Linear Convection - Solving 1D Linear Equation for Different Grid Sizes

Objective:This work attempts to create a custom icoFoam based solver to solve for a temperature equation and validate the solver through a benchmarking example. Introduction: OpenFOAM (Open Field Operation and Manipulation) is a popular open-source CFD toolbox known for its flexibility and extensibility. Unlike…

Aim:- To simulate pipe flow using Wedge and Symmetry boundary conditions for different angles and validate them with the Hagen–Poiseuille equation.   Objective:- Validate Hydro-dynamic length with the numerical result Validate the fully developed flow velocity profile with its analytical profile Validate maximum…

Objective: - Create an OpenFOAM solver for scalar transport.   Methodology: - We will be creating an OpenFOAM solver by customizing an already existing solver called icoFoam. We will be adding additional scalar equations to the solver and validating it with a case pertaining to the same. The post-processing of the…

Objective: - Write a Reaction Reduction Mechanism code for the GRI 3.0 mechanism, The code should do the following, First, it should run a reference Ignition delay and Max temperature Analysis for the unreduced mechanism for a set of state values (Temperature, Pressure and Equivalence ratio). Then, a sensitivity analysis…

Objective: - In this assignment, we will be writing a Python/Cantera Code to find the 10 most sensitive reactions out of all the reactions in GRI mechanism.The main parameters of this assignment are given below: - Write a code that takes all the reactions from the GRI mechanism and calculates 10 most sensitive reactions…

Aim:- To simulate flow through a pipe using OpenFoam and validate the results.   Objective:- Validate Hydro-dynamic length with the numerical result Validate the fully developed flow velocity profile with its analytical profile Validate maximum velocity and pressured drop for fully developed flow Post-process Shear…

Objective: - In this challenge, we will write a Cantera code to calculate the Auto Ignition time for Methane under various conditions. This challenge is divided into two parts. Part 1: - Simulate the Auto Ignition Time for Methane under the following conditions, (simulation time = 10 secs) Plot the variation of Auto…

Objective: - In this challenge, we will be writing a code for a Stoichiometric Combustion Calculator for Alkane, Alkene and Alkyne fuels using Matlab.   Solution: - Stoichiometric or Theoretical Combustion is the ideal combustion process where fuel is burned completely.When fuel and air mix they produce products  as…

When does an ODE become unstable? Stability means that errors at any stage of the computation are not amplified but are attenuated as the computation progresses. In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close…

Objective: - In this project, we are going to solve the two dimensional Heat Conduction for steady and unsteady states. For the Steady state, we have to solve it using three iterative approaches namely, Jacobi Method Gauss-Seidel Method and Successive Over Realxation Method   For Transient state, we have to solve…

Objective: - In this challenge, we will perform a 1D Flame Speed Analysis for Methane and Hydrogen mechanisms using Python and Cantera. We will be trying to answer some questions devised to understand the significance of this assignment. What type of analysis is this, steady-state or transient? Why do we need a separate…

Objective: - In this challenge, we are going to write a code solve the Quasi 1D supersonic nozzle flow equations using the Macormack Method implementing both the conservative and non-conservative forms of the governing equations and also perform a grid dependence test.   Solution: - For the current challenge, we are…

Objective: -  The objective is to write a program in Matlab to simulate the motion of a simple pendulum.   Solution: - For this current challenge, we are going to solve the following ODE to simulate the motion of a pendulum. `(d^2theta)/dt^2 + (b/m)((d theta)/dt) + (g/l)sintheta = 0` Where, b = damping coefficient…

Objective: - In this challenge, we are going to derive the following 4th order approximations of the second order derivative 1. Central difference 2. Skewed right sided difference 3. Skewed left sided difference Write a program in Matlab to evaluate the second order derivative of the analytical function `exp(x)*cos(x)`…

Objective: - For both implicit and explicit methods, look at the effect of time step on the stability of the solution. 1. How does an unstable solution look like? 2. At what CFL number does your solution become unstable ? 3. Write a detailed explanation for your results, plots or codes without explanation will awarded…

Objective: - Compare your fastest steady state simulation vs your fastest transient simulation and explain your results.   Solution: -  The link for the original Steady and Unsteady 2-D Heat Conduction is given below: https://projects.skill-lync.com/projects/Solving-the-Steady-and-Unsteady-2D-Heat-Conduction-Problem-29271…

In this challenge, we are going to determine the efficiency of a Heat Recuperator device. Problem Statement: - A furnace is used to burn Methane and Air Mixture. A Recuperator is connected to this furnace to recover some energy. What is the effect of preheating on the adiabatic flame temperature? Change pre-heating temperature…

Objective: -   In the Current challenge, we will be creating a Data Visualizer tool. The objectives of this challenge are as follows: - 1. Data Visualizer: - The script should take the column numbers as inputs and plot them respectively. 2. Compatibility Check: - The code should exit gracefully when given a wrong…

Objective: - Write a MATLAB code for 1D Linear convection equation for the following parameters. Domain Length L = 1 meter Initial Velocity Profile velocity is 2 m/s for x = 0.1 to 0.3, velocity is 1 m/s rest of the length domain Use First order forward differencing for the time derivative Use first order backward/rearward…

Objective: - IN the current challenge, we are going to write a MATLAB program that compares the first, second and fourth order approximations of the first derivative against the analytical and exact derivative.   Solution: -  The function given for the approximation is as follows `f(x) = (Sinx)/x^3` we are going…

Objective: - Write a function that accepts time step as an argument and solves the same problem comparing the effect of time step on the numerical solution for the follwoing parameters. Grid size n = 80 Time steps = 1e-4, 1e-3, 1e-2, and 1e-1 Simulation time = 0.4 seconds   Solution: -  The given 1D linear equation…

 Problem Statement: - In determining the minimum cushion pressure needed to break a given thickness of ice using an air cushion vehicle, Muller (“Ice Breaking with an Air Cushion Vehicle”) in Mathematical Modeling: Classroom Notes in Applied Mathematics, SIAM 1987 derived the equation `p^3(1-beta^2) +…

Objective: - In the current challenge, we are going to be programming a Multivariate Newton Rhapson Solver for the following system of equations.                                 …