*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. |

Governance by those who do the work.

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!

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

I am the one in the bright blue jacket. |

Labels:
fluid mechanics,
Ink Marbling,
Mathematics,
Oseen flow

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.

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

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.

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.

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.

Labels:
correlation,
experiment,
L2-norm,
L4-norm,
Mixed Convection,
Wind-tunnel

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?

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?

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.

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.

Using an auto-transformer to reduce the voltage to the wind-tunnel fan

in order to reduce its speed didn't work below 45 r/min (it ran for a

while and stopped). So I modified The Convection Machine to toggle

the fan power with a solid-state relay controlled by micro-processor.

Consider the shaft of the wind-tunnel fan. Every full rotation of the

fan results in 3 micro-processor interrupts. A phase-accumulator

register is incremented by the desired rotation rate (in r/min) 1200

times a second and decreased by 24000 every time a fan blade crosses a

light beam. If the fan rotates at the desired rotation rate, then the

average phase-accumulator value is constant. If the fan is too slow,

then the phase-accumulator value increases with time; if it is too

fast, then the phase-accumulator value decreases with time.

Phase-locking is the process of controlling the fan-speed so that the

average phase-accumulator value is 0. Such feedback systems are

tricky to stabilize. My fan controller operates in one mode when the

desired speed is less than 400 r/min and a different one otherwise.

At high speeds, the fan speed is roughly proportional to the average

voltage applied, which is proportional to the duty cycle of applied

voltage. The phase accumulator operates as described above. Its

instantaneous value is compared with a variable which decrements from

the upper phase range bound to 0 ten times a second. If greater, the

fan is turned on, otherwise it is turned off. Some of the

instabilities of the fan speed may be due to a centripetal switch

disconnecting the starter capacitor and hooking in the running

capacitor, which increases the loop gain of the system. The change in

gain causes the system to overshoot and undershoot the desired r/min

with long settling times.

At low speeds each pulse of power incrementally increases the fan

speed while friction continually slows it. The solid-state relay has

"zero-crossing" control, so only complete half-cycles of 60 Hz power

are applied to the fan motor. The combination of the motor windings

and phase capacitor stores energy, so the acceleration of the rotor is

delayed from the application of power. At low speeds the rotational

inertia of the rotor introduces 90 degrees of phase shift. The

microprocessor clock is not synchronized to the line voltage, so the

minimum pulse width varies with the relative phase, another source of

loop gain variation.

This photo shows the new fan-speed control. The number on the

7-segment displays is the rotation rate in r/min measured every

second. The right 3 dial switches set the desired rotation rate.

This video which shows the phased-locked fan in

operation at a variety of speeds. The low light level was necessary

so that the stroboscopic interaction of camera shutter with the

scanned 7-segment display didn't render the numbers unreadable. If

you turn up the audio volume you can hear the fan chugging as its

power is switched on and off.

in order to reduce its speed didn't work below 45 r/min (it ran for a

while and stopped). So I modified The Convection Machine to toggle

the fan power with a solid-state relay controlled by micro-processor.

Consider the shaft of the wind-tunnel fan. Every full rotation of the

fan results in 3 micro-processor interrupts. A phase-accumulator

register is incremented by the desired rotation rate (in r/min) 1200

times a second and decreased by 24000 every time a fan blade crosses a

light beam. If the fan rotates at the desired rotation rate, then the

average phase-accumulator value is constant. If the fan is too slow,

then the phase-accumulator value increases with time; if it is too

fast, then the phase-accumulator value decreases with time.

Phase-locking is the process of controlling the fan-speed so that the

average phase-accumulator value is 0. Such feedback systems are

tricky to stabilize. My fan controller operates in one mode when the

desired speed is less than 400 r/min and a different one otherwise.

At high speeds, the fan speed is roughly proportional to the average

voltage applied, which is proportional to the duty cycle of applied

voltage. The phase accumulator operates as described above. Its

instantaneous value is compared with a variable which decrements from

the upper phase range bound to 0 ten times a second. If greater, the

fan is turned on, otherwise it is turned off. Some of the

instabilities of the fan speed may be due to a centripetal switch

disconnecting the starter capacitor and hooking in the running

capacitor, which increases the loop gain of the system. The change in

gain causes the system to overshoot and undershoot the desired r/min

with long settling times.

At low speeds each pulse of power incrementally increases the fan

speed while friction continually slows it. The solid-state relay has

"zero-crossing" control, so only complete half-cycles of 60 Hz power

are applied to the fan motor. The combination of the motor windings

and phase capacitor stores energy, so the acceleration of the rotor is

delayed from the application of power. At low speeds the rotational

inertia of the rotor introduces 90 degrees of phase shift. The

microprocessor clock is not synchronized to the line voltage, so the

minimum pulse width varies with the relative phase, another source of

loop gain variation.

This photo shows the new fan-speed control. The number on the

7-segment displays is the rotation rate in r/min measured every

second. The right 3 dial switches set the desired rotation rate.

This video which shows the phased-locked fan in

operation at a variety of speeds. The low light level was necessary

so that the stroboscopic interaction of camera shutter with the

scanned 7-segment display didn't render the numbers unreadable. If

you turn up the audio volume you can hear the fan chugging as its

power is switched on and off.

Labels:
Phase-Lock,
Phased-Lock-Loop,
software PLL,
Wind-tunnel

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