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

SYMMETRY-INDUCED HIGH-MOMENT TURBULENT SCALING LAWS OFA SPATIALLY EVOLVING TURBULENT ROUND JET

Cat Tuong Nguyen
Chair of Fluid Dynamics, Department of Mechanical Engineering
Technical University of Darmstadt
64287 Darmstadt, Germany
nguyen@fdy.tu-darmstadt.deMartin Oberlack
Chair of Fluid Dynamics, Department of Mechanical Engineering
Technical University of Darmstadt
64287 Darmstadt, Germany
oberlack@fdy.tu-darmstadt.de

ABSTRACT

variant solutions, which in this context is equivalent to turbulent scaling laws. However, this approach was limited tothe mean velocity, meaning the statistical moment of first order, at that time. This has been extended with Oberlack &Rosteck (2010), where they recognized that the MPMEs admit Lie symmetries that do not appear in either the Euler or
the Navier-Stokes equations and are therefore called statisticalsymmetries. Usually, when dealing with turbulence statistics
the Reynolds decomposition Ui = U i + ui is used where U i denotes the mean velocity and ui the turbulent fluctuations. In the
Reynolds-averaged Navier-Stokes (RANS) model the instantaneous moment UiU j is decomposed into U iU j + ui u j whereui u j is called the Reynolds stress tensor. However, focusing
on the MPMEs based on the instantaneous velocities, an infinite set of linear equations are received where the coupling
between the equations is only between ”neighboring” equations. By introducing Hi j = UiU j and Ri j = ui u j , the omissionof the Reynolds decomposition shall be called the H-approachand the classical approach is called the R-approach. Additionally, the instantaneous moments of an arbitrary order n canbe introduced with Hi{n} and the correlation at one point with
0Hi{n}= Uin . By finding symmetries of the MPMEs, invariant
solutions and therefore turbulent scaling laws can be derived
rigorously. In the work of Sadeghi et al. (2018) and Sadeghi
et al. (2021), this method has been used successfully for temporally evolving turbulent plane jets for not only velocity moments but also passive scalar moments. In addition to that, justrecently in Hoyas et al. (2022) and Oberlack et al. (2022), they
were able to derive turbulent scaling laws with this method formoments of arbitrary order in two regions of a turbulent channel flow and validated them with a new DNS of a Poiseuillechannel flow at a friction Reynolds number of 10000.

A direct numerical simulation (DNS) of a spatially evolving
turbulent round jet has been conducted at Re0 = 3500 in a verylarge box using a fully turbulent pipe flow as an inlet to analyzeclassical and new scaling laws from the near up to the far-field.
The primary object of this study is to deepen our knowledge on
turbulent jet flows employing the symmetry based turbulence
theory using the multi-point moment equations (MPME). With
the DNS data the scaling laws derived from Lie symmetry
analysis are validated and extended up to statistical momentsof arbitrary order n. Further investigations show that when fitting the instantaneous moments to a Gaussian function of thenormalized radius r/z, the prefactors of the exponents show
a linear behavior in n. This suggests the existence of an additional statistical symmetry which, if confirmed, can only befound in the MPME rather than in the Navier-Stokes equations.

Introduction
Until today only a few DNS have been conducted for studying the fundamentals of a spatially developing turbulent roundjet. The earliest work using DNS was done by Boersma et al.
(1998) where the research group studied the effect of inflow
conditions on the self-similarity scaling up to z/D = 45 fora Reynolds number of Re0 = 2400 based on the orifice diameter. In this work, the Navier-Stokes equations are solved
numerically using the Nek5000 code developed by Fischer
et al. (2008). The box of the DNS shall extend axially up toz/D = 75 to properly investigate not only the near-field but
also the far-field behavior of self-preservation. In other roundjet DNS like Boersma et al. (1998), Babu & Mahesh (2004)
and Taub et al. (2013), top-hat profiles with small disturbances
have been used. However, the transition to turbulence of thejet usually takes around 10D downstream of the orifice. That
is why in the present DNS a turbulent pipe is used as an inlet condition to properly capture physics, which to the author’s
knowledge has not yet been reported to study round jets in thecontext of self-similarity.
It might have been in Oberlack (2001), where symmetries
have been used in turbulence for the first time to generate in-Governing equations
The governing equations in this work are the Navier-Stokes
equations for an incompressible flow consisting of the continuity equation∇·U = 01

(1)

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

Figure 1: Schematic view of a round jet flow. Fluid is blown through a nozzle with diameter D. The mean axial centerline
velocity at the orifice is denoted by U 0 and the axial centerline velocity of the jet is denoted by U z,cand the momentum balance equations∂U
1 2+ (U · ∇) U = −∇P +
∇ U.
∂t
Re0

defined as follows
Re0 =

(2)

Ubulk D
ν(3)

where Ubulk refers to the bulk average velocity at the inlet
in streamwise direction, D to the pipe diameter and therefore
also to the diameter of the inlet nozzle and ν to the viscosity.
Re0 = 3500 has been chosen to lie in a range of a fully turbulent pipe flow and still being computationally feasible withthe available computational resources. The cross-section of
the mesh contains 33 cells. Further, the mesh has 30 cells inaxial direction and extends in that direction for 5D. Factoringin the N = 7 GLL points, the pipe mesh has around 2 million
grid points. Therefore we have a 1D × 5D (diameter × axial
length) mesh. The boundary conditions at the wall are no-slip
and impermeable wall while at z = −5D and z = 0 periodicity
has been employed.
To generate a turbulent velocity field in the pipe a small disturbance in all three direction is added to the initial condition ofa laminar pipe profile

with U, P, x, t being respectively velocity vector, pressure, thespatial coordinate and time. Further, the Reynolds number isdefined as Re0 = U0 D/ν with D and U0 being a length andvelocity scale to be defined below.

DNS of the turbulent jet flow
The Navier-Stokes equations (1) and (2) are solved numerically using Nek5000 developed by Fischer et al. (2008).
Nek5000 is a computational fluid dynamics solver based on
spectral element method (SEM), which is a subclass of theweighted residuals method and therefore part of the Galerkinschemes. A thorough description of the numerical method canbe found e.g. in Patera (1984). It uses a weak formulation of
the Navier-Stokes equations and is solved on a hexahedral element mesh. SEM uses piecewise high-order Lagrange polynomials as basis functions similar to the finite element method.
To integrate over each element, the Gauss-Lobatto-Legendre(GLL) integration is used. Besides accuracy, one of the main
reasons for using Nek5000 in this work is the very efficientparallel implementation needed for the large scale DNS conducted. For optimal efficiency the polynomial order N = 7 isused as suggested by Fischer et al. (2008). Further, the BDF2
scheme has been used as a timestepping method. The adaptive
timestep is controlled by the Courant number.

Uz = 6(r2 − R2 ), with R = 0.5.

(4)

Pressure gradient and viscosity are set, such that the bulk velocity is 1 and Re0 = 3500. After running the simulation untila fully developed turbulent pipe flow has been reached, thevelocities at the cross section z = 0 are interpolated onto theinlet of the main computational domain at each timestep. The
main computational domain is a truncated cone with the axial length 75D. At the inlet z/D = 0 the diameter is 4D andat the outlet z/D = 75 the diameter is 64D. This ensures thatthe box is large enough to capture the spreading of the jet andthat there is no interaction with the lateral far-field boundaries.
The main computational mesh consists of two parts: The inner part contains the domain of a pipe and the outer part isa symmetrical cylindrical mesh which are scaled linearly inradial direction when moving axially. In axial direction thecells of both the pipe and the outer part are stretched geometrically with the factor of 1.006 with 234 cells. Around theinner part, the cells are stretched geometrically in transverse
direction with the factor of 1.06 with 25 cells. Considering theN = 7 GLL points, the main computational mesh has around180 thousand cells which amounts to 240 million degrees of
freedom (DOFs). During the simulation, the periodic pipe atz = 0 is then interpolated onto the main computational domain
at z = 0 so that a fully turbulent pipe inflow is achieved for thejet. Figure 2 shows isosurfaces of the q-criterion of the conducted jet flow.

Computational domain of the DNS
Previous round jet DNS such as of Boersma et al. (1998),
Babu & Mahesh (2004) and Taub et al. (2013) have employed
top-hat profiles extended by small velocity disturbances as
inlet profiles. The transition to a fully turbulent behaviorof these jet usually take around 10D downstream of theinlet. For this reason in the present DNS we use a fully
developed turbulent pipe as an inlet condition. Self-similarity
is predicted to occur closer to the inlet and secondly, the inlet
condition is physically more realistic. To accomplish that, thesimulation has been split into two domains. One is that of aperiodic pipe flow to generate the inlet condition and the otheris the main computational domain to capture the turbulent jet
flow.
The presently employed bulk Reynolds number has beenchosen to be Re0 = 3500. The bulk Reynolds number is212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2: A cross section of q-criterion isosurfaces of the conducted jet DNS colored with the velocity magnitude ranging

from 1.5 (red) to 0 (blue).
Table 1: Parameters of the mean axial centerline velocity

15

compared to other experiments.

Present DNS
Boersma et al.
Babu & MaheshTaub et al.

U 0 /Uz,c10

Reference

Re0

Bu

z0

Boersma et al. (1998)

2400

5.9

4.9

Taub et al. (2013)

2000

5.4

1.3

Present

3500

5.29

−1.3

5

Symmetry based scaling laws
0

0

20

40
z/D

The symmetries needed for the derivation of the scaling laws
are derived from the Euler equations and can be transferredto the MPMEs with the limit of Re0 → ∞. The MPMEs arederived by multiplying equation (2) with n − 1 velocities at n −1 different locations xi with a subsequent statistical averaging.
This results in the MPMEs with order n60

Figure 3: Inverse mean axial centerline velocity plotted

over the distance from the orifice. The present DNS at
Re0 = 3500 is compared to various other DNS and experiments.

∂ Hi{n}
∂t



n

+∑
l=1

h
i
∂ Hi{n+1} [i(n) 7→k(n) ] x(n) 7→ x(l)∂ xk(l)Classical scaling laws
In classical theory the mean velocities of a self-preserving jet
are scaled with the centerline velocity

Uz,c (z)Bu D=,z − z0U0−+

2
1 ∂ Hi{n}Re0 ∂ xk(l) ∂ xk(l)∂ Ii{n−1} [l]∂ xi(l)
#
=0

(6)

where

(5)

Hi{n} = Ui(1) (x(1) ) · ... ·Ui(i) (x(i) )where U 0 refers to the mean axial centerline velocity at theorifice, Bu to the decay constant and z0 to the virtual origin.
The inverse of the mean axial centerline velocity compared toother DNS and experiments can be depicted in Figure 3. Qualitatively, it can already be observed that the present DNS begins decaying earlier than any experiment conducted before.
In Table 1 the parameters of the present DNS and other experiments are linked. It can be seen that the decay rate of thepresent DNS is slightly lower compared to the other experiments conducted. Additionally, the virtual origin lies behind
the inlet which is due to early mixing because of the turbulent
inlet.

(7)and Ii{n−1} [l] includes the correlations with the pressure. A
deeper insight into MPMEs can be gained in Oberlack & Rosteck (2010).
In the present work, the focus is set on three symmetries. The
first symmetry is a translation in space

Tx :t ∗ = t,xi∗ = xi + axi ,ri∗ = ri ,∗Ui = Ui ,Hi∗{n} = Hi{n}3(8)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 2: Parameters of equation (15) for each direction

and the second symmetry is a scaling in space and timer, ϕ, z.
Tt :t ∗ = exp(aSt )t,xi∗ = exp(aSx )xi ,∗Ui = exp(aSx − aSt )Ui ,ri∗ = exp(aSx )ri ,Hi∗{n} = exp(n(aSx − aSt ))Hi{n} ,P∗ = exp(2(aSx − aSt ))P.

(9)

The third symmetry is a statistical symmetry, emerging fromthe linearity of the MPMEs reading asTs :t ∗ = t,xi∗ = xi ,ri∗ = ri ,(10)

The symmetries in equation (8)-(10) form the characteristic
condition for a spatially developing turbulent round jet where
subsequently we only consider one-point statistics, i.e. Uin .
Since a spatially round jet is symmetrical in azimuthal direction the characteristic condition gives

dUin
dUi
= ... =.
[aSx − aSt + aSs ]Ui[n(aSx − aSt ) + aSs ]Uinrz + z0e (η) = U i (r, z)Ui(z + z0 )−λ1

(11)

(12c)

fn (η) =UiUin (r, z)n(λ(z + z0 ) 1 −λ2 )−2λ1 +λ2

(12d)

0.133

1.270

z

0.683

2.284

αu,i eβu,i n.
(z + z0 )n(λ2 −λ1 )+2λ1 −λ2

fn (η)
U2z= e−γn η ,
g
Un

(15)

(16)

(17)

z,c

a
a
λ2 = 2 St − Ss − 2.
aSx aSx(13)
The constant n refers to the order of the velocity moments. If
n = 1 or n = 2 are respectively introduced into (12d), the mean
velocity scaling in equation (12b) or the second order velocity
moments in equation (12c) are obtained. The free parameters
λ1 and λ2 are the exponents of the first and second velocity
moment. The self-preservation of three axial moments can beseen in Figure 4.

aSta− Ss − 1,aSx aSx

where, when comparing with (12), (15) and (16), we havegnβu,z . The DNS data up to the tenth axialnUz,c = Cz,n = αu,z e
moment are presented in Figure 6 where, due to a logarithmic scaling, a parabolic exponent of the moments suggests that
equation (17) fits quite well. A very good linear fitting in n ofγn for the axial moments can be taken from Figure 7.
Equation (2) in z-direction can be statistically averaged in its
conservative form and taking the limit of Re0 → ∞ gives∂UrUz ∂Uz2
+
= 0.
∂r
∂z

Validation of the symmetry based scaling laws
In order to validate the symmetry-induced scaling laws of thevelocities we set r = 0 and the equation (12d) needs to be rearranged toUin (r = 0, z) =ϕ

There have been many works e.g. Panchapakesan & Lumley (1993), noticing that the radial profile of the mean axial
velocity (see Figure 4) behaves similar to a Gauss-type profile. However, we have yet to understand where this behavior comes from and if this can be derived from the underlying
equations. Comparing the radial profiles of the higher instantaneous axial moments leads to the conclusion that also thescaled higher axial moments in (12) can be fitted to a Gaussianwhich leads to(12b)

Ui2 (r, z)(z + z0 )−λ2

λ1 =

1.273

Radial scaling of a turbulent round jet(12a)f
Ui2 (η) =

az
,
aSx

0.135

Uin (r = 0, z) =

where
z0 =

r

quite accurately with R2 = 0.99 for all fits. The constants forthe r and ϕ-direction only differ by around 1.5% as seen in Table 2 which is due to symmetry. In conclusion, this means thatthe scaling of all moments of arbitrary order can be determinedwith

By integrating equation (11) the following scaling laws can bederived:η=βu,iCi,n = αu,i enβu,i ,dUiU jdrdz==
aSx z + az
aSx r[2(aSx − aSt ) + aSs ]UiU j=αu,ifn (η = 0) = Ugnwhere Uii,c = Ci,n . The virtual origin z0 = 1.276
can be determined by the spreading rate of the first moment
while λ1 = 1.036 and λ2 = 2.068 can be received by fitting theaxial centerline to the first and second moment in z-direction.
All higher moments determine Ci,n which show the relation as
seen in Figure 5. These constants follow the relation∗

Ui = exp(aSs )Ui ,Hi∗{n} = exp(aSs )Hi{n} .

i

(18)

If then, equations (12) and (17) are implemented in equation(18) for n = 2 and also using a similar scaling law for UrUz we−λ2 , and finally receive
obtain Ug
r Uz (η) = UrUz (r, z)/(z + z0 )Ci,n, for n ≥ 1 & i = r, ϕ, z,(z + z0 )n(λ2 −λ1 )+2λ1 −λ2
(14)



2
∂ Ug
r Uz
Ug
= C1 η λ2 − 2γ2 η 2 e−γ2 η ,r Uz + η
∂η

4

(19)

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

0.5

0

0

0.1

0.2

z = 25z = 35z = 45z = 55z = 65

1

0.5

0

0.3

0

0.1

η

0.2

z = 25z = 35z = 45z = 55z = 65

1
10
Uz10 /Uz,cz = 25z = 35z = 45z = 55z = 652
Uz2 /Uz,cUz /Uz,c

1

0.5

0

0.3

0

0.1

η

(a)

0.2

0.3

η

(b)

(c)

Figure 4: The radial profiles of the (a) 1st , (b) 2nd and (c) 10th axial moment normalized with the scalings in (12).

600
109

γn
γn fitCr
Cr fitCϕCϕ fitCz
Cz fit

107
105
C

400
γn

103
200
101
10−1

0
0

2

4

8

6

10

2

4

8

10

Figure 7: Constants γn from equation (17) are shown for

Figure 5: Constants Ci,n according to the equation (15)each moment up to order n = 10 determined by fitting to
the DNS yielding the following fit: γn = 48.0n + 59.2.

are shown for each moment up to order n = 10 for eachdirection and determined by fitting to the DNS.

n
Uzn /Uz,c

6
n

n

100

leading to the solution

10−3

g

i
2 h
Uz,c
2
Ug
2γ2 η 2 − λ2 + 2 e−γ2 η + (λ2 − 2) ,r Uz (η) =
2γ2 η
(20)
g
2
where C1 = Uz,c and the integration constant C2 = [C1 (λ2 −2)]/(2γ2 ) have been derived from the boundary conditionlimUg
U (η) = 0. Plotting the solution and the DNS data

10−6

η→0

10−9

10−12

r z

of UrUz in Figure 8 one can see, that they behave similarly.
Following up on this, that could mean that there is another statistical symmetry hidden in the MPMEs that we have yet tofind analytically and which could explain the Gaussian behavior of the axial moments.

0.05

0.1

0.15

0.2

η
Conclusion and Outlook
Figure 6: The radial profiles of the 1st (top) up to the 10th

A DNS of a spatially evolving turbulent round jet has been
conducted with a turbulent pipe flow as an inlet at Re0 = 3500.
Contrary to works in the past, a turbulent pipe flow with
Re0 = 3500 has been used as an inlet to shorten the transition region. Further, new scaling laws for a spatially turbulent(bottom) axial moment, normalized with the scaling inequation (12) and shown in a semi-logarithmic plot at
z = 45.
5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
·10−2

4

z = 25z = 35z = 45z = 55z = 65Eq. (20)

3
2
UrUz /Uz,cscaling laws for the PDF functions. Here, it is intended to construct symmetry invariant PDF up to its two-point version andcompare this to the PDF generated from the DNS data.

2

REFERENCES
Babu, P. C. & Mahesh, K. 2004 Upstream entrainment in numerical simulations of spatially evolving round jets. Physics
of Fluids 16 (10), 3699–3705.
Boersma, B. J., Brethouwer, G. & Nieuwstadt, F. T. M. 1998
A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jet. Physics of
Fluids 10 (4), 899–909.
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G. 2008 nek5000
web page.
Hoyas, Sergio, Oberlack, Martin, Alcántara-Ávila, Francisco,
Kraheberger, Stefanie V. & Laux, Jonathan 2022 Wall turbulence at high friction reynolds numbers. Phys. Rev. Fluids
7 (1), 014602.
Oberlack, Martin 2001 A unified approach for symmetries inplane parallel turbulent shear flows. J. Fluid Mech. 427,
299–328.
Oberlack, Martin, Hoyas, Sergio, Kraheberger, Stefanie V.,
Alcántara-Ávila, Francisco & Laux, Jonathan 2022 Turbulence statistics of arbitrary moments of wall-bounded shear
flows: A symmetry approach. Phys. Rev. Lett. 128 (2),024502.
Oberlack, Martin & Rosteck, Andreas 2010 New statisticalsymmetries of the multi-point equations and its importancefor turbulent scaling laws. Discrete & Continuous Dynamical Systems - S 3 (3), 451–471.
Panchapakesan, N. R. & Lumley, J. L. 1993 Turbulence measurements in axisymmetric jets of air and helium. part 1. airjet. 0022-1120 246, 197–223.
Patera, A. T. 1984 A spectral element method for fluid dynamics: Laminar flow in a channel expansion. Journal ofComputational Physics 54 (3), 468–488.
Sadeghi, H., Oberlack, M. & Gauding, M. 2018 On new scaling laws in a temporally evolving turbulent plane jet using lie symmetry analysis and direct numerical simulation.
0022-1120 854, 233–260.
Sadeghi, H., Oberlack, M. & Gauding, M. 2021 New
symmetry-induced scaling laws of passive scalar transport
in turbulent plane jets. 0022-1120 919.
Taub, G. N., Lee, Hyungoo, Balachandar, S. & Sherif, S. A.
2013 A direct numerical simulation study of higher order
statistics in a turbulent round jet. Physics of Fluids 25 (11),115102.

1

0
0

0.1

0.2

0.3

η
Figure 8: The radial profiles of UrUz at different distancesz from the orifice compared to the solution in equation(20).

round jet have been derived for the velocity using Lie symmetry analysis. The scaling laws are then validated by the DNS.
It has been shown that, the velocity moments are dependent of
7 different parameters. By determining the parameters using
the DNS, all moments up to an arbitrary order are determinedby the scaling laws.
Similarly to previous work, the radial moment scaling has been
fitted to a Gaussian distribution. This also works very well forthe higher moments where a linear behavior of the constantin the exponent has been detected. With the assumption ofa Gaussian radial scaling and the invariants derived using Liesymmetry analysis, the radial scaling of UrUz has been derived
which behaves similarly to the DNS data.
The next steps would be to conduct additional DNS with ahigher Reynolds number, at different co-flow rates and different inlet conditions (e.g. synthetic jets) to investigate how thefree parameters of the symmetry based scaling laws are influenced. For that, DNS data is needed which can be generated
by modifying the current DNS setup. Also, the analysis of theradial scaling might suggest the existence of an additional statistical symmetry that can explain the Gaussian distribution.
Therefore, a deeper analysis of the MPMEs should be undertaken regarding the existence of additional symmetries. Additionally, as all symmetries of the moment equations have theircounterpart in the Lundgren-Novikov-Monin probability density function (PDF) approach, we can also construct turbulent

6