Dynamic evolution of a transient supersonic trailing jet induced by a strong incident shock wave

The dynamic evolution of a highly underexpanded transient supersonic jet at the exit of a pulse detonation engine is investigated via high-resolution time-resolved schlieren and numerical simulations. Experimental evidence is provided for the presence of a second triple shock configuration along with a shocklet between the reflected shock and the slipstream, which has no analog in a steady-state underexpanded jet. A pseudo-steady model is developed, which allows for the determination of the postshock flow condition for a transient propagating oblique shock. This model is applied to the numerical simulations to reveal the mechanism leading to the formation of the second triple point. Accordingly, the formation of the triple point is initiated by the transient motion of the reflected shock, which is induced by the convection of the vortex ring. While the vortex ring embedded shock move essentially as a translating strong oblique shock, the reflected shock is rotating towards its steady-state position. This results in a pressure discontinuity that must be resolved by the formation of a shocklet.

Figure 1

Dominant features of a starting transient underexpanded jet. A time series of six $\overline{\frac{\partial \rho }{\partial y}}$ schlieren images shows the early stage of the exhaust flow for $\text{Ms}=1.76$. The $x$ and $y$ coordinates are normalized by the tube diameter, $D$. The origin of the axis corresponds to the point on the tube centerline at the tube exit.

Figure 2

Overview of the main flow features at $\tau =6.13.$

Figure 3

$\overline{\frac{\partial \rho }{\partial y}}$ schlieren images showing the pairing of two eddies for $\text{Ms}=1.76$.

Figure 4

Sketch of the experimental setup showing (a) the pulse detonation engine, pressure sensors, and ion probes, (b) the schlieren setup, and (c) the total pressure probe.

Figure 5

Mean and standard deviation of total pressure for four test runs with $\text{Ms}=1.968$, 1.977, 1.983, and 1.983 at $x/D=-0.3$.

Figure 6

Total pressure over time for $Ms=2.15$ at $x/D=-0.3$.

Figure 7

Formation of the shocklet based on experimental $\overline{\frac{\partial \rho }{\partial x}}$ schlieren images for $\text{Ms}=2.15$ (left column) numerical $\overline{\frac{\partial \rho }{\partial x}}$ schlieren images (center), and Mach number contour plots at $z=0$ (right column).

Figure 8

Schematic illustration of the jet structure for the transient and steady underexpanded jet.

Figure 9

$\overline{\frac{\partial \rho }{\partial y}}$ schlieren images for $\text{Ms}=1.76$ showing the evolution of the second triple point configuration.

Figure 10

Sequence of pressure contours determined from numerical simulations for a Riemann problem with $\text{Ms}=1.71$. The yellow dashed line indicates the formation of a pressure gradient that potentially leads to the formation of the shocklet.

Figure 11

Schematic illustration of the transient oblique shock (TOS) model. Shown are a transient shock wave at two time instances and the quantities used for the calculation of the postshock flow conditions of the particle at $t_2$.

Figure 12

(a) Pressure distribution at $\tau =2.66$ as derived from the numerical simulations at constant inflow conditions with $\text{Ms}=1.71$. ${\text{TP}}_1$ and O represent the first triple point and the tail of the VRES. The TOS model is applied for $x_1, x_2$, and $x_3$. The result of the TOS analysis is given in Table 1. (b) Pressure distribution at $\tau =3.15$ shows a uniform pressure distribution downstream of the reflected shock.

Figure 13

Shock propagation velocity for (a) orthogonal and (b) rotational shock displacement.

Figure 14

(a) Pressure distribution at $\tau =3.88$ as derived from the numerical simulations at constant inflow conditions with $\text{Ms}=1.71$. ${\text{TP}}_1$ and O represent the first triple point and the tail of the VRES. The TOS model is applied for $x_1$ to $x_4$. The result of the TOS analysis is given in Table 2. (b) Numerical pressure distribution at $\tau =4.36$, showing a pressure gradient on the downstream face of the reflected shock.

Figure 15

(a) Pressure distribution at $\tau =4.60$ as derived from the numerical simulations at constant inflow conditions with $\text{Ms}=1.71$. ${\text{TP}}_1$ and O represent the first triple point and the tail of the VRES. The TOS model is applied for $x_1$ to $x_7$. The result of the TOS analysis is given in Table 3. (b) Pressure distribution at $\tau =5.09$.

Figure 16

(a) Pressure distribution at $\tau =5.81$ as derived from the numerical simulations at constant inflow conditions with $\text{Ms}=1.71$. ${\text{TP}}_1$ and O represent the first triple point and the tail of the VRES. The TOS model is applied for $x_1$ to $x_7$. The result of the TOS analysis is given in Table 4. (b) Pressure distribution at $\tau =6.30$.