Rainfall drop velocity from a given height (not terminal)

[uluru]

hi everybody,

I posted this in an engineering forum but I think it's more relevant here, because it's really just a question of fluid and Newtonian mechanics.

I'm working on a project where I'm trying to measure raindrop parameters, and one thing I'm looking at is the sub-terminal speed of drops released from a certain height. The equation that I'm using is from http://staff.science.uva.nl/~jboxel/Publications/PDFs/Gent_98.pdf

The gist of the equation that I was considering is:

F = g*ρ_w*∏*d^3/6 - 3*∏*d*μ*V*C_t*C_d

where Ct = 1+0.16*Re^(2/3)

and Re = ρVD/μ;

and Cd = 1+a(We+b)^c - ab^c

where a,b,c are empirically derived constants and We = ρ*V^2*d/σ

Basically, when I put everything together and try to calculate fall velocity, I get stuck with a disgusting integral, because I use

V(t)=∫a(t) = (1/m)*∫F(t)

Does anybody have suggestions for how to approach this? I just want to make a model in matlab.. it seems like I could do some kind of step approach, because I looked at the integral and it's really nasty, but I don't know what to do, because I have V(t) on both sides...

Or if anybody knows of a simpler model presented in a paper, I could use that too. I just want to compare my data with a preexisting model; it's not critical to my project, but I think it's important.

 

[gomunkul51]

As I see it you need to write it in the from of a differential equation and Matlab will solve it (numericaly or otherwise):
Sum( Force( V(t) ) ) = mV'(t)
i.e. the sum of all the forces acting on the drop (drag ect.) - you need to write the those forces as a function of the drops velocity, equals the mass of the drop times acceleration (derivative of velocity with time).

Cheers.Roman.

 

[haruspex]

The most obvious numerical solutions might accumulate significant errors.
If you're interested in the way it approaches terminal velocity, you should try working with the dependent variable being the difference between V and Vt. You might then be able to make suitable approximations to obtain an analytic solution for the asymptotic behaviour.
But it would require knowing the values of the constants and figuring out what terms can be ignored.

 

[uluru]

Sweet, i think I figured it oUt. Thanks

 

[PJC01]

I completely understand your desire to compare your data with a preexisting model. It is always important to validate your findings and make sure they align with established theories and equations. In this case, your equation for calculating rainfall drop velocity from a given height is based on fluid and Newtonian mechanics, which are well-established principles in the scientific community. However, the integral you are facing may seem daunting and difficult to solve.

One approach you could take is to break down the integral into smaller, more manageable parts. This can be done by using numerical methods, such as the trapezoidal rule or Simpson's rule. These methods involve dividing the integral into smaller sections and approximating the area under the curve.

Another approach could be to use a computer program, such as MATLAB, to solve the integral for you. MATLAB has built-in functions and tools for solving integrals, making it a useful tool for scientists and engineers.

Additionally, you could also consider looking for alternative equations or models in existing literature that may be simpler and easier to use. This could save you time and effort in trying to solve the integral.

In summary, there are various approaches you can take to solve the integral and calculate rainfall drop velocity. It may require some trial and error, but with determination and the right tools, you will be able to compare your data with a preexisting model and further contribute to the understanding of raindrop parameters.