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

SYMMETRY-BASED EDDY-VISCOSITY MODELS: A HANDS-ON
APPROACH

Dario Klingenberg
Chair of Fluid Dynamics
Technical University Darmstadt
Otto-Berndt-Straße 2, 64287 Darmstadt
klingenberg@fdy.tu-darmstadt.deMartin Oberlack
Chair of Fluid Dynamics
Technical University Darmstadt
Otto-Berndt-Straße 2, 64287 Darmstadt
oberlack@fdy.tu-darmstadt.de

ABSTRACT

where Ui stands for the velocity, P the pressure divided by thedensity, t and xi are temporal and spatial coordinates, respectively, and ν is the kinematic viscosity. However, despite theadvent of supercomputers, a full solution of (1) remains too
computationally expensive for most real-life applications, andthe need for efficient and accurate turbulence models still remains. RANS models occupy an important place in the spectrum of turbulence models, because they are computationallyefficient while often being accurate enough. Their starting
point are the RANS equations, which can be derived from averaging (1) and readIn the present work, we revisit the classical problem of
Reynolds-Averaged Navier–Stokes (RANS) modeling by taking into account additional constraints arising from the Liesymmetries that govern turbulence. Symmetries are definedas variable transformations that when inserted into an equation leave this equation form invariant. Even though originally a mathematical concept, it turns out that in the context of equations describing physical phenomena, symmetries
usually carry physical meaning themselves. Generally speaking, it is desirable to incorporate this physical meaning intoa model, so that symmetries can be used to infer constraints
on model equations in a mathematically rigorous and algorithmic manner. The symmetries that are important for turbulence
fall into two categories: First, classical symmetries, which areobserved throughout classical mechanics and are found in theoriginal Navier–Stokes equations as well as any full statisticaldescription of turbulence, and, second, statistical symmetries,
which are only observed once a statistical view of turbulence
is adopted, and, therefore, cannot be found in the unaveraged
Navier–Stokes equations. Since the implications of the classical symmetries are fairly trivial, they are usually accounted
for in complete turbulence models. The statistical symmetries,
on the other hand, are a relatively recent development, and are
generally overlooked in turbulence modeling. The goal of thiswork is to present the development of a modified version of
the k-ε-model, which hopefully illustrates the steps needed tobuild these additional symmetry constraints into a model. First
results show a promising performance of the modified model.

∂ Ūi ∂ Hi j
∂ P̄∂ 2Ūi
∂ Ūi
= 0,
+
=−
+ν
∂ xi
∂t
∂xj
∂ xi
∂ x j∂ x j(2)Here, the bar denotes averaging, and Hi j = UiU j . The velocity
and pressure are usually decomposed into a mean and a fluctuating value (Reynolds, 1895),

Ui = Ūi + ui , P = P̄ + p

(3)

∂ Ūi
DŪi ∂ Ūi∂ P̄∂ 2Ūi∂ Ri j=+ Ū j
=−
+ν
−
Dt
∂t
∂xj
∂ xi
∂ x j∂ x j ∂ x j(4)leading towhere the new unknown term Ri j = ui u j , also known as the
Reynolds stress tensor, arises. For this unknown term, an exact
equation can be derived, however, this equation again contains
unknown correlations. This is known as the famous closure
problem of turbulence. In order to obtain a solution to (4),empirical closure relations, i.e. turbulence models, have to be
introduced at some level. Presently, we consider the simplest
closure of eddy-viscosity models, which rely on inserting amodel for the unknown Ri j in (4). If we denote this model with

INTRODUCTION
The best known turbulence model for incompressible flow
is given by the Navier–Stokes equations

DUi ∂Ui
∂Ui
∂P
∂ 2Ui
∂Ui
= 0,
=
+U j
=−
+ν
= 0 (1)
∂ xi
Dt
∂t
∂xj
∂ xi
∂ x j∂ x j

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
This paper is structured as follows: We first give a brief
introduction to Lie-symmetries and discuss the relevant symmetries of turbulence. Then, we discuss how conclusions fromLie-symmetry analysis allow enhancing existing turbulencemodels, and show some results of simple flows obtained witha modified version of the k-ε-model. We end with a short conclusion.

R̃i j to highlight its approximate nature, (4) as used in eddyviscosity models reads
DŪi ∂ Ūi
∂ Ūi
∂ P̄
∂ 2Ūi∂ R̃i j=+ Ū j
=−
+ν
−
Dt
∂t
∂xj
∂ xi
∂ x j∂ x j ∂ x j(5)

The simplest model for R̃i j is given by the Boussinesq approximation (Boussinesq, 1877)

R̃i j = −νt∂ Ūi ∂ Ū j+
∂ x j ∂ xi!

2
+ kδi j3

LIE-SYMMETRIES OF TURBULENCE
In the present context, symmetries are defined as transformations that leave a given equation such as (1) form invariant.
Note the analogy to geometrical symmetries, which can be defined as geometric transformations (e.g. rotation) that leave geometric objects (e.g. a circle) invariant. The symmetries of theunaveraged Navier–Stokes equations (1) were first calculatedby (Bytev, 1972) and, assuming ν = 0, read(6)

where the task of modeling six independent components of the
Reynolds stress tensor is reduced to modeling a turbulent (oreddy) viscosity νt and the turbulent kinetic energy k.
In this work, we restrict our attention to the classical k-εmodel (Jones & Launder, 1972), though the method can easily
be applied to other two-equation models as well. Note, however, that the application to Reynolds-stress models (RSMs)
is more complicated, so that we exclude this class of models
from the present discussion.
The classical k-ε-model solves a transport equation for k,which is based on the exact equation that can be derived forthe turbulent kinetic energy and reads

Tt :Trotα :



∂k
νt
ν+
σk ∂ x j

xi∗ = xi

Ui∗ = Ui ,P∗ = P∗[α]xi∗ = x j Qi jt = t,[α]

TGali :

Ui∗ = U j Qi j ,P∗ = P∗xi∗ = xi + fGali (t)′′P∗ = P − x j fGal(t)j∗xi = xiP∗ = P + fP (t)xi∗ = xi eaSc,I∗
2aSc,It = t,′
Ui∗ = Ui + fGal
(t),
i

TP :
Dk
∂ Ūi
∂
= −R̃i j
−ε +
Dt
∂xj
∂xj

t ∗ = t + aT ,∗t = t,
Ui∗ = Ui ,!
(7)

TSc,I :t ∗ = t,Ui∗ = Ui eaSc,I ,TSc,II :Herein, the first term on the right-hand side (the productionterm) appears in closed form, whereas the second one (dissipation) and the last one (diffusion) require modeling. For
the diffusion term, a simple gradient-diffusion hypothesis has
been introduced in the development of (7), and the dissipation
is obtained from the transport equation

P = Pet ∗ = teaSc,II ,xi∗ = xi

Ui∗ = Ui e−aSc,II ,P∗ = Pe−2aSc,II



1
0
0
Q = 0 cos arot1 − sin arot1 
0 sin arot1 cos arot1

cos arot2 0 − sin arot210Q[2] =  0
sin arot2 0 cos arot2

cos arot3 sin arot3 0
Q[3] = − sin arot3 cos arot3 0
0
0
1

(8)

which is entirely empirical and whose structure is based on(7). The constants σk ,Cε,1 ,Cε,2 and σε are model parameters,
and establishing appropriate values for them is accomplishedby calibrating the model against a number of canonical flows.
Apart from entering (7), ε also serves as a scale-providing
variable that is needed to formulate a dimensionally correct
expression for νt . The k-ε-model usesk2νt = Cµ
ε

(11)
(12)
(13)
(14)
(15)

where the constant rotational matrices Q[α] are given by

[1]

Dε
∂ Ūi
ε
ε2
= −Cε,1 R̃i j
−Cε,2
Dt
k
∂xj
k
!


∂
∂ε
νt
+
ν+
∂xj
σε ∂ x j

(10)

(16)

(17)

(18)

respectively. The ai stand for arbitrary real-valued constants,
and fP (t) and fGali (t) are free functions of time. Noticethat in the case of nonvanishing viscosity, due to a so called
symmetry-breaking effect of ν, (14) and (15) combine toTSc,ns : t ∗ = te2aSc,ns , xi∗ = xi eaSc,ns , Ui∗ = Ui e−aSc,ns ,(9)P∗ = Pe−2aSc,ns

where the additional model parameter Cµ appears.
A major issue associated with the k-ε-model and similar
models is their lack of universality, i.e. when simulating a new
flow problem, it is not generally clear if a given model will be
sufficiently accurate. In the following, we explore how we canaddress this issue using symmetry constraints.

(19)

The symmetries (10)–(15) and (19), which we also refer to asclassical symmetries, each have an intuitive physical interpretation. The time translation symmetry (10) is connected to theprinciple that there is no absolute origin in time, so that time
shifts cannot have an impact on physical processes. The analogous concept for the spatial coordinates is encoded in (12) if

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
we assume the special case fGali (t) = const. Equation (11) is
the rotational symmetry, which encodes the principle that theorientation of the coordinate system must be arbitrary. Thisis directly connected to the principle of correct tensor formulation. Note that the similar transformation of time-dependent
rotation, i.e. (11) with arotα = arotα (t), is not a symmetry of theNavier–Stokes equations, because inertial effects do play a role
here. The generalized Galileian symmetry (12) is also well′ (t) = const, this symknown. If we restrict ourselves to fGalimetry essentially states that a linear movement at a constantvelocity cannot affect physical processes, a principle foundthroughout classical mechanics. For the special case of incompressible flow, we may even allow acceleration, because
its effect can be absorbed by the pressure field. Another symmetry that is specific to incompressible flow is given by (13),which states that the absolute value of the pressure does notmatter, but only spatial pressure differences. Finally, the scaling symmetries (14) and (15) (or (19) in the viscous case) areconnected to the principle of dimensional correctness. Thisconnection becomes clear if one realizes that changing the system of measurement, which must obviously not have any effect on the equations, is nothing but a particular rescaling of
the appearing variables.
So far, nothing particularly surprising has been revealedby this symmetry analysis. In fact, the principles encoded by
the symmetries (10)–(15) and (19) are essentially equivalent
to the principles of invariant modeling (Donaldson & Rosenbaum, 1968). Roughly speaking, two-equation models andmore complex ones fulfill all of these symmetries. Only simple special-purpose models, such as the mixing length model(Prandtl, 1925), whose restriction to particular flow types leads
to a violation of one or more of the principles listed here, breaksome of the symmetries discussed so far. In general, it must beemphasized that symmetry-breaking only restricts the universality of a model and says nothing about its usefulness for any
one particular flow.
However, additional symmetry constraints can be foundby looking at a complete statistical description of turbulence,such as that given by (4) and the infinite hierarchy of equations
for the higher moments. Oberlack & Rosteck (2010); Rosteck& Oberlack (2011) found that in this framework, the symmetries

the Hopf functional approach (Hopf, 1952). In the PDF picture, it also becomes possible to develop a physical interpretation of the symmetries (20)–(22), and Waclawczyk et al.
(2014) were able to establish a connection to intermittency andnon-Gaussianity, two frequently observed features of turbulent
statistics.

MODIFYING THE k-ε -MODEL USING STATISTICAL SYMMETRY CONSTRAINTS
Whereas the classical symmetries have generally been accounted for in turbulence modeling, the statistical symmetries
have so far been overlooked. In fact, it turns out to be difficult to include them into a turbulence model. In the context
of eddy-viscosity models, this can be understood by realizingthat the exact mean momentum equation (4) only fulfills (21)due to the complicated transformation of the Reynolds stresstensor. Once a model for it is inserted, (21) essentially becomes a translation symmetry in Ūi . In order to fulfill it (without relying on Ri j ), the velocity must not appear explicitly, but
only its derivatives. Clearly, this is violated by the convective
term. Similarly, once the exact Reynolds stress tensor is replaced by a model R̃i j , the statistical scaling symmetry (20)is broken by the nonlinearity of the convective term of (4),and, as can be shown, also by the production and dissipation
terms of (7) and (8). In a preliminary study, Klingenberg et al.
(2020) were able to show that in order for reasonable model
equations to be possible subject to the constraints introducedby the classical symmetries (10)–(15) (with the last two combining to (19) in the viscous case) and the statistical symmetries (20)–(22), auxiliary velocity and pressure fields, which
we call Ûi and P̂, respectively, must be introduced. These
fields must behave like the mean velocity (or, respectively, themean pressure) under all classical symmetries while being invariant under all statistical symmetries. If this auxiliary velocity then replaces the mean velocity in the convective terms ofall model equations, the first issue discussed above is solved.
In order to also address the second issue associated with thescaling of the source terms, a third scale-providing variable
has to be introduced. The model equations for Ûi and P̂ arequite strongly constrained by the assumed symmetry behaviorof the two variables, and we use

TSc,stat : t ∗ = t, xi∗ = xi , Ūi∗ = Ūi eaSc,stat , P̄∗ = P̄eaSc,statR∗i j = (Ri j + ŪiŪ j )eaSc,stat − ŪiŪ j e2aSc,stat

∂ Ûi
=0
∂ xi
D̂Ûi ∂ Ûi
∂ Ûi
∂ P̂
=
+ Û j
=−
∂t
∂xj
∂ xi
D̂t

(20)

TTr,stat,1 : t ∗ = t, xi∗ = xi , Ūi∗ = Ūi + aTr,stat,I,i , P̄∗ = P̄R∗i j = Ri j − Ū j aTr,stat,I,i − Ūi aTr,stat,I, j− aTr,stat,I,i aTr,stat,I, j∂(ν + ινt )
+
∂xj

(21)

TTr,stat,2 : t ∗ = t, xi∗ = xi , Ūi∗ = Ūi , P̄∗ = P̄R∗i j = Ri j + aTr,stat,II,i j∂ Ûi ∂ Û j+
∂ x j ∂ xi

(23)

!!

2 ε̂ ∂ k− ι
3 ε ∂ xi

(24)

(22)
Herein, some freedom exists in the choice of the source
terms in that additional source terms would be possible, andthe terms containing the model parameter ι are not mandatory from a symmetry perspective. The particular form (23)and (24) is motivated by the assumption that Ūi and Ûi should
not behave too differently, and, thus, their equations should
be structurally similar. The third scale-providing variable ε̂,which appears in the last term of (24), is assumed to behave
like ε under all classical symmetries (10)–(15) and (19) andthe statistical translation symmetries (21) and (22) while scaling linearly under the statistical scaling symmetry (20). Furthermore, we assume that k behaves like the turbulent kinetic
energy under all classical symmetries, is invariant under thewhich have no counterpart in the unaveraged Navier–Stokes
equations, appear. Note that these symmetries affect the entire
infinite hierarchy of statistical moments, but in order to keep
the notation simple, we only write out variables that are important for our present purposes. For more details on thesesymmetries, we refer to Rosteck (2013). Waclawczyk et al.
(2014) were able to show that these symmetries, which we refer to as statistical symmetries in the following, also appear
in other complete statistical descriptions of turbulence, suchas the Lundgren–Monin–Novikov (LMN) hierarchy (Lundgren, 1967) based on probability-density functions (PDFs) and3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
The equations (1), (6), (9), (23), (24), (28), (30) and (31) and(5) with D̂Ūi /D̂t instead of DŪi /Dt then form a turbulencemodel that is invariant under all classical and statistical symmetries. For more details on the derivation of the model equations, we refer to Klingenberg & Oberlack (2022).

statistical translation symmetries and scales linearly under thestatistical scaling symmetry. The model variable ε is assumed
to behave like the turbulence dissipation rate under all classical symmetries, to be invariant under the statistical translation
symmetries and to scale quadratically under the statistical scaling symmetry. Note that these assumptions are made because
the behavior of the turbulent kinetic energy and the turbulent
dissipation rate under the statistical symmetries is very complicated, and it would not be feasible to incorporate them into themodel. At the same time, the main purpose of eddy-viscosity
models is to predict the mean velocity as accurately as possible, and the interpretation of k as the turbulent kinetic energy
and of ε as the turbulent dissipation rate should not be taken
too seriously. In classical turbulence modeling, this insightperhaps most obviously manifests itself in the development ofthe ε-equation (8), which is entirely empirical, and basing iton the exact equation for the turbulent dissipation rate would
make little sense (e.g. Pope, 2000). To summarize, the symmetries (14), (15) and (20) with the auxiliary model variables
read

RESULTS OF THE MODIFIED k-ε -MODEL
The model development presented in the previous section
is largely based on mathematical arguments. In order to assess
the performance of the resulting model in practice and to establish appropriate values for the appearing model constants,
it has to be applied to a range of canonical flow cases. The
steps for the model calibration partly overlap with those used
for the classical k-ε-model, but, as it turns out, the modified
model can incorporate a wider variety of flows. More details
on the model calibration are given in Klingenberg & Oberlack(2022).
First, the case of homogeneous turbulence, where all spatial gradients vanish, is invoked. Due the resulting simplification of the model equations, an analytical solution for the
temporal evolution of k can be found, which reads

TSc,I : t ∗ = t, xi∗ = xi eaSc,IŪi∗ = Ūi eaSc,I , P̄∗ = P̄e2aSc,Ik∗ = ke2aSc,I , ε ∗ = εe2aSc,I , ε̂ ∗ = ε̂e2aSc,I
∗

1

(25)

k(t) ∝ t 1−Ĉε,2, xi∗ = xi
−aSc,II∗

TSc,II : t = teŪi∗ = Ūi e∗

k = ke2aSc,II, P̄ = P̄e−2aSc,II, ε ∗ = εe3aSc,II , ε̂ ∗ = ε̂e3aSc,II

A comparison with experimental data leads to the choiceĈε,2 = 1.92, which is analogous to the constraint arising fromhomogeneous turbulence for the classical k-ε-model.
A slightly more complicated case arises from homogeneous shear turbulence, i.e. the case where one component of
the velocity gradient, say ∂ Ū1 /∂ x2 , takes a constant nonzero
value, but all other spatial gradients vanish as before. Under
these circumstances, turbulence is sustained, and experiments
show that the ratio k/ε becomes constant for large times, andthe ratio of production and dissipation also approaches a constant value determined by Cε,1 . Since the evolution equationof k/ε is exactly the same as for the classical k-ε-model, thesame conclusion of Cε,1 = 1.44 can be obtained for the modified model.
Furthermore, the modified model should be capable of accurately predicting the logarithmic law of the wall. Constraints
on the model parameters can be inferred by inserting the classical ansatz(26)

TSc,stat : t ∗ = t, xi∗ = xi , Ūi∗ = Ūi eaSc,stat , P̄∗ = P̄eaSc,statk∗ = keaSc,stat , ε ∗ = εe2aSc,stat , ε̂ ∗ = ε̂eaSc,stat(27)

The other symmetries (10)–(13), (21) and (22) are not repeatedhere because k, ε and ε̂ are invariant under them.
Using these assumptions, a transport equation for ε̂ canbe formulated in analogy to the ε-equation (8)
D̂ε̂
D̂t

= − Ĉε,1+(32)

aSc,II

∂
∂xj

!
∂ Ūi
∂ Ûi
ε̂ 2
ε
ε̂ 2
R̃i j
+ γ R̂i j− Ĉε,2
kε
∂xj
k
∂xj
k
!


∂ ε̂
νt
ν+
(28)
σ̂ε ∂ x j

whereR̂i j = −νt∂ Ûi ∂ Û j+
∂ x j ∂ xiuτ
log x2 + B, Û1 = Ĉ log x2 + B̂κ
k = Ck x2nk , ε = Cε x2nε , ε̂ = Cε̂ x2nε̂

Ū1 =

!
(29)

Here, the additional model parameters Ĉε,1 , γ, Ĉε,2 and σ̂ε appear. Note that the γ-term is not strictly necessary from a symmetry perspective, but was found to improve the numericalstability of the model. The model equations for k and ε cannow be modified according to the statistical scaling symmetry,leading to
D̂k

∂ Ūi
∂
ε̂
= − R̃i j− ε̂ +
ε
∂xj
∂xj
D̂t

!


∂k
νt
ν+
σk ∂ x j

into the model equations (5), (28), (30) and (31). The requirement that all x2 -terms must cancel then leads to the result
nk = 0, nε = nε̂ = −1, which is also observed in the classicalk-ε-model. In analogy to the classical model, the k-equation in
conjunction with experimental results implies Cµ = 0.09. Furthermore, assuming that we can neglect the small γ-term, theequations for ε and ε̂ yield the constraint(30)

Cε,1 −Cε,2σ̂ε=σεĈε,1 − Ĉε,2

D̂ε

∂ Ūi
ε̂
ε ε̂
= −Cε,1 R̃i j
−Cε,2
k
∂xj
k
D̂t
!


∂
∂ε
νt
+
ν+
∂xj
σε ∂ x j(33)

Note that it is possible to take the γ-term into account, but then,due to the more complicated form of the equations, a simple
constraint on the model parameters cannot be inferred.

(31)

4

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

At this point in the discussion, we deviate from the line ofargumentation used for calibrating the classical k-ε-model forthe first time, because we now consider a newly found coreregion scaling law (Oberlack et al., 2022). This scaling law
matches direct numerical simulation (DNS) data in a large central region of channel flow. Inserting it for the mean velocity
along with a power-law ansatz for the other variablesÛ1 = Ĉ1 x2σ̂1 + Ĉ, Ū1 = C1 x2σ1 +C
k = Ck x2nk , ε = Cε x2nε , ε̂ = Cε̂ x2nε̂1v̂˜′ = û˜ + η û˜′2 2 ′1˜k̃ ˜′û˜û + η û˜′ + v̂˜û˜′ = ιCµ
û
2
ε̃


 2 ′1k̃ ′ũ + η ũ′ + v̂˜ũ′ = Cµ
ũ
û˜
2
ε̃

k̃2 ε̂˜û˜ k̃ + η k̃′ + v̂˜k̃′ = Cµ 2 ũ′2ε̃ 2 ′Cµ k̃ ′k̃− ε̂˜ +
σk ε̃


5
k̃ε̂˜
û˜
ε̃ + η ε̃ ′ + v̂˜ε̃ ′ = Cµ Cε,1 ũ′2
2
ε̃

′
˜
ε̃ ε̂ Cµ k̃2 ′
ε̃
−Cε,2
+
σε ε̃
k̃
!


˜2
˜û 5 ε̂˜ + η ε̂˜ ′ + v̂˜ε̂˜ ′ = Cµ Ĉε,1 k̃ε̂ ũ′2 + γ k̃û˜′2
2
ε̃ 2

′
ε̂˜ 2 Cµ k̃2 ˜ ′ε̂
− Ĉε,2 +σ̂ε ε̃
k̃

(34)

leads to the exponents being restricted to nk = 1, nε = σ1 ,nε̂ = −σ1 + 1 and σ̂1 = 0. DNS data suggests an asymptotic value of σ1 = 2 for large Reynolds numbers, which isalso predicted by the modified model if the model parameters
fulfill the constraint arising from the log law (33). It is remarkable that no additional constraint arises from this flow,which highlights that more symmetries in a model increase its
universality. The classical k-ε-model, on the other hand, does
not perform well for this calibration case, because it predicts avalue of σ1 = 1/2, which contradicts numerical evidence.
Another flow that is not generally taken into account forthe calibration of the k-ε-model is shear-free one-dimensional
turbulence. Here, we assume that the velocity gradient vanishes, but allow a gradient of the scalar variables in one direction, which leads to a balance of dissipation and diffusion.
This flow is also practically interesting, because it serves as asimple model for a quiet body of water in which turbulence isintroduced by surface waves (Umlauf et al., 2003). The ordinary differential equation (ODE) system arising from inserting these simplifications into the model equations is difficult
to solve analytically, but experimental evidence suggests thepower-law ansatz (Umlauf et al., 2003)

k = Ck x2nk , ε = Cε x2nε , ε̂ = Cε̂ x2nε̂

1
2

x1
k=

, Û1 =û˜1
2

, Û2 =

x1

ε̃
ε̂˜
k̃
, ε = 5 , ε̂ = 5
x1
x12
x12

(39)

(40)

(41)

(42)

σk = 0.91, ι = 2.0
Cε,1 = 1.44, Cε,2 = 1.92, σε = 1.3
Ĉε,1 = 1.42, γ = 0.05, Ĉε,2 = 1.92, σ̂ε = 1.25

(43)

for the model parameters not only fulfills all constraints developed above, but also yields very good agreement for the planejet. The numerical results for the mean velocity are shown in
Fig. 1. Evidently, the modified model performs slightly better than the classical model. In particular, it is interesting thatthe performance at the intermittent edge is slightly superior,which might be related to the fulfilling of the statistical symmetry (20), since this symmetry has also been connected tointermittency. However, more research is needed before thiscan be asserted with confidence.

(35)

CONCLUSION
In the present work, symmetry-based modifications areapplied to the classical k-ε-model. The resulting model istested against a range of canonical flow cases and found to
be more universal than the classical k-ε-model. To some degree, this increased generality can be expected simply becauseof the additional model parameters appearing in the modified
model, but it must be stressed that the additional statisticalsymmetries fulfilled by the model also contribute to the increased universality. This is perhaps most clearly observed
for the core-region scaling law, which the modified model predicts accurately as long as the model parameters are calibrated
against the log law. This result is promising, because it could
be an indication that the modified model performs well in otherflows it was not originally calibrated against, which is something that the classical k-ε-model and similar models tend to
struggle with.

As a final test case, we consider the self-similar plane turbulent jet issuing into x1 -direction. Inserting the classical similarity ansatz

ũ

(38)

Herein, primes denote derivation with respect to the similarity
variable η = x2 /x1 .
It was found that choosing

The resulting nonlinear system of algebraic equations is solved
numerically, and, for the choice of model parameters givenby (43), the exponent nk is determined to be −3.7, whichis reasonably close to the experimentally observed range of−1.7 . . . − 3.0 (Nokes, 1988). On the other hand, the classical k-ε-model with standard model parameters predicts nk ≈−4.97, and changing the model parameters to yield more accurate results immediately leads to an inaccurate performancein the log region. The modified model, on the other hand, canproduce reasonable results in both cases.

Ū1 =

(37)

v̂˜
1

x12
(36)

into the model equations (5), (23), (24), (28), (30) and (31)512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

1.0

Jones, W.P. & Launder, B.E. 1972 The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat
Mass Transfer (15), 301–314.
Klingenberg, Dario Simon & Oberlack, Martin 2022 Statistically invariant eddy viscosity models. Physics of Fluids .
Klingenberg, Dario Simon, Oberlack, Martin & Pluemacher,
Dominik 2020 Symmetries and turbulence modeling.
Physics of Fluids 32 (2).
Lundgren, T. S. 1967 Distribution functions in the statisticaltheory of turbulence. Physics of Fluids 10 (5), 969.
Nokes, R. I. 1988 On the entrainment rate across a density
interface. Journal of Fluid Mechanics 188, 185–204.
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, 024502.
Oberlack, M. & Rosteck, A. 2010 New statistical symmetriesof the multi-point equations and its importance for turbulent
scaling laws. Discrete and Continuous Dynamical System
Series S 3 (3).
Pope, S. B. 2000 Turbulent Flows. Cambridge University
Press.
Prandtl, L. 1925 Über die ausgebildete turbulenz. Z. Angew.
Math. Mech. (5), 136–139.
Reynolds, Osborne 1895 On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philosophical Transactions of the Royal Society of
London. A 186, 123–164.
Rosteck, Andreas 2013 Scaling laws in turbulence - a theoretical approach using lie-point symmetries. Dissertation,
Technical University of Darmstadt.
Rosteck, A. & Oberlack, M. 2011 Lie algebra of the symmetries of the multi-point equations in statistical turbulence.
Jounal of Nonlinear Mathematical Physics 18 (1).
Umlauf, L, Burchard, H & Hutter, K 2003 Extending thek–ω turbulence model towards oceanic applications. Ocean
Modelling 5 (3), 195–218.
Waclawczyk, M., Staffolani, N., Oberlack, M., Rosteck, A.,
Wilczek, M. & Friedrich, R. 2014 Statistical symmetries of
the Lundgren-Monin-Novikov hierarchy. Physical Review(90), 1–11.
Wygnanski, I. & Fiedler, H. 1969 Some measurements in the
self-preserving jet. Journal of Fluid Mechanics 38 (3), 577–
612.

experimental dataclassical k-ε model
modified k-ε model

0.8

ũ

0.6
0.4
0.2
0.0
0.00

0.05

0.10

0.15

0.20

0.25

η

Figure 1: Plane jet experimental data (Wygnanski &Fiedler, 1969) compared with classical and modified,statistically invariant k-ε-model
ACKNOWLEDGEMENT
The work of Dario Klingenberg is supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Excellence Computational
Engineering at Technical University Darmstadt and by theDeutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Project number OB 96/54-1.

REFERENCES
Boussinesq, J. 1877 Essai sur la théorie des eaux courantes.
Mémoires présentés par divers savants à l’Académie desSciences (XXIII, 1), 1–680.
Bytev, V. O. 1972 Group properties of the Navier-Stokes equations. Chislennye metody mehaniki sploshnoy sredy 3, 13–
17.
Donaldson, C. duP. & Rosenbaum, H. 1968 Calculation of the
turbulent shear flows through closure of the reynolds equations by invariant modeling. Aeronaut. Res. Assoc. Princeton (127).
Hopf, Eberhard 1952 Statistical hydromechanics and functional calculus. Journal of rational Mechanics and Analysis
1, 87–123.

6