摘要
数值计算方法是燃气轮机研究的一种重要手段，空间离散精度对数值计算结果的准确性及可信度意义重大。为适应燃气轮机内部复杂冷却结构，本文基于在非结构网格上可实现任意高阶精度的间断有限元方法发展了一套高精度数值计算工具，并采用该程序探讨了间断有限元方法在燃气轮机冷却系统中的应用效果。
本文发展的数值计算工具基于非结构网格，低阶精度计算采用MUSCL重构的有限体积离散方法，高阶精度采用间断有限元离散方法。其中低阶精度有限体积求解器采用隐式求解，计算稳定性强、计算效率高；高阶精度求解器采用显式求解，具备实现任意高阶精度的计算能力。该计算工具兼容Roe，HLLC，AUSM+等多种通量格式，包含SA，SST，SST-γ-Reθ多种湍流模型，可用于定常/非定常，可压/不可压，高Ma数/低Ma数，流动/导热/流热耦合等问题的并行计算。
为提高数值计算工具的实用性：本文推导了高计算效率、矩阵形式的Modal型间断有限元离散公式，通过对虚拟时间项离散进行合理简化，提出了一种可减少计算量及内存的实现方法；将预处理矩阵方法引入间断有限元框架，数值算例表明预处理矩阵+间断有限元的实现方法可用于三维粘性低Ma数流动及不可压流动的计算，能保持间断有限元的离散精度，且收敛速度几乎与Ma数无关；在间断有限元框架引入Wang修正的Venkatakrishnan限制器，有效改善了BJ限制器收敛滞止现象；提出了一种利用低阶精度虚网格边界值在边界进行限制的限制器边界处理方法，一定程度可避免壁面附近的不合理限制。本文采用高阶精度显式求解、一阶精度隐式求解的p型多重网格，该方法可避免高阶精度隐式计算的大内存开销并对本文算例达到8左右的加速比。此外本文还对p型多重网格方法进行了傅立叶线性稳定性分析，分析结果表明：低阶精度隐式迭代算子在二阶精度计算中能总体改善计算稳定性，向前Euler方法在p型多重网格计算中条件稳定。
对MarkII流热耦合问题、轴向通流旋转盘腔热浮升力驱动流动、转静腔室流动计算结果表明：采用DG+γ-Reθ转捩模型对MarkII及盘腔的传热预测结果与实验吻合较好；DG+SST模型对转静腔室计算的传热误差在10%以内。这较好的检验了本文程序的准确性及实用性，并展现了DG方法在燃气轮机叶片冷却及二次空气系统中广阔的应用前景。
Nowadays CFD is one of the most important methods for gas turbine investigation,the accuracy of the spatial discretisation methods exerts fundamental influence on theaccuracy and credibility of the numerical results. To adapt the complex geometry'scalculation of the gas turbine, a parallel discontinuous finite element numericalcomputational tool, which can fulfill any high-order accuracy on the unstructured grid,was developed in this thesis; furthermore, this newly developed tool had been applied tothe calculation of the cooling flow of gas turbine to test its efficiency and credibility.
The finite volume method with MUSCL Reconstruction was adopted for thelow-order accuracy calculations, while the discontinuous galerkin method was used forthe high-order accuracy calculations in the in-house unstructured numerical tool. TheLow-order FV Solver is much more robust and efficient than the High-order DG Solver,as the numerical dissipation of the former is much higher and the implicit schemes forthe discretisation in time are applied. However, the High-order DG Solver can realizeany order accuracy. Moreover, this tool is compatible with Roe, HLLC, AUSM+andother numerical inviscid flux schemes, and contains SA, SST, SST-γ-Reθturbulencemodels, and can be used for the calculation of steady/unsteady, compressible/incompressible, high Ma number/low Ma number，flow/heat conduction/conjugate heattransfer problems.
To improve the practical performance of the numerical tool developed, some workhas been accomplished on the discontinuous galerkin method: a matrix formula aboutmodal type discontinuous finite element's discrete equation with highercomputational efficiency was derived, a reasonable simplification was carried outon a discretization of virtual time derivative term, and an implementationmethod was proposed to reduce the requirement on memory and CPU time. Thefinite volume preconditioning method was introduced to the discontinuous finiteelement method for three-dimensional viscid low Mach number flows. The testcases verified that the preconditioning discontinuous finite element method wassuitable for viscid low Mach number flow, and at meantime retained the originaldiscrete accuracy of the discontinuous finite element method, and also indicated that the converge was independent of the Mach number. To avoid convergencestall phenomenon of the BJ Limiter, Wang's modified type Venkatakrishnanslope limiter was introduced to the discontinuous finite element method. Toavoid the stability problems caused by unreasonable slope limiter on boundaryelements, a method making use of the virtual mesh's boundary value wasproposed, and the boundary value at virtual mesh was calculated by FV methodto limit the boundary elements. In this thesis, p-multigrid method, which solvehigh-order DG discrete equation by explicit method and low-order FV Discreteequation by implicit method, was adopted to accelerate the convergence, and thetime cost can reduce to1/8of the original one. Besides, the linear waveequation's Fourier analysis of p-multigrid method was carried out. And theresults showed the p-multigrid method would be more stable than the originalone, and the forward Euler time scheme was conditionally stable when it wasused with p-multigrid
In this thesis, the MarkII conjugate heat transfer problem, the bouyancy-inducedflow in axial through-flow rotating cavity, and the flow in rotor-stator cavity wereapplied to verify the performance of DG method on gas turbine's simulation. The resultusing γ-Reθtransition model agreed well with the experimental data for MarkII androtating cavity, and the error of heat transfer coefficient was less than10%for therotate-stator cavity. These cases validated our numerical tool and showed that the DGmethod could be suitable for blade cooling and flow of secondary air system in gasturbine.
引文
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