Lectures on CFD

Fundamental Equations

Dr. Sreenivas JayantiDepartment of Chemical Engineering

IIT-Madras

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

Governing Equations for Incompressible, Constant Property Flow

• Continuity equation :

• Momentum conservation equation:

• Energy conservation equation:

0 x

u

i

i

2

i2

1 g

i ji i

j i j

u uu up

t x x x

2v

T(uT) T p pC C k

t

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

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

THE STAGGERED GRID

P - cell

V - cell

U - cell

i I i+1I-1i-1I-2 I+1

J

J-1

J+1

j

j+1

CASE STUDY (Hand-calculation)

Fully Developed Flow through a Triangular Duct

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

Cz

p

y

w

x

w

1

2

2

2

2

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

Cz

pw

1

dVCdSwndVwCVCSCV )(

VolCjy

wi

x

wjSiSdSwn

sidesideyx

CS

.)(

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

The CFD Solution: Discretization of Equation

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

The CFD Solution: Set of Algebraic Equations

• Put in a matrix form Aw =b and solve using standard methods to get wi

The CFD Solution

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

The CFD Solution: Variants• To get more accuracy, divide into more number of cells and apply the same

template of CFD solution

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!