| In recent decades, turbomachinery is increasingly required to work under higher speed, temperature and load situations. It poses an austere challenge to designers and researchers in that the conventional design can hardly satisfy these requirements. This paper starts with a status quo investigation of the topics on the inverse design and optimum design of turbomachines and airfoils home and abroad. The merits and demerits of the existing methods are discussed in detail. The topic in this dissertation is selected on the basis of the key National Natural Science Foundation item "Research on a new generation of inverse and optimum design problems in turbomachine aerodynamics and aerothermoelastodynamics". It covers the unsteady pitching problems and the single point inverse problems. Specifically they are as the following.1 For unsteady variational problems, time dimension can never be regarded as a space dimension. Looking them as the same will lead to nothing but divergence. So, space-time finite element method should be taken in discretization. But unfortunately, the existing space-time finite element method is extremely complicated and inconvenient. In addition, gradients are frequently asked for in finite element method. Its precision is directly related to the calculational precision of the primary variables. In consideration of this situation, it is of great importance to discuss existing ways of getting gradients. Particularly, a new way using higher order shape functions to obtain gradients is proposed. Through comparison of the precision by different ways, a supportive selection is made. On the basis of these considerations, two problems are exemplified to evaluate the new space-time FEM. The first is about the incompressible flow in a tube, the second about the steady flow in a syphon by a time-marching method. All the calculations lead to satisfactory results, proving that our new space-time finite elementmethod can be further generalized to solve unsteady airfoil pitching problems.2 Based on variable-domain variational finite element method, the inverse problems about airfoils under steady compressible flows are reconsidered. Previously, the variational principles for inverse problems are obtained by assuming the variational calculus is in y direction. But for the unsteady problem of the pitching airfoil, we will take a new generalized form to consider all the actual situations. Here, the variational calculus is assumed to be in the y0 direction when the airfoil is at idle. Then, the discretized forms by variable-domain variational finite element method are obtained. The complete variable-domain variational finite element equation, instead of the simple difference quotient equation, is coupled with pseudo-unsteady treating way to solve the inverse problems. This complete formulation has greatly boosted the convergent speed. The generalization makes the method much more complete and applicable.3 A detail deduction of the direct and inverse problem variable-domain variational principles for pitching airfoil is followed. Previously, the principles for unsteady inverse problems are under the assumption of small pitching angle. And, it is assumed that the variable-domain variational calculus is, for the foil itself, in different direction at different attack angle. A new complete and reasonable principle is proposed in this paper. From the detail derivation of the functional, we should say that the new one has more practical significance. The principle is then further generalized to get its sub-generalized, generalized principles and the general forms of the generalized principles.4 The unsteady pitching flow of airfoils is analyzed by variational finite element method. The wake vortex streamline is for the first time obtained numerically during the flow field analysis. The application of unsteady Kutta condition is discussed. The results prove that the outflow of the pitching airfoil is along the tangential direction of...
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