1. Objective
Numerical methods
Finite volume

2. Finite Volume Method - Conservation of f for the finite volume
Divide the whole computation domain into sub-domains
One dimension: n h W P dx E dx w e s Dx l e w
- 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. f

3. Convection term dxw P dxe W E Dx – Central difference scheme:
- Upwind-scheme:
If Vx>0
and
If Vx<0
and

4. Diffusion term W
dxw P
dxe E Dx w e

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

6. 1D example - uniform mesh
After substitution a), b) and c) into I):
We started with partial differential equation:
same
and developed algebraic equation:
We can write this equation in general format:
Unknowns
Equation coefficients

7. 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

8. 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 larger

9. Iteration method
Alternative to use matrix solver tool is to use iterations
You can use excel if you are not familiar with matrix solver tools
General Iteration Procedure:
1) Express equation in explicit form
2) Guess initial values
3) Substitute initial values and calculate new values
4) Substitute new values and calculate newer values
6) Repeat step 4) until convergence is achieved
example
Iterations -residual
Value: T1
Residual
initial guess 22
iteration 1 23
iteration 2
23.25
iteration 3
iteration 4
iteration 5 ……
---
Iteration 98 2 2 2
Difference of value
between two iteration 2 2

10. Numerical instability divergency
divergence
variable
solution
convergence
iteration

11. Navier Stokes Equations
Continuity equation
This velocities that constitute advection coefficients: F=rV
Momentum x
Momentum y
Momentum z
Pressure is in momentum equations
which already has one unknown
In order to use linear equation solver we need to solve two problems:
find velocities that constitute in advection coefficients
2) link pressure field with continuity equation

12. Pressure and velocities in NS equations
How to find velocities that constitute in advection coefficients?
For the first step use Initial guess
And for next iterative steps use
the values from previous iteration

13. Pressure and velocities in NS equations
How to link pressure field with continuity equation?
SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm W Dx P Dx E Dx Aw Ae
Aw=Ae=Aside
We have two additional equations for y and x directions
The momentum equations can be solved only when the pressure field is given or is
somehow estimated. Use * for estimated pressure and the corresponding velocities

14. SIMPLE algorithm
Guess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L
1) For this pressure field solve system of equations: x: y:
………………..
……………….. z:
Solution is:
2) The pressure and velocity correction
P = P* + P’
P’ – pressure correction
For all nodes E,W,N,S,…
V = V* + V’
V’ – velocity correction
Substitute P=P* + P’ into momentum equations (simplify equation) and obtain
V’=f(P’)
V = V* + f(P’)
3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V
4) Solve T , k , e equations

15. SIMPLE algorithm start Guess p* p=p*
Step1: solve V* from momentum equations
Step2: introduce correction P’ and express V = V* + f(P’)
Step3: substitute V into continuity equation solve P’ and then V
Step4: Solve T , k , e equations no
Converged
(residual check)
yes
end

16. Other methods SIMPLER SIMPLEC variation of SIMPLE PISO
COUPLED - use Jacobeans of nonlinear velocity functions to form
linear matrix ( and avoid iteration )

17. Relaxation Relaxation with iterative solvers:
When the equations are nonlinear
it can happen that you get divergency
in iterative procedure for solving considered
time step
divergence
variable
solution
convergence
Solution is Under-Relaxation:
Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor
For our example Y*in iteration 101=f·Y(100)+(1-f) ·Y(99)
f = [0-1] – under-relaxation -stabilize the iteration
f = [1-2] – over-relaxation - speed-up the convergence
iteration
Value which is should be used for the next iteration
Under-Relaxation is often required when you have nonlinear equations!

18. Example of relaxation (example from homework assignment)
Example: Advection diffusion equation, 1-D, steady-state, 4 nodes
1) Explicit format: 1 2 3 4
2) Guess initial values: 3)
Substitute and calculate: 4)
Substitute and calculate:
Substitute and calculate:
………………………….