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.