导及波束波导天线作为实例,构建了几何及电磁模型,并进行了包括远区方向图、近区功率密度分布在内的数值模拟,对该技术的正确性和通用性进行了验证。结果表明,该技术边界拟合准确,内存消耗低,且对馈电模式能予以准确反映,适用于HPM传输发射的高效高精度模拟。     

Characteristics and generation of elastic turbulence in a three-dimensional parallel plate channel using direct numerical simulation

Direct numerical simulations(DNSs) of purely elastic turbulence in rectilinear shear flows in a three-dimensional(3D) parallel plate channel were carried out,by which numerical databases were established.Based on the numerical databases,the present paper analyzed the structural and statistical characteristics of the elastic turbulence including flow patterns,the wall effect on the turbulent kinetic energy spectrum,and the local relationship between the flow motion and the microstructures' behavior.Moreover,to address the underlying physical mechanism of elastic turbulence,its generation was presented in terms of the global energy budget.The results showed that the flow structures in elastic turbulence were 3D with spatial scales on the order of the geometrical characteristic length,and vortex tubes were more likely to be embedded in the regions where the polymers were strongly stretched.In addition,the patterns of microstructures' elongation behave like a filament.From the results of the turbulent kinetic energy budget,it was found that the continuous energy releasing from the polymers into the main flow was the main source of the generation and maintenance of the elastic turbulent status.