# Temperature profile for a boundary layer

So I was reading the textbook today, and I see this picture for the temperature profile through the boundary layer.

And I was confused on why the shape looks like this? Should it not look more like this where the temperature is hotter closer to the surface, and gradually cools down as we move away from the surface? • Shock waves also affect the gradient, see Why does the temperature rise in the boundary layer after a shock wave? So: 1) please share the context of the images (what do the references say about each?) 2) Why and how questions are always better, so consider turning the title into a question, e.g. How do the two different gradients compare? and revise the body accordingly.
– user14897
May 16 at 5:46
• Does "the textbook" (which one?) give any boundary conditions? I would think the temperature profile only looks like this when the surface temperature ($T_w$) is lower than ambient ($T_b$)? The textbook profile looks like it does increase towards the surface (higher speed gradient, so more "friction" and heat production, just like you suggest in the second plot?), but then cooling towards the wall? May 16 at 12:59
• @Raketenolli, the textbook I’m referring to is “fundamentals of aerodynamics”. I think the only condition it gave was no slip. I don’t understand why the temperature would cool down close to the wall. May 16 at 13:59
• What if the wall is much cooler than the gas? May 16 at 14:16

Look at the positions of $$T_w$$ and $$T_b$$ relative to the $$y$$-axis - $$T_w$$ is lower than $$T_b$$.

So the question then, is why does the temperature increase in the middle of the boundary layer in the first diagram and not for the second?

The second diagram is graphing stagnation temperature, not static temperature. When turbulence/viscosity converts kinetic energy to thermal, it increases the static temperature but the stagnation temperature remains constant. (Stagnation temperature is the total of thermal energy + kinetic energy). So the bulge in the first image is because the static temperature increases from turbulence/viscosity converting the kinetic energy, no bulge in the second because it's showing stagnation temperature which already includes the kinetic energy.

• Thank you. So the first diagram is static temperature, and it’s zero at the wall because the kinetic energy there is zero, and it increases due to friction. And then it falls due to the cooler air outside of the boundary layer. Does that sound correct? May 17 at 3:25
• It's not zero at the wall, it's equal to $T_w$ at the wall - the distance of the line from the y-axis represents static temperature, and the dot marked $T_w$ isn't on the y-axis. $T_w<T_b$, so a plot of stagnation temperature would decrease continuously towards the wall - looking like the velocity profile with a constant value added. Because it's static temperature, it increases in the middle of the boundary layer where the conversion of kinetic to thermal energy is strongest
– sqek
May 17 at 9:36
• thank you very much. I understand it now. May 17 at 13:36

The wall temperature determines t$$_w$$ due to heat transfer between wall and the air close to it.

The boundary temperature t$$_b$$ is simply the temperature of air outside and at the outer limit of the boundary layer. So much is obvious.

In between, friction will heat up the air, and the steeper the speed gradient is, the more friction heat will be set free. This means that the boundary temperature will increase only slowly as one moves into the boundary layer from the outside. Closer to the wall frictional heating will increase and have its peak close to the wall, where cooling from contact to the wall is still small. Directly at the wall this cooling effect from contact with the wall will become dominant, so there is again a steep drop in air temperature.

Only when the wall has been heated up by the boundary layer such that there is no cooling effect, the temperature profile in the bottom graph will occur. This is the case for a surface of little heat capacity and an equilibrium condition with no more heat transfer between air and wall.