Equations for fluid flowing down a slope

[BLKX-Darkreaper]

I want to be able to calculate the change in velocity of a fluid flowing down a slope using the drag equation as the source of friction. However the drag equation uses velocity squared which is constantly changing as the fluid accelerates/decelerates. How do I integrate the drag equation into those used for determining velocity with an non-constant acceleration?
Ideally I'd also like to integrate fluid shear into these equations as well which would mitigate flow drag with increasing depth

 

[boneh3ad]

The problem is that the drag equation would be a very bad way to go about this. For one, how would you determine a drag coefficient and how would it change as the shear changed? The source of "drag" in this case is the shear, so treating them separately doesn't make a whole lot of sense to me. Generally, if you want to get a decent answer, you'll likely have to just use the Navier-Stokes equations or one of the various models approximating them.

 

[BLKX-Darkreaper]

Drag is produced by the surface that the fluid is flowing over and therefore the shear as well. There are existing drag coefficients for different types of water channels, which is what i would use

 

[boneh3ad]

The drag and shear are one in the same. The shear stress at the wall is exactly the drag. Water channel drag coefficients don't necessarily make sense because you'll have a free surface present.

 

[Andy Resnick]

I want to be able to calculate the change in velocity of a fluid flowing down a slope using the drag equation as the source of friction. <snip>

How did you incorporate the no-slip boundary condition at the fluid-solid interface, and how did you incorporate contact line motion at the 3-phase line?

 

[BLKX-Darkreaper]

How did you incorporate the no-slip boundary condition at the fluid-solid interface, and how did you incorporate contact line motion at the 3-phase line?

As to the first question, I havent. Still trying to work out a set of equations.
As for the second question I am not familiar with that issue, could you elaborate?

 

[BLKX-Darkreaper]

The problem is that the drag equation would be a very bad way to go about this. For one, how would you determine a drag coefficient and how would it change as the shear changed? The source of "drag" in this case is the shear, so treating them separately doesn't make a whole lot of sense to me. Generally, if you want to get a decent answer, you'll likely have to just use the Navier-Stokes equations or one of the various models approximating them.

Not sure how to apply the Navier-Stokes equations in this case. Would you be able to provide a simplified version of the equations to help solve this problem?

 

[Andy Resnick]

As to the first question, I havent. Still trying to work out a set of equations.
As for the second question I am not familiar with that issue, could you elaborate?

To be honest, this was the answer I expected. They are still open questions, FWIW. Elizabeth Dussan (E. B. Dussan V.) co-wrote several seminal papers, for example:

http://www.annualreviews.org/doi/abs/10.1146/annurev.fl.11.010179.002103?journalCode=fluid

Good luck...

 

[BLKX-Darkreaper]

I'm not looking for something perfect. Just a good approximation for use in a game

 

[boneh3ad]

Yeah the problem is that even "good approximations" of that flow are quite complex, and trying to use the drag equation is not a suitable solution. It is not a model that was designed with such a situation in mind.