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Two-dimensional critical nozzle flows at low Reynolds numbers are visualized by the rainbow schlieren deflectometry. Experiments have been performed in a region of overexpanded nozzle flow. The variation of the shock structure against the back pressure ratio can be clearly visible with color gradation. Static pressure rises due to the shock-induced flow separation are compared with the previous theories. The unsteady characteristics of overexpanded critical nozzle flows at low Reynolds numbers are quantitatively and qualitatively visualized using laser schlieren and Mach-Zehnder interferometer systems combined with a high-speed digital camera. It was found that an oscillating normal shock wave appears inside the nozzle, and that the shock wave has a specified dominant frequency. Also the time-history of the oscillating shock wave is obtained from both the systems and compared with each other.

A critical nozzle provides a reliable and precise way of measuring mass flow rate and it has an important role in many industrial applications including mass flow controllers, calibration standards for other gas flow meters, and pressure isolators. When the ratio of back pressure downstream of a critical nozzle to the upstream stagnation pressure is maintained below a critical threshold known as the critical back pressure ratio, the mass flow rate through the nozzleremains constant and it is entirely independent of any changes in flow conditions downstream of the nozzle throat. In the ISO Standard 9300, the design and usage of critical nozzles are well described over a range of throat Reynolds numbers from 10^{5} to 10^{7} and the divergent section just downstream of the nozzle throat plays a key role functioning as a pressure recovery element that extends the flow range capabilities of the device by increasing the minimum critical back pressure ratio required to reach a choked flow condition. However, in critical nozzles with significantly smaller throat diameter, relatively large fluctuations in the mass flow through the nozzles have been observed, making the calibration of these nozzles difficult. In the past, although flow properties through supersonic nozzles have been extensively studied in the research areas of rocket propulsion, the geometrical size, and therefore the Reynolds numbers of these nozzles were significantly higher than considered in the present study.

Von Lavante et al. [^{5} to 1.0 × 10^{7}. At the throat Reynolds numbers of 1.0 × 10^{5} and 5.0 × 10^{5} with a back pressure ratio for a normal shock wave to stay in the nozzle evaluated under the isentropic flow conditions except the shock, they numerically showed that pressure disturbances can propagate upstream beyond the nozzle throat, even above the critical back pressure ratio, thus leading to fluctuating mass flow rate through the nozzle. Kim et al. [_{th} = 4000 and that the local minimum value is around Re_{th} =10,000 due to the change in characteristics of the boundary layer in the divergent section of the nozzle. Also, they demonstrated that the critical back pressure ratio is a function of the theoretical Reynolds number only, not dependent on the throat diameter. Furthermore, the change of the discharge coefficients occurs abruptly when the back pressure ratio approaches the critical value, and the distribution of the discharge coefficient against the back pressure ratio has at least one local maximum and one local minimum when the back pressure ratio is above the critical value and they explained the phenomena in terms of the interaction between the shock wave and the boundary layer in the diffuser, as Von Lavante et al. [

Lower back pressure ratios to fulfill a purely supersonic flow in the divergent section downstream of a nozzle throat will stabilize the mass flow rate through the nozzle. Ideally, when a nozzle is operated with overexpansion, the flow in the nozzle is supersonic over the entire region downstream of the throat, and the static pressure at the exit plane is uniformly lower than the back pressure. However, a detailed and deep comprehension of critical nozzle flows operating under low Reynolds numbers with overexpansion remains to be fully elucidated and little is known in current literature because of technical difficulties in measurements. The real nozzle flows with overexpansion at higher Reynolds numbers often involve the flow separation in a section between the throat and the nozzle exit as a result of the interaction of a shock wave with a boundary layer, because the boundary layer cannot very frequently endure the abrupt pressure gradient due to the shock wave. Under such conditions the flows expand inside the nozzle to a location just upstream of the shock wave and then compressed to a specified back pressure and the theory to predict the pressure rise due to the shock-in- duced boundary layer separation proposed by Arens et al. [

The objective of the present study is to investigate experimentally flow behavior in a critical Venturi nozzle by optical flow visualization and the interaction of a shock wave with a boundary layer at low Reynolds numbers are investigated by a comparison of the present experiments by the rainbow schlieren deflectometry with the past theories including that of Arens et al. [

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 × 10^{4} to 8.9 × 10^{4} 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

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. [

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 × 10^{4} to 1.1 × 10^{5}. These conditions are lower than the range of the Reynolds number covered by ISO 9300. As illustrated in

As shown in

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. [

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

between reference and test beams and the wavelength of the laser light used. The intensity profile

with the phase shift _{0} = 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_{0 }when the interval between fringes is sufficiently small, the Fourier spectra in Equation (4) are separated by the wave number k_{0} and have the three independent peaks, as schematically shown in _{0} on the wave number axis toward the origin to obtain

From Equation (5), we can obtain the phase shift

where L = 12 mm is the spanwise length of the test section, ρ_{a} is the atmospheric density, λ_{0} is the wavelength of a light source in vacuum, and K = 2.2587 ´ 10^{−4} m^{3}/kg is the Gradstone-Dale constant. Considering the two-dimensional refractive index field, i.e., independent of the z axis, the density

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 _{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

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

compared with the past theories. _{1} along the nozzle centerline (x axis) from the nozzle throat against the back pressure ratio p_{b}/p_{os} where the x_{1} 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 _{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 _{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.

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 [

The separation pressure ratio p_{1}/p_{s} is plotted against the separation Mach number M_{1e} in _{1e} is calculated on the basis of isentropic expansion from plenum chamber to the shock location x_{1}, and p_{1} is the static pressure at the x_{1}, 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. [_{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}/p_{1} or p_{1}/p_{s} can be obtained when the static pressure p_{1} just upstream of the shock or the corresponding freestream Mach number at the shock location is known.

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

_{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.

The PSD shows that the oscillating shock has a maximum value of 3 kHz with a narrowband spectrum. _{b}/p_{os} = 1.0 (background image) and p_{b}/p_{os} = 0.65, respectively. A uniform brightness image can be seen in ^{5} 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

Overexpanded nozzle flows at low Reynolds numbers were visualized by a rainbow schlieren optical system. Schlieren pictures show that the shock gradually moves toward the nozzle exit from the location just downstream of the nozzle throat with decreasing back pressure ratio and its structure progressively changes from nearly normal shock to shock train. Also, the separation pressure ratio for a single normal shock is in good agreement with the theoretical solution for a strong oblique shock, while that for shock trains remains a constant value. Critical nozzle flows at low Reynolds numbers have been experimentally investigated in a region of overexpanded nozzle flow by using laser schlieren and Mach- Zehnder interferometer systems combined with a high-speed digital camera. As a result, a nearly normal shock wave stands at the location just downstream of the nozzle throat and it oscillates at a dominant frequency of around 3 kHz at a Reynolds number of 7.5 × 10^{4}. Furthermore, the unsteady two-dimensional characteristics of the critical nozzle flows can be obtained by the present Mach- Zehnder interferometer system. As a result, two characteristics of critical nozzle flows at low Reynolds numbers were obtained from the present work. Firstly, a nearly normal shock wave appears in the overexpanded nozzle flow and it oscillates with a dominant frequency of around 3 kHz. Secondly, a shock train appears for 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 [

The authors would like to gratefully acknowledge the assistance of graduate students Shota GOTO, Kazuki KITAMURA, and Atsushi MATSUYAMA of the University of Kitakyushu for their invaluable support to the experimental work and exceptional skills in the fabrication of the facility.

Yagi, S., Inoue, S., Nakao, S., Ono, D. and Miyazato, Y. (2017) Optical Measurements of Shock Waves in Critical Nozzles at Low Reynolds Numbers. Journal of Flow Control, Measurement & Visualization, 5, 36-50. https://doi.org/10.4236/jfcmv.2017.52003