Transverse jet injection into a supersonic turbulent cross-flow

Z. A. Rana, B. Thornber, and D. Drikakis

Citation: Physics of Fluids 23, 046103 (2011);

View online: https://doi.org/10.1063/1.3570692
View Table of Contents: http://aip.scitation.org/toc/phf/23/4
Published by the American Institute of Physics

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 PHYSICS OF FLUIDS 23, 046103 共2011兲

Transverse jet injection into a supersonic turbulent cross-flow

Z. A. Rana,a兲 B. Thornber, and D. Drikakis
Department of Fluid Mechanics and Computational Sciences, School of Engineering, Cranfield University,
Cranfield MK43 0AL, United Kingdom
共Received 13 December 2010; accepted 10 February 2011; published online 28 April 2011兲
Jet injection into a supersonic cross-flow is a challenging fluid dynamics problem in the field of
aerospace engineering which has applications as part of a rocket thrust vector control system for
noise control in cavities and fuel injection in scramjet combustion chambers. Several experimental
and theoretical/numerical works have been conducted to explore this flow; however, there is a dearth
of literature detailing the instantaneous flow which is vital to improve the efficiency of the mixing
of fluids. In this paper, a sonic jet in a Mach 1.6 free-stream is studied using a finite volume
Godunov type implicit large eddy simulations technique, which employs fifth-order accurate
MUSCL 共Monotone Upstream-centered Schemes for Conservation Laws兲 scheme with modified
variable extrapolation and a three-stage second-order strong-stability-preserving Runge–Kutta
scheme for temporal advancement. A digital filter based turbulent inflow data generation method is
implemented in order to capture the physics of the supersonic turbulent boundary layer. This paper
details the averaged and instantaneous flow features including vortex structures downstream of the
jet injection, along with the jet penetration, jet mixing, pressure distributions, turbulent kinetic
energy, and Reynolds stresses in the downstream flow. It demonstrates that Kelvin–Helmholtz type
instabilities in the upper jet shear layer are primarily responsible for mixing of the two fluids. The
results are compared to experimental data and recently performed classical large eddy simulations
共LES兲 with the same initial conditions in order to demonstrate the accuracy of the numerical
methods and utility of the inflow generation method. Results here show equivalent accuracy for
1 / 45th of the computational resources used in the classical LES study. © 2011 American Institute of
Physics. 关doi:10.1063/1.3570692兴

I. INTRODUCTION into the air. Schetz and Billig15 explored the transverse jet
penetration in a supersonic free-stream using a solid body
In recent years, large eddy simulation 共LES兲 technique
drag model and presented an analytical method for the pre-
has made significant contributions toward understanding the
diction of jet penetration. They introduced the jet-to-cross-
dynamics of certain flows for which it is very difficult to
flow momentum flux ratio 共J兲 as the most important param-
carry out experiments. This is mainly due to efficiency of the
eter in order to determine the jet penetration in the cross-
LES codes and the computational resources available today.
flow, as shown in
One such flow is the jet injection into a free-stream cross-
flow, where the free-stream flow could be subsonic or super- ␳ jV2j ␥ j P j M 2j
sonic. For the subsonic flow case, an important example is a J⬅ = , 共1兲
␳cV2c ␥cM cM 2c
jet emerging through a hole in a gaseous tank 共e.g., hydro-
gen兲 at high pressure. Important examples for a supersonic where ␳, V, ␥, P, and M represent density, velocity, ratio of
free-stream flow could be part of a missile thrust vector con- specific heats, pressure, and Mach number, respectively; the
trol system,1,2 noise control in cavities during flight,3–7 and subscripts j and c represent jet and cross-flow, respectively.
the combustion chamber of a typical scramjet engine.8–11 Further experimental studies13,16,17 were carried out for the
Both of these flows require understanding of the flow jet penetration in a supersonic flow at various Mach numbers
mechanics/physics for proper design of the equipment. For and used the correlations provided by Schetz and Billig.15
both 共subsonic and supersonic兲 examples, the underpinning Cohen et al.,16 in 1971, devised an empirical correlation to
knowledge of the jet entering into a transverse flow is simi- determine the height of the jet penetration into the transverse
lar; therefore, most of the experimental/theoretical studies of flow as in

冤冉 冊
冥 冋
this phenomenon started with subsonic main flow and ex- ␥j − 1 2 0.25
panded to include supersonic free-stream flows.12,13
In 1959, Adamson and Nicholls14 presented the internal
structure of an underexpanded jet into quiescent air in order
Hmid
D
=
2 1+
2
Mj

␥2j M j共␥ j + 1兲
⫻
1.25共1 + ␥c兲␥cM 2c
共1 − ␥c兲 + 2␥cM 2c
册 0.5

to study the structure of the jet and discussed a method to ⫻ J, 共2兲

calculate the position of the Mach disk as the jet expanded
where Hmid / D represents the height of the midpoint of the
a兲
Author to whom correspondence should be addressed. Electronic ad- Mach disk nondimensionalized by the diameter of the jet
dresses: z.a.rana@cranfield.ac.uk and zeeshan.rana@yahoo.com. hole. These two correlations have been utilized in this work

1070-6631/2011/23共4兲/046103/21/$30.00 23, 046103-1 © 2011 American Institute of Physics

046103-2 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

to determine the corresponding values of jet-to-cross-flow

momentum flux ratio 共J兲 and jet penetration.
Over the years, several more experimental studies18–23
have had been conducted which have resulted in a good un-
derstanding of jet injection into a supersonic cross-flow
共JISC兲 today. There have been several computational
studies24–26 carried out as well to determine various averaged
flow features at various Mach numbers. As most of the com-
putational studies were based upon Reynolds averaged
Navier–Stokes methodology which provides averaged flow
properties, hence there is still opportunity to understand the
instantaneous flow features and instabilities in JISC which
cause the flow mixing. This is important as it can help im-
prove the mixing efficiency of air and fuel inside the com-
bustion chamber of a scramjet engine. Recently, Santiago
and Dutton27 carried out experiments on a sonic jet of air
injected into a cross-flow of air at Mach 1.6 and measured all
three velocity components 共u , v , w兲 and five Reynolds
stresses using laser Doppler velocimeter. Further experi-
ments using the same conditions were carried out to deter-
mine the pressure distributions on the flat-plate using
pressure-sensitive-paint 共PSP兲 共Ref. 28兲 and mixing of the jet
with the supersonic free-stream.29 As the measurements were
taken in several planes, and the accuracy of the measurement
technique is considered very good, it provides an excellent
opportunity for computational fluid dynamics 共CFD兲 code
validation.
Figure 1 presents a well understood schematic diagram FIG. 1. Schematic diagram of a sonic jet injection into a transverse stream
for a typical underexpanded sonic jet injected in a supersonic of supersonic flow. 共a兲 Instantaneous view of middle plane to show various
transverse stream along with a three-dimensional schematic two-dimensional flow features. 共b兲 Three-dimensional schematic diagram to
showing the structure of the shock waves generated when the show major averaged flow features 共Refs. 18 and 19兲. Reprinted with per-
mission from A. Ben-Yakar et al., Phys. Fluids 18, 026101 共2006兲. ©2006,
jet interacts with the cross-flow in Fig. 1共b兲. As the underex- American Institute of Physics.
panded jet enters the cross-flow, it expands through a
Prandtl–Meyer expansion fan and at the same time deflects
and turns along the main flow. Due to the difference in ve- The comprehensive results from these experiments27–29
locities, the jet acts as an obstruction to the main supersonic provide an opportunity for validation/verification of state-of-
cross-flow and generates a bow shock ahead of the injection the-art CFD research codes employing time-evolving LES or
hole. The incoming supersonic turbulent boundary layer direct numerical simulations 共DNS兲 methodology. One im-
共STBL兲 starts to separate just ahead of the bow shock and a portant aspect of these experiments is the supersonic turbu-
small separation zone is created which results in a smaller lent boundary layer in the free-stream. Recently, Génin and
weak shock, called a lambda shock, that interacts with the Menon30 presented a LES study to understand the dynamics
stronger bow shock. The lambda shock and the size of the of JISC using similar initial conditions as in experiment27
separation zone are mainly dependent upon the momentum and also further expanded the study to include various jet-to-
of the main stream flow.15 The jet emerges from the orifice cross-flow momentum flux ratios and Mach numbers, but did
and expands to the atmospheric pressure at the jet boundary. not provide comparison of the upstream STBL. Kawai and
This constant pressure on the jet boundary causes it to bend Lele31,32 performed a comprehensive LES of the same JISC
toward the axis of flow and the barrel shock emerges. Due to experiment27 including the STBL upstream of the jet plume
the high pressure ratio of the flow, the barrel shock does not and their results were found to be in very good agreement
meet at the axis of flow but instead a normal shock 共Mach with the experiment. It was also elaborated that the free-
disk兲 is generated which has its center at the axis of flow. A stream STBL is vital to capture the correct flow physics and
small recirculation/separation zone is also visible immedi- mixing as compared to free-stream laminar flow.
ately downstream of the jet. There is a horseshoe vortex In order to numerically introduce the turbulent boundary
which wraps around the jet column and forms wake vortices layer in the flow field, there have been several options used
in the flow. Further downstream the jet boundary takes the for various applications. The obvious solution would be to
form of a pair of counter-rotating vortices 共CRVs兲 and some simulate laminar flow and allow it to develop turbulence
trailing counter-rotating vortices 共TCRVs兲. All these separa- over a long computational domain for a long period of time.
tion zones, shocks, and vortex structures give rise to a very But of course this method is not a reasonable one consider-
complex flow downstream of the jet which is helpful for the ing the computational costs involved. An alternative would
mixing of the two fluids. be to utilize a rescaling-reintroduction method as presented
046103-3 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

in Ref. 32 as well. This could require a comparatively U = 关␳, ␳u, ␳v, ␳w,E兴T ,
smaller computational domain but still require a long period
of time for the STBL to grow to a required size and hence F = 关␳u, ␳u2 + p, ␳uv, ␳uw,u共E + p兲兴T ,
could be computationally expensive. Another alternative to
this is synthetic turbulence data generation33 based upon a G = 关␳v, ␳vu, ␳v2 + p, ␳vw, v共E + p兲兴T ,
digital filter and correlation functions that generate turbulent
data for the inflow plane at every time step and can be a very
H = 关␳w, ␳wu, ␳wv, ␳w2 + p,w共E + p兲兴T ,
efficient method.
In this paper, a finite volume 共FV兲 Godunov type im-
plicit large eddy simulations 共ILESs兲 method is utilized to L = 关0, ␶xx, ␶xy, ␶xz,u␶xx + v␶xy + w␶xz − q̇x兴T ,
study the JISC flow and compare the result with the experi-
mental data27–29 and classical LES.32 The method employs M = 关0, ␶yx, ␶yy, ␶yz,u␶yx + v␶yy + w␶yz − q̇y兴T ,
fifth-order accurate MUSCL 共Monotone Upstream-centered
Schemes for Conservation Laws兲 scheme with modified vari- N = 关0, ␶zx, ␶zy, ␶zz,u␶zx + v␶zy + w␶zz − q̇z兴T ,
able extrapolation for spatial discretization34 and a three-
stage second-order strong-stability-preserving Runge–Kutta where ␳ is the density, 共u , v , w兲 are the components of veloc-
共SSPRK兲 共Ref. 35兲 scheme for temporal discretization. A ity, E is the total energy, p is the pressure, ␶ is the stress, and
digital filter based turbulent inflow data generation q̇ is the rate of heat transfer. The Cartesian equations are
method36,37 is implemented in order to capture the physics of converted to the nondimensional form and transformed to
the incoming STBL. Section II details the governing equa- curvilinear coordinates using the approach described by
tions and numerical methods utilized in this article. Section Drikakis and Rider,38 and then the inviscid flux vectors are
III explains the computational domain selected for the simu- solved using the method of lines in each direction and the
lations and the initial conditions employed. The results and viscous fluxes are solved using a second order central differ-
findings from this work are presented in Sec. IV and a dis- encing scheme. The total energy 共E兲 is the sum of internal
cussion is provided as to elaborate the findings. Finally, in energy 共i兲 and kinetic energy 共KE兲,
the end, Sec. V provides a conclusion for the work presented. E = ␳共i + KE兲,
共4兲
E = ␳关i + 0.5共u2 + v2 + w2兲兴,
II. COMPUTATIONAL FRAMEWORK
and finally, the system of equations is closed using an equa-
The computational study is based on the CFD code
tion of state,
CNS3D.38–41 The code includes different Riemann
solvers,38,42 including flux vector splitting methods, a p = ␳RT = ␳i共␥ − 1兲, 共5兲
characteristic-based scheme, and the HLLC 共Harten-Lax-
vanLeer-Contact兲 Riemann solver43 within the framework of where R is the gas constant, T is the temperature, and ␥ is the
high-resolution methods 共HRMs兲.44 In the present study, the ratio of specific heats.
HLLC Riemann solver is used, which assumes a three-wave
structure of the Riemann problem solution, allowing for two B. Numerical methods
intermediate states enclosed by the two fastest waves. The CNS3D utilizes a FV Godunov51 type method where the
HLLC Riemann solver does not use linearization of the initial and boundary values of the solution vector are speci-
equations and works well for low-density problems and fied at the start of the simulation. Higher order accuracy is
sonic points without any fixes. It has successfully been used obtained by employing a fifth-order accurate MUSCL
to simulate a variety of flows in conjunction with the CNS3D scheme,52
code.34,45–50
L
Ui+1/2 = Ui + 21 关␾lim共rlim,L兲共Ui − Ui−1兲兴,
A. Governing equations 共6兲
The basic governing equations of a Newtonian fluid
R
Ui+1/2 = Ui+1 − 21 关␾lim共rlim,R兲共Ui+2 − Ui−1兲兴,
flow, i.e., Navier–Stokes equations 共NS-equations兲, have where integer i represents the cell numbers and the ratio of
been employed in this study. For computational purpose, the the slopes 共r兲 is defined as
complete set of NS-equations 共in three dimensions兲 can be
written in the Cartesian matrix form as below, Ui+1 − Ui Ui+1 − Ui
rlim,L
i = , rlim,R
i = . 共7兲
⳵U ⳵F ⳵G ⳵H ⳵L ⳵M ⳵N Ui − Ui−1 Ui+2 − Ui+1
+ + + = + + , 共3兲
⳵t ⳵x ⳵y ⳵z ⳵x ⳵y ⳵z The fifth-order limiter employing one-dimensional
implementation of Kim and Kim53 used above is as follows:
where U represents a vector of conservative variables and
contains the required variables to be solved for 共solution vec- lim,L
− 2/ri−1 + 11 + 24rlim,L − 3rlim,L lim,L
ri+1
i i
tor兲, F, G, H are vectors of inviscid fluxes, and L, M, N are ␾ⴱlim
M5,L = ,
30
vectors of viscous fluxes in x, y, and z directions, respec-
tively, as below, 共8兲
046103-4 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 2. One-dimensional 共a兲 spatial and 共b兲 temporal

correlations obtained by the digital filter based turbulent
inflow data generator showing a reasonable match with
the corresponding target exponential functions.

lim,R
− 2/ri−1 + 11 + 24rlim,R
i − 3rlim,R
i
lim,R
ri+1 the role of an implicit subgrid scale 共SGS兲 model in the
␾ⴱlim
M5,R = , numerical scheme. As no explicit SGS model is employed in
30
the code, this class of high-resolution scheme can be termed
and monotonicity is maintained by as ILESs.57
Finally, the time integration is obtained by an explicit
␾lim , ␾ⴱlim three-stage second-order accurate SSPRK35,58 scheme which
M5,L兲兴,
lim,L
M5,L = max关0,min共2,2ri
extends the stability of the method up to a
共9兲
Courant–Friedrichs–Lewy59 number of 2,
␾lim lim,R
M5,R = max共0,min共2,2ri , ␾ⴱlim
M5,R兲兲.

In 2004, Guillard et al.54 established the incorrect pressure 1 ⌬t

U1i = Uni + f共Uni 兲,
difference scaling for low Mach numbers for standard Go- 2 ⌬x
dunov schemes. In 2008, Thornber et al.47 presented a theo-
retical analysis demonstrating that this is caused by the large
velocity jumps at the cell interfaces. They adopted a low 1 ⌬t
U2i = Uni + 关f共U1i 兲兴, 共11兲
Mach treatment for this excessive numerical dissipation and 2 ⌬x
proposed that the velocity jumps at the cell interfaces can be

冉 冊
modified by a function z which gives the reconstructed ve-
locities u as follows:34 1 ⌬t
Un+1
i = 2U2i + Uni + 关f共U2i 兲 + f共U1i 兲兴 .
3 ⌬x
uL + uR uL − uR
uL,M5+LM = +z ,
2 2 C. Synthetic turbulent inflow data generation
共10兲
uL + uR uR − uL Digital filter based synthetic turbulent inflow data gen-
uR,M5+LM = +z . eration is a novel, yet simpler, technique which is useful
2 2
when limited turbulence data are available. In 2003, Klein33
It was also demonstrated that with this modification the developed a technique of generating artificial velocities syn-
leading-order kinetic energy dissipation is proportional to thetically for the inflow data which was based upon the as-
u3 / ⌬x, which is similar to that proposed by Kolmogorov55 sumption that for homogeneous turbulence the two-point
for the decaying turbulence and validated this approach for a correlation takes the Gaussian form. On the other hand, re-
deep cavity48 and ship analysis.56 This dissipation rate plays cently, Xie and Castro36 argued that the correlation takes the
046103-5 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 3. 共Color online兲 Schematic diagram of the com-

putation domain selected for the JISC simulations.
Dash-dot-dashed black line is X / D = −5 position where
velocity profile is matched between the experiment and
the CFD.

form which is closer to exponential rather than Gaussian. Lund et al.,60

This technique has been recently used by Touber and
Sandham37 for the LES of turbulent shock-induced separa-

冤 冥冤 冥
u共0,y,z,t兲 具u共0,y,z,t兲典
tion bubble. This method of turbulent inflow data generation
has been implemented in the CNS3D code to synthetically v共0y,z,t兲 = 具v共0,y,z,t兲典
generate STBL at Mach 1.6 for the free-stream flow in JISC. w共0,y,z,t兲 具w共0,y,z,t兲典
The two-point correlation function used is defined36 as be-

冤 冥
冑R11 0 0
low,
+ R21/冑R11 冑R22 − 共R21/冑R11兲2 0
0 0 冑R33
冉 冊 ␲x

冤 冥
R共xk + x兲 = exp − , 共12兲 ␺u共y,z兲
2Ix
⫻ ␺v共y,z兲 , 共13兲
␺w共y,z兲
where xk is a point of reference, x is the point some distance
away from the reference point, and Ix is defined as the inte- where u共0 , y , z , t兲, v共0 , y , z , t兲, and w共0 , y , z , t兲 are the veloc-
gral length scale. In order to determine the fluctuations in the ity profile with fluctuations, values within 具 典 are averaged
prescribed velocity profiles, we used the relation given by profiles, Rxx are the Reynolds stresses obtained from the

TABLE I. Computational mesh used for the simulation of the STBL and JISC using ILES and digital filter
based turbulent inflow data generator. Also, in the bottom part of the table, the grid sizes for the computational
domain used in LES 共Ref. 32兲 have been provided for comparison only.

Grid Nx 共STBL+ JISC兲 Ny Nz Total 共STBL+ JISC兲 共⫻106兲 Ly Lz ⌬y+

Coarse 274 共127+ 147兲 85 85 2.0 共0.9+ 1.1兲 0.5 0.2 14

Medium 404 共187+ 217兲 101 115 4.7 共2.2+ 2.5兲 0.5 0.2 14
Fine 522 共242+ 280兲 116 151 9.2 共4.3+ 4.9兲 0.5 0.2 14

Coarsea 552 共251+ 301兲 131 87+ 115 7.4 共2.9+ 4.5兲 ¯ ¯ 14.5
Mediuma 772 共361+ 411兲 187 101+ 154 18.6 共6.8+ 11.8兲 ¯ ¯ 20.5
Finea 912 共361+ 551兲 243 120+ 204 37.8 共10.5+ 27.3兲 ¯ ¯ 29.0
a
Reference 32.
046103-6 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

TABLE II. Averaged stagnation inflow conditions from experiment. The

subscripts c and j refer to cross-flow and jet properties, respectively.

Property Value Units

Mach number 共c兲 1.6

Mach number 共j兲 1.0
Stag. pressure 共c兲 241 kPa
Stag. pressure 共j兲 476 kPa
Stag. temperature 共c兲 295 K
Stag. temperature 共j兲 295 K
Average velocity 共c兲 446.1 m/s
Reynolds number 2.4⫻ 10+04

experiment,27 and ␺ is the velocity field as described in Ref.

37. For further details of this method, please refer to Xie and
Castro36 and Touber and Sandham.37

FIG. 5. Grid sensitivity analysis. 共a兲 rms of stream-wise velocity compo-

nent. 共b兲 rms of wall-normal velocity component. 共c兲 rms of wall-parallel
velocity component 共Ref. 63兲.

In case of compressible turbulent flow, the density varia-

tions also play an important role; therefore, it is important at
this stage to include density fluctuations in the inflow data as
well. The density fluctuations are added to the STBL density
profile using the following relation:
␳共0,y,z,t兲 = 具␳共0,y,z兲典 + ␺共y,z兲, 共14兲
FIG. 4. Grid sensitivity analysis. 共a兲 Mean stream-wise velocity in semi-
logarithmic scale 共Ref. 61兲. 共b兲 Averaged velocity profile compared with the where ␳共0 , y , z , t兲 is the STBL density profile with fluctua-
experimental results 共Ref. 27兲 at three grid resolutions. tions. The resultant turbulent inflow data is a two-
046103-7 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 6. 共Color online兲 Instantaneous snapshot of incoming STBL using digital filter based turbulent inflow data generator and the injection of a sonic jet
creating a complex flow structure downstream the jet plume; density gradient contours 共vertical plane兲 and velocity contours 共horizontal plane兲. The line
indicates the Mach 1.5 position to demonstrate the location of lambda shock just upstream of the jet plume.

dimensional slice which is implemented on the computa- temporal correlations obtained from the turbulent inflow data
tional grid as input for the JISC simulations. Figure 2 generator. Although the spatial correlation data fit the tar-
presents the actual/target exponential plots in space and time. geted exponential function, there is a discrepancy in the tem-
Also plotted are the exponential trends in the spatial and poral data fit; nevertheless, it exhibits the same exponential

FIG. 7. 共Color online兲 Typical shocks and flow features

are identified as the sonic jet mixes with transverse
共Mach 1.6兲 supersonic flow; 共a兲 Two-dimensional JISC
flow structures at the midline transverse plane 共Z / D
= 0兲, and 共b兲 three-dimensional iso-surfaces for the
Q-criterion.
046103-8 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 8. 共Color online兲 Time averaged contours of various flow properties on the wall-normal midplane 共Z / D = 0兲. 共a兲 Mach number. 共b兲 Passive scalar. 共c兲
Turbulent kinetic energy. 共d兲 Reynolds shear-stress.

trend as expected from the two-point correlation function. levels showing a reasonable match with the DNS data.61,62
The STBL velocity profile from the experiment at X / D = −5
III. COMPUTATIONAL DOMAIN AND INITIALIZATION is matched with the STBL velocity profile at X / D = −5 of the
computational domain, as shown in Fig. 4共b兲. Although the
The experiments27–29 were carried out on a flat-plate Reynolds number used is small, the thickness of the turbu-
with a circular injection port that allowed the jet of air to lent boundary layer 关␦99 / D = 0.775共3.1 mm兲兴 has been
emerge into the supersonic free-stream air flow. The compu- matched at the X / D = −5 position in the experiment. The mo-
tational domain for this geometry comprises of a solid sur- mentum flux ratio 共J兲 is calculated to be 1.7, which also
face that represents the flat-plate with a circular hole as the matches the experimental data. Figure 5 presents urms, vrms,
injection port. Figure 3 is a schematic diagram of the com- and wrms in comparison with the DNS data,63 where, al-
putational domain selected for the ILES showing all the though coarse and medium grids are under resolved, the fine
boundary conditions associated with the domain. Three grid grid is showing a reasonable match with DNS data.
levels 共coarse, medium, and fine grids兲 are used in these
simulation details of which are presented in Table I. The
initial conditions prescribed for the simulations are the same IV. RESULTS
as the stagnation conditions used for the experiment which
A. Jet penetration
are tabulated in Table II. Note that the Reynolds number used
is six times smaller compared to the experiment. This is to Instantaneous overview of the STBL and JISC simula-
allow reasonable resolution of the computational domain for tion is shown in Fig. 6, where the velocity is on the horizon-
ILES and also to match the initial conditions used for the tal plane and density gradient on the vertical middle plane.
LES by Kawai and Lele.31,32 Figure 7 represents the JISC flow structure generated when a
The velocity profile from the experiment, obtained at transverse sonic jet of fluid emerges into a stream of Mach
X / D = −5 for a fully developed supersonic turbulent bound- 1.6 turbulent flow. Figure 7共a兲 shows all the major flow
ary layer, is applied at the X / D = −8 position in the compu- structures 共bow, barrel, and lambda shocks, Mach disk, and
tational domain, as shown in Fig. 3. The long upstream do- recirculation zones兲 captured using ILES as they are gener-
main is to allow the numerical expansion fan developed at ally understood 共fine grid results shown兲. For the given ini-
the start of the computational domain which was unavoid- tial conditions from the experiment,27 it is understood that
able. there should be three recirculation zones in the flow which
Figure 4共a兲 presents the nondimensional velocity versus are successfully captured as “R1,” “R2,” and “R3.” As the jet
the distance from the flat-plate on log-scale for three grid emerges into the free-stream flow, it expands and turns along
046103-9 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 9. 共Color online兲 Time averaged contours of various flow properties on the wall-parallel plane 共Y / D = 1兲. 共a兲 Mach number. 共b兲 Passive scalar. 共c兲
Turbulent kinetic energy. 共d兲 Reynolds shear-stress 共contours legend same as in Fig. 8兲.

FIG. 10. 共Color online兲 Time averaged contours of various flow properties on the cross-view plane 共X / D = 1兲. 共a兲 Mach number. 共b兲 Passive scalar. 共c兲
Turbulent kinetic energy. 共d兲 Reynolds shear-stress 共contours legend same as in Fig. 8兲.
046103-10 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 11. 共Color online兲 Time averaged contours of various flow properties on the cross-view plane 共X / D = 3兲. 共a兲 Mach number. 共b兲 Passive scalar. 共c兲
Turbulent kinetic energy. 共d兲 Reynolds shear-stress 共contours legend same as in Fig. 8兲.

FIG. 12. 共Color online兲 Time averaged contours of various flow properties on the cross-view plane 共X / D = 5兲. 共a兲 Mach number. 共b兲 Passive scalar. 共c兲
Turbulent kinetic energy. 共d兲 Reynolds shear-stress 共contours legend same as in Fig. 8兲.
046103-11 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

the main flow at the same time, as shown by the Prandtl– direction to the major CRVs which further enhances the mix-
Meyer expansion fan in Fig. 7共a兲. It is clear that maximum ing in the region below the major CRVs. Viti et al.65 men-
Mach number is inside and toward the top end of the jet tioned another small pair of TCRVs just on top of the major
plume just before the Mach disk. CRVs for a Mach 4 JISC, but for Mach 1.6 JISC this has not
The boundary of the jet that forms the barrel shock and been observed in the experiment,27 the classical LES,32 or
the Mach disk meets at a point referred to as the triple point, current investigations. Therefore, it can be deduced from this
where reflected shocks are also visible. There are also that the major CRVs are a common feature of JISC but the
present in the flow structure a horseshoe vortex, a pair of TCRVs are dependent upon the free-stream Mach number.
CRVs, and a pair of TCRVs, which will be discussed later Figure 8共b兲 shows the contour map for the passive scalar
关Fig. 7共b兲兴. Using Eqs. 共1兲 and 共2兲, the height of the midpoint 共jet fluid兲. It is observed from the contours that after injection
of Mach disk nondimensionalized by the diameter of the jet the jet fluid quickly mixes with the free-stream fluid 共X / D
hole 共i.e., jet penetration兲 has been measured to be ⬇1.4 D = 1兲 and then gradually dilutes further downstream 共X / D = 3
关as shown in Fig. 7共a兲 by H / D兴. Abramovich64 presented a and 5兲. The important factor to note here is that the mixing
correlation, as shown in Eq. 共15兲, based upon the trajectory starts immediately after the jet injection and most of it occurs
of maximum injectant concentration, in the thicker shear layer on the windward side of jet plume,
as shown in Fig. 10共b兲, which is also shown by the stream
Y/D = 共P j/Pc兲0.434共X/D兲0.333 , 共15兲 lines in Fig. 8共a兲. Downstream from the jet injection, the
mixing region increases its size as the CRVs increase in size
where P represents the dynamic pressure. This has been used 关Figs. 11共b兲 and 12共b兲兴.
by Orth and Funk17 in their experiments to study the jet Turbulent kinetic energy nondimensionalized by the
penetration in supersonic flow where they demonstrated that free-stream velocity 关TKE= 共具u⬘u⬘典 + 具v⬘v⬘典 + 具w⬘w⬘典兲 /
Eq. 共15兲 agrees “reasonably well with the experimental val- 共2U⬁兲兴 is presented in Figs. 8共c兲 at various planes in the
2
ues” for X / D ⱕ 8. This correlation has been plotted in Fig. computational domain. There are three regions of high TKE
7共a兲 for the jet trajectory of maximum jet concentration. It just upstream and downstream of the jet plume, also identi-
can be noticed that the trajectory path follows the jet plume fied in previous LES 共Ref. 32兲 results and shown in Fig. 8共c兲.
deflection reasonably well and passes through the Mach disk; The high TKE region just upstream of the jet plume is the
however, there is a slight discrepancy in determining the region of shock/boundary-layer interaction. Also the barrel
midpoint of the Mach disk. This small discrepancy was also shock region displays high TKE due to the presence of shear
observed in the experiment.17 layer between the inner region of the jet plume and the cross-
flow. The third high TKE region just downstream of the jet
B. Averaged flow analysis
plume is around the third recirculation zone 关R3 as shown in
Time averaged analysis has been presented for flow vi- Fig. 7共a兲兴. This is the region where the CRVs are originating,
sualization of various properties for the fine grid in Figs. the low-pressure recirculation zone just below the CRVs is
8–12 showing the contours of Mach number, passive scalar generated, and the CRVs create a very active mixing zone.
共jet fluid兲, turbulent kinetic energy 共TKE兲, and Reynolds Further downstream of the jet plume, the TKE dissipates
shear-stress at various locations in the domain along with gradually, as shown in Figs. 10共c兲.
stream lines. These stream lines clearly show three recircu- Figure 8共d兲 represents the dimensionless Reynolds
lation zones in Fig. 8共a兲. It can be noted from Fig. 8共a兲 that shear-stress 共RS兲 nondimensionalized by the free-stream ve-
most of the jet fluid is passing through the windward side of locity 关RS= 共具u⬘v⬘典兲 / 共U⬁2 兲兴 in various planes. There are two
the jet plume and the Mach disk and then diverts toward the high RS zones identified just upstream of the jet plume
direction of free-stream flow where mixing occurs. Also Fig. shown in Fig. 8共d兲. These two locations show high RS in
8共a兲 reveals that the lambda shock is a weak shock as repre- opposite directions 共i.e., +ve and ⫺ve兲, which is also identi-
sented by small change in contour colors. Figure 9共a兲 shows fied in the experiment.27 As shown earlier, this is the region
the stream lines for the flow when it is obstructed by the jet of shock/boundary-layer interaction and high TKE is also
plume on the wall-parallel plane 共Y / D = 1兲. At this point, due observed in this section produced by Reynolds shear-
to low pressure, a recirculation zone is created which runs stresses; both of these high values are due to the fact that this
around the jet plume and just above the flat-plate to create a region presents high mean velocity gradients. Figures 10共d兲
horseshoe vortex, as shown in Fig. 7共b兲. show the RS in the cross-view planes 共X / D = 1, 3 and 5兲
Figures 10共a兲 represent the stream lines of the flow which represent the diffusion of TKE and RS further down-
downstream the jet plume. At location X / D = 1, a pair of stream. In fact, it is clear from the RS contours that further
CRVs appears due to the interaction of the free-stream flow downstream the majority of the flow consists of only ⫺ve
with the jet fluid. This pair of CRVs then grows in size fur- RS.
ther downstream 共X / D = 3 and 5兲 and provides a major area Further analysis is carried out to compare the plots of the
where mixing of the two fluids occurs. As this pair of CRVs mean stream-wise and wall-normal velocities in the flow
starts to grow in size 关as in Fig. 11共a兲兴 another pair of small field with the experiment and the recent LES results using
TCRVs 共trailing CRVs兲 is created. This small pair is due to the three grid resolutions explained earlier. Figures 13 and 14
the low-pressure recirculation zone 关R3 as in Fig. 7共a兲兴 and show the comparison of the mean stream-wise and wall-
the suction action of the major pair of CRVs. Close analysis normal velocities 共nondimensionalized by the mean free-
of Fig. 12共a兲 reveals that the TCRVs rotate in the opposite stream velocity兲 at one upstream location 共X / D = −1.5兲 and
046103-12 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 13. Mean stream-wise velocity profiles are compared with the experimental and previous LES results on three grid levels. X / D = −1.5 position is just
upstream of the jet plume which is important for comparing the effect of upstream STBL. The downstream position compared is at X / D = 2, 3, 4 and 5 on the
midplane 共Z / D = 0兲 in 共b兲, 共c兲, 共d兲, and 共e兲, respectively.

further four downstream locations 共X / D = 2, 3, 4, and 5兲 at 14共b兲–14共e兲 and compared with the experiment and the LES
the wall-normal midplane 共Z / D = 0兲. It must be mentioned results. The stream-wise velocity profiles are slightly under-
here that the results presented in the current work are ob- predicted compared to the experiment at the X / D = 2 posi-
tained with one-third of the grid points in the classical LES tion, but better agreement with experiment has been found
共Ref. 32兲 共fine grid comparison兲 using 1/45th of computa- compared to classical LES. Moving further downstream the
tional resources. jet plume, the ILES results tend toward the experimental
It has been shown32 that the upstream velocity profiles data, whereas the classical LES is slightly offset from the
共at location X / D = −1.5兲 are influenced by the incoming experiment. In Fig. 13共e兲, however, the profile above the
STBL. The STBL affects the growth of the separation zone Y / D = 1.5 position is close to the classical LES, but below
关R1 in Fig. 7共a兲兴, i.e., restricts its growth and the lambda this position they are matching with the experiment again.
shock is developed. On the other hand, with laminar incom- The wall-normal velocity profiles in the downstream direc-
ing flow this separation zone grows too large, and thus no tion are all following the trends presented in classical LES,
lambda shock is visible, which is an unphysical behavior.32 which are slightly overpredicted from the experiment 关Figs.
This highlights the importance of simulating this flow with 14共b兲–14共d兲兴; however, at location X / D = 5 关in Fig. 14共e兲兴 all
the STBL. Using digital filter based turbulent inflow data the the three methods show same profiles. Although like experi-
required thickness 共as in experiment27兲 of STBL is generated; mental uncertainties the CFD results can also be influenced
a good match has been found for the velocity profiles at by the computational errors, high-resolution scheme can re-
X / D = −1.5 position for both the stream-wise and wall- duce the computational errors considerably. The best way
normal velocities. It is noted that with increased resolution ahead would be to analyze the experiment and CFD together
共fine grid兲 the results are getting even better and are tending in order to get a better understanding of the problem under
toward the LES results 关Fig. 14共a兲兴. investigation for which the results shown in Figs. 13 and 14
Downstream velocity profiles are shown in Figs. 13 and are an excellent example.
046103-13 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 14. Mean wall-normal velocity profiles are compared with the experimental and previous LES results on three grid levels. 共a兲 X / D = −1.5 position is just
upstream of the jet plume which is important for comparing the effect of upstream STBL. The downstream position compared is at X / D = 2, 3, 4, and 5 on
the midplane 共Z / D = 0兲 in 共b兲, 共c兲, 共d兲, and 共e兲, respectively.

C. Pressure distributions pressure increases considerably, and from Fig. 11共a兲 it is ob-
Mean pressure distributions normalized by the free- served that this is the location around where the TCRVs start
stream pressure 共P / P⬁兲 on the wall have been analyzed, up- to emerge, which cause an increase in the wall pressure.
stream and downstream the jet injection hole, for three levels In Fig. 16 the mean wall pressure distributions are com-
of grid resolution. The experiment was conducted by Everett pared with the experiment and classical LES at the wall
et al.28 using PSP. Figure 15共a兲 represents the pressure dis- 共Y / D = 0兲 and Z / D = 0, 1 and 2 locations. Figure 16共a兲 shows
tributions on the wall-normal midline plane 共Z / D = 0兲 show- a slight increase in the wall pressure just ahead and decrease
ing very high pressure behind the bow shock, comparatively in the pressure just after the injection hole. Similar results are
low pressure in the upstream recirculation zones 关R1 and R2, found in Figs. 16共b兲 and 16共c兲. However, in Fig. 16共c兲 the
Fig. 7共a兲兴 and very low pressure in the recirculation zone results are slightly under-resolved from the current study at
behind the jet plume 关R3, Fig. 7共a兲兴. Figure 15共b兲 shows wall location Z / D = 2, which could be due to the coarse grid res-
pressure distribution contours 共Y / D = 0兲 indicative of the olution away from the midline plane 共Z / D = 0兲, where the
small rise in the wall pressure just upstream of the jet plume, bow shock thickness is affected, which has results in discrep-
but downstream the wall pressure undergoes a sudden de- ancy in results at Z / D = 2 location. But overall the results in
crease in the recirculation zone R3. A more quantitative Fig. 16 show that the ILES results concur the experimental
analysis of the wall pressure downstream the jet plume and classical LES investigation.
关Fig. 15共c兲兴 enhances this understanding, where normalized
D. Turbulent kinetic energy and Reynolds stresses
wall pressure is plotted at various X / D locations. It is shown
that the wall pressure along the X / D = 1 line on the flat-plate A qualitative analysis has been presented above for the
is very low just behind the jet plume. Moving further turbulent kinetic energy and Reynolds shear-stress. Figures
downstream, the wall pressure at the X / D = 2 lines starts to 17 and 18 present a quantitative perspective into the analysis
gradually increase. Along the X / D = 3, 4 and 5 lines the wall of turbulent kinetic energy and Reynolds shear-stress in the
046103-14 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

E. Flow mixing
Further analysis has been carried out to understand the
mixing of the two fluids. Kawai and Lele32 presented a com-
parison of passive scalar 共jet fluid兲 for the incoming turbulent
flow versus the laminar flow and highlighted the importance
of the turbulent flow for the mixing of the two fluids using
the quantitative analysis of passive scalar and the rms of
passive scalar. Figure 19 shows the contour plots for the
passive scalar and rms of passive scalar for the fine grid at
wall-normal midline plane 共Z / D = 0兲. It is clear from Fig.
19共a兲 that there is a rapid mixing occurring on the windward
side of the jet plume and the jet fluid progressively dilutes
further downstream. This is further understood by analyzing
Figs. 10共b兲 where progressive dilution is clearly seen in the
cross-view planes 共X / D = 1, 2 and 3兲. Figure 19共b兲 shows the
contours of rms of passive scalar which highlighted the same
observation. Kawai and Lele32 highlighted that this progres-
sive mixing is a feature of incoming turbulent flow and is not
present as clearly in the case of incoming laminar flow and
presented a quantitative comparison of these two different
incoming flows. Figure 20 presents the passive scalar distri-
butions at various locations along the X / D axis on the mid-
line 共Z / D = 0兲 plane and compares the results with previous
LES with turbulent incoming flow for the fine grid case.
Similarly, Fig. 21 presents the rms of passive scalar at the
same locations as in Fig. 20 and compares the results with
those from previous LES.

F. Instantaneous flow and instabilities

It has been understood in the discussion above that ma-
jority of the jet fluid is passing through the windward jet
boundary and the Mach disk and maximum mixing of the
two fluids is occurring in this region. In order to understand
what is happening in this region, an instantaneous flow field
analysis is carried out for the pressure in the region in Fig.
FIG. 15. 共Color online兲 Pressure distributions. 共a兲 Midplane 共Z / D = 0兲. 共b兲 22. Figure 22共a兲 presents the time-history of the nondimen-
Wall-normal 共Y / D = 0兲. 共c兲 Pressure profiles plotted on the flat-plate 共Y / D
= 0兲 at various X / D locations. sional pressure data inside the upstream recirculation zone at
point X / D = −0.75, Y / D = 0.5, and Z / D = 0 for a nondimen-
sional time 共␶兲 from 20 to 145. Figure 22共b兲 presents a
zoomed-in version for instantaneous pressure data versus the
nondimensional time 共␶兲 89.5–100 with various locations in
JISC flow field. A comparison of these two has been pre-
time 共shown as a , b , . . . , o兲. Figures 23共a兲–23共o兲 show the
sented with the previous LES 共Ref. 32兲 showing concord
negative instantaneous divergence of the velocity 共line con-
between the two methods. It has been noted earlier that the
tours兲 along with passive scalar 共gray contours at the back-
jet fluid is mainly passing through the windward side of the
ground兲 at each time instance which correspond to the pres-
jet plume and the Mach disk. The TKE plots in Fig. 17 show
that the maximum TKE is present in the region between sure fluctuation points in Fig. 22共b兲.
X / D = 1 and X / D = 3. Also it can be noticed that the majority The instantaneous flow field represents the unsteadiness
of the TKE is present below Y / D = 3, which is the region of in the windward side of the jet plume and the process of the
most of the mixing 共discussed later兲. Similarly, the Reynolds start of Kelvin–Helmholtz 共KH兲 instabilities in this region,
shear-stress plots in Fig. 18 show negative Reynolds shear- whereas the Leeward side of the barrel shock does not show
stresses, which is again representing that the barrel shock on any major instabilities. With the fluctuations in the pressure,
the upstream side of the jet plume is a very unstable region the jet shear layer on the windward side of the jet plume
resulting in better mixing of the fluids. On the Leeward side fluctuates, a local weak shock is developed, and a kink is
and further downstream, the Reynolds shear-stress is positive visible in the jet shear layer 关Fig. 23共a兲兴. This local shock is
pointing to the large recirculation zone and area of less attached to the barrel shock and the pressure difference
mixing. across the local shock is responsible for the kink in barrel
046103-15 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 16. Mean pressure profiles on the flat-plate 共Y / D = 0兲 are compared with the experimental and previous LES results on three grid levels at locations
Z / D = 0, 1, and 2 shown in 共a兲, 共b兲, and 共c兲, respectively.

shock. With time advancement this local shock grows in size shock at the triple point, but the KH instability is clearly seen
and entrains the jet fluid. The velocity near the shear layer is as growing in size even after the local shock disappears. The
higher than the velocity away from it, this difference in ve- KH instabilities result in large scale vortex structures on the
locities results in a KH type instability on the windward side windward and top sides of the jet plume which are rotating
of the barrel shock which also grows as the local shock ad- counterclockwise in Figs. 23共a兲–23共o兲. The large scale vorti-
vances downstream in time. At the junction of barrel shock ces are rotating but also moving in the downstream direction,
and Mach disk, there is a reflected shock, as shown in Fig. which result in ⍀ shaped intermittent circumferential rollers
7共a兲. The local shock grows and merges with the reflected on the windward side, which run around the jet plume on the

FIG. 17. Turbulent kinetic energy 关TKE= 共具u⬘u⬘典 + 具v⬘v⬘典 + 具w⬘w⬘典兲 / U⬁2 兴 profiles are compared with the previous LES results on three grid levels at X / D
= 0, 1, 2, 3, and 4 locations on the midplane 共X / D = 0兲 shown in 共a兲, 共b兲, 共c兲, 共d兲, and 共e兲, respectively.
046103-16 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 18. Reynolds stress 共RS= u⬘v⬘ / U⬁2 兲 profiles are compared with the previous LES results on three grid levels at X / D = 0, 1, 2, 3, and 4 locations on the
midplane 共X / D = 0兲 shown in 共a兲, 共b兲, 共c兲, 共d兲, and 共e兲, respectively.

jet shear layer 共as described in Ref. 18兲 creating an area of

very active mixing. With the turbulence in the free-stream
flow, this mixing activity is enhanced as compared to laminar
free-stream flow.32
On the leeward side of the jet plume, the instantaneous
flow analysis shows that there is very less activity, but still
there is some mixing occurring in this region. As the local
shock develops on the windward side of the barrel shock, it
creates a pressure differential due to which there is an inward
kink on the upstream side of the local shock. At the same
time, at the downstream side of the local shock, the barrel
shock kink develops in the outward direction, shown clearly
in Figs. 23共e兲–23共g兲. As the barrel shock kinks in the out-
ward direction, it pulls the whole jet plume with it resulting
in a very small deflection of the leeward side of barrel shock.
This small disturbance over the time results in fluctuations in
the leeward side of the barrel shock. These fluctuations and
existing low pressure in this region result in some mixing in
this region, shown by light gray contours in Figs.
23共a兲–23共o兲. These fluctuations are the cause of high TKE
zones on the leeward side of the barrel shock, as well as
shown in Figs. 8共c兲 and 9共c兲. Closer to the flat-plate there is
less activity, but away from the flat-plate the fluctuation/
activity increases, and thus gradual increase in the mixing is
shown by increased darkness of gray contours in the leeward
side of the jet plume.
Figure 24共a兲 represents the instantaneous view of the FIG. 19. 共Color online兲 Contours of the jet passive scalar and its corre-
density gradient on the wall-normal midplane 共Z / D = 0兲 sponding rms on the midplane 共Z / D = 0兲.
046103-17 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

showing the generation of the KH instabilities on the wind- explore and understand the JISC flow by detailed analysis of
ward side of the jet plume due to the upstream recirculation the flow features with STBL in order to improve the design
zone and STBL, as discussed in Fig. 23. Figure 24共b兲 shows optimization process of a scramjet combustion chamber, ac-
the entrainment of the jet fluid in the upstream recirculation tive flow control in cavities during supersonic flights, and
and the presence of the jet fluid on the leeward side of barrel thrust control system in rockets.
shock. On the leeward side of the jet plume, the fluctuating In order to achieve these targets, simulations have been
kinetic energy is responsible for mixing of the two fluids and carried out using ILES technique for a sonic transverse jet
Figs. 24共a兲 and 24共b兲 show the fluctuations on the leeward injection into a supersonic cross-flow in order to understand
side of the barrel shock and the mixing, as discussed in Fig. the flow physics, jet penetration, fluid mixing, and turbulent
23. On the online version of this article, Figs. 24共a兲 and kinetic energy. HRM is employing a fifth-order spatially ac-
24共b兲 can be used to access the animations for the density curate modified MUSCL scheme47,52 and an explicit three-
gradient and jet passive scalar on the wall-normal midplane stage second-order accurate SSPRK scheme35,58,66 for time
共Z / D = 0兲 that would help enhance the understanding of the integration. A digital filter based turbulent inflow data
discussion provided in this section. generator33,36,37,60 has been implemented for the simulation
of incoming STBL. The initial conditions were based upon
V. CONCLUSIONS the experiment carried out by Santiago and Dutton,27 which
The objectives of this work were threefold. First, we are the same as used for LES by Kawai and Lele.32 It has
wanted to validate the digital filter based turbulent inflow been demonstrated that the ILES investigations concur the
data generator implemented for the generation of STBL in results obtained from the experiment27,29 and previous LES
the CNS3D code. Second, we aimed to accurately simulate a 共Ref. 32兲 which points to the accuracy of the current meth-
complex high speed flow such as JISC using Godunov type odology. Better understanding of the JISC flow features has
fifth-order spatially accurate MUSCL scheme and an explicit been obtained using the time-averaged and instantaneous
three-stage second-order accurate SSPRK scheme using flow analysis simultaneously. The simulations on the fine
HRMs within the framework of ILES. Finally, we wanted to grid were performed using nearly 1800 core hours, which is

FIG. 20. Jet passive scalar at the midplane 共Z / D = 0兲 at various locations on the X / D axis showing mixing of the two fluids downstream of the jet plume. Plots
at locations X / D⫽1, 2, 3, 4, and 5 are shown in 共a兲, 共b兲, 共c兲, 共d兲, and 共e兲, respectively.
046103-18 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

FIG. 21. rms of jet passive scalar at the midplane 共Z / D = 0兲 at various locations on the X / D axis showing mixing of the two fluids downstream of the jet
plume. Plots at locations X / D⫽1, 2, 3, 4, and 5 areshown in 共a兲, 共b兲, 共c兲, 共d兲, and 共e兲, respectively.

FIG. 22. 共Color online兲 Time history of instantaneous pressure measurement. 共a兲 Pressure plot for dimensionless time 20–145. 共b兲 Focused view of pressure
plot against dimensionless time 89.5–100 shown at various points 共a兲–共o兲, shown on the pressure history plot. These points are described in details in Fig. 23.
046103-19 Transverse jet injection Phys. Fluids 23, 046103 共2011兲

FIG. 23. Instantaneous contours of passive scalar 共background兲 obtained at the midline plane 共Z / D = 0兲 along with the negative divergence of velocity
contours which describe the behavior of the flow at different time instants, as shown by various points in Fig. 22共b兲.

nearly 45 times less computational time compared with the

previous classical LES. It must be mentioned here that these
simulations were carried out to a nondimensional time 共␶兲 of
nearly 150 compared to 100 as in Kawai and Lele.32
The following are the main findings from the work pre-
sented in this article:
共1兲 The HRMs provide a fast method for CFD simulations
involving discontinuities, using which complicated
flows can be captured accurately. JISC is a very complex
flow exhibiting various discontinuities and flow struc-
tures that involve shock/boundary-layer interactions.
共2兲 Incoming STBL plays an important role in enhancing
the mixing of the two fluids as it “stirs-up” the fluids
after jet injection. The momentum in the STBL also acts
to help contain the size of the recirculation zone up-
stream of the jet injection which entrains the jet fluid in
it. This entrained jet fluid in the recirculation zone can
be of help in case of fuel injection as if the temperature
high enough can autoignite and start the combustion
process.
共3兲 KH instabilities are generated on the jet shear layer on
the windward side of the jet plume which produce large
scale eddies, which results in better mixing of the two
FIG. 24. Instantaneous views on the wall-normal midplane 共Z / D = 0兲 of 共a兲 fluids, and mixing process starts rapidly in the thick
density gradient showing the generation of KH instabilities on the windward shear-layer section of the jet plume.
side of the jet plume and fluctuating kinetic energy on the leeward side and
the large scale vortex structures along with various other features of JISC. 共4兲 On the leeward side of the jet plume, slight fluctuations
共b兲 Jet passive scalar showing the entrainment of jet fluid in the upstream in the barrel shock result in some mixing in the recircu-
recirculation zone and mixing on the windward and leeward sides of the lation zone just downstream of the jet plume which can
barrel shock due to KH instabilities and fluctuating kinetic energy, respec-
tively 共enhanced online兲. 关URL: http://dx.doi.org/10.1063/1.3570692.1兴 be helpful in case of fuel 共e.g., hydrogen兲 injection for
关URL: http://dx.doi.org/10.1063/1.3570692.2兴 combustion in the boundary layer.
046103-20 Rana, Thornber, and Drikakis Phys. Fluids 23, 046103 共2011兲

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