Computational Turbulent Incompressible Flow

Abstract

This is made possible by the amazing power of CC using only basic tools of calculus combined with computing.We hope the reader will have a good productive time reading the book and also trying out the G2 FEniCS software on old and new challenges. For inspiration a vast material of G2 simulations of turbulent flows is available on the web page of the book at www.bodysoulmath.org.The authors would like to thank the participants of the 2006 Geilo Winter School in Computational Mathematics, who offered valuable comments on the manuscript, and who helped in tracking down some of the mistakes.The first author would like to acknowledge the joint work with Prof. Jonathan Goodman at the Courant Institute in developing the mesh smoothing algorithm of Section 32.5.The main source of mathematicians pictures is the MacTutor History of Mathematics archive, other pictures are taken from what is assumed to be the public domain, or otherwise the sources are stated in the picture captions.

Blow up of incompressible Euler solutions

“…In this note we present evidence of blowup for the incompressible Euler equations drawing from our recent work [13] and references therein, widening the study to both before and after blowup. We compute specific viscosity solutions by EG2 and we discover wellposedness of mean-value outputs such as drag and lift (coefficients), showing that EG2 solutions are representative Euler solutions.…”

“…We summarize the evidence in [13] Since the potential solution is resolvable with finite mesh size, there is strong evidence that the turbulent blowup detection is correct.…”

“…This is motivated by the experimentally observed fact that the skin friction from non-zero tangential velocities on the boundary, tends to zero as the viscosity tends to zero. This connects back to Eulers original idea of the Euler equations with slip as a useful model of slightly viscous flow, as developed in detail in [13], in contrast to the legacy of Prandtl discarding Euler solutions because they do not satisfy no-slip boundary conditions, thus requiring the resolution of thin boundary layers of Navier-Stokes solutions, which however is impossible. We find that EG2/slip allows simulation of flows with large Reynolds numbers (Re ≥ 10 6 say) on meshes with 10 6 mesh points, rather than 10 16 as required for no-slip according to state-of-the-art CFD [16].…”

“…A basic question is what quality requirements should be put on computed solutions of differential equations to allow them to replace exact solutions. We shall see that an affirmative answer can be given in terms of Hadamard's notion of wellposedness [9] and residual-based a posteriori error estimation [13]. We first recall some basic facts of general nature and we then turn to the Euler blowup problem, starting with compressible flow and then addressing the Millennium problem of incompressible flow under vanishing viscosity.…”

“…This is analogous to the existence proof by Descartes based on the a posteriori observation: Cog-ito, ergo sum or I think, therefore I exist, (as compared to an analytical/logical a priori proof of the existence of God attempted by many philosophers without clear success). A computational foundation of thermodynamics based on EG2 for compressible flow is developed on [14].…”