Curvature of along a streamline

[Trenthan]

Hey Guys/Girls and thanks in advance
Not quite sure this is in the correct forum since its not a homework question, more private study and curiosity lolz!

Im trying to evaluate the curvature along the streamline within a hydrodynamic potential field (fluid flow). I have no issue calculating the streamline and plotting it along with several others within the flow field. Now the problem is determining the curvature along it!

Lets assume I am trying to calculate the curvature at two points very close together on the streamline for starters. All i know about point 1 is:

"u and v" components of velocity
"x and y" position
"tangential direction"

Now using some of the information at point 1, i can find the location of point 2 (Using EULERS method with a step size of 1/128) and determine its tangential direction using the velocity components at point 1point (using arctan(v/u) )


I thought calculating the curvature would be quite straight forward however when searching the web and my shelf of textbooks, every method involves the use of the second derivative of the line segment, (streamline in my case) which i don't know.

Most methods seems to be similar to what is presented on Wolfram MAthWorld
http://mathworld.wolfram.com/Curvature.html"

Equations (1) - (7) are perfectly fine. However equation (8) is redundant in my case since the second derivative of my x and y position's at point 1 is unknown!

I'm just curious to know if anyone knows of any methods, or suggestions to determine the curvature knowing the information i know at point 1, above**

Cheers Trent

 

[hunt_mat]

Can you write down what you're doing as this sounds very familiar to the kind of stuff I am looking at currently. I imagine you're wring down the streamline equations in terms of the potentials and then trying to examine the curvature in terms of the derivatives in the potentials?

Are you doing this in 2 or 3 dimensions?

 

[Trenthan]

Can you write down what you're doing as this sounds very familiar to the kind of stuff I am looking at currently. I imagine you're wring down the streamline equations in terms of the potentials and then trying to examine the curvature in terms of the derivatives in the potentials?

Are you doing this in 2 or 3 dimensions?

Working in two dimensions at the moment.

Using the potential field consisting of a sink. source, and the circle theorem to model circular obstacles within it.

Taking the derivative of the potential field results in the velocities. dw/dz = u + i*v

I than use Eulers method to determine the next location along the streamline with a time step of 1/128.



Problem is i cannot calculate acceleration from the potential field;all methods use during the calculation of curvature from what I've seen. Ideally i want a mathematical expression that i can just plug the variables into from the potential field (source, sink, obstacle locations, strength, object radius) to determine the streamline curvature at that point or the next rather than a numerical approach. (I can determine the curvature currently using a numerical approach but ideally i want a mathematical one!)

I know I'm asking a lot but yea.

Any educated guess's, suggestions are welcome. They don't need to lead to a solution but if your unsure I'm happy to do the leg work and determine if it does!


btw hunt_mat, what approach are you using to determine the streamline curvature?


Cheers and Thanks
Trent

 

[hunt_mat]

So would you want to calculate the plane curvature for the streamlines?

The method that I would take in this case is to write down the streamlines:
$$\frac{dx}{ds}=\frac{\partial\phi}{\partial x}$$
along with the other equation and I would try and compute the second order derivatives:
$$\frac{d^{2}x}{ds^{2}}$$
The way to do this is to use the chain rule:
$$\frac{d}{ds}=\frac{dx}{ds}\frac{\partial }{\partial x}+\frac{dy}{ds}\frac{\partial }{\partial y}$$
This will allow you to get the second order derivatives of x and y in terms of partial derivatives of the velocity potential $\phi$. This will allow you to compute the plane curvature in terms of partial derivatives of the second order partial derivatives of the velocity potential. I will say now that I think it will be messy...