Proceedings of the 11th International and 51st National Conference on Fluid Mechanics and Fluid Power (FMFP)

December 21-23, 2024, AMU Aligarh, Uttar Pradesh, India

FMFP2024-04-007
Large Eddy Simulation of Isothermal and Heated Supersonic
Round Jets at high Reynolds number

Asmita M. Rahatgaonkar1,* and Somnath Ghosh2

1
Research Scholar, Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
2
Associate Professor, Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
* Corresponding Author, email: asmita.rahatgaonkar@gmail.com

ABSTRACT 106 and applied Lighthill’s analogy for far field noise calcu-
Large eddy simulations (LES) of spatially developing lations; however, they did not provide quantitative compar-
supersonic isothermal and heated round jets are performed isons with experimental results. Bodony et al.[4] compared
at high Reynolds number using high-order compact finite the noise emitted by a fully expanded jet (Mj = 1.95)
difference schemes. An explicit filtering method based on and an underexpanded jet (Mj = 2.2) using LES without
the approximate deconvolution model (ADM) is used for implementing any shock capturing schemes. They found that
LES. These jets are perfectly expanded (Pj /P∞ = 1) and the underexpanded jet had a longer potential core but lower
the Reynolds number for the isothermal (Tj /T∞ = 1) and axial velocity fluctuation levels as compared to the fully
heated (Tj /T∞ = 2) jets are 5 × 105 and 1.55 × 105 , expanded jet. Moreover, the sound radiation is also stronger
respectively. The isothermal jet has an inflow Mach number for the underexpanded jet indicating presence of shock-
of Mj = 1.8 and for the heated jet, inflow Mach number is associated noise. Lo et al.[5] performed LES to investigate
Mj = 2.2. The mean and turbulent characteristics of these the effect of inflow conditions on mean flow, turbulence
jets are compared with the reference (experimental and intensity levels and far-field noise of unheated supersonic
LES) data and are found to agree well with the behavior (Mj = 1.95) jets. The far-field noise was computed by the
reported in literature. Ffowcs Williams-Hawkings (FWH) surface integral method.
They concluded that reduction of inflow momentum thick-
Keywords: Large eddy simulation, supersonic jet, heated jet ness leads to decrease in the velocity decay rate and lower
turbulence intensities along the jet centerline. Removing the
lower forcing modes consequently led to increase in the
I. INTRODUCTION centerline velocity decay rate, velocity fluctuation levels and
Prediction of noise emitted by heated supersonic jets far-field OASPL.
has been the topic of numerous investigations. In particular, From the above discussed literature it can be noted that,
Senier et al.[1] performed benchmark experiments to under- heated supersonic jets at high Reynolds number have been
stand the effect of temperature (1.124 ≤ Tj /T∞ ≤ 5.422) rarely studied using LES. In this regard, the present study
on Mach wave radiation for hot supersonic jets (1.497 ≤ is focused on large eddy simulations of isothermal (Tj /T∞
Mj ≤ 2). With an increase in jet temperature, Seiner et = 1 and Mj = 1.8) and heated (Tj /T∞ = 2 and Mj = 2.2)
al. observed that higher noise was radiated by the jets, supersonic round jets at a Reynolds number of 5 × 105 and
and the peak noise level shifted downstream, at a higher 1.55 × 105 , respectively, in order to understand the effects of
angle relative to the flow direction. Recently Georgiadis et temperature on mean flow and turbulence statistics of jets.
al.[2] performed experiments for supersonic jets (Mj =
1.36, 1.63 and 2) at multiple temperature ratios and used II. METHODOLOGY
modern nonintrusive diagnostic methods including particle A. Governing equations and discretization schemes
image velocimetry (PIV) for mean and root-mean-square The governing equations (Eqs. 1-5) for compressible flow
(rms) velocity, rotationally resolved Raman spectroscopy to are solved in generalized curvilinear coordinates (cylindrical
acquire mean and rms temperature data, and Background- in the present case) in a characteristic form [6] involving the
oriented schlieren (BOS) to visualize the density gradients primitive variables pressure (p), velocity (u, v and w) and
in the jet flow. They reported that Reynolds stresses increase entropy (s). These equations are solved along with the ideal
with increasing jet temperatures, resulting in faster mixing. gas equation p = ρRT , with constant Cp , Cv and Prandtl
However, with increasing Mach number, turbulence is sup- number (P r = 0.7). The dynamic viscosity is assumed to
pressed as indicated by reduction in jet mixing rate and lower vary with temperature according to the power law: µ ∝ T 0.7 .
turbulence intensity levels. Bulk viscosity effects are neglected. X ± ,Y ± ,Z ± can be
The LES approach has also been used by several re- interpreted
p as acoustic p waves propagating p with velocities
searchers for supersonic jet noise analysis. Mankbadi et u ± g 11 c, v ± g 22 c and w ± g 33 c. X s , Y s , Z s
al.[3] were one of the first to use LES as an investigation are entropy waves traveling with velocities u, v and w
tool for jet noise prediction. They performed LES for a fully while X v,w , Y u,w , Z u,v are vorticity waves traveling with
expanded jet (Mj = 1.5) with Reynolds number of 1.27 × velocities u, v and w. The expressions for X ± , X s , and

1
X v,w are shown in Eqs. 6–9. The viscous stress tensor and The vortex ring-based inflow forcing method of Bogey et
the dissipation rate are given by Eq 10 and 11 respectively. al.[16] is used to promote transition of the initially laminar
l
∂ξ l ∂
g lm = ξ,il ξ,im and ξ,il denote ∂ξ ∂
xi such that ∂xi = ∂xi ∂ξ l ,
shear layer to turbulence. The maximum value of the fluctu-
where xi are Cartesian coordinates. ations added near the inflow is 3.3% of u0 . Characteristics-
ρc based inflow conditions as proposed by Lodato et al.[17]
pt = [X + + X − + Y + + Y − + Z + + Z − ] are used. In this method, incoming characteristic waves are
2 calculated using the known values of primitive variable at
p
+ [st + X s + Y s + Z s ] (1) inflow boundary (equation 14-18). The small variation in
Cv
primitive variable is allowed, to avoid stiff implementation
ξ,i1 ξ,j
l
p
g 11 + ∂τij of boundary condition.
ut = − [X + X − ] − Y u − Z u + (2)
2 ρ ∂ξ l
ρc2 (1 − M 2 )
l
ξ,i2 ξ,j X + = η1 (u − uex ) (15)
p
g 22 + ∂τij
vt = − [Y + Y − ] − X v − Z v + (3) Lx
2 ρ ∂ξ l X − = −X + (16)
ξ,i3 ξ,jl
p
g 33 + ∂τij v c
wt = − [Z + Z − ] − X w − Y w + (4) X = η2 (v − vex ) (17)
2 ρ ∂ξ l Lx
1

∂

∂T
  c
st = −Xs − Ys − Zs + − ξil l − λξil l + Ψ X w = η3 (w − wex ) (18)
ρT ∂ξ ∂ξ Lx
c
(5) X s = η4 (s − sex ) (19)
p p ξ u ξ
Lx
X ± ≡ (u ± g 11 c)[ ± p ] (6) In equation 14-18 subscript ex stands for expected value
ρc g 11
of particular variable at the inflow. η1 , η2 , η3 and η4 are
X s ≡ usξ (7) constants, whose value depend on particular simulation. In
X v ≡ uvξ (8) present study η1 = η2 = η3 = η4 = 3.58. Partially
w non-reflecting boundary condition proposed by Poinsot and
X ≡ uwξ (9)

1
 Lele[18] is applied at the radial outflow. At the outflow in
τij = 2µ sij − skk δij (10) the axial direction, a sponge zone is specified which starts
3 at x/r0 = 55 and ends at x/r0 = 60 where partially non-
 
1 ∂ui ∂uj reflecting condition is applied. The outflow sponge used here
sij = + (11)
2 ∂xj ∂xi is similar to the one used by Sandberg et al.[19]. These
Ψ = τij sij (12) boundary conditions have been successfully implemented by
Thaker and Ghosh[12].
6th order compact central finite difference scheme of Lele[7]
is used to discretize the spatial derivatives. Time integration C. Computational domain and jet parameters
is carried out using a low-storage third-order Runge-Kutta The computational domain is 60r0 × 20r0 in the axial
scheme[8]. An explicit filtering variant of the approximate (x) and radial (r) direction, respectively, where r0 is the
deconvolution method [9] is used for LES. The method given jet radius at inflow. The grid consists of 512 points in the
by Mohseni and Colonius[10] is used for the axis singularity axial direction, 128 points in the azimuthal direction, and
treatment. This formulation has been extensively used for 120 points in the radial directiom. The grid is stretched in
the LES of compressible pipe, nozzle, diffuser flows[11] the radial direction using a sinh function but is uniform in
and compressible round jets[12]. For the heated jet, an the axial and azimuthal directions. The grid has a minimum
adaptive filtering technique developed by Bogey et al.[13] radial spacing of ∆r/r0 ≈ 0.05 (which is the same as that
is implemented for additional filtering in region dominated of Keiderling et al.[14]) at the jet centerline and the spacing
by shocks. increases gradually towards the radial boundary, reaching
a maximum value of ∆r/r0 ≈ 0.5. Details of the flow
B. Inflow and boundary conditions parameters are mentioned in Table 1. The Reynolds number
For the present simulations, a hyperbolic tangent velocity is calculated based on jet diameter (2r0 ) and centerline
profile (Eq. 13) used by Keiderling et al.[14] is specified velocity at the inflow (uj ). The Mach number at inflow is
at the inflow. The remaining two velocity components (v given as
and w) are set to zero at inflow. The momentum thickness p
at inflow is θ0 = 0.05r0 . For the heated jet, temperature Mj = uj / γRTj (20)
profile (Eq. 14) at the inflow is determined from the Crocco- where Tj is the temperature of jet at inflow. The simulations
Busemann relation[15]. are performed over a non-dimensional time of Tsim = tu0 /r0
u 1
 
r0

r
 = 800. The statistics are computed over 300 non-dimensional
= 1 + tanh 1− (13) times.
uj 2 2θ0 r0
   2   III. RESULTS AND DISCUSSION
T T∞ T∞ u γ−1 u u u
= − −1 + 1− An instantaneous snapshot of axial velocity contours
Tj Tj Tj uj 2 aj uj uj along with the pressure fluctuations (P − P∞ ) in an axial-
(14) radial (x-r) plane is shown in Fig. 1 for the isothermal

2
Table 1: Jet parameters for present LES and reference
(LES and experimental) studies

Mj Tj /T∞ Re xc /r0
Isothermal jet 1.8 1 5 × 105 17.4
Heated jet 2.2 2 1.55 × 105 20.4
Seiner et al.[1] 2 1.798 ≈ 1.6 × 106 22.76
Georgiadis et al.[2] 2 1 ≈ 3.4 × 106 22.15
Georgiadis et al.[2] 2 ≈ 1.33 ≈ 6.7 × 106 21.6
Bodony et al.[4] 1.95 1 3.36 × 105 21.69
Lo et al.[5] 1.95 0.568 3.36 × 105 22

Figure 3: Vorticity magnitude in the radial-azimuthal (r-

θ) plane for the isothermal jet at different axial locations.
Contour level range is up to |ω| = 5uj /r0 .

jet. From Fig. 1 it can noted that the simulation is free

from any spurious effects at the boundaries. The potential
core region followed by the fully-turbulent region is shown.
The acoustic waves moving in radially outward direction
are clearly visible. Instantaneous vorticity magnitude for the
isothermal jet is shown in Fig. 2, in an axial-radial (x-r)
plane. The reduction in vorticity and spreading of the jet
in the axial direction is evident in Fig. 2. In Fig. 3, the
Figure 1: Contours of axial velocity in the x-r plane over- vorticity magnitude in radial-azimuthal (r-θ) planes for the
laid upon corresponding pressure field for the isother- isothermal jet is shown at seven different axial locations.
mal jet. The colored contour levels shows u/uj values. At x/r0 = 0, vorticity is concentrated in a circular ring
Gray contour levels show pressure fluctuations within (annular shear layer), and as we move downstream this
−500 ≤ P − P∞ ≤ 500P a. initially laminar shear layer starts to distort. Finally merging
of shear layers can be seen at the end of potential core region
(x/r0 = 17.4). Figure 4 shows instantaneous temperature
contours in an axial-radial (x-r) plane and corresponding
pressure fluctuations (P − P∞ ) for the heated jet. Acoustic
waves are emitted predominantly in the vicinity of the end
of potential core region, and are propagated in the radially
outward direction at a steeper angle (with respect to jet
centerline) as compared to the isothermal jet. Instantaneous
vorticity magnitude for the heated jet is presented in Fig. 5.
When compared to the isothermal jet, the shear layer for the
heated jet develops earlier as can be seen in Fig. 5.
To validate the present LES results, the mean axial
velocity (normalized with the centerline velocity at inflow,
uj ) along the centerline for both, isothermal and heated jets,
are plotted against the axial coordinate (normalized with the
jet radius at inflow, r0 ) along with results from reference
experiments and LES in Fig. 6. Jet flow parameters for the
reference experimental and LES studies are mentioned in
Table 1. In Fig. 6, the reference profiles and the heated jet
Figure 2: Vorticity magnitude in an axial-radial (x-r)
velocity profile (present LES) are shifted in the axial direc-
plane for isothermal jet. Contour level range is up to
tion to match the potential core length with the isothermal
|ω| = 8uj /r0 .
jet (present LES) so that proper comparison can be made.

3
 1.2

0.8

ucl /uj
0.6

0.4

0.2

0
0 10 20 30 40 50 60
x/r0

Figure 6: Centerline axial velocity variation with axial

distance. Isothermal jet ( ), Heated jet ( ), Seiner et
Figure 4: Visualization of instantaneous temperature al.[1] ( ), Georgiadis et al. (isothermal)[2] ( ), Georgiadis
contours in the x-r plane and corresponding pressure et al. (heated)[2] ( ), Bodony et al.[4] ( ) and Lo et
fluctuations (P − P∞ ) in the heated jet near-field. The al.[5] ( )
colored contour levels shows T /T∞ values. Gray contour
levels show pressure fluctuations within −5000 ≤ P − 1.2
P∞ ≤ 5000P a.
1

0.8
u/ucl

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3
r/r0.5

Figure 7: Axial velocity variation in radial direction at

three axial locations. Isothermal jet: x/r0 = 35 ( ),
x/r0 = 40 ( ), x/r0 = 45 ( ) and Heated jet: x/r0 =
Figure 5: Vorticity magnitude in an axial-radial (x-r) 35 ( ), x/r0 = 40 ( ), x/r0 = 45 ( )
plane for heated jet (Tj /T∞ = 2). Contour level range is
up to |ω| = 5uj /r0 .
It can be noted that the heated jet has a longer potential
core than the isothermal jet (due to higher Mach number
For the isothermal jet, it is noted that the velocity decay at inflow) and the centerline velocity decay rate is also
rate is similar to the reference LES (Bodony et al.[4] and higher for the heated jet (refer Fig. 6). Self-similar behavior
Lo et al.[5]). The centerline mean axial velocity decay is of the jets is shown in Fig. 7. The results are plotted at
more rapid in the numerical simulations including the present three different axial locations x/r0 = 35, 40 and 45. The
LES as compared to the experimental data and this trend axial velocity is normalized using the local centerline axial
has also been reported by Lo et al.[5]. The deviation from velocity (ucl ) and the radial coordinate is normalized using
the reference data can be attributed to different Reynolds the jet half-width (r0.5 ). It should be noted that the profiles
number and temperature ratios at the inflow (refer Table 1). nearly collapse onto each other along the entire jet radius
Table 1 shows potential core length, xc (normalized with the for both the jets. A comparison of centerline temperature
jet radius at inflow, r0 ) for present LES and reference cases. variation (normalized with the jet temperature at inflow, Tj )

4
 1.2 0.18

1.1 0.15

u /uj
1 0.12


Tcl /Tj

u]′′
0.9 0.09

′′
q
0.8 0.06

0.7 0.03

0.6 0
0 10 20 30 40 50 60 0 10 20 30 40 50 60
x/r0 x/r0

Figure 8: Centerline temperature variation with axial dis- Figure 10: Variation of RMS axial velocity fluctuations
tance. Heated jet ( ) and Georgiadis et al.(heated)[2] along the jet centerline. Isothermal ( ), Heated jet
() ( ), Georgiadis et al. (heated)[2] ( ), Georgiadis et al.
(isothermal)[2] ( ), Bodony et al.[4] ( ) and Lo et al.[5]
5 ( )

0.15
4
0.125
3
r0.5 /r0

w w /uj

0.1


2
^ ′′

0.075
′′
q

1 0.05

0 0.025
0 10 20 30 40 50 60
x/r0 0
0 10 20 30 40 50 60
Figure 9: Jet half-width variation with axial distance. x/r0
Isothermal jet ( ) and Heated jet ( )
Figure 11: Variation of RMS radial velocity fluctuations
along the jet centerline. Isothermal ( ), heated jet
with reference experimental data (Georgiadis et al.[2]) for ( ), Georgiadis et al. (heated)[2] ( ), Georgiadis et al.
the heated jet is shown in Fig. 8. The heated jet simulation (isothermal)[2] ( ) and and Lo et al.[5] ( )
was perfomed at a higher temperature ratio (Tj /T∞ = 2)
as compared to the reference data (refer Table 1) and hence
a higher centerline temperature decay rate is noted for the fluctuation levels are higher for the heated jet however, the
heated jet. The jet half-width (normalized with the jet radius isothermal jet shows higher radial velocity fluctuations along
at inflow, r0 ) variation in the downstream direction is shown the jet centerline. Overall, the streamwise variation of axial
in Fig. 9. It can be observed that the jet spreading rate and radial velocity fluctuations is similar to the reference
increases with rise in jet temperature. The RMS axial and experiments (Georgiadis et al.[2]) and LES (Lo et al.[5]) for
radial velocity fluctuations along the jet centerline (r/r0 = the heated as well as the isothermal jet.
0) are shown in Fig. 10 and 11 respectively. It is important
to mention that Bodony et al.[4] have performed LES IV. CONCLUSIONS
using inflow momentum thickness 0.09r0 while for present Large eddy simulations of supersonic isothermal (Tj /T∞
LES inflow momentum thickness is 0.05r0 . Thicker inflow = 1, Mj = 1.8) and heated (Tj /T∞ = 2, Mj = 2.2) jets are
momentum thickness results in higher velocity fluctuation performed at high Reynolds number. The initial comparisons
levels[5] as can be seen from Fig. 10. The axial velocity between present LES and reference (both, experiments and

5
LES) data are very encouraging in terms of mean flow [11] Ghosh, S., Sesterhenn, J., and Friedrich, R., “Large-eddy simulation
characteristics as well as turbulence statistics. Furthermore, of supersonic turbulent flow in axisymmetric nozzles and diffusers,”
increasing the jet temperature resulted in rapid jet mixing International Journal of Heat and Fluid Flow, Vol. 29, 2008,
pp. 579–590.
as indicated by higher mean axial velocity decay rate and
[12] Thaker, P. and Ghosh, S., “Large eddy simulation of compressible
higher axial turbulence intensity levels. round jets with coflow,” International Journal of Heat and Fluid
Flow, Vol. 87, 2021.
ACKNOWLEDGEMENTS [13] Bogey, C., De Cacqueray, N., and Bailly, C., “A Shock-Capturing
The computations were performed on the Paramshakti Methodology Based on Adaptative Spatial Filtering for High-
Order Non-Linear Computations,” Journal of Computational Physics,
supercomputing facility at Indian Institute of Technology Vol. 228, No. 5, 2009, pp. 1447–1465.
Kharagpur. The authors acknowledge the support from [14] Keiderling, F., Kleiser, L., and Bogey, C., “Numerical study of eigen-
SERB under the grant CRG/2022/000377. mode forcing effects on jet flow development and noise generation
mechanisms,” Physics of Fluids, Vol. 21, 2009.
[15] Pineau, P. and Bogey, C., “Temperature effects on the genera-
tion of steepened waves by supersonic temporal round jets,” 25th
NOMENCLATURE AIAA/CEAS Aeroacoustics Conference, 2019.
uj Jet centerline velocity at inflow [m/s] [16] Bogey, C., Bailly, C., and Juvé, D., “Noise investigation of a high
subsonic, moderate Reynolds number jet using a compressible large
ucl Local centerline velocity [m/s] eddy simulation,” Theoretical and Computational Fluid Dynamics,
r0 Jet radius at inflow [m] Vol. 16, 2003, pp. 273–297.
r0.5 Jet half-width [m] [17] Lodato, G., Domingo, P., and Vervisch, L., “Three-dimensional
θ0 Momentum thickness at inflow [m] boundary conditions for direct and large-eddy simulation of com-
Pj Jet pressure at inflow [Pa] pressible viscous flows,” Journal of Computational Physics, Vol. 227,
P∞ Ambient pressure [Pa] 2008, pp. 5105–5143.
Tj Jet temperature at inflow [K] [18] Poinsot, T. J. and Lele, S. K., “Boundary conditions for direct
simulations compressible viscous flows,” Journal of Computational
T∞ Ambient temperature [K] Physics, Vol. 101, 1992, pp. 104–129.
Tcl Local centerline temperature [K] [19] Sandberg, R. D., Sandham, N. D., and Suponitsky, V., “Boundary
Re Reynolds number – conditions for direct simulations compressible viscous flows,” Inter-
M
q j Mach number at inflow – national Journal of Heat and Fluid Flow, Vol. 35, 2012, pp. 33–44.
] ′′ ′′
qu u RMS axial velocity fluctuations –
′′
^
w w′′ RMS radial velocity fluctuations –

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