Fractal Scaling of Population Counts Over Time Spans

It's been 7 months since my last post.  The process of downsizing to a smaller home last winter put my projects on hold.

Although fractals have stubbornly refused to appear in my investigation of self-similar surface roughness, they have shown up at my day job as a data scientist at Digilant.

Investigating the possibility of combining weekly counts of unique user IDs, I discovered that the L^p-norm does so with surprisingly good accuracy on digital advertising datasets.  The L^p-norm implies a scaling law.  My son Martin (who also works at Digilant) noticed that the scaling law exponent is a fractal dimension.  The L^p-norm and scaling law are implied by the Pareto distribution of lifetimes in a population.  This link between the L^p-norm and fractal dimension should have application beyond counting populations.

We wrote a paper about these results at https://arxiv.org/abs/1806.06772

The Physics of Marbling

Last week I spoke on The Physics of Marbling at the Form in art, toys and games workshop at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge!

I am the one in the bright blue jacket.

The four day workshop had many fascinating presentations on a wide variety of topics.  Videos for most of the talks are available.

Mixed Convection from an Isothermal Rough Plate

In March 2017 the roughness of the aluminum plate was reduced from 3mm to 1mm.  I installed it in the wind-tunnel and started running experiments.  The forced convection measurements were nearly 30% higher than expected!

I examined nearly every aspect of the physical device and its mathematical model.  The fan-speed calibration was found to be sensitive to the distance between the test surface and the wind-tunnel wall.  Conditioning the rpm-to-speed conversion on the plate's orientation improved the earlier data taken with the plate with 3mm roughness.

When the plate is not parallel to the wind-tunnel, the forced and mixed measurements are affected.  With the 3mm roughness plate, the alignment had been controlled within a couple of millimeters over the plate's 305mm length.  The 1mm roughness plate seemed to require stricter tolerances.  Using a caliper, I am able to control the alignment to better than 1mm.

The primary cause of the measured excess was that, when the height of the posts had been reduced, the size and spacing of the posts had not been reduced.  At high wind-speeds the convection from the flat post tops was exceeding the "fully-rough" mode of convection.  The model incorporating this phenomena is developed in the "Rough to Smooth Turbulence Transition" section of my "Mixed Convection from an Isothermal Rough Plate" paper.

Writing a paper forces one to revisit all the questions and anomalies that occurred during research and experiment.  Understanding, resolving, and testing all of these issues has taken months.  I would appreciate any proof-reading or critiquing that others might provide before I submit it for publication.

Oseen Flow in Ink Marbling

Mathematical modeling of ink marbling has long been a fascination of mine.  My Ink Marbling web pages have presented emulations of a number of marbling techniques.  But the raking techniques modeled were either paths across the whole tank or circular paths.

Pictorial ink marbling designs are created using short strokes, where a stylus is inserted into the tank; moved a short distance; then extracted.  There seems to be no way to adapt the line or circle draws to short strokes with endpoints.

Having bought a copy of Boundary-layer theory (Hermann Schlichting et. al.) for my convection project, I started reading from the beginning.  It didn't take long until I found a description of Oseen flow on page 115 (chapter IV, very slow motion).  Its streamline figure looked very promising.  After further research I have written: Oseen Flow in Ink Marbling arXiv:1702.02106 [physics.flu-dyn].

Unlike a cylinder in a 2-dimensional flow, the velocity field induced by an infinitesimally thin stylus can be exactly solved in closed form.  In this sense, marbling is the purest form of Oseen flow.

The partial differential equations solved include conservation of mass (divergence=0), but not conservation of momentum (Navier-Stokes).  It's not clear how much momentum is imparted by the stylus, or how that imparting momentum changes with time.

Liquid marbling is sensitive to the speed of a stylus moving through the tank.  At low speeds the induced flow is laminar; at high speeds the flow becomes turbulent.  Both are used by marblers, but only the laminar flow is possible to solve in closed form.

Mixed Convection from a Rough Plate

Its been a long time since my last blog entry; there are new developments.

After completing the vertical convection measurements, I returned the Convection Machine to its horizontal orientation and did some runs to make sure everything was as before.  But things were not the same.

With the rough side facing down, the transition where mixed convection dropped below the linear asymptote had disappeared.  Varying parameters did not restore the dip.  Changing inclination of the plate; tilting the wind-tunnel; resealing the cardboard around the fan; nothing I tried restored the dip.  Something has permanently changed in the wind-tunnel or the plate.

So I reran the measurements of horizontal upward and downward facing mixed convection.  The new curves match simple L2 and L4 norms of the forced and natural convection components.  My guess of what changed has to do with the suspension of the plate.  The vertical suspension was a single long wire which, after hooking around the a top corner post, wrapped across the back and hooked around the other top corner post.  Wrapping around the back compressed the back sheet and insulation against the back side of the rough plate.  Perhaps the pressure closed gaps in the glue between aluminum and insulation.

Having data for horizontal forced flow with 3 plate orientations, it was time to measure downward forced flow with a vertical plate. Because vibration of the plate had caused excess convection with the single wire suspension, I added two wrap-around wires pulling in opposite directions to the plate suspension.  This new suspension is quite rigid and works with the wind-tunnel in any orientation.

I added four legs to support the wind-tunnel upright with the fan drawing downward.  I was expecting either L2-norm mixing or for convection to drop below the natural level when the natural and forced components were equal.  But it was neither!  I devised a model which transitions between L2 and L4 norm that matched the measurements well; it is detailed in my paper.

Because the opposed mixing was unexpected, aided mixing had to be tried.  It also turned out to involve a transition between L2 and L4 norms, but with a gentler transition.

I have finished writing the article and put it and the supplementary data on http://people.csail.mit.edu/jaffer/convect

As described in the paper, the next step is to shave 4.mm off the rough side of the plate and repeat the measurements.

Mixed Convection from a Vertical Rough Surface

I turned the wind-tunnel on its side and hung the plate vertically as shown in the photograph.



These graphs show that my mixed convection model is successful from natural through forced convection for horizontal and vertical rough plates with forced flow perpendicular to the natural flow.  The leftmost red dot in each graph is the natural convection (Re=0) for that orientation; it is placed at Re=1000 so that it can appear on the graph.


The L4-norm for downward convection indicates that the interaction between the downward mode and forced convection is more competitive than the fairly cooperative L2-norm of the vertical and upward modes.

Horizontal downward and vertical natural convection from the rough plate match that expected from a smooth plate.  Horizontal upward convection matches assuming that the upper (rough) surface convection is reduced by 93% of the non-forced convection from the four adjoining sides.  In order to test if horizontal upward convection is the same for rough and smooth, I will cover the rough surface with a flat sheet of aluminum and repeat the test.

I have started writing a paper titled "Mixed Convection from a Rough Plate".  Which journal should I submit it to?

Fan Windspeed

With the wind-tunnel fan being phase-locked now, the speed variability which plagued earlier speed measurements should be reduced or eliminated.

The process of conducting the measurements for these graphs finds this to be the case.  Although some variability remains above 3 m/s (1000 r/min), at slower speeds the anemometer readings are steady after the phase-lock-loop settles.  The measured traces are in blue; the black curve is that used in convection calculations.  This first graph is for the wind tunnel with horizontal plate.


 

This second graph is with the wind tunnel laying on its side with the plate vertical.



An earlier post found that the kink just above 2 m/s is due to the anemometer; it remains.  Friction in the anemometer makes the measurements below fan speeds of 700 r/min unreliable; below fan speeds of 200 r/min the anemometer reads 0.0.