1. Lectures on CFD Fundamental Equations
Dr. Sreenivas Jayanti
Department of Chemical Engineering
IIT-Madras

2. Equations Solved in CFD
Conservation of mass †
Conservation of linear momentum†
Conservation of energy
Equation of state
Initial and boundary conditions
† Mass and momentum conservation equations together are usually called Navier-Stokes equations

3. Governing Equations for Incompressible, Constant Property Flow
Continuity equation :
Momentum conservation equation:
Energy conservation equation:

4. Outline of the Finite Volume Method
The CFD approach
Discretization of the governing equations
Converts each partial differential equation into a set of coupled algebraic equations

5. THE CFD APPROACH Assembling the governing equations
Identifying flow domain and boundary conditions
Geometrical discretization of flow domain
Discretization of the governing equations
Incorporation of boundary conditions
Solution of resulting algebraic equations
Post-solution analysis and reformulation, if needed

6. THE STAGGERED GRID i I i+1 I-1 i-1 I-2 I+1 J J-1 J+1 j j+1 P - cell
V - cell
U - cell i I
i+1
I-1
i-1
I-2
I+1 J
J-1
J+1 j
j+1

7. CASE STUDY (Hand-calculation)
Fully Developed Flow through a Triangular Duct

8. A Simple Example
Fully developed laminar flow in a triangular duct of irregular cross-section
Flow governing equation is known:
boundary condition: w =0 on walls
Analytical solution not available for an arbitrary triangle
Velocity field can be readily obtained using CFD approach

9. A Simple Example: The CFD Solution
Governing equation put in conservation form:
Domain divided into triangles and rectangles
GE integrated over a control volume and is converted into a surface integral using Gauss’ Divergence Theorem:
Apply to each cell:
Each cell gives an algebraic equation linking the cell value with those of the neighbouring cells

10. The CFD Solution: Spatial Discretization
Divide the domain into cells and locate nodes at which the velocity has to be determined
The example below 20 nodes out of which 8 are boundary nodes with zero velocity; velocity at the other 12 needs to be calculated

11. The CFD Solution: Discretization of Equation

12. The CFD Solution: All the Equations
Application to all the cells gives a set of algebraic equations
In this case, 12 simultaneous linear algebraic equations

13. The CFD Solution: Set of Algebraic Equations
Put in a matrix form Aw =b and solve using standard methods to get wi

14. The CFD Solution
Solution of Aw =b, say, using Gauss-Seidel iterative method, gives the required velocity field.

15. The CFD Solution: Variants
To get more accuracy, divide into more number of cells and apply the same template of CFD solution

16. A Simple Example: The CFD Solution
At some point, the CFD solution becomes practically insensitive to further refining of the grid and we have a grid-independent CFD solution
Using the hydraulic diameter concept, one would have obtained a Reynolds number of 500 => an error of 23.8% if one goes by hydraulic diameter!