代做EG501V Computational Fluid Dynamics (AY 2023/24) Tutorial 5帮做R程序

Tutorial EG501V Computational Fluid Dynamics (AY 2023/24)

Tutorial 5. Building a system matrix

Two-dimensional fluid flow can be described by means of a stream function φ(x, y) that obeys the following elliptic

PDE: Consider the two-dimensional contraction as shown in the figure. The left panel of the figure is a cartoon of the streamlines.

The right panel defines the flow geometry and boundary conditions: at the inlet (left) and at the outlet (right); φ = 0  on the entire lower wall; φ = 1 on the upper wall. The figure also defines the discretization. We use dimensionless quantities throughout this problem.

Q1

From the discretization (with  Δx = 1 and Δy = 0.5 ) of the PDE, and from the boundary conditions determine the 10×10 matrix [A] and the 10-dimensional vector b such that the 10-dimensional vector φ containing φk , k = 1…10  satisfies  [A]φ = b . Number the unknowns φk as indicated in the figure.

Q2

The fluid velocity in x andy-direction ( ux and uy ) is related to the stream function according to and The solution to = [0.2322 , 0.2049 , 0.4781 , 0.4542 ,

0.3513 , 0.3389 , 0.7359 , 0.7233 , 0.6800 , 0.6716]. Given this solution, determine ux in points 3 and 6, and determine uy in points 2 and 5 based on central differences approximations.