scholarly journals Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems

2021 ◽  
Author(s):  
François Hamel ◽  
Nikolai Nadirashvili
Keyword(s):  
Free Boundary ◽  
Euler Equations ◽  
Free Boundary Problems ◽  
Two Dimensional ◽  
Boundary Problems ◽  
Annular Domains ◽  
Circular Flows

scholarly journals Free boundary problems in shock reflection/diffraction and related transonic flow problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140276 ◽  
Author(s):  
Gui-Qiang Chen ◽  
Mikhail Feldman
Keyword(s):  
Free Boundary ◽  
High Speed ◽  
Free Boundary Problems ◽  
Fluid Flows ◽  
Shock Reflection ◽  
Two Dimensional ◽  
Open Problems ◽  
Boundary Problems ◽  
Flow Problems ◽  
Concave Corner

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

Unsteady two-dimensional flows with free boundaries. - I. General theory

1956 ◽  
Vol 235 (1202) ◽  
pp. 375-381 ◽  
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Steady State Solution ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Boundary Problems ◽  
Initial Development ◽  
Bernoulli’S Equation ◽  
Hodograph Plane ◽  
Two Dimensional Flows

The paper contains a summary of relevant earlier work on free-boundary problems (§1) and then considers the initial development of steady two-dimensional flows. The motion of an incompressible in viscid fluid with free boundaries is considered (§2) by transforming into the hodograph plane (In q 0 / U , θ 0 ) of the steady flow. The equation of the free boundary and the velocity potential are expanded in powers of e- λt . Thus ϕ ( x, y, t ) = ϕ 0 ( x, y ) + e - λt ϕ 1 ( x, y ) + ..., where ϕ 0 is the known steady-state solution and ϕ 1 is to be determined. The exact boundary condition, which is the unsteady form of Bernoulli’s equation, is applied on the free boundary which is not taken as a streamline. A general discussion of the validity of the approach is given (§3). It is foreshadowed that for jet flow through a slit the predicted shape of the jet will probably have a kink at the nose; this is consistent with the assumptions made in the analysis.

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