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NPTEL

Computational Gas Dynamics

NPTEL via Swayam

Overview

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ABOUT THE COURSE: This course provides a comprehensive study of gas dynamics and its computational modeling, focusing on high-speed compressible flows. The first part of the course builds the foundation by covering the physics of shock waves, expansion fans, nozzle flows, and quasi-one-dimensional theory. The second part introduces numerical methods to solve the governing equations of compressible flow using shock-capturing schemes and Riemann solvers in finite difference and finite volume frameworks. Students will learn to implement basic and advanced discretization techniques and simulate practical problems such as shock tubes. INTENDED AUDIENCE: UG and PG students in Mechanical/Aerospace engineering departmentsPREREQUISITES: Fluid Mechanics, Thermodynamics, Engineering MathematicsINDUSTRY SUPPORT: HAL, NAL, ISRO, DRDO and Private Automobile and Aerospace Industries (Airbus, Boeing, Tata Motors etc.)

Syllabus

Week 1: Review of Thermodynamics & Fluid Mechanics
  • Governing laws: mass, momentum, energy conservation
  • Thermodynamic relations for ideal gases
  • Speed of sound and Mach number
  • Classification of flows: Incompressible, compressible, subsonic, supersonic

Week 2:Compressible flows and Isentropic relations
  • Steady 1D and quasi-1D flow equations
  • Isentropic flow relations
  • Area-Mach number relation
  • Isentropic flow through nozzles

Week 3:Normal and Oblique Shock Waves
  • Normal shock relations and properties
  • Oblique shock theory and θ–β–M relation
  • Prandtl-Meyer expansion fans
  • Shock-expansion theory

Week 4:Quasi-1D Flow and Nozzle Dynamics
  • Flow with friction (Fanno flow) and heat addition (Rayleigh flow)
  • Analysis of converging and converging-diverging (C-D) nozzles

Week 5:Hyperbolic Systems and Conservation Laws
  • Properties of hyperbolic PDEs
  • Characteristics, eigenvalues, and wave speeds
  • Integral, Conservation and Characteristic forms
  • Advection and Diffusion Equations
  • Burgers' Equations
  • Euler equations in 1D and 2D

Week 6:Basic Discretization Methods
  • Finite difference methods: Taylor series approach
  • Central difference schemes (1st and 2nd order)
  • Upwind schemes: first-order upwind
  • Consistency, stability, and convergence
  • CFL condition

Week 7:Numerical Methods for Conservation Laws
  • Lax-Friedrichs, Lax-Wendroff, MacCormack methods
  • Concept of total variation and oscillations
  • Flux Splitting Methods

Week 8:Riemann Solvers
  • Riemann problem for hyperbolic systems
  • Exact vs. approximate Riemann solvers
  • Godunov's method and its limitations
  • Roe's approximate Riemann solver
  • HLL and HLLC solvers

Week 9:Shock-Capturing and TVD Methods
  • Nonlinear instability and need for limiters
  • Total Variation Diminishing (TVD) schemes
  • Flux and slope limiters (minmod, superbee, van Leer)
  • Entropy fix

Week 10:Finite Volume Method
  • Finite volume formulation
  • MUSCL scheme and slope reconstruction
  • Flux calculation and reconstruction
  • Multidimensional extension to 2D Euler equations

Week 11:Boundary Treatments and Solver Development
  • Reflective, inflow/outflow, wall boundaries
  • Characteristic boundary conditions
  • 1D and 2D Euler solvers

Week 12:Applications and Advanced topics
  • Numerical simulation of shock tube
  • Oblique shock and expansion fan simulations
  • Verification and validation of CFD codes
  • Brief intro to Navier-Stokes solvers

Taught by

Prof. Rajesh Ranjan, Prof. Niranjan Sahoo

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