PM Che 5321 Project 1 Final Draft Fall 2022

10422 152 Pm Che 5321 Project 1 Final Draft Fall 2022localhost888

Analyze the system of linear equations related to the flow of a liquid film down an inclined surface. Use the finite difference method to compute the velocity profile of water flowing down a surface at an angle, considering the fluid's properties and the given parameters. Develop functions to generate the coefficient matrix and the right-hand side vector for the finite difference equations, solve for the velocity distribution across the grid points, plot the velocity profiles for different grid resolutions, compare with the analytical solution, and interpret the effects of grid density on the results.

Paper For Above instruction

The analysis of fluid flow down inclined surfaces has vital applications in natural scenery, industrial processes, and environmental engineering. In this context, understanding the velocity profile of a liquid film flowing down an inclined plane provides insight into the flow behaviors and aids in designing efficient systems. This paper presents a comprehensive methodology to compute the velocity distribution using finite difference methods, compares numerical results with analytical solutions, and discusses the influence of grid resolution on the computed velocity profiles.

To begin with, the problem involves modeling the steady, fully developed flow of water along an inclined plane at an angle α relative to vertical. The physical parameters involve the fluid's density ρ, viscosity μ, the inclination angle α, and the film’s thickness H = 100 mm. The purpose is to compute the velocity profile v(y) across the film depth, where y extends from the bottom to the top of the film (0 to H). The governing equation is derived from the Navier-Stokes equations, simplified under steady-state, laminar flow assumptions, leading to a second-order ordinary differential equation which can be discretized using finite differences.

Finite Difference Method for Velocity Profile

The finite difference method transforms the differential equation into a system of algebraic equations. Discretizing the vertical axis y into n equally spaced grid points yields a set of equations that can be represented in matrix form A * v = B, where A is the coefficient matrix, v the velocity vector, and B the known source terms incorporating the gravitational component and boundary conditions. Generating matrix A involves creating a tridiagonal matrix reflecting the second derivative approximation, with boundary conditions applied at y=0 (no-slip: v=0) and y=H (free surface or specified shear conditions).

Function Development

First, a function to generate the coefficient matrix A is implemented. It takes the number of grid points as input and returns the matrix, considering the boundary conditions. Next, a function to generate the vector B is created, which accounts for the gravitational body force component and boundary conditions. These functions streamline the process of solving the finite difference equations for any specified grid resolution.

Solving and Visualizing Velocity Profiles

Using the matrix A and vector B, the linear system is solved via numerical methods, such as the numpy.linalg.solve function in Python. The resulting velocity vector v provides the velocity at each grid point in the fluid domain. Velocity profiles are plotted for different grid resolutions—specifically for 3, 10, and 50 grid points—to observe the effects of grid density on the solution accuracy. As the number of grid points increases, the numerical solution should converge toward the true velocity distribution, displaying finer resolution and less numerical diffusion.

Furthermore, the velocity profile for 100 grid points is generated and compared against the analytical solution, which is given by v(y) = (ρg cos α H²) / (2 μ) [1 - (y/H)²]. This analytical expression serves as the benchmark for assessing the accuracy of the finite difference approximation. Overlaying the finite difference result with the analytical profile demonstrates the method’s convergence properties and the adequacy of the grid resolution.

Discussion on Grid Resolution Impact

With increasing grid points, the velocity profile computed via finite difference becomes smoother and more closely aligns with the analytical solution. Small grid sizes, such as 3 or 5 points, provide overly coarse approximations, often failing to capture the parabolic shape accurately. Larger grids, like 50 or 100 points, offer high fidelity, reducing numerical dispersion and approximating the true velocity distribution effectively. This behavior stems from the convergence properties of finite difference schemes, where increasing discretization density enhances numerical accuracy but at the expense of computational cost.

Implications and Conclusion

Understanding the effects of grid resolution is critical in computational fluid dynamics modeling. Fine discretizations produce more accurate results but demand greater computational resources. Optimum grid density balances accuracy with efficiency. The implementation of matrix generation functions, solving algebraic systems, and comparing numerical solutions with analytical benchmarks collectively illustrate the robustness and limitations of finite difference methods in modeling fluid flow phenomena. Such insights aid in designing practical computational schemes for more complex flow scenarios, extending beyond idealized assumptions.

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