Symplectic Geometric Algorithms For Predicting Dynamic Behavior Of Gas Turbine Engines

This is an interdisciplinary thesis, a combination of predicting dynamicbehavior of gas turbine engines and symplectic geometric algorithms, newfrontiers of computational mathematics.As every one knows, the gas turbine engine plays more and moreimportant role in transportation, power generation, national defense etc. So,the industry of gas turbine becomes one of the strategic high and new-techindustries in China. The dynamic performance is an important part of gasturbine engines. And the numerical solutions influence the prediction ofdynamic behavior. The main topic is to improve the accuracy and evaluatethe reliability of the numerical solution.A lot of experience has been accumulated to predict the dynamicperformance of gas turbine engines, such as improved Euler methods andRunge-Kutta methods. But how to check the accuracy and reliability of thecalculation results before engine testing, are still problems, not yet beingsolved well.In fact, the traditional numerical methods, improved Euler methods andRunge-Kutta methods, have one thing in common, that is when constructingschemes only local point-wise accuracy at each step is taken into account.Every one knows that high local precision is not equal to high qualitysimulation of overall structure. For traditional methods, there is always energy dissipation due to the truncation errors in each point. Anisotropic energy dissipation makes the errors in the calculation of the total energy of investigated systems, moreover, the mentioned errors will accumulate from one computational step to another. Eventually, it will distort the dynamic performance of the engine.Symplectic geometric algorithms, different from traditional numerical methods, have no dissipation and can preserve the symplectic structure of Hamiltonian system of dynamic evolutions. It is believed that the systematic structure-preserving would produce far-reaching effect for the performance prediction of gas turbine engines.However, the dynamic performance of gas turbine is still expressed with Newtonian system. In order to apply symplectic geometric algorithms and check the accuracy and reliability of the calculation results before testing, it is necessary to express the dynamic performance with so-called Hamiltonian canonical system. With the help of the supervisors, a Hamiltonian system for the dynamic performance of gas turbine is built as follows: φ=1/Jzλ where H is Hamiltonian function and φ is the angle of rotation, A, is the angular momentum of the rotor, in which Jz is the moment inertia,△M(φ) is the torque difference between compressor and matched turbine.As knowing from theory of ’manifold’, the symplectic geometric algorithms are the only reliable way to solve Hamiltonian systems. Thus, toselect an appropriate symplectic geometric algorithm to solve the dynamicproblem of gas turbine engines is important.A fractional-step composing symplectic geometric scheme (abbr. FSJS)was proposed. The FSJS method is able to keep the structure, not only at eachstep but also at every fractional-step, and energy conservation in dynamicprocess of gas turbine engines. And all five-step third-order FSJS schemes(abbr. FSJS3) are expressed by the general solution formula with one freevariable. Therefore, by one condition, a specific scheme can be designed tomatch responsible requirements for specific problems in engineering.But due to the ‘phase error’, the accuracy of numerical accuracy is not yetfully improved. Of course, this also happens in traditional methods. With thetheory of phase errors, the coefficients of five-step third-order FSJS schemeswith minimal phase error (abbr. FSJS3-N3) was given by analytical formula,Moreover, a series of methods for correcting drifts were proposed, such asnarrowing step, using third order schemes, and using the minimal phase errorschemes, etc. And there is a point to emphasize that correcting drifts is toimprove point-wise accuracy in time domain.For checking the effectiveness of the presented FSJS3method, two specialdynamic models of one-shaft gas turbine engine with analytic solutions aredesigned. The results show that the new schemes are much better thanforth-order Runge-Kutta method and the common third-order symplecticpartitioned-Runge-Kutta scheme.The presented FSJS method has been used to investigate the dynamicperformance of three-axial ‘Spey’ marine gas turbine. The results obtainedalso verifies that FSJS3-N3scheme not only have physical meaning in each step, but also overcome the defects of conventional methods in integrity,stability, long-term tracking ability, energy-and structure-preservation.In a word, symplecetic geometric methods open up a new prospect fornumerical simulation of dynamic process of gas turbine. The law of dynamicevolution of engines could only be discovered in the Symplectic Space!...