### Credits:

3L-0T-0P-0A (9 credits)

### Brief Syllabus:

Introduction: Governing equations for fluid flow and heat transfer, classifications of PDE, finite difference formulation, various aspects of finite difference equation, error and stability analysis, dissipation and dispersion errors, modified equations; Solutions of simultaneous equations: iterative and direct methods, TDMA, ADI; Elliptic PDE: One- and Twodimensional steady heat conduction and their solutions, extension to three-dimensional; Parabolic PDE: Unsteady heat conduction, explicit and implicit methods, solution of boundary layer equation, upwinding; Solution of incompressible N-S equation: Stream function and vorticity formulation, primitive variable methods: MAC and SIMPLE.

### Objectives:

The primary objective of the course is to teach fundamentals of computational method for solving linear and non-linear partial differential equations (PDE). The course offers introductory concepts about solving PDE mainly in the finite difference (FD) framework though some amount of finite volume (FV) concept has also been introduced.

### Prerequisites:

Knowledge of undergraduate heat transfer and fluid mechanics.

### Lecture-wise break-up (lecture duration: 50 minutes)

 Topics No. of suggested lectures 1 Introduction: Brief introduction of ODE (IVP and BVP) and PDE, Initial and Boundary conditions, classification of PDE, various methods to solve PDE numerically along with their advantages and disadvantages, 3 2 FDM: Taylor series expansion, Finite difference equations (FDE) of 1st, and 2nd order derivatives, Truncation errors, order of accuracy. 3 3 Application of FDM: Steady and unsteady one- and two-dimensional heat conduction equations, one-dimensional wave equations,General method to construct FDE 2 4 Aspects of FDE: Convergence, consistency, explicit, implicit and C-N methods. 2 5 Solution of simultaneous equations: direct and iterative methods; Jacobi and various Gauss-Seidel methods (PSOR, LSOR and ADI), Gauss-elimination, TDMA (Thomas), Gauss-Jordan, other direct and indirect methods. 5 6 Errors and Stability of FDE: Diffusion and dispersion errors Stability of 1D and 2D diffusion equation, 1D wave equation (FTCS, FTBS and FTFS). 4 7 Modified equations of FD formulation:Diffusion and dispersion errors of modified equation (wave equation) having second and third order derivatives, modified wave number and modified speed. 3 8 Upwinding: Upwinding of convective terms and its significance, Transportiveand conservative properties.Upwind biased difference schemes and its significance. 3 9 FDE in other coordinate systems: Cylindrical and polar coordinate systems, 2 10 FVM: Two approaches of generating Cartesian grids, Solution of fin problem in FVM, Handling of BCs in FVM; Generalized FVM approach for orthogonal grids (complex geometry).. 3 11 Stream function-vorticity approach: Derivation of stream function and vorticity equations; derivation pressure Poisson equation. Application 2-3 problems 2 12 Primitive variable approach: Grid system (Staggered vs collocated grids); their advantages and disadvantages; control volumes for continuity and N-S equations. MAC method; derivation of pressure correction equations; discretization of GDE and BCs for channel flow; solution algorithm; stability constraints.  Projection/Fractional step method; solution algorithm; difference with MAC. SIMPLE and SIMPLER method (FVM): derivation of pressure and pressure-correction and velocity correction equations. Discretization and solution algorithm. 8 Total Number of Lectures 40

### Suggested reference books:

• Computational Fluid Flow and Heat Transfer, Second Editionby K. Muralidhar, T. Sundararajan(Narosa), 2011.

• Computer Simulation of Flow and Heat Transfer by P. S. Ghoshdastidar (4th Edition, Tata McGraw-Hill), 1998.

• Numerical Computation of Internal and External Flows by Hirch C., Elesvier 2007.

### Suggested reference books:

• Numerical Heat Transfer and Fluid Flow by S. V. Patankar(Hemisphere Series on Computational Methods in Mechanics and Thermal Science)

• Essential Computational Fluid Dynamics by Zikanov.O., Wiley 2010.

• Computational Fluid Dynamics by Chung T. J., Cambridge University Press, 2003.