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

TURBULENCE ENHANCEMENT IN BODY FORCE OPPOSED FLOWSStephen D Jackson
Department of mechanical engineering
University of Sheffield
Sir Fredrick Mappin Building, Mappin Street, S1 3JD
sdjackson1@sheffield.ac.uk

Shuisheng He
Department of mechanical engineering
University of Sheffield
Sir Fredrick Mappin Building, Mappin Street, S1 3JD
s.he@sheffield.ac.uk

ABSTRACT

spective on the mechanism by which turbulence is increased.
Petukhov et al. (1988) showed that the effect of buoyancy canbe split into two parts, the external (indirect) effect where thechange in density causes a change to the mean velocity profile
which in turn leads to a change in turbulence production, andthe structural (direct) effect where density fluctuations directly
cause velocity fluctuations. Here we choose to investigate only
the direct effect. In conjunction with the Boussinesq approximation these conditions allows the use of idealised body force
profiles.

Turbulence in vertical pipe flows changes strongly wheninfluenced by non-uniform body forces. A motivating example is buoyancy acting in mixed convection flows. Here we
run Direct Numerical Simulations (DNSs) of flows subjected
to a set of idealised streamwise body forces opposing the main
flow direction. By systematically varying body force profile
shape we report the resulting changes to turbulence characteristics and coherent structures. The conventional understanding is that by introducing an opposing non-uniform body force
whilst maintaining an Equal Flow Rate (EFR), turbulence will
increase. Here we show that by instead maintaining an Equal
Pressure Gradient (EPG), the key turbulence characteristics do
not change. As a result the increased turbulence observed in
the conventional theory can be explained by an increased ’apparent Reynolds number’. On top of this we show how the
turbulent shear stress, and velocity profiles can be easily predicted simply from the non-uniform body force profile and
EPG reference data despite the non-equilibrium nature of theflows.

METHODOLOGYThe DNSs are conducted using an in-house Fortrancode CHAPSim (Seddighi-Moornani (2011), He & Seddighi(2013)) which uses a fractional step method with a second
order central difference scheme for the spatial discritisation.
The temporal terms are discritised with an explicit third order
Runge-Kutta scheme for the non-linear terms, and an implicit
Crank-Nicholson scheme for the linear terms. All cases use apipe flow domain of length 20R where R is the pipe radius.
The non-uniform streamwise body force profiles are chosen based on physical density profiles, with linear profiles
for flows at sub-critical pressure (Groups A and B), and stepchange profiles for those at supercritical pressure (Group C)
which show sudden strong density changes across the pseudocritical temperature (Figure 1). We non-dimensionalise using
the radius R∗ , bulk velocity Ub∗ , density ρ ∗ , and viscosity µ ∗INTRODUCTION
The use of non-uniform body forces, while initially seeming abstract, is a useful generalisation since it can underpin
various physical phenomena. For example there are flow control techniques, such as Kühnen et al. (2018) who applied
pseudo body forces with streamwise fluid injection in order toforce flow laminarisation. Other examples using actual bodyforces occur frequently in magnetohydrodynamics and mixed
convection. Here we focus our attention on opposing nonuniform body forces which act against the main flow direction,
causing increased turbulence. This is particularly important inmixed convection cooling systems where the changes in turbulence may lead to heat transfer enhancement or deterioration. In particular Hall & Jackson (1969), proposed that thebuoyancy causes a redistribution of the shear stress, and in thebuoyancy aided case the shear stress may be reduced to zero in
the near-wall region which causes strong laminarisation whilst
the shear is increased in the buoyancy opposed case leading
to stronger turbulence. Here we aim to provide a new per-so that Re =

ρ ∗Ub∗ R∗
f ∗ R∗2P∗µ ∗ , f = ρ ∗Ub∗2 , P = ρ ∗Ub∗2 .

The body force profiles are chosen such that groups A
and C have increasing coverage, which is to say that the bodyforce acts further into the pipe, whereas group B has an increasing amplitude at the wall. In general all the cases havea base Reynolds number of 2650, with the exception beingcase L2 which has a lower base Reynolds number of 1000, but
the same profile shape as case B2. We also use three physical profiles P1 − P3 extracted from You et al. (2003) temperature profiles, hence decoupling the flow and thermal fields andthus only considering the indirect effect. In order to determine
the strength of each body force profile, we use the integral,1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022(a) ×10−2(b)

ity analysis.

0−1

RESULTS

f −2
−3
−4

(a) u+

B1
B2
L2

A1
A2

(c) ×10−2(b) u′ v ′ ×10−3
6

20

(d)

15

4

0

Reτ =180
P1
P2
P3
B1
C1
L2

10

−1

5

f −2

0

−3

100

−4

(c) k
3

0.0

0.5

1.0

0.0

y

Figure 1.

0.5

0

102

0.0

0.5

y+

P1
P2
P3

C1
C2

101

2

1.0

y
×10−2

1.0

y

2

Opposing non-uniform body force profiles .
1

Table 1.

0

DNS case table.

0.0

0.5

1.0

y

Case

Reτ p

Reτ

F

EFR reference

180

180

0

A1, A2201-229169-174

0.0035-0.0063

B1, B2

208-237

181-182

0.0029-0.0066

C1, C2230-277

113-182

0.0066-0.013

P1, P3

198-260

174-180

0.002-0.011

L2

94.6

75.3470.0066

Figure 2. Velocity profiles, Turbulent shear stress and Turbulent kinetic energy for various cases.

We first show a brief overview of how turbulence characteristics change with different body force profiles. The velocity profiles (Figure 2a) change mildly, with increasing centerline velocity and reduced near wall velocity with the exception of very low coverage cases such as A1. Turbulence
Shear Stress (TSS) in Figure 2b and Turbulence Kinetic Energy (TKE) in Figure 2c show strong increases with increasing
body force strength. Essentially from the conventional viewpoint turbulence increases, which is inline with previous studies.
By comparing the eddy viscosity profiles (Figure 3) we
see from the conventional viewpoint (left) the eddy viscosity
increases with stronger body forces, which is to be expected
since turbulence should be increasing. However, from the alternative perspective (right), the eddy viscosity profiles collapse, except the case L2 which has the lower base Reynolds
number. Even still, the eddy viscosity profiles are virtually
the same as their EPG reference flows, particularly in the nearwall region as indicated in the plot. (Here only the EPG flows
for cases L2, A1, and C2 are shown for simplicity, however
agreement is strong for all cases).
More than this, Figure 4 shows the RMS fluctuations ineach direction using the standard scaling (a, c), and the alternative EPG scaling (b, d). In the usual scaling and compared totheir EFR reference case, the addition of the body force causes
increased turbulence in all directions. (the spanwise fluctuations act similarly to the wall normal fluctuations and so havebeen ommitted from the plot, although sometimes still discussed) There is a small increase of 14% in the streamwise direction but a much larger increase of 41% and 50% in the wall
normal and spanwise directions respectively for the strongest
case P3. The fact that the changes to v′ and w′ are stronger
than u′ suggests that opposing body forces cause the flow to

F = 2 01 r f dr as shown in Table 1.
Here we extend the approach of He et al. (2016) for flow
laminarisation to explain turbuence enhancement. That is, weuse an alternative perspective where the body force influenced
flows are compared to their equal pressure gradient reference
flows rather than their equal flow rate reference flows. In order
to compare flow characteristics in this framework, we use analternative scaling similar to the inner scaling, however while
the usual wall scaling is based upon the wall shear stress, τw ,we instead use the wall shear stress of the EPG flow (or equivalently the pressure gradient of the flow concerned), τwp anddenote this using a ”+1” notation such that
R

y+1 =

y
ν

r

τwp
.
ρ

This perspective has proved useful in various studies buildingon the analysis in He et al. (2016). For example a detailed analysis of the laminarisation mechanisms for supercritical heatedpipe flows using pseudo-body forces from He et al. (2021).
While Marensi et al. (2021) shows how the new perspective
can also give new insight in the critical Reynolds number fortransition / reverse transition in comparison with linear stabil-2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 202220ντ∗ /ν ∗15

A1Reτ =201.
A2
B1
B2
L2
Reτ =94.6
C1
C2
Reτ =277.
P1
P2
P3

10
Reτ =180
A1
A2
B1
B2
L2
C1
C2
P1
P2
P3

5

0
0

25

50

75

y
Figure 3.

100

125

150

0

25

50

+

75

y

100

125

150

+1

Eddy viscosity plotted against y scaled by standard inner scaling (left) and EPG scaling (right).

become more isotropic. The behaviour is different in the EPG
framework, there is very little change to the v′ and w′ fluctuations. For cases P2 and P3, which have strong coverage, there
is a slight reduction of around 12%. These are particularly significant fluctuations in relation to turbulent mixing and hence
heat transfer. In general this means that the turbulence in bodyforce opposed flows tends to act the same as their EPG flows,
and hence can be thought of as acting according to an apparent
Reynolds number which is the Reynolds number of its EPG
flow (Reτ p in Table 1). Interestingly, comparing case B2 to
L2, which have the same non-dimensional body force, but L2
has a lower Reynolds number there is a much smaller reduction in the streamwise velocity fluctuations. This suggests thatfor lower Reynolds number flows there is less deviation awayfrom its corresponding EPG flow.

the fluctuations in Figure 4d, however it is interesting to notethat the eddy viscosity profiles still show good agreement.

(a)

u′+
rms

(b)

u′+1
rms

3.0
2.5
2.5
2.0

2.0

1.5

1.5
1.0

1.0

0.5

0.5
0.0

0.0
0
(c)

50

100

y+

′+
vrms

0

150
(d)

50

100

150

100

150

y +1

′+1
vrms

1.2

The behaviour of the RMS fluctuations can be explainedusing the Reynolds stress budgets. Using case A2 as an example, Figure 5a shows the streamwise Reynolds stress budgets
in the conventional scaling. As body forces are imposed, theproduction term (I) increases as more energy is provided tothe streamwise fluctuations. The pressure strain term (IV ) alsostrongly increases as streamwise streaks break down into v′
and w′ fluctuations and this acts as an energy source for wall
normal fluctuations, increasing all wall normal budget terms asshown in Figure 5c. This is also related to the increased v′ fluctuations in Figure 4c. The story is different in the EPG framework. The production decreases, representing a reduction in
the energy contained in the streamwise fluctuations. However
the key point is that the pressure strain term is exactly the same
suggesting that the body force does not change how the energy
is transferred from streamwise fluctuations into the wall normal and spanwise components. This is shown to be related toa weakening of the streaks, but no significant changes in streak
breakdown. While only case A2 is shown here, the same behaviour is observed in all cases with a low coverage. For rarecases with a larger coverage, such as P2 and P3 the pressurestrain is slightly reduced, corresponding with the behaviour of

0.8

1.0

0.6

0.8
0.6

A1
A2
B1
B2
L2
C1
P1

0.4
0.2
0.0
0

50

P2
P3
Reτ =94.6
Reτ =201.
Reτ =236.
Reτ =180

0.2
0.0

100

y+

0.4

150

0

50

y +1

Figure 4. Root mean square turbulent fluctuations plotted instandard inner scaling ((a) & (c)), and EPG scaling ((b), (d)).

Streaks
In Figure 6 we show correlations of fluctuating velocities in the streamwise and spanwise directions, with the aim
to characterise the streak length and spacing. Figures 6a andb show the spanwise correlations with the minimum points indicated by the markers. In the conventional scaling the streak312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 5. Reynolds stress budget terms in the streamwise direction (a, b), and the wall normal direction (c, d). The terms shown are
Production (I), Dissipation (II), Viscous Diffusion (III), Pressure Strain (IV ), Turbulent Transport (V ), and Pressure Diffusion (V I).
The reference flow is EFR in plots with conventional ’+’ scaling and EPG in the ’+1’ scaling.
spacing reduces by δ (rθ )+ ≈ 20 compared to their EFR reference flow due to the effect of the body force. On the other
hand, in the new scaling the flows the spacings are in generalmore clustered and agree better with their EPG flows. For the
streamwise correlations, Figure 6c shows how the body force
influenced flows become less correlated in a shorter streamwise distance. This, in conjunction with the increased pressurestrain and enhanced v′ and w′ fluctuations strongly suggeststhat the body forces cause stronger streak breakdown. However, using the alternative scaling in Figure 6d the streamwise
correlations collapse to each other and show very little shortening of the streaks compared to their EPG flow. In this sense it is
as if the streaks in the body force influence flow have the samestructure as that of their corresponding EPG flow. However,taking into account the reduction in u′ in Figure 4b, the conclusion is that the streaks become weakened (note the reduced
production), however maintain the same structure as their EPG
flow.

(a)
1.0

By using the fact that whilst maintaining constant pressure gradient the eddy viscosity is unchanged by the additionof a body force (νt = νt p ), where subscript p denotes that it iscorresponding to the EPG flow, the turbulent shear stress andthe velocity profiles can be predicted simply from the EPG reference flow, and the body force profile. Following the methodfrom He et al. (2016), the body force influenced RANS flow,
can be written as,


 

dP
1 ∂
1 ∂ uz0= −+r− u′z u′r + f .
dx p r ∂ r
Re ∂ r(1)

The total flow can be split into its EPG flow, which is often
easily available from reference data

 


1 ∂
1 ∂ (uz ) pdP+r− (u′z u′r ) p
0= −dx p r ∂ r
Re ∂ r

(2)

(b)
A1
A2
B2

0.8
0.6
Ruu

Predictions

A1
A1 epg
A2
A2 epgB2
B2 epgReτ = 180

0.4

and a perturbation flow which is simply the difference between
the body force influenced flow (1) and EPG flow (2). Notingνt ∂ uz
if one applies the eddy viscosity hypothesis u′z u′r = − Re∂r
and using the fact that the pressure gradient and eddy viscosity
are the same between the two flows (νt = νt p ), the perturbation
flow can be written as (1) - (2),

0.2
0.0
−0.2
0
(c)
1.0

20

40 60
δ(rθ)+

100

0
(d)

20

40 60
δ(rθ)+1

A1
A2
B2

0.8
0.6
Ruu

80

80

100

0=
A1
A1 epg
A2
A2 epgB2
B2 epgReτ = 180

0.4
0.0
−0.2
200 400 600 800 1000
δz +

0

(3)

The subscript b is used to represent the perturbation suchthat (u′z u′r )b = u′z u′r − (u′z u′r ) p and (uz )b = uz − (uz ) p . By solving Equation 3 one can obtain (u′z u′r )b and (uz )b , which whencombined with the EPG reference flow can be used to solve
for the total flow.
Figure 7 shows profiles of the turbulent shear stress. In
the usual inner scaling the shear stress increases highlighting
the non-equillibrium nature of the flows which are typicallydifficult to predict. However by using these new ideas the turbulent shear stress of the idealised profiles can be predictedvery well (Figure 7a). The physical cases P1 − P3 are also
shown to have excellent predictions, and the shear stress profiles from You et al. (2003) are also included. In these cases
the flow and energy equations are coupled and the direct effect

0.2

0

 

1 ∂
1 ∂ (uz )b νt p ∂ (uz )b
r
+
+ f.
r ∂r
Re ∂ rRe ∂ r200 400 600 800 1000
δz +1

Figure 6. Spanwise and streamwise correlations taken at awall normal distance of y+(1) = 12. Markers in (a) and (b)indicate the minimum point.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
• The increase in turbulence in the conventional understanding can be explained using an apparent Reynolds
number, which is the Reynolds number of its EPG reference flow.
• Streak breakdown is not enhanced by applying bodyforces whilst maintaining a constant pressure gradient.
• By exploiting the unchanging eddy viscosity, usually difficult to predict quantities such as turbulent shear stress,
velocity profiles and skin friction can be predicted well
simply from reference data and a given body force profile.

is considered. Since the profiles are close, this lends weight tothe idea that the indirect effect dominates in these flows.
Next we use FIK expression introduced by Fukagata et al.
(2002) to determine different dynamical contributions to theskin friction, in particular the laminar contributions and turbulent contributions split into the EPG and perturbation flows
using the decomposition u′z u′r = (u′z u′r )b + (u′z u′r ) p . This gives
the following terms:
1
1
16
+ 16
2r2 (u′r u′z ) p dr + 16
2r2 (u′r u′z )b drRe0|{z} | 0
{z
} |
{z
}

Z

Cf =

Z

i

ii

Z 1

iii

2

+ 16 (r − 1)( f − F)r dr
{z
}
| 0

Acknowledgements

iv

This work makes use of DNS code CHAPSim, which
is currently maintained by CCP-NTH (EP/T026685/1). Also
grateful acknowledgments to Dr W.Wang for insights into
CHAPSim and to M. Falcone and Dr K. Chinembiri for many
useful discussions.

These terms represent different physical contributions,terms i & iv represent the laminar contributions, while terms
ii and & iii are the turbulent contributions from the EPG and
perturbation flows respectively, the contributions are shown in
Figure 7c. The term i does not change between cases, since
they are considered in the EFR framework and have the same
Reynolds number. However each case individually has a different EPG flow and hence different apparent Reynolds numbers. This is linked to the change in term ii where the turbulence shear stress of the EPG flow, (u′r u′z ) p , is changed due tothe EPG acting at a different Reynolds number. By using thepredicted turbulent shear stress shown above, we obtain a prediction of term iii, and hence a prediction of C f . This is shownin Figure 7c (grey dot), agreeing very strongly with the actual
DNS values of C f (black cross).
It should be noted that the body force influenced flow does
not act exactly like its EPG reference flow. The effect of the
body force clearly reduces the streamwise fluctuations, (Figure 4 (d)) which is thought to have links to some aspects ofthe coherent structures, namely a weakening of the streamwise
streaks. Importantly though, this change does not influence theturbulence mixing of the flow. Interestingly, comparing caseB2 to L2, which have the same non-dimensional body force,
but L2 has a lower Reynolds number there is a much smaller
reduction in the streamwise velocity fluctuations. This suggests that for lower Reynolds number flows there is less deviation away from its corresponding EPG flow.

REFERENCES
Fukagata, Koji, Iwamoto, Kaoru & Kasagi, Nobuhide 2002
Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Physics of Fluids 14 (11), L73–
L76.
Hall, WB & Jackson, JD 1969 Laminarization of a turbulent pipe flow by buoyancy forces. ASME-AMER SOC MECHANICAL ENG .
He, J., Tian, R., Jiang, P.X. & He, S. 2021 Turbulence in aheated pipe at supercritical pressure. Journal of Fluid Mechanics 920, A45.
He, S., He, K. & Seddighi, M. 2016 Laminarisation of flow at
low Reynolds number due to streamwise body force. Journal of Fluid Mechanics 809, 31–71.
He, S. & Seddighi, M. 2013 Turbulence in transient channel
flow. Journal of Fluid Mechanics 715, 60–102.
Kühnen, Jakob, Song, Baofang, Scarselli, Davide, Budanur,Nazmi Burak, Riedl, Michael, Willis, Ashley P., Avila,
Marc & Hof, Björn 2018 Destabilizing turbulence in pipe
flow. Nature Physics 14 (4), 386–390, publisher: Springer
US.
Marensi, Elena, He, Shuisheng & Willis, Ashley P. 2021 Suppression of turbulence and travelling waves in a verticalheated pipe. Journal of Fluid Mechanics 919, A17.
Petukhov, BS, Polyakov, AF & Launder, BE 1988 Heat transfer in turbulent mixed convection .
Seddighi-Moornani, Mehdi 2011 Study of Turbulence andWall Shear Stress in Unsteady Flow over Smooth andRough Surfaces p. 236.
You, Jongwoo, Yoo, Jung Y. & Choi, Haecheon 2003 Direct
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CONCLUSIONS• Flows influenced by opposing non-uniform body forces
have much stronger turbulence characteristics, including
eddy viscosity, turbulent shear stress, turbulent kinetic energy. Turbulence also becomes more isotropic.
• In contrast, when the pressure gradient is kept constant,as opposed to the flow rate (used in bullet point 1), theadditional body force does not change the key turbulence
characteristics.

5

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

Figure 7.

Turbulent shear stress profiles and skin friction with their predictions calculated using EPG reference data.

6