Equal Wall Temperatures

Here we assume that Te = T„; that is, he = h„. The enthalpy profile for this case, from Equation (16.16), is

. 2

or

hw + – Pr u]

In terms of temperature, this becomes

Note that the temperature varies parabolically with у, as sketched in Figure 16.5. The maximum value of temperature occurs at the midpoint, у = D/2. This maximum value is obtained by evaluating Equation (16.33) at у = D/2.

The heat transfer at the walls is obtained from Equations (16.24) and (16.25) as

At у = 0: qw = r-‘-^ [16.35]

At у = D:

Equations (16.35) and (16.36) are identical; the heat transfers at the upper and lower walls are equal. In this case, as can be seen by inspecting the temperature distribution shown in Figure 16.5, the upper and lower walls are both cooler than the adjacent fluid. Hence, at both the upper and lower walls, heat is transferred from the fluid to the wall.

Question: Since the walls are at equal temperature, where is the heat transfer coming from? Answer: Viscous dissipation. The local temperature increase in the flow as sketched in Figure 16.5 is due solely to viscous dissipation within the fluid. In turn, both walls experience an aerodynamic heating effect due to this viscous dis­sipation. This is clearly evident in Equations (16.35) and (16.36), where qw depends on the velocity ue. Indeed, qw is directly proportional to the square of ue. In light of Equation (16.9), Equations (16.35) and (16.36) can be written as

[16.37]

which further emphasizes that qw is due entirely to the action of shear stress in the flow. From Equations (16.35) to (16.37), we can make the following conclusions that reflect general properties of most viscous flows:

1. Everything else being equal, aerodynamic heating increases as the flow velocity increases. This is why aerodynamic heating becomes an important design factor in high-speed aerodynamics. Indeed, for most hypersonic vehicles, you can begin to appreciate that viscous dissipation generates extreme temperatures within the boundary layer adjacent to the vehicle surface and frequently makes aerodynamic heating the dominant design factor. In the Couette flow case shown here—a far cry from hypersonic flow—we see that qw varies directly as u2e.

2. Everything else being equal, aerodynamic heating decreases as the thickness of the viscous layer increases. For the case considered here, qw is inversely proportional to D. This conclusion is the same as that made for the above case of negligible viscous dissipation but with unequal wall temperature.