Turbulent Fluid Flow daVinci [1510]

Examples Turbulent votices separating from a cylinder wake "False color image of the far field of a submerged turbulent jet" by C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands - Own work. Licensed under CC BY 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:False_color_image_of_the_far_field_of_a_submerged_turbulent_jet.jpg#mediaviewer/File:False_color_image_of_the_far_field_of_a_submerged_turbulent_jet.jpg Pyroclastic flow in Indonesia Vortices visualized with laser fluoresence Vortices in a rising jet Mixing layer https://www.tumblr.com/search/wake%20turbulence http://www.grc.nasa.gov/WWW/k-12/airplane/downwash.html http://www.dailymail.co.uk/news/article-2550079/Volcano-eruption-kills-14-people-Indonesia-ash-sent-spewing-miles-air.html http://coewww.rutgers.edu/www2/vizlab/node/82

Reynold’s Number

Karmen Vortices http://www.aps.org/units/dfd/pressroom/gallery/2009/kumar09.cfm http://envsci.rutgers.edu/~lintner/teaching.html http://mydev.info/karman.html http://nylander.wordpress.com/2005/01/11/von-karman-vortex-street/

Laminar – Turbulent Flow Regimes Free stream plume Blue dye injected into a clear pipe at different flow regimes Boundary layer obstruction Laminar – Turbulent transition with distance An album of fluid motion, Milton Van Dyke http://blog.nialbarker.com/252/slow_is_faster http://www.sciencedirect.com/science/article/pii/S0021999109001119

Flow velocity in a tidal channel Velocity in all directions = mean + variation Milne et al. [2013] http://rsta.royalsocietypublishing.org/content/371/1985/20120196

Velocity variations over 60s at a point in a channel Milne et al. [2013]

Some characteristics Flows become unstable at high Re Laminar flow becomes perturbed Perturbation damped (low Re) Perturbation grows (high Re) Vortices/eddies form, wide range of scales Rapid mixing, momentum, mass, heat Large vortices break up into smaller vortices Energy dissipation Largesmall vortex  molecular motion heat

How to characterize turbulent flows Empirical Laws Manning (channels) Darcy-Weisbach (conduits) Izbash (porous) Forchheimer (porous) dh/dx q

Darcy-Weisbach Eqn. Pressure drop in pipes is fluid density, v is average velocity, d is pipe diameter, and f is the friction factor. Low Re

Analysis Methods http://www.bakker.org/dartmouth06/engs150/10-rans.pdf

DNS Simulation LES Simulation

RANS Reynolds Average Navier Stokes

Strain Change in length/original length Change in angle

Momentum Eqn. Constitutive law for fluid Einstein Notation Navier-Stokes for Incompressible Fluid

Reynolds’ averaging, Mass Mean + fluctuation Substitute Take average Averaging rules Result

Reynolds’ averaging, Momentum Starting eq. Substitute Focus on one term Other terms Result Note the fluctuating terms

Closure Problem 6 unknowns

Turbulent Viscosity Boussinesq (1892) Turbulence dissipates energy in a way that is analogous to viscous dissipation In Turbulent flow

Classical models based on RANS Zero equation model: mixing length model. One equation model: Spalart-Almaras. 3. Two equation models: k- ε style models (standard, RNG, realizable), k- ω model 4. Seven equation model: Reynolds stress model.

k-e Method k: Turbulent kinetic energy e: Turbulence dissipation rate Cm: constant Need equations for k, e Assume k, e are conserved, use standard approach A= vc =vkr

k-e Method

Implementation

Boundary Conditions Inlet, Outlet, Wall, Open, other No slip, wall = default Specify inlet and outlet Need to specify pressure somewhere Dirichlet (specified pressure) Neumann (specify velocity) n unit vector normal to boundary u flux vector C1 known function

Options for boundary conditions Velocity (uniform) =0.001m/s Laminar Inflow = 0.001 m/s 1m entrance length Pressure = 1 No viscous stress Pressure=0 No viscous stress Need to calculate Pressure in top two cases, calculate velocity in bottom case

Wall Conditions Need b.c. for k and e Represent steep gradients at wall in turbulence http://www.bakker.org/dartmouth06/engs150/11-bl.pdf http://www.efluids.com/efluids/gallery/gallery_pages/MWSmith_3.jsp