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Detailed flow physics of the supersonic jet interaction flow field

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The supersonic jet interaction flow field generated by a sonic circular jet with a pressure ratio of 532 exhausting into a turbulent MACH 4.0 cross flow over a flat plate was investigated using numerical simulations. The simulations made use of the three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations coupled with Wilcox's 1998 k-omega turbulence model. The numerical solution was validated with experimental data that include the pressure distribution on the flat plate, with an empirical formula for the height of the barrel shock, and with the Schlieren pictures showing the location and shape of the main shock formations. The simulations correctly captured the location and shape of the main flow features and compared favorably with the experimental pressure distribution on the flat plate. The validated numerical simulation was used to investigate in detail the flow physics. The flow field was found to be dominated by the shock formations and their coupling with the strong vortical structures. Three primary shock formations were observed: a barrel shock, a bow shock, and a separation-induced shock wave. While the general structure of the barrel shock was found to be similar to that of the underexpanded jet exhausting into a quiescent medium, two unique features distinguished the flow field: the concave indentation in the leeside of the recompression (barrel) shock and the folding of the windward side of the barrel shock due to an inner reflection line. The presence of the steep pressure gradients associated with the shocks creates strong vortical motions in the fluid. Six primary vortices were identified: (i) the well-known horseshoe vortex, (ii) an upper trailing vortex, (iii) two trailing vortices formed in the separation region and, aft of the bow shock wave, (iv) two more trailing vortices that eventually merge together into one single rotational motion. The low-pressure region aft of the injector was found to be generated by the combined effect of the concave indentation in the leeside of the barrel shock and the lower trailing vortices. The trailing vortices were found to be the main mechanism responsible for the mixing of the injectant with the freestream fluid.
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Detailed flow physics of the supersonic jet interaction flow field
Valerio Viti,
1
Reece Neel,
2
and Joseph A. Schetz
3
1
Department of Mechanical Engineering, University of Kentucky, Lexington, Kentucky 40506, USA
2
AeroSoft, Inc., Blacksburg, Virginia 24060, USA
3
Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia 24060, USA
Received 20 March 2008; accepted 26 February 2009; published online 16 April 2009
The supersonic jet interaction flow field generated by a sonic circular jet with a pressure ratio of 532
exhausting into a turbulent
MACH 4.0 cross flow over a flat plate was investigated using numerical
simulations. The simulations made use of the three-dimensional Reynolds-averaged Navier–Stokes
RANS equations coupled with Wilcox’s 1998 k-
turbulence model. The numerical solution was
validated with experimental data that include the pressure distribution on the flat plate, with an
empirical formula for the height of the barrel shock, and with the Schlieren pictures showing the
location and shape of the main shock formations. The simulations correctly captured the location
and shape of the main flow features and compared favorably with the experimental pressure
distribution on the flat plate. The validated numerical simulation was used to investigate in detail the
flow physics. The flow field was found to be dominated by the shock formations and their coupling
with the strong vortical structures. Three primary shock formations were observed: a barrel shock,
a bow shock, and a separation-induced shock wave. While the general structure of the barrel shock
was found to be similar to that of the underexpanded jet exhausting into a quiescent medium, two
unique features distinguished the flow field: the concave indentation in the leeside of the
recompression barrel shock and the folding of the windward side of the barrel shock due to an
inner reflection line. The presence of the steep pressure gradients associated with the shocks creates
strong vortical motions in the fluid. Six primary vortices were identified: i the well-known
horseshoe vortex, ii an upper trailing vortex, iii two trailing vortices formed in the separation
region and, aft of the bow shock wave, iv two more trailing vortices that eventually merge together
into one single rotational motion. The low-pressure region aft of the injector was found to be
generated by the combined effect of the concave indentation in the leeside of the barrel shock and
the lower trailing vortices. The trailing vortices were found to be the main mechanism responsible
for the mixing of the injectant with the freestream fluid. © 2009 American Institute of Physics.
DOI: 10.1063/1.3112736
I. INTRODUCTION
The jet interaction flow field is the name given to the
fluid dynamics phenomenon produced by a jet exhausting in
a cross flow. This flow field can be found in several techno-
logical applications and, due to the presence of separated
flows, vortical motions, turbulence, and, if the flow is super-
sonic shocks and expansion fans, is a formidable fluid dy-
namics problem. The AGARD conference proceedings
1
give
an ample and detailed review of the range of possible appli-
cations. Examples range from the low-speed regimes of a
chimney plume in a cross flow to the very high-speed re-
gimes of scramjet combustion and missile control systems,
from the low mass flow cases of boundary layer control sys-
tems and gas-turbine blade cooling to the high mass flow
cases of a landing V/STOL vehicle. The basic problem of a
fluid injected into a cross flow has several variables depend-
ing on its intended application: injector yaw and pitch angle,
jet flow conditions subsonic, sonic, and supersonic,
freestream conditions subsonic, supersonic, laminar, and
turbulent, not to mention the phase and the chemical com-
position of the injectant single or multiphase, nonreacting or
reacting mixture, etc..
The present study focuses on the case of sonic, normal
injection of a perfect gas through a circular injector into a
MACH 4.0 turbulent cross flow over a flat plate. The ratio of
the jet total pressure to the freestream static pressure, defined
as the pressure ratio, is 532 as defined by Cubbison et al.
2
This configuration is representative of a typical reaction con-
trol system installed on a hypersonic vehicle. In reaction
control systems, normal injection is usually chosen over
angled injection because it maximizes the lateral force pro-
duced by the thrust of the jet. Two primary mechanisms con-
tribute to the production of the lateral force.
3
The first con-
tribution comes purely from the thrust produced by the jet.
The second contribution is produced by the complex interac-
tion of the jet with the cross flow. The injected gas acts as an
obstruction to the primary flow and, as such, produces a
shock wave in the primary flow see Fig. 1. The shock wave
produces an adverse pressure gradient that causes the bound-
ary layer on the wall to form a separation region ahead of the
injector. The high pressures typical of recirculated flows see
Refs. 46 augment the lateral force produced by the thrust
of the jet. Therefore, a jet operating in a cross flow over a flat
surface at zero angle of attack will produce a larger force
than if it was exhausting into a quiescent medium.
3
However,
concurrent with the separation region, a large wake with a
PHYSICS OF FLUIDS 21, 046101 2009
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low-pressure region forms aft of the injector, as described by
Spaid et al.
4
The low-pressure region has two main effects
on the forces and moments produced by the jet on the sur-
rounding surface. The first effect is to decrease the normal
force on the plate.
7
The low-pressure region effectively cre-
ates a suction behind the jet and, even though the suction is
not strong it acts over a large area aft of the injector thus
creating a strong upward force. The second, and in many
aspects most detrimental effect is the coupling with the high-
pressure region ahead of the jet and the formation of a nose-
down moment about the injector. The contribution to the
nose-down moment from the low-pressure region is particu-
larly high since this region extends far aft of the injector.
5,8
This shift in the center of pressure of the vehicle has to be
corrected through the use of an attitude control system that
actuates counterbalancing jet thrusters. The region of low-
pressure aft of the injector corresponds, in part, to the wake
behind the injector. The flow field in the wake is dominated
by the presence of strong vortical motions that are formed in
the boundary layer separation and by the barrel shock and
that are convected downstream by the free stream. The de-
tachment of the barrel shock from the surface of the flat plate
forces these trailing vortices closer together and toward the
solid surface, thus enhancing the longitudinal rotation of the
fluid aft of the injection.
While in the low-speed jet interaction case the flow field
can be largely modified by changing the injector geometry,
9
in the high-speed jet interaction regimes the hole geometry
does not have a strong influence on the flow field.
10
The
undesirable effects created by the jet interaction flow field
can be mitigated by designing the surface around the injector
in such a way as to modify the local flow field. A properly
designed surface requires detailed knowledge of the flow
field and of the flow structures responsible for the generation
of the low- and high-pressure regions. Once these structures
are well understood, they can be altered or removed to im-
prove the functionality and performance of the whole injec-
tion system. A number of investigations aimed at the devel-
opment of thrust vector control systems were carried out in
the 1960s to study the pressure distribution in the region
around the injector and the resulting normal force and pitch-
ing moment.
4,11,12
Several researchers
10,11,1318
analyzed this
flow field through analytical models and experiments. How-
ever, these efforts have been only partially successful due in
large part to the difficulty of experimentally measuring the
local flow without disrupting it and in part, due to the inher-
ent complexity of the flow physics involved. Byun et al.
19
attempted to decrease the area of low pressure by inserting a
solid ramp aft of the injector. Conversely, Viti et al.
8
sug-
gested that the same effect as a solid ramp could be obtained
by using a concept similar to the “aeroramp,”
20,21
which con-
sists in inserting smaller secondary injectors in the region aft
of the main injector.
The present work aims at producing a detailed physical
analysis of the supersonic jet interaction flow field through
the use of computational tools. Such an analysis can improve
the understanding of the relevant flow structures responsible
for the generation of the pressure field and for the mixing of
the injectant with the cross flow, ultimately improving
present-day jet-thruster configurations and contributing to
the understanding of scramjet fuel injection systems.
II. GOVERNING EQUATIONS, COMPUTATIONAL GRID,
AND BOUNDARY CONDITIONS
The governing equations of a compressible turbulent
flow can be written using time-averaged Reynolds-
averaged, indicated by an overbar values of the density,
pressure, and mass-weighted Favré-averaged, indicated by a
tilde averages for the velocity components and temperature.
Following, the governing equations used in this study are
presented in their differential form.
Conservation of mass,
¯
t
+
u
i
x
i
=0. 1
Conservation of momentum,
¯
u
˜
i
t
+
x
i
¯
u
˜
i
u
˜
i
+ p
¯
ij
=
x
i
˜
ij
+
¯
ij
x
i
¯
u
i
u
j
. 2
Conservation of energy,
¯
e
˜
o
t
+
x
i
¯
e
˜
o
u
˜
i
+ pu
i
+
¯
e
o
u
i
=
x
i
ij
u
j
q
i
x
i
, 3
where
e
˜
o
= C
¯
v
T
˜
+
1
2
u
˜
i
u
˜
i
+
1
2
u
i
u
i
. 4
The perfect gas law is used to close the system,
p
¯
=
¯
RT
˜
. 5
The Reynolds-stress tensor is defined as
ij
=−
¯
u
i
u
j
. 6
In the above expressions, the tensor u
i
represents the x,
y, and z components of the velocity in a Cartesian coordinate
system, T is the static temperature, C
v
is the specific heat at
constant volume, R is the gas constant 286.7 kJ/ kg K,
is
the fluid density,
is the fluid laminar viscosity, and q is the
heat flux.
The numerical calculations performed in this study used
Wilcox’s k-
1998 turbulence model.
22,23
This model was
FIG. 1. Color online Schematic of the flow field along the tunnel center
line. The definition of the jet PR proposed by Cubbison et al. Ref. 2,is
used throughout this paper.
046101-2 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
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chosen because of its good ability in predicting separation
and in dealing with adverse pressure gradients and separated
flows compared to other two-equation models
2325
and to
Wilcox’s Reynolds-stress transport model.
26
In particular,
when compared to the more advanced eddy-viscosity model
of Menter Menters shear stress transport model
27
,itap-
pears that at least for the case of compressible jet interaction
flow fields, the Wilcox model has better predicting
capabilities.
24
The numerical solver used in this study is AeroSoft’s
GASP version 4.0. GASP was chosen because it is a mature
program with a proven reliability record in simulations of
turbulent flows,
28
vortical flows,
29
jets,
30
shock-vortex
interaction,
31
and jet interaction flows.
7,8,32
GASP solves the
discretized integral form of the time-dependent Reynolds-
averaged Navier–Stokes RANS equations over a structured
grid.
33,34
The solution was driven to a steady-state using the im-
plicit Gauss–Seidel scheme
35
and a Courant–Friedrich–Levy
CFL number of 0.75. The relatively low CFL number was
used in order to converge the solution without convergence
problems which were observed during the initial iterations.
The convective fluxes were computed using the flux-vector
splitting of Roe with third order spatial upwind-biased accu-
racy using the Min-Mod limiter. The viscous terms were dis-
cretized using a second-order-accurate central differencing
scheme. An exception to this flux combination was the re-
placement in the radial direction of the C-type zone that
surrounds the injector of the Roe flux with the Van Leer flux
leaving all the other parameters unchanged in order to avoid
the “carbuncle effect.”
36
The computational grid used in this
work is a combination of H-type and C-type grids shown in
Fig. 2 that allows an optimal cell clustering around the in-
jector. The grid size was dictated by the need to find a bal-
ance between the refinement of the grid and the CPU re-
sources available for these runs. The grid was created using
FIG. 2. Color online Isometric view of the structured computational grid
composed of a combination of C-type and H-type grid topologies for a total
of 13 zones. The inset shows detail of the C-type grid wrapping around the
primary injector. Total number of cells is 1.5410
6
cells, the surface mesh
shows every other computational cell.
FIG. 3. Blow-up sequence showing the mesh close to the solid surface of
the flat plate.
046101-3 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
GRIDGEN version 13.3.
37
Care was taken to ensure that the
cells closest to the solid surface would lie below a y
+
of 1.0.
One-dimensional hyperbolic tangent stretching
38
was used in
all regions with a different stretching parameter to smoothly
distribute the cells without steep changes in cell size. An
example of this distribution close to the injector is given in
Fig. 3. The first cell height was 1.8 10
−6
m and the ratio of
the second to first cell height was in the order of 2.0. The
injector was simulated by cells on the surface of the flat plate
with an imposed pressure and velocity equal to the jet total
conditions. To help with convergence rate, the grid was se-
quenced twice by eliminating every other cell in the three
spatial directions. The sequencing procedure generated a nu-
merical solution on three grids with the same topology but
different number of cells.
The computational domain for the flat plate with normal
injection consisted of a six-sided box, 27.69 cm long, 15.24
cm wide, and 11.43 cm high, as shown in Fig. 2 and as
described in Table I. The plate dimensions are listed in Table
II and a full set of jet and freestream conditions can be found
in Table III. The lower plane, i.e., the plane defined by y / d
=0.0, corresponds to the solid surface of the flat plate. Adia-
batic wall
T/
y =0.0, no-slip conditions u=
v
=w= 0.0
were imposed on the flat plate. The adiabatic wall condition
is an approximation for the low-heat flux measured during
experimental runs in the wind tunnel. The circular injector is
cut flush in the surface of the flat plate and sonic conditions
were applied at the cells simulating the jet Ma
J
=1.00,
J
=
, u
J
=w
J
=0.0 m/ s,
v
J
=
v
, and p
J
= p
. The jet pressure
ratio, PR= P
j,t
/ P
, was 532 and the momentum flux, q
¯
=p
M
2
j
/ p
M
2
, was 17.4. The jet was assumed to have
a step profile, i.e., no boundary layer profile in the nozzle
was simulated. The area of the simulated jet is smaller than
the jet used in the experiments and the ratio of the two areas
is equal to the nozzle discharge coefficient Cd
J
, which was
estimated through the use of numerical simulations to be
0.78.
6
As a consequence, the injector in the tunnel had a
diameter of 4.76 mm and the one in the present computations
4.12 mm, the two diameters related by d
j,CFD
=Cd
j
0.5
d
j,expt
.
By doing this, the viscous effects inside the nozzle were
taken into consideration, and the mass flow of the simulated
jet was the same as the real jet. Previous work on the effect
of a velocity profile for the choked nozzle showed little or no
effect on the shock formations in the cross flow.
6
The flow
upstream of the injector is supersonic, and a turbulent bound-
ary layer is present. All the dependent variables at the inlet
outside the boundary layer were assigned their respective
freestream value corresponding to a
MACH 4.0. The initial
freestream turbulence intensity TI was assumed to be 5%
since no turbulence measurements were available. This value
was thought to be a reasonable assumption given the tunnel
conditions. From this value and the assumption that the ini-
tial turbulent viscosity,
t
, is 1/10th the laminar viscosity, it
was possible to calculate the initial turbulent kinetic energy
k=
3
2
TI· U
2
and turbulent frequency
=C
k/
T
.
Considering Wilcox’s k-
sensitivity to the freestream con-
ditions, the forces and moments on the flat plate might have
been affected by fixing the inlet turbulence level.
39
However,
only the initial inlet turbulence level was specified. That is
TABLE I. Computational domain dimensions.
Parameter Dimensions
Streamwise length, x 27.69 cm 58x / d
j
Height, y 15.24 cm 32y / d
j
Width, z 11.43 cm 24z/ d
j
TABLE II. Flat plate and injector dimensions.
Parameter Dimensions
Flat plate entry length, x
0
7.62 cm
Injector diameter, d
j
0.476 cm
Injector effective diameter, d
j,e
0.412 cm
x
0
/ d
j
16.0
TABLE III. Summary of freestream and jet conditions.
Parameter Conditions
a Free stream
Gas Air, perfect gas
=1.40, Pr=0.72, R = 286.7 J kg K
M
4.0
P
,t
1120 kPa
P
7.1 kPa
T
70.3 K
Inlet
1.65 cm
b Jet conditions
Gas Air, perfect gas
=1.40, Pr=0.72
M
j
1.0
P
j,t
3797 kPa
P
j
2006 kPa
T
j
261 K
P
j,t
/ P
PR 532
Momentum ratio 17.4
Jet mass flow 0.116 kg/s
Jet thrust 37.5 N
TABLE IV. Grid convergence study results, normal force coefficient, C
Fy
top, and pitching moment coefficient, C
Mz
bottom.
Grid sequence No. of cells C
Fy
Difference
% Normalized C
Fy
Coarse 24 127 1.01 6.6 0.94
Medium 193 012 1.06 1.6 0.99
Fine 1 544 098 1.07 0.6 1.00
f
exact, Richardson
= 1.08 0 1.01
Grid sequence No. of cells C
Mz
Difference Normalized C
Mz
Coarse 24 127 11.76 7.0 0.93
Medium 193 012 12.51 1.1 0.99
Fine 1 544 098 12.64 0.1 1.00
f
exact, Richardson
= 12.64 0 1.00
046101-4 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
the inlet TI was fixed only during the very first iteration, and
then the inlet turbulence level was extrapolated from the in-
terior turbulence quantities. In this way the inlet TI was not
preset and could adjust and relax to the proper level. In light
of this approach, the final solution is not affected by the
initial freestream TI. Due to restrictions in computational
resources, a sensitivity analysis of forces and moments to the
initial freestream TI was not performed. No TI was measured
during the experiment and therefore it was not possible to
make a more precise assumption or a direct comparison of
the test and CFD turbulence levels. The entry boundary layer
thickness,
, was obtained from the Schlieren pictures of the
tunnel flow, and the boundary layer velocity profile was as-
sumed to follow the 1/7th power-law relationship. The as-
sumption of the turbulent boundary layer profile combined
with the length of the computational domain ahead of the
separation region allows the boundary layer to develop to its
proper equilibrium state before it separates.
The symmetry plane is represented by the x- y plane. The
three remaining sides of the computational domain the
downstream exit plane, the top surface, and the longitudinal
plane opposite the symmetry plane do not represent any
physical surface. The top surface and the sidewall of the
wind tunnels were assumed to be distant enough from the
injector not to interfere with the flow field of interest. Fol-
lowing this assumption the computational domain was
smaller than the wind tunnel cross section and a first-order
extrapolation boundary condition was applied to the top and
side boundaries of the computational box as well as to the
downstream exit plane.
The iterative convergence of the calculations was deter-
mined by checking the variation over time of the residuals of
the five RANS equations and of the turbulent equations plus
several flow parameters. Convergence was declared when the
residuals, normal force, axial force, pitching moment, pres-
sure distribution, and skin friction coefficient along the cen-
ter line ahead of the injector were steady or showing a small-
amplitude periodic behavior about a fixed value.
5
The
discretization error of the computations was calculated using
the “mixed first+ second order Richardson extrapolation” de-
scribed by Roache
40
and Roy.
41
The procedure made use of
the solution and of the ratio of the number of cells on the
three grid sequences to estimate the discretization error. The
results of the grid-convergence study performed on the com-
putational mesh of this work are tabulated in Table IV, in-
cluding the “exact” solution computed via the Richardson
extrapolation, and the same data are plotted in Fig. 4. The
plot shows the change in normal force coefficient and pitch-
ing moment coefficient as the grid is refined from a coarse
grid level with 2.410
4
cells to the medium grid level,
1.93 10
5
cells to the fine grid level, 1.54 10
6
. The change
in the results from one grid level to the next is an indication
of the error given by the discretization of the computational
domain. The discretization error on the fine grid was esti-
mated to be 0.6% for the normal force and 0.1% for the
pitching moment see Table IV. It should be noted that the
mesh topology shown in Fig. 2 was the final result of an
iterative mesh-optimization process in which the mesh den-
sity was increased or decreased according to the flow gradi-
ents obtained on a previous mesh topology. This process was
repeated several times during the initial stages of the present
work, starting from an initial multiblock Cartesian mesh and
ending with the efficient mesh topology and cell distribution
shown in Fig. 2. Complete details of the mesh-optimization
process and of the estimation of the uncertainty can be found
in Ref. 6.
Depending on the inlet conditions during tests, the flow
FIG. 4. Results of the grid-convergence study. The moment and force coef-
ficients are normalized using the results from the fine grid 1.56 10
6
cells.
FIG. 5. Color online Mach contours on the plane of symmetry of the jet.
Part a shows large-scale view and part b shows the detail of the flow field
around the injector with the main flow features highlighted with solid lines.
The solid lines are sketches indicating the recognizable flow patterns typical
of the underexpanded jet exhausting in a quiescent medium.
046101-5 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
field is not steady and shows periodic asymmetries about the
jet centerline.
4244
However, the RANS simulations did not
capture the flow unsteadiness also when running the full
three-dimensional 3D domain and perturbing the inlet con-
ditions. Therefore, it seemed appropriate to assume a steady-
state flow field and make use of a symmetry boundary con-
dition along the domain center line. While these two
assumptions would not be adequate for extracting detailed
time-accurate information, they are an adequate assumption
for capturing and analyzing the main flow features.
III. RESULTS
This section presents the results and discussion based on
the numerical simulation of the jet interaction flow field pro-
duced by normal sonic injection into a
MACH 4.0 cross flow
with a jet pressure ratio PR of 532 see Table III. A general
description of the jet interaction flow field based on the work
of other researchers was given in Sec. I and some of its basic
characteristics were schematically shown in Fig. 1. In Secs.
III A–III E the flow field is analyzed more in depth with
focus on the compressible features and the vortical structures
which are the main mechanisms responsible for the forma-
tion of the pressure field on the solid surface surrounding the
injector and for the mixing of the jet fluid with the cross
flow.
A. Main flow features of the supersonic jet interaction
flow field
A general view of the main features that characterize the
supersonic jet interaction flow field is provided by the map-
ping of the Mach number contours on the plane of symmetry
of the computational domain, Fig. 5a. The sonic jet ex-
hausting at a right angle into the supersonic cross flow pro-
duces an inclined barrel shock that, due to the jet being
highly underexpanded, terminates in a Mach disk. A reflected
shock is formed downstream of the barrel shock wave and it
impinges on the flat plate. The barrel shock acts as a blunt
body obstruction to the incoming flow thus forming a de-
tached bow shock. A fully developed turbulent boundary
layer is present at the upstream inlet and, as it approaches the
adverse pressure gradient created by the bow shock wave, it
separates from the tunnel flow, see contours of TI in Fig. 6.
Figure 5b is a detailed view of the Mach contours around
the injection location. The superimposed black lines help
identifying the main structures that are typically found in an
underexpanded sonic jet exhausting in a quiescent medium
see Ref. 45. However, different from the case of the sonic
jet exhausting in a quiescent medium the backpressure is not
uniform around the expanding jet due to the presence of the
cross flow, the backpressure being higher on the windward
side than on the leeward side of the plume. This nonunifor-
mity of the backpressure causes the jet plume to trail down-
stream and to lose its axial symmetry. Looking at the interior
volume of the barrel shock, a large expansion fan is present
with its boundaries defined by a recompression shock that
ends with a Mach disk. The Mach disk is essentially a nor-
mal shock that slows down the highly supersonic flow inside
the plume to subsonic. The subsonic flow that is generated
by the Mach disk forms a slip surface with the supersonic
fluid flowing around and past the barrel shock. The slip sur-
face is clearly visible in the Mach contours of Fig. 5b.
The two streams eventually mix together into a highly turbu-
lent flow further downstream. According to Woodmansee
et al.,
45,46
a sonic line should envelope the barrel shock on its
sides. Because of the mixing with the cross flow and the
presence of the bow shock, it is difficult to identify the sonic
line and the outer shear layer of the jet plume as described by
Woodmansee et al. The windward side of the barrel shock
appears to have less resemblance to the underexpanded jet
flow field than the leeward side mainly because of the strong
influence of the bow shock. A smeared sonic recompression
line can be seen on the windward side of the barrel shock,
generating from the windward side of the injector and ex-
tending past and above the barrel shock. A sonic recompres-
sion line does not form on the leeward side of the barrel
shock due to the presence of the solid wall. The location
where the downwind side of the barrel shock intersects the
Mach disk is known as the triple point. A reflected shock
extends downstream from this point and it impinges on the
surface of the flat plate at x / d = 15.0. This location can be
clearly identified by the sudden pressure increase in the C
p
plot of Fig. 7b. The adverse pressure gradient produced by
FIG. 6. Color online TI contours a on the plane of symmetry and b as
seen in an isoview of the detailed area at the inlet. The colors on the surface
of the flat plate represent pressure coefficient and are used for illustration
only in this caption.
046101-6 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
the impingement of the reflected shock on the flat plate
causes the boundary layer to thicken suddenly, as indicated
by the plot of the Mach contours. On the upstream side of the
barrel shock, the triple point can be easily located but the
reflected shock extending from this location is barely identi-
fiable. As mentioned before, this is a result of the strong
interference created by the cross flow and the bow shock.
The strength of the bow shock varies depending on its
location relative to the barrel shock. The bow shock is stron-
gest along the plane of symmetry upstream of the barrel
shock, where it is basically a normal shock. Away from this
location, the bow shock curves downstream in both the lat-
eral and vertical directions, thus forming a wrapping surface
around the barrel shock. Immediately aft of the normal shock
section, local regions of subsonic flow are formed, and this
flow is accelerated back to supersonic speeds by mixing with
the supersonic cross flow fluid that has passed through the
oblique sections of the bow shock.
Figure 5a places in evidence the lambda shock as it is
often referred in literature. The Mach number contours along
the plane of symmetry show that the two shocks never
merge. This observation is contrary to what can be observed
in shadowgraphs and Schlieren pictures where the two
shocks appear to merge. The merging of the shocks observed
in the experiments is likely due to the optical “collapse” of a
3D flow field on the two-dimensional plane of the photo-
graphs. This region has been studied by several works due to
the complexity of the microflow structures that form between
the two shocks see Ref. 47.
Figure 6a shows the contours of the TI
TI
=
2
3
k·
0.5
/ U
on the plane of symmetry. As expected, the TI
is particularly high in the areas with high velocity gradients,
such as in the separation region, across and downstream of
the shocks, and in the wake of the barrel shock where strong
vortical structures are present and most of the mixing is oc-
curring. At the inlet plane a turbulent boundary layer has
developed from the initial guess of i a power-law velocity
profile and ii a uniform 5% TI profile, as showed in the
mappings of Fig. 6b. The turbulent boundary layer is un-
disturbed in the region away from the center line while close
to the center line, its thickness rapidly increases due to the
presence of the adverse pressure gradient created by the pres-
ence of the jet. The TI distribution observed in the boundary
layer mapping is typical of that for the flat plate with the
locus of maximum turbulence level located at a distance
above the solid surface.
23
This is clearly seen in Fig. 8a
where the TI profiles at the inlet plane are plotted for differ-
ent spanwise locations with z / d = 0.0 representing the center
line. The vertical axis is normalized using the measured
FIG. 7. Color online兲共a Pressure coefficient distribution along the tunnel
centerline and b pressure coefficient mapping on the surface of the flat
plate.
FIG. 8. Color online Converged inlet boundary layer profiles at different cross flow locations for a TI and b velocity. z/ d = 0.0 corresponds to the center
line.
046101-7 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
boundary layer thickness of 1.65 cm see Table III. From
these profiles it is clear that the initial estimate of a uniform
TI of 5% has adjusted accordingly to the flow solution inside
the domain and the maximum TI is now 3.2% along the
center line. As noticed in the mappings of Fig. 6b the re-
gion of maximum turbulence in the boundary layer gets
closer to the solid surface with the distance form the center
line due to the decrease in the effect of the adverse pressure
gradient created by the bow shock. Also, the freestream TI
from the initial estimate of 5% has dissipated to a uniform
value of 0.014%. The velocity profiles for the same loca-
tions, shown in Fig. 8b, corroborate the finding that the
propagation of the effects of the jet-induced separation to the
inlet plane is limited to the region next to the center line and
shows that the velocity profiles away from the center line
remain practically undisturbed.
B. Validation of the numerical solution
The experimental data available for the case under inves-
tigation see Ref. 48 are limited and, more importantly for
this work, it was affected by large uncertainties due to the
pressure-sensitive paint PSP used in the measurements. As
a consequence of the limited data available, it was not pos-
sible to conduct an exhaustive quantitative validation of the
numerical simulation and a limited qualitative validation
study is conducted by comparing the CFD solution to the
experimental Schlieren photographs of the flow field.
5,6
The
Schlieren photograph is shown in Fig. 9. The picture pro-
vides a means to draw an outline of the main flow features
visible in the experiment such as the barrel shock, the bow
shock and the separation-induced shock. Schlieren photo-
graphs depict the first spatial derivative of the density. There-
fore, this derivative can be computed from the CFD simula-
tions and the flow field features visible in the photograph of
Fig. 9 can be superimposed on the numerical mapping. It is
important to remember that while the Schlieren picture is a
two-dimensional representation of a 3D flow, the CFD solu-
tion shown is an actual real two-dimensional slice through
the 3D flow field. For this reason, some of the flow features
visible in the Schlieren photographs that may appear to lie on
the symmetry plane in actuality do not lie on it and cannot be
directly compared to the CFD mappings on the symmetry
plane. Further, the Schlieren picture is an instantaneous snap-
shot of the flow field while the CFD picture represents a
time-averaged solution. The comparison of the Schlieren
photograph to the numerical solution is shown in Fig. 10.
The CFD simulation correctly predicted the location of the
separation-induced shock near the location where it im-
pinges on the bow shock, the location and shape of the bow
shock, and of the barrel shock. Also, the Mach disk height
over the flat plate, h, see Fig. 10, is in agreement with the
measurements of Schetz et al.,
12
which uses the concept of
equivalent backpressure, P
eb
=0.8P
t,2
, where P
t,2
is the total
pressure behind a normal shock, for correlating the penetra-
tion height of a highly underexpanded jet to the Mach disk
height. In the present case, the ratio P
j
/ P
eb
was calculated to
be 16.5, which correlates to a Mach disk height of 4.3h / d
j
,
while the CFD predicted a Mach disk height of approxi-
mately 4.5h / d
j
. A comparison of the pressure field predicted
by CFD with the experimental results is presented in Figs.
11a and 11b. Figure 11a shows the mapping of the pres-
sure coefficient extracted from the PSP data at the top half of
the picture to the computed one, at the bottom half of the
picture. The comparison highlights the qualitative agreement
between the experiment and the CFD. However the PSP data
present i a high level of experimental noise as evidenced by
the fragmented isolines and ii a lack of resolution, shown
by the lack of the high-pressure region in the separation
ahead of the injector. The latter point can help explain the
large discrepancy between the PSP and the CFD solution in
the region immediately in front of the jet, −3.0 x / D 0.5.
Other CFD studies of the supersonic jet interaction flow field
with more accurate surface pressure experimental data see
Refs. 24 and 49 have observed a pressure distribution which
resembles very closely that predicted by the present numeri-
cal simulations. It must be noted that both the Tam and
Chenault cases had much lower jet pressure ratios than the
present work with a consequently lower absolute overpres-
sure. Cubbison et al.
2
measured via pressure orifices in the
flat plate just ahead of the injection pressure coefficients up
to 0.70 for the jet interaction flow field with a freestream
Mach number of 3.0 and a PR of 677. Also, the pressure
distribution measured in the same experiment resembles very
closely that predicted by the present numerical simulation,
FIG. 9. Experimental Schlieren photograph of the jet interaction flow
field, Ma = 4.0, PR = 532 see Viti et al. Ref. 8 and Wallis
Ref. 48兲兴.
FIG. 10. Color online Comparison of the Schlieren picture with the CFD
solution on the plane of symmetry. The CFD contours represent the magni-
tude of the first-derivative of the density with respect to space,
.
046101-8 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
with a high plateau corresponding to the separation region
followed by a sharp peak created by the bow shock. Part of
the discrepancy is attributable to the weaknesses associated
with an eddy-viscosity model in which the assumption of
isotropic turbulence might not hold true for the region with
very high-pressure gradients and highly rotating flows. Un-
fortunately the lack of more accurate experimental data for
the present case prevents a more complete validation of the
numerical procedure. While these comparisons do not quan-
titatively validate the numerical solution, they provide a level
of confidence necessary to proceed with the qualitative
analysis of the flow field.
C. Vortical structures of the supersonic jet interaction
flow field
A valuable insight of the jet interaction flow field and its
vortical structures is provided by the isometric view of the
flow near the injector, as shown in Fig. 12. This snapshot
shows the Mach number contours mapped on the plane of
symmetry compare with Fig. 5a, the C
p
contours on the
surface of the flat plate and the vorticity magnitude contours
on the cross plane aft of the barrel shock. The paths of the
trailing vortices are highlighted by streamlines that follow
the vortex core. The interpretation of the flow features of Fig.
12 is enhanced by the use of the two-dimensional pressure
plots of Fig. 7. Following the flow along its path as indicated
by the arrow, the first flow conditions to be encountered are
those produced by the undisturbed freestream, region 1 of
Fig. 7a. The inlet boundary layer is clearly visible at the
extreme left of Fig. 5a where the Mach number on the
surface of the flat plate is zero, and it gradually increases
until it reaches the freestream conditions. The turbulent
boundary layer is allowed to grow freely along the flat plate
surface to the location of the separation. Separation see Fig.
10 and region 2 of Figs. 7a and 7b is caused by the
shock-boundary layer interaction. The strong adverse pres-
sure gradient caused by the bow shock propagates upstream
through the subsonic region of the boundary layer. In Fig.
7a, the C
p
plot along the center line shows the onset of
separation as a region where the pressure increases steeply
region 2, then it plateaus and decreases again region 3.
The C
p
contours of Fig. 7b show the separation as a well-
defined lobe near the plane of symmetry corresponding to
regions 2 and 3 of Fig. 7a that extends downstream and
away from the tunnel center line. Region 3 is also where the
core of the horseshoe vortex forms and is shed sideways
from the symmetry plane as highlighted by the streamlines of
Fig. 12. On the plane of symmetry the core of the horseshoe
vortex appears as the upstream vortex of a pair of counter-
rotating vortices, see Fig. 13. The progression of the horse-
shoe vortex as it trails downstream is also evident in the
cross sectional mappings of the vorticity shown in Fig. 14.In
these mappings the vortex is shown as a localized region of
high-vorticity intensity close to the bottom surface and mov-
FIG. 11. Color online Comparison of the experimental and CFD pressure
coefficient. a Mappings on surface of flat plate and b along the tunnel
center line. The experimental data were obtained through PSP. Ma= 4.0,
PR=532 Viti et al.Ref. 26兲兴.
FIG. 12. Color online Isometric view of the flow around the injector with
streamlines highlighting the main vortical structures. Mach number contours
on symmetry plane, C
p
contours on surface of flat plate, vorticity magnitude
contours on cross plane.
046101-9 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
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ing away from the plane of symmetry with downstream dis-
tance. The Mach number contours of Fig. 5a show the pres-
ence of a separation-induced shock. This oblique shock is not
as strong as the bow or barrel shock as it is generated by the
sudden thickening of the separated boundary layer, and it
impinges on the upstream side of the bow shock. The bound-
ary between regions 3 and 4, where the pressure along the
center line decreases x / d = −4.0, defines the stagnation lo-
cation between the two counter-rotating vortices both of
which are clearly visible through the streamlines of Fig. 13.
They rotate in opposite directions and, on the center plane,
their vorticity is normal to the incoming cross flow. How-
ever, as both vortices move away from the center line, their
vorticity is realigned in the streamwise direction by the cross
flow. The two vortices are divided by an attachment line
region 5 in Fig. 7a, indicated as a peak in the C
p
plot. The
rotation of the second downstream vortex is dictated by the
direction of the injectant flow as it exhausts from the up-
stream rim of the orifice. Note the symmetry in the trends of
the C
p
distribution about region 5 in Fig. 7a. Upstream
region 4 and downstream region 6 of region 5 the pres-
sure drops rapidly, and then it recovers to some level in
regions 3 and 7. The pressure drop corresponds to the accel-
eration of the fluid moving away from the attachment line
and the formation of the core of the two counter-rotating
vortices. The pressure rise corresponds to the fluid moving
away from the attachment line while being slowed down and
turned around either by the incoming boundary layer fluid, as
in the case for the upstream vortex in region 5, or by the
barrel shock as for the downstream vortex, aft of region 5.
The pressure peaks in regions 6 and 7 of Fig. 7a are also
visible in the pressure mapping of Fig. 7b as the two small
lobes with the highest C
p
values just in front of the injector.
The two high-pressure lobes merge together as they move
away from the centerline and trail downstream to form the
footprint of the bow shock on the flat plate. In their numeri-
cal analysis of the two-dimensional jet interaction flow field,
Chenault and Beran
49
reported a tertiary vortex in the sepa-
ration region, rotating counterclockwise and located between
the core of the horseshoe vortex and the flat plate. In the
present study, no tertiary vortex was present in the separation
region. This discrepancy could be due to the fact that the
tertiary vortex is a feature of the two-dimensional jet inter-
action flow field only. In fact, the same authors did not report
the existence of this vortex for the 3D numerical simulation
of the jet interaction flow field.
50
As discussed above, the first of the two counter-rotating
vortices in the separation region create one strong vortical
structure that is the horseshoe vortex. The second counter-
rotating vortex does not generate one single coherent struc-
ture but rather it generates several smaller vortical structures
that trail downstream and around the barrel shock. One of
these trailing vortices stemming from the separation region is
the upper trailing vortex. This vortex is formed by the recir-
culating fluid close to the plane of symmetry, and it follows
the leading edge of the barrel shock away from the solid
surface. The core of this vortex is clearly visible in Fig. 12
and with more detail in the close-up of Fig. 13. As this vor-
FIG. 14. Color online Cross plane mappings of vorticity magnitude left
and Mach number right.
FIG. 13. Color online Detail of the isometric view of the oblique barrel
shock with two groups of streamlines highlighting the flow in the recircula-
tion region. Mach numbers contours are plotted on the cross plane and
plane of symmetry, C
p
contours on the flat plate surface. Velocity vectors
y-z projection superimposed on the cross plane.
046101-10 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
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tex trails downstream, it moves away from the solid surface
and away from the plane of symmetry, as shown in Fig. 14.
The rest of the fluid in the second counter-rotating vortex
is convected downstream sideways, close to the surface of
the flat plate and around the footprint of the barrel shock to
form the surface trailing vortex. As shown by Fig. 12, the
fluid that forms the core of the surface trailing vortex moves
away from the symmetry plane as the barrel shock expands
around the injector. When the barrel shock detaches from the
surface of the flat plate, the surface trailing vortex moves
toward the center line and into the low-pressure region be-
hind the injector. Due to its proximity to the solid surface,
the trailing vortex entrains large quantities of low-
momentum boundary layer fluid, as is evident from Fig. 15.
This presence of the trailing vortex and its behavior are in
agreement with the observations of Palekar et al.
51
However
these authors did not report finding any vortical formation
that resembles the upper trailing vortex and presently it is not
clear why there exists this discrepancy between the two sets
of results.
While the present work did not focus on the mixing of
the injectant with the freestream, we can infer that such mix-
ing is enhanced by the action of four distinct pairs of
counter-rotating trailing vortices. The cores of the four vor-
tices are highlighted in Fig. 15 through the plot of the vor-
ticity magnitude on a cross flow plane at 15 jet diameters
downstream of the injection location. The surface trailing
vortex was discussed earlier, and it was shown that it origi-
nates from the second counter-rotating vortex of the separa-
tion region and is energized by the shear layer of the barrel
shock. Almost all of the fluid contained in the core of this
vortex is freestream fluid. The trailing vortex 1 and trailing
vortex 3 are a couple of counter-rotating vortices formed as
the slow-moving injectant fluid comes in contact with the
high-speed cross flow aft of the Mach disk, as shown in Figs.
12 and 14 for x / d of 6.00 and 12.00. Most of the fluid con-
tained in these two vortices is injectant fluid, with small
quantities of freestream fluid being entrained from the shear
layer between the barrel shock and the freestream. Relatively
little mixing with the freestream occurs until a location 30
diameters downstream of the injection location. The fourth
vortex shown in Fig. 15 is trailing vortex 2. This vortex
forms in the shear layer region existing between the wind-
ward side of the injector and the second of the two counter-
rotating vortices. Part of the vortex fluid is injectant fluid
entrained from the windward side of the barrel shock. The
vortex core forms on the center line, and it is convected
downstream and upward along the sharp angle in the barrel
shock, as shown in Fig. 16. Figure 16 also shows the mecha-
nism that moves the surface trailing vortices toward the cen-
ter line. As the barrel shock detaches from the solid surface
FIG. 15. Color online Cross plane mappings of vorticity magnitude left
and Mach number right with velocity vectors superimposed at a location of
x/ d = 15.00 downstream of the injector. The flow is into the plane of the
page.
FIG. 16. Color online Cross plane mappings of vorticity magnitude left
with projected velocity vectors and Mach number right with velocity vec-
tors superimposed at a location of x / d = 3.5 downstream of the injector. The
flow is into the plane of the page. The dashed box represents the flow region
that is magnified in Fig. 18a.
046101-11 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
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of the flat plate, it creates the low-pressure region which
occupied by the vortex. Initially, trailing vortex 2 is bounded
by the plate surface and the bottom side of the barrel shock.
As shown in Fig. 14, the three upper vortices trailing vorti-
ces 1, 2, and 3 rotate with respect to each other around a
common longitudinal axis see Figs. 14 and 15, x/ d = 20.0.
As they trail downstream, they merge into a single vortex
see Fig. 14, x / d =35.0 that is the main mechanism driving
the mixing of the freestream fluid with the injectant. The
horseshoe vortex and the trailing upper vortices continue to
be convected downstream along their trajectories and do not
contribute to the mixing of the injectant with the freestream;
see, for example, Figs. 7 and 14 at locations x/ d = 35.0 and
40.0.
A summary of the vortical structures found in the present
study of the supersonic jet interaction flow field is shown in
Fig. 17. This figure shows a schematic of the cross flow
section at a location aft of the barrel shock. A system of five
pairs of counter-rotating vortices forms in the recirculation
region ahead of the injector, along the barrel shock wave and
immediately downstream of the Mach disk. Of these ten vor-
tices, eight form in the recirculation region, and the other
pair is formed by the recompression of the jet fluid passing
through the Mach disk. This vortex is generally referred to as
the kidney-shaped vortex, see Ref. 52 for details. The horse-
shoe vortex and the trailing upper vortex systems form and
immediately move away from the centerline of the plate. The
horseshoe vortex moves horizontally along the solid surface
and away from the symmetry plane while the upper vortex
moves vertically along the symmetry plane and away from
the flat plate surface see Fig. 17. The longitudinal vortices
form in the recirculation region and gain in strength as they
are convected downstream and upwards along the barrel
shock plume. The trailing lower vortices also form in the
recirculation region, but they remain close to the surface and
to the plane of symmetry. The kidney-shaped counter-
rotating vortices form downstream of the jet plume and are
the major contributors to the mixing of the injectant with the
freestream, mainly by entrainment of the freestream in the
vortices. Both the horseshoe and the upper vortex systems
trail downstream isolated from the other vortex systems. The
upper vortex is weaker than the other systems hence more
difficult to identity and to follow in the cross sectional map-
pings. It appears clearly defined in the vorticity mappings of
Fig. 15 and as the streamlines of Fig. 12. The lower trailing
vortex remains attached to the solid surface as it entrains
fluid from the surrounding boundary layer. The other two
vortex systems, the longitudinal, and kidney-shaped vortex
systems, merge aft of the Mach disk into a single vortex that
trails downstream along a constant cross plane location. This
system of three trailing vortices was also reported in the
numerical study of Tam and Gruber.
24
It is of interest to notice the major differences between
the vortical formations observed in the subsonic and in the
supersonic jet interaction flow field. In the subsonic jet inter-
action flow field the main mechanism responsible for the
formation of the longitudinal trailing vortices is the realign-
ment of the vorticity present in the injector boundary layer.
These vortices are shed intermittently and form a double-
deck structure with the pair of stable vortices stacked above
them.
9
In the supersonic flow field, the majority of the vor-
tical structures are formed by the shock waves and the sepa-
ration region ahead of the injector. Although in the present
study a boundary layer was not simulated inside the injector,
the high expansion of the injectant fluid suggests that the
flow field inside the barrel shock is dominated by inviscid
rather than viscous phenomena. The assumption of a step
profile for the injector, corrected for viscous effects through
the discharge coefficient, is a common practice in the nu-
merical study of chocked nozzles exhausting either in a qui-
escent medium or in a cross flow.
24,2831,49,50
Further, in the
supersonic flow field the largest contribution to the genera-
tion of vorticity is primarily due to the entropy changes gen-
erated by the shocks rather than the direct interaction of the
injectant with the cross flow boundary layer.
D. Features of the barrel shock
Two prominent features differentiate the barrel shock
formed by an underexpanded sonic jet exhausting in a qui-
escent medium from the case with a cross flow. These two
features are a the barrel shock indentation created by the
reflection of the shock itself on the flat plate, and b the
inner shock reflection line caused by the folding of the wind-
ward side of the barrel shock into itself. The barrel shock
indent was introduced previously in the analysis of Figs. 13
and 16. The latter clearly shows the sharp angle formed by
the reflected shock penetrating into the main shock. The
shock reflection is caused by the downstream tilt of the bar-
rel shock axis. Due to the tilt, the injectant on the down-
stream side of the barrel shock does not have space to ex-
pand and recompress through the barrel shock to the correct
local pressure. For this reason, the barrel shock is attached to
the surface of the flat plate just downstream of the injector,
as shown in the side view of Fig. 5b and in the cross sec-
tion of Fig. 14 x/ d
j
=0.0. The presence of the solid surface
creates a reflection of the barrel shock that moves back in-
ward into the barrel shock. Due to the curvature of the barrel
shock, the shock boundary tangential to the surface of the flat
plate is reflected first, thus creating the concave triangular
indent observed in Figs. 14 and 16 at x/ d
j
=6.0. A closer
FIG. 17. Color online Schematic of the flow field at a transverse section
aft of the barrel shock.
046101-12 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
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view of the indent is shown in Fig. 18a, which represents a
detail of the dashed box of Fig. 16. The vectors represent the
density gradient, and the contours represent the magnitude of
the density gradient. The right side of the mapping shows the
curved cross section of the barrel shock, with injectant fluid
on its inside top half of picture and freestream fluid on the
outside lower half of picture. The concave indentation is
located in the proximity of the plane of symmetry since it is
at this location only that the barrel shock is in contact with
the flat plate. The shape of the indent resembles in thickness
and curvature an inverted continuation of the barrel shock.
The movement of the recompression shock away from the
solid surface creates the region of low pressure in the prox-
imity of the center line. The footprint of this low-pressure
region on the flat plate, as shown in Fig. 7b, is a result of
the indent in the barrel shock. The low-pressure lobes con-
tours B and C in Fig. 7b that appear to extend along a
radial line from the injector correspond to the inflection lines
of the barrel shock cross section shown in Fig. 18a. The
effect of the indentation on the general shape of the barrel
shock is clearly shown by the isosurface of Fig. 18b, where
the surface corresponding to a Mach number of 5.0 is high-
lighted. The background mapping is colored with the magni-
tude of vorticity on a cross plane at x / d
j
=1.0. The isosurface
highlights the three dimensionality of the indent that forms a
channel in the leeward side of the barrel shock. The presence
of the concave channel creates a local region of low pressure
that makes the surface trailing vortex move closer to the
center line. Again, the footprint of the low-pressure region in
Fig. 7b is correlated with the indent channel and inflection
lines. Also the indent channel clips the lower side of the
Mach disk. The relationship between the concave channel in
the barrel shock created by the reflection of the shock from
the solid surface and the low-pressure in the region aft of the
jet is relevant to jet-thruster control system applications. Ac-
cording to the present analysis, the low-pressure region could
be minimized by allowing the injectant to equalize its pres-
sure to the local freestream pressure without the interference
of the solid surface. This could be achieved by designing the
surface of the flat plate immediately aft of the jet as a con-
cave surface that would accommodate without interference
the volume of the barrel shock. This design philosophy is
opposite to that pursued by Byun et al.
19
and Viti et al.
8
who
attempted to decrease the low-pressure region by using a
protrusion in the solid surface, either in the form of a 3D
solid ramp or an array of secondary jets to create and aero-
dynamic ramp. The design with a concave surface would
have the advantage of being low-drag and simple to imple-
ment with no actuating or moving parts.
Figure 18b shows the second feature that distinguishes
the barrel shock formed by an underexpanded jet in a quies-
cent environment from that with a cross flow, i.e., the inter-
nal reflection line. The internal reflection line is created by
the folding of the windward side of the barrel shock onto
itself due to the localized high backpressure that exists due to
the presence of the bow shock on this side of the injector.
The expansion fan in Fig. 5b shows that the injectant ex-
pands symmetrically in the region near the nozzle. However,
on the upstream side of the nozzle, the high pressure gener-
ated by the compression of the freestream fluid passing
through the bow shock, causes the expanding injectant to
recompress earlier than on the downstream side of the
nozzle. The recompression shock on the windward side of
the barrel shock is pushed downstream by the incoming
freestream flow, thus breaking the symmetry of the expand-
ing jet. Notice in Fig. 5b how the injectant can expand to
much lower pressure and higher Mach numbers on the lee-
ward side of the barrel shock where the local backpressure is
lower than the windward side. The deformation of the barrel
shock due to the internal reflection line is clearly shown by
the
MACH 5.0 isosurface of Fig. 19, which is a side cross
section along the plane of symmetry of Fig. 18b. The back-
ground contours represent the Mach number on a longitudi-
nal plane at z/ d
j
=5.0. The use of the Mach number isosur-
face allows the analysis of the 3D features found in the
interior of the barrel shock. Inside the barrel shock, the first
MACH 5.0 surface is visible enveloping the injector. This sur-
face appears to be symmetrical about the injector, and it is
formed by the expansion of the sonic jet. The second isosur-
face represents the approximate boundary of the barrel shock
as it denotes the location at which the injectant is recom-
pressed to the local static pressure. The internal reflection
line is clearly visible as a straight line that starts upstream at
the location where the expanding injectant loses its symme-
try and ends at the downstream side of the barrel shock.
Notice also the presence of the indent line that does not
FIG. 18. Color online Downstream view of the indent in the barrel shock
created by the reflection of the compression wave on the surface of the flat
plate downstream of the injection location. The flow is out of the plane of
the page. a Detailed view of the indent. Density gradient contours on a
cross plane at x / d
j
=3.5. b Downstream view of the barrel shock repre-
sented by the MACH 5.0 isosurface. Cross plane is colored by vorticity
magnitude.
046101-13 Detailed flow physics of the supersonic jet Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
appear in the contour plots on the plane of symmetry of Fig.
5 due to these plots being purely two dimensional. The in-
cline angle of the inner reflection line is a function of the
momentum flux ratio and of the local backpressure created
by the freestream and the bow shock. The presence of the
inner reflection line influences the flow field outside and
around the barrel shock since a strong shear layer is gener-
ated by the injectant fluid expanding around the reflection
line. When visualized through a Mach isosurface, the inner
reflection lines appear as finlike structures that extend along
the length of the barrel shock, as shown in Fig. 19b. Notice
that the inner reflection line is visible in the Schlieren picture
of Fig. 9 on the windward side of the barrel shock.
E. Oil surface-flow results
Relevant information on the mechanisms that create the
pressure field on the flat plate can be obtained by the oil
surface-flow visualization shown in Fig. 20. In this figure,
streamlines are drawn just above the surface so as to high-
light the projection of the two-dimensional velocity field
above the surface. It is important to bear in mind that this is
a two-dimensional representation of a 3D flow and therefore,
there are velocity components that are moving into or out
of the plane of these streamlines. The major flow structures
such as the bow shock, the separation and the barrel shock
are clearly visible as thicker oil lines. The freestream appears
undisturbed until the bow shock. Behind the bow shock the
freestream assumes a lateral velocity component to compen-
sate for the volume occupied by the barrel shock. In the
separation region, the fluid is turned around by the two
counter-rotating vortices and flows in the opposite direction
as the freestream. The local pressure is higher than the
freestream. As discussed before, the pair of horseshoe vorti-
ces is shed from the most upstream of the two counter-
rotating vortices in the separation region. The core of the
horseshoe vortices can be traced by following the low-
pressure lobe on the solid surface see also the mapping of
Fig. 7b. Immediately aft of the injector, there is a small
region where the plume is attached to the solid surface. The
oil-flow shows the footprint of the concave indentation in the
leeside of the barrel shock, analyzed in Sec. III D. The foot-
print of the barrel shock is clearly visible on the surface as
are the attachment lines of the surface trailing vortices. At
the location at which the plume becomes detached from the
solid surface, a low-pressure region forms, and the surface
trailing vortices are pulled together toward the plane of sym-
metry. Further downstream, the reflected shock from the
Mach disk impinges on the solid surface. The local increase
in pressure along the centerline see the C
p
plot of Fig. 7a,
regions 10 and 11 causes the surface trailing vortices to
move away from the symmetry plane. Once past this loca-
tion, the surface trailing vortices return to move parallel to
the symmetry plane and the pressure recovers to the
freestream value. This flow pattern is similar to that observed
by Palekar et al.
51
through the use of 3D streamlines. In their
analysis, the impingement of the shock on the flat plate is
clearly indicated by a lateral movement in the path of the
streamlines, a similar behavior to that observed in Fig. 20.
IV. CONCLUSIONS
Numerical simulations of the 3D jet interaction flow
field produced by a sonic circular jet exhausting normally
into a turbulent supersonic cross flow over a flat plate were
performed to study the time-averaged flow features that char-
acterize this fluid-dynamic problem. The numerical compu-
tations made possible a detailed analysis of the prominent
features that dominate the flow field. Through comparison
FIG. 19. Color online兲共a Side view of the inside of the barrel shock
represented by
MACH 5.0 isosurfaces also shown in Fig. 18b. The colored
contours represent Mach number on a plane at z / d
j
=5.0 from the plane of
symmetry. b Isometric view of the MACH 5.0 isosurface. The contours on the
plane of symmetry represent Mach number, on the flat plate pressure coef-
ficient and on the cross plane vorticity magnitude.
FIG. 20. Color online Streamlines above the flat plate simulating oil
surface-flow visualization with pressure coefficient mapping superimposed.
046101-14 Viti, Neel, and Schetz Phys. Fluids 21, 046101 2009
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
with experimental data, the solution was found to capture the
typical shock formations such as the bow shock, the barrel
shock wave, and the separation-induced shock wave. These
compressible flow features were found to be closely coupled
with a complex system of vortical structures that dominate
the flow field. In particular, the trailing vortices were found
to be generated by the cross flow that, after being com-
pressed by the bow shock, has to move around the barrel
shock and mix with the expanding injectant fluid. The pres-
sure distribution on the flat plate was correlated with the
aforementioned flow features. The nose-down pitching mo-
ment typical of the jet interaction flow field was found to
result from the coupling of the high pressure in the separa-
tion region ahead of the injector with the low-pressure region
aft of the injector. The high-pressure region corresponding to
the separation exhibits localized pressure maxima and
minima. These local peaks in pressure are generated by the
presence of two counter-rotating vortices that impinge on the
surface of the flat plate, the pressure peaks corresponding to
local stagnation conditions and the pressure troughs to the
vortical flow moving away from the surface. The low-
pressure region aft of the injector was found to be created
primarily by the reflection of the barrel shock on the solid
surface of the flat plate. This reflection creates a concave
indent in the leeward side of the barrel shock that promotes
the lowering of the local pressure. The footprint of the low-
pressure region on the flat plate with its two prominent lobes
extending far downstream was correlated with the 3D con-
cave channel that the shock reflection creates in the back side
of the barrel shock. The lack of symmetry in the backpres-
sure, the windward pressure being higher than the leeward
side also creates an inner reflection plane in the barrel shock.
In particular, the inner reflection was found to be generated
by the folding of the windward side of the barrel shock into
itself, thus creating a truncated and leaning barrel shock for-
mation. The inner reflection line was observed to appear as a
finlike structure on the lateral sides of the barrel shock and it
promotes the formation of one major vortical structure, trail-
ing vortex 2 and the mixing of the injectant with the
freestream fluid.
ACKNOWLEDGMENTS
The authors would like to thank Dr. William McGrory
for his invaluable advice and for granting the authors access
to the computational resources and software of AeroSoft,
Inc. Funding for the present study was granted by the
Air Force Research Laboratory under Contract No.
AFR-2T-3014-AOS.
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