Lectures on CFD Fundamental Equations Dr. Sreenivas Jayanti Department 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:

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 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

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

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

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!