摘要
本文由三部分组成：(一)系统地建立了理论上严密、适用范围普遍的三维无粘有旋流动模型——赝势函数模型；(二)建立了二维定常有旋流动正、反命题的赝势-涡势函数变分原理及引入映象平面，导出了赝势函数在映象平面的方程；(三)建立了三维涡势函数三拟变分形式。
本论文的内容概括如下：
依据缩项法则以张量的形式引入了三维可压缩流动的赝势函数-涡势函数模型。这一模型不仅保持了势函数的主要特点，而且适用于有旋流动，同时还具有物理上的相容性，是势函数向有旋流动的合理推广。除此之外它还克服了由于忽略跨声速流动中激波后的熵增而引起的守恒型势函数解的不唯一性。
详细的推导了在任意曲线坐标下赝势函数、涡势函数和滞止焓函数表示的气动力学方程组，在圆柱坐标系下各有关方程的具体形式可作为特例而得出。为了适应物理界面形状的复杂性而引入了贴体曲线坐标系及相应的映象空间，推出了各方程在该空间中的具体形式。
建立了二维定常流动的赝势函数-涡势函数模型的正命题和反命题的严格的完整的变分原理，从而为求解二维定常有旋流动(特别是跨声速流动)奠定了严密的理论基础。
详细探讨了三维涡势函数方程的三种拟变分形式，并就第三种情况提出了涡势函数边界条件。所有这些为其进一步的应用奠定了坚实的理论基础。
The present dissertation consists of three parts: (1) The pseudo-potential function model for 3-D inviscid rotational rotor flow is systematically established , which is theoretically rigorous and generally applicable; (2) The Variational Principles for 2-D compressible rotational flow using pseudo-potential model encompassing pseudo-potential function and vorticity-potential function are established ,and the basic aerodynamic equations in terms of the pseudo-potential function are derived in image-plane; (3) Three kinds of quasi-variational formulation for 3-D vorticity-potential model are established.
The main work of the thesis can be outlined as follow : The pseudo-potential function and vortex potential functions are derived by means of the term-condensing method ,this model not only remains the basic advantages of potential function model ,but also applies to the rotational flow. It is a physically self-consistent model and is a rational extension of the potential function to rotational flow. It removes the non-uniqueness of the potential function solutions at transonic speeds due to ignoring entropy gradient behind the shock.
The general form of the fundamental equations of aerodynamics in terms of the pseudo-potential function, the vortex potential function and the enthalpy are derived in the general non-orthogonal
curvilinear coordinate system by means of both tensor calculus and Jacobians. To facilitate the numerical solution, an image-space based on a body-filled curvilinear coordinate system is introduced .
Using the pseudo-potential function, Variational principles for direct and inverse problems of 2-D steady rotational compressible flow are developed rigorously.
Three alternative formulations for the vortex-potential equations are provided.
All these lay a firm theoretical basis for its application to computational fluid dynamics.
引文
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