We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the...We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.展开更多
WT5”BZ]A numerical simulation was performed to study the roll up of vortex sheet behind a slender delta wing and the evolution of vortex layer shed by a plunging airfoil using the Multhopp superconvergence scheme g...WT5”BZ]A numerical simulation was performed to study the roll up of vortex sheet behind a slender delta wing and the evolution of vortex layer shed by a plunging airfoil using the Multhopp superconvergence scheme given in Part I. Mathematical models, which are regular in the sense of Lighthill′s lemma, were proposed to simulate the vortex sheet evolution for side slip and roll motion of wings, and solved numerically by using the superconvergence vortex method. An approach to investigate the vortex evolution in the near wake of the oscillating airfoil was also given. Our calculations have confirmed that the approach can reasonably predict the nonlinear evolution of vortex sheet.展开更多
The traditional Kelvin-Helmholtz notion of studying the shear instability is not suitable for the case associated with shear line with the strong wind shear in the vortex sheet. Since then, the shear instability becom...The traditional Kelvin-Helmholtz notion of studying the shear instability is not suitable for the case associated with shear line with the strong wind shear in the vortex sheet. Since then, the shear instability becomes theinstability of the vortex sheet. If the velocity is induced by the vortex sheet, the inequalities (1? R r + Ri d)> 0 and U(v,t)> U(A(t)) become the criterion of the vortex sheet instability. This criterion indicates that 1) the disposition of environment field restrains the disturbance developing along the shear line. 2) There exist multi—scale interactions in the unstable process of the shear line. The calculation of the necessary condition for the instability is also presented in this paper. Key words Shear line - Induced velocity - Instability of the vortex sheet This work was supported by the project on the study of the formative mechanism and predictive theory of the significant climate and weather disaster in China under Grant G 1998040907 and by the key project on the Dynamic Study of Severe Mesoscale Covective Systems sponsored by the National Natural Science Foundation of China under Grant No.49735180.展开更多
A new vortex sheet model was proposed for simulating aircraft wake vortex evolution.Rather than beginning with a pair of counter-rotating cylindrical vortices as in the traditional models, a lift-drag method is used t...A new vortex sheet model was proposed for simulating aircraft wake vortex evolution.Rather than beginning with a pair of counter-rotating cylindrical vortices as in the traditional models, a lift-drag method is used to initialize a vortex sheet so that the roll-up phase is taken into account. The results of this model report a better approximation to a real situation when compared to the measurement data. The roll-up induced structures are proved to influence the far-field decay.On one hand, they lead to an early decay in the diffusion phase. On the other hand, the growth of linear instability such as elliptical instability is suppressed, resulting in a slower decay in the rapid decay phase. This work provides a simple and practicable model for simulating wake vortex evolution, which combines the roll-up process and the far-field phase in simulation. It is also proved that the roll-up phase should not be ignored when simulating the far-field evolution of an aircraft wake vortex pair, which indicates the necessity of this new model.展开更多
基金supported in part by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1 and EP/L015811/1
文摘We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.
基金Project supported by the National Natural Science Foundation of China. ( No.1 9372 0 60 )
文摘WT5”BZ]A numerical simulation was performed to study the roll up of vortex sheet behind a slender delta wing and the evolution of vortex layer shed by a plunging airfoil using the Multhopp superconvergence scheme given in Part I. Mathematical models, which are regular in the sense of Lighthill′s lemma, were proposed to simulate the vortex sheet evolution for side slip and roll motion of wings, and solved numerically by using the superconvergence vortex method. An approach to investigate the vortex evolution in the near wake of the oscillating airfoil was also given. Our calculations have confirmed that the approach can reasonably predict the nonlinear evolution of vortex sheet.
基金This work was supported by the project on the study of the formative mechanism and predictive theory of the significant climat
文摘The traditional Kelvin-Helmholtz notion of studying the shear instability is not suitable for the case associated with shear line with the strong wind shear in the vortex sheet. Since then, the shear instability becomes theinstability of the vortex sheet. If the velocity is induced by the vortex sheet, the inequalities (1? R r + Ri d)> 0 and U(v,t)> U(A(t)) become the criterion of the vortex sheet instability. This criterion indicates that 1) the disposition of environment field restrains the disturbance developing along the shear line. 2) There exist multi—scale interactions in the unstable process of the shear line. The calculation of the necessary condition for the instability is also presented in this paper. Key words Shear line - Induced velocity - Instability of the vortex sheet This work was supported by the project on the study of the formative mechanism and predictive theory of the significant climate and weather disaster in China under Grant G 1998040907 and by the key project on the Dynamic Study of Severe Mesoscale Covective Systems sponsored by the National Natural Science Foundation of China under Grant No.49735180.
基金supported by the Boeing-COMAC Aviation Energy Conservation and Emissions Reduction Technology Center (AECER)
文摘A new vortex sheet model was proposed for simulating aircraft wake vortex evolution.Rather than beginning with a pair of counter-rotating cylindrical vortices as in the traditional models, a lift-drag method is used to initialize a vortex sheet so that the roll-up phase is taken into account. The results of this model report a better approximation to a real situation when compared to the measurement data. The roll-up induced structures are proved to influence the far-field decay.On one hand, they lead to an early decay in the diffusion phase. On the other hand, the growth of linear instability such as elliptical instability is suppressed, resulting in a slower decay in the rapid decay phase. This work provides a simple and practicable model for simulating wake vortex evolution, which combines the roll-up process and the far-field phase in simulation. It is also proved that the roll-up phase should not be ignored when simulating the far-field evolution of an aircraft wake vortex pair, which indicates the necessity of this new model.
