Qualitative Considerations

We consider in a qualitative way the influence of the thermal state of the surface on some viscous flow properties. These are the viscous thermal surface effects. The results hold in general, i. e., for two-dimensional and for three­dimensional flow.

We treat first—with the help of the reference-temperature concept—the influence of the wall temperature Tw at flat surface portions. Basically we collect a few results from Section 7.2.2 They are ordered as they appeared in Table 1.3 of Section 1.4. It is assumed that all temperatures are above 200 K, such that the viscosity can be approximated with ш = 0.65, eq. (4.15), Sub-Section 4.2.2.

The reference temperature T* is composed of the static temperature at the outer edge of the boundary layer Te, the wall temperature Tw, and the recovery temperature Tr, eq. (7.62) in Sub-Section 7.1.6. The wall temper­ature with 50 per cent has the largest share. We therefore formulate the observations made from Table 10.1 in terms of Tw.

The boundary-layer thickness 6 increases with increasing wall tempera­ture, for the laminar boundary layer stronger than for the turbulent one. The same holds for the displacement thickness 61. The characteristic boundary – layer thickness A, which governs the skin friction and the heat flux in the gas at the wall, for laminar flow is the boundary-layer thickness 6 and for turbulent flow the thickness of the viscous sub-layer 6vs [3]. We note that 6vs increases with increasing Tw much stronger than the laminar boundary-layer thickness 6.[171] [172]

The skin friction is affected according to the increase of the characteristic boundary-layer thickness A. To understand this better, we approximate the skin friction relation by

This approximation shows that an increase of A reduces the skin friction. Because the thickness of the viscous sublayer rises much stronger than that of the laminar boundary layer, the increase of Tw lowers the turbulent skin friction stronger than the laminar one. (Note that the viscosity pw in both cases is the same.)

The relation of the heat flux in the gas at the wall qgw can be treated similarly. We look only at the radiation-adiabatic wall and see that an increase of Tw lowers both qgw and hence also the radiation-adiabatic temperature Tra. Again the influence is much stronger for turbulent than for laminar flow.

Now we consider the tangential velocity profile u(y), Fig. 7.4 c). We ask for the influence of the temperature gradient in the gas normal to the vehi­cle surface dT/dngw. A This influence is similar to that of the stream-wise pressure gradient or that of wall-normal suction or injection (blowing). The discussion of the wall-compatibility conditions in Sub-Section 7.1.5 gives us the clue to that. (We consider the effect of the temperature gradient isolated. If other effects are present, they may increase or weaken it.)

Consider eq. (7.53) for the ж-direction and there the last term on the right-hand side. For the case of a radiation cooled surface, the heat flux in the gas is directed toward the wall (cooling of the boundary layer), hence we have dT/dygw > 0, Sub-Section 4.2.1. Because du/dy > 0 (attached viscous flow), and dp/dT > 0 (we consider air), the heat flux has the same effect as a favorable pressure gradient (dp/dx < 0): the second derivative d2u/dy2y=0 is negative, the boundary-layer profile is fuller than the Blasius profile, case 2 in Fig. 7.4 c). Due to the inverse of the viscosity ahead of the terms on the right-hand side, a high wall temperature reduces the effect, a low one enlarges it.

The negative temperature gradient also affects the stability behavior of the boundary layer. First modes are damped, second modes are amplified, Sub-Section 8.1.4.

If the heat flux is directed away from the wall, dT/dygw < 0 (heating of the boundary layer), the heat flux has the same effect as an adverse pressure gradient has. A point of inflection appears, the boundary layer becomes prone to separation. The effect is stronger for a low wall temperature than for a high one.

Of course also of interest is the distribution of the density p across the boundary layer. In attached viscous flow past surfaces with small curvature at high Reynolds numbers, the pressure p is nearly constant across the boundary layer (first-order boundary layer [3]):

P (y) ~ Pe ~ const., (10.2)

i. e., the pressure is approximately equal to that at the boundary-layer edge. Consequently in the boundary layer

pT = peTe = const., (10.3)

and

poc^. (10.4)

This means that at a hot wall the density is small at and above the wall, and at a cold wall vice versa. The average tangential momentum flux < pu2 > in the boundary layer is affected. At a cold wall this flux is larger than at a hot wall and the boundary layer can better negotiate, for instance, an adverse pressure gradient than a hot one. Also the three-dimensionality of the flow is affected [3].