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Instantaneous three-dimensional (3D) density distributions of a shock-cell structure of perfectly and imperfectly expanded supersonic microjets escaping into an ambient space are measured. For the 3D observation of supersonic microjets, non-scanning 3D computerized tomography (CT) technique using a 20-directional quantitative schlieren optical system with flashlight source is employed for simultaneous schlieren photography. The 3D density distributions data of the microjets are obtained by 3D-CT reconstruction of the projection’s images using maximum likelihood-expectation maximization. Axisymmetric convergent-divergent (Laval) circular and square micro nozzles with operating nozzle pressure ratio 5.0, 4.5, 4.0, 3.67, and 3.5 have been studied. This study examines perfectly expanded, overexpanded, and underexpanded supersonic microjets issued from micro nozzles with fully expanded jet Mach numbers
*M*
* _{j}* ranging from 1.47 - 1.71, where the design Mach number is

*M*= 1.5. A complex phenomenon for free square microjets called axis switching is clearly observed with two types “upright” and “diagonal” of “cross-shaped”. The initial axis-switching is 45 ° within the first shock-cell range. In addition, from the symmetry and diagonal views of square microjets for the first shock-cells, two different patterns of shock waves are viewed. The shock-cell spacing and supersonic core length for all nozzle pressure ratios are investigated and reported.

_{d}Visual observation and investigation of fluid flow provide useful and comprehensive insight into a flow field under inspection and help derive quantitative data from the observed flow patterns. Fluids are mostly transparent, and therefore, invisible; there are some good techniques that can visualize flowing fluids with non-uniform density. A Schlieren technique is very sensitive to density changes and it can visualize invisible flows in transparent media. This technique has been widely implemented in fluids and combustion. In fluid dynamics, the schlieren imaging technique is a common tool to illustrate detailed descriptions of air flows, shock waves, compressions, and expansion fans in supersonic jet flows for use in aerodynamical studies [

It is very difficult and unsatisfactory to investigate microjets when using measurement devices because of the very small dimensions of the micro nozzles, especially those close to the micro nozzle exit. In the present study, to provide a suitable technique for the 3D observation of a supersonic microjet (non-uniform density flow), a non-scanning 3D-CT technique using a multi-directional quantitative schlieren technique was employed [

Air microjets are employed to deliver pressure to a workpiece in many industrial activities using simple sonic nozzles [

Aniskin et al. [

Supersonic microjets have diverse applications in various fields of engineering and industries; for example, in the cooling of micro-electro mechanical systems (MEMS) components, fine particle deposition and removal, and as actuators to control the ground effect in short take-off and vertical landing (STOVL) aircraft when hovering [

Supersonic free jets issued from circular, square, rectangular with different aspect ratios, and regular triangular nozzles are visualized using a laser-induced fluorescence method proposed by Teshima [_{throat} of 6 mm and a nozzle wall thickness of 1.5 mm. Noncircular jets were identified as an efficient technique for passive flow control [

There are very few experimental studies on supersonic microjets, especially for the square cross-section, in the literature. However, with a few exceptions (e.g. [

In the present study, quantitative measurements and qualitative investigations are combined, and supersonic microjets from circular and square micro nozzles have been successfully CT-reconstructed with a custom-made 20-directional schlieren optical system. The results show instantaneous 3D density distributions downstream of the micro nozzles exit. Further, shock-cell spacing, supersonic core length, axis switching, and two different shocks patterns of the symmetry and diagonal views of square microjets are clearly observed and investigated.

A circular and square cross-section convergent-divergent (Laval) micro nozzle, with a design Mach number M_{d} = 1.5, is studied ( [_{e}/A_{th} = 1.18; the square nozzle has a throat size of 850 × 850 μm^{2}, an exit size of 1000 × 1000 μm^{2}, an area ratio A_{e}/A_{th} = 1.38. The convergent and divergent section lengths are 6000 μm and 1300 μm, respectively. The center of the nozzle exit is selected as the origin of the xyz-coordinate system; the nozzle exits are then located downward in the negative z-axis. The x-axis is located

between camera No. 10 (θ = −4.5˚) and camera No. 11 (θ = +4.5˚); the details of the camera system are described in the next section. Detailed descriptions of micro nozzles are reported in [

Figures 2(a)-(c) illustrate the concept of a 20-directional schlieren camera. In the camera system, the target supersonic microjet/non-uniform density field is observed from a 180° direction using numerous schlieren optical systems simultaneously. Twenty systems capture multi-directional views in the 20 angle positions from θ = −85.5˚ to +85.5˚ at intervals of 9˚. Here, angle θ is defined as the

horizontal angle from the x-axis. For pre-investigation and for time-series observation (high-speed schlieren movies) of the target flow, a high-speed camera (HSC) is used simultaneously with the 20-direction schlieren photographing apparatus (between cameras No. 13 and No. 14,

From a technological point of view, this experimental method (CT non-scanning,

Density, deviation density, and density thickness values with an asterisk sign (*) express actual values, and those without the sign represent derived values from the CT-reconstruction process. _{a}*(= 1.2 kg/m^{3}) on the periphery of the observed range of radius R. Figures 3(b)-(k) shows observations in the direction of θ from the x-axis in the inclined coordinates denoted by X(θ) and Y(θ).

The first goal is obtaining the density thickness of deviation density (Dt*(X(θ))), as shown in ^{3})(m). The density thickness of deviation density Dt*(X(θ)) is obtained automatically from schlieren observation using spatial integration of deviation density Δρ*(X(θ),Y) along the line of sight. In other words, the density distribution of target flow is reconstructed using a 2D distribution (image) of “density thickness” as a projection in the CT-reconstruction process.

In the practice for obtaining the density thickness of deviation density (

of images, “with target” and “without target” (without any disturbance in the test section) with a horizontal brightness gradient in the x-direction. Two sets of images are presented by schlieren observation as B(X) and B_{nj}(X) (brightness of schlieren image in no-jet condition). To obtain the density thickness of deviation density Dt(X(θ)) from B(X) and B_{nj}(X), both as shown in Figures 3(f)-(i). As indicated in

Δ B = B ( X ) − B n j ( X ) (1)

is scaled to d(Dt)/dX by

d ( D t ) / d X = ( 1 / K ) ( Δ s / f ) [ Δ B ( X ) / B n j ( X ) ] (2)

where K is the Gladstone–Dale constant for air (K = 2.26 × 10^{−4} m^{3}/kg), Δs (= 1 mm × 2 mm (Hor. × Ver.)) is the transparent width of the light source image on schlieren stop location and f (= 300 mm) is the focal length of the convergent lens. Deviation density thickness Dt(X(θ)) is, therefore, reproduced by the transverse integration of d(Dt)/dX from schlieren images, as shown in

D t ′ ( X ( θ ) ) = D t ( X ( θ ) ) + 2 ( R 2 − X 2 ( θ ) ) 1 / 2 ( ρ a ∗ − ρ r e f ) (3)

In the present study, ρ_{ref} = 0.8 is selected as a reference density because the minimum jet density is above this amount. Using non-deviation treatment for projections of CT by employing reference density guarantees that the CT-reconstructed deviation density ( Δ ρ ( X ( θ ) , Y ) + ( ρ a ∗ − ρ r e f ) ) is non-negative and the CT-reconstruction is possible.

Density thickness images are used for CT-reconstruction by maximum likelihood-expectation maximization (ML-EM) [_{ref}, in which one step (

ρ ( x , y ) = Δ ρ ( x , y ) + ρ a ∗ (4)

The reconstruction was performed cross-section by cross-section and then the cross-sections were stacked to form a three-dimensional density distribution. Therefore, a 2D distribution ρ(x, y) is accumulated in layers to form the 3D-CT distribution ρ(x, y, z). In the present study, the density thickness projections images of 50 (H) × 375 (V) pixel (2 × 15 mm^{2}) are used for CT-reconstruction to produce 3D data 50 (x) × 50 (y) × 375 (z) pixel (2 × 2 × 15 mm^{3}). The voxel size is 0.04 mm in each direction.

A jet is referred to as underexpanded if jet pressure ratio (JPR), P_{e}/P_{b} ˃ 1, overexpanded if P_{e}/P_{b} < 1, and perfectly expanded or pressure matched if P_{e}/P_{b} = 1. In the present work, axisymmetric convergent–divergent (Laval) circular and square micro nozzles with operating nozzle pressure ratios (NPR = P_{o}/P_{b}) of 5.0, 4.5, 4.0, 3.67, and 3.5 have been studied.

This study examines perfectly expanded (ideally, correctly expanded) and imperfectly expanded (overexpanded and underexpanded) supersonic microjets issued from micro nozzles with fully expanded jet Mach numbers M_{j} ranging from 1.47 - 1.71. The design Mach number of these micro nozzles is M_{d} = 1.5. The isentropic nozzle-flow relation [

Owing to page limitations, only sample sets of images in some directions and some positions for some NPR (nozzle pressure ratio) are shown in the subsequent figures and sections.

The images in

Quantity | Symbol | Circular Nozzle | Square Nozzle |
---|---|---|---|

Ambient temperature | T_{b} | 300.35 K (27.2˚C) | 296.55 K (23.4˚C) |

Back (ambient) pressure | P_{b} | 0.101 MPa | 0.102 MPa |

Nozzle design Mach number | M_{d} | 1.5 | 1.5 |

Nozzle exit diameter or exit size | D or h × b | 1000 μm (1 mm) | 1000 × 1000 μm^{2} |

Quantity [Unit] | Symbol | Nozzle | Nozzle Pressure Ratio NPR (P_{o}/P_{b}) | ||||
---|---|---|---|---|---|---|---|

5.0 | 4.5 | 4.0 | 3.67 | 3.5 | |||

Absolute (Stagnation) pressure [MPa] | P_{o} | Circular | 0.505 | 0.455 | 0.404 | 0.371 | 0.354 |

Square | 0.510 | 0.459 | 0.408 | 0.374 | 0.357 | ||

Gauge pressure [MPa] | P_{G} | Circular | 0.404 | 0.354 | 0.303 | 0.270 | 0.253 |

Square | 0.408 | 0.357 | 0.306 | 0.272 | 0.255 | ||

Jet pressure ratio (JPR) | P_{e}/P_{b} | Circular | 1.36 | 1.23 | 1.09 | 1.0 | 0.95 |

Square | |||||||

Exit pressure [MPa] | P_{e} | Circular | 0.138 | 0.124 | 0.110 | 0.101 | 0.096 |

Square | 0.139 | 0.125 | 0.111 | 0.102 | 0.097 | ||

Fully expanded jet Mach number | M_{j} | Circular | 1.709 | 1.638 | 1.559 | 1.5 | 1.467 |

Square | |||||||

Fully expanded jet diameter / height [μm] | D_{j} | Circular | 1070 | 1044 | 1018 | 1000 | 991 |

h_{j} | Square | ||||||

Reynolds number | Re | Circular | 29875 | 29083 | 28140 | 27410 | 26991 |

Square | 29978 | 29182 | 28237 | 27504 | 27084 | ||

Operating condition | - | Circular / Square | Underexpanded | Full expanded | Over expanded |

_{s}” and “supersonic core length L_{c}” are depicted by the dashed line in

Using the density thickness images (

reconstruction, 3D instantaneous density distributions of supersonic microjets have been successfully obtained for z = 0 to 15 mm.

The values of the densities are displayed in light and dark based on the grayscale level under images. The darker parts are related to lower density areas and the brighter parts are related to higher density points. The corresponding density contour diagrams of each cross-section are depicted as well (

because of the reflection of the expansion fans as compression waves from the jet boundary. The compression waves converge and form intercepting shocks (inception, incident, or barrel shocks). Then, the intercepting shocks intersect and the reflected shocks are formed; this intersects with the jet boundary and reflects as expansion fans again; however, this process is repeated.

Based on the viewing angle, two types of schlieren images (_{circular} = 31 pixel = 1.24 mm, D_{square_symmetry}_{ }= 32 pixel = 1.28 mm, and D_{square_diagonal} = 35 pixel = 1.4 mm. The diagonal of a square is equal to D s q = 2 S = 1.41 S = 1.41 mm , where S = 1 mm equals one side length of the square. Therefore, it is straightforward that the diagonal view must be wider than the symmetry view as shown in

reported by some research papers.

By comparing cross-sections near the nozzle outlet and the near end of the first shock-cell, we can see microjet axis-switching clearly as sketched on the left side of

with two types “upright” (

Recently, Cabaleiro and Aider [

For better evaluation of the reconstructed density values, radial and axial density distributions diagrams are illustrated in

Shock-cell parameters are important to characterize supersonic jets such as

“shock-cell spacing (length) L_{s}” and “supersonic core length L_{c}”. Note that shock-cell spacing L_{s} for all shock-cells can be obtained from schlieren images. The shock-cell spacing for the first five shock-cells based on our experimental data and results is depicted in _{s}) in supersonic circular [

L s = π D j ( M j 2 − 1 ) 1 / 2 / 2.40483 (5)

L s = h j [ 2 ( M j 2 − 1 ) ] 1 / 2 (6)

where M_{j}, D_{j}, and h_{j} are the fully expanded jet Mach number, diameter (circular jets), and width (square jets), respectively. For the first L_{s}, Mehta and Prasad [_{s}; no attempt was made to compute the lengths of the second and later cells for microjets.

L s = D ( 0.57 M j 2 − 0.15 ) (7)

where D is the nozzle diameter.

We compare the shock-cell spacing (L_{s}) determined from our experimental results and estimated using Tam, Phalnikar and Mehta’s correlations, as shown in _{s} in this study and Tam’s correlation. However, the measured values of the L_{s} in our studies a little differ from those estimated by Tam’s correlation. The L_{s} in the present study, about 9% - 13% for circular and 4% - 8% for square microjets, are smaller than those predicted by Tam’s correlation (Equations (5) and (6)). Similar comparisons were also made by Hu and McLaughlin [_{s} of about 10% was smaller than that predicted by Tam’s correlation (Equation (5)). Further, they mentioned that the difference is attributed to a thicker boundary layer caused by the low Reynolds number for the microjets, as the boundary layer thickness would be inversely proportional to that of Reynolds number, and the Reynolds number directly proportional to the nozzle outlet size [_{x} is the Reynolds number. For circular and rectangular microjets, NPR = 5.0, the boundary layer thickness within a distance equal to the nozzle diameter is almost one-pixel (0.04 mm) or rarely two-pixel (0.08 mm) and based on Schlichting equation [

Phalnikar et al. [_{s}.

Further, “supersonic core length L_{c}” is usually used to characterize supersonic jets. It is defined [_{c} as the length of the conical region surrounding shock-cells from schlieren images. Phalnikar et al. [_{c} for circular microjets.

L c = D [ ( 1.8 × N P R ) + 2.9 ] (8)

where D is the nozzle diameter and NPR = nozzle pressure ratio (P_{o}/P_{b}). We used CT-reconstructed density data for estimating L_{c} and compared it with Equation (8), as shown in _{c} for square microjets is less than that for circular microjets (

spacing L_{s} (

By employing the non-scanning 3D-CT technique using a 20-directional quantitative schlieren optical system, 3D-CT reconstructions of instantaneous density distributions of the shock-cells structure of circular and square supersonic microjets were obtained successfully. The results clearly demonstrated the flow pattern downstream of the micro nozzle exit. The multiple reflected waves configuration achieved a diamond-like form throughout the exhaust microjets.

A complex phenomenon for free square microjets, which is called axis switching, is clearly observed with two types “upright” and “diagonal” of “cross-shaped”. In addition, for the first shock-cells from the symmetry and diagonal views of square microjets, two different patterns of shock waves were observed.

The shock-cell spacing and supersonic core length for all nozzle pressure ratios were investigated and reported. Comparisons with experimental and theoretical computations (particularly those of Tam and Tanna [_{s} and L_{c} in this study and other correlations. As microjets have low Reynolds number, some acceptable differences exist between the compared results.

I thank our colleagues from the Institute of Environmental Science and Technology, the University of Kitakyushu who provided the micro nozzles that greatly assisted the research. The supports are gratefully acknowledged for the JSPS Grants-in-Aid for Scientific Research (C)16K06118.

The authors declare no conflicts of interest regarding the publication of this paper.

Nazari, A.Z., Ishino, Y., Ishiko, Y., Ito, F., Kondo, H., Yamada, R., Motohiro, T., Miyazato, Y. and Nakao, S. (2020) Multi-Schlieren CT Measurements of Supersonic Microjets from Circular and Square Micro Nozzles. Journal of Flow Control, Measurement & Visualization, 8, 77-101. https://doi.org/10.4236/jfcmv.2020.83005

A: Cross section area [m^{2}]

B: Brightness of schlieren image [-]

B_{nj}: Brightness of schlieren image in no-jet (no disturbance) condition in the test section [-]

b: Nozzle width [m]; rectangular nozzle

D: Nozzle exit diameter [m]

D_{j}: Fully expanded jet diameter [m]

Dt: Density thickness of deviation density [(kg/m^{3}) (m)]; derived value

Dt': Density thickness [(kg/m^{3}) (m)]; derived value

Dt^{*}: Density thickness of deviation density [(kg/m^{3}) (m)]; actual value

f: Focal length of convergent lens in schlieren system [m]

h: Nozzle height [m]; rectangular nozzle

h_{j}: Fully expanded jet height [m]; rectangular nozzle

JPR: Jet pressure ratio (= P_{e}/P_{b}) [-]

K: Gladstone-Dale constant of air [m^{3}/kg]

L_{c}: Supersonic core length [m]

L_{s}: Shock-cell spacing (length) [m]

M_{d}: Nozzle design Mach number [-]

M_{j}: Fully expanded jet Mach number [-]

n: Shock-cell number [-]

NPR: Nozzle pressure ratio (=P_{o }/P_{b}) [-]

P_{o}: Plenum (absolute, stagnation) pressure [MPa]

P_{b}: Back pressure [MPa]

P_{e}: Exit pressure [MPa]

P_{G}: Gauge pressure [MPa]

R: Radius of reconstruction area [m]

Re: Reynolds number [-]

T_{b}: Ambient temperature [K]

x, y, z: Cartesian coordinates system of reconstruction volume [m]

X(θ), Y(θ): Inclined coordinates by angle θ [m]

ΔB: Deviation brightness on schlieren image [-]

Δs: Transparent width of the light source [m]

Δρ: Deviation density [kg/m^{3}]; derived value

Δρ^{*}: Deviation density [kg/m^{3}]; actual value, ( Δ ρ ∗ = ρ ∗ − ρ a ∗ )

δ: Boundary layer thickness [m]

θ: Angle of observation [˚ (degree)]

ρ: Density [kg/m^{3}]; derived value, ( ρ = Δ ρ + ρ a ∗ )

ρ^{*}: Density [kg/m^{3}]; actual value

ρ a ∗ : Ambient density of air [kg/m^{3}]; actual value

ρ_{ref}: Reference density [kg/m^{3}]