Objective:

The primary objective of the course is to teach fundamentals of computational method for solving non-linear partial differential equations (PDE) primarily in complex geometry. The emphasis of the course is to teach CFD techniques for solving incompressible and compressible N-S equation in primitive variables, grid generation in complex geometry, transformation of N-S equation in curvilinear coordinate system and introduction to turbulence modellin.

Prerequisite:

Knowledge of undergraduate heat transfer and fluid mechanics, basic computational fluid dynamics.

Brief Syllabus:

Introduction: Finite difference (FDM) and Finite volume (FVM) methods, elliptic, parabolic and hyperbolic equations, Navier-Stokes (N-S) and energy equations, explicit and implicit methods, higher order schemes Solutions of simultaneous equations: iterative and direct methods, Gauss-Seidel iteration, CGS, Bi-CGSTAB and GMRES (m) matrix solvers, different acceleration techniques; Incompressible flow: N-S equation using explicit methods: MAC and SMAC (staggered and collocated grids), semi-implicit methods: SIMPLE and SIMPLER, projection method, higher order discretisations, Compressible Flow: solution of compressible N-S equation, finite volume formulations, geometric flexibility, Jameson's, MacCormack's, Steger and Warming schemes in FVM, flux splitting scheme and upwinding, different acceleration technique, multigrid method; Grid generation: grid generation using algebraic and partial differential equations; N-S equations in irregular geometry: transformation of N-S equation in curvilinear coordinate system, non-orthogonal grid, Uncertainty of numerical results: Sources of uncertainties, independence studies on grid, time-step, domain and initial condition. Turbulence modeling: scales of turbulence, concept of turbulence modeling, different eddy viscosity based models, introduction to large eddy simulation (LES) and direct numerical simulation (DNS).

Lecturewise Breakup (based on 50min per lecture)

I. Introduction: (3 Lectures)

• Brief introduction of boundary layer flow, incompressible and compressible flows, finite difference and finite volume method, example of parabolic and hyperbolic systems and time discretization technique, explicit and implicit methods, upwind and central difference schemes, stability, dissipation and dispersion errors.

   

II. Solution of Simultaneous Equations: (4 Lectures)

• point iterative/block iterative methods, Gauss-Seidel iteration (concept of central coefficient and residue, SOR), CGS, Bi-CGSTAB and GMRES (m) matrix solvers, different acceleration techniques.

III. Incompressible Flow: (11 Lectures)

• Higher order upwind schemes: second order convective schemes, QUICK.

• Solution of NS equations: Solution of incompressible N-S equation (Explicit time stepping, Semi–explicit time stepping).

• SMAC method for staggered grid: Predictor - Corrector step, discretization of N-S and continuity equations, Pressure correction Poisson's equation, boundary conditions (no-slip, moving wall, slip boundary and inflow conditions), outflow (zero gradient/Orlanski) boundary conditions for unsteady flows, algorithm for the SMAC method, stability considerations for SMAC method.

• Semi–implicit method (SIMPLE): Comparison with the SMAC and fully – implicit methods, algorithm for semi–implicit method, discussion on SIMPLE/SIMPLER and SIMPLEC. Discretization of governing equations and boundary conditions in FVM framework.

• SMAC method for collocated grid: Pressure–velocity coupling, N- S equations on a collocated grid, concept of momentum interpolation to avoid pressure velocity decoupling, discretization of governing equations using the concept of momentum interpolation.

IV. FDE in complex geometries: (4 Lectures)

• Transformation of governing equation in ξ η− plane, transformation of Laplace equation, introduction to geometrical parameters and the accuracy of the solution, basic facts about transformation, grid transformation on complex geometries. N-S equations in transformed plane, matrices and Jacobians.

V. Compressible Flow: (7 Lectures)

• N-S and energy equations, properties of Euler equation, linearization.

• Solution of Euler equation: Explicit and implicit treatment such as Lax-Wendroff, MacCormark, Beam and Warming schemes, Upwind schemes for Euler equation: Steger and Warming, Van Leer's flux splitting, Roe's approximate Riemann solver, TVD schemes.

• Solution of N-S equations: MacCormack, Jameson algorithm in finite volume formulation and transformed coordinate system.

VI. Grid system: (5 Lectures)

• Historical aspects of the various grids, Body fitted grids in complex geometries, orthogonal grids, mapping functions, staggered/collocated and structured/unstructured, various methods of grid generations (Algebraic, Transfinite, Poisson equation methods).

VII. Uncertainty of numerical results: (2 Lectures)

• Sources of uncertainties, studies on grid independence, time-step independence, domain independence, initial condition dependence.

VIII. Turbulence modelling: (4 Lectures)

• Introduction to turbulence, scales of turbulence, Reynolds Averaged Navier Stokes (RANS) equation, closure problem, eddy viscosity model, k-ε and k-ω model, introduction to large eddy simulation (LES) and direct numerical simulation.

References:

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

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

3. Computational Fluid Dynamics, Tapan K. Sengupta, University Press, 2005

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

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

6. Essential Computational Fluid Dynamics, Zikanov. O., Wiley 2010.

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