# Miss

Topics: Equations, Fundamental physics concepts, Nonlinear system Pages: 7 (847 words) Published: May 18, 2013
COMPUTATIONAL FLUID DYNAMICS (MKM411)
Dr. M. Sharifpur mohsen.sharifpur@up.ac.za

April 2nd
Department of Mechanical and Aeronautical Engineering

University of Pretoria

2013

Test 1 Perusal Friday April 5th 12:00 to 14:00
CFD Training Courses The schedule is already posted on clickUP (find your group in the clickUP) ANSYS-FLUENT (Group1) FLoEFD ANSYS-FLUENT (Group2) STAR CCM+ (Group 1) STAR CCM+ (Group 2) April 3rd 9:00-17:00 April 10th 9:00-17:00 April 11th 8:00-10:15 and then

11:30 -17:00

April 17th 9:00-17:00 April 24th 9:00-17:00

Finite Difference Method (FDM)
Computational Solution

   

Finding; - Governing (Differential) Equations - Initial & Boundary Conditions

Discretization of the Domain

Changing Differential Eqs. to Algebraic Eqs.

Internal Nodes and Boundary Nodes

Solving the system of Algebraic equations

Example:

Example: A chimney
Inside: Convection Ti  300 oC , hi  70 W/m 2 .o C Outside: Convection + Radiation TO  20 oC , hO  21 W/m 2 .o C

TO , hO Tsky

Tsky  260 K Inside Flow Section: 20 cm x 20 cm
Wall Thickness: 20 cm

TO , hO Tsky

Ti , hi

TO , hO Tsky

if x  y  10 cm
k  1.4 W/m. o C and   0.9
Using the advantage of symmetry to find:

TO , hO Tsky

Temp. at all Nodes by FDM

In the Representative Section: How many internal nodes do we have? Only one; Node 4 It can change to 3; Nodes 3,4 & 5 How many nodes do they have the same conditions ? Only Two; Nodes 7 and

8

Node 1

All sides

Q  eVelement  0

Taking x  y  l

Node 2

All sides

Q  eVelement  0

Taking x  y  l

Nodes 3,4 and 5 (Interior nodes)

Node 4 : T3  T5  T7  T2  4T4  0 Node 3 : T4  T4  T1  T6  4T3  0 Node 3 : 2T4  T1  T6  4T3  0 Node 5 : T4  T8  T4  T8  4T5  0 Node 5 : 2T8  2T4  4T5  0

Node 6

All sides

Q  eVelement  0

Taking x  y  l

Nonlinear equation

(T6  273) 4

Node 7

All sides

Q  eVelement  0

Taking x  y  l

Node 8

All sides

Q  eVelement  0

Same conditions of node 7 45, 67, 78 & 89

Node 7

Node 8

Node 9

All sides

Q  eVelement  0

Taking x  y  l

Whenever we have radiation then the Temp. must be in Kelvin for radiation part (T  273)

Is this a system of Linear or Nonlinear equations?

Two ways for solving; First, because of the number of the nodes is 9 We can use an equation solver

Tair = 20 ºC & Tsky = -13 ºC

Tair = 20 ºC Tsky = -13 ºC

300 ºC

Tair = 20 ºC Tsky = -13 ºC

Tair = 20 ºC & Tsky = -13 ºC

Second Way , changing the nonlinear equations
to linear and then with iteration finding the answer

For Example for Node 7

(T74 ) K  (T7  273) 4  (T7  273)(T7  273)3

(T )  (T7  273)  (T7  273) (T7  273)     4 7 K 4 3

A  (T7  273)
P 4 7 K

3

From Pr evious iteration

(T )  (T7  273)  A (T7  273)
4 P

Now , we have a system of linear equations and then we can solve it with Gauss-Seidel iteration You should iterate till no changes in the answers,  new previous      Ti  Ti   i 1   n

First choose a bigger

Iterative method;
Node 0 

a0T0  b0T1  S 0

S 0  b0T1 T0  a0

Node 1 

a1T0  b1T1  c1T2  S1 a2T1  b2T2  c2T3  S 2

T1  F (T2 )
T2  F (T3 )

Node 2 

. . .
Node M 

TM 1  F (TM )
aM TM 1  bM TM  S M

TM  ...

Whenever the number of nodes are a lot we will do the iteration for x-direction first and then for y and z. After z again x, y, z ….. Converge.

HW4 (30 Marks)

0.5

Using the advantage of symmetry and FDM to find Temp. at all Nodes by making the nonlinear equations to linear ones and using Iteration .

(due date April 25th , 2013 at 12:30 in class )
Inside: Convection Ti  300 oC , hi  70 W/m 2 .o C Outside: Convection + Radiation

TO , hO Tsky

TO  20 oC , hO  21 W/m 2 .o C Tsky  260 K Inside Flow Section: 20 cm x 20 cm Wall Thickness:...