scholarly journals Global existence of shock for the supersonic Euler flow past a curved 2-D wedge

2013 ◽  
Vol 254 (5) ◽  
pp. 2076-2127 ◽  
Author(s):  
Dian Hu
Keyword(s):  
Global Existence ◽  
Euler Flow ◽  
Flow Past

The physical interpretation of the lift discrepancy in Lanchester–Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow

2007 ◽  
Vol 463 (2085) ◽  
pp. 2257-2275 ◽  
Author(s):  
Edmund Chadwick ◽  
Ali Hatam
Keyword(s):  
Flow Field ◽  
Physical Interpretation ◽  
Lift Force ◽  
Physical Explanation ◽  
Trailing Edge ◽  
Viscous Term ◽  
Euler Flow ◽  
Flow Past ◽  
Euler Model ◽  
Real Flow

Consider uniform, steady potential and incompressible flow past a fixed thin wing inclined at a small angle to the flow. An investigation is conducted into the physical interpretation and consequences of the revision by Chadwick (Chadwick 2005 Proc. R. Soc. A 461 , 1–18) of the Lanchester–Prandtl lifting wing theory in Euler flow. In the present paper, the lift is evaluated from the pressure distribution over the top and bottom surfaces together with a contribution across the trailing edge of the wing. It is shown that this contribution across the trailing edge has previously been erroneously omitted in the standard approach but confirms and provides a physical explanation for the discrepancy in the lift calculation found by Chadwick. This results in a reduction of the lift by a half, but this reduction in lift from the additional calculation is not the right answer, and instead arises from a mathematical discrepancy with the physically observed lift. The discrepancy is due to the pressure becoming singular at the trailing edge in the Euler model. The physical explanation is that in real flow the pressure is regularized by the action of viscosity and so is not singular at the trailing edge. So this lift force at the trailing edge is present in the Euler model but not in a real flow. In a real flow, the viscous effects prevent the pressure becoming singular and so there is no lift force, and consequently no large torque, concentrated at the trailing edge. That the lift force at the trailing edge has been ignored in the Lanchester–Prandtl theory in Euler flow has led to fortuitous agreement with the experimental results on real flows. This shows that the Euler model does not properly predict forces for this problem in which there are singularities (vorticity) within the flow field. We propose a revision to the Euler model by allowing a counterbalancing singular viscous velocity term to reside on the trailing vortex sheet, which is derived from the lift oseenlet. This viscous term ensures that the pressure and velocity are not singular in the flow field. The consequences for the flow due to the inclusion of this term for extending triple-deck and similar asymptotic theories to the case for flow past wings rather than aerofoils are discussed, as well as for the (ideal) high Reynolds number limit and for slender body lift.

GLOBAL MULTIDIMENSIONAL SHOCK WAVE FOR THE STEADY SUPERSONIC FLOW PAST A THREE-DIMENSIONAL CURVED CONE

2006 ◽  
Vol 04 (02) ◽  
pp. 101-132 ◽  
Author(s):  
ZHOUPING XIN ◽  
HUICHENG YIN
Keyword(s):  
Shock Wave ◽  
Boundary Condition ◽  
Supersonic Flow ◽  
Global Existence ◽  
Three Dimensional ◽  
Flow Equation ◽  
Flow Past ◽  
Existence And Stability ◽  
Whole Space ◽  
Steady Potential

In this paper, we establish the global existence and stability of a multidimensional conic shock wave for three-dimensional steady supersonic flow past an infinite cone. The flow is assumed to be hypersonic and described by a steady potential flow equation. Under an appropriate boundary condition on the curved cone, we show that a pointed shock attached at the vertex of the cone will exist globally in the whole space.

scholarly journals Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls

2017 ◽  
Vol 49 (2) ◽  
pp. 818-873 ◽  
Author(s):  
Gui-Qiang G. Chen ◽  
Jie Kuang ◽  
Yongqian Zhang
Keyword(s):  
Two Dimensional ◽  
Euler Flow ◽  
Flow Past

scholarly journals Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

2010 ◽  
Vol 199 ◽  
pp. 151-181 ◽  
Author(s):  
Gang Xu ◽  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Strong Shock ◽  
Weak Shock ◽  
Vertex Angle ◽  
Transonic Shock ◽  
Euler Flow ◽  
Flow Past ◽  
Curve Method ◽  
Sharp Cone ◽  
Supersonic Shock

AbstractIn this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

scholarly journals Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

2010 ◽  
Vol 199 ◽  
pp. 151-181 ◽  
Author(s):  
Gang Xu ◽  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Strong Shock ◽  
Weak Shock ◽  
Vertex Angle ◽  
Transonic Shock ◽  
Euler Flow ◽  
Flow Past ◽  
Curve Method ◽  
Sharp Cone ◽  
Supersonic Shock

AbstractIn this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the bookSupersonic Flow and Shock Wavesby Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

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