Hello. First, this thread was in "homework", but I thought this probably wasn`t a "homework" because is a lot more conceptual , and maybe deeper, than a simple exersise. Not more difficult though. I apologize if someone feels bad because I copied the thread here.
I have some problems analysing the results of a simulation of Rayleigh-Benard problem. I`m confused with a energy balance. Since it`s more a general question, I`ll describe first the situation. Please, be patient, I`m going to describe the problem carefully in order to help the understanding :
In a vertical rectangle (width 2, heigh 0.5), you have air inside. It´s a bidimensional problem. Side walls are adiabatic (insulated), always.
On t=0, air is at temperature T=0 (non dimensional temperature), and top wall is at temperature T=0 too. At the same time, on bottom wall is imposed a temperature T=1, that remains. So the bottom wall temperature is constant. Air temperature, upper wall temperature, and side walls are not. Rayleigh number is enough to make convection possible.
So as time elapses, there will be heat transpor, and in some time, it will be done by convection (creating some regular patterns of flux), and the system will reach an steady state (where convection exists)
When I plot Nusselt number on top wall, and on bottom wall, I see they converge to a value , i.e they have the same numerical value, as the system goes to a steady state. Nusselt number is defined as (H/k)*(q/(T1-T0), that`s , the heigh, divided by the fluids thermal conductivity, multiplied by the heat flux, divided by the temperature difference betweeen walls (dimensional temperature).
If Nusselt numbers (at bottom and top wall) are the same, that should imply that the heat fluxes are identical too. So all the energy that enters by the bottom wall, leaves by the upper surface (top wall). Now, where does the energy necessary to maintain convection (the velocity of the fluid and the patterns) come from? In other way, to make the fluid get some velocity, and get regular patterns, you need work. But where the system gets it, if all the energy that`s supplied, leaves.
Thank for your time