Stagnation points for flow around a cylinder

[SaveFerris]

i was just wondering how you'd go about finding the location of stagnation points for an incompressible flow around a cylinder if you know what the stream function is?

Thanks!

 

[mrmiller1]

You have to come up with representative nodes to solve ur equation for velocity and then see which points have no flow velocity. I usually use finite differences at each node to solve the PDE (Navier-Stokes equation). Remember that molecules on the surface have zero velocity.

 

[cjl]

If you know the stream function, it's easy. Just look for points on the surface of the cylinder with zero velocity. Typically, the velocity components can be found from the streamfunction in a fairly straightforward way:

$$u = d\psi/dy$$
$$v = -d\psi/dx$$

At the stagnation point, u and v will be zero.

 

[SaveFerris]

Thank you so much.
This is a great help!

 

[pmacias]

I would approach this question by first understanding the concept of stagnation points and their significance in fluid dynamics. Stagnation points are locations in a flow where the velocity of the fluid is zero. In the case of flow around a cylinder, these points are important as they determine the separation of the flow from the surface of the cylinder.

To find the location of stagnation points for an incompressible flow around a cylinder, one can use the stream function method. The stream function is a mathematical function that is used to describe the flow pattern of a fluid. It is defined as the flux of the fluid through a line perpendicular to the direction of flow.

To find the location of stagnation points using the stream function, we can use the following steps:

1. Write the stream function equation for the flow around the cylinder, which is given by ψ = U(rsinθ) where U is the free stream velocity, r is the radial distance from the center of the cylinder, and θ is the angle measured from the horizontal axis.

2. Differentiate the stream function equation with respect to r to find the velocity components in the radial and tangential directions.

3. Set the velocity components to zero to find the location of stagnation points. This will give us two equations, one for the radial direction and one for the tangential direction.

4. Solve the two equations simultaneously to find the values of r and θ at the stagnation points.

5. Once the values of r and θ are known, the location of the stagnation points can be determined by using the polar coordinates (r, θ).

It is important to note that the stream function method assumes a two-dimensional flow and does not take into account the three-dimensional effects near the surface of the cylinder. Therefore, the calculated location of the stagnation points may not be exact but can provide a good estimation.

In conclusion, the location of stagnation points for an incompressible flow around a cylinder can be found by using the stream function method. This method utilizes the stream function equation and the knowledge of the free stream velocity to determine the location of these important points in the flow.