A Numerical Solution to One-Dimensional Euler equation, Shock tube Problem

It is expensive to use physical experiments to get essential engineering data for designing is quite expensive. Getting engineering data of a fluid in motion, using Computational Fluid Dynamics (CFD) Simulations are relatively inexpensive. Costs are likely to be decreased as computational power has increased. So, using numerical methods to solve various flow problems is effective. In this project, you are going to solve a 1D Shock tube problem using Finite Volume Method.

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A shock tube is a device in which a high density, high-pressure fluid is separated from a low-density low-pressure fluid using a thin diaphragm. When we rupture the diaphragm, the high-density fluid will move towards low-density fluid which will result in the change in properties of the mixture. We need to compute this change in the properties of fluid inside Shock tube. This change in properties is very difficult to determine by performing the physical experiment as this change is very rapid in nature. That's why we are going for the numerical solution of this problem. To implement this project, you need to have knowledge of FVM and little coding skills. You need to solve this 1D shock tube problem by implementing FVM using a central solver scheme known as the Lax-Friedrichs scheme to get a stable and accurate solution.

Problem Description:

A shock tube of length one meter is extended from -5 to +5 in X domain. In the left side of the shock tube, high pressure high density fluid and right side of the shock tube low pressure low density fluid is there. In the left side pressure is 10e5 pascale, density is 1.0 Kg/m^3 and right side pressure is 10e4 pascale, density is 0.125 Kg/m^3. Take a domain of -5 to +5 and compute the value of pressure, density and velocity at a time step equal to 0.01 sec. Take ?= 1.4 and while calculating dt take relaxation factor as 0.25.

Project Description:

• Euler equation: The 1st simplification to Navier-Stokes equation is the Euler equation. This equation is obtained by neglecting the effect of viscous and conductive heat transfer in the Navier-stokes equation. In this project, you are going to simulate the above problem using 1D Euler equation.

• Finite Volume Method (FVM): The finite volume method is a method for representing and evaluating the partial differential equation in the form of algebraic equations. Here finite volume (cell) refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contains divergence terms are converted to surface integrals using Gauss divergence theorem. These terms are the evaluated as fluxes at the surfaces of each finite volume. These are conservative in nature since the amount of flux leaving a surface is equal to the amount of flux entering into the finite volume.
• Lax Friedrich’s scheme: Several schemes are implemented on the basis of the finite volume method, one of which is Lax Friedrich’s scheme. To avoid the dependency of the solution on the direction of information flow, a central solver can be preferred. Lax-Friedrich’s scheme is one of the central solvers which can be used to solve a flow problem. Use Local Lax-Fridrich scheme for a better result.

Project Implementation:

1. First, you need to discretize both space (X domain) and time by writing logic using C programming. Then you need to properly specify all the initial conditions and boundary conditions as it is given in the problem.
2. Then by using Local Lax-Fridrich scheme, you need to solve the 3-setoff equation present in the Euler equation. To solve the system of differential equation present in the Euler equation you have to write a few logics too.
3. Interact the solution for a number of times until you get an accuracy of 10e-4 and you reached the time step of 0.01 sec. After this save the result data in a text file.
4. Plot the result data using Gnuplot or Minitab. For plotting contours use Miniplot for ease and convenience.
5. You can verify your solution by clicking on this link, https://depts.washington.edu/clawpack/clawpack-4.3/applications/euler/1d/shocktube/www

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Project Brief: By doing this project you can observe that, In the Density plot, Pressure plot and Velocity plot there will be expansion at (-5 to 0) and shuck at 5. If you run the interaction for a higher time step, you can observe more property changes.

Software requirements:

1. DevC++: You will be needing DevC++ software to write logic and interact the solution for a number of times.

2. Gnuplot: Also, you will be needing plotting software such as Gnuplot to plot the result data and compare the solution.

Programming language: C language

• DevC++
• Minitab

• Mechanical
• CFD
• Solution to Euler equation
• shock absorbers

A Numerical Solution to One-Dimensional Euler equation, Shock tube Problem