Turbulent Fluid Flow da Vinci 1510 Examples Turbulent

Turbulent Fluid Flow da. Vinci [1510]

Examples Turbulent votices separating from a cylinder Vortices visualized with laser fluoresence wake Pyroclastic flow in Indonesia Mixing layer Vortices in a rising jet

Reynold’s Number

Karmen Vortices http: //www. aps. org/units/dfd/pressroom/gallery/2009/kumar 09. 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 obstruction An album of fluid motion, Milton Van Dyke http: //blog. nialbarker. com/252/slow_is_faster Laminar – Turbulent transition with distance http: //www. sciencedirect. com/science/article/pii/S 0021999109001119 Boundary layer

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 60 s 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 r r Low Re is fluid density, v is average velocity, d is pipe diameter, and f is the friction factor.

Analysis Methods http: //www. bakker. org/dartmouth 06/engs 150/10 -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 1. Zero equation model: mixing length model. 2. 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 C 1 known function

Options for boundary conditions Velocity (uniform) =0. 001 m/s Pressure=0 No viscous stress Laminar Inflow = 0. 001 m/s 1 m entrance length Pressure=0 No viscous stress 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/dartmouth 06/engs 150/11 -bl. pdf http: //www. efluids. com/efluids/gallery_pages/MWSmith_3. jsp