An Introduction to Fluid Dynamics and Numerical Solution of Shock Tube Problem by Using Roe Solver

TL;DR

This paper introduces numerical methods for solving fluid dynamics problems, focusing on the Sod's Shock Tube problem using Roe solver, and demonstrates the application of these methods through Python code.

Contribution

It presents a detailed numerical approach to solving the Sod's Shock Tube problem using Roe solver, including derivation of governing equations and implementation in Python.

Findings

Successful numerical simulation of shock tube problem

Comparison of exact and approximate solutions

Implementation of Roe solver in Python

Abstract

The study of fluids has been a topic of intense research for several hundred years. Over the years, this has further increased due to improved computational facility, which makes it easy to numerically simulate the fluid dynamics, which was beyond the analytical reach before. In this project, we discuss the general transport theorem for moving control volume system, apply this theory to control mass system which gives the continuity equation, and on momentum conservation from which we get the Navier-Stokes equation and energy conservation. By approximating the three equations for the ideal gas flow, we get one dimensional Euler equation. These equations are non-linear, and their analytic solutions are highly non-trivial. We will use Riemann solvers to get the exact solution numerically. In this project, we have considered only the Sod's Shock Tube problem. Finally, we focus on the exact and approximate solution of density, pressure, velocity, entropy, and Mach number using Python code.