Blow up of incompressible Euler solutions

Hoffman, Johan; Johnson, Claes

doi:10.1007/s10543-008-0184-x

Cited by 18 publications

(12 citation statements)

References 15 publications

(39 reference statements)

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“…A posteriori error estimation underlies our adaptive algorithm, in the form of a bound on the error in a functional of the solution. For a turbulent solution, pointwise convergence of an approximate solution to an exact solution cannot be expected, but mean value functionals of the solution can still be well posed (see, e.g., ).…”

Section: Adaptive Methodsmentioning

confidence: 99%

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

Nazarov

Hoffman

2012

Numerical Methods in Fluids

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SUMMARY In this paper, we present a finite element method with a residual‐based artificial viscosity for simulation of turbulent compressible flow, with adaptive mesh refinement based on a posteriori error estimation with sensitivity information from an associated dual problem. The artificial viscosity acts as a numerical stabilization, as shock capturing, and as turbulence capturing for large eddy simulation of turbulent flow. The adaptive method resolves parts of the flow indicated by the a posteriori error estimates but leaves shocks and turbulence under‐resolved in a large eddy simulation. The method is tested for examples in 2D and 3D and is validated against experimental data. Copyright © 2012 John Wiley & Sons, Ltd.

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Section: Adaptive Methodsmentioning

confidence: 99%

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

Nazarov

Hoffman

2012

Numerical Methods in Fluids

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“…We argue that the proper mathematical objects for assessing well-posedness in DFS are functionals of the computed weak solutions [37]. To represent the error in a functional of a dissipative weak solution we introduce a dual (adjoint) linearized problem, which opens for a posteriori estimation of the error in outputs of interest, such as the drag and lift of an airplane.…”

Section: Direct Finite Element Simulationmentioning

confidence: 99%

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Hoffman

Jansson

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Highlights• We present a parameter-free method for high Reynolds number turbulent flow.• The method is based on adaptive mesh optimization using adjoint techniques.• We connect the numerical approximation to dissipative weak solutions.• Turbulent boundary layers are left unresolved, no boundary layer mesh is needed.• We highlight benchmark problems for which the method is validated. AbstractThis article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier-Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form, to report on recent progress, and to discuss open problems that are subject to ongoing research.

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“…We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations [11] with slip boundary conditions initiated as potential flow develop into turbulent solutions, which are wellposed with respect to drag/lift and show substantial drag/lift. Additional evidence is presented in [25] and the related work on blowup of Euler solutions [28], separation in inviscid flow [27], and drag/lift of airplanes and cars [30,29].…”

Section: Introductionmentioning

confidence: 99%

Resolution of d’Alembert’s Paradox

Hoffman¹,

Johnson²

2008

J. Math. Fluid Mech.

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We propose a resolution of d'Alembert's Paradox comparing observation of substantial drag/lift in fluids with very small viscosity such as air and water, with the mathematical prediction of zero drag/lift of stationary irrotational solutions of the incompressible inviscid Euler equations, referred to as potential flow. We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations with slip boundary conditions initiated as potential flow, develop into turbulent solutions which are wellposed with respect to drag/lift and which show substantial drag/lift, in accordance with observations.

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Blow up of incompressible Euler solutions

Abstract: Abstract.We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L2-residuals of computed solutions under decreasing mesh size.

Cited by 18 publications

References 15 publications

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Resolution of d’Alembert’s Paradox

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