Amplitude Blowup in Radial Isentropic Euler Flow

2020 ◽  
Vol 80 (6) ◽  
pp. 2472-2495
Author(s):  
Helge Kristian Jenssen ◽  
Charis Tsikkou
Keyword(s):  
Euler Flow

On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex

1998 ◽  
Vol 374 ◽  
pp. 285-303 ◽  
Author(s):  
D. W. MOORE ◽  
D. I. PULLIN
Keyword(s):  
Finite Difference ◽  
Compressible Flows ◽  
Numerical Solutions ◽  
Flow Direction ◽  
Fluid Velocity ◽  
Major Axis ◽  
Sonic Point ◽  
Euler Flow ◽  
Compressible Euler ◽  
Spherical Vortex

We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M∞=U∞/C∞, where U∞, C∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh–Janzen expansion in powers of M2∞ to order M4∞. When M∞>0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M∞≈0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

scholarly journals Numerical simulation of modification of non-metallic inclusions by calcium treatment in the argon-stirred ladle

2019 ◽  
Vol 116 (5) ◽  
pp. 515
Author(s):  
Edgar Ivan Castro-Cedeno ◽  
Alain Jardy ◽  
Benjamin Boissiere ◽  
Jean Lehmann ◽  
Pascal Gardin ◽  
...  
Keyword(s):  
Flow Model ◽  
Fluid Dynamic ◽  
Steel Ladle ◽  
Local Concentration ◽  
Two Phase ◽  
Euler Flow ◽  
Metallic Inclusions ◽  
Dissolved Species ◽  
Secondary Steelmaking ◽  
Technological Challenge

Nowadays, depending on the steel grade, Ca treatment with the aim of modifying the morphology and melting temperature of non-metallic inclusions is performed in the secondary steelmaking process. The addition of calcium to steel melts rises a technological challenge because at steelmaking temperatures Ca has the tendency to vaporize from the ladle. Efforts are actively pursued in developing solutions that increase Ca yield and improve repeatability of results from treatment to treatment. This work presents a two-phase Euler-Euler flow model of a steel ladle with gas stirring through bottom porous plugs. The model considers that before gas exits through the ladle top, some Ca is transferred from the gas to the liquid steel. The yield is thus defined as the ratio between the Ca transferred to the steel and the total calcium injected into the ladle. The fluid-dynamic calculations are coupled with ArcelorMittal thermodynamic software CEQCSI to get the evolution of the local concentration of dissolved species and non-metallic inclusions assuming local thermodynamic equilibrium. Industrial trials have been performed at one of ArcelorMittal’s facilities with the aim of obtaining data to validate the model. Samples of steel were taken before, during, and after the Ca injection treatment. The total Ca content and the inclusion populations in the steel samples can be compared against the results given by the model, as well as the measured and calculated Ca yield.

The mean electromotive force generated by elliptic instability

2012 ◽  
Vol 707 ◽  
pp. 111-128 ◽  
Author(s):  
K. A. Mizerski ◽  
K. Bajer ◽  
H. K. Moffatt
Keyword(s):  
Electromotive Force ◽  
Analytical Techniques ◽  
Accretion Discs ◽  
Dynamo Action ◽  
Flow Models ◽  
Euler Flow ◽  
Resonant Instability ◽  
Astrophysical Objects ◽  
The Mean ◽  
Tidal Deformations

AbstractThe mean electromotive force (EMF) associated with exponentially growing perturbations of an Euler flow with elliptic streamlines in a rotating frame of reference is studied. We are motivated by the possibility of dynamo action triggered by tidal deformation of astrophysical objects such as accretion discs, stars or planets. Ellipticity of the flow models such tidal deformations in the simplest way. Using analytical techniques developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312) in the limit of small elliptic (tidal) deformations, we find the EMF associated with each resonant instability described by Mizerski & Bajer (J. Fluid Mech., vol. 632, 2009, pp. 401–430), and for arbitrary ellipticity the EMF associated with unstable horizontal modes. Mixed resonance between unstable hydrodynamic and magnetic modes and resonance between unstable and oscillatory horizontal modes both lead to a non-vanishing mean EMF which grows exponentially in time. The essential conclusion is that interactions between unstable eigenmodes with the same wave-vector $\mathbi{k}$ can lead to a non-vanishing mean EMF, without any need for viscous or magnetic dissipation. This applies generally (and not only to the elliptic instabilities considered here).

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