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.

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?

Mixed Convection

As my blog post Upward Natural Convection details, there is no guarantee that natural convection of my plate with the rough surface facing up behaves in a manner which can be modeled. The air warmed by the insulated back and sides rises adjacent to the heated rough surface which draws air towards its center. So how strong is this effect? Measured with peak temperature differences of 15 K, 10 K, and 5 K, the measured upward natural convection is about half that predicted. But looking at the natural convection components, the deficit is roughly equal to sum of the back and side convective heat flows!
With the rough surface facing down, the mix of convective and radiative heat loss from the four sides matters little because both are subtracted from the overall heat loss. But when the rough surface faces upward it does matter; side convection reduces the rough surface convection while thermal radiation does not. Creating a plot of downward natural convection at a range of temperature differences allows evaluation of simulated mixtures. As the fraction of simulated radiative heat loss increases, the slope connecting the measured points increases. The graph below shows the fit when the effective radiative height of the side is 41% of its actual height, and the effective convective surface area is adjusted to fit:

With this rough estimate of the relative strengths of convective and radiative heat loss, we are now ready to see whether the effect of the sides on the top surface can be reasonably modeled.
The plot below compares upward convection correlations with total non-radiative heat flow minus 77% of the (modeled) sides and back natural convection. There is less variation from point to point because upward natural convection is three times stronger than downward natural convection. "Horizontal Hot Top" is my generalization of the four conventional upward convection correlations.

The unexpectedly close match above lends support to the idea that air heated by the sides is drawn over the upper surface of the plate, reducing the effective temperature difference between the plate and air, and can be modeled as a reduction of upward convection by an amount proportional to the side convection.
Now that I have convection measurements at low fan speeds which are comparable in magnitude to natural convection, the next step is to evaluate mixed convection (with the rough surface facing upward).
Just as there was no guarantee of a workable model for the interaction of the sides with the top in still air, there is no guarantee of a model of that interaction in forced air. Needed is a generalization matches the model developed for V=0 and matches the forced correlation as V grows.
The graph below shows the correlation I have arrived at for upward convection (see simulations). It assumes that mixed convection for the four short sides is the L⁴-norm of the natural and forced convections and that 77% of only the natural component of the back and sides is absorbed by the convection of the (upward-facing) rough surface.

If the L⁴-norm mixing for the short sides is instead L², the increase in natural convection from the sides reduces the top surface convection, spoiling the upward-convection match with L²-norm (gray dashed line) and other L-norm exponents (best is about L^2.3).
Here are the calculated values of convection from all six sides of the plate at windspeeds from 0 to 4 m/s and ΔT=11K:

insulated back	+ 2 ⋅	parallel side	+ 2 ⋅	windward leeward	=	total	vs	rough	@	windspeed
53.1mW/K		55.3mW/K		55.3mW/K		0.272W/K		0.467W/K		0.0m/s
55.2mW/K		55.4mW/K		55.4mW/K		0.274W/K		0.484W/K		0.12m/s
60.4mW/K		55.5mW/K		55.4mW/K		0.280W/K		0.520W/K		0.25m/s
65.0mW/K		55.9mW/K		55.9mW/K		0.286W/K		0.666W/K		0.50m/s
68.5mW/K		57.3mW/K		57.6mW/K		0.296W/K		1.18W/K		1.0m/s
71.0mW/K		61.5mW/K		63.4mW/K		0.318W/K		2.22W/K		2.0m/s
72.6mW/K		72.4mW/K		78.4mW/K		0.371W/K		4.36W/K		4.0m/s

I had expected L⁴-norm mixing for upward convection. So these results will require modifications to my theory of mixed convection.