谱方法用于非定常流动计算的隐式求解

本文对谱方法用于周期性非定常流动的隐式求解方法进行了探讨,分析了影响计算稳定性和收敛速度的因素.提出了结合多重网格的隐式求解方法并对算法进行了验证,初步计算表明本文算法具有良好的稳定性和收敛速度.对于周期性非定常流动,结合本文提出的隐式求解的时域谱方法可以达到很高的精度且具有良好的计算效率.     

多群辐射扩散方程组的分裂迭代算法收敛分析

分析了多群辐射扩散方程组的分裂迭代算法的收敛速度,证明其收敛特性,给出迭代矩阵谱半径的解析公式.对谱半径进行数值计算与分析,揭示算法的收敛速度与辐射系数之间的依赖关系,数值算例验证了理论结果,给出了该算法的适用条件.     

求解流动与传热问题的一种高效稳定的分离式算法-IDEAL

本文提出了一种求解流动与传热问题的高效稳定的分离式算法-IDEAL(Inner Doubly-iterative EfficientAlgorithm for Linked-equations).在IDEAL算法中每个迭代层次上对压力方程进行两次内迭代计算,第一次内迭代过程用于克服SIMPLE算法的第一个假设,第二次内迭代过程用于克服SIMPLE算法的第二个假设.这样在每个迭代层次上充分满足了速度和压力之间的耦合,从而大大提高了计算的收敛速度和计算过程的稳定性.本文通过2个三维不可压缩流动和传热的算例对IDEAL算法与其它三个被广泛使用的算法(SIMPLER、SIMPLEC和PISO)进行了比较.通过分析比较得出IDEAL算法在收敛性和健壮性上均优于SIMPLER、SIMPLEC和PISO算法.在这2个算例中IDEAL算法几乎可以在任意的松弛因子下获得收敛的解,并且IDEAL算法所需最短计算时间较SIMPLER算法减少12.9%～52.6%;较SIMPLEC算法减少48.3%～79.1%;较PISO算法减少10.7%～46.5%.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

不可压缩流的并行两水平稳定有限元算法

提出一种数值求解定常不可压缩Stokes方程的并行两水平Grad-div稳定有限元算法。首先在粗网格中求解Grad-div稳定化的全局解, 再在相互重叠的细网格子区域上并行纠正。通过对稳定化参数、粗细网格尺寸恰当的选取, 该方法可得到最优收敛率, 数值结果验证了算法的高效性。     

非定常动态演化伴随优化设计方法

研究了基于定常流动解和伴随方程定常解基础上的传统的伴随方法.在此基础上对定态飞行气动外形的优化设计提出了基于非定常流动控制方程瞬态解和非定常伴随方程瞬态解的新的优化方法,称之为动态演化伴随方法.这种新的优化方法保留了传统伴随方法适用于具有大数量设计变量的气动优化问题,而且比传统的伴随方法可节省大量的计算时间.大量算例计算结果表明,新方法与传统方法具有相同的精度.     

封闭圆内开缝圆自然对流的非线性特性研究

本文通过数值计算对封闭圆内开缝圆自然对流的非线性特性进行了研究。数值计算以整个流场为计算区域,采用非稳态数学模型和具有QUICK差分格式的SIMPLE算法。计算结果表明,在不同参数下流动和换热存在稳态定常解、周期性振荡解、拟周期性振荡解和非周期性振荡解。稳态定常解的相图是一个点;周期性振荡解的相图是一个极限环,对应功率谱含一个基频及其谐波;拟周期性振荡解的相图为环面,对应功率谱含两个不相关的频率及它们的线性组合频率;非周期性振荡解的功率谱为无规则的宽带连续谱。     

一种隐式预处理方法及其在定常和非定常流动数值模拟中的应用

将Choi-Merkle矩阵预处理方法与LU-SGS隐式方法、双时间法以及多重网格方法结合,发展适用于绕飞行器定常和非定常粘性流动的高效隐式预处理计算方法和程序.介绍一种针对定常和非定常流动的LU-SGS隐式预处理方法的统一表述方法.在不改变流动解的前提下,对Navier-Stokes方程的伪时间导数项实施Choi-Merkle矩阵预处理,从而改善可压缩控制方程在低速情况下的系统刚性,使基于LU-SGS时间推进格式的数值模拟方法同时适用于从极低马赫数到可压缩范围内的数值模拟.对Jameson中心格式的人工粘性进行相应的修改,以提高低速流动的计算精度.翼型、机翼以及翼身组合体绕流的数值模拟研究表明,隐式预处理方法获得了很高的计算效率,可使马赫数0.1左右的低速流动计算时间减少50%以上;通过对现有可压缩计算程序进行小量改动,便可使其均匀覆盖整个低速流动范围,提高CFD程序在飞行器绕流数值模拟中的实用性.     

时间谱方法的扩稳与加速技术研究

时间谱方法作为目前应用最广泛的频域计算方法,很容易在现有的定常求解器上进行扩展而实现。但是当非定常流组分的频率足够大时,时间谱源项的刚性不仅会影响求解的收敛速度,甚至会引起求解的不稳定性。块雅可比方法通过采用迭代的方法对时间谱源项进行隐式处理,不仅能够简化离散方程的隐式求解,也能很好地起到稳定计算结果的作用。同时通过对通量雅可比矩阵的近似LU分解,减少LU分解的复杂度,也可以加快计算收敛。本文将采用不同的算例,研究块雅可比方法及近似LU分解对周期非定常流时间谱计算的稳定及加速特性的影响。     

利用加权预测的图像迭代盲解卷积

为解决大气湍流造成的图像退化问题,本文鉴于现有的盲解卷积算法收敛性不稳定,计算量大等特点,提出了一种基于加权预测的迭代盲解卷积算法。对目前性能优秀的用迭代实现盲解卷积的L-R算法进行优化,在每次迭代结束后通过加权方法求出预测值,根据预测值计算方向加速算子,从而大大提高算法的收敛速度。实验表明：该算法不仅可对模糊退化图像进行很好的复原,同时与L-R算法相比收敛速度提高约43.8倍,其迭代速度快的特点决定了算法具有较高的工程实用价值。     

Spectral/hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L₂ least-squares functional and L₂ error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.     

Spectrum evolutions of spontaneous and pump-depleted stimulated Brillouin scatterings in liquid media

A theoretical model for calculating spontaneous and stimulated Brillouin scattering(SBS) spectra is described. An empirical formula for the Stokes output spectral linewidth, a function of spontaneous Brillouin linewidth and the exponential gain coefficient, is obtained by the calculated data fitting. The formula holds true for two cases involving pump undepletion and depletion. The lineshape change from spontaneous to highly pump-depleted SBS spectra is also investigated. The result shows that for the pump power below the SBS threshold, the Stokes output spectral lineshape evolves from Lorentzian to approximately Gaussian as the pump power increases. For the pump power near or beyond the threshold, the SBS spectrum is in the form of a steady Gaussian profile, and the spectral linewidth comes to a certain value about 7 times narrower than the spontaneous one. The theoretical results are experimentally demonstrated by using several common liquid media.     

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier–Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows.     

Topology optimization of unsteady incompressible Navier–Stokes flows

This paper discusses the topology optimization of unsteady incompressible Navier–Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier–Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier–Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier–Stokes flows.     

A high order Discontinuous Galerkin – Fourier incompressible 3D Navier–Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order ? 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.     

Spectral element formulations on non-conforming grids: A comparative study of pointwise matching and integral projection methods

Pointwise matching (PM) and integral projection (IP) methods are two widely used techniques to extend the classical weak formulations to include non-conforming grids. We present spectral element formulations on polynomial (p-type) and geometric (h-type) non-conforming grids using both the PM (also known as the Constrained Approximation) and IP (also known as the Mortar Element) methods. We systematically compare the convergence characteristics of PM and IP methods for diffusion, convection, and convection–diffusion equations. Consistency errors due to the non-conforming formulations of the diffusion equation result in convergence problems for the PM method using the maximum rule. Both non-conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values, causing convergence problems due to the consistency errors. However, small “physical” diffusion in the convection–diffusion equation eliminates these problems, resulting in spectral convergence for both methods. Encouraged by this, we present spectral element formulations for incompressible Navier–Stokes equations using PM and IP methods on p-type and h-type non-conforming grids, and demonstrate spectral convergence for unsteady and steady test cases. Results for two-dimensional lid-driven cavity flow at Re = 1000 are also presented.     

Boundary integral solutions of coupled Stokes and Darcy flows

An accurate computational method based on the boundary integral formulation is presented for solving boundary value problems for Stokes and Darcy flows. The method also applies to problems where the equations are coupled across an interface through appropriate boundary conditions. The adopted technique consists of first reformulating the singular integrals for the fluid quantities as single and double layer potentials. Then the layer potentials are regularized and discretized using standard quadratures. As a final step, the leading term in the regularization error is eliminated in order to gain one more order of accuracy. The numerical examples demonstrate the increase of the convergence rate from first to second order and show a decrease in magnitude of the error. The coupled problems require the computation of the gradient of the Stokes velocity at the common interface. This boundary condition is also written as a combination of single and double layer potentials so that the same approach can be used to compute it accurately. Extensive numerical examples show the increased accuracy gained by the correction terms.     

Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.     

Improved Local Projection for the Generalized Stokes Problem

We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the methods is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although the stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This makes it much simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.     

A 3D pseudospectral method for cylindrical coordinates. Application to the simulations of rotating cavity flows

The present work proposes a collocation spectral method for solving the three-dimensional Navier–Stokes equations using cylindrical coordinates. The whole diameter $-R?r?R$ is discretized with an even number of radial Gauss–Lobatto collocation points and an angular shift is introduced in the Fourier transform that avoid pole and parity conditions usually required. The method keeps the spectral convergence that reduces the number of grid points with respect to lower-order numerical methods. The grid-points distribution densifies the mesh only near the boundaries that makes the algorithm well-suited to simulate rotating cavity flows where thin layers develop along the walls. Comparisons with reliable experimental and numerical results of the literature show good quantitative agreements for flows driven by rotating discs in tall cylinders and thin inter-disc cavities. Associated to a spectral vanishing viscosity [E. Séverac, E. Serre, A spectral vanishing viscosity for the LES of turbulent flows within rotating cavities, J. Comp. Phys. 226 (2007) 1234–1255], the method provides very promising LES results of turbulent cavity flows.