Journal articles: 'Beltrami flow'

1

Nadirashvili, Nikolai. "Liouville theorem for Beltrami flow." Geometric and Functional Analysis 24, no. 3 (May 3, 2014): 916–21. http://dx.doi.org/10.1007/s00039-014-0281-8.

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Zuykov, A. L., G. V. Orekhov, and V. V. Volshanik. "ANALYTICAL MODEL OF GROMEKA — BELTRAMI FLOW." Vestnik MGSU, no. 4 (April 2013): 150–59. http://dx.doi.org/10.22227/1997-0935.2013.4.150-159.

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3

Mahajan, S. M., and Z. Yoshida. "Double Curl Beltrami Flow: Diamagnetic Structures." Physical Review Letters 81, no. 22 (November 30, 1998): 4863–66. http://dx.doi.org/10.1103/physrevlett.81.4863.

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4

González, Rafael, Gustavo Sarasua, and Andrea Costa. "Kelvin waves with helical Beltrami flow structure." Physics of Fluids 20, no. 2 (February 2008): 024106. http://dx.doi.org/10.1063/1.2840196.

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Zeng, Yong, and Zhibing Zhang. "Applications of a formula on Beltrami flow." Mathematical Methods in the Applied Sciences 41, no. 10 (March 25, 2018): 3632–42. http://dx.doi.org/10.1002/mma.4851.

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Dascal, Lorina, and Nir A. Sochen. "A Maximum Principle for Beltrami Color Flow." SIAM Journal on Applied Mathematics 65, no. 5 (January 2005): 1615–32. http://dx.doi.org/10.1137/s0036139903430835.

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7

Wong, Tsz Wai, and Hong-kai Zhao. "Computing Surface Uniformization Using Discrete Beltrami Flow." SIAM Journal on Scientific Computing 37, no. 3 (January 2015): A1342—A1364. http://dx.doi.org/10.1137/130939183.

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8

IQBAL, M., and P. K. SHUKLA. "Beltrami fields in a hot electron–positron–ion plasma." Journal of Plasma Physics 78, no. 3 (February 6, 2012): 207–10. http://dx.doi.org/10.1017/s0022377812000050.

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AbstractA possibility of relaxation of relativistically hot electron and positron (e − p) plasma with a small fraction of hot or cold ions has been investigated analytically. It is observed that a strong interaction of plasma flow and field leads to a non-force-free relaxed magnetic field configuration governed by the triple curl Beltrami (TCB) equation. The triple curl Beltrami (TCB) field composed of three different Beltrami fields gives rise to three multiscale relaxed structures. The results may have the strong relevance to some astrophysical and laboratory plasmas.

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González, R. "CHANDRASEKHAR-KENDALL MODES ANALYSIS OF AN HELICAL BELTRAMI FLOW." Anales AFA 23, no. 3 (September 17, 2013): 21–24. http://dx.doi.org/10.31527/analesafa.2013.23.3.21.

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10

Kwang-Hua Chu, Z. "Ball-like structures of the Beltrami flow field." ZAMM 85, no. 2 (February 25, 2005): 147–51. http://dx.doi.org/10.1002/zamm.200310161.

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11

Lui, Lok Ming, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau. "Optimization of Surface Registrations Using Beltrami Holomorphic Flow." Journal of Scientific Computing 50, no. 3 (July 1, 2011): 557–85. http://dx.doi.org/10.1007/s10915-011-9506-2.

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12

Iqbal, M. "Relaxed states in electron-depleted electronegative dusty plasmas with two-negative ion species." Journal of Plasma Physics 80, no. 1 (December 13, 2013): 59–65. http://dx.doi.org/10.1017/s0022377813000925.

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AbstractThe relaxation of an electron-depleted electronegative dusty plasma with two-negative ions is investigated. When the ratio of canonical vorticities to corresponding flows of all the plasma species is the same and all inertial and non-inertial forces are present, the relaxed state appears as a double Beltrami magnetic field which is the superposition of two force-free relaxed states. The numerical results show that highly diamagnetic relaxed magnetic fields can be obtained by controlling the flow and vorticities through a single Beltrami parameter. The study is useful to investigate the creation of diamagnetic plasma configurations which are considered to be very important in the context of nuclear fusion.

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13

Labropulu, F. "Generalized Beltrami flows and other closed-form solutions of an unsteady viscoelastic fluid." International Journal of Mathematics and Mathematical Sciences 30, no. 5 (2002): 271–82. http://dx.doi.org/10.1155/s0161171202109185.

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We study flows of an unsteady non-Newtonian fluid by assuming the form of the vorticity a priori. The two forms that have been considered are∇2ψ=F(t)ψ+G(t), which is known as the generalized Beltrami flow and∇2ψ=f(t)ψ+g(t)y.

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Słomka, Jonasz, Piotr Suwara, and Jörn Dunkel. "The nature of triad interactions in active turbulence." Journal of Fluid Mechanics 841 (February 26, 2018): 702–31. http://dx.doi.org/10.1017/jfm.2018.108.

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Generalised Navier–Stokes (GNS) equations describing three-dimensional active fluids with flow-dependent narrow spectral forcing have been shown to possess numerical solutions that can sustain significant energy transfer to larger scales by realising chiral Beltrami-type chaotic flows. To rationalise these findings, we study here the triad truncations of polynomial and Gaussian GNS models focusing on modes lying in the energy injection range. Identifying a previously unknown cubic invariant for the triads, we show that their asymptotic dynamics reduces to that of a forced rigid body coupled to a particle moving in a magnetic field. This analogy allows us to classify triadic interactions by their asymptotic stability: unstable triads correspond to rigid-body forcing along the largest and smallest principal axes, whereas stable triads arise from forcing along the middle axis. Analysis of the polynomial GNS model reveals that unstable triads induce exponential growth of energy and helicity, whereas stable triads develop a limit cycle of bounded energy and helicity. This suggests that the unstable triads dominate the initial relaxation stage of the full hydrodynamic equations, whereas the stable triads determine the statistically stationary state. To test whether this hypothesis extends beyond polynomial dispersion relations, we introduce and investigate an alternative Gaussian active turbulence model. Similar to the polynomial case, the steady-state chaotic flows in the Gaussian model spontaneously accumulate non-zero mean helicity while exhibiting Beltrami statistics and upward energy transport. Our results suggest that self-sustained Beltrami-type flows and an inverse energy cascade may arise generically in the presence of flow-dependent narrow spectral forcing.

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15

Kwang-Hua Chu, R. "Possible Ball-like Formations of the Beltrami Flow Field." Meccanica 39, no. 2 (April 2004): 181–86. http://dx.doi.org/10.1023/b:mecc.0000005117.57782.90.

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16

Prugger, Artur, and Jens D. M. Rademacher. "Explicit superposed and forced plane wave generalized Beltrami flows." IMA Journal of Applied Mathematics 86, no. 4 (June 29, 2021): 761–84. http://dx.doi.org/10.1093/imamat/hxab015.

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Abstract We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier–Stokes equations on ${{{\mathbb{R}}}}^n$, as well as the rotating Boussinesq equations on ${{{\mathbb{R}}}}^3$. We cast these solutions are superpositions of certain linear plane waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave vectors and flow directions. For $n\leqslant 3$, these are explicit examples for generalized Beltrami flows. We show that forcing terms of corresponding plane wave type yield explicit solutions by linear variation of constants. We work in Eulerian coordinates and distinguish the two situations of vanishing and of gradient nonlinear terms, where the nonlinear terms modify the pressure. The methods that we introduce to find explicit solutions in nonlinear fluid models can also be used in other equations with material derivative. Our approach offers another view on known explicit solutions of different fluid models from a plane wave perspective and provides transparent nonlinear interactions between different flow components.

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17

Balan, Vladimir, and Jelena Stojanov. "Anisotropic image evolution of Synge-Beil type." Filomat 33, no. 4 (2019): 1071–79. http://dx.doi.org/10.2298/fil1904071b.

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The anisotropic Beltrami framework is introduced as an extended promising tool in image processing. In this framework, image surface evolution is governed by an anisotropic flow determined by an energy Lagrangian of Polyakov type. The Synge-Beil flow is derived, and applicative aspects illustrate the developed theoretical results.

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18

Abolarinwa, Abimbola. "Eigenvalues of the weighted Laplacian under the extended Ricci flow." Advances in Geometry 19, no. 1 (January 28, 2019): 131–43. http://dx.doi.org/10.1515/advgeom-2018-0022.

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Abstract Let ∆φ = ∆ − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e−φ dν on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.

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19

Liang, Zhizheng, and Youfu Li. "Beltrami flow in Hilbert space with applications to image denoising." Journal of Electronic Imaging 21, no. 4 (December 12, 2012): 043019. http://dx.doi.org/10.1117/1.jei.21.4.043019.

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20

Ming Lui, Lok, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing Tung Yau. "Detection of shape deformities using Yamabe flow and Beltrami coefficients." Inverse Problems & Imaging 4, no. 2 (2010): 311–33. http://dx.doi.org/10.3934/ipi.2010.4.311.

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21

Wong, Tsz Wai, and Hong-kai Zhao. "Computation of Quasi-Conformal Surface Maps Using Discrete Beltrami Flow." SIAM Journal on Imaging Sciences 7, no. 4 (January 2014): 2675–99. http://dx.doi.org/10.1137/14097104x.

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22

Lui, Lok Ming, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau. "Erratum to: Optimization of Surface Registrations Using Beltrami Holomorphic Flow." Journal of Scientific Computing 51, no. 1 (September 16, 2011): 258. http://dx.doi.org/10.1007/s10915-011-9541-z.

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23

Dascal, Lorina, Adi Ditkowski, and Nir A. Sochen. "On the Discrete Maximum Principle for the Beltrami Color Flow." Journal of Mathematical Imaging and Vision 29, no. 1 (October 6, 2007): 63–77. http://dx.doi.org/10.1007/s10851-007-0025-6.

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24

DZIUK, GERHARD. "CONVERGENCE OF A SEMI-DISCRETE SCHEME FOR THE CURVE SHORTENING FLOW." Mathematical Models and Methods in Applied Sciences 04, no. 04 (August 1994): 589–606. http://dx.doi.org/10.1142/s0218202594000339.

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Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in L∞((0, T), L2(ℝ/2π)) ∩ L2((0, T) H1(ℝ/2π)). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature flow which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can be easily solved.

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25

Groves, M. D., and J. Horn. "A variational formulation for steady surface water waves on a Beltrami flow." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2234 (February 2020): 20190495. http://dx.doi.org/10.1098/rspa.2019.0495.

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This paper considers steady surface waves ‘riding’ a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be formulated as two equations for two scalar functions of the horizontal spatial coordinates, namely the elevation η of the free surface and the potential Φ defining the gradient part (in the sense of the Hodge–Weyl decomposition) of the horizontal component of the tangential fluid velocity there. These equations are written in terms of a non-local operator H ( η ) mapping Φ to the normal fluid velocity at the free surface, and are shown to arise from a variational principle. In the irrotational limit, the equations reduce to the Zakharov–Craig–Sulem formulation of the classical three-dimensional steady water-wave problem, while H ( η ) reduces to the familiar Dirichlet–Neumann operator.

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26

Nishiyama, Takahiro. "A relaxation method for constructing a Beltrami flow in a bounded domain." Journal of Mathematical Physics 46, no. 8 (August 2005): 083102. http://dx.doi.org/10.1063/1.1996440.

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27

Zheligovsky, V. A. "A kinematic magnetic dynamo sustained by a Beltrami flow in a sphere." Geophysical & Astrophysical Fluid Dynamics 73, no. 1-4 (December 1993): 217–54. http://dx.doi.org/10.1080/03091929308203629.

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28

Ng, Tsz Ching, Xianfeng Gu, and Lok Ming Lui. "Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow." Journal of Scientific Computing 60, no. 2 (October 20, 2013): 249–75. http://dx.doi.org/10.1007/s10915-013-9791-z.

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29

Ershkov, Sergey V. "About existence of stationary points for the Arnold–Beltrami–Childress (ABC) flow." Applied Mathematics and Computation 276 (March 2016): 379–83. http://dx.doi.org/10.1016/j.amc.2015.12.038.

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30

ELCRAT, ALAN R., BENGT FORNBERG, and KENNETH G. MILLER. "Steady axisymmetric vortex flows with swirl and shear." Journal of Fluid Mechanics 613 (October 1, 2008): 395–410. http://dx.doi.org/10.1017/s002211200800342x.

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A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moffatt, tubes of vorticity extending to infinity and Beltrami flows. When there is a spherical obstacle we have found multiple solutions for each set of parameters. Flows are found by numerically solving the Bragg–Hawthorne equation using a non-Newton-based iterative procedure which is robust in its dependence on an initial guess.

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31

Abolarinwa, Abimbola. "Differential Harnack estimates for conjugate heat equation under the Ricci flow." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550063. http://dx.doi.org/10.1142/s1793557115500631.

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We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

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32

Al-Saif, Abdul-Sattar J., and Assma J. Harfash. "A New Approximate Analytical Solutions for Two- and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier-Stokes Equations." Journal of Applied Mathematics 2019 (January 1, 2019): 1–19. http://dx.doi.org/10.1155/2019/3084394.

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In this work, the kinetically reduced local Navier-Stokes equations are applied to the simulation of two- and three-dimensional unsteady viscous incompressible flow problems. The reduced differential transform method is used to find the new approximate analytical solutions of these flow problems. The new technique has been tested by using four selected multidimensional unsteady flow problems: two- and three-dimensional Taylor decaying vortices flow, Kovasznay flow, and three-dimensional Beltrami flow. The convergence analysis was discussed for this approach. The numerical results obtained by this approach are compared with other results that are available in previous works. Our results show that this method is efficient to provide new approximate analytic solutions. Moreover, we found that it has highly precise solutions with good convergence, less time consuming, being easily implemented for high Reynolds numbers, and low Mach numbers.

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33

Luo, Jiawen, Long Chen, Kuan Li, and Andrew Jackson. "Optimal kinematic dynamos in a sphere." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2233 (January 2020): 20190675. http://dx.doi.org/10.1098/rspa.2019.0675.

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A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudo-vacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 Phys. Rev. E 88 , 031001 ( doi:10.1103/physreve.88.031001 )). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere’s centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound.

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34

Chai, J., T. Wu, and L. Fang. "Single-scale two-dimensional-three-component generalized-Beltrami-flow solutions of incompressible Navier-Stokes equations." Physics Letters A 384, no. 34 (December 2020): 126857. http://dx.doi.org/10.1016/j.physleta.2020.126857.

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35

Galanti, B., N. Kleeorin, and I. Rogachevskii. "Ampère and Hall nonlinearities in the magnetic dynamo in the Arnol’d–Beltrami–Childress (ABC) flow." Physics of Plasmas 2, no. 11 (November 1995): 4161–68. http://dx.doi.org/10.1063/1.871040.

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36

Davies-Jones, Robert. "Can a Descending Rain Curtain in a Supercell Instigate Tornadogenesis Barotropically?" Journal of the Atmospheric Sciences 65, no. 8 (August 1, 2008): 2469–97. http://dx.doi.org/10.1175/2007jas2516.1.

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Abstract This paper investigates whether the descending rain curtain associated with the hook echo of a supercell can instigate a tornado through a purely barotropic mechanism. A simple numerical model of a mesocyclone is constructed in order to rule out other tornadogenesis mechanisms in the simulations. The flow is axisymmetric and Boussinesq with constant eddy viscosity in a neutrally stratified environment. The domain is closed to avoid artificial decoupling of a vortex from the storm-scale circulation. In the principal simulation, the initial condition is a balanced, slowly decaying, Beltrami flow describing an updraft that is rotating cyclonically at midlevels around a low pressure center surrounded by a concentric downdraft that revolves cyclonically but has anticyclonic vorticity. The boundary conditions are no slip on the tangential wind and free slip on the radial or vertical wind to accommodate this initial condition and to allow strong interaction of a vortex with the ground. Precipitation is released through the top above the updraft and falls to the ground near the updraft–downdraft interface in an annular curtain. The downdraft enhancement induced by the precipitation drag upsets the balance of the Beltrami flow. The downdraft and its outflow toward the axis increase low-level convergence beneath the updraft and transport air with moderately high angular momentum downward and inward where it is entrained and stretched by the updraft. The resulting tornado has a corner region with an intense axial jet and low pressure capped by a vortex breakdown and a transition to a broader vortex aloft (a tornado cyclone). A clear slot of subsiding air with anticyclonic vorticity surrounds the vortex. The vertical kinetic energy of the entire circulation declines dramatically prior to tornado formation.

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37

Schulz, Volker, and Martin Siebenborn. "Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization." Computational Methods in Applied Mathematics 16, no. 3 (July 1, 2016): 485–96. http://dx.doi.org/10.1515/cmam-2016-0009.

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AbstractWe compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace–Beltrami type metrics are compared with Steklov–Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.

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38

Słomka, Jonasz, and Jörn Dunkel. "Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids." Proceedings of the National Academy of Sciences 114, no. 9 (February 13, 2017): 2119–24. http://dx.doi.org/10.1073/pnas.1614721114.

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Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicity-driven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.

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Viúdez, Álvaro. "The Vorticity–Velocity Gradient Cofactor Tensor and the Material Invariant of the Semigeostrophic Theory." Journal of the Atmospheric Sciences 62, no. 7 (July 1, 2005): 2294–301. http://dx.doi.org/10.1175/jas3482.1.

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Abstract A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted Ξ̃) is derived. The specific potential Ξ̃, that is, Ξ̃ in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential Ξ̃. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and f-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential Ξ̃) differs conceptually from the Beltrami–Rossby–Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the f-plane approximation and on the horizontal nature of the geostrophic velocity.

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Madhura, K. R., D. S. Swetha, and S. S. Iyengar. "The impact of Beltrami effect on dusty fluid flow through hexagonal channel in presence of porous medium." Applied Mathematics and Computation 313 (November 2017): 342–54. http://dx.doi.org/10.1016/j.amc.2017.06.016.

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41

Sakajo, Takashi. "Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2224 (April 2019): 20180666. http://dx.doi.org/10.1098/rspa.2018.0666.

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A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.

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42

Dobrokhotov, S. Yu, and A. I. Shafarevich. "Tunnel splitting of the spectrum of the Beltrami-Laplace operators on two-dimensional surfaces with square integrable geodesic flow." Functional Analysis and Its Applications 34, no. 2 (April 2000): 133–34. http://dx.doi.org/10.1007/bf02482427.

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43

Davies-Jones, Robert. "Roles of Streamwise and Transverse Partial-Vorticity Components in Steady Inviscid Isentropic Supercell-Like Flows." Journal of the Atmospheric Sciences 74, no. 9 (August 31, 2017): 3021–41. http://dx.doi.org/10.1175/jas-d-16-0332.1.

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Abstract Investigations of tornadogenesis in supercells attempt to find the origin of the tornado’s large vorticity by determining vorticity generation and amplification along trajectories that enter the tornado from a horizontally uniform unstable environment. Insights into tornadogenesis are provided by finding analytical formulas for vorticity variations along streamlines in idealized, steady, inviscid, isentropic inflows of dry air imported from the environment. The streamlines and vortex lines lie in the stationary isentropic surfaces so the vorticity is 2D. The transverse vorticity component (positive leftward of the streamlines) arises from imported transverse vorticity and from baroclinic vorticity accumulated in streamwise temperature gradients. The streamwise component stems from imported streamwise vorticity, from baroclinic vorticity accrued in transverse temperature gradients, and from positive transverse vorticity that is turned streamwise in cyclonically curved flow by a “river-bend process.” It is amplified in subsiding air as it approaches the ground. Streamwise stretching propagates a parcel’s streamwise vorticity forward in time. In steady flow, vorticity decomposes into baroclinic vorticity and two barotropic parts ωBTIS and ωBTIC arising from imported storm-relative streamwise vorticity (directional shear) and storm-relative crosswise vorticity (speed shear), respectively. The Beltrami vorticity ωBTIS is purely streamwise. It explains why abundant environmental storm-relative streamwise vorticity close to ground favors tornadic supercells. It flows directly into the updraft base unmodified apart from streamwise stretching, establishing mesocyclonic rotation and strong vortex suction at low altitudes. Increase (decrease) in storm-relative environmental wind speed with height near the ground accelerates (delays) tornadogenesis as positive (negative) ωBTIC is turned into streamwise (antistreamwise) vorticity within cyclonically curved flow around the mesocyclone.

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44

Moffatt, H. K., and M. R. E. Proctor. "Topological constraints associated with fast dynamo action." Journal of Fluid Mechanics 154 (May 1985): 493–507. http://dx.doi.org/10.1017/s002211208500163x.

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The conjecture of Vainshtein & Zel'dovich (1972) concerning the existence of a fast dynamo (i.e. one whose growth rate is independent of magnetic diffusivity η in the limit η → 0) is discussed with particular reference to (i) the stretch–twist–fold cycle which can double the strength of a magnetic flux tube, and (ii) the space-periodic Beltrami flow of maximal helicity, which has been shown to be capable of space-periodic dynamo action with the same period as the velocity field, by Arnold & Korkina (1983) and by Galloway & Frisch (1984). The topological constraint associated with conservation of magnetic helicity is shown to preclude fast dynamo action unless the scale of the magnetic field is almost everywhere of order η½ as η → 0; in this case, the field structure is severely singular in the limit. A steady incompressible velocity field, quadratic in the space variables, is shown to mimic the action of the stretch–twist–fold cycle, and is proposed as a plausible candidate for fast dynamo action.

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45

Mininni, P. D., D. Rosenberg, and A. Pouquet. "Isotropization at small scales of rotating helically driven turbulence." Journal of Fluid Mechanics 699 (April 13, 2012): 263–79. http://dx.doi.org/10.1017/jfm.2012.99.

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AbstractWe present numerical evidence of how three-dimensionalization occurs at small scale in rotating turbulence with Beltrami ($\mathit{ABC}$) forcing, creating helical flow. The Zeman scale ${\ell }_{\Omega } $ at which the inertial and eddy turn-over times are equal is more than one order of magnitude larger than the dissipation scale, with the relevant domains (large-scale inverse cascade of energy, dual regime in the direct cascade of energy $E$ and helicity $H$, and dissipation) each moderately resolved. These results stem from the analysis of a large direct numerical simulation on a grid of $307{2}^{3} $ points, with Rossby and Reynolds numbers, respectively, equal to $0. 07$ and $2. 7\ensuremath{\times} 1{0}^{4} $. At scales smaller than the forcing, a helical wave-modulated inertial law for the energy and helicity spectra is followed beyond ${\ell }_{\Omega } $ by Kolmogorov spectra for $E$ and $H$. Looking at the two-dimensional slow manifold, we also show that the helicity spectrum breaks down at ${\ell }_{\Omega } $, a clear sign of recovery of three-dimensionality in the small scales.

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Tomek, Lukáš, and Karol Mikula. "Discrete duality finite volume method with tangential redistribution of points for surfaces evolving by mean curvature." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (October 18, 2019): 1797–840. http://dx.doi.org/10.1051/m2an/2019040.

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We propose a new discrete duality finite volume method for solving mean curvature flow of surfaces in ℝ3. In the cotangent scheme, which is widely used discretization of Laplace–Beltrami operator, a two-dimensional surface is usually approximated by a triangular mesh. In the cotangent scheme the unknowns are the vertices of the triangulation. A finite volume around each vertex is constructed as a surface patch bounded by a piecewise linear curve with nodes in the midpoints of the neighbouring edges and a representative point of each adjacent triangle. The basic idea of our new approach is to include the representative points into the numerical scheme as supplementary unknowns and generalize discrete duality finite volume method from ℝ2 to 2D surfaces embedded in ℝ3. To improve the quality of the mesh we use an area-oriented tangential redistribution of the grid points. We derive the numerical scheme for both closed surfaces and surfaces with boundary, and present numerical experiments. Surface evolution models are applied to construction of minimal surfaces with given set of boundary curves.

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DAVIDSON, P. A. "An energy criterion for the linear stability of conservative flows." Journal of Fluid Mechanics 402 (January 10, 2000): 329–48. http://dx.doi.org/10.1017/s002211209900693x.

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We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

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Davies-Jones, Robert. "Growth of Circulation around Supercell Updrafts." Journal of the Atmospheric Sciences 61, no. 23 (December 1, 2004): 2863–76. http://dx.doi.org/10.1175/jas-3341.1.

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Abstract A formula is derived for the rate of change of circulation around an updraft perimeter at a constant elevation. This quantity depends on the continuous propagation of points on the edge, so an expression for local propagation of the edge is obtained from Petterssen's formula for the motion of an isopleth and the vertical equation of motion. On the edge of an updraft in inviscid anelastic flow, the local propagation velocity along the outward normal is equal to the local nonhydrostatic vertical pressure-gradient force (NHVPGF) divided by the magnitude of the local vertical-velocity gradient. Circulation around an updraft perimeter increases at a rate equal to the line integral around the edge of vertical vorticity times the outward propagation velocity. Formulas are also found for the propagation of an updraft's centroid at a given height and for the acceleration of an updraft's vertical helicity. All of the formulas are tested on exact Beltrami-flow solutions of the governing equations. The relevance of two paradigms of supercell dynamics to local edge propagation and circulation growth of updrafts is evaluated by decomposing the NHVPGF into linearly and nonlinearly forced parts and examining results of supercell simulations in different types of shear. Propagation across the shear and rate of increase of circulation depend mostly on the nonlinear part of the NHVPGF (as in the vertical-wind-shear paradigm) for updrafts in nearly unidirectional shear and on the linear part (as in the helicity paradigm) for updrafts in shear that turns markedly with height.

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Migdal, Alexander. "Clebsch confinement and instantons in turbulence." International Journal of Modern Physics A 35, no. 31 (November 10, 2020): 2030018. http://dx.doi.org/10.1142/s0217751x20300185.

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The turbulence in incompressible fluid is represented as a field theory in 3 dimensions. There is no time involved, so this is intended to describe stationary limit of the Hopf functional. The basic fields are Clebsch variables defined modulo gauge transformations (symplectomorphisms). Explicit formulas for gauge invariant Clebsch measure in space of generalized Beltrami flow compatible with steady energy flow are presented. We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the “instantons and intermittency” program we started back in the 1990s.1 These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier–Stokes equation smears this delta function to the Gaussian with width [Formula: see text] at [Formula: see text] with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation [Formula: see text] around fixed loop [Formula: see text] in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations2 including pre-exponential factor [Formula: see text]. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions [Formula: see text] can be fitted at small [Formula: see text] to the parabola, coming from the Liouville theory with two parameters [Formula: see text], [Formula: see text]. At large [Formula: see text] the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of [Formula: see text] in agreement with DNS.

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Kartashov, E. M. "Model representations of heat shock in terms of dynamic thermal elasticity." Russian Technological Journal 8, no. 2 (April 14, 2020): 85–108. http://dx.doi.org/10.32362/2500-316x-2020-8-2-85-108.

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This article is devoted to mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to the specific conditions of intensive heating and cooling of solids. A scheme is proposed for deriving the compatibility equation in voltages for dynamic problems, which generalizes the well-known Beltrami-Mitchell relation for quasistatic cases. The proposed relation can be used to consider numerous special cases in the theory of thermal shock in Cartesian coordinates for both bounded canonical bodies and partially bounded ones. As a detailed study, the latter case was considered under conditions of abrupt temperature heating and cooling, thermal heating and cooling, and medium heating and cooling. Numerical experiments were carried out, and the wave nature of the propagation of thermoelastic waves was described. The effect of relaxation of the solid boundary on sudden heating and sudden cooling, which has been little studied in thermomechanics, is described. It is established that this effect influences maximum of internal temperature stresses, which depend on the parameters characterizing the elastic and thermal properties of materials, as well as the heating time and cooling time. A “compatibility equation” in displacements was proposed to study the problem of thermal shock in cylindrical and spherical coordinate systems in bodies with a radial heat flow and central symmetry. The formulation of a generalized problem in the theory of thermal shock is formulated, which is of practical and theoretical interests for many areas of science and technology.

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Journal articles on the topic 'Beltrami flow'

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