High Precision Algorithms For Dynamics Problems And Their Exploration In Gas Turbine Engineering

This paper is an interdisciplinary research achievement of gas turbine engineering and computational mathematics.Gas turbines and aeroengines are widely used in many fields.They not only are the key equipments of national defense,but also play an irreplaceable strategic role in power,energy exploitation and transportation,and distributed energy systems in the national economy.Dynamics is an important theoretical basis for gas turbines and aeroengines.In gas turbines,processes include the dynamic characteristics of engines,unsteady flow fields in compressor and turbine,unsteady heat and mass transfer processes of high temperature components,chemical and physical processes related to combustion in combustion chambers involve dynamics,and the rational organization of these important processes must be guided by dynamics.Dynamics is also a field that mathematicians attach great importance to and contribute to.In order to solve the dynamic problems by “scientific calculation” effectively and reliably,the dynamic problems expressed in Newton's mechanical system are transferred to Hamilton's mechanical system and symplectic geometric algorithms are proposed.In this paper,according to the need of gas turbine dynamics(category of engineering thermophysics),new high-precision algorithms are proposed for solving Hamilton system with the help of finite element method.Although many symplectic algorithms for numerical solution of linear Hamilton systems can guarantee the structural characteristics of the system,the phase and energy errors are still large.In this paper,a weighted discontinuous Galerkin method without phase error(WDG-PDF)is proposed for linear Hamilton system.WDG-PDF method utilizes the discontinuous characteristic in nodes.The weight without phase error is given,and symplectic preservation can be realized by processing the transfer matrix.In this paper,the proof of symplectic preservation and phase-free error of WDG-PDF method are given.WDG-PDF method achieves symplectic preservation and has no phase error,meanwhile,Hamilton function error reach computer rounding error magnitude.Therefore,for linear Hamilton system,weighted discontinuous time finite element method without phase error is the best choice.In this paper,a time adaptive finite element method(A-TFEM)is proposed for nonlinear Hamilton systems.In recent years,adaptive and efficient algorithms have been widely used in solving dynamic problems,but the existing adaptive algorithms for solving Hamilton systems often fail to guarantee the inherent characteristics of Hamilton systems(energy conservation,symplectic structure,etc.).The A-TFEM method uses the posterior error estimate of the time finite element method to determine the adaptive index ?.When the adaptive index ? is larger than the upper bound of the preset error range,the calculation step size will be automatically reduced,and when the adaptive index ? is smaller than the lower bound of the preset error range,the calculation step size will be automatically increased.The energy-preserving and symplectic-preserving properties of A-TFEM method are proved theoretically.Some typical nonlinear Hamiltonian equations have been calculated,using A-TFEM method,to verify the effectiveness of presented method.The calculation of dynamic processes of gas turbines are usually carried out in Newton mechanics system.The mathematical model of dynamic processes of gas turbine engines has been transferred successfully to Hamilton System by our research group.The study shows that the “A-TFEM” mentioned above is very suitable for the numerical calculation of the dynamic process of gas turbine,and improves the calculation efficiency obviously.The numerical results show that the “A-TFEM” is much better than the previous “FSJS” for solving the model in the sense of energy conservation and calculation accuracy.Many dynamic problems in gas turbine engineering must be described by “partial differential equation”.The most typical one is Navier-Stokes equation,which is the governing equation of fluid.Mathematicians have done a lot of research work and constructed many numerical models and algorithms.To avoid the saddle point problem in numerical solution of Navier-Stokes equation,mathematicians have proposed different decoupling methods.The Gauge method is a well-known decoupling algorithm based on Hamilton form of Navier-Stokes equation.However,there are still many problems,which have to be solved in the calculation practice.In order to solve these problems,a modified Gauge method(MGM)is proposed in this paper.MGM method is an innovation in the numerical solution of Navier-Stokes equation.On the one hand,the stability analysis of MGM method and the error estimation of velocity and pressure are given,which proves the effectiveness of the algorithm in theory.On the other hand,a large number of calculation practices of classical models in fluid dynamics are used to evaluate the effectiveness.The results of theoretical analysis are verified by numerical experiments.The MGM method is not only applicable to Navier-Stokes equations,but also to some more complicated partial differential equations,such as Boussinesq equations.