Message-ID: <1025623053.12194.1594632059379.JavaMail.confluence@full.dc.univ-lorraine.fr> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_12193_304813597.1594632059379" ------=_Part_12193_304813597.1594632059379 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html ENGS9AA.F TRANSITION TO TURBULENCE

# ENGS9AA.F TRANSITION TO TURBULENCE

 = ENGS9Aa.F<= /p>Transition to turbulence in thermoconvectionand aerodynamics= Credits: 2 ECT= S= Duration: 21 hou= rs Semester: S9 Person in cha= rge: Emmanuel Plaut, professor Keywords: nonlinear phenomena, stability, instability, bifur= cation theory Prerequisites= : fluid dynamics elementary course, numerical methods= Objectives= : learn relevant non= linear phenomena in hydrodynamics with heat transfers (thermoconvecti= on) and aerodynami= cs; learn some methods to model these phenomena. Program and contents: The transition to spatio-tempora= l complexity and turbulence in fluid dynamics= , which is intrinsically nonlinear, is studied by focusing on two families of systems. This is also an occasio= n to enrich the knowledge and know-how of the students in general fluid mec= hanics. The families of systems= studied are: Natural convection= or `thermoconvection'The emphasis is on the `Rayleigh-B=C3=A9nard' configuration in = extended geometry, where convection r= olls set in under the influence of a vertical do= wnward temperature gradient, through an instabilit= y. This is an occasion to introduce the thermal buoyancy, = the Oberbeck - Boussinesq approximation,  the methods of the linear<= /strong> and weakly nonlinear stability analy= ses, and to evidence a = supercritical pitchfork bifurcation that leads t= o increased heat transfers. The secondary instabilities are also briefly discussed. The chaos is introduced both with the (historical) Lorenz mo= del and the (more realistic) example of the Rayleigh-B=C3=A9nard convection in a square cell<= /span>, where chaotic large-scale flow reversals occur. Other = geometries and systems ar= e also briefly discussed, for instance, the `differential heating' configuration, where the b= asic temperature gradient is horizontal, therefore, thermoconvection sets i= n directly, as it is often the case for heating in buildings. <= strong>Open shear flowsT= he emphasis is on the Tollmienn-Schlichting waves<= /em> that set in through an = instability of channel flows. In this different context, the linear and weakly nonlinear stability analyses already introduced are performed now with numerical computations (spectral meth= od), to evidence a subcritical Hopf bifurcation. The further transition t= o turbulence is also briefly discussed, for channel flows, and also boundary layer flows and airfoils. Openings concerning aerodynamics and wind energy are finally prese= nted.Importantly, the stability analyses methods and the theory of= bifurcations (or `catastrophes') introduced here are, in fac= t, relevant for any nonlinear deterministic model; applications also exist = in other domains of mechanical engineering, in physics, etc... Please check = the web page of this module on htt= p://emmanuelplaut.perso.univ-lorraine.fr/t2t : it sketches the planning of this module, gives the lect= ure notes and instructions, etc... In particular, you will use Mathematica= to perform formal (symbolic) and numerical computations on your laptop. Abilities : Levels Description and operational vocabulary Know Understand <= /strong> Apply Analyze Summarize Assess Evaluation: Written test Continuous assessment Oral presentation Project Written report
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