How big is the influence of color on the stability of the boundary layer?

Question

I'm not sure if this is the best place, or physics.SE would a better place. I guess the aerospace engineers might be on both.

In class we were talking about stability of the boundary layer, and the influence of the temperature on the boundary layer.

The governing equation is given by Wiki boundary layer:

$$ u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}=-{1\over \rho} {\partial p \over \partial x}+{\nu}{\partial^2 u\over \partial y^2} $$

If we evaluate this $y=0$, where both $u$ and $v$ are zero:

$${1\over \rho} {\partial p \over \partial x}={\nu}{\partial^2 u\over \partial y^2} $$

We then deduced that, since $\nu=f(T)$, the instabilitiy of a boundary layer is thus a function of the temperature.

One of us mentioned that this was the reason that the Pepsi-livery Concorde had a lower maximum operating speed, since it is not white, but blue: This would increase the temperature at the wing, making the boundary layer less stable.

I did so a search, which brought me to this Concorde SST website, which says that the wings were kept white, to reduce the temperature of the wing, but because of the fuel temperature. As the wings are white, the effects we were talking about are not relevant anymore, but still it made me wonder.

Could the color of a surface have so much influence on the stability of the boundary layer?

Answer 1

Technically, yes. Practically, no.

Boundary layer stability

You are right that boundary layer stability depends on viscosity, but it depends much more on the pressure gradient. Accelerating flow exhibits extreme stability: Even if there is movement orthogonal to the flow direction (due to Tollmien-Schlichting waves), their speed becomes an ever smaller fraction of flow speed in accelerating flow. This kind of flow is always found right after the stagnation point, when the suction area over the mid section of the wing accelerates flow into it.

Conversely, decelerating flow has lower stability: Now any crossflow becomes relatively stronger simply because flow speed is decreasing. This kind of flow is found aft of the suction peak, when the flow has to manage the long and arduous pressure rise to the trailing edge. It does not take long, and transition into turbulent flow happens.

On the Concorde at high angle of attack, lift was created by a strong vortex. The boundary layer became turbulent right after the leading edge.

In cruise, the flow would hit the leading edge at a small angle, so the speed distribution over the wing was less peaky. The long chord length and high flow speed would still cause an early transition, however, so most of the wing had turbulent flow. Indeed, cooling the wing would lengthen the laminar portion, and heating would shorten it, but the difference in friction drag would be small when looking at the full picture.

Air temperature over the wing

In supersonic flow you always have both a temperature boundary layer and a viscous boundary layer. Air is heated when compressed and cools when it expands, and in supersonic flow these effects become too large to be ignored. Also, friction in the boundary layer heats up the air near the wing's surface.

The stagnation temperature increase at Mach 2 is 173°C. The leading edge temperature can be expected to be around 120°C (standard atmosphere air temperature being at -56°C in FL 600). Past the stagnation point, the flow accelerates into a suction field in which air temperature drops below its ambient value. At the bottom of the boundary layer, however, the temperature stays almost as high as at the stagnation point due to frictional heating. In a turbulent boundary layer on a flat plate the frictional effect would increase temperature by 152°C. Without the pressure distribution over the wing I cannot give a value for the absolute temperature, though.

Due to heat transfer with the wing, the air is cooled over much of the wing in the vicinity of the wing's surface. A dark coating will add a few degrees and result in a higher equilibrium temperature over much of the wing. A hotter boundary layer is thicker due to the expansion of air with temperature and will have higher viscosity. The most simple explanation why viscosity increases with temperature is that the mean molecular velocities, which are directly related to temperature, facilitate more transfer of shear forces between molecules. However, when it comes to wall friction, the effect of lower density with temperature outweighs the increase in viscosity, and wall friction is lower in a hotter boundary layer.

Thus, a heated boundary layer shows earlier transition and lower friction drag. The dark paint should actually help to lower the friction losses, and since the influence of temperature on the transition point is small, the change in friction due to lower density at the wall should dominate.

Maximum wing temperature

The maximum speed of the Concorde was limited by the stagnation temperature of air. If we assume that a switch from white to dark blue paint increases the equilibrium temperature of the wing leading edge by 10°C (which is not unreasonable), this increase could be compensated by a reduction of the maximum flight Mach number from 2.0 to 1.94. A restriction of the Pepsi Concorde to Mach 1.7 seems to me to be the result of extreme (and unreasonable) caution.

My verdict

I did not run the numbers, but I would expect that a different paint job would not change wing temperature much. At the leading edge and over much of the wing it adds a little to aerodynamic heating. A warmer wing would translate into less friction, but the difference is extremely small.

On the ground, in the absence of the strong convective heat transfer during supersonic flight, the temperature difference between white and dark wings is much more pronounced (in the order of 30°C to 50°C), and in this case a white wing is very effective in keeping the fuel temperature down and avoiding gas bubbles in the fuel line.