Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body

2016 ◽  
Vol 221 (2) ◽  
pp. 559-602 ◽  
Author(s):  
Chao Chen ◽  
Lili Du ◽  
Chunjing Xie ◽  
Zhouping Xin
Keyword(s):  
Symmetric Body ◽  
Two Dimensional ◽  
Euler Flows

On the application of the Kelvin–Arnol'd energy principle to the stability of forced two-dimensional inviscid flows

1998 ◽  
Vol 356 ◽  
pp. 221-257 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Magnetic Field ◽  
Nonlinear Stability ◽  
Broad Class ◽  
Mhd Flow ◽  
Two Dimensional ◽  
Energy Principle ◽  
Inviscid Flows ◽  
Virtual Displacement ◽  
Euler Flows ◽  
Universal Procedure

Arnol'd developed two distinct yet closely related approaches to the linear stability of Euler flows. One is widely used for two-dimensional flows and involves constructing a conserved functional whose first variation vanishes and whose second variation determines the linear (and nonlinear) stability of the motion. The second method is a refinement of Kelvin's energy principle which states that stable steady Euler flows represent extremums in energy under a virtual displacement of the vorticity field. The conserved-functional (or energy-Casimir) method has been extended by several authors to more complex flows, such as planar MHD flow. In this paper we generalize the Kelvin–Arnol'd energy method to two-dimensional inviscid flows subject to a body force of the form −ϕ∇f. Here ϕ is a materially conserved quantity and f an arbitrary function of position and of ϕ. This encompasses a broad class of conservative flows, such as natural-convection planar and poloidal MHD flow with the magnetic field trapped in the plane of the motion, flows driven by electrostatic forces, swirling recirculating flow, self-gravitating flows and poloidal MHD flow subject to an azimuthal magnetic field. We show that stable steady motions represent extremums in energy under a virtual displacement of ϕ and of the vorticity field. That is, d1E=0 at equilibrium and whenever d2E is positive or negative definite the flow is (linearly) stable. We also show that unstable normal modes must have a spatial structure which satisfies d2E=0. This provides a single stability test for a broad class of flows, and we describe a simple universal procedure for implementing this test. In passing, a new test for linear stability is developed. That is, we demonstrate that stability is ensured (for flows of the type considered here) whenever the Lagrangian of the flow is a maximum under a virtual displacement of the particle trajectories, the displacement being of the type normally associated with Hamilton's principle. A simple universal procedure for applying this test is also given. We apply our general stability criteria to a range of flows and recover some familiar results. We also extend these ideas to flows which are subject to more than one type of body force. For example, a new stability criterion is obtained (without the use of Casimirs) for natural convection in the presence of a magnetic field. Nonlinear stability is also considered. Specifically, we develop a nonlinear stability criterion for planar MHD flows which are subject to isomagnetic perturbations. This differs from previous criteria in that we are able to extend the linear criterion into the nonlinear regime. We also show how to extend the Kelvin–Arnol'd method to finite-amplitude perturbations.

scholarly journals Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

2019 ◽  
Vol 266 (7) ◽  
pp. 4337-4376 ◽  
Author(s):  
Feimin Huang ◽  
Jie Kuang ◽  
Dehua Wang ◽  
Wei Xiang
Keyword(s):  
Contact Discontinuity ◽  
Two Dimensional ◽  
Compressible Euler ◽  
Euler Flows

scholarly journals Two-dimensional Euler flows with concentrated vorticities

2010 ◽  
Vol 362 (12) ◽  
pp. 6381-6381 ◽  
Author(s):  
Manuel del Pino ◽  
Pierpaolo Esposito ◽  
Monica Musso
Keyword(s):  
Two Dimensional ◽  
Euler Flows

Structure and stability of solutions of the Euler equations: a lagrangian approach

1990 ◽  
Vol 333 (1631) ◽  
pp. 321-342 ◽  
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Saddle Points ◽  
Relaxation Processes ◽  
Two Dimensional ◽  
Scalar And Vector Fields ◽  
Euler Flows ◽  
Relaxation Procedure ◽  
Unbounded Fluid

This paper reviews methods that are essentially lagrangian in character for determination of solutions of the Euler equations having prescribed topological characteristics. These methods depend in the first instance on the existence of lagrangian invariants for convected scalar and vector fields. Among these, the helicity invariant for a convected or ‘frozen-in’ vector field has particular significance. These invariants, and the associated topological interpretation are discussed in §§1 and 2. In §3 the method of magnetic relaxation to magnetostatic equilibria of prescribed topology is briefly described. This provides a powerful method for determining steady Euler flows through the well-known exact analogy between Euler flows and magnetostatic equilibria. Stability considerations relating to magnetostatic equilibria obtained in this way and to the analogous Euler flows are reviewed in §4. In §5 the related relaxation procedure is discussed; for two-dimensional and axisymmetric situations this technique provides stable solutions of the Euler equations for which the vorticity field has prescribed topology. The concept of flow signature is described in §6: this is the relevant topological characteristic for two-dimensional or axisymmetric situations, which is conserved during frozen-field relaxation processes. In §§7 and 8, the formation of tangential discontinuities as a normal part of the relaxation process when saddle points of the frozen-field are present is discussed. Section 9 considers briefly the application of these ideas to the theory of vortons, i.e. rotational disturbances that propagate without change of structure in an unbounded fluid. The paper concludes with a brief discussion, with comment on the possible development of the results in the context of turbulence.

scholarly journals Stability of transonic jet with strong shock in two-dimensional steady compressible Euler flows

2015 ◽  
Vol 258 (7) ◽  
pp. 2572-2617 ◽  
Author(s):  
Vaibhav Kukreja ◽  
Hairong Yuan ◽  
Qiuju Zhao
Keyword(s):  
Strong Shock ◽  
Two Dimensional ◽  
Compressible Euler ◽  
Euler Flows
Sign in / Sign up

Export Citation Format

Share Document