Results for Compressible Couette Flow

Some typical results for compressible Couette flow are shown in Figure 16.9 for a cold wall case, and in Figure 16.10 for an adiabatic lower wall case. These results are

u_ A = (y – l )PrM;

ue

(a) (b)

A = (y- )PrM}

obtained from White (Reference 43); they assume a viscosity-temperature relation of ц/цтах = ("/’/ ‘/’ref)2/3> which is not quite as accurate for a gas as is Sutherland’s law [Equation (15.3)]. Recall from Section 15.6 that a compressible viscous flow is governed by the following similarity parameters: the Mach number, the Prandtl number, and the ratio of specific heats, y. Therefore, we expect the results for compressible Couette flow to be governed by the same parameters. Such is the case, as illustrated in Figures 16.9 and 16.10. Here we see the different flow-field profiles
for different values of the combined parameter A = (у – l)Pr М]. In particular, examining Figure 16.9 for the equal temperature, cold wall case, we note that;

1. From Figure 16.9a, the velocity profiles are not greatly affected by compress­ibility. The profile labeled A = 0 is the familiar linear incompressible case, and that labeled A = 30 corresponds to Me approximately 10. Clearly, the velocity profile (in terms of u/ue versus y/D) does not change greatly over such a large range of Mach number.

2. In contrast, from Figure 16.9b, there are huge temperature changes in the flow; these are due exclusively to viscous dissipation, which is a major effect at high Mach numbers. For example, for A = 30 (Me ~ 10), the temperature in the middle of the flow is almost five times the wall temperature. Contrast this with the very small temperature increase calculated in Example 16.1 for an incompressible flow. This is why, on the scale in Figure 16.9b, the incompressible case (A = 0) is seen as essentially a vertical line.

For the adiabatic wall case shown in Figure 16.10, we note the following:

1. From Figure 16.10a, the velocity profiles show a pronounced curvature due to compressibility.

2. From Figure 16.1 Ofo, the temperature increases are larger than for the cold wall case. Note that, for A = 30 <M,, ~ 10), the maximum temperature is over 15 times that of the upper wall. Also, note the results, familiar from our discussion in Section 16.3, that the temperature is the largest at the adiabatic wall; that is, Taw is the maximum temperature. As expected, Figure 16.1 Oh shows that Taw increases markedly as Me increases.

In summary, in a general comparison between the incompressible flow discussed in Section 16.3 and the compressible flow discussed here, there is no tremendous qual­itative change; that is, there is no discontinuous change in the flow-field behavior in going from subsonic to supersonic flow as is the case for an inviscid flow, such as discussed in Part 3. Qualitatively, a supersonic viscous flow is similar to a subsonic viscous flow. On the other hand, there are tremendous quantitative differences, es­pecially in regard to the large temperature changes that occur due to massive viscous dissipation in a high-speed compressible viscous flow. The physical reason for this difference in viscous versus inviscid flow is as follows. In an inviscid flow, informa­tion is propagated via the mechanism of pressure waves traveling throughout the flow. This mechanism changes radically when the flow goes from subsonic to supersonic. In contrast, for a viscous flow, information is propagated by the diffusive transport mechanisms of // and к (a molecular phenomenon), and these mechanisms are not basically changed when the flow goes from subsonic to supersonic. These statements hold in general for any viscous flow, not just for the Couette flow case treated here.

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