femr2 wrote:Did a quick test with Full scale and 1:400 scale, and flow rate accross the boundary comes out with the same value for each...
Probably not, but I think we're approaching things differently. I chose not to scale all
lengths identically for a few reasons. While the tower size model is supposed to be as much like a real tower as practical, it is also a model in my scenario. Some aspects differ because of practical necessity, others to illustrate the desired point.
The surface to volume ratio is independent of scale, and it's also independent of vertical displacement of a 'plunger' through the interior space. If the two models have all the same proportions, all geometric ratios and things having dependence on relative lengths will be the same by definition. Something had to be different (but it probably clouded the issue to have too many differences).
The important differences are velocity and side width to story height ratio.
For a moment, forget about the core and consider large and small models identical except for (square) footprint width, story height and story count. Let y
be the vertical displacement of a monolithic debris zone, upper block, slab, plunger, what have you. Let x
be the generalized position coordinate of the leading edge of a volume of incompressible fluid extruded through only the exterior via a surface of one story height. The width of a side and the height of a story are w
. The boundary condition is the value of y
By considering only a small vertical displacement, the change in exit surface area during the displacement can be ignored without substantial error, so I'll focus the situation when the slab first enters a given story volume, something I didn't do earlier with the small model. We can see what the instantaneous changes are at that point in time using the constant exit surface of the four wall planes (where there are no walls). Moreover, changes of velocity over a small distance can be neglected.
This is not precisely the mechanics of air flow and nothing like a real tower, but applicable to the small model I have in mind. That's also why I used the word 'crude' three times in one sentence.
The change in volume for a vertical displacement Δy
is ΔV = w²Δy
The (constant) cross-sectional area through which the fluid initially escapes isA = 4wh
The way I've framed the problem dictates solving for the velocity in the x direction as a function of the givens which include velocity in the y direction. So, here goes, with dot notation used as shorthand for the velocities.(edit: ignore the double equals signs; typo)
The sensible thing to do, before plugging in real values for either model, is examine the dependencies above and the relation between the horizontal velocities in two systems. The horizontal velocity varies in proportion to the width and vertical velocity, and inversely with the story height, as one might expect. In taking the ratio of two systems:
you see identically proportioned models will only have a different exit velocity if the vertical velocity differs, entirely length-scale independent. If this is why you get the same result for both of your models, it makes sense and perhaps you can see why I chose different proportions to illustrate the point. Besides, I'm not considering a perfect replica of the tower in the small because that's not what's been done or proposed.
If you used different absolute velocities but got the same exit velocity, I think you made a mistake.
The ratio of story height to width, as well as the relative velocities, determine the disparity between exit velocities. With the real tower or a model of comparable dimensions, the velocity can exceed the theoretical limit of a freefall descent for a small model by having even a modest terminal velocity (hence a lowballed figure from the actual collapse). While it would be simple to build a small slab model with the same overall height:width ratio as a real tower, it's not the total height that matters here but story height, and incorporating 110 stories would be no minor effort. Plus it would provide little delta PE between collisions!
A manageable small model must have at least 10 stories to qualify for a non-issue like the Heiwa Axiom, which is frequently an issue despite the absurdity. Excluding slab height, that's a 10cm story in a 1m high model - barely tabletop size already despite having only 10 floors with a modest drop between each. Hence the 10cm height I chose for the small model. The same figure for the base makes it a model like what might be constructed using the bathroom tiles I showed towards the beginning of the thread. Simply a practical choice, nothing special.