Chapter V: Optical Measurements in a Shock/Boundary-Layer Interaction
5.4 Two-Temperature Calculations of Oblique Shock System
Streamlines traveling through a system of multiple oblique shocks experience ther- mal non-equilibrium due to the long time scales for vibrational relaxation compared to the local flow time. In addition, the temporal evolution of the system of oblique shocks causes changes to the extent of temperature relaxation. To understand the effects of thermal non-equilibrium in the oblique shock portion of the flow, the Landau-Teller calculation with a two temperature model of chemical reaction rates can be used to compute flow properties. Within the Shock and Detonation Toolbox [51], the Landau-Teller relaxation calculation is only configured for normal shocks.
In order to solve for the reaction zone along streamlines in an oblique shock system, a mapping of the normal shock calculation is applied assuming no curvature exists in the shock and no viscosity. For each wave, the normal component of the gas
velocity relative to the shock is used for the jump condition. The upstream condition can be in thermochemical non-equilibrium. The resulting reaction zone calculation is mapped to lab frame coordinates by assuming that the velocity component paral- lel to the wave is conserved after the shock jump. Geometry of the oblique shock system is measured from schlieren videos of M7-H8-A presented in Fig. 5.4.
Two-Shock System
Early in test time, double wedge flow forms as a two-shock system consisting of the lead oblique shock and the reattachment shock at the location of the hinge. From the schlieren movies,t = 45 µs is used to obtain measurements of the wave angles for a two-temperature reactive Landau-Teller calculation. The lead oblique shock is measured to have a wave angle ofβL = 38.6◦±0.5◦and the reattachment shock is measured to be βR = 42.1◦±0.5◦. The perfect gas freestream condition is used as the inflow state for the lead oblique shock wave.
(a) Streamline Path (b)T,Tv
(c) Pressure and Density (d) Mole Fractions
Figure 5.23: Calculation of a two-shock system using a two-temperature reactive Landau-Teller relaxation model. Wave angles are taken fromt = 45 µs in schlieren images.
to the supersonic velocity of the gas, it is observed that the gas remains in thermal non-equilibrium before arriving at the reattachment shock. Once another shock jump occurs, the translational-rotational temperature increases a second time to a value of 5106 K. Vibrational temperature rises faster after the second oblique shock, eventually becoming in equilibrium withT approximately 10 mm downstream the second shock. A plot of pressure and density is also shown, where pressure jumps to a calculated value of 68.0 kPa immediately downstream of the reattachment.
Finally, a plot of the species mole fraction is shown in Fig. 5.23d. As expected, little dissociation occurs downstream of the lead oblique shock due to low post- shock temperatures. Calculated values of NO(A) mole fraction are on the order of X = 10−16 in this region. At the reattachment shock, species dissociation begins to occur, as the mole fractions of minor species begin to increase. Values of NO(A) reach over X = 10−10 downstream the reattachment shock, over two orders of magnitude lower than the normal shock case with M7-H8-A. Therefore, it is expected that measurements behind the reattachment shock will require more exposure time and camera gain for low signals.
Three-Shock System
After flow startup, the boundary layer separates and the separation region grows in length. This generates an additional separation shock to the geometry, turning the flow a second time before arriving to the reattachment shock. It is possible to extend the calculation to add a third shock wave. To represent the final state of the shock structure at the end of test time, a frame taken at a timet = 150 µs will be used to extract the wave angles. Measured wave angles of βL = 38.6◦±0.5◦, βS =26.5◦±0.5◦, and βR = 40.0◦±0.7◦were obtained for the lead, separation, and reattachment shocks, respectively. This will serve as a representation for the final time interval used in radiation measurements downstream the reattachment shock.
Figure 5.24 shows a two-temperature reactive Landau-Teller calculation of a stream-
(a) Streamline Path (b)T,Tv
(c) Pressure and Density (d) Mole Fractions
Figure 5.24: Calculation of a three-shock system using a two-temperature reactive Landau-Teller relaxation model. Wave angles are taken fromt =150 µs in schlieren images.
line processed by three oblique shocks. The same initial conditions for the streamline are used in this calculation to compare with the two-shock case. Similar to the two- shock calculation, curvature in the streamline is observed in the immediate vicinity of the lead oblique shock before becoming approximately straight. At the separation shock location, the flow turns upward. A jump in translational-rotational temper- ature of 3809 K is observed, while pressure jumps to 28.2 kPa. Once the gas is processed by the reattachment shock, the flow turns a third time upward. The frozen translational-rotational temperature is calculated to be 4373 K, while pressure jumps to 53.0 kPa. The mole fractions of the minor species show that little dissociation continues to occur downstream of the separation shock, meaning the reattachment shock is expected to have the most NOγsignal in this portion of the flow. However, the magnitude of NO(A) mole fraction is considerably less than in the two-shock case.
(a) Two-Shock Contour ofTv
(b) Three-Shock Contour ofTv
Figure 5.25: Contours of vibrational temperature for a two-shock and three-shock system. Streamlines are shown as solid red lines, shock locations are dashed lines, and wedge geometry is shown as solid black lines.
For a comparison, multiple streamlines have been computed for both the two-shock and three-shock systems. Figure 5.25 shows the contours of the vibrational tem- perature of multiple streamlines. Red solid lines are two representative streamline paths with the same initial coordinates between the two cases. It can be observed that the rate of vibrational relaxation is different with the addition of the separation shock. In the three-shock case, the streamline originating at the origin shows a re- gion representing the separation bubble, indicated by the absence of contour color.
The separation region ends at the surface of the second wedge. Note that viscosity is
not taken into account in these calculations and may not represent the true behavior of the unsteady separation region. In both cases, the streamlines are spaced 10 mm along the lead shock wave. After multiple compressions, the streamlines eventu- ally coalesce and are spaced close together downstream of the reattachment shock.
Gradients inTv occur perpendicular to the streamline direction. This is a result of streamlines being processed by the shock waves at different times in the relaxation of the gas.
The most notable differences between the two-shock and three-shock cases are the translational-rotational temperature and pressure immediately downstream of the reattachment shock. With the addition of the separation zone, the flow is allowed to gradually turn a second time upstream of the hinge. As a result, both the frozen translational-rotational temperature and pressure downstream of the reattachment shock decrease later in test time.
From the experimental studies of Davis and Sturtevant [20], a scaling for the sepa- ration length was proposed to be of the form:
Lsep
x1 ∝ p3− p2
p1 3/2
, (5.1)
where Lsep/x1 is the normalized separation length and p1, p2, p3 are the post shock pressure of the lead, separation, and reattachment shock, respectively. Recall that the scaling is based on a momentum balance proposed by Roshko [82] for supersonic base flows. A rise in pressure p3 requires a longer separation length to impart the necessary shear stress along the dividing streamline to maintain the momentum balance. The scaling shows that the separation length is proportional to (p3− p2)3/2. From Fig 5.8, the measured separation front shows a decrease in upstream velocity with time. Within the same time frame, the reattachment shock experiences a reduction in βR that slows in the later portions of test time in Fig. 5.9. We observe similar time scales in the evolution of the separation and reattachment shock, consistent with the idea that these features are linked, as described in the base flow model of Roshko. In addition, the dissociation behind the reattachment shock leads to a reduction in shock angle. A perfect gas calculation of the reattachment shock results in a shock angle of βR = 48.0◦, significantly larger than what was measured. Due to the larger wave angle, the perfect gas pressure is 89.1 kPa downstream of the reattachment shock. Therefore, the two-temperature reactive Landau-Teller calculation shows that dissociation behind the reattachment
Figure 5.26: Schlieren image att = 110 µs. A red line 9 mm long represents the orientation of the spectrometer slits in the flow.
shock decreases the post-shock pressure relative to the perfect gas case due to the reduction in shock angle. These observations will later be confirmed with emission experiments, where direct measurements of the NO(A) species will interrogate the state of the gas.
5.5 Emission Measurements in Post-Lead Oblique Shock and Reattachment