1. Lecture Objectives: Newton-Raphson method
Discuss Airpak and other CFD software
Boundary Conditions

2. Newton-Raphson method (example of Jacobean solver)
Faster convergence
Used in many professional tools (MathCAD, EES, MatLab, Mathematica, etc)
More complex for programming
Requires linear solver
Based on Taylor-Series Expansion
You need first derivative for each function to create the Jacobean matrix
Equations in the form where all side are on one side of equality sign
Our simple example: X-Y/2= → X-Y/2+1=0
X2-Y= → X2-Y+3=0

3. Newton-Raphson method
Section 6.11 of handouts
Our simple example:
f1 = X-Y/2+1=0
f2 = X2-Y+3=0
Steps:
0) Find derivatives d(f1)/dX = , d(f1)/dY =-1/2
d(f2)/dX =2X , d(f2)/dY =-1
1) Initial guess: Y(0)=2, X(0)=2
2) Find f1(Y(0),X(0))=2-2/2+1=2
f2(Y(0),X(0))=22-2+3=5
3) Using derivatives and guess values find the Jacobean matrix
4) Solve the matrix using linear solver and find DX and DY
5) Find Y(1)=Y(0)+ DY, X(1)=X(0)+ DX,
Repeat step (2) with Y(1) and X(1) … Follow the procedure till convergence
Unknowns
(correction Dxi)
Jacobean matrix
Function values
for guessed
variables

4. Surface boundaries wall functions
Wall surface
Introduce velocity
temperature and concentration
Use wall functions to model the micro-flow in the vicinity of surface
Using relatively large mesh (cell) size.

5. Surface boundaries wall functions
Course mesh distribution in the vicinity of surface Y
Wall surface
Velocity in the first cell will depend on the distance y.

6. Surface boundary conditions and log-wall functions
E is the integration constant and
y* is a length scale
Friction velocity
u+=V/Vt
y*=(n/Vt)
y+=y/y*
k- von Karman's constant
The assumption of ‘constant shear stress’ is used here.
Constants k = 0.41 and E = 8.43 fit well to a range of boundary layer flows.
Surface cells
Turbulent profile
Laminar sub-layer

7. K-e turbulence model in boundary layer
Wall shear stress
Eddy viscosity V
Wall function for e
Wall function for k

8. Modeling of Turbulent Viscosity in boundary layer
forced convection
natural convection

9. Temperature and concentration gradient in boundary layer
Depend on velocity field
Temperature
q=h(Ts-Tair)
Concentration
F=hc(Cs-Cair/m)
m=Dair/Ds
m- segregation coefficient
h = f(V) = f(k,e)
Tair Ts
Into source term
of energy equation
hC = f(V, material prop.)
Cair Cs