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 one-dimensional problem of interacting blast waves using the Finite Volume Method (FVM).

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This one-dimensional test problem was introduced to illustrate the strong relationship between the accuracy of the overall flow solution and the thinness of discontinuities on the grid. It involves multiple interactions of strong shocks and rarefactions with each other and with contact discontinuities.

The governing differential equation of this problem is the Euler equation. Hear the problem will be solved using Local-Lax Friedrich’s scheme. Here the initial condition and application of boundary condition are a bit different than the usual fluid flow problem. Read out the next paragraph to understand the boundary conditions.

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.

**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.

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- 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.
- 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.
- 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.
- Plot the result data (Density and Velocity) using Gnuplot and observe the shock wave interactions in time intervals such as 0.01, 0.016, 0.026, 0.028, 0.03, 0.032, 0.034 and 0.038.

**Project Brief: **It’s true that this test case evaluation is quite complex But you can observe that with optimized discretization and application of Lax-Fridrich’s scheme we can accurately capture all the features and shock interaction in it.

**DevC++:**You will be needing DevC++ software to write logic and interact the solution for a number of times.**Gnuplot:**Also, you will be needing plotting software such as Gnuplot to plot the result data and compare the solution.

**Programming language: **C language

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