Analysis of Turbulence in a Two Dimensional Cavity Flow

Turbulent flows have an infinite variety ranging from the flow of blood in our body to the atmospheric flows. Everyday life gives us an intuitive knowledge of turbulence in fluids; during air travel, one often hears the word turbulence generally associated with the fastening of seat-belts. The flow passing an obstacle or an airfoil creates turbulence in the boundary layer and develops a turbulent wake which will generally increase the drag exerted by the flow on the obstacle. The majority of atmospheric or oceanic currents cannot be predicted accurately and fall into the category of turbulent flows, even in the large planetary scales. Galaxies look strikingly like the eddies which are observed in turbulent flows such as the mixing layer, and are, in a way of speaking, the eddies of a turbulent universe. Numerous other examples of turbulent flows arise in aeronautics, hydraulics, nuclear and chemical engineering, oceanography, meteorology, astrophysics, and internal geophysics. A clear understanding of this physical phenomena is one of the most essential and important problems of applied science.

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The governing differential equations which you need to solve to achieve the result are the momentum equations along X and Y direction which are present in the legendary Navier-Stokes equation, Continuity equation, and Pressure Poisson 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.

Problem Description:

In the project, you have to find a solution for different Reynolds number i.e. from 1 to 10000 for the lid-driven cavity on a collocated grid and at the end, you have to compare the result with the fine grid solutions available in the literature.

Project Description:

• Lid-Driven Cavity: The lid-driven cavity flow is most probably one of the most studied fluid problems in computational fluid dynamics field. The simplicity of the geometry of the cavity flow makes the problem easy to code and apply boundary conditions and etc. Even though the problem looks simple in many ways, the flow in a cavity retains all the flow physics with counter-rotating vortices appear at the corners of the cavity. Driven cavity flow serve as a benchmark problem for numerical methods in terms of accuracy, numerical efficiency and etc.

• Continuity Equation: Continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity such as momentum.
• FDM: This is one of the discretization techniques used in numerical analysis to solve the differential equation by approximating the derivatives. Here we discretize both space and time into N no of data points, where we store the data and keep updating as we move forward in time.

Project Implementation:

1. The computational algorithm for simulating fluid flow is based on discretizing the governing equation and then converted into the transcendental equation or algebraic equation from its original partial differential equation. You have to perform the same by using FDM during the initial stage of this project.
2. The discretized algebraic equations are typically solved on discretized computational domains, known as meshes or grids. You have to create this grid by writing a set of logic using the C programming language.
3. Then use the discretized equation and write a set of codes to solve them by computation.
4. Interact the solution for a number of times until you get an accuracy of 10e-4 and achieving study state.
5. After achieving solution plot various contours at different Reynolds no. capturing the central vortex, bottom right vortex and primary bottom left vortex at the appropriate location.
6. Compare your result with the existing solution in available research papers.

Project Brief: At the end of this project you will learn how to simulate the turbulent fluid flow field based on different grid size.

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++
• Gnuplot

• Numerical Methods
• FDM
• Turbulence modeling

Analysis of Turbulence in a Two Dimensional Cavity Flow