Application of Rainbow Schlieren Deflectometry for Jets from Round Laval Nozzles Followed by Cylindrical Ducts

The jet from a round Laval nozzle followed by a cylindrical duct with an inner diameter of 10 mm and a length of 50 mm is investigated experimentally. The Laval nozzle has a design Mach number of 1.5. Quantitative flow visualization of the jet issued from the duct exit is performed over a range of nozzle pressure ratios from 2.0 to 4.5 using the rainbow schlieren deflectometry combined with the computed tomography to investigate the jet three-dimensional structure. The flow features of the near-field shock systems in the jets are displayed with the density contour plot at the cross-section including the jet centerline. Effects of the nozzle pressure ratio on the density profile along the jet centerline are clarified quantitatively. In addition, a comparison between the present experiment and the previous one with a conventional Laval nozzle for jet centerline density profiles is carried out to examine the effect of the cylindrical duct. Furthermore, the three-dimensional structures of overexpanded and underexpanded jets are demonstrated with the isopycnic surfaces to visualize the internal flow features.

Journal of Flow Control, Measurement & Visualization tion and ability to observe easily such structures of shock waves, Prandtl-Meyer compression and expansion waves in supersonic flows. However, conventional schlieren techniques have been extensively applied for qualitative flow visualization because they take considerable effort to extract quantitative properties. On the other hand, recent methods for visualizing jet density fields quantitatively include laser-based interferometry such as Twyman-Green interferometry [2], Mach-Zehnder interferometry [3] [4] [5] or schlieren based optical techniques such as the background oriented schlieren (BOS) [5] [6], calibrated schlieren [7] [8], and rainbow schlieren deflectometry [7] [9] [10] [11]. Among them, the rainbow schlieren deflectometry would presumably be the simplest optical technique to acquire density fields of jets quantitatively.
The rainbow schlieren deflectometry is a modified form of the conventional schlieren such as the greyscale schlieren with a knife edge and color schlieren with a tricolor filter [1] [12] and it is different from the conventional schlieren in that the rainbow schlieren can capture density fields of flows with variable refractive index. In addition, the rainbow schlieren deflectometry can measure density fields of three-dimensional free jets by combining it with computed tomography [9] [10] [12]. A comprehensive and thorough review of the rainbow schlieren technique is presented by Agrawal and Wanstall [12], where methods to acquire quantitative density data including mathematical relationship, numerical algorithms, system design criteria, hardware issues, calibrations of rainbow filters, and so on are summarized. Recently, Ezoe et al. [13] applied the rainbow schlieren deflectometry for leek peeler nozzle jets to improve the nozzle peeling performance and developed a new nozzle with higher removing potential for peeling leeks in comparison with the conventional nozzle.
In the present study, the rainbow schlieren deflectometry is applied for jets from a round Laval nozzle followed by a cylindrical duct. Such a Laval nozzle followed by a long duct attracts special attentions on applications for the cold spray technology [14] [15], which includes a deposition process in which small particles in the solid state accelerate to high velocities in a supersonic gas jet and deposit on the substrate material. It has been widely recognized that the quality of the coating depends significantly on the gas flow velocity and stagnation temperature. The higher gas velocity makes the particle velocity increased, resulting in highly tough and dense coatings on a solid surface. To achieve optimal conditions for deposition, the quantitative information of the supersonic jets is required. However, there are little quantitative experimental data about the gas dynamics of the cold splay technology. In addition, the design of the spray gun has been primarily empirical and based upon engineering intuition. Therefore, as a first step of an application for the cold spray technology, the effects of nozzle pressure ratios on flow features of the jet from a supersonic nozzle followed by a long duct are described in this paper. Furthermore, the previous experimental data [10] on a jet issued from a conventional Laval nozzle without a long duct is used for a comparison with the present experimental data.

Experimental Apparatus and Method
A schematic drawing of experimental apparatus with a rainbow schlieren system is shown in Figure 1. A blowdown wind tunnel with a high-pressure tank (2 m 3 ) was used to provide the air flow to a cylindrical plenum chamber connected to a test nozzle. The high-pressure dry air from the tank is stagnated in the plenum chamber and then discharged into the atmosphere through the test nozzle. The total temperature in the plenum chamber was equal to the room temperature, and the plenum pressure was controlled and maintained constant during the testing by a valve. The test nozzle is shown schematically in Figure 2. It is made of acrylic and consists of a round Laval nozzle with a design Mach number of 1.5 followed by a constant-area straight duct with an inner diameter of 10 mm and a length of 50 mm. The wall contour of the Laval nozzle has a sinusoidal curve over a range of parts A to B and the contour between the throat (part B) and exit (part C) is designed by the axisymmetric method of characteristics [16] to provide uniform and parallel flow in the nozzle exit plane at the design condition.
The jet issued from the nozzle was visualized by the rainbow schlieren deflectometry over a range of nozzle pressure ratios (NPR = p os /p b ) from 2.0 to 4.5 where p os is the plenum pressure and p b the back pressure (=101.8 kPa) or atmospheric pressure, and T b (= 295.5 K) the ambient temperature. The rainbow schlieren system consists of rail-mounted optical components  and 500 mm focal length, a rainbow filter, and a digital camera (Nikon D7100) with a 30 mm diameter focusing lens of 600 mm focal length. Figure 3(a) shows a rainbow filter used in the present experiment and the corresponding calibration curve is displayed in Figure 3  The detailed description for the reconstruction process is given in Awata et al. [17]. The jet three-dimensional density field was reconstructed using both of the Abel inversion method based upon the assumption of axisymmetric jets and the convolution back-projection (CBP) method. However, only those obtained from the CBP method are demonstrated in the present paper because the density fields obtained by the Abel inversion method produced some noises on the jet centerline. The principle of the rainbow schlieren deflectometry combined with the CBP method is also given by Takano et al. [9], Maeda et al. [10], and Agrawal et al. [18].

Rainbow Schlieren Pictures
Jets from a Mach 1.5 round Laval nozzle followed by a cylindrical duct were visualized using a rainbow schlieren system as shown in Figure 4. The flow is from left to right. Schlieren pictures were taken at an exposure time of 1/8000 s with continuous schlieren light source. In addition, a rainbow filter was placed at the cut-off plane in parallel with respect to the z axis and its orientation is illus-

Density Contour Plots
The schlieren pictures of Figure 4 show only qualitative flow features of the jet.

Density Profiles along Jet Centerline
Effects of the nozzle pressure ratio on jet centerline density profiles are shown in

Effect of Cylindrical Duct
Effects of the cylindrical duct on the density profile along the jet centerline for NPR = 4.0 are shown in Figure 7. The black line indicates the same experimental data as Figure 6(e) and the blue one shows the experimental results conducted by Maeda et al. [10].   [20] states that the maximum possible length where the flow is not choked at the duct exit is 85 mm for a constant-area duct with an inlet Mach number of 1.5, constant Fanning friction factor of 0.004 [21], and duct diameter of 10 mm. Therefore, for the present Laval nozzle with a cylindrical duct, the average Mach number over the cross-section of the duct exit is beyond unity.

Three-Dimensional Jet Structure
The three-dimensional structures of two typical shock-containing jets are illustrated in Figure 8   seen clearly with suddenly varying colour. Figure 8(b) shows that the oblique shocks in shock-cells are reflected at the opposite jet free boundaries to form bicone structure. The structure gradually becomes smaller in shape toward downstream.

Concluding Remarks
The density fields of jets from a round Laval nozzle followed by a cylindrical duct were measured by the rainbow schlieren deflectometry. The three-dimensional density fields of the jets were reconstructed by the convolution back-projection (CBP) method to investigate the effects of the nozzle pressure ratio on the jet structure. A quantitative comparison between the centerline density profiles As a result, it was found that the jet from the Laval nozzle with the cylindrical duct reaches a shock-free state at a nozzle pressure ratio lower than the design condition because of the wall friction along the duct wall. However, the freestream Mach number at the cylindrical duct exit is almost the same as that calculated based upon the assumption of the isentropic flow over the whole flow field.
The cylindrical duct causes the average Mach number at the exit to be reduced, but not change the freestream Mach number at the duct exit and it smooths the density profile when compared with that for the jet issued from the conventional Laval nozzle. Two types of shock-containing jets showing overexpanded and underexpanded states were displayed with the respective isopycnic surface and it was found that the overexpanded jet produces a Mach disk following by successive weak shocks with a circular shape, while the underexpanded jet forms a bicone structure composed of an oblique shock and expansion waves in each shock-cell, which gradually becomes smaller in shape toward downstream.