편집위원회

The unsteady CFD simulations were performed for a two-dimensional model as shown in Fig. 3. The 2D simulation can replicate the characteristics of the shock system at the center plane of the test section. The stagnation pressure at the inlet of the isolator is 1.65 bar, and the stagnation temperature is 300 K. At the operating condition, the ratio of stagnation pressure at the inlet to exit is set to be 1.65. The boundary conditions employed in the simulation is also presented in Fig. 3.

Fig. 3 
The geometry details and boundary conditions of the flow field.

The symmetric plane containing the shock train system is considered for the computational analysis, and the flow field is simulated by commercial software FLUENT 19.2. The boundary conditions at inlet and exit are specified as pressure inlet and pressure outlet. Non-slip walls and symmetry boundary conditions have been imposed at the walls and symmetry surface, separately. The steady, two-dimensional, compressible Navier-Stokes equations have been incorporated to simulate the flow field in the duct. The structured grid system is employed in the flow domain with a y+ value of 1.5. The numerical simulations were carried out in the finite volume solver with density based solver with second order upwind spatial discretization. The k-omega SST turbulence model was used to simulate the physical problem. The working fluid is taken as an ideal gas. The simulations were carried out with a time step size of 1e-06 seconds.

To validate the numerical model, the data obtained from the steady state experimental results are used. The obtained results are plotted in Fig. 4. It can be seen from the figure that the results match very well with a small variation in the shock angle which was formed primarily due to the variation of boundary layer thickness in the flow upstream of the first shock wave. The upper part of the figure represents the numerical Schlieren of the flow field while the bottom one is the corresponding experimental result. The same regions in the flow field were considered for both numerical as well as the experimental results and the flow is from left to right.

Fig. 4 
Comparison of experimental and computational results.

The Schlieren images of multiple shock waves and comparison of surface pressure measurements between experiment and computation are shown in Fig. 5. The nozzle throat location is taken as the reference point for measurements. Fig. 5a represents the position of shock train in the rectangular duct while Fig. 5b is the comparison of wall static pressure on the lower wall for experiment and computation.

Fig. 5 
(a) Schlieren image from experiment and; (b) comparison of wall static pressure between experiment and computation.

The pressure on the wall is normalized with inlet stagnation pressure, and shock location is normalized with the throat height of the nozzle. The starting shock location is the same for both computation and experiment. In the real case, the shock will not be stationary, but oscillating even if we maintained a constant pressure ratio across the geometry.

The reliability of unsteady results can be obtained by comparing the obtained numerical results with the experimental results. Fig. 6 represents the comparison of unsteady results.

Fig. 6 
Comparison of unsteady numerical results with the experimental results.

The upper part indicates the numerical results, while the lower part indicates the experimental results. The first image in Fig. 6 is taken as the reference image for the comparison. It is represented at a time of Δt = 0 ms. The next image Ⅱ is taken at a time difference of 0.025 ms for both experimental and numerical schlieren. In other words, both numerical and experimental images are taken at the same interval of time from the reference image. It can be observed that there is a great matching for the shock locations for the upper half and lower half of each image.

Fig. 7 shows the numerical Schlieren of the flow field obtained from the simulation. The multiple shock wave patterns are present in the duct and it is visible. For a constant pressure ratio of 1.65, the shock system is oscillatory and it oscillates about its mean position. Since we are not imposing any changes in the boundary conditions, the oscillations are termed as self-excited oscillations. The characteristics of oscillations are elaborated in the following sessions.

Fig. 7 
Numerical Schlieren showing the flow field with multiple shock waves.

Fig. 8 shows the time variation of the shock locations obtained from the numerical Schlieren images. To obtain the shock locations from images, each image at every time interval is processed using a Matlab image processing code. This code gives the shock location based on the intensity of pixels in the images. Thus the shock locations are obtained by taking an appropriate region in the images. The mean location of each shock wave is presented as dotted lines of the same color. It is observed that the distance between mean locations of first and second shock is larger, while the distances between subsequent shocks are reduced.

Fig. 8 
Variation of the first four shock locations with time. The dotted line in the same color represents the mean value of shock location.

The fluctuating components or amplitude of each shock system is represented in Fig. 9. The y-axis is obtained by subtracting the mean value of each shock location to its absolute value. The positive value represents that the shock is downstream of its mean position while the negative value corresponds to upstream conditions. It is also noted that the first shock has the maximum amplitude of oscillation, while the other shocks have a lower amplitude of oscillation. It occurs since the second and subsequent shocks are affected by the pressure fluctuation created by the first shock.

Fig. 9 
The variation of the amplitude of the first four shock oscillations to its mean location.

To analyze the response and dependence of each shock with each other, initially cross correlation analysis has conducted. The cross correlation function describes the dependence which one signal has on another, as a function of the delay time between signals. It is determined by taking the average product of Px(t) at a time t and of Py(t) at a time t+τ over a period T, as T approaches infinity.

One of the most important applications for the cross correlation functions is to determine the time required for a signal source to pass from one point in space to another. The cross correlation between the shock system is presented in Fig. 10. The ordinate represents the value of the correlation between the shock locations. The first and second shocks are well correlated in its amplitude. This means that there is no phase difference in the oscillatory characteristics between them. But the first and third shock correlation suggests that there is a phase difference of –0.005 ms. Or in other words, the third shock lags the first at 0.005 ms. The delay between the first shock location and the fourth shock location is –0.047 ms. The delay time corresponds to both the cases are negative. So it may be said that in the low speed region between first shock and the third shock wave, pressure fluctuations propagate in an upstream direction, due to the existence of flow separation region between them. The same trend applies to the first shock and the fourth shock wave. The change in the delay time of the fourth shock arises from the fact that the shock is affected by the pressure fluctuations of upstream shock waves.

Fig. 10 
The cross correlation of shock location with respect to the first shock wave.

The mean wall static pressure during the simulation is presented in Fig. 11. The lower wall pressure distribution is considered in the present analysis. We can see that there is a pressure increment at x-120 mm which corresponds to the foot of the lambda shock. The pressure rises continuously after the shock train region which is at 200 mm, which indicates the mixing region in the flow field. To analyze the pressure fluctuations in detail, six monitor points were considered and are presented in Fig. 12. The first point P1 at 115 mm is in the undisturbed flow region and the last point P6 at 250 mm is at far downstream location after the mixing region. The other four points at 133 mm, 155 mm, 168 mm, and 180 mm were considered at the mean shock locations.

Fig. 11 
Wall mean static pressure distribution of the flow field. The lower wall pressure variation is considered.

Fig. 12 
Pressure fluctuation monitor points on the lower wall of the test section.

The variation of wall static pressure in the monitoring points were studied in detail. Fig. 13 shows the time variation of wall static pressure distributions at these monitoring points. The vertical axis is normalized with the upstream stagnation pressure and only fluctuating components are presented with arbitrary units.

Fig. 13 
Time variation of lower wall static pressure at different monitoring points.

The black line corresponding to point P1 does not show any fluctuating component, which means that the shock wave never crosses the point P1. Therefore we can consider this point as the point in the undisturbed flow. The larger values of normalized pressure fluctuations are observed at point P2, which corresponds to the first shock location. These values are reduced for the following monitoring points.

It is also noted that there is an increment in pressure when the shock wave crosses the monitor point while moving upstream direction. The sudden increment in pressure is observed at this time. Similarly, when shock wave moves downstream of the monitor point, there is a decrement in the wall static pressure at that time.

Fig. 14 shows the FFT plot obtained from the wall static pressure variation for the four shock locations, namely at P1, P2, P3 and, P4. The dominant frequencies of oscillations are in the range of 100 to 350 Hz, which is consistent with the frequency value mentioned in the literature for the self-excited oscillations. It is also observed that wall pressure fluctuations contain strong oscillations of low frequency. Therefore it can be said that wall pressure fluctuations are induced by each shock wave oscillation composed of multiple shock waves.

Fig. 14 
The FFT plot obtained from pressure variation at first four mean shock locations corresponding to points P1, P2, P3, and P4.

The streamwise variation of rms values of pressure in the flow field is presented in the Fig. 15. This variation along the lower wall is presented in this figure. The y axis is normalized with the inlet stagnation pressure. The maximum rms value is observed at the first shock location, where the intermittency due to shock oscillation is very large. The rms values at other shock locations also inhibit a peak in the plot. This shows that the shock oscillations are intermittent. The fluctuation levels downstream of the shock are very likely to be profoundly influenced by the presence of shock-induced separation, and far downstream, the levels are nearly constant.

Fig. 15 
The streamwise root-mean-square distribution in static pressure on the lower wall.

The first peak is observed at x/d = 12.5 where the first shock wave is present. Ideally, the second peak shock occurs at x/D =15.5 that corresponds to the mean location of the second shock wave. But we could observe that there is a small peak at x/D =14.2. This arises from the presence of shock-induced separation between first and second shock wave. The effect of flow separation is more profound in between the first and second shock waves. The flow separation and hence the induced oscillations are not very evident after the second shock wave. All the pressure fluctuations and the corresponding peaks after the second shock is originated from the oscillation of shock waves alone.