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- To learn fundamentals of computational methods like FDM and FVM for solving linear and non-linear partial differential equations related to fluid dynamics and heat transfer.

- Assess the principles of numerical analysis and concepts of consistency, stability, and convergence.
- Evaluate finite difference/volume schemes on model problems of computational fluid dynamics.
- Construct program-code using Python to obtain numerical solutions of partial differential equations, relevant to Computational Fluid Dynamics.

**MODULE ****I**

**Introduction to CFD**: Basics of computational fluid dynamics, Definition and overview of CFD- need, advantages, problem areas, Governing equations of fluid dynamics – Continuity, Momentum and Energy equations, Non-Dimensional form of these governing equations, Classifications of PDE: Elliptic, Parabolic and Hyperbolic equations.

**MODULE ****II**

**Finite Difference Method (FDM)**: Derivation of Finite difference equations (FDE) of 1^{st }and 2^{nd} order derivatives using Taylor series expansion. Explicit method-FTCS Method,Implicit method-BTCS Method, Crank-Nicholson method, Error, Convergence and stability analysis of above numerical Scheme, Keller Box Method.

**MODULE III**** **

**Solution of Simultaneous Equations:** Direct and Iterative methods; Gauss-elimination, Gauss-Jordan, Gauss-Jacobi and Gauss-Seidel methods, Tri Diagonal Matrix Algorithm(TDMA) (Thomas)

**Practice 1:** Gauss-elimination method using Python

**Practice 2:** Gauss-Seidel method using Python

**Practice 3:** Tri Diagonal Matrix Algorithm using Python

**Project 1:** Solution of Simultaneous Equations using Gauss-Jordan method.

**Project 2:** Solution of Simultaneous Equations using Gauss-Jacobi method.

**MODULE ****IV**

**Application of FDM:**** **Solutions of

Elliptic PDE: One-and Two-dimensional steady heat conduction, Laplace’s Equation, Poisson’s equation.

Parabolic PDE: Unsteady heat conduction, Stoke’s 1^{st} & 2^{nd} Problems.

Hyperbolic PDE: One-dimensional wave equation.

**Practice 4:** Solution of One-dimensional steady heat conduction using Python.

**Practice 5:** Solution of Laplace’s equation using Python.

**Practice 6:** Solution of Unsteady heat conduction using Python.

**Practice 7:** Solution of One-dimensional wave equation using Python.

**Project 3: **Solution of Burger’s equation.

**Project 4: **Solution of Poisson’s equation.

**MODULE ****V**

**Finite Volume Method (FVM):**

Fundamentals of FVM, Integral Form of 1-D Conservation equation, Finite Volume Method in 2-D

**MODULE ****VI**

**Application of FVM: **Solutions of 1-D steady state Diffusion and Convection equations.

**Project 5: **Solutions of 1-D steady state Diffusion equation.

**MODULE ****VII**

**Application of FVM:** Solutions of 2-D steady state Diffusion and Convection equations.

**Project 6: **Solutions of 2-D steady state Convection equation.

**Text Books:**

- Computational Fluid dynamics by John D.Anderson,Jr
- Computational Fluid dynamics and Heat Transfer , by John C. Tannehill , Dale A.Anderson ,Richard H. Pletcher
- Introduction to finite elements in engineering, by Tirupathi R. Chadraupala, Ashok D.Belegundu.
- An introduction to computational fluiddynamics, by HKVersteeg and W Malalasekera.

Basics of computational fluid dynamics, Definition and overview of CFD- need, advantages, problem areas,Governing equations of fluid dynamics – Continuity, Momentum and Energy equations.

Non-Dimensional form of these governing equations, Classifications of PDE: Elliptic, Parabolic and Hyperbolic equations.

Derivation of Finite difference equations (FDE) of 1^{st} and 2^{nd} order derivatives using Taylor series expansion.

Explicit method-FTCS Method,Implicit method-BTCS Method.

Crank-Nicholson method, Error, Convergence and stability analysis of above numerical Scheme.

Keller Box Method.

Gauss-elimination method.

**Practice 1:** Gauss-elimination method using Python

Gauss-Jordan method.

**Project 1:** Solution of Simultaneous Equations using Gauss-Jordan method.

Gauss-Jacobi method.

**Project 2:** Solution of Simultaneous Equations using Gauss-Jacobi method.

Gauss-Seidel method.

**Practice 2:** Gauss-Seidel method using Python.

Tri Diagonal Matrix Algorithm.

**Practice 3:** Tri Diagonal Matrix Algorithm using Python

Solutions of One-dimensional steady heat conduction.

** Practice 4:** Solution of One-dimensional steady heat conduction using Python.

Solutions of Two-dimensional steady heat conduction.

Solution of Laplace’s Equation.

**Practice 5:** Solution of Laplace’s equation using Python.

Solution of Poisson’s equation.

**Project 3: **Solution of Poisson’s equation.

Solution of Unsteady heat conduction.

**Practice 6:** Solution of Unsteady heat conduction using Python.

Solution of Stoke’s 1^{st} & 2^{nd} Problems.

**Project 4: **Solution of Burger’s equation.

Solution of One-dimensional wave equation.

**Practice 7:** Solution of One-dimensional wave equation using Python.

Fundamentals of FVM, Integral Form of 1-D Conservation equation.

Finite Volume Method in 2-D.

.

Solutions of 1-D steady state Diffusion equation.

**Project 5: **Solutions of** **1-D steady state Diffusion equation.

Solutions of 1-D steady state Convection equation.

Solutions of 2-D steady state Diffusion equation.

Solutions of 2-D steady state Convection equation.

**Project 6: **Solutions of** **2-D steady state Convection equation.

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