P-Multigrid Discontinuous Finite Element Method And Application For Gas Turbine’s Cooling Flow

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-multigridIn 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.