Analytical Solutions And Applied Research For The Flow Field Induced By Concentrated Vortices Of Helical Shape

The concentrated vortex with helical shape is a kind of special vortices with three-dimensional structure, and it is widespread in technological and mechanical devices with swirl flow. In some industry equipment, the work efficiency depends on the generation and operation of helical vortex. But, for other equipment, the helical vortex is considered to be very harmful, and it gives rise to severe vibration and oscillation of rotating blade, solid wall and flow field around it. In some military installations, such as, in the wake flow of airplanes and submarines, the helical vortex influences not only their propulsion efficiency, but also stability and silencing of noise in motion. Therefore, the problems brought by helical vortex in flow field arouse interest widely, and the investigations of helical vortex possess important theoretic significance and utility value.By the analytical calculation method, the paper focuses on researching systematically flow field induced by helical vortex on aspects of theory and application. The theoretic research includes: deducing out the analytical solutions for the flow field induced by cylindrical helical vortex with all features and the solution's expressions with finite items (36 formulas), proving five stream functions gained from the paper, discussing all the relations among ten constants in the flow field and acquiring their analytical solutions (including self-induced velocity). The application research includes: offering a method in calculation of engineering flow field by helical vortex model, that is, selecting vortex parameters based on the curve of the average velocity, analytically calculating three flow field in engineering and its results in accord with the existent experiments. Based on theory analysis and numerical calculations, some valuable conclusions are put forward, and some problems in early investigations are found out. Therefore, the results enrich the helical vortex theory and also clear the way from the helical vortex theory to its engineering application. In contrast to the works of former investigators, the present paper represents several features as follows:The paper presents the new method of analyzing the flow fields, which is mainly to combine the non-inertial coordinate system with the helical symmetrical technique, and deduces out the governed equations in relative coordinate system with the helical symmetrical and independent-time flow field. Thus, it solves a problem about including period item in analytical formulas of flow field induced by uniformly moving helical vortex, and explains the difference between Hardin's velocity formula and Okulov one. By solving velocity and vorticity equation, it also presents an analytical solution of the velocity component VB in moving flow field and finds the component VB to be only equal to a constant for static helical vortex.By constructing a generalized function"helical solitary vortex function", the paper solves the stream function of the helical vortex tube with the help of vortex filament one, and obtains some analytical expressions for large single-vortex core and small multi-vortex core. According to Borissov's viewpoint, the stream functions of helical vortex tube not only are the middle results of solving velocity and pressure field, but also express directly temperature field induced. By substituting stream function into differential equation, its validity is proved in mathematics and the analytical expression of helical solitary vortex function is also obtained. The discovery of this function will conduce to analytical solutions for other problems in physics.The paper deduces out the analytical solutions for the velocity and pressure field induced by cylindrical helical vortex with all features and the solution's expressions with finite items. For the first time, the problem of pressure field is discussed. Utilizing the solutions above, Okulov's supposition with same average value about angle and about time in the velocity field is attested. All the analytical solutions complete the theory of flow field induced by cylindrical helical vortex, and the remainder is to seek the more exact expressions with finite items about these analytical solutions.The paper gives a detailed discussion about all the relations among the eleven constants of the flow field induced by cylindrical helical vortex with all features, and some analytical solutions are acquired (including self-induced velocity). The paper deduces out the analytical relationships between U0 and V0, andΩrespectively and also obtains analytical formula of Swirl number. Moreover, it gives a condition of self-induced velocity with stability for helical vortex with arbitrary vorticity distributing.The paper presents the new methods of calculating the flow fields by helical vortex model, and discuses the flow fields of draft tube, the tip vortex as well as Ranque tube in detail. All their analytical solutions are obtained. Based on the calculative result of tip vortex flow field, it puts forward a different viewpoint from those of former investigators. The paper also improves Borissov's calculative model of the Ranque tube (considering time item in formulas). Moreover, it also puts forward a new calculative model of Ranque tube, that is, the nested helical vortex model. All analytical solutions of velocity and temperature fields are deduced out.As mentioned above, by systematic research on helical vortex theory in the present paper, the primary theory with features of static state, single filament and core radius vanished have been developed into new one with motion, multi-vortex and finite-core. The research result fill the blank spaces on the helical vortex theory, and also builds a bridge between the theoretic study and engineering application. Furthermore,it supports the theoretical investigation on stability problem of helical vortex, In application research, the presented results of three actual flow fields in engineering are significant and helpful to explain the mechanism about helical-vortex's generation, analyze boundary conditions of the flow fields in inlet and outlet, design rotating blade optimally, control helical vortex, and so on. Similarly, these research results also possess important theoretic significance and utility value for analytical calculation of other flow field with helical vortex in engineering.