Search results for "Prandtl’s equation"

showing 4 items of 4 documents

Singularity formation for Prandtl’s equations

2009

Abstract We consider Prandtl’s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method. We classify Van Dommelen and Shen’s singularity as a cubic root singularity. We introduce a class of initial data, uniformly bounded in H 1 , which have a dipole singularity in the complex plane. These data lead to a solution blow-up whose time can be made arbitrarily short within the class. This is numerical evidence of the ill-posedness of the Prandtl equations in H 1 . The presence of a small viscosity in the streamwise direction changes the behavior of the singularities. They stabilize at a distanc…

Complex singularitiePrandtl numberFOS: Physical sciencesRegularizing viscositySeparationPhysics::Fluid Dynamicssymbols.namesakeViscosityMathematics - Analysis of PDEsSingularityFOS: MathematicsUniform boundednessSpectral methodSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematicsMathematical analysisStatistical and Nonlinear PhysicsMathematical Physics (math-ph)Condensed Matter PhysicsPrandtl–Glauert transformationPrandtl’s equationsymbolsGravitational singularitySpectral methodComplex planeAnalysis of PDEs (math.AP)Blow–up timePhysica D: Nonlinear Phenomena
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High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex

2013

Abstract We compute the solutions of Prandtl’s and Navier–Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl’s equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier–Stokes solution at different Reynolds numbers Re = 103–105, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale interaction, which is visible only for moderate-high Re numbers Re = 104–105. We also investi…

D'Alembert's paradoxGeneral Computer SciencePrandtl numberMathematics::Analysis of PDEsFOS: Physical sciencesPhysics::Fluid Dynamicssymbols.namesakeMathematics - Analysis of PDEsHagen–Poiseuille flow from the Navier–Stokes equationsFOS: MathematicsSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematicsMathematical analysisGeneral EngineeringFluid Dynamics (physics.flu-dyn)Reynolds numberPhysics - Fluid DynamicsMathematical Physics (math-ph)Non-dimensionalization and scaling of the Navier–Stokes equationsBoundary layersymbolsTurbulent Prandtl numberReynolds-averaged Navier–Stokes equationsBoundary layer Unsteady separation Navier Stokes solutions Prandtl’s equation High Reynolds number flows.Analysis of PDEs (math.AP)
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Singularity tracking for Camassa-Holm and Prandtl's equations

2006

In this paper we consider the phenomenon of singularity formation for the Camassa-Holm equation and for Prandtl's equations. We solve these equations using spectral methods. Then we track the singularity in the complex plane estimating the rate of decay of the Fourier spectrum. This method allows us to follow the process of the singularity formation as the singularity approaches the real axis.

Essential singularityNumerical AnalysisCamassa–Holm equationApplied MathematicsComplex singularitieMathematical analysisPrandtl numberPrandtl’s equationsSingularity functionPrandtl–Glauert transformationComputational Mathematicssymbols.namesakeSpectral analysiSingularitysymbolsCamassa–Holm equationSpectral methodComplex planeMathematicsBoundary layer separation
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Singularities for Prandtl's equations.

2006

We used a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our toll is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformely bounded in H^1.

Prandtl’s equations Separation Spectral methods Complex singularities Blow–up time Regularizing viscosity.Settore MAT/07 - Fisica Matematica
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