The experiments were carried out in a blowdown facility with a rainbow schlieren optical system, and performed for a range of nozzle back pressure ratios from 0.55 to 0.65, which are included in a range of overexpanded nozzle flows. The Reynolds number employed in the present paper was varied from 7.5 × 10⁴ to 8.9 × 10⁴ and it is defined based on the assumption of the ideal flow through a nozzle under choked flow conditions as Re_th = ρ^*u^*H_th/μ_os where ρ^* and u^* are the density and velocity at the nozzle throat, respectively, H_th is the throat height, and μ_os is the coefficient of viscosity evaluated under stagnation conditions at nozzle inlet. It is referred to as the throat Reynolds number in the research areas on critical nozzle flows. The Reynolds numbers in the present study are lower than the range of the Reynolds number covered by ISO 9300. As illustrated in Figure 1, the present nozzle has the same wall contour as the toroidal throat Venturi nozzle recommended by ISO 9300 and a constant width of 12 mm over the whole length from the inlet to the exit. The rainbow schlieren system consists of rail-mounted optical components including a 50 μm diameter pinhole,

two 100 mm diameter, 500 mm focal length achromatic lenses, a computer generated 35 mm wide slide with color gradation in a 1.4 mm wide strip, and a digital camera with variable focal length lens. A continuous 250 W metal halide light source connected to a 50 μm diameter fiber optic cable provides the light input at the pinhole through a 16.56 mm focal length objective lens. The camera output in the RGB format is digitalized by a personal computer with 24 bit color frame grabber. A detailed description for a calibration method of the rainbow filter is contained in the paper by Takano et al. [7] .

The experiments have been carried out in a blow down facility with a laser schlieren optical system and Mach-Zehnder interferometers, and performed for a range of nozzle operating conditions from p_b/p_os = 0.45 to 0.65 which are included in a region of overexpanded nozzle flow where p_os is the plenum pressure and p_b is the back pressure. The Reynolds number of the present study was varied from 7.5 × 10⁴ to 1.1 × 10⁵. These conditions are lower than the range of the Reynolds number covered by ISO 9300. As illustrated in Figure 1, the wall contour of the present nozzle is the same shape as the toroidal throat Venturi nozzle recommended by ISO 9300.

As shown in Figure 2, a He-Ne laser is used as a light source for laser schlieren and Mach-Zehnder interferometer systems and these systems are combined with a high-speed digital camera with 30 k frames per second. Visualization of the critical nozzle flows for each operating pressure ratio was performed and recorded in 2000 pictures. The effective spatial resolution of the present imaging system is 25 pixel/mm or 0.04 mm/pixel. Using these systems for two-dimen- sional nozzle flows including a shock wave, the instantaneous shock position, the frequency and amplitude of the shock wave, and the two-dimensional unsteady flow field can be measured with ease. For each laser schlieren image, the shock wave is defined to be positioned at the steepest negative grey scale gradient. Since a shade level of video output is essentially dependent on the intensity of schlieren light source and cut off of the light at knife edge, the shock image with

comparatively constant intensity were obtained at the height of the nozzle centerline in all the present test conditions where the shapes of the shock wave were maintained nearly normal to the streamwise direction.

On the other hand, Mach-Zehnder interferometers can estimate the unsteady two-dimensional characteristics of the whole flow field in the nozzle including the shock displacement by analyzing fringe shift from the reference image. Since the wedge fringe method can measure density more minutely than the infinite fringe method by analyzing a fringe shift, in the present work the wedge fringe method is employed. The analytical procedure of an interferogram is divided into two steps, which are the fringe shift analysis and density calculation from the fringe shift. The former is done using the Fourier transform method, which is presented by Takeda et al. [8] and can obtain a fringe shift two-dimensionally with a high space resolution. The latter is done under the assumption that a flow field is two-dimensional. The quantitative methodology of extracting the two- dimensional density field in the test section is described below.

In the wedge fringe method, the mirrors and beam splitters are deliberately misaligned to produce a background fringe pattern of straight lines. In the pre- sent experiment, the background fringes were made perpendicular to the nozzle axis by appropriate adjustment of the beam splitter 2 shown in Figure 2. When a flow with a refractive-index disturbance is introduced in the test section, the background fringes are changed into deformed fringe patterns. On a screen or recording media for a fixed time, the typical profiles showing background and deformed fringe patterns are illustrated in Figure 3 as lines of constant phase. The parallel and equally spaced fringes shown as red solid lines in Figure 3 are also referred to as wedge fringes. This interval b (see Figure 3) between two successive crests of the background fringes is a function of the intersection angle

between reference and test beams and the wavelength of the laser light used. The intensity profile of the deformed fringe pattern can be given by

with the phase shift containing the desired information on the density field in the test section where and represent unwanted irradiance variations arising from the nonuniform light reflection or transmission when the test beam is passing through the test section, and k₀ = 2π/b. The coordinates x and y form the vertical plane, which is perpendicular to the test beam propagation direction, and the z-axis is taken as the direction in which the test beam propagates.

Equation (1) is rewritten in the following expression

with

where i is the imaginary number and the asterisk * denotes a complex conjugate.

The Fourier transform of Equation (2) with respect to x is given by

where the capital letters denote the Fourier transforms of the primitive functions, respectively and k is the spatial wave number in the x direction. Since the spatial variations of, , and are slow compared with the spatial frequency k₀ when the interval between fringes is sufficiently small, the Fourier spectra in Equation (4) are separated by the wave number k₀ and have the three independent peaks, as schematically shown in Figure 4(a). We make use of either of the two spectra on the carrier, say, and translate it by k₀ on the wave number axis toward the origin to obtain, as shown in Figure 4(b). The unwanted background variation has been filtered out in this stage by a pertinent band pass filter. We compute the inverse Fourier transform of with respect to the k to obtain, defined by Equation (3). Then we calculate a complex logarithm of Equation (3) as follows:

From Equation (5), we can obtain the phase shift in the imaginary part completely separated from the unwanted amplitude variation in the real part. The phase shift is related to the density field through the Gradstone-Dale formula as follows:

where L = 12 mm is the spanwise length of the test section, ρ_a is the atmospheric density, λ₀ is the wavelength of a light source in vacuum, and K = 2.2587 ´ 10⁻⁴ m³/kg is the Gradstone-Dale constant. Considering the two-dimensional refractive index field, i.e., independent of the z axis, the density in the flow field is expressed by

Since Equation (7) shows that the phase shift at a fixed time is directly proportional to the flow density at the same time, characteristics of an unsteady flow field including a shock wave can be evaluated by examining the time history of the fringe shift.

An image of the flow field in the test section by the Mach-Zehnder interferometer is formed onto the CMOS sensor of a high-speed digital camera (Photron, FASTCAM SA1.1) which records a JPEG RGB image (24-bit each color) at a resolution of 640 × 272 square pixels. The plane of focus is located in the middle of the test section. The RGB image is then turned into an 8-bit grayscale image by a linear transformation. Therefore, the distributions of background and deformed fringes with 256 different possible intensities can be calculated from the Mach-Zehnder images for the density-field in the test section.

Rainbow schlieren pictures of overexpanded nozzle flow are shown in Figure 5 with the flow from left to right where the rainbow filter is vertically set to view the shape of the shock waves. The ratio p_b/p_os of back pressure p_b (p_b is the same value as the atmospheric pressure) to plenum pressure p_os was decreased in steps of 0.05 from p_b/p_os = 0.65 to 0.50. Generally, no shock wave exists inside the nozzle for the overexpanded flow condition based upon the assumption of one-dimensional isentropic theory except for a normal shock wave. However, as can be seen in Figure 5(a), a nearly normal shock wave stands at a location just downstream of the nozzle throat. Proceeding from top to bottom, the shock moves toward the nozzle exit and the shock structure progressively varies from nearly normal shock to X-type shock train which implies that the boundary layer flow separation occurs at the foot of the shock.

To verify the shock-induced flow separation, the experimental data are

compared with the past theories. Figure 6 shows the variation of the shock location x₁ along the nozzle centerline (x axis) from the nozzle throat against the back pressure ratio p_b/p_os where the x₁ for a shock train was measured as the distance parallel to the x axis from the throat to the foot of the leading shock of the first shock. The density gradient across a shock wave increases in the flow direction when the flow passes through the shock wave. Since a large density gradient leads to the large transverse displacement of the light ray at the rainbow filter plane, the position of the leading shock from the nozzle throat was estimated from the streamwise hue variations using the calibration curve for the rainbow filter used in the experiment.

The dash-dotted line in Figure 6 indicates the theoretical solution derived from the Rankine-Hugoniot relation for a normal shock location in the present nozzle where the back pressure ratios when a normal shock just stands at the nozzle exit and at the throat become 0.654 and 0.837, respectively. The dashed and solid lines show the exact solutions on the static pressure ratio across an oblique shock based upon the assumption that the flow separation occurs at the flow deflection angle equal to the nozzle wall half-angle and the static pressure p_s just behind the shock is equal to the back pressure p_b. They correspond to weak and strong shock solutions, respectively. In addition the dotted line in Figure 6 shows the theoretical curve proposed by Arens and Spiegler [5] where the static pressure p_s after the shock-induced flow separation is assumed to be the same as the back pressure p_b. The symbols A to D correspond to the schlieren pictures of Figures 5(a)-(d), respectively. Figure 6 shows that the experimental values for a nearly single normal shock shown at Points A and B are in good agreement with the strong shock solution. The theoretical solutions shown in Figure 6 don’t

provide any exact prediction about the experimental values for shock trains at Points C and D. The reason why these differences appear seems to come from the boundary layer condition. Arens and Spiegler [5] attempt to predict the turbulent shock wave-boundary layer separation. They found that the best agreement with the experimental data for the shock pressure ratio required to separate a turbulent boundary layer is obtained by using the assumption that the pressure rise must be sufficient to stagnate a characteristic velocity in the boundary layer. This theoretical curve agrees well with the experimental data caused by the turbulent shock wave-boundary layer separation. However, the throat Reynolds numbers in the present study is quite smaller than those of the previous research. Therefore, the differences between the theoretical curve proposed by Arens and Spiegler and the experimental plots obtained from the present study may come from the estimation of the velocity profile in the boundary layer.

The separation pressure ratio p₁/p_s is plotted against the separation Mach number M_1e in Figure 7 where M_1e is calculated on the basis of isentropic expansion from plenum chamber to the shock location x₁, and p₁ is the static pressure at the x₁, and p_s is the static pressure after the compression due to the shock-induced flow separation. Although the separation pressure p_s cannot be obtained in the present experiment, the p_s is assumed to be the same as the back pressure based on the assumption of Arens et al. [5] . The validation is performed by a comparison between the present experimental results and the theory of Arens et al. The theory of Arens et al. shown in Figure 6 and Figure 7 has extensively been used in research areas of rocket propulsion to predict the exact location of the shock-induced boundary layer separation in a supersonic nozzle with a diffuser divergent angle held almost constant in the downstream direction. Their theory assumes that the p_s is the same as the back pressure downstream of the nozzle, i.e., no pressure recovery occurs after the separation.

In our experiments, the p_s itself cannot be directly obtained , but the ratio p_s/p₁ or p₁/p_s can be obtained when the static pressure p₁ just upstream of the shock or the corresponding freestream Mach number at the shock location is known.

Figure 7 shows that the flow separations at Points A and B are deemed to be responsible at the adverse pressure gradient by the shock wave only, not the interaction between the shock wave and boundary layer. The flow separation in the case depends only on the free stream Mach number just upstream of the shock wave because the shock strength is a function of the Mach number only. On the other hand, the flow separations at Points C and D are caused by the interaction between the shock wave and the boundary layer because a shock train appears as a consequence of the interaction between a normal shock wave and a boundary layer [9] . The separation pressure ratios for shock trains remain a constant value regardless of the separation Mach numbers. According to a lot of experimental research in the past, a shock train is produced in a confined duct for the free stream Mach numbers beyond around 1.5 [9] . However, the present study shows a shock train appears for the free stream Mach numbers below 1.5 and it may be considered as an effect of the lower Reynolds number, as compared to that of the previous research, in other words, a thinner velocity profile in the boundary layer at the location of the shock. This leaves room for further investigation.

Only the result for a nozzle operating condition of 0.65 is discussed in detail in the present paper. Typical laser schlieren image of an instantaneous flowfield in a critical nozzle at p_b/p_os = 0.65 is shown in Figure 8 where spatial resolution of the shock position is 0.04 mm in the streamwise direction. Generally, no shock wave exists inside the nozzle for overexpanded flow conditions. However, as shown in Figure 8, a nearly normal shock wave stands at the location just downstream of the throat. A normal shock wave was observed in much of the 2000 successively-recorded schlieren images.

Figure 9 displays the time history of a shock wave location in case of p_b/p_os = 0.65. The light intensity from the laser schlieren is directly proportional to the density gradient in the flow. In the present laser schlieren, an increase in density

gradient can be seen as a drop in light intensity and vice versa. In other words, whenever a shock wave is present in the flow field, a large drop followed by an increase in light intensity is observed. The time history of an oscillating shock can be obtained from such light intensity variation in the flow field. The x_s = 0 shows the time-mean location of an oscillating shock in the nozzle where the origin of the time averaged shock wave location is identical with a location of x = 3.17 mm. The shock wave vibrates across a time-mean position. Figure 10 shows the power spectrum density distribution (PSD) of the shock displacement calculated using the experimental data of Figure 9. The area under the curve of the PSD is identical to the variance of the shock displacement along the nozzle centerline.

The PSD shows that the oscillating shock has a maximum value of 3 kHz with a narrowband spectrum. Figure 11(a) and Figure 11(b) show typical interferogram of nozzle flow in the case of p_b/p_os = 1.0 (background image) and p_b/p_os = 0.65, respectively. A uniform brightness image can be seen in Figure 11(a). In Figure 11(b), the fringes near the shock wave and upper and bottom walls are distorted notably. It is known that fringe shift depends only on density in the wedge fringe method. Therefore, the time history of the two-dimensional density field can be calculated by analyzing the fringe shift obtained from Figure 11. Figure 12 shows the power spectrum density distribution (PSD) of the fringe shift fluctuation at the time-mean shock location in Figure 11. As can be seen in Figure 12, the dominant frequency is 3 kHz and it is in good quantitative agreement with that obtained from the laser schlieren. Von Lavante et al. [1] [2] numerically solved the unsteady behavior of a critical Venturi nozzle flow at a Reynolds number of 1.0 × 10⁵ and showed that pressure fluctuations at the downstream of the nozzle exit propagate upstream beyond the position of the

nozzle throat. As a result, an intermittent decrease of the mass flow rate occurs because of periodical unchoking at the nozzle flow. As can be seen in Figure 10 and Figure 12, the present study shows a shock wave oscillating across a mean position with a dominant frequency of 3 kHz.