Numerical study of a sonic jet in a

supersonic crossflow over a flat plate

Cite as: Phys. Fluids 32, 126113 (2020); https://doi.org/10.1063/5.0026214
Submitted: 25 August 2020 . Accepted: 27 November 2020 . Published Online: 15 December 2020

Imran Rasheed, and Debi Prasad Mishra

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Phys. Fluids 32, 126113 (2020); https://doi.org/10.1063/5.0026214 32, 126113 © 2020 Author(s).

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Numerical study of a sonic jet in a supersonic
crossflow over a flat plate
Cite as: Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214
Submitted: 25 August 2020 • Accepted: 27 November 2020 •
Published Online: 15 December 2020

Imran Rasheed and Debi Prasad Mishraa)

AFFILIATIONS
Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India

a)
Author to whom correspondence should be addressed: mishra@iitk.ac.in

ABSTRACT
A sonic circular injector discharging into a Mach 1.6 freestream over a flat plate with a jet to crossflow momentum flux ratio of
1.73 is investigated numerically using a three-dimensional Reynolds-averaged Navier–Stokes equation. Menter’s shear stress
transport k–ω turbu lence model is employed to understand the complex flow features associated with jet–freestream
interaction. The validation of the numerical solution is achieved by comparing the velocity and flat plate surface pressure
measurements from the experiments, and the numerical solu tion shows good agreement with the experimental data. The
present work emphasizes the flow field studies that include the identification of shocks, recirculation zones, and vortex
structures. The Omega (Ω) vortex visualization method is employed for identifying the vortex structures. Comparison with high
Mach number freestream conditions (M∞ = 2 and 4) shows that the vortex structures remain the same, irrespective of the
freestream Mach number. A close analysis of the jet near-field shows several new vortex structures, including a secondary
surface trailing vortex. The formation of each of these vortices lacks clarity to date. Considering the complex three-dimensional
nature of the flow field, an attempt has been made to trace the formation of the vortex structures associated with a jet in
supersonic crossflow.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0026214
.,s
appears due to the blocking of the incoming supersonic
freestream by the jet. The adverse pressure gradients, due to
the bow shock, force the incoming boundary layer to
I. INTRODUCTION separate and lead to the forma tion of a weak shock called
the lambda shock. This separation also results in a
The need for high-speed applications, ranging from horseshoe vortex, which remains close to the flat plate. A
missile technology to space applications, has drawn pair of counter-rotating vortices (CRVP) is formed away from
significant interest in hypersonic propulsion research over the the injector. The CRVP plays a significant role in the mixing
past few decades. The scramjet or supersonic combustion of the jet and freestream. Another pair of vortices called
ramjet is one of the most pre ferred technologies for surface trailing vor tices (STVs) originates from the
hypersonic propulsion. It is crucial to achieve proper mixing separation zone at the jet leeward side and propagates close
of fuel and air inside the combustor for efficient com bustion. to the flat plate surface.
However, because of the engine requirements, very little The jet in supersonic crossflow has been studied
time, of the order of a few milliseconds (1 ms–3 ms), is extensively in the past using the Reynolds-averaged
available to mix the fuel with air.1,2 Moreover, high Navier–Stokes (RANS4) equa tion, Large Eddy Simulation
temperature inside the com bustor can lead to structural (LES5,6), Direct Numerical Simulation (DNS7), and hybrid
degradation of the combustor walls. As a result, fuel injection RANS/LES.8 While LES and DNS are essential to identify the
methods play an essential role in penetration, mixing, and instantaneous structures, RANS is useful in extract ing mean
combustion of fuel inside the combustor. Transverse injection
flow features. Viti et al.4employed RANS simulation for a
is one of the simple and suitable methods for fuel injec tion in
supersonic flows and offers excellent penetration and sonic jet injected into a freestream flow of Mach number
efficient near field mixing.3 The complex three-dimensional (M∞) 4.0 and jet to crossflow momentum flux ratio (J) of
flow structures associated with the jet in supersonic 17.4. They iden tified several vortex structures near-field of
crossflow (JISCF) include recir culation zones, which are the injector, including three new trailing vortices that merge to
low-velocity pockets for the flame to stabi lize. The highly form the widely desig nated counter-rotating vortex pairs
expanded jet results in the formation of barrel shock, which (CRVPs) and an upper trailing
terminates in a Mach disk. A three-dimensional bow shock
Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214 32, 126113-1 Published under license by AIP Publishing
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understand the complex flow features associated with a jet in
supersonic crossflow, there is still a lack of clarity.
In the present study, a numerical simulation is carried
vortex (UTV). In another LES study, Xue et al.9identified out based on the experimental works of Santiago and
three trailing vortices similar to those of the work of Viti et Dutton,13 VanLerberghe et al.,14 and Everett et al.15
al.,4 with a freestream Mach number of 4.0. At the same The freestream Mach number (M∞) is selected as 1.6 with a
time, Rana et al.,6for a nominal M∞ of 1.6 and J of 1.73, jet to crossflow momentum flux ratio (J) of 1.73 to validate
did not identify any trailing vor tices reported by Viti et the numerical model. The objective of the present work is
al.4and concluded that the results are caused due to the divided into two parts: (i) Study the flow field structures
associated with jet–freestream interaction and investigate
effect of a higher freestream Mach number. Kawai and
their role in mixing the jet with freestream. The freestream
Lele,5for conditions similar to those of Ref. 6, reported very Mach number of 1.6 with a jet to crossflow momentum flux
faint upper trailing vortices but did not report any other ratio (J) of 1.73 is used for this part of the study. (ii) Identify
trailing vortices near-field of the injector. A recent study by whether the free stream Mach num ber (M∞) affects the
Chai et al.10 also did not report any trailing vortices vortex structures. Three different freestream Mach numbers,
reported by Viti et al.4 At the same time, for a jet in M∞ = 1.6, 2, and 4, with J = 1.73, are com pared to identify
subsonic crossflow, Broadwell and Breidenthal11 noted that the effect the freestream Mach number on vortex structures.
far-field mixing is independent of the Reynolds number for
high values of the flow Reynolds number. This could imply
that the vortex structures associated with jet and freestream II. COMPUTATIONAL DETAILS
interaction are independent of the flow field Mach number.
Recently, Sun and Hu12 performed a detailed study of surface A. Physical model
trailing vortices in a flow field with an M∞ of 2.7 and J of 2.3 The present numerical study has been carried out using
and 5.5. The authors claimed that the surface trailing vortices ANSYS Fluent V18.0. The three-dimensional domain for the
originate from the jet leeward side separation and not from study is shown in Fig. 1. The model comprises a rectangular
the secondary separation zone at the jet windward side, as domain where the sonic air is injected through a circular port
reported by Viti et al.4In the same study, the authors also of diameter (D) 4 mm into a freestream air of Mach number
reported the formation of a secondary surface trailing vortex, 1.6. The details of the computational domain are given in
which is not identified in any other JISCF studies. The above Table I.
analysis shows that despite the decade-long efforts to
 FIG. 1. (a) Generated three-dimensional mesh. Half part of the flat plate is modeled, assuming that the flow is symmetric. (b) Mesh distribution in the symmetric center
plane with a denser mesh near the fuel injector. (c) Meshes at the circular injector.

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y+value is close to 1.
For reducing computational costs, many standard
methods can be utilized to simulate an incoming boundary
TABLE I. Details of the computational domain. layer, such as the recycling-rescaling method16 and synthetic
boundary layer gener ation method.6Peterson and Candler17
Height of air inlet (Hd) 50 mm Flat plate length (Ld) 90 mm suggested that for circular
Flat plate half width (Wd) 45 mm Fuel injector diameter (D) injectors aligned perpendicular with freestream, a mean
4 mm Fuel injector center from the air inlet (L1) 30 mm RANS pro file of the boundary layer can be considered as a
Boundary layer thickness (δ) at X/D = −5 3.1 mm good representation of the incoming boundary layer,
provided that the jet penetrates well above the boundary
layer and the jet exit is adequately resolved. For the present
simulation, a separate 3D RANS simulation of the flat plate is
B. Computational grid carried out with the same incoming flow conditions as those
The three-dimensional mesh for the current study is of the experiment. The velocity profile is then extracted from
gener ated using Ansys Meshing. Considering the the RANS simulation and is given as input to the main jet in
computational time and resources, half of the flat plate is supersonic crossflow simulation, such as to keep δ = 3.1 mm
modeled for the study. A fine mesh is placed near the injector at X/D = −5.
region and close to the flat plate to accu rately capture the
boundary layer and complex flow features near the injector. C. Boundary conditions
Initially, a relatively coarse mesh with 836 500 cells is pre
pared for the study, which is then extended to a medium For the flow field study of a jet in supersonic crossflow
mesh with 1 269 100 cells. Finally, grid adaption based on over the flat plate, the incoming free stream air is given at a
the pressure gradient is done to capture the regions with Mach num ber (M∞) of 1.6 with a stagnation pressure (P0∞)
higher pressure gradients such as bow shock, which resulted and temperature (T0∞) of 241 kPa and 295 K, respectively.
in a fine mesh with 1 501 367 cells. The fine mesh is used to The static pressure (P∞) of the incoming air is 57 kPa. The jet
generate the results presented in this paper. The height of
to crossflow momentum flux ratio (J) is 1.73. Later, two
the first cell is kept as 1.6 × 10−6m, which ensured that the
freestream Mach numbers, M∞ = 2 and 4, are studied for
 comparison with the M∞ = 1.6 flow field. The bound ary the present study.
conditions for the present study are shown in Fig. 2. The
upper boundary is at a sufficient height from the plate surface
such that it is unaffected by the major flow features and is
defined as a symmet ric or freestream boundary. The flow
conditions are summarized in Table II.

D. Governing equations
For compressible flows, the density is not constant;
hence, the Reynolds Averaged Navier–Stokes equations
(RANS) become

FIG. 2. Boundary conditions adopted for

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TABLE II. Flow conditions for the present simulation.

Freestreamb
Flow propertiesa Case A Case B Case C Jetc

Mach number 1.6 2 4 1

Stagnation temperature (K) 295 295 295 295
Stagnation pressure (kPa) 241 284 1373.925 476
Velocity (m/s) U∞ = 446 513 672 Uj = 315
Pressure ratio, P0j/P∞ 8.4 13.1 52
Turbulent intensity, I 5%
μt
dynamic viscosity ), μ10
turbulent viscosity
Viscosity ratio (
Jet to crossflow momentum flux ratio, J 1.73
a
Subscript 0 denotes total flow conditions.
 b
Subscript ∞ denotes the freestream properties.
c
Subscript j denotes the jet properties.
The time-averaged values are represented using an
overbar (−), and mass (or Favre) averaged values are
complicated because of density fluctuations. In these cases, ˜
density weighted averaging or Favre averaging is applied to represented using a tilde (∼). ρ¯, p¯, uĩ , and H represent
certain quanti ties in the RANS equation. The quantities such the mean density, pressure, velocity, and total enthalpy. T is
as pressure and den sity are averaged using Reynolds the static temperature, and Cp is the spe cific heat at
averaging, while variables such as velocity, enthalpy, internal constant pressure. kt is the thermal conductivity of air. τ˜ij −
energy, and temperature undergo Favre averaging. A Favre denotes the mean total stress tensor, which is a
averaged variable can be written as ρ¯u′′iu′′j
combi
nation of molecular and turbulent stress. τ˜ij is the
Φi = Φĩ + Φ′′i, mean molecular (laminar) viscous stress, given by
′′
where Φĩ is the mean value and Φ iis the fluctuating part. The ∂xj+∂uj̃
aver
∂xkδij), (6)
age of the fluctuation part is Φ̃ i= 0. From τ˜ij = μ(∂uĩ
′′

the definition of Reynolds 2

∂xi− 3∂uk ̃
and Favre averaging, Φ′′i≠ 0, but ρΦ′′i= 0.
Conservation of mass: where μ is the dynamic viscosity and is computed using
Sutherland’s law.
∂(ρ¯uĩ ) The Reynolds stress tensor (τijT) is given by
∂xi= 0. (1)
τijT= −ρ¯ui′′uj′′. (7)
Conservation of momentum:
The molecular heat flux is given by
∂xj(ρ¯uj̃ uĩ ) = −∂p¯
∂ ∂
∂xi+ ∂xj(τ˜ij − ρ¯ui′′uj′′). (2)
q˜ = −kth∂T˜
Conservation of energy: ∂xj. (8)
The turbulent heat flux is given by
˜ ∂
∂xj(uj̃ ρ¯H ) = ∂xj(kth∂T˜
∂
∂xj− ρuj′′h′′ + ui′′τij − ρuj′′k) 2, (5)
∂ ˜ ˜
+ ∂xj[uĩ (τ̃ij − ρ¯ui uj )], (3) where the specific enthalpy is h = CpT .
′′ ′′

˜
ρuj′′h′′. (9) The turbulent transport of k is
˜
The molecular diffusion of k is ρuj′′k. (11)

ui′′τij. (10) In order to achieve complete closure of Eqs. (1)–(3), it is

neces sary to model the terms in Eqs. (8) and (10)–(12).
where kth is thermal conductivity of air and k is the
The Reynolds stress tensor (τijT) can be modeled using
turbulent kinetic energy. The total enthalpy (H) and
turbulent kinetic energy (k) are given as Boussinesq eddy viscosity hypothesis,
2 ˜
τijT= −ρ¯ui′′uj′′ = μtSij − 3ρ¯ kδij, (12)
˜ ˜ ˜ 1
ρ¯H = ρ¯ h + ρ¯ k + 2ρ¯uĩ uĩ , (4)
˜
ρ¯ k =ρ¯uĩ ′′ui′′

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∂xj+∂uj̃
where β1 =0.075 is constant. d1 = 1.6 × 10−6 m is the height
where μt is the turbulent eddy viscosity. Sij is the strain rate of the first node from the wall.
and is given by The second-order upwind scheme is used to discretize the gov
2
Sij = (∂uĩ ∂xi− 3∂uk ̃ erning equations, and the Roe
∂xkδij). (13) flux-difference splitting scheme is
 used to compute the convective fluxes. The solution was driven
The turbulent heat flux is modeled by eddy-diffusivity the Courant–Friedrichs–Levy (CFL) number is kept at 0.75 to
approxima tion, ensure
to steady-state with the implicit Gauss–Seidel scheme, and
ρuj′′h′′ = −Cpμt Prt ∂xj, (14) methods are available from the
stability.
∂T˜ Several vortex identification
˜
where Prt is the turbulent Prandtl number, which is taken most popular being the Δ method,27 Q criteria,28 and λ2
as 0.85 in the present simulation. method.29 One of the major drawbacks of these methods is
˜ the need for identification of a threshold value to visualize the
The molecular diffusion and turbulent transport of k are vortices, and there is no proper procedure to select the
solved by an approximation correct threshold value. Higher threshold values might filter
out the weak vortices and lower threshold val ues might
ui′′τij − ρuj′′k = (μ +μt result in complicated features, which makes it challenging to
identify the vortices accurately. Recently, Liu et al.30
σk)∂k
introduced another method, the Omega (Ω) vortex
∂xj, (15) identification method. The major difference from the earlier
where σkis a model coefficient and is taken as 1. methods lies in the selection of the threshold value. Liu et
Menter’s Shear Stress Transport (SST) k–ω turbulence al.30 and Dong et al.31 demonstrated that for a
model18 is used to close the Reynolds stress term. The SST threshold value of Ω = 0.52, the method successfully
k–ω turbulence model combines the Wilcox k–ω model captured weak and strong vortices irrespective of different
and standard k–ε model using a blending function. The input conditions. In contrast, the Q criteria and λ2 method
Wilcox k–ω model is used in the near-wall region, and the required different thresh old values for different cases to
standard k–ε model is employed away from the wall. The visualize vortices accurately. The Ω vortex identification
SST k–ω turbulence model performed well in adverse method is employed in the present study to visualize the
pressure gradients and was used to study transverse injector vortices.
flow fields in the past.19,20 Huang et al.21 compared the
Spalart–Allmaras, RNG k–ε, and SST k–ω models for a
transverse slot injector and con cluded that the wall pressure
profile was more accurately predicted by the SST k–ω III. RESULT AND DISCUSSION
model. Furthermore, Gao and Lee,19 for a circular injector, A. Grid independence and validation of numerical
achieved close results with experiments using the SST k–ω results
turbulence model. Lee22 validated the computational code for
The numerical results are validated by comparing with
dual injector configuration using the SST k–ω turbulence
model, where the model achieved good agreement with the the experimental measurements of streamwise and
wall-normal velocity components from Ref. 13 and wall
Gruber experiment.23 Recently, Sharma et al.24 employed
pressure measurements from Ref. 15. Figure 3 presents the
the SST k–ω turbulence model in their dual injector studies streamwise velocity (U) normalized with the incoming free
and achieved good agreement with the experimental results.
stream flow velocity (U∞) at the flat plate sym metric
The model uses the Boussinesq hypothesis, and two
centerline plane at five different streamwise locations and for
additional transport equations18 for the turbulent kinetic three different computational grid sizes. All the grid sizes
energy (k) and spe cific dissipation rate (ω) are solved. The show satis factory results and are in good agreement with the
model constants are fixed as the values recommended by experimental val ues. The medium and fine grid size
Menter.18,25 The computation of tur bulent viscosity (μt) is displayed only a slight improve ment in predicting the velocity
then computed as a function of k and ω. For the inlet fields, as evident from Fig. 3(e). The numerical solution
freestream flow and injector, the turbulent intensity (I) is slightly under predicts the results close to the plate surface
selected as 5%. The viscosity ratio, μtμ, is selected as 10, as but also shows excellent agreement at X/D = −1.5. Figure 4
recommended. The inlet turbulent kinetic energy (k) and
26 shows the velocity component normal to the flow direc tion
specific dissipation rate (ω) are then calculated as (V) normalized with the incoming free stream velocity (U∞).
The numerical solution is slightly shifted compared to the
3 experi mental values. However, the numerical solution seems
k= 2 n(U ⋅ I ) 2
, (16)
k t −1 to agree with the experimental results as the X/D is
ω = ρ μ(μ μ) , (17) increased from −1.5 to 5. The slight variation of the numerical
solution can be attributed to uncertainties in the velocity
where Un = U∞ for freestream air and Un = Uj for the measurements owing to the complex three-dimensional flow
injected air jet. field near the injector.
The turbulent kinetic energy (k) near the solid wall is Figure 5 presents the static pressure distribution over the
taken as zero. The specific dissipation rate near the solid flat plate for three different spanwise locations, Z/D = 0, 1,
wall is given by and 2. The pressure drops before and after the injector due
to the presence of recirculation zones. The pressure rise due
ω = 106μ to the lambda shock and bow shock is also clear, shown as
the first two peaks in Fig. 5(a). The
ρβ1(d1 ), (18)
2

past literature to identify vortices in turbulent flow fields, the

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FIG. 3. Streamwise velocity component (U) at the flat plate symmetric centerline plane normalized by the incoming free stream velocity (U∞) at five streamwise locations:
13
(a) X/D = −1.5, (b) X/D = 2, (c) X/D = 3, (d) X/D = 4, and (e) X/D = 5 and for three different mesh sizes and comparison with the experimental data.
pressure contour is shown in Fig. 5(d). The
increase in pressure due to shock structures and drop due to
numerical solution under predicts the pressure near the jet recir culation regions are visible. The trail of the horseshoe
wind ward recirculation region but overall shows good vortex can be identified by the lower pressure region before
agreement with the experimental data. The streamline the bow shock and is seen to extend around the jet.
distribution over the flat plate surface overlaid with the
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FIG. 4. Wall normal velocity component (V) at the flat plate symmetric centerline plane normalized by the incoming free stream velocity (U∞) at five streamwise locations: (a)
13
X/D = −1.5, (b) X/D = 2, (c) X/D = 3, (d) X/D = 4, and (e) X/D = 5 and for three different mesh sizes and comparison with the experimental data.
in the formation of several complex shocks and vortex
structures.
B. Mean flow field properties The Mach number contour at the symmetric center plane for
case A is shown in Fig. 6. The sonic air jet is injected
1. Jet centerline plane perpendicular to the supersonic freestream. An inclined
A sonic jet discharging into supersonic crossflow results barrel shock is formed, which acts as an obstruction to the
incoming supersonic flow and results in the formation of a
 bow shock. A normal shock wave is present at the

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FIG. 5. Static pressure at the flat plate surface normalized by freestream pressure at three different spanwise locations: (a) Z/D = 0, (b) Z/D = 1, and (c) Z/D = 2 for three
15
levels of mesh refinement and comparison with experiments. (d) Pressure contour over the flat plate surface (Y/D = 0) overlaid with velocity streamlines.
main reflected shocks are visi ble near the Mach disk, and
the points of intersection of three shocks
(Mach disk, windward/leeward barrel shock, and reflected
end of the barrel shock, termed Mach disk, due to which the shock) are termed triple points. The Mach number contour
flow Mach number reduces to subsonic speeds. As the flow shows a higher shock strength near the injectors, and the
progresses, the flow Mach number again reaches supersonic shock strength reduces toward the oblique section of the bow
speed due to inter action with supersonic crossflow. Two shock. The incoming bound ary layer separates due to the
 influence of the bow shock, thus giving

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Physics of Fluids scitation.org/journal/phf FIG. 6. (a) Mach number contour and (b) numerical schlieren (a function of
 density gradient) of the flow field.

rise to a recirculation zone and a relatively weak shock called gradients caused by the bow shock and gives rise to
the lambda shock, as evident from Fig. 6. The numerical recircula tion zone 1. Recirculation zone 1 also entrains the
schlieren and Mach number contour shows that the lower slow-moving freestream fluid coming out of the bow shock.
reflected shock inter acts with the separation zone behind the Recirculation zone 2 is a combined effect of the under
injector at X/D = 2.8. Viti et al.4reported a sonic expanded jet and the bow shock. The fluid elements of
recompression line between the bow shock and the recirculation zone 1, which are close to the windward side of
windward side of the barrel shock, which is not observed in the barrel shock, get accelerated in the transverse direction.
this case. The numerical schlieren also suggests a very As the velocity of the fluid elements close to the jet is higher,
complex flow field near the injector, most of which require a it starts to curl inwards, giving rise to recirculation zone 2.
three-dimensional understanding. The incoming high-speed crossflow flows around the jet and
The mean velocity magnitude of JISCF is shown in Fig. merges at the leeward side of the barrel shock, giving rise to
7. For Santiago and Dutton,13 the maximum velocity was 589 recirculation zone 4. Recirculation zone 3 is observed at the
jet leeward side as an effect of the fluid particles from
m/s and was obtained near the Mach disk at about X/D =
recirculation zone 4 getting dragged by the discharging jet.
1.25 and Y/D = 1.38. In the present simulation, the injected
jet expands, increasing its veloc ity and reaching a maximum
velocity of 695 m/s, which is similar to the velocity reported 2. Jet cross-sectional planes
from the LES study by Genin and Menon16 for the same flow
For the current study, the entire flow field is divided into
conditions. Four recirculation zones are visible, as two parts, near-field and far-field. The distance up to X/D =
highlighted in Fig. 7, and are denoted as recirculation zone 3 falls into
1–4. The incoming boundary layer separates due to the
adverse pressure

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Physics of Fluids scitation.org/journal/phf FIG. 7. Mean velocity magnitude at the jet center plane combined with the
 velocity streamline.

near-field, and X/D > 3 is termed far-field. It should be noted to accurately explain the flow features, these contours are
that the present classification is only intended to simplify the examined along with the isosurface of vortex structures
analysis of the complex flow field associated with the jet shown in Fig. 9. At X/D = 0, velocity streamlines and the Ω
interaction. criterion contour show a vortex pair close to the injector,
named as vortex 1 in Fig. 8. These vortices are similar to the
a. Injector near-field. Figure 8 shows “hanging vortices” in a jet in subsonic crossflow.32,33 At X/D
the contours of veloc ity magnitude combined with velocity = 1, a second vortex pair, vortex 2, is visible in the Ω contour.
streamlines and Ω criterion for cross-sectional planes at The corresponding velocity streamlines show only one
X/D = 0, 1, 2, and 3, respectively. The magnitude of Ω recirculation region, which symbolizes that these two vortices
shows the strength of the vortex structure. In order

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 FIG. 8. Mean velocity magnitude contours combined with velocity streamlines (left) and Ω criterion contours (right) at (a) X/D = 0, (b) X/D = 1, (c) X/D = 2, and (d) X/D = 3
cross-sectional planes.

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 FIG. 9. Isosurface of Ω = 0.52: (a) overall structure; (b)–(d) shown with the cross-sectional plane of Ω contours at X/D = 1, 2, and 3 respectively, to identify the propagation
and merging of vortices with the streamwise distance.
vortex cores. A part of vortex 2 is still evident, while another
part combines with vortex 1 and vortex 3 to form a
single vortex, leading to the formation of a counter-rotating
vortex pair (CRVP). Three other vortices were also evident in
may be a combined vortex structure. Another vortex pair, Figs. 8(c) and 8(d). The first one is called a surface trailing
vortex 3, is visible between the jet plume and the flat plate. vortex (STV), generated close to the flat plate, and another
This vortex origi nates near the separated region at the one is vortex 5 or the upper trail ing vortex (UTV), which
leeward side of the barrel shock and propagates alongside propagates along the top side of the barrel shock. A
the barrel shock under the influence of the emerging jet. At horseshoe vortex is also seen wrapping around the jet, which
X/D = 2, the vortex structures look more compli cated with originates from jet windward side separation, recirculation
two new vortices, first of which is vortex 4. Viti et al.4 also zone 1.
reported vortex structures similar to vortex 4 and vortex 3 for
a higher J of 17.4 and for M∞ = 4. Still, it was not reported b. Injector far field. At X/D > 3, most
of the vortices reported in Sec. III B 2 a start to merge and
previ ously for lower J and M∞.5,6,10 The second vortex is
form a single counter-rotating vortex pair (CRVP). Figure 10
vortex 6, which forms in the shear layer between the jet shows the velocity magnitude con tours and Ω contour for
escaping the barrel shock and crossflow. At X/D = 3, most X/D = 4 and 5. The corresponding
of the vortices combined and reduced to two prominent

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FIG. 10. Mean velocity magnitude contours combined with velocity streamlines (left) and Ω criterion contours (right) at (a) X/D = 4 and (b) X/D = 5 and isosurface of Ω =
0.52 with the cross-sectional plane of Ω contours at (c) X/D = 4 and (d) X/D = 5.
reflected shock from the lower triple point induces this
second vor tex, and its strength reduces as it moves along
isosurface of Ω is also shown to visualize the structures. The with the flow. Sun and Hu12 also observed a secondary
main jet core has detached entirely from the boundary layer. surface trailing vortex in their recent study, but other detailed
At X/D = 4, the streamlines show a second vortex core, studies of JISCF4–6did not report any such
surface trailing vortex 1 (STV1), next to the primary surface
trailing vortex (STV). The
Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214 32, 126113-13 Published under license by AIP Publishing
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in the mixing process.
1. Vortex 1, vortex 2, and horseshoe vortex
Figure 12 presents vortex 1, vortex 2, and horseshoe
vortex. Vortex 1 and vortex 2 reappear downstream, which vortices with the help of the three-dimensional volumetric
led us to conclude that these vortices do not always merge stream traces col ored with the velocity magnitude to
with the main flow and only shed into the stream distinguish the origin of the vortices as the vortices are not
intermittently, as evident in Figs. 9(a) and 10(d). At X/D = 5, occurring in an orderly manner. The horseshoe vortex is
two main cores of vortices are visible, which are formed by visible and is originating from recirculation zone 1.
merging of the vortices discussed before. This pair of vortices Recirculation zone 1 does not consist of any injected jet fluid,
is termed counter-rotating vortex pairs (CRVPs), as shown in and as a result, the horseshoe vortex does not augment the
Fig. 9(a). mixing of the jet with the crossflow. Vortex 1 (or the hanging
vortices) originates from recirculation zone 2 [Figs. 9(b) and
12] and not between the flat plate and boundaries of the
3. Jet transverse planes laterally expanding jet as reported by Génin and Menon.16
Figure 11 shows different planes in the jet transverse The flow velocities reduce at the foot of bow shock, and there
locations from Y/D = 0 to 0.5, colored by static pressure. will be large velocity gradients in the mixing layer between
The trail of the horse shoe vortex is identified as the the recirculation zone 2 fluid and the crossflow as the highly
low-pressure region that is originating from recirculation zone deflected crossflow accelerates around the jet. The Kelvin–
1. The streamlines do not show any evidence of the Helmholtz or K–H instabilities due to these large velocity
horseshoe vortex until Y/D = 0.2 in Fig. 11(c), indicating that gradients in the mixing layer further energize vortex 1. This
the horseshoe vortex is not attached to the flat plate surface. vortex can be considered as the primary source contributor
The trail of the three-dimensional bow shock is visible as the for CRVP formation downstream and is very evident from
high-pressure region aft of the horseshoe vortex trail. Figs. 8 and 10. Vortex 1 is mostly crossflow fluid, with only a
Streamlines originating from recirculation zone 2 show the small fraction of the injected jet trapped in recirculation zone
trail of the hanging vortex that devel ops very close to the 2. Vortex 2 originates from the injected jet stream and
injected jet and will be discussed in detail in Secs. III C 1 and comprises jet fluid elements originating from injec tor sides
III C 3. that are aligned with the crossflow. The jet fluid elements
from the injector sides form a mixing layer with the freestream
fluid flowing around the injector. Since the part of the mixing
C. Formation of vortices layer close to the freestream has higher velocities, K–H
The complex three-dimensional structures associated instabilities develop, which forms vortex 2. At X/D = 2, both
with JISCF play a vital role in the subsequent mixing of the jet the stream traces and Ω contours show the beginning of
with high-speed crossflow. It is crucial to identify the mixing of vortex 1 and vortex 2 and the entrainment of the
formation of these structures in order to understand their role crossflow fluid.
 FIG. 11. Velocity streamlines at different transverse locations given by (a) Y/D = 0, (b) Y/D = 0.1, (c) Y/D = 0.2, (d) Y/D = 0.3, (e) Y/D = 0.4, and (f) Y/D = 0.5 colored by
pressure. The velocity magnitude is mapped onto the symmetric (Z/D = 0) and cross-sectional (X/D = 1) planes.

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FIG. 12. Stream traces colored with

the velocity magnitude. Ω contours are
mapped onto the cross sections at (a)
X/D = 1 and (b) X/D = 2. The mean
velocity magnitude is shown at the sym
metric center plane.
The reflected shock wave from the bottom triple point
impinges the separation zone at X/D = 2.8, as evident from
Fig. 6(c), and pushes the stream lines at this section toward
the spanwise direction, giving rise to the secondary surface
trailing vortex, STV1. This type of secondary
vortex was reported recently by Sun and Hu12 for a higher
2. Surface trailing vortex (STV) and vortex 3 crossflow Mach number of 2.7 and J of 2.3 and 5.5. Still, this
vortex was not reported in any other detailed flow field
The surface trailing vortices are a pair of vortices that studies for the same flow conditions employed in the present
prop agate close to the flat plate surface. The incoming study.4–6,16 Viti et al.4reported that the primary surface
freestream flow deflects around the jet and merges at the trailing vortex (STV) originates from recir culation zone 2.
leeward side of the jet. The fluid elements roll up and form However, the streamline analysis from the present study
recirculation zone 4, shown as the low-velocity region at the shows that STV1 and STV are originating from recirculation
jet leeward side in Fig. 13. Three impor tant mechanisms are zone 4. Because of the proximity of recirculation zone 2 with
happening here: (i) The jet particles roll up and fall under the the deflected flow, a minor amount of fluid particles from
suction of vortex 1 and vortex 2, which imparts a rotational recirculation zone 2 is also carried with the deflected flow.
motion to the fluid particles. (ii) The negative veloc ity Figure 14 shows vortex 3, and as discussed above, the
induced in the separated region causes some fluid particles vortex consists of fluid particles from the jet leeward
to flow back, which falls under the influence of the emerging separation zone. Com parison with a jet in subsonic
jet and is forced to move along the jet, creating vortex 3. (iii)
 crossflow case,32 vortex 3 resembles that of a “wake vortex.” the injected jet fluid and named it as “trailing vortex 3.”
Viti et al.4for a high crossflow Mach number of 4 and J of However, our current study shows that vortex 3 is made of
17.4 observed a vortex similar to vortex 3, which is made of fluid particles from

FIG. 13. Velocity magnitude isosurface corresponding to 150 m/s combined with velocity streamlines.

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3. Vortex 4, vortex 5 (upper trailing vortex),

and vortex 6
Figure 8(c) shows two vortices that are in proximity to vor
tex 1, vortex 2, and vortex 3. These two vortices are vortex 4 and
vortex 6. Figure 15(a) shows vortex 4, and it mainly consists of the
injected jet fluid coming out of the Mach disk, as reported by Viti et
al.4and Sun and Hu.34 However, our observation shows that a small
amount of fluid from recirculation zone 4 is also entrained into the
vortex. This is due to the effect of vortex 3, which is propagating
close to vortex 4. Vortex 6 is not previously reported and is made of
the injected jet fluid coming out of the windward side of the barrel
shock, as shown in Fig. 15(b). These jet fluid particles form a mix
ing layer between the less deflected freestream flow from the oblique
section of the bow shock. The lateral side of the mixing layer gets
slightly accelerated due to the deflected flow, leading to a weak vortex
motion.
Vortex 5 or the upper trailing vortex has been reported in many
studies before.4,5 Yet the origin of the vortex is still a disputable
topic. Viti et al.4located the source of the vortex as recirculation
zone 2.4
al.6did not report the vortex and claimed that the
FIG. 14. Stream traces colored with the velocity magnitude showing vortex 3. The
cross-sectional plane is mapped with Ω contours and velocity magnitude contour discrepancy from Ref. 4 might be due to the higher crossflow
assigned to the symmetric center plane. Mach number.
In the present study, analysis of the streamlines is
undertaken to identify the origin of the upper trailing vortices
recirculation zone 4, with only a small amount of the injected and is shown in Fig. 16. The stream traces are colored with
fluid. At the same time, for a lower M∞ and J, Rana et static temperature to differentiate the source of the vortex.
 The vortex stays close to the symmetric center plane and tours and velocity magnitude contour
consists of fluid particles from the upper part of separation mapped to the jet center plane.
zone 2 and jet particles escaping through the bottom part of
the windward barrel shock. The jet fluid imparts a transverse
velocity to the low-velocity fluid from recirculation zone 2, as
apparent from the elongated shape of recirculation zone 2
from

FIG. 16. (a) Stream traces showing vor

tex 5 or the upper trailing vortex. The
cross-sectional plane is mapped with Ω
contours and velocity magnitude contour
FIG. 15. Stream traces showing (a) vor assigned to the symmetric center plane.
tex 4 and (b) vortex 6. The cross (b) UTV merging with other vortices.
sectional plane is mapped with Ω con

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Physics of Fluids scitation.org/journal/phf FIG. 17. Jet passive scalar at (a) symmetric centerline (Z/D = 0) (b) flat plate surface (Y/D =

0).

Fig. 7. The fluid particles from recirculation zone 2 close to from the jet exit is less deflected, resulting in a mixing layer
the flat plate are less affected by the injected jet velocity. The with small velocity gradients, which results in the formation of
high-velocity freestream flow after passing through the bow a weak upper
shock deflects around the jet; the deflection is higher near the trailing vortex. The upper trailing vortex then slowly merges
jet exit. The highly deflected freestream flow, which with the remaining vortices and is absorbed into the CRVP,
accelerates around the jet exit, creates a skewed mixing as shown in Fig. 16(b).
layer with recirculation zone 2 fluid particles close to the flat
plate and results in the formation of vortex 1 (hanging vortex).
The fluid particles from the upper part of recirculation zone 2 D. Role of vortices in mixing
are dragged by the discharging jet. The freestream fluid away From the above analysis, it is evident that not all vor
 tices contribute to the mixing of the injected jet with absence of injector jet fluid and one indicates the presence of
freestream. Figure 17(a) shows the passive scalar at the jet only the injected jet fluid. For a steady flow,
symmetric centerline plane, where zero indicates the

FIG. 18. Isosurface of Ω = 0.52 colored by the jet passive scalar.

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around the
injected jet. Figure 17(b) shows small traces of jet fluid
particles near the flat plate surface and around the periphery
passive scalar analysis of the mean flow field can be used to of the injector. As the flow deflects around the jet, some of
visualize the mixing of the freestream and jet fluid. Since the the jet fluid particles escap ing the sides of the barrel shock
flow field in the present investigation is steady, the passive are also carried by the freestream into recirculation zone 4.
scalar is used to indicate the mixing of the freestream and As a result, STV and STV1 promote jet and freestream
injected jet. Most of the fluid par ticles from the injector mixing close to the plate surface. Figure 18 depicts the
escape the barrel shock from the windward side. The primary isosurface of Ω combined with a jet passive scalar to
and secondary surface trailing vortices, STV and STV1, visualize jet and freestream mixing. The horseshoe vortex
originate from recirculation zone 4 at the jet’s leeward side, originates from
which is mostly made of the freestream fluid that flows
 FIG. 19. Velocity magnitude contours for the M∞ = 2 case mapped with velocity streamlines at cross-sectional planes on (a) X/D = 0, (b) X/D = 2, (c) X/D = 4, and (d) X/D =
4.5.

Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214 32, 126113-18 Published under license by AIP Publishing
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in Fig. 18.
Vortex 2, vortex 4, and vortex 6 have jet fluid as the
major ity constituent. Vortex 3 originates from recirculation
recirculation zone 1 and shows zero value of the passive zone 4 and contains freestream fluid and a small amount of
scalar, implying an absence of an injected jet. As a result, the the injected jet fluid. It is observed that these vortices merge
vortex does not contribute to the mixing process. Among the as the flow progresses, leading to the formation of CRVP,
near-field vortices, vortex 1 or hanging vortex originates from thus augmenting the mixing process. The decrease in the
recirculation zone 2 and consists of mainly the freestream passive scalar caused by the intermin gling of vortices
fluid and a small amount of the jet fluid. As the flow signifies jet particles’ mixing with the freestream fluid. The
progresses, more jet fluid particles are entrained into vortex 1 upper trailing vortex (UTV) is a combination of the jet and
as evident from the passive scalar distribution along vortex 1 freestream fluid and contributes to the mixing process.
 Stream line analysis shows that UTV gets absorbed into the are shown in Fig. 19. Comparing with the M∞ = 1.6 flow field
CRVP down stream of the injector. It can be safely concluded in Figs. 8 and 10, all the vortices present in the M∞ = 1.6
that all the vortices observed in the jet near the injector field
case are present for the M∞ = 2 case as well. The secondary
contribute to the mixing process.
surface trailing vortex, STV1, is visible at X/D = 4.5, unlike
the M∞ = 1.6 case where STV1 is vis ible at X/D = 4, as
E. Effect of freestream Mach number on vortex shown in Fig. 10(a). The difference is due to the impact
structures location of the reflected shock wave on the separated bound
ary layer. The vortices present in Figs. 9 and 10 for the M∞ =
The effect of freestream Mach number on the vortex
structures prevailing in a jet in supersonic crossflow is 1.6 case are clearly visible in the isosurface of Ω = 0.52 for
explored in this sec tion. Rana et al.,6for a freestream the M∞ = 2 case shown in Fig. 20. The UTV is more evident
Mach number of 1.6, did not report any trailing vortices or the than the lower Mach number case. Vortex 6, on the other
hand, is showing the opposite trend compared to UTV.
upper trailing vortices reported by Viti et al.4for a higher
freestream Mach number case. How ever, Rana et
al.6addressed the same and concluded that CRVP is a
common feature of JISCF, while the trailing vortices are a 2. Freestream Mach number, M∞ = 4
function of the freestream Mach number. A recent DNS The velocity magnitudes at four different cross sections
simula tion by Sun and Hu34 addressed a trailing vortex are shown in Fig. 21. Although the flow field shows similarity
similar to trail ing vortex 1 reported in Viti et al.4(vortex 4 to that of M∞ = 1.6 and 2.0, certain minor differences are
in our study). Sun and Hu34 also employed a high freestream also observed. Vortex 3 and vortex 4 are visible at X/D = 3,
Mach number of 2.7. unlike M∞ = 1.6 and 2.0 cases where the vortices are visible
In this section, the flow fields using two different Mach num at X/D = 2. The reason for this dissimilarity can be
bers (M∞ = 2 and 4) are selected to compare with the M∞ = explained using the streamwise flow field in Fig. 22. For the
1.6 case discussed in Secs. III B-III D. The objective was to M∞ = 2 case, the tip of the barrel shock is located at X/D =
esti mate whether the trailing vortices are a function of the 1.8. Vortex 3 and vortex 4 result from flow variations behind
freestream Mach number (M∞). the Mach disk, as discussed in Secs. III C 2 and III C 3. As a
result, the vortices are visible at X/D = 2 for the M∞ = 2
case. At the same time, for the M∞ = 4 case, the barrel shock
1. Freestream Mach number, M∞ = 2 appears tilted and is extended until X/D = 2.5.
The velocity magnitudes at four different cross sections

FIG. 20. Isosurface of Ω = 0.52 colored by the velocity magnitude for the M∞ = 2 case.

Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214 32, 126113-19 Published under license by AIP Publishing
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where the vortex is visible at X/D = 4.5. The difference is
due to the change in the impact loca tion of the reflected
shock wave from the Mach disk, as evident from Fig. 22.
As a result, vortex 3 and vortex 4 are visible at the cross Figure 23 shows the isosurface of Ω = 0.52 for the M∞ =
section after X/D = 2.5. The secondary surface trailing 4 case. The horseshoe vortex propagates closer to the
vortex (STV1) is vis ible at X/D = 5, unlike the M∞ = 2 case, injected jet as compared to the lower Mach number cases.
 The UTV is clearly vis ible, and as discussed before, the numbers. At the same time, the strength of the vortex
vortex seems to originate from the structures depends highly on the freestream Mach number.
upper part of recirculation zone 2. On the other hand, vortex Vortex 5 or UTV is seen to gain strength with the freestream
6 is not visible. Mach number. The origin location of the vortex structures can
Comparing the vortex structures for M∞ = 1.6, 2, and 4, change with the freestream Mach number. The differ ence in
the vortex structures remain the same except in the case of the position of vortex 3 and vortex 4 for different freestream
vortex 6, which is not visible at higher freestream Mach

FIG. 21. Velocity magnitude contours for the M∞ = 4 case combined with velocity streamlines at cross-sectional planes on (a) X/D = 0, (b) X/D = 2, (c) X/D = 3, and (d) X/D
= 5.

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 ARTICLE
Physics of Fluids scitation.org/journal/phf FIG. 22. Numerical schlieren of the jet symmetric centerline for (a) M∞ = 2 and (b) M∞ = 4.

FIG. 23. Isosurface of Ω = 0.52 colored by the velocity magnitude for the M∞ = 4 case.
The hanging vortex originates from the core of recirculation
zone 2 and is energized by the K–H instabilities due to large
velocity gra dients in the mixing layer between the jet and
Mach numbers is due to the barrel shock tip location. The deflected freestream. The upper trailing vortex originates
secondary surface trailing vortex, STV1, also depends on the from the upper part of recircula tion zone 2 and gets
barrel shock tip location. convected above the barrel shock by the injected jet. The
upper trailing vortices also include jet fluid escaping from the
IV. CONCLUSION lower windward side of the barrel shock. Recirculation zone 4
contributes to the formation of four vortices, STV, STV1,
Numerical analysis of a sonic jet discharging into a vortex 3, and vortex 4. STV1 is a secondary surface trailing
super sonic freestream Mach number of 1.6 is performed vortex that forms when the reflected shock from the lower
using three dimensional Reynolds averaged Navier–Stokes triple point incidents recir culation zone 4. On the other hand,
equations. The pri mary emphasis of this study is to identify vortex 3 comprises recirculation zone 4 fluid elements that
the vortices, their formation, and their role in the mixing fall under the upwash of the injected jet. In a similar way,
process. A comparatively new vortex identification method, Ω vortex 4 also includes recirculation zone 4 fluid ele ments
criterion, is employed in this study to identify the complex from jet upwash, but the major portion includes slow jet fluid
vortex structures. The Ω vortex method can predict both coming out of the Mach disk. The jet fluid passing through the
weak and strong vortices at a fixed thresh old value of 0.52 Mach disk combines with the high-speed flow around the
as compared to other classical methods, which lacks a fixed barrel shock and imparts vorticity to the flow, creating vortex
threshold value. 4. Vortex 2 and vortex 6 originate from the injected jet. Vortex
Recirculation zone 2 contributes to the formation of two 2 is a fairly strong vortex
vor tices, vortex 1 (hanging vortex) and vortex 5 (upper
trailing vortex).

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of the barrel shock comes in contact with the slightly
deflected freestream fluid passing through the oblique section
of the bow shock, resulting in a weak vortex 6.
originating at the lateral side of the injector due to the velocity The vortex structures present in the higher freestream
gradients in the mixing layer between the jet and the Mach number cases are also identified for the lower
deflected crossflow and propagates very close to vortex 1. freestream Mach num ber cases. However, vortex 6 reported
The injected jet fluid escaping from the upper windward side in the lower freestream Mach number case is not visible at a
higher Mach number. The vortices formed due to the skewed
 mixing layer between the jet and crossflow at the injector crossflow,” J. Turbul. 11, N4 (2010).
17
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W. Huang, J. Liu, L. Jin, and L. Yan, “Molecular weight and injector
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Phys. Fluids 32, 126113 (2020); doi: 10.1063/5.0026214 32, 126113-22 Published under license by AIP Publishing