Introduction & conservation laws of fluid motion: Models of the fluid flow, Substantial
derivative, Divergence of the velocity, Laws of Conservation - Continuity Equation,
momentum Equation, Energy Equation, Dimensionless forms of Equations, Simplified
Mathematical Model, Mathematical Classification of flows, Physical Boundary conditions.
Basics of numerics: Components of the Numerical solution Methods – Mathematical
model, Discretization method, Co-ordinates and Basic Vector systems, Numerical
Grid, Finite approximations, Solution Method, Convergence criteria. Properties of
Numerical Solution Methods – Consistency, Stability, Convergence, Conservativeness,
Boundedness, Realizability. Discretization Approaches – FEM, FDM, FVM.
Finite difference method: Approximation of the first Derivative – Taylor series
expansion, Polynomial Fitting, Compact Schemes, Non-Uniform Grids. Approximation
of the second derivative, Approximation of the mixed derivative, Explicit and Implicit
approaches, Errors and Analysis of stability.
Spectral analysis and grid generation: Spectral Analysis of numerical Schemes,
Higher order methods, High accuracy compact schemes. General transformation
of the equations, Matrics and Jacobians, Stretched grids, Boundary fitted Coordinate systems, Elliptic grid generation, unstructured grids.
Computational heat transfer: Steady one & two dimensional heat conduction,
Unsteady one-dimensional heat conduction, over-relaxation and under-relaxation.
One dimensional steady convection and Diffusion.
Computational Fluid Flow: Solution methods for incompressible flows - collocated
and staggered grid, Pressure correction equations, SIMPLE and SIMPLER Algorithm.
Examples in simple geometries such as flow in channel, lid driven cavity flow and validation. Solution methods for compressible flows - Importance of conservation
and upwinding. Simple artificial dissipation methods, pressure-correction methods
for arbitrary Mach numbers. Applications to inviscid compressible flows.