A Numerical Solution to Quasi-One-Dimensional Nozzle

Analytical solutions give exact solutions that can be used to study the behavior of the system with varying properties. Unfortunately, very few practical systems lead to analytical solutions, and again the use of analytical solutions is of limited use because of its difficulty. That’s why we use a numerical approach to make a close answer to practical result with ease and convenience. Recently there is a lot of research going on in this field. Doing a project in this field will give you an advantage over others. In this project, you are going to write your own code to solve a one-dimensional nozzle problem using numerical approach and you have to find out the properties of fluid flowing through the nozzle along its length.

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To implement this project, you need to have knowledge of Fluid Mechanics, FVM and little coding skills. You need to find the Analytical solutions to, governing equations for a one-dimensional nozzle in differential form. To do show you have to write your own logic and Interact the solution for no of times to get an accurate result.

Project Description:

1. Nozzle: A Nozzle is a mechanical device of varying cross-sectional area which increases the velocity or K.E. energy of a fluid at the expanse of its pressure drop. These are frequently used to control the rate of flow, speed, direction, the pressure of the fluid that emerges from them.
2. Quasi-One-Dimensional Flow: In a nozzle area varies in the direction of flow then flow should be treated as three-dimensional flow. However, if such a variation is gradual then we can neglect the changes taking place in cross-stream directions. Such flows are called quasi-one-dimensional flows. So, for solving this problem assume that flow is Quasi-static in Nature.
3. 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.
4. 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 to 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.
5. Area-Mach Number Relation: The three-governing equation of nozzle is not enough to solve the problem. To make things easier we require Area-Mach number relation.

Project Implementation:

1. Take a convergent-divergent nozzle problem from your text book.
2. Understand the problem fully, then start solving the problem using numerical methods.
3. At 1st write C code to discretize the domain and fix the area of the Nozzle. The variation of the area is along x and should be parabolic in nature.
4. Give proper Initial boundary conditions according to the problem.
5. Then derive require a system of differential equations for solving a Nozzle problem.
6. Using Local Lax-Fridrich scheme solve the three-governing equation for Nozzle along with Area-Mach Number relation to find out the Mach number.
7. Using Mach number at one point calculate pressure, density, temperature and Mach number at the same point. Repeat the solution for all the grid points and interact the solution to get accuracy up to 10e-5.
8. With the result data, plot various graph such as Mach no. variation, Pressure variation, Temperature variation along the length of the Nozzle.
9. Study the result data and see how various properties of the fluid are varying along the length.

Project Brief: By doing this project you can observe that there will be shock and expansion along the length of the Nozzle. Another thing you can observe that, at higher Mach no. a nozzle acts as a diffuser.

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.

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Programming language: C language

• DevC++
• Minitab

• quasi static flow
• nozzle
• FVM
• Lax Friedrichs scheme

A Numerical Solution to Quasi-One-Dimensional Nozzle