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# 640 Outline for M E 521 at PSU

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 38 views.

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Date Created: 02/06/15

Detailed Outline M E 522 Foundations of Fluid Mechanics II Prepared by Professor J M Cimbala Penn State University continued from M E 521 VII Laminar Boundary Layers F 2D Laminar Free Shear Layers 1 Introduction 2 2D laminar jet a integral CV analysis b differential analysis c similarity solution 1 example problem e discussion G AXisymmetric Boundary Layers 1 Introduction 2 AXisymmetric boundary layer equations 3 The Mangler transformation a introduction b derivation c example 4 General axisymmetric boundary layers a introduction b axisymmetric BL equations 5 The thin axisymmetric jet H ThreeDimensional Boundary Layers 1 Introduction 2 Boundary layer equations 3 How to choose a 3D coordinate system a developable surface b intrinsic coordinates c computer generation 4 Summary Why 3D BLs are harder than 2D BLs a outer ow b coordinate system c BL equations 1 secondary ow 5 Secondary ow a de nition b example problem at plate with transverse pressure gradient c class demonstration the teacup experiment 1 practical application of secondary ow the cyclone separator I Numerical Solutions 1 Introduction 2 G1id options a structured b unstructured c hybrid 3 Convergence a residuals b underrelaxation factors c grid adaption Latest revision 02 January 2008 VIII Instability A Introduction 1 Examples of instability a ball on the wall b unstable ows 1 pipe ow 2 jets 3 Blasius boundary layer 4 ow over a circular cylinder 5 freely rotatable cylinder with splitter plate 6 thermal instability 7 surface tensiondriven Benard convection 8 TaylorCouette ow 9 KelvinHelmholtz instability 2 Notation B Linear Stability Theory 1 De nition 2 Procedure for linear stability analysis 3 Simple example C Method of Normal Modes 1 Introduction 2 Thermal instability the Benard problem 3 Other examples a centrifugal instability D Stability of Locally Parallel Flows 1 Introduction 2 Linear stability analysis 3 Squire s threorem 4 OrrSommerfeld equation a derivation b boundary conditions c solutions and examples 5 The Rayleigh equation a derivation b conditions for instability c typical solutions d example far wake of a circular cylinder E Transition IX Turbulence The Final Frontier A Introduction What is Turbulence 1 De nition of turbulence 2 Characteristics of turbulence a randomness or irregularity b nonlinearity c diffusivity d 3D uctuating vorticity at wide range of scales e dissipation 3 Other features of turbulence a turbulence is a continuum phenomenon b turbulence is a property of uid ows not of uids B Notation and Terminology 1 Notation 2 Types of averages a time average b ensemble average c phase average 3 Other de nitions and rules a the ensemble average of any uctuating component is zero b commutative rules for ensemble averages c commutative rules do not work for multiplication of uctuations d statistical de nitions e turbulence intensity f turbulent kinetic energy tke g intermittancy h homogeneous turbulence i isotropic turbulence j inhomogeneous turbulence C Equations of Motion for Turbulent Flow 1 NavierStokes equations 2 Reynolds decomposition 3 Comparison of laminar and turbulent mean ows 4 Some comments about the Reynolds stress tensor a it is a 9component tensor b it is symmetric c sign of the normal components d special case isotropic turbulence e eg turbulent shear ow f notation 5 The closure problem a mean ow quantities b Reynolds stress transport equation 6 Mean and turbulent kinetic energy equations a mean kinetic energy equation b turbulent kinetic energy equation c comments about the tke equation D Order of Magnitude Analysis of Turbulence 1 Introduction a notation b example laminar at plate boundary layer 2 Eddy scales and the energy cascade a largest eddies b intermediatesized eddies c smallest eddies 3 Kolmogorov s contribution a Kolmogorov s universal equilibrium hypothesis b Kolmogorov microscales 4 How to estimate the turbulent dissipation rate a order of magnitude estimate b some examples 5 The turbulent Reynolds number a de nition b examples 6 Can turbulent ows be solved exactly E Turbulence Models 1 Introduction a Why do we need turbulence models b equations of motion c desired attributes of a turbulence model d classi cation of turbulence models 1 zeroequation models 2 oneequation models 3 twoequation models 35 algebraic stress models ASM 4 Reynolds stress models RSM 5 large eddy simulation LES 2 The eddy Viscosity a eddy Viscosity for 2D boundary layer ows b eddy Viscosity for general 3D ows 3 Closure for algegraic zeroeq turbulence models a how to nd the mixing length b examples of mixing length models 1 free shear ows 2 wallbounded shear ows 3 von Karman s mixing length 4 BaldwinLomax Model c summary of zeroequation eddy Viscosity mixing length turbulence models 3a Closure for halfequation turbulence models 4 Closure for oneequation turbulence models a introduction b the modeled tke equation 5 Closure for twoequation turbulence models a introduction b exact transport equation for dissipation rate c modeled transport equation for dissipation rate d the near wall region 6 Closure for Reynolds stress turbulence models a introduction b equations c summary of RSM F Turbulent Flow Solutions 1 Turbulent channel ow a analytical analysis b approximate analysis using layers c wall functions d solutions using turbulence models 2 Free shear ows a introduction b equations of motion c turbulent jets d turbulent far wake

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