12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

TURBULENT DIFFUSION OF SCALAR AND HEAT IN AN OFF-SOURCEHEATED STEADY ROUND JET

Rohit Singhal
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru, Karnataka-560012,India
rohitsinghal@iisc.ac.in

S. Ravichandran
Nordic Institute for Theoretical Physics
KTH Royal Institute of Technology
and Stockholm University
Stockholm, SE-106 91, Sweden
ravichandran@su.se

Sourabh S. Diwan
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru, Karnataka-560012,India
sdiwan@iisc.ac.in

Table 1. Experimental (Exp) and numerical (DNS/LES)
studies of off-source heated jet with their respective locationsof heat injection zone (zb -zt ) and heating numbers G (see text
for definition).

ABSTRACT
We study, using direct numerical simulation (DNS), theproperties of a steady jet heated externally in a heat-injection
zone, the rate of heat addition being proportional to the local
concentration of a scalar. We examine the turbulent diffusivities of the scalar concentration, the temperature and the axial
momentum, and compare their values with those for unheatedjets and plumes. We also include results from the available literature and compare them with our results where possible. We
find that external heat addition decreases the turbulent scalardiffusivity, leading to a decrease in the ratio of scalar to velocity flow widths relative to the unheated jet to a value that islargely insensitive to the precise details of heat addition, expectfor one study. We also find that while the turbulent thermal
diffusivity also decreases, the ratio of temperature to velocity
flow widths is similar to those for unheated plumes. Our results suggest that further studies of off-source heated jets are
warranted.

Study

Type

zb -ztG

Present

DNS

15d-35d

0.37

Bhat & Narasimha (1996)

Exp

133d-213d

0.007-0.16

Venkatakrishnan (1997)

Exp

53d-83d

0.06

Agrawal & Prasad (2004)

Exp

200d-265d

0.2

Agrawal et al. (2005)

DNS

23d-31d

12

Aspden et al. (2017)

LES

10d-14d0.075-0.8

Kewalramani et al. (2022)

LES

30d-42d0.075-0.8

heat addition (as is the case with velocity distribution), andthat the growth rate of the scalar-profile width reduces uponheat addition. The scalar concentration here is analogous to
the water vapour concentration in clouds, which controls therate of condensation-heating.
In this study, we aim to better understand the diffusion of
the scalar concentration and temperature in externally heatedjets. To this end, we perform DNS of an off-source heated jet
and find, in agreement with earlier observations, that the ratioof scalar- to velocity-profile widths reduces upon heat addition. We find, interestingly, that this ratio falls in a narrow
range beyond the heating zone for most of the reported cases,
largely independent of the details of heat addition. Our results
show that the turbulent diffusivity of the scalar (defined below)
decreases upon heat addition, leading to a greater turbulent
Schmidt number, especially within the core of the heated jet.
The turbulent Prandtl number also increases in the core, but isassociated with a qualitatively different radial profile of temperature with peaks away from the jet axis. Lastly, we showthat the acceleration of the flow by off-source heat addition results in the total scalar content (scalar concentration integrated
over the horizontal plane) remaining nearly constant with axial
distance.

INTRODUCTION
Cumulus clouds play a key role in the transport of heat
and water vapour in the Earth’s atmosphere. Yet, the current
understanding of cumulus convection, and in particular the dynamics of entrainment, remains incomplete. The release oflatent heat by the condensation of water vapour in cumulus
clouds makes them qualitatively different from jets and plumes(e.g. Narasimha et al., 2011). Previous studies have examined
the role of such ‘off-source’ heat addition in modifying the dynamics of plumes (e.g. Venkatakrishnan, 1997) and jets (e.g.
Bhat & Narasimha, 1996), finding that external heating fundamentally alters the dilution and diffusion characteristics ofthe flow. Previous experimental, direct numerical simulation(DNS) and Large eddy simulation (LES) studies of steady jetwith off-source heat addition are listed in Table 1.
These studies, and in particular that of Bhat & Narasimha(1996), attempted to characterise the anomalous entrainment
in externally heated flows, and hence focused on the turbulent
diffusion of momentum. They also found that the scalar distribution attains a Gaussian radial profile even with external

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 1. A contour plot showing the passive scalar concentration on a vertical planethrough the centreline of an off-source
heated jet. Heat is added to the flow in theregion zb = 15d < z < zt = 35d.

Figure 2. Axial variation of (a) the centerline velocity Uc , (b) centerline passive scalar
concentration Cc (c) the 1/e width of velocity profile bUe , and (d) the ratio of scalar
to velocity flow widths ξC = bCe /bUe for heated and unheated jets. The dashed lines
represent the extent of the HIZ.

G = 0.37, is obtained with a pre-set parameter gβT αc d 2 /Uo3 (=0.0008) for the simulation reported here, is comparable tothose in earlier studies (see Table 1).

NUMERICAL SIMULATION DETAILSThe governing equations are the Boussinesq-NavierStokes equations, coupled with transport equations for thetemperature and the scalar concentration through the buoyancy
term, as described in Aspden et al. (2017),

The equations (1) are solved using the finite volume
solver Megha-5, which has been extensively used and validated, including in simulations of unheated jet & plume in
Singhal et al. (2021) and dry & moist thermals in Vybhav& Ravichandran (2021). The boundary conditions and othernumerical details are similar to those used in Singhal et al.
(2021) for a canonical jet, except that heat is added externally.
We compare the heated jet with the unheated jet obtained by
setting αc = 0 in equations (1). Hereinafter, we refer to thecanonical round turbulent jet as the “unheated jet” (UJ), andthe flows with heat added externally that we study here as “offsource heated jets” (OSHJ). We report results of the UJ andOSHJ averaged over 1440 and 1120 flow units respectively,
which are sufficiently long for the statistics to converge (seefigures 2, 8, 9, 10 and 11). Averages are performed over asmaller time interval in the OSHJ since the local eddy-turnovertimes (∝ bue /Uc ) are smaller for this case.

∇ · u = 0 , Du/Dt = −∇p + (1/Re)∇2 u + (gβT d/Uo2 )Tez ,DT /Dt = [1/(Re · Pr)]∇2 T + (αC d/Uo )H C,
DC/Dt = [1/(Re · Sc)]∇2C.

(1)

These equations have been nondimensionalised using theorifice diameter (d), the orifice velocity (Uo ) and orifice scalar
concentration (Co ) as the scales. The non-dimensional temperature (T ) at the orifice is chosen to be zero, and only becomes non-zero when the flow is externally heated; the flowis thus entirely momentum-driven at the orifice. The source
term (αC d/Uo )H C, where C represent scalar concentration
field and αC is the maximum heating rate, controls the rate ofheat addition. The Heaviside function H = 1 in the heat injection zone (HIZ, zb = 15d < z < zt = 35d in the present study),
and vanishes elsewhere (see figure 1(a)). The scalar concentration which controls the rate of heating and thus the buoyancygβT d/Uo2 of the flow, where g is gravitational acceleration and
βT is thermal expansion coefficient, is itself advected by theflow. This coupling is responsible for the intricacy of the resulting dynamics.
The flow is governed by the nondimensional Reynolds
number Re = 2400; the Prandtl and Schmidt numbers Pr = 1
and Sc = 1 respectively; and the nondimensional heating rate
which is quantified, as in earlier studies (Bhat & Narasimha,
1996; Aspden et al., 2017), using the “heat-release number”(or the bulk Richardson number)G=

gβT αc d 2
U03



 Z Z
dUo3 1 zt ∞2πrC dr dz,
bbUb3 d 3 zb 0

The axial variation of the characteristic scales in unheatedand heated jets is shown in figure 2. The decay rate of thecenterline velocity Uc , and the spread rate of the width (bue ,defined as u(z, r = bue ) = Uc (z)/e) are found to be 6.18 and
0.114 respectively for UJ and lie within the range observed
in Panchapakesan & Lumley (1993). Similarly, for UJ, thecenterline scalar concentration Cc decays as Cc ∼ 1/z, withthe ratio of scalar to velocity widths ξC = bCe /bue ≈ 1.21, inagreement with Turner (1986); Kaminski et al. (2005). These
results act as further validation of the numerical methods employed here. In the OSHJ, we see (i) a slower decay or mild
increase of Uc (z), (ii) jet width undergo an initial increase followed by a low spread rate compared to UJ, (iii) a faster decay of Cc (z), and (iv) a smaller value ξC relative to the UJ. In
both the UJ and the OSHJ, we find that the radial profiles of
the temporally and azimuthally averaged velocity and scalar
concentration are Gaussian (see figure 8), in agreement withearlier studies, whereas the radial profiles of the temperature
are not Gaussian in beginning of HIZ.

(2)

which is the ratio of total buoyancy added in the HIZ to flow
inertia at zb , with the overbar in C(r, z) denoting temporal andazimuthal averaging. This quantity is an output from the dynamics, and must be determined a posteriori. The value for2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the ratio ξT rapidly decreases upon heat addition, attaining a
final value ξT ≈ 0.92 < 1. Note that this value falls within
the range ξT ∈ [0.88, 1.07], typically found in the self-similar
region of a (classical) plume (Kaminski et al., 2005). Interestingly, the behaviour of ξT for Kewalramani et al. (2022)
matches well with that for others unlike that of ξC . WhileξC and ξT quantities are available in Agrawal et al. (2005)(see their figure 8), they are not plotted here in figure 3. Thisis due to very large heating number, G = 12 (denoted as Ri∗
in Agrawal et al. (2005)), which makes it difficult to directlycompare their results with those of others.

Figure 3. (a) The ratio of scalar to velocity flow widths ξC(see text for explanation). (b) The ratio of temperature to velocity flow widths, ξT . (c) The deviation ∆ of the temperature
profile from the self-similar Gaussian profile. The horizontal
axis in each graph is the scaled axial length in which 0 < z̃ < 1is the heat-injection zone.

Figure 4. (a) The turbulent Schmidt number Sct in the UJ,
and (b, c) Sct and the turbulent Prandtl number Prt in the OSHJ
on a vertical plane containing the flow axis.

The profiles of T peak away from the jet cenerline at the
beginning of the HIZ (shown in figure 8(c), and also seen in
Aspden et al. (2017); Kewalramani et al. (2022)). In figure3(c), we quantify the RMS deviation ∆ from the self-similar
Gaussian profile, showing that the OSHJ becomes self-similar(in its first-order statistics) for z̃ > z̃ss ≈ 0.5 (subscript “ss”
for self-similar). We again see that z̃ss tends to decrease as
G increases. From figures 3(a-c), we see that the width ratios
ξC,T reach their steady post-HIZ values nearly at the same axial location where the temperature profiles become Gaussianand self-similar. Thus z̃ss represents the location of onset ofanother regime of self-similar behaviour after heat addition,
especially with regard to the scalar and temperature distributions. We also see that while some properties of the OSHJ,
such as the ratio ξT (here) and the entrainment coefficient (Aspden et al., 2017) are similar to those of a classical plume,other features such as the turbulent Prandtl number Prt are distinct from those in a classical plume (as discussed below).
The turbulent transport of scalar concentration and temperature are quantified using the Schmidt and Prandtl numbers,
Sct = νt /Dt , and Prt = νt /αt , where νt , Dt and αt are the turbulent diffusivities of axial momentum, scalar concentrationand temperature:

RESULTS AND DISCUSSION
A prominent feature of heated jets is the reduction of thegrowth rate of the scalar flow-width bCe . We observe, in agreement with earlier studies, that off-source heat addition leads toa reduction in the ratio ξC . In figure 3(a), we plot the variation of ξC with the scaled axial distance z̃ = (z − zb )/(zt − zb ).
In agreement with previous studies (except Kewalramani et al.
(2022)), we see that ξC ≈ 1.2 for z̃ < 0, and rapidly drops toξC ≈ 0.83 at z̃ ≈ 0.5. We see by comparing the data from Bhat
& Narasimha (1996) and Aspden et al. (2017) with the present
OSHJ (figure 3(a)) that the final value of the width ratio attained is ξC ≈ 0.83. The location zb of the base of the HIZ, its
extent, and the amount of heat added are quite different in thesestudies (see Table 1). In view of this, the observation that approximately the same value of ξC ≈ 0.83 is realised after heat
addition is interesting, although its cause is not immediately
clear. We may also infer that for larger heating rates (larger
G), the scaled distance at which ξC reaches its final value decreases. Thus, in Aspden et al. (2017) & Bhat & Narasimha(1996), the value ξC ≈ 0.83 is attained above the HIZ (z > zt ).
In contrast, in the present OSHJ study, we see that ξC briefly
decreases to a lower value before reaching its final value atz̃ ≈ 0.5. We also note that the numerical value ξC = 0.83 isapproximately the inverse of the value ξC,UJ ≈ 1.21 in UJs.
For the LES of Kewalramani et al. (2022), for G = 0.075,
ξC ≈ 0.83 is realized for z̃ > 1 but for the higher heating rates
ξC reaches a value less than 0.8 (figure 3(a)). It is worth notingthat for Kewalramani et al. (2022), ξC < 1 for z̃ = 0 which is asomewhat unexpected behaviour.
The analogous ratio ξT = bTe /bUe for the temperature isplotted in figure 3(b), and compared against data from Aspden
et al. (2017) and Kewalramani et al. (2022). Again, we see that′νt = −

′

′
′
′
′
ur uz
uC
uT
; Dt = − r; αt = − r.
∂ uz /∂ r∂ T /∂ r∂C/∂ r(3)

The correlations of fluctuating quantities in equation (3)are averaged temporally and azimuthally, and their normalizedradial variations plotted later in figure 9, 10 and 11, respectively. The effect of off-source heating on the Sct is shownin figure 4, where we see that Sct for the OSHJ is larger thanthat for the UJ in the core of the jet for z ≳ zt , and decays
more rapidly with the radius for r ≳ bUe . Thus in comparison

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 7. (a) Total scalar flux and (b) total scalar present at
different locations in unheated and off-source heated jet.

z ∈ (20, 40) and r/bUe ∈ (0.42, 1.41), comparable with thevalue of 0.72 found by Reeuwijk et al. (2016). The radial
range 0.42 < r/bUe < 1.41, however, does not capture the behaviour of Sct for z ≳ zt seen in figure 4(b). We therefore
average over a narrower radial range 0.35 < r/bUe < 1.17, obtained by multiplying the earlier range by a factor of ξC = 0.83.
This curve, also shown in figure 5(b), better reflects the behaviour seen in figure 4(b). Next, we plot Prt for the OSHJ
in figure 5(b), where we find that Prt takes on large values
for z ≈ zb before reaching a steady value approximately equal
to the value of Sct . This value, Prt > 0.75 near z ≈ zt , islarger than the range of values Prt,plume ∈ (0.62, 0.67) found insteady turbulent plumes (Wang & Law, 2002; Reeuwijk et al.,2016), although the ratio ξT of temperature to velocity widths
is similar in OSHJs and plumes, as seen earlier (figure 3b).
The radial profiles of the azimuthally, axially and temporally averaged Sct for the OSHJ are compared with the Sct
for a UJ in figure 6(a). The large deviations close to the axis
(the grey semi-transparent region; r/bUe < 0.2 ) are due tothe smaller number of grid points available and due to thesmall numerical values of the radial derivative ∂ /∂ r. For
r/bUe > 0.2, the variations are smoother; the small wiggles
observed, which are also seen for the curve of Reeuwijk et al.
(2016), are due to the sensitive dependence of ∂C/∂ r on the
precise radial variation of C and not due to inadequate statistical averaging. The averages in the OSHJ are performed
over an axial range 32d < z < 43d where the Sct in the coreof OSHJ is largest (see figure 4). We find that, in the interval
0.2 < r/bUe ≲ 0.8, Sct is larger for the OSHJ than for the UJ.
We also see a steeper decrease in Sct in the OSHJ away fromthe axis. To examine why Sct is larger in the OSHJ, we haveplotted normalized νt and Dt separately in figure 6(b), showing that heat addition reduces Dt of the scalar in the region
r/bUe ≲ 0.8, while νt is much less affected. The lower valueof Dt is responsible for the reduction of ξC in the OSHJ. Ev′′
idently, the larger turbulent radial flux urC in the OSHJ (seefigure 10(b)), is offset by the large gradient ∂ C̄/∂ r.
Lastly, we discuss total scalar flux and total scalar mass(mC (z)) per unit axial distance available at different z locations, quantified using the Gaussian profile parameters

Figure 5. (a) Sct for the UJ and OSHJ. In the OSHJ, the radial
averages are also performed over a smaller radial extent since
we know from figure 4 that the scalar flow-width is smaller (b)Prt and Sct in the OSHJ.

Figure 6. (a) The radial variation of the turbulent Schmidt
number in the self-similar region of the UJ and the post-HIZ
region (32d −43d) in the OSHJ. (b)The turbulent diffusivity ofaxial velocity (νt ) and scalar (Dt ) for the UJ and OSHJ. In both(a) and (b), azimuthal averages of Dt , νt and Sct are unreliablein the shaded region (r/bUe < 0.2) near the jet’s axis.

with UJ, heat addition in OSHJ produces two different regionswith a large difference in Sct (see also figure 6(a)). The region
close to axis with higher Sct , the ‘core’, could have qualitatively different features with respect to the region near edge
as suggested in the Bhat & Narasimha (1996). They also observed an enhanced mixing within the core of the heated jet.
The implications of our results towards this observation is a
topic of future work.
We plot the axial profiles of the radially and temporally
averaged Sct for the unheated and heated jet in figure 5(a).
For UJ, we find that Sct has an average value of 0.69 over

Z ∞
0

b2 b2
ξC2
2
2πruCdr = 2πCcUc 2 Ce Ue2 = 2πCcUc bUe;
bCe + bUeξC2 + 1Z ∞and mc (z) =

4

0

2
2
2πrCdr = πCc bCe= πCc bUe
ξC2 .

(4)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Kewalramani, G, Pant, C S & Bhattacharya, A 2022 Energy
consistent gaussian integral model for jet with off-source
heating. Phys. Rev. Fluids 7 (1), 013801.
Narasimha, R, Diwan, S S, Duvvuri, S, Sreenivas, KR &Bhat, GS 2011 Laboratory simulations show diabatic heating drives cumulus-cloud evolution and entrainment. PNAS
108 (39), 16164–16169.
Panchapakesan, N R & Lumley, J L 1993 Turbulence measurements in axisymmetric jets of air and helium. part 1. air jet.
J. Fluid Mech. 246, 197–223.
Reeuwijk, M V, Salizzoni, P, Hunt, G R & Craske, J 2016
Turbulent transport and entrainment in jets and plumes: a
dns study. Phys. Rev. Fluids 1 (7), 074301.
Singhal, R, Ravichandran, S, Diwan, S S & Brown, G L 2021
Reynolds Stress Gradient and Vorticity Fluxes in Axisymmetric Turbulent Jet and Plume. In Proceedings of 16th
Asian Congress of Fluid Mech. (ed. L. Venkatakrishnan etal.). Bangalore.
Turner, J S 1986 Turbulent entrainment: the development of
the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.
Venkatakrishnan, L 1997 Development of a plume with offsource volumetric heating. PhD thesis.
Vybhav, GR & Ravichandran, S 2021 Entrainment in dry andmoist thermals. Phys. Rev. Fluids (in press) .
Wang, H & Law, A W 2002 Second-order integral model for around turbulent buoyant jet. J. Fluid Mech. 459, 397–428.

In the fully developed unheated jet, the total scalar flux
is axially invariant and the total scalar mass increases linearlywith axial distance, as expected from the relations Uc ,Cc ∝1/z & be ∝ z, and shown in figure 7. The external addition ofheat changes these profiles. From the radial balance of scalar
concentration, which reads
d
dz

Z ∞
0



Z ∞
d
′
′
ruCdr +
ruzC dr = 0.
dz 0

(5)

′

′

we see that the behaviour of the turbulent scalar flux uzC determines the profile of the mean scalar flux. Heat addition′′
leads to positive values of ∂ uzC /∂ z in the HIZ (see figure 10),and thus a reduction in the scalar flux (see figure 7(a)), as wellas ξC (figure 3(b)). The total scalar mass mC (z) in the OSHJ isconstant with axialR distance, despite the increase of the mass
flow rate (mu (z) = 0∞ 2πrudr) (see Bhat & Narasimha (1996);
Aspden et al. (2017)); the decaying centreline scalar concentration and increasing bCe (figure 2) evidently offset each other.
The relationship between the entrainment and the scalar dilution in an OSHJ can have direct implications for cumulus cloudflows. The connection of the staistical behaviour reported herewith the coherent motions in the off-source heated jet is another fruitful line of investigation worth pursuing.

CONCLUSION
In summary, our DNS results suggest that the OSHJ attains a nearly self-similar state, some distance above zb , in thecore of which the turbulent Schmid number Sct is larger, andfor which the ratio ξC lower, than in an unheated jet. The turbulent Prandtl number Prt in this self-similar heated jet is also
larger, but does not seem to lead to a concomitant decrease in
the ratio ξT beyond the heating zone. The flow thus has a coreannular structure, with Prt and Sct attaining greater values, anddecaying more rapidly with radius, than in an unheated jet.
We also found that the linear growth in the total passive scalarcontent with axial distance is arrested by heat addition. These
differences and their possible implications for cumulus clouddynamics make the off-source heated jet a valuable model for
future study.
S.S.D. acknowledges support from IISc Bengaluru towards running the simulations at its supercomputing facility.
S.R. is supported under Swedish Research Council grant no.
638-2013-9243. We thank the reviewers for their useful comments.

REFERENCES
Agrawal, A, Boersma, B-J & Prasad, A K 2005 Direct numerical simulation of a turbulent axisymmetric jet with buoyancy induced acceleration. Flow, turbulence and combustion 73 (3-4), 277–305.
Agrawal, A & Prasad, A K 2004 Evolution of a turbulent jet
subjected to volumetric heating. J. Fluid Mech. 511, 95–
123.
Aspden, AJ, Nikiforakis, N, Bell, JB & Dalziel, S B 2017 Turbulent jets with off-source heating. J. Fluid Mech. 824, 766–
784.
Bhat, GS & Narasimha, R 1996 A volumetrically heatedjet: large-eddy structure and entrainment characteristics. J.
Fluid Mech. 325, 303–330.
Kaminski, E, Tait, S & Carazzo, G 2005 Turbulent entrainment
in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–
376.

Figure 8. Radial profile of first order statistics of field variables: (a) axial velocity (u), (b) scalar concentration (C), and(c) Temperature (T ). The arrow sign in (c) represents the evolution of temperature profile with respect to downstream axial
direction.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

′

′

′

′

Figure 9. (a) Radial profiles of uz2 (b) axial variation of center-line axial turbulent intensity uz2 , and (c) radial profiles of uz ur in
unheated and off-source heated jet. In OSHJ case, we see decrement in velocity correlations with a later recovery in axial distance;
direction of increment or decrement is shown by arrows.

Figure 11. Radial self-similar profiles of second-order quan′′′′′tities (a) uz T , (b) ur T and (c) T 2 in unheated and off-source
heated jet. The arrow sign represents the evolution of profiles
with respect to downstream axial direction. Here, the negative
correlation signifies that low velocity regions undergo a higher
heating (hence, temperature).

Figure 10. Radial self-similar profiles of second-order quan′′′′′tities (a) uzC , (b) urC and (c) C 2 in unheated and off-source
heated jet. Unheated jet shows collapse of all the profile to aself-similar curve. In OSHJ case, we can see an increase invelocity scalar correlation with increase in axial distance after
starting of HIZ; direction of increase is shown by arrow on red
curves.

6