Theoretical analysis of quasi-one-dimensional compressible magnetohydrodynamic channel flow
Introduction
As researches and explorations in the aerospace field continue to advance, the role of Magnetohydrodynamics (MHD) is becoming more prominent. In the last century, the AJAX program was proposed by Russia, using the MHD energy-bypass technology to broaden the operating Mach number of air-breathing scramjet and reduce the total pressure loss [1], [2], [3]. Currently, some research institutes such as NASA, Air Force Research Laboratory (AFRL), and Central Aerohydrodynamic Institute (TsAGI) developed MHD acceleration wind tunnels and have achieved certain results [4], [5], [6], [7], [8], [9]. Theoretically speaking, the aforementioned air-breathing hypersonic vehicles with MHD energy-bypass technology and the hypersonic MHD acceleration wind tunnel can be abstracted as a compressible channel flow of gas with low conductivity in an electromagnetic field. Consequently, it is of significance to deeply study the characteristics of MHD channel flow.
The quasi-one-dimensional (quasi-1D) MHD channel flow at low magnetic Reynolds number has been studied since the 1950s [10]. Rosa [11] and Chekmarev [12] performed integrations on the governing equations for constant-cross section in quasi-1D, steady, inviscid channel flow. For the case of variable cross-section but maintaining a constant density, Resler and Sears [13] gave analytical expressions for the velocity, temperature, and pressure along the flow direction. In the meantime, Resler and Sears investigated the variation patterns of the velocity and Mach number along the flow direction under the conditions of constant cross-section and constant density. It has been concluded that the relative magnitudes between the flow velocity and three critical velocity values (, and in the literature) would change the asymptotic properties of the flow significantly [13], [14]. In addition, the phenomenon of choke occurs when the velocity crosses the speed of sound even in a constant cross-section channel flow, but some special tunnels still enable a continuous transonic process [13], [15]. Tang et al. [16] obtained three modes of operation for a supersonic MHD generator through theoretical analysis of a constant cross-section channel flow. The modes of supersonic flow at the exit of the channel depend on the relations between the applied electric field and the critical electric field . Unfortunately, the corresponding conclusions and further theoretical analysis for variable cross-section channels are still not available.
All the above analyses were performed to study the variation of gas parameters along the flow direction. However, Culick [17] considered this problem from a pathbreaking perspective: he analyzed the MHD channel flow by studying the phase trajectory consisting of the allowed constant cross-sectional flow parameters in the velocity-Mach number phase plane, whose direction is marked by the second law of thermodynamics. In this way, the critical values of the velocity that affect the extreme and asymptotic characteristics of the flow parameters will be visualized as curves in the velocity-Mach number phase plane. Moreover, Powers et al. [18], and Wu [19] plotted the critical velocity in the dimensionless current density-electric field plane as a limiting circle, which can describe the MHD accelerator/generator and the supersonic/subsonic flow simultaneously. However, the lack of physical quantities related to the gas dynamics makes it difficult to consider the coupling effect of the electromagnetic field with the gas’s thermodynamics in a variable cross-section. Recently, Makinde et al. [20], and Hamrelaine et al. [21] obtained approximate solutions for conductive viscous incompressible channel flow (MHD Jeffrey-Hamel flow) under a uniform magnetic field by the perturbation method. Nevertheless, such approximate solutions are difficult to consider the compressible flow.
From the above review, it can be recognized that, on the one hand, the existing analysis of quasi-1D MHD channel flow has been carried out for constant cross-section, but the results have not been extended to the case of variable cross-section. Since most of the cases required in practical engineering are variable cross-section channel flows, and the coupling effects caused by variable cross-section with the electromagnetic field will produce very complex phenomena, deeper investigations of constant/variable cross-section channel flow in the electromagnetic field are of theoretical and engineering value. On the other hand, for the mathematical methodology adopted in theoretical analysis of MHD channel flow, although the direction of the phase trajectory in the phase plane can be obtained by employing the entropic increase principle , other features such as the asymptotic properties, turning points, and fixed points [22] are related to the adjacency of the acceleration/deceleration regions. These characteristics can be easily expressed using graph theory in mathematics. To the authors’ knowledge, no work has been done to study the phase plane and phase trajectory properties using graph theory in MHD channel flow. In consequence, this paper will adopt mathematical tools such as graph isomorphism and equivalence class to study nonlinear second-order dynamical system describing the flow of MHD constant/variable cross-section channel flow to analyze the properties in a more comprehensive and in-depth manner.
Based on the previous results, this paper will study the equations describing the quasi-1D, steady, inviscid, compressible MHD constant/variable cross-section channel flow in the dimensionless velocity-Mach number phase plane. Then, mathematical methods such as labeled graph and equivalence class will be adopted to discuss the characteristics of channel flow. This paper is organized as follows: Part 2 will derive the dimensionless MHD equations under the assumption of low magnetic Reynolds number by novelly selecting dimensionless parameters, and then obtain the second-order nonlinear dynamical system that can comprehensively consider the coupling effects of compressibility and electromagnetism even under the variable cross-section conditions; In Part 3, by analyzing the second-order dynamical system, the distribution of different regions in the phase plane is obtained which can reflect the effects of acceleration/deceleration of velocity and Mach number; Finally, Part 4 will establish equivalence classes based on the isomorphism of labeled graphs, and study several cases through equivalence relations to cover all possible cases of MHD constant/variable cross-section channel flow.
Section snippets
Dimensionless equations for quasi-1D MHD channel flow
For flow problems in aerospace, the low electric conductivity of the gas will result in a dimensionless magnetic Reynolds number Rem much less than one [10]. The magnetic Reynolds number Rem is physically characterized by the induced magnetic field ratio to the applied magnetic field. Thus, the induced magnetic field can be ignored, and the magnetic field is considered as the applied magnetic field for low Rem problems. In this section, based on the simplified MHD equations under low Rem
Theoretical analysis of MHD channel flow
A detailed analysis of the dynamical system is of particular importance to obtain the evolution behavior of the MHD channel flow parameters. There is almost no possibility to integrate a nonlinear dynamical system analytically. Even if analytical solutions exist for a small bunch of nonlinear systems, the systems’ properties and characteristics are usually implicit in the solutions. For general nonlinear systems, Poincaré started to analyze the phase trajectories in the phase plane to acquire
Phase planes classification
In the case of MHD wind tunnels or MHD generators in practical applications, according to the definition of the interaction number, will vary continuously even under the same operating conditions. Hence it is important to study the variation of the acceleration/deceleration regions in the phase plane with . The analysis in the previous section shows that there are three critical values ( and ) and two limiting values (0 and ) of the interaction number. When transition across
Conclusions
In this paper, a second-order dimensionless dynamical system with respect to the velocity and the Mach number with a parameter of the magnetic interaction number is obtained to theoretically analyze quasi-1D, steady, inviscid, compressible, MHD channel flow at a low magnetic Reynolds number with constant or variable cross-section. The flow behaviors of MHD channel are acquired for all possible cases by investigating the characteristics of fixed points, phase trajectories and
Acknowledgements
This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China through grant 11721202 and the National Numerical Wind Tunnel Project through Grant NNW2018-ZT3A05.
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