Open Journal of Fluid Dynamics, 2013, 3, 85-91 Published Online July 2013 (
Structure and Oscillation of Underexpanded Jet
Hiromasa Suzuki1, Masaki Endo2, Yoko Sakakibara2
1Graduate School of Advanced Science and Technology, Tokyo Denki University, Saitama, Japan
2Division of Electronic and Mechanical Engineering, Tokyo Denki University, Saitama, Japan
Received June 3, 2013; revised June 10, 2013; accepted June 17, 2013
Copyright © 2013 Hiromasa Suzuki et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, flow characteristics are experimentally and numerically discussed, especially concerning shock-cell struc-
ture and oscillatory phenomena of supersonic jet. The jet becomes underexpanded when the nozzle pressure ratio of
convergent nozzle exceeds the critical value. The underexpanded jet is not uniform because of the presence of the ex-
pansion wave, the compression wave and the shock wave formed in it. Many vortices are induced in jet boundary region
by shearing stress generated between supersonic jet flow and atmospheric air. They interact with shock waves compos-
ing cell node of jet, which is closely related to a noise radiating from the jet. In experiment, flow field is visualized and
the shape of cell structure is examined. Furthermore, frequency of vortex generation is measured. The jet is simulated
using TVD scheme and the results are compared with the experimental ones. Distortion of cell structure caused by three
dimensional oscillation of jet and b ehavior of vortices are discussed.
Keywords: Underexpanded Jet; Plate Shock; Incident Shock; Visualization; TVD Scheme
1. Introduction
Underexpanded jet is one of the supersonic jets; when the
nozzle pressure ratio of the convergent nozzle is higher
than the critical value, the jet is underexpanded. In case
of air, the critical pressure ratio is about 1.893. The ex-
pansion waves, which are generated at the nozzle lip,
propagate in the jet and reflected at jet boundary as the
compression waves. They converge to form a shock
wave, and the expansion waves are generated again when
the shock wave reaches the jet boundary. The jet is not
uniform because these phenomena repeat and typical
shock cell structure of the underexpanded jet is formed.
When the flat plate is placed perpendicularly to the jet,
the strong wave called “plate shock” appears in front of
the plate. The air passes through the plate shock, and its
velocity becomes subsonic. The flow changes its direc-
tion and the wall jet is formed along the plate.
The structure of underexpanded impinging jet is
shown in Figure 1. Such a flow field is often observed in
the assist gas of laser cutting [1], the cooling jet of glass
tempering process [2,3] and the exhaust jet of VSTOL
aircraft or rocket. The flow field causes several issues,
for example, behavior of jet disturbed by vortex of the jet
boundary and high frequency noise generation called
“screech tone” [4-7].
In this study, shadowgraph method and Schlieren
method were used to visualize the flow field. Further-
more, a period of generation cycle of vortex near the jet
boundary was measured using a device comprising a
laser and a photoelectric sensor.
The behavior of jet and vortices near the jet boundary
were examined. In addition, under the same conditions,
the numerical study was carried out using TVD scheme.
2. Experimental Apparatus and Numerical
2.1. Experimental Considerations
The piping system of the experimental apparatus is
shown in Figure 2. The air is compressed by screw
Figure 1. Schematic of underexpanded impin ging jet.
opyright © 2013 SciRes. OJFD
Figure 2. Piping system.
compressors and passes through the air dryer, surge tank,
and oil mist separator. The compressed air is supplied
into the plenum chamber to which convergent nozzle is
attached in the soundproof room. The stagnation pr essure
of the air in the plenum chamber can be manually regu-
lated by two gate valves and is measured by a digital
manometer. An acoustic absorbent sheet is glued to the
exit plane of the convergent nozzle. In this study, the
convergent nozzle is adopted. The diameter of the con-
vergent nozzle exit is D = 10 mm and the radius of the
internal surface of the nozzle R = 20 mm, which is well
finished. And a flat plate is placed to be perpendicular to
the jet axis, downstream of the nozzle.
The experiment is carried out under the condition of
the nozzle pressure ratio p0/pa = 3.0 to 3.8 with 0.1 steps
and the nozzle plate distance l/D = 1.5 to 2.5 with 0.1
steps, where p0 is the pressure in the plenum chamber
and pa the atmospheric pressure. The behavior of jet,
vortex moving along j et boundary and plate shock above
plate was examined under the conditions as mentioned
The flow fields were visualized using Schlieren pho-
tography and shadowgraphy. The spark bulb is employed
as source of light, which emits flushing light duration a
period of 18 0 n s .
Furthermore, the advective motion of vortex near the
jet boundary is measured using a device composed of a
laser and a photoelectric sensor, i.e. a simple Schlieren
system. A semi-conductor type laser is used the beam
width being lw = 3 mm. The vortex measurement system
is shown in Figure 3. The knife edge is installed to cause
it to block out the light. Going through the vortex, the
laser beam is refracted in a direction away from the vor-
tex core, and then, the illumination of beam reachable the
sensor increases. Thus, the sensor detects the moving
vortex. The laser is set at the position of antinode of sec-
ond cell where the vortex starts to be organized. The
antinode position is measured by means of the Schlieren
Figure 3. Vortex measurement system.
2.2. Numerical Scheme
The underexpanded jet impinging on the plate without an
elastic deformation is numerically simulated. The com-
putational flow field is assumed to be axisymmetric and
Euler equations are employed as the governing equations.
They are solved using 2nd-order TVD scheme proposed
by Yee [8], with minmod function as a stable limiter
function [9].
The computational region used in this study is covered
with a structured mesh. The radial distance of the region
is 7.5 diameters of the converging nozzle, where 150
computational grids are clustered towards the jet axis.
Along the jet axis there are 100 grids at uniform intervals.
The critical condition estimated from the stagnant prop-
erties in the plenum chamber is specified at the exit plane
of the nozzle. The boundary condition is given to be re-
flective at the jet axis. The nozzle wall and the plate are
slip wall boundary and, at the other boundaries, the Rie-
mann variables are kept at constant.
3. Results and Discussion
3.1. Behavior of Free Jet
Figure 4 shows the flow field visualized using Sch lieren
method. These pictures were taken at random under the
same condition. In these figures, the underexpanded free
jet issues from the nozzle in the lower side of the picture
at the pressure ratio p0/pa = 3.5. The knife edge is placed
in the direction normal to jet axis so that the expansive
region is dark and compressive region bright. As can be
seen in these pictures, the dark region and the bright re-
gion are deformed and the deformation becomes stronger
as further away from the nozzle. Thus, the jet is found to
be unsteady and oscillate laterally or helically.
In order to quantitatively evaluate the unsteady be-
havior of the jet, the statistical analysis is conducted
through the visualization of the flow field. 50 pictures
were taken at random at each pressure ratio, and the in-
stantaneous shape of the jet was measured, e.g. the cell
length lc and the inclination of cell node αn as shown in
Figure 5. The cell node between the 1st and 2nd cell, i.e.
end node of cell, is defined as 1st cell node, and so on.
3.2. Cell Length
Figure 6 shows deviations of cell length of 1st, 2nd and
3rd cell at each pressure ratio. The empty plots show the
Copyright © 2013 SciRes. OJFD
(a) (b) (c)
(d) (e)
Figure 4. Schlieren photograph (p0/pa = 3.5).
Figure 5. Measurement points.
Figure 6. Deviation of cell length.
lengths of individu al cell, and the solid ones to tal lengths
between the nozzle exit and the end node of corresp ond-
ing cell. Overall, these plots gradually rise as the pressure
ratio increases and these take extreme values at p0/pa =
3.4 and 3.6. Concerning the individual cell length, the
deviation for downstream cell is seen to swell two-fold.
Thus, the amplitude of the oscillation in length becomes
larger downstream.
Comparing the deviations between the individual and
3.3. Cell Node Angle
n of cell node angles αn of 1st,
3.4. Behavior of Impinging Jet
he same condition
hows the deviation against the pressure ratio.
the deviation of αp also has high value at
e total lengths, the deviation of total length is smaller
than that of individual length regardless of the pressure
ratio. This means that the 3rd cell shrinks if the 2nd cell
stretches, that is, the changes of length caused by oscilla-
tion of the neighboring cells are cancelled as a whole.
Observing the difference between the deviations at each
pressure ratio carefully, the oscillation at p0/pa = 3.6 may
have the same phase.
Figure 7 shows deviatio
2nd and 3rd cell. This deviation is related to three-di-
mensional oscillation because αn means the angle of node
against jet axis as shown in Figure 5. The value of the
deviation almost linearly rises with the increase in the
pressure ratio and becomes larger downstream, e.g., σ1 =
0.9, σ2 = 2.2 and σ3 = 4.3 at p0/pa = 3.4. Hence, this
means that the hydrodynamic instability along the jet
boundary grows downstre am.
The experiment is carried out under t
of p0/pa as the free jet and the pictures are statistically
analyzed. 50 pictures were taken at random in each noz-
zle plate distance. The geometric characteristics of the jet
structure were measured, e.g. the positions of vertex and
base of the plate shock (lvp and lbp), the plate shock di-
ameter dp and the inclination of plate shock αp as shown
in Figure 8.
Figure 9 s
e deviation σ in the ordinate is normalized by the de-
viation σs at l/D = 1.5. It can be seen that the deviation of
αp undergoes abrupt change from l/D = 1.6 to 1.8 and
that the fluctuation of lvp stands out at l/D = 1.8. These
comparatively high values represent that the jet strongly
oscillates under this condition. Especially at l/D = 1.8,
deviation of lvp stands out and this means that the center
part of plate shock strongly oscillates in axially-sym-
metric pattern.
= 1.7, and this means that lateral or helical motion
occurs simultaneously. Thus, the plate shock intensely
Figure 7. Deviation of angle of cell node.
Copyright © 2013 SciRes. OJFD
Figure 8. Measurement pots. in
Figure 9. Deviation of plate shock.
scillates at a certain nozzle plate distance.
p0/pa = 3.4.
oFigure 10 shows averages of lvp and lbp at
e dashed line shows the position of 1st cell node ob-
tained from pictures of free jet. Position of plate shock
moves rapidly downstream to approach the plate at be-
tween l/D = 1.6 and l/D = 1.7. Distance between the ver-
tex and base of the plate shock increase in the range from
l/D = 1.7 to l/D = 1.8. These phenomena seem to be af-
fected by location of the vertex of plate shock; whether
the vertex is in the compression region or not. In the
compression region, the plate shock crosses the incident
shocks from the nozzle lip before they reach the jet axis,
while in the expansion region, the incident shocks inter-
sect each other on jet axis as shown in the next.
Figure 11 shows shadowgraph pictures of im
t at various nozzle plate distances from l/D = 1.5 to 2.5.
The jet issues from the nozzle exit on the bottom of each
picture. Plate shock crosses the incident shocks at l/D =
1.5 and 1.6, and at l/D > 1.7, the intersection of incident
shock appears upstream of plate shock. In addition, a
weak shock wave can be seen in the region between the
plate shock and the plate. This weak shock appears up to
l/D = 1.6 and disappears at l/D = 1.7. At l/D > 1.8, the
plate shock is in the expansion region of second cell.
Details of shock structure can be seen in the res
tained by the numerical simulation. Figures 12 show
the density contours (left-side half of each figure) and the
pressure contours (right-side half of each figure) at dif-
ferent nozzle plate distance between l/D = 1.4 and 2.0.
The flow in the region between the plate shock and the
plate is seen to be complex with l/D smaller than l/D =
1.6. Under such conditions, because the intersection of
the incident shocks in the jet is located downstream of
Figure 10. Distance of plate shock from nozzle.
(a) (b) (c)
(d) (e) (f)
Figure 11. Shadow0 a l/D =
e plate shock, the strength of plate shock near the jet
position of plate shock chang es into second cell at
graph photograph (p/p = 3.4). (a)
1.5; (b) l/D = 1.6; (c) l/D = 1.7; (d) l/D = 1.8; (e) l/D = 1.9; (f)
l/D = 2.0.
axis is stronger than that in the surrounding region.
Therefore, contact surface is generated downstream of
the plate shock and the iteration of reflection of waves
between jet boundary and the contact surface occurs.
Weak shock wave seen in Figures 11(a) and (b) is con-
sidered to be the shock wave which consists of con-
verged compression wave during this iteration. The weak
shock wave disappears at l/D = 1.7, where the intersec-
tion of the incident shocks is upstream of plate shock
(see Figure 11(c) and Figure 12(e)). In Figure 12(f),
plate shock forms close to the incident shock and they
are partly piled up. This can be seen also in Figure
rger l/D.
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Copyright © 2013 SciRes. OJFD
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 12. Numerical result ( = 1.6; (e) l/D = 1.7; (f) l/D = 1.8; (g)
3.5. Frequency of the Vortex Near the Free Jet
Figu the output signal history measured by
change of the dominant frequencies against the pressure
p0/pa = 3.4): (a) l/D = 1.3; (b) l/D = 1.4; (c) l/D = 1.5; (d) l/D
l/D = 1.9; (h) l/D = 2.0.
re 13 shows
the photoelectric sensor at p0/pa = 3.0. The sharp drop of
the voltage means that a vortex passed through a laser.
The vortices are seen to intermittently go through the
measuring point. To search the periodicity of the vortex
movement, the output signal is analyzed using FFT. The
rise is shown in Figure 14. The empty plot denotes the
dominant frequency obtained though the FFT analysis of
output data as in Figure 13 at each pressure ratio. The
solid plots are obtained from the experimental results of
the sound pressure wave, which comes from the paper by
S. Tamura and J. Iwamoto [10]. The sound pressure
wave has two components of the frequencies. Those
dominant frequencies lie around 15 kHz and 25 kHz,
Figure 13. Output signal wave (p0/pa = 3.0).
Figure 14. Frequency of vortex generation and sound wave.
igher frequency jumps at p0/pa = 3.3 and lower one
oscillatory phenomena of underex-
ow field was visualized using
the downstream region
frequency of vor-
[1] J. Fieret, M. J.“Overview of Flow
k Waves on
ass,” Journal of
slightly decreasing with the rise of pressure ratio. The
drops at 3.1. These discontinuous changes are known to
be caused by the structure of the underexpanded jet. As
can be seen in Figure 14, the change of lower frequency
quantitatively agrees well with that of the vortex. It is
evident that the sou nd emitted form the jet is cause by the
behavior of the vortex moving along the jet boundary.
4. Conclusions
To investigate the
panded jet, the flthe
schlieren photography, and shadowgraphy and the be-
havior of vortex near the jet boundary was observed. As
a result, the following conc lusions were drawn:
The deviation of cell node angle linearly increases
with rise of pressure ratio .
The deviation of cell length takes maxima and min-
ima at some pressure ratios.
The weak shock is formed in
of plate shock, when the vertex of plate shock locates
near the end of expansion region.
Good agreement is found between
tex generation and that of sound wave.
Terry and B. A. Ward,
Dynamics in Gas-Assisted Laser Cutting,” SPIE High
Power Lasers, Vol. 801, 1987, pp. 243-250.
[2] S. Aratani and N. Ojima, “Effect of Shoc
Fracture and Quenching in Tempered Glass,” Proceedings
of the 18th International Symposium of Shock Waves,
Sendai, 21-26 July 1991, pp. 1277-1282.
[3] J. M. Barsom, “Fracture of Tempered Gl
the American Ceramics Society, Vol. 51, No. 2, 1968, pp.
75-78. doi:10.1111/j.1151-2916.1968.tb11840.x
[4] A. Powell, Y. Umeda and R. Ishii, “Observation of the
Oscillation Modes of Choked Circular Jets,” Journal of
the Acoustical Society of America, Vol. 92, No. 5, 1992,
pp. 2823-2839. doi:10.1121/1.404398
[5] A. Powell, “The Sound-Producing Oscillations of Round
Underexpanded Jets Impinging on Normal Plates,” Jour-
nal of the Acoustical Society of America, Vol. 83, No. 2,
1988, pp. 515-533. doi:10.1121/1.396146
[6] C. K. W. Tam, “Broadband Shock Associated Noise from
Supersonic Jets Measured by a Ground Observer,” AIAA
Journal, Vol. 30, No. 10, 1992, pp. 2395-2401.
[7] J. Panda, “Shock Oscillation in Underexpanded Screech-
ing Jets,” Journal of Fluid Mechanics, Vol. 363, 1998, pp.
173-198. doi:10.1017/S0022112098008842
[8] H. C. Yee, “Upwind and Symmetric Shock-Capturing
umerical Study of Os-
Scheme,” National Aeronautics and Space Administration
(NASA), Washington DC, 1987.
[9] Y. Sakakibara and J. Iwamoto, “N
cillation Mechanism in Underexpanded Jet Impinging on
Plate,” Journal of Fluids Engineering, Vol. 120, No. 3,
1998, pp. 477-481. doi:10.1115/1.2820687
[10] S. Tamura and J. Iwamoto, “Study on the Noise Gen-
erated from Underexpanded Free and Impinging Jet,”
Transactions of the Japan Society of Mechanical En-
gineers, Series B, Vol. 60, No. 579, 1994, pp. 283-289.
Copyright © 2013 SciRes. OJFD
zle exit
e shock
xit and flat plate
e of plate sh ock
lw: Width of laser beam
pa: Atmospheric pressure a plenum chamber
f free jet
of 2nd cell of free jet
D: Diameter of noz
dp: Diameter of plat
f: Frequency
l: Distance between nozzle e
lc: Length of c
lan: Position of antinode of second cell
lbp: Position of bas
l: Average of position of base of plate shock
lvp: Positio n of verte x o f p l a te shock
l: Average of position of vertex of plate shock
p0: Stagnation pressure in
R: Radius of convergent no
αn: Angle of cell node
αp: Angle of plate shock
σ: Deviation
σ1: Deviation of 1st cell o
σ2: Deviation
σ3: Deviation of 3rd cell of free jet
σs: Deviation at l/D = 1.5
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