Thermal penetration depth animation (rev October 1996)



First frame is just the required Los Alamos disclaimer.



Second frame shows a central cross section of a long resonator,

with plots of pressure and velocity of the fundamental mode 

in the resonator.



Third frame adds lines in the gas in the resonator, showing how 

elements of gas move in the standing wave.  Imagine tracer particles.

Also displays temperature throughout the resonator instead of

velocity.



But this temperature plot shows a nonzero oscillating temperature at 

the walls, which is not true if the walls are massive and hence have

substantial heat capacity.  This coarse view has assumed a negligibly 

small thermal penetration depth. (or perfectly insulating walls!)  

Yellow circles in fourth frame show region to be magnified in the 

next frame.



Fifth frame shows solution to 2-dimensional heat-transfer equation, 

in low-amplitude approximation, assuming these boundary conditions:  

an isothermal, stationary wall at the left side, and uniform oscillating 

pressure with no motion of the gas in the vertical direction. 

Each horizontal tic mark is one thermal penetration depth.

The moving, wavy line is the temperature in the gas as a function of

position and time.  The moving horizontal line is the pressure, and the

moving vertical lines are tracer-particle lines in the gas.



The pressure oscillations in this thermal boundary layer are lossy;  the

"spring" of the compressible gas is hysteretic, because the adiabatic and

isothermal compressibilities are different.  To see this, imagine in the 

sixth frame that one of the tracer lines is replaced by a piston, and

ask how much work does that piston do on all the gas to its left.  



The work per cycle is the cyclic integral of pressure times differential

in volume p dV.  Since the intersection of the pressure line and the 

piston-face line mark pressure and volume at each instant of time, the area 

traced out by that intersection, shown in the seventh frame, is the desired 

integral.  Direction of rotation around the loop gives the sign of the work.



We could have put our imaginary piston at some other location, and obtained

a different value for the work done on all gas to its left.  Bottom trace 

in the eighth frame shows that work as a function of location of the

imaginary piston.



The derivative of that work with respect to y is displayed in the ninth 

frame. This shows the dissipation per unit volume as a function of y.  It is

greatest about one thermal penetration depth from the wall, which is where

the gas is farthest from both of the two ideal, reversible cases: adiabatic

and isothermal oscillations.  Note also the interesting fact that the gas 

roughly 4 penetration depths from the surface does work on its 

surroundings, rather than dissipating work.