Fluid mechanics, turbulent flow and turbulence ... - Chalmers

1 Fluid mechanics, turbulent flow and turbulencemodelingLars DavidsonDivision of Fluid DynamicsDepartment of Mechanics and Maritime SciencesChalmers University of TechnologySE-412 96 G oteborg, 19, 2022 AbstractThis course material is used in two courses in the International Master s pro-grammeApplied Mechanicsat Chalmers . The two courses areTME226 Mechanics of fluids(Chapters 1-10, part of Chapter 11), andMTF271 turbulence Modeling(part ofChapter 9 and 10, Chapters 11-27). MSc students who follow these courses aresupposed to have taken one basic course in Fluid eBook can be downloaded lada/ Fluid courses in the MSc programme are presented lada/ MSc programme is presented Motion, Eulerian, Lagrangian, material derivative.

2 What is the difference betweendv2dtand v2 t?.. Viscous stress, pressure.. Strain rate tensor, vorticity.. Product of a symmetric and antisymmetric tensor.. Deformation, rotation.. Irrotational and rotational flow.. Ideal vortex line.. Shear flow.. Eigenvalues and eigenvectors: physical interpretation.. 252 Governing flow The Navier-Stokes equation.. The continuity equation.. The momentum equation.. The energy equation.. Transformation of energy.. Left side of the transport equations.. Material particle vs. control volume (Reynolds Transport Theorem). 323 Solutions to the Navier-Stokes equation: three The Rayleigh problem.

3 Flow between two plates.. Curved plates.. Flat plates.. Force balance, channel flow.. Balance equation for the kinetic energy.. Two-dimensional boundary layer flow over flat plate.. Momentum balance, boundary layer.. 434 Vorticity equation and potential Vorticity and rotation.. The vorticity transport equation in three dimensions.. The vorticity transport equation in two dimensions.. Boundary layer thickness from the Rayleigh problem.. Potential flow.. The Bernoulli equation.. Complex variables for potential solutions of plane flows.. zn.. flow.. flow.. over a wedge and flow in a concave corner.

4 Analytical solutions for a line source.. Analytical solutions for a vortex line.. Analytical solutions for flow around a cylinder.. Analytical solutions for flow around a cylinder with Magnus effect.. The flow around an airfoil.. 705 Introduction.. turbulent scales.. Energy spectrum.. The cascade process created by vorticity.. 786 turbulent mean Time averaged Navier-Stokes.. Boundary-layer approximation.. Wall region in fully developed channel flow.. Reynolds stresses in fully developed channel flow.. Boundary layer.. 917 Probability density functions938 Transport equations for turbulent kinetic Rules for time averaging.

5 What is the difference betweenv 1v 2andv 1v 2?.. What is the difference betweenv 21andv 12?.. Show that v1v 21= v1v 21.. Show that v1= v1.. The ExactkEquation.. Spectral transfer dissipation vs. true viscous dissipation, The ExactkEquation: 2D Boundary Layers.. Spatial vs. spectral energy transfer.. The overall effect of the transport terms.. The transport equation for vi vi/2.. 1069 Transport equations for Reynolds Source terms.. Reynolds shear stress vs. the velocity gradient.. 11310 Two-point correlations.. Auto correlation.. Taylor s hypothesis of frozen turbulence .. 12011 Reynolds stress models and two-equation Mean flow equations.

6 Flow equations.. Temperature equation.. The exactv iv jequation.. The exactv i equation.. Thekequation.. The equation.. The Boussinesq assumption.. Modeling assumptions.. Production terms.. Diffusion terms.. Dissipation term, ij.. Slow pressure-strain term.. Rapid pressure-strain term.. Wall model of the pressure-strain term.. Thek model.. The modeledv iv jequation with IP model.. Algebraic Reynolds Stress Model (ASM).. Explicit ASM (EASM or EARSM).. Derivation of the Explicit Algebraic Reynolds StressModel (EARSM) The linear assumption ofaij.. Boundary layer flow.. Wall boundary conditions.. Wall Functions.

7 Low-Re Number turbulence Models.. Low-Rek Models.. Wall boundary Condition fork.. Different ways of prescribing at or near the wall.. 15912 Reynolds stress models vs. eddy-viscosity Stable and unstable stratification.. Curvature effects.. Stagnation flow.. RSM/ASM versusk models.. 16613 Two-component limit.. 16814 Non-linear Eddy-viscosity Models17015 The V2F Modified V2F model.. Realizable V2F model.. To ensure thatv2 2k/3.. 17716 The SST Model17817 Overview of RANS models18318 Large Eddy Time averaging and filtering.. Differences between time-averaging (RANS) and space filtering (LES) Resolved & SGS scales.

8 The box-filter and the cut-off filter.. Highest resolved wavenumbers.. Subgrid model.. Smagorinsky model vs. mixing-length model.. Energy path.. SGS kinetic energy.. LES vs. RANS.. The dynamic model.. The test filter.. 2D filtering.. 3D filtering.. Stresses on grid, test and intermediate level.. Numerical dissipation.. Scale-similarity Models.. The Bardina Model.. Redefined terms in the Bardina Model.. A dissipative scale-similarity .. Forcing.. Numerical method.. RANS vs. LES.. One-equationksgsmodel.. Smagorinsky model derived from theksgsequation.. A dynamic one-equation model.. A Mixed Model Based on a One-Eq.

9 Model.. Applied LES.. Resolution requirements.. 20619 URANS: Unsteady turbulence Modeling.. Discretization.. 21020 DES: DES based on two-equation models.. DES based on thek SST model.. DDES.. 21621 Hybrid Momentum equations in hybrid LES-RANS.. The one-equation hybrid LES-RANS model.. 21922 The SAS Resolved motions in unsteady.. The von K arm an length scale.. The second derivative of the velocity.. Evaluation of the von K arm an length scale in channel flow.. 22223 The PANS PANS as a hybrid LES-RANS model.. The interface conditions at the RANS-LES interface.. Zonal PANS: different treatments of the RANS-LES interface.

10 The Interface Condition.. Modeling the Interface.. Interface Model 1.. Interface Model 2.. Interface Model 3.. Interface Model 4.. A new formulation offkfor the PANS model.. from the equivalence criterion.. The IDD-PANS model.. LES length scale.. The IDDES model.. Equivalence between PANS and DES/IDDES.. PANS equations.. IDDES equations.. PITM equations.. Deriving an expression forfkfor PANS.. Deriving an expression forfkfor PITM.. turbulence with convection.. PANS/PITM equations.. DES equations.. 24224 The PITM RANS mode.. LES mode.. 24425 Hybrid LES/RANS for Introduction.. Reynolds-Averaging Navier-Stokes equations: RANS.