Does aerodynamic heatings at the wall depend on the material of the wall itself?

I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. He said:

"The slope of the temperature profile at the wall is very important; it dictates the aerodynamic heating to or from the wall. Let $(dT/dy)_{y=0}$be definedas the temperature gradient at the wall. Then the aerodynamic heating rate (energy per second per unit area) at the wall is given by: $$q_{w} = -k\left(\frac{dT}{dy} \right)_{y = 0}$$ where k is the thermal conductivity of the gas, and the minus sign connotes that heat is conducted from a warm region to a cooler region, in the opposite direction as the temperature gradient. We note that k is a physical property of the fluid, and is a function of temperature"

It seems like the aerodynamic heating does not depend on the material of the wall, which in normal sense, it should depend. Why was that ?

• I'm more confused as to why the temperature profile increases through the boundary layer and then decreases again. Surely temperature doesn't normally 'bunch up' like that? – Time4Tea Aug 15 '18 at 0:54
• @Time4Tea I strongly believe that curve was arbitrary. The important point here is the author want to express the term dT/dy at the wall. – Dat Aug 15 '18 at 15:19
• You are probably right that it is arbitrary. I think it's very confusing though, to use such a non-physical diagram as an example in a text book. It implies that heat is being generated within the boundary layer and flowing in both directions to the free stream and the wall. – Time4Tea Aug 15 '18 at 15:55

1 Answer

This just describes the heat flux and temperature gradient on the air side of the well. There is also heat conduction occurring on the solid side of the wall (and a similar equation can be written for that), and both the heat flux and temperature must match (i.e., be continuous) at the interface.

• It should be the heat flux on the air side. But keep reading the solutions for Couette flow, I think the author does not think that. avionicsengineering.files.wordpress.com/2016/11/… Please go to page 937, below the equation (16.23) you will see that the heat flux from air into the wall or vice versa – Dat Aug 15 '18 at 15:29
• Yes. I saw the equation. So? – Chet Miller Aug 15 '18 at 15:53
• So the equation does not include any information about the property of the material of the wall but it is still the heat flux from the air into the wall ? – Dat Aug 15 '18 at 15:55
• Well, like I said in my answer, it has to couple to the heat flow within the wall material, through matching both the temperature at the wall and the heat flux at the wall. The temperature of the wall material and the heat flux of the wall material has to match the air wall temperature and the air wall heat flux. – Chet Miller Aug 15 '18 at 17:30
• Do you mean: $q_w = k_{property of the air}(dT/dy)_{air side} = k'_{property of the material of the wall}(dT/dy)_{wall side}$ ? Because k $\neg$ k' so dT/dy is discontinuous at the wall ?? Am I right? – Dat Aug 16 '18 at 2:38