Lecture Objectives: Review discretization methods for advection diffusion equation Accuracy Numerical Stability Unsteady-state CFD Explicit vs. Implicit method HW2 Turbulence http://network.bepress.com/physical-sciences-and-mathematics/physics/fluid-dynamics/ http://www.transportation.anl.gov/engines/multi_dim_model_les.html

Steady–state 1D example I) X direction If Vx > 0, If Vx < 0, Convection term - Upwind-scheme: W P dxw dxe E and a) and Dx Diffusion term: w e b) When mesh is uniform: DX = dxe = dxw Assumption: Source is constant over the control volume c) Source term:

Advection diffusion equation 1-D, steady-state Dx N Dx N+1 Different notation: Dx General equation

1D example multiple (N) volumes N unknowns 1 2 3 i N-1 N Equation for volume 1 N equations Equation for volume 2 …………………………… Equation matrix: For 1D problem 3-diagonal matrix

3D problem Equation in the general format: H N W P E S L Wright this equation for each discretization volume of your discretization domain A F 60,000 elements 60,000 cells (nodes) N=60,000 x = 60,000 elements 7-diagonal matrix This is the system for only one variable ( ) When we need to solve p, u, v, w, T, k, e, C system of equation is very larger

Boundary conditions for CFD application - indoor airflow Real geometry Model geometry Where are the boundary Conditions?

CFD ACCURACY Depends on airflow in the vicinity of Boundary conditions 1) At air supply device 2) In the vicinity of occupant 3) At room surfaces Detailed modeling limited by computer power

Surface boundaries thickness 0.01-20 mm for forced convection Wall surface W use wall functions to model the flow in the vicinity of surface Using relatively large mesh (cell) size.

Airflow at air supply devices momentum sources Complex geometry - Δ~10-4m We can spend all our computing power for one small detail

Diffuser jet properties High Aspiration diffuser D D L L How small cells do you need? We need simplified models for diffusers

Simulation of airflow in In the vicinity of occupants How detailed should we make the geometry? Peter V. Nielsen

General Transport Equation Unsteady-state H N Equation in the algebraic format: W P E S L We have to solve the system matrix for each time step ! Transient term: Are these values for step  or + ? Unsteady-state 1-D If: -  - explicit method - + - implicit method

General Transport Equation unsteady-state 1-D Fully explicit method: Implicit method: Value form previous time step (known value) Make the difference between - Calculation for different time step - Calculation in iteration step