Rain falling on a tree
I have always be fascinated by the problem of rain falling on a tree in Summer. The issue is that you can stay dry under the tree even though it is raining hard. However, one has the suspicion that this is a temporary phenomenon and that eventually the rain must "saturate" the tree and start to drip through onto you.
The general question that intrigues me is - how long does it take for the tree to start having water drip through, and how much of the rain is directed down the branches and to the base of the trunk?
That is the starting point for this problem. I think we will need to discuss this in more detail before moving to a mathematical description.
Additional Discussion can be added:
One of the first questions that I have when I started to think about this, was - at what level would I need to model this ? Should I think about the dynamics of the individual rain drops striking a leaf ? Or can the focus be on the tree itself with an aggregate flow of water on to it ? Or should I split the tree into bits and think about those ?
Please post your own thoughts here:
Here are my thoughts:
If you think about the individual raindrops falling on the leaves, you have a lot of different modeling to do with disturbances. Rain does not fall at a constant rate, nor is it predictable. But say that forcasters were finally able to do what they have been trying to do for centuries and you can predict the rate at which the raindrops fall. Then as you know, rain does not always fall straight. Since it does not always fall straight, that would give preference to which leaf the rain hits and slides down upon. But for this instance, let's say that the rain falls straight down and we have a constant flow rate out of the clouds.
Then the next question possed is, how should it be modeled? If one split the tree into parts of branches, the model would be of a series of buckets empting onto a conveyor and finally into one large dump (the tree stump). If you then took into account some sort of "splash over" that would model the rain that actually falls straight down and does not slide down the branch to the trunk. One issue I think you might have is the branches do not all have the same angel in which they join with the tree and also water slides off the branch before it hits the tree.
Those are some of my initial thoughts. What do you think? Mary B.
So first, you have identified that we need to be concerned with two dynamic processes. The first is the rainfall itself, which could be very complicated. However, as you point out, assuming the rain is straight down and constant is probably reasonable - why ? Because intuitively it would be very surprising if it was the dynamics of the rain that eventually decided the question of why someone can stand underneath a tree and not get wet for some period of time after the rain starts.
Having dismissed the dynamics of the rainfall itself (other than the fact that it has a beginning) as the explanation for why one can stand under the tree and not get wet that leads to the dynamics of the rain falling on and through the tree. The proposition suggested by Mary is to have a "series of buckets" emptying onto a conveyor that moves the water to the stump. I am not quite sure how to begin modeling this:
- at what speed does the conveyor move ?
- how do I describe the "buckets dumping" ?
If I can interpret the spirit of your suggestions (and feel free to reinterpret) you want some finite capacity, represented by the buckets that overflow into a channel, and which may provide flow to other channels as well. The channels themselves would need to combine as we go down the tree. This would lead to two basic dynamic units that we need to describe.
- the bucket and its overflowing
- the channel and its flow
I think the first of these should be quite straightforward, consider the overflow from a distillation tray as the basic prototype, and you get a wier equation which comes down to the flow being proportional to the square root of the free height of liquid above the top of the bucket.
The second model is flow in an open channel, this is the sort of thing that civil engineers know more about, but it should be possible to find the equations in a basic fluid mechanics text.
This is only one interpretation of the above comments and there is still room for other competing models. Let's keep the discussion going and try to converge on a model.
Dr. R.
Dr. R,
You did interpret what I was saying. Thanks for putting it in an easier way. I will work some on the modeling equations. My roommate is a civil engineer so maybe she can help me out with the open channel equation. Once you get these two models, would one put them in series to help describe the flow?
Thanks,
Mary B.
First, I would try the arrangement as you described, a single "tray" connected in series with an open channel, study how the flow out of the channel varies as a function of time with a step change of the input of "rain" into the "tray". Then I think there are a number of directions in which one could go. Note this sounds simple but has an issue with how to handle the "distributed parameter" open channel model.
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