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Sungik Sohn - One of the best experts on this subject based on the ideXlab platform.
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self similar roll up of a Vortex Sheet driven by a shear flow hyperbolic double spiral
Physics of Fluids, 2016Co-Authors: Sungik SohnAbstract:In this paper, we consider the roll-up of an infinite Vortex Sheet and investigate its self-similar behavior. We address the question of whether the unsteady double spiral produced by the curvature singularity in finite time exhibits self-similar behavior. We find a self-similar solution of the double-spiral Vortex Sheet, which in fact, is a hyperbolic spiral. The radius of the spiral asymptotically grows with time and is proportional to the inverse of the angle from the spiral center. The curvature singularity plays the role of triggering spiral formation, but the source of vorticity for forming the spiral is the initial vorticity of the Sheet. We show analytically that the self-similar solution satisfies the Birkhoff-Rott equation asymptotically. Numerical validation is also given by applying the blob-regularization model to the Vortex Sheet with a periodic perturbation. We examine various asymptotic relations among primitive variables for the spiral turns and find agreement of numerical results of the ...
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self similar roll up of a Vortex Sheet driven by a shear flow hyperbolic double spiral
Physics of Fluids, 2016Co-Authors: Sungik SohnAbstract:In this paper, we consider the roll-up of an infinite Vortex Sheet and investigate its self-similar behavior. We address the question of whether the unsteady double spiral produced by the curvature singularity in finite time exhibits self-similar behavior. We find a self-similar solution of the double-spiral Vortex Sheet, which in fact, is a hyperbolic spiral. The radius of the spiral asymptotically grows with time and is proportional to the inverse of the angle from the spiral center. The curvature singularity plays the role of triggering spiral formation, but the source of vorticity for forming the spiral is the initial vorticity of the Sheet. We show analytically that the self-similar solution satisfies the Birkhoff-Rott equation asymptotically. Numerical validation is also given by applying the blob-regularization model to the Vortex Sheet with a periodic perturbation. We examine various asymptotic relations among primitive variables for the spiral turns and find agreement of numerical results of the inner turns of the Vortex Sheet with the analytic solution. Our study clarifies contrasting results on the existence of the self-similar double-spiral of a large structure in the previous studies. Our solution also suggests the possibility of bifurcation of the self-similar solution of the double-spiral as the Sheet strength varies.
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two Vortex blob regularization models for Vortex Sheet motion
Physics of Fluids, 2014Co-Authors: Sungik SohnAbstract:Evolving Vortex Sheets generally form singularities in finite time. The Vortex blob model is an approach to regularize the Vortex Sheet motion and evolve past singularity formation. In this paper, we thoroughly compare two such regularizations: the Krasny-type model and the Beale-Majda model. It is found from a linear stability analysis that both models have exponentially decaying growth rates for high wavenumbers, but the Beale-Majda model has a faster decaying rate than the Krasny model. The Beale-Majda model thus gives a stronger regularization to the solution. We apply the blob models to the two example problems: a periodic Vortex Sheet and an elliptically loaded wing. The numerical results show that the solutions of the two models are similar in large and small scales, but are fairly different in intermediate scales. The Sheet of the Beale-Majda model has more spiral turns than the Krasny-type model for the same value of the regularization parameter δ. We give numerical evidences that the solutions of the two models agree for an increasing amount of spiral turns and tend to converge to the same limit as δ is decreased. The inner spiral turns of the blob models behave differently with the outer turns and satisfy a self-similar form. We also examine irregular motions of the Sheet at late times and find that the irregular motions shrink as δ is decreased. This fact suggests a convergence of the blob solution to the weak solution of infinite regular spiral turns.
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Simple and efficient numerical methods for Vortex Sheet motion with surface tension
International Journal for Numerical Methods in Fluids, 2013Co-Authors: Suyeon Shin, Sungik Sohn, Woonjae HwangAbstract:SUMMARY We present two simple and efficient explicit methods for the Vortex Sheet with surface tension. The first one is the standard point Vortex method, which has been known to be unstable in the presence of surface tension, due to spurious growth of waves of high modes. We show, for the first time, that the standard point Vortex method is able to calculate the Vortex Sheet motion with surface tension by employing a Fourier filtering. The second method is a modification of the Pullin method using central differences for numerical differentiations. This method is more convenient to implement than other spectral methods and is free from the aliasing instability. We give a linear stability analysis for the numerical methods and show results for the long-time evolution of the Vortex Sheet. We also propose a new redistribution procedure to control point clustering, by setting limits of minimum and maximum distances between neighboring points. This procedure is found to be very efficient for long-time computations of the explicit methods. Copyright © 2013 John Wiley & Sons, Ltd.
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singularity formation and nonlinear evolution of a viscous Vortex Sheet model
Physics of Fluids, 2013Co-Authors: Sungik SohnAbstract:We study Dhanak's model [J. Fluid Mech. 269, 265 (1994)]10.1017/S0022112094001552 of a viscous Vortex Sheet in the sharp limit, to investigate singularity formations and present nonlinear evolutions of the Sheets. The finite-time singularity does not disappear by giving viscosity to the Vortex Sheet, but is delayed. The singularity in the sharp viscous Vortex Sheet is found to be different from that of the inviscid Sheet in several features. A discontinuity in the curvature is formed in the viscous Sheet, similarly as the inviscid Sheet, but a cusp in the Vortex Sheet strength is less sharpened by viscosity. Exponential decay of the Fourier amplitudes is lost by the formation of the singularity, and the amplitudes of high wavenumbers exhibit an algebraic decay, while in the inviscid Vortex Sheet, the algebraic decay of the Fourier amplitudes is valid from fairly small wavenumbers. The algebraic decay rate of the viscous Vortex Sheet is approximately −2.5, independent of viscosity, which is the same rate a...
Marcus Herrmann - One of the best experts on this subject based on the ideXlab platform.
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a eulerian level set Vortex Sheet method for two phase interface dynamics
Journal of Computational Physics, 2005Co-Authors: Marcus HerrmannAbstract:A Eulerian fixed grid approach to simulate the dynamics of two-phase interfaces in the presence of surface tension forces is presented. This level set/Vortex Sheet method consists of a simplified system of equations that contain individual source terms describing the relevant physical processes at the phase interface explicitly. Hence, this approach provides a framework that will allow for a simplified subsequent modeling of phase interface dynamics in turbulent environments. In the presented level set/Vortex Sheet method, the location and the motion of the phase interface are captured by a level set equation. Topological changes of the interface, like breakup or merging, are thus handled automatically. Assuming that all vorticity is concentrated at the phase interface, the phase interface itself constitutes a Vortex Sheet with varying Vortex Sheet strength. The Eulerian transport equation for the Vortex Sheet strength is derived by combining its Lagrangian formulation with the level set equation. The resulting differential equation then contains source terms accounting for the stretching of the interface and the influence of surface tension, thus allowing for a detailed study of each effect individually. The results of three test problems, namely the roll-up of a Vortex Sheet without surface tension, the growth of the Kelvin-Helmholtz instability in the linear regime, and the long-time evolution of the Kelvin-Helmholtz instability are presented.
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modeling primary breakup a three dimensional eulerian level set Vortex Sheet method for two phase interface dynamics
Center for Turbulence Research Annual Research Briefs 2003, 2003Co-Authors: Marcus HerrmannAbstract:This paper is divided into four parts. First, the level set/Vortex Sheet method for three-dimensional two-phase interface dynamics is presented. Second, the LSS model for the primary breakup of turbulent liquid jets and Sheets is outlined and all terms requiring subgrid modeling are identified. Then, preliminary three-dimensional results of the level set/Vortex Sheet method are presented and discussed. Finally, conclusions are drawn and an outlook to future work is given.
Takashi Sakajo - One of the best experts on this subject based on the ideXlab platform.
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harnessing the kelvin helmholtz instability feedback stabilization of an inviscid Vortex Sheet
Journal of Fluid Mechanics, 2018Co-Authors: Bartosz Protas, Takashi SakajoAbstract:In this investigation, we use a simple model of the dynamics of an inviscid Vortex Sheet given by the Birkhoff–Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback control. As actuation, we consider two arrays of point sinks/sources located a certain distance above and below the Vortex Sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff–Rott equation linearized around the flat-Sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next, we design a state-based linear-quadratic regulator stabilization strategy, where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff–Rott equation and the chosen form of actuation, an issue that is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of non-negligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyse the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid Vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers for relatively large initial perturbations, provided that the actuation has sufficiently many degrees of freedom.
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harnessing the kelvin helmholtz instability feedback stabilization of an inviscid Vortex Sheet
arXiv: Fluid Dynamics, 2017Co-Authors: Bartosz Protas, Takashi SakajoAbstract:In this investigation we use a simple model of the dynamics of an inviscid Vortex Sheet given by the Birkhoff-Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback control. As actuation we consider two arrays of point sinks/sources located a certain distance above and below the Vortex Sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff-Rott equation linearized around the flat-Sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next we design a state-based LQR stabilization strategy where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff-Rott equation and the chosen form of actuation, an issue which is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and of the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of nonnegligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyze the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid Vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers for relatively large initial perturbations, provided the actuation has sufficiently many degrees of freedom.
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harnessing the kelvin helmholtz instability feedback stabilization of an inviscid Vortex Sheet
arXiv: Fluid Dynamics, 2017Co-Authors: Bartosz Protas, Takashi SakajoAbstract:In this investigation we use a simple model of the dynamics of an inviscid Vortex Sheet given by the Birkhoff-Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback. First, we demonstrate using analytical computations that the Birkhoff-Rott equation linearized around the flat-Sheet configuration is in fact controllable when a pair of point vortices located on both sides of the Sheet is used as actuation. On the other hand, this system is not controllable when the actuation has the form of a pair of sinks/sources with zero net mass flux. Next we design a state-based LQR stabilization strategy where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff-Rott equation and the chosen form of actuation, an issue which is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals a range of different behaviors varying from a purely exponential decay of the perturbation energy to its significant transient growth as the number of actuators is reduced. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of nonnegligible amplitude can be effectively stabilized when a sufficient number of actuators is used in an aligned arrangement. We also thoroughly analyze the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid Vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers, provided the actuation has sufficiently many degrees of freedom and the initial perturbations are not too large.
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Motion of a Vortex Sheet on a sphere with pole vortices
Physics of Fluids, 2004Co-Authors: Takashi SakajoAbstract:We consider the motion of a Vortex Sheet on the surface of a unit sphere in the presence of point vortices fixed on north and south poles. Analytic and numerical research revealed that a Vortex Sheet in two-dimensional space has the following three properties. First, the Vortex Sheet is linearly unstable due to Kelvin–Helmholtz instability. Second, the curvature of the Vortex Sheet diverges in finite time. Last, the Vortex Sheet evolves into a rolling-up doubly branched spiral, when the equation of motion is regularized by the Vortex method. The purpose of this paper is to investigate how the curvature of the sphere and the presence of the pole vortices affect these three properties mathematically and numerically. We show that some low spectra of disturbance become linearly stable due to the pole vortices and thus the singularity formation tends to be delayed. On the other hand, however, the Vortex Sheet, which is regularized by the Vortex method, acquires complex structure of many rolling-up spirals.
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motion of a Vortex Sheet on a sphere with pole vortices
Preprint Series of Department of Mathematics Hokkaido University, 2003Co-Authors: Takashi SakajoAbstract:We cons i der the motion of a Vortex Sheet on the surface of a unit sphere in the presence of point vortices xed on north and south poles.Analytic and numerical research revealed that a Vortex Sheet in two-dimensional space has the following three properties.First,the Vortex Sheet is linearly unstable due to Kelvin-Helmholtz instability.Second,the curvature of the Vortex Sheet diverges in nite time.Last,the Vortex Sheet evolves into a rolling-up doubly branched spiral,when the equation of motion is regularized by the Vortex method.The purpose of this article is to investigate how the curvature of the sphere and the presence of the pole vortices a
Katsunobu Nishihara - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear motion of a current-Vortex Sheet in MHD Richtmyer-Meshkov instability
Journal of Physics: Conference Series, 2016Co-Authors: Chihiro Matsuoka, Katsunobu Nishihara, Takayoshi SanoAbstract:Nonlinear motion of a Vortex Sheet applied an initially homogeneous magnetic field is investigated using the Vortex method. We show that the existence of a Vortex Sheet induces a strong magnetic field and non-uniform current at the fluid interface. This system can be a model of MHD Richtmyer-Meshkov instability (MHD RMI). We present some interfacial profiles in MHD RMI and discuss the difference between the pure Vortex Sheet in ordinary RMI and the current-Vortex Sheet in MHD RMI, in which it is shown that the magnetic field suppresses the roll-up of the Vortex Sheet. We also find that the strong magnetic field amplification is caused by the stretch of the interface.
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analytical and numerical study on a Vortex Sheet in incompressible richtmyer meshkov instability in cylindrical geometry
Physical Review E, 2006Co-Authors: Chihiro Matsuoka, Katsunobu NishiharaAbstract:Motion of a fluid interface in the Richtmyer-Meshkov instability in cylindrical geometry is examined analytically and numerically. Nonlinear stability analysis is performed in order to clarify the dependence of growth rates of a bubble and spike on the Atwood number and mode number $n$ involved in the initial perturbations. We discuss differences of weakly and fully nonlinear evolution in cylindrical geometry from that in planar geometry. It is shown that the analytical growth rates coincide well with the numerical ones up to the neighborhood of the break down of numerical computations. Long-time behavior of the fluid interface as a Vortex Sheet is numerically investigated by using the Vortex method and the roll up of the Vortex Sheet is discussed for different Atwood numbers. The temporal evolution of the curvature of a bubble and spike for several mode numbers is investigated and presented that the curvature of spikes is always larger than that of bubbles. The circulation and the strength of the Vortex Sheet at the fully nonlinear stage are discussed, and it is shown that their behavior is different for the cases that the inner fluid is heavier than the outer one and vice versa.
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fully nonlinear evolution of a cylindrical Vortex Sheet in incompressible richtmyer meshkov instability
Physical Review E, 2006Co-Authors: Chihiro Matsuoka, Katsunobu NishiharaAbstract:Fully nonlinear motion of a circular interface in incompressible Richtmyer--Meshkov instability is investigated by treating it as a nonuniform Vortex Sheet between two different fluids. There are many features in cylindrical geometry such as the existence of two independent spatial scales, radius and wavelength, and the ingoing and outgoing growth of bubbles and spikes. Geometrical complexities lead to the results that nonlinear dynamics of the Vortex Sheet is determined from the inward and outward motion rather than bubbles and spikes, and that the nonlinear growth strongly depends on mode number.
Chihiro Matsuoka - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear motion of a current-Vortex Sheet in MHD Richtmyer-Meshkov instability
Journal of Physics: Conference Series, 2016Co-Authors: Chihiro Matsuoka, Katsunobu Nishihara, Takayoshi SanoAbstract:Nonlinear motion of a Vortex Sheet applied an initially homogeneous magnetic field is investigated using the Vortex method. We show that the existence of a Vortex Sheet induces a strong magnetic field and non-uniform current at the fluid interface. This system can be a model of MHD Richtmyer-Meshkov instability (MHD RMI). We present some interfacial profiles in MHD RMI and discuss the difference between the pure Vortex Sheet in ordinary RMI and the current-Vortex Sheet in MHD RMI, in which it is shown that the magnetic field suppresses the roll-up of the Vortex Sheet. We also find that the strong magnetic field amplification is caused by the stretch of the interface.
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nonlinear motion of non uniform current Vortex Sheets in mhd richtmyer meshkov instability
Physica Scripta, 2014Co-Authors: Chihiro MatsuokaAbstract:Nonlinear motion of a Vortex Sheet initially applied a homogeneous magnetic field is investigated using the Vortex blob method. We show that the existence of a Vortex Sheet induces a strong magnetic field and non-uniform current at the fluid interface. This system can be a model of incompressible magnetohydrodynamic Richtmyer–Meshkov instability (MHD-RMI). We present some interfacial profiles in MHD-RMI and discuss the difference between the pure Vortex Sheet in standard RMI and the current–Vortex Sheet in MHD-RMI, in which it is shown that the magnetic field suppresses the roll-up of the Vortex Sheet. We also find that the strong magnetic field amplification is caused by the stretch of the interface.
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analytical and numerical study on a Vortex Sheet in incompressible richtmyer meshkov instability in cylindrical geometry
Physical Review E, 2006Co-Authors: Chihiro Matsuoka, Katsunobu NishiharaAbstract:Motion of a fluid interface in the Richtmyer-Meshkov instability in cylindrical geometry is examined analytically and numerically. Nonlinear stability analysis is performed in order to clarify the dependence of growth rates of a bubble and spike on the Atwood number and mode number $n$ involved in the initial perturbations. We discuss differences of weakly and fully nonlinear evolution in cylindrical geometry from that in planar geometry. It is shown that the analytical growth rates coincide well with the numerical ones up to the neighborhood of the break down of numerical computations. Long-time behavior of the fluid interface as a Vortex Sheet is numerically investigated by using the Vortex method and the roll up of the Vortex Sheet is discussed for different Atwood numbers. The temporal evolution of the curvature of a bubble and spike for several mode numbers is investigated and presented that the curvature of spikes is always larger than that of bubbles. The circulation and the strength of the Vortex Sheet at the fully nonlinear stage are discussed, and it is shown that their behavior is different for the cases that the inner fluid is heavier than the outer one and vice versa.
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fully nonlinear evolution of a cylindrical Vortex Sheet in incompressible richtmyer meshkov instability
Physical Review E, 2006Co-Authors: Chihiro Matsuoka, Katsunobu NishiharaAbstract:Fully nonlinear motion of a circular interface in incompressible Richtmyer--Meshkov instability is investigated by treating it as a nonuniform Vortex Sheet between two different fluids. There are many features in cylindrical geometry such as the existence of two independent spatial scales, radius and wavelength, and the ingoing and outgoing growth of bubbles and spikes. Geometrical complexities lead to the results that nonlinear dynamics of the Vortex Sheet is determined from the inward and outward motion rather than bubbles and spikes, and that the nonlinear growth strongly depends on mode number.
