Apertura de distintas puertas sin control en la extracción

In this section, the influence of vibrational relaxation on the stability properties of carbon dioxide flows is investigated. Nonequilibrium flow effects in CO2 are consider-

ably more important than in air, because CO2 exhibits a faster vibrational relaxation

time and stores a larger fraction of its total internal energy within vibrational modes. This enables both a tuning of the relaxation time to the disturbance frequency and a large damping rate of acoustic waves.

Following the same procedure used for air in the preceding section, we first inves- tigate the effect of the base flow calculation on the stability results. This is done by performing a nonequilibrium stability analysis on three different base flows obtained using frozen, equilibrium, and nonequilibrium models. The resulting growth rates are

0 500 1000 1500 2000 2500 3000 0 50 100 150 F [kHz] − α i [m − 1] R = 400 R = 600 R = 1000 R = 2500 e

Mean: Fr., Dist: Neq. Mean: Neq. Dist: Neq. Mean: Eq. Dist: Neq.

0 1 2 3 4 0 2 4 6 8 10 Rex×10−6 N F a ct o r

Mean: Fr., Dist: Neq. Mean: Neq. Dist: Neq. Mean: Eq. Dist: Neq.

a) b)

Figure 5.22: a) Stability diagrams for CO2,M = 5,Te∗ =T

∗

v,e = 1000 K,P

∗

e = 20 kPa.

Three different models of the mean flow: frozen, nonequilibrium, and equilibrium. Disturbances are nonequilibrium. b) N factor diagram corresponding to a).

given in Figure 5.22a and the N factors in Figure 5.22b.

The results show that, as one might expect, the nonequilibrium base flow solution with finite rates of energy transfer yields results that are very close to the equilibrium solution, except near the leading edge. This indicates that the base flow is close to equilibrium, which is confirmed by looking at the base flow profiles that were shown already in Figure3.5. That figure showed that the nonequilibrium profiles are very close to the equilibrium one downstream of about R = 600. The N factors in Figure5.22b also show good agreement between the equilibrium and nonequilibrium results.

Interestingly, stability analysis of the frozen base flow yields an enormous damping of the disturbances and complete stabilization of the boundary layer as one moves downstream. The same result was observed by Johnson et al. (1998), who applied a chemically reacting stability analysis to a non-reacting base flow of CO2 and also

found complete stabilization. The reason for this behavior is that the vibrationally frozen base flow profiles, which resemble Figure 3.5a at all R, feature a large, arti- ficial separation between the mean vibrational and translational temperatures since vibrational energy transfer was eliminated. Such a separation is non-physical, since the high rate of vibrational energy exchange in reality would rapidly bring the mean

0 500 1000 1500 2000 2500 3000 0 50 100 150 200 F [kHz] − α i [m − 1] R = 400 R = 600 R = 1000 R = 2500 e

Mean: Neq. Dist: Fr. Mean: Neq. Dist: Neq. Mean: Neq. Dist: Eq.

0 1 2 3 0 2 4 6 8 10 12 Rex×10−6 N F a ct o r

Mean: Neq. Dist: Fr. Mean: Neq. Dist: Neq. Mean: Neq. Dist: Eq.

a) b)

Figure 5.23: a) Stability diagrams for CO2,M = 5,Te∗ =T

∗

v,e = 1000 K,P

∗

e = 20 kPa.

Three different models of the disturbances: frozen, nonequilibrium, and equilibrium. Mean flow is nonequilibrium. b) N factor diagram corresponding to a).

temperature distributions together. By artificially freezing the base flow but allow- ing vibrational energy exchange in the stability analysis, one produces an enormous stabilizing source term in the linearized stability equations.

The influence of vibrational relaxation in the disturbance modeling can be assessed by calculating a fully nonequilibrium base flow (See Figure 3.5 for the profiles) and running three different stability analyses on the same base flow. The three stability analyses employ nonequilibrium, equilibrium, and frozen disturbance models. The resulting growth rates and N factor distributions are given in Figure 5.23.

In this figure, one observes that the disturbances are highly stabilized by vibra- tional nonequilibrium. Stabilization is only achieved in the nonequilibrium situation; if the vibrational relaxation time is too small (approaching the equilibrium result) then the relaxation time and the inverse frequency of disturbances are no longer com- parable, and damping does not occur. It is interesting to note that for high Reynolds numbers, the “tail” on the RHS of the growth rate curve is much larger when the disturbances are in equilibrium. As discussed in Section 5.3, this feature exists when unstable modes travel supersonically with respect to the freestream. It is well known that in a gas which has a single dominant vibrational relaxation timescale, the sound speed decreases monotonically as one moves from frozen to equilibrium flow (See

Vincenti and Kruger(1967), Chapter VIII). Because of this reduction in sound speed at equilibrium conditions, the unstable disturbances are able to travel supersonically and hence develop the features mentioned above.

The results above have shown that vibrational nonequilibrium affects the stability characteristics in two ways. First, vibrational relaxation leads to a change in the mean temperature profile, which generally has a destabilizing effect relative to the frozen base flow. Second, acoustic absorption of the second mode waves reduces the growth rate, an effect which is quite small for air but can be significant for carbon dioxide. The net result of these two phenomena can be either stabilizing or destabilizing compared to a completely frozen flow, depending on the flow conditions. The relative importance of these two processes will be investigated further in the sections that follow.