Lecture Objectives: - Numerics

Finite Volume Method - Conservation of  for the finite volume w e w e l h n s P E W xx xx xx - Finite volume is a fixed space in the flow domain with imaginary boundaries that allow the fluid to flow in and out. - Integral conservation of the quantities such as mass, momentum and energy. Divide the whole computation domain into sub-domains One dimension: 

General Transport Equation -3D problem steady-state W E N S H L P Equation for node P in the algebraic format:

1-D example of discretization of general transport equation Steady state 1dimension (x): Point W and E represent the cell center of the west and east neighbors of cell P and w, e the neighboring surfaces. Integrating with Gaussian theorem on this control volume gives: e w P E W xx xwxw xexe To obtain the equations for the value at point P, assumptions are used to convert the surface values to the center values.

Steady–state 1D example e w P E W xx xwxw xexe Upwind scheme: X direction and Diffusion term: Convection term When mesh is uniform:  X =  x e =  x w Central-difference scheme:

1D example After substitution a), b) and c) into I): same We started with partial differential equation: and developed algebraic equation: We can write this equation in general format: Equation coefficients Unknowns

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

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

Boundary conditions in CFD application in 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 Wall surface W use wall functions to model the micro-flow in the vicinity of surface Using relatively large mesh (cell) size mm for forced convection thickness

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

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

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

AIRPAK Software

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