A Unified Theory of Hybrid Problems for Fully Three-Dimensional Incompressible Rotor Flow Based on Variational Principles With Variable Domain

Abstract

Making use of functional variations with variable domain and natural boundary conditions, this paper presents a unified theory of various hybrid problems (being a unification as well as a generalization of direct and inverse problems) for three-dimensional incompressible potential flow in a rotor blading. Three families of variational principles (VPs) have been established and provide a series of new rational ways for blade design and a sound theoretical basis for the finite element method (FEM). This theory can be extended to compressible and rotational flows and also constitutes an important part of the optimum design theory of bladings [8].

Advances in research on inverse and hybrid problems of turbomachinery aerothermodynamics in China

“…Perhaps a very attractive merit of this variable-domain approach is that it can be straightforwardly extended to fully 3-D flow [12]. …”

“…on the blade surface are combined with those on the annular walls, it is necessary to employ some properly defined compound symbols to designate them as proposed in Refs. [12,13]. For instance, the symbol (I x HA) designates such a hybrid problem type in which an inverse problem is posed on the blade surface, while a HA-problem is posed on the annular walls ( Fig.…”

“…The variable-domain variational approach [I 1,301 has been extended by Liu to hybrid problems for fully 3-D incompressible [12], compressible [I 3) and transonic 1141 flows in rotors. Only one of the VPs is given below for reference: where the boundary integral term L takes different forms for different problem types.…”

Variable-Domain Finite Element Method Based on Variational Principles for Solving Hybrid Problems of Fully 3-D Compressible Rotor-Flow

“…M^ti.g-^)///-, These equations should be solved together with some boundary conditions (BC), which depend upon the hybrid problem types on the blade and annular walls. In the present paper two hybrid problem types H A X D and [He + D] X D are solved numerically, which, according to the definition proposed previously (Liu, 1986(Liu, , 1988(Liu, , 1995a mean that 1) (H a X D)-problem: a hybrid problem of type A (H a ) is posed on the blade surface (i.e. on some portion of the blade surface the wall shape is given, while on the remaining portion the pressure distribution is specified) and a direct problem (D) is posed on the hub and casing walls.…”

“…The hybrid problem turns out to be much more general and versatile than the conventional direct and inverse problems, being a unification as well as a generalization of them. Then, a unified variational theory with variable domain of hybrid problems for fully 3-D incompressible flow in impeller was developed by Liu (1986) and the corresponding numerical solutions were carried out by Yan and Liu (1994) by introduciing a new self-adjusting (deforming) finite element (FE) for numerically realizing the functional variation with variable domain. This unified variational theory of hybrid problems was extended to fully 3-D compressible rotor-fl· w by Liu (1988 and1995a) and based thereupon in the present paper two types of the hybrid problem (H A X D) and (Hc+D) X D for fully 3-D compressible flow are solved also by the selfadjusting finite element method (Yan and Liu, 1994b) with some modifications.…”