Plotting Stream lines (Fluid Mechanics)

[kingjames]

Plotting Stream lines! (Fluid Mechanics)

Homework Statement

In this project you are required to investigate two types of potential flow solution:
a. Flow over a vertical plate (flow is perpendicular to the plate i.e.  = 90)
b. Flow over an inclined plate ( = 135).


You are required to do the following:
1. Find the expression of the stream function for these types of flow.
2. Derive the expression for the velocity profile.
3. Sketch the streamlines for different values of the stream function using your favorite computer software (Matlab, Excel …).

Homework Equations

i found the equations of stream functions for both cases:

Case of a flow over a vertical plate: ψ⁴ + ψ² U²_∞(x²-y²+a²) = U⁴_∞x²y²

Case of a flow over an inclined plate ψ=(-1)/√2 q₀ c sinhζ (cos η-sin η)

The Attempt at a Solution

IF u can help me with codes to plot these functions using MATLAB or excel or any other program!

 

[KataKoniK]

Hi there! As a scientist who specializes in fluid mechanics, I would be happy to assist you with plotting streamlines for these two types of potential flow solutions.

First, let's start with the expression for the stream function for each case:

1. Flow over a vertical plate:

The stream function for this case is given by the equation ψ = U∞(x2-y2+a2) where U∞ is the free stream velocity and a is the distance between the plate and the origin.

2. Flow over an inclined plate:

The stream function for this case is given by the equation ψ = (-1)/√2 q0 c sinhζ (cos η-sin η) where q0 is the free stream velocity, c is the chord length of the plate, ζ is the distance along the plate from the leading edge, and η is the distance normal to the plate.

Next, let's derive the expression for the velocity profile for each case:

1. Flow over a vertical plate:

The velocity components can be found by taking the partial derivatives of the stream function with respect to x and y. This gives us u = U∞(2x) and v = -U∞(2y). Therefore, the velocity profile is given by the equation V = √(u^2+v^2) = 2U∞√(x^2+y^2).

2. Flow over an inclined plate:

The velocity components can be found by taking the partial derivatives of the stream function with respect to ζ and η. This gives us u = (-1)/√2 q0 c coshζ (cos η-sin η) and v = (-1)/√2 q0 c sinhζ (-sin η-cos η). Therefore, the velocity profile is given by the equation V = √(u^2+v^2) = (-1)/√2 q0 c √(cosh^2ζ+sinh^2ζ) = q0 c.

Now, to plot the streamlines for different values of the stream function, we can use MATLAB or any other software that allows us to plot graphs. Here's how we can do it in MATLAB:

1. Flow over a vertical plate:

a. Define the variables U∞ and a (you can choose any values you want).
b