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Lecture Objectives: Review discretization methods for advection diffusion equation Accuracy Numerical Stability Unsteady-state CFD Explicit vs. Implicit method HW2 Turbulence

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**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:

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**Advection diffusion equation 1-D, steady-state**

Dx N Dx N+1 Different notation: Dx General equation

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**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

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**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

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**Boundary conditions for CFD application - indoor airflow**

Real geometry Model geometry Where are the boundary Conditions?

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**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

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Surface boundaries thickness 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.

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**Airflow at air supply devices**

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

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**Diffuser jet properties**

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

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**Simulation of airflow in In the vicinity of occupants**

How detailed should we make the geometry? Peter V. Nielsen

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**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

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**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

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