How can I predict the convective updraft speed above a stove element? I want to know the rate at which air will ascend above a stove element.
Is there some relationship between ambient air temperature, element temperature, and speed of the resulting updraft?
If nothing definite, is there anything approximate that I could use?
 A: In this book they first non-dimensionalize the NS equations and then, assuming terminal velocity, small temperature differences, and using scaling arguments they arrive at the following relationship for the terminal speed of a particle moving buoyantly in a stratified flow:
$V = \frac{g \alpha \Delta T r^2}{6 \pi \nu}$


*

*$\alpha$: Coefficient of thermal expansion

*$\Delta T$: Temperature difference

*$g$: Gravitational acceleration

*$r$: Radius of spherical fluid element under consideration

*$\nu$: Kinematic viscosity
This is roughly what happens, I assume a lot of things as it is necessary for a sane understanding of the subject. Initially, a (stationary) stably stratified homogeneous fluid is above the stove element. There are roughly three stages, generation, evolution, steady state.
In the first stage, the element is turned on, fluid particles nearest to the element experience a change in temperature (positive), kinetic energy increases and nearby fluid parcels propagate such information upwards (can't go down, forget sides) via diffusive heat transfer (in the initial stages). At this point in time particles have barely moved from the surface (considering the total timescale we are interested in). However, this process effectively gives rise to considerable upwards fluid velocity in fluid that is adjacent to the heat element.
In the next stage, some fluid has accelerated to the point that we can now physically discern fluid motion. Diffusion is no longer important (it never really was), at this point advection takes the lead. A fluid parcel along the bottom, the warmest one with respect to all the other fluid parcels along the bottom, will, at this stage, begin to feel a little different from its neighbors. Since the fluid parcel is warmer, it is also lighter than all the fluid around it, so, according to Archimedes it must go up. This is very similar to the way the sun heats up the earth every morning and generates thermals. There will be lots of different convection cells owing to imperfections in the heating element, which will in turn give rise to eddying motion and turbulence.
At steady state, the fluid will be moving away from the element at a steady pace in a way that you can average out all the nuances of fluid dynamics, so that you can arrive at that formula. Sometimes though, I just wish we could give someone a million dollars.
