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

INVARIANT MOMENT AND PDF SOLUTIONS OF WALL-BOUNDEDTURBULENT SHEAR FLOWS

Dominik Plümacher
Martin Oberlack
Department of Mechanical Engineering
Department of Mechanical Engineering
Technical University of Darmstadt
Technical University of Darmstadt
Otto-Bernd-Str. 2, 64287 Darmstadt, Germany Otto-Bernd-Str. 2, 64287 Darmstadt, Germany
pluemacher@fdy.tu-darmstadt.deoberlack@fdy.tu-darmstadt.deSergio Hoyas
Instituto Universitario de Matemática Pura y Aplicada, UP València,
Camino de Vera, 46024 València, Spain
serhocal@mot.upv.es

ABSTRACT

was not remedied in the following decades of researchin modified versions of the log-law, see e.g. Barenblatt(1993). One of the first works that derived a scaling
law as a self-similar solution is the work of von Kármán& Howarth (1938) on the decay of isotropic turbulence.
Subsequently, this approach was applied very successfully to a large number of flows.
The idea of applying the methods of symmetry
theory for a systematic derivation of “scaling laws”
via self-similar solutions was probably first proposed
by Oberlack (2001). Further work followed even for
more complex flows with rotation, wall transpiration,and many more, see Avsarkisov et al. (2014); Oberlack
et al. (2015); Sadeghi et al. (2018). However, all theseapproaches were limited to low order moments. For allthese scaling laws statistical symmetries played an essential role. These were first observed by Oberlack &Rosteck (2010) in the infinite sequence of the multipoint
moment equations (MPME). Furthermore, it was shown
by Waclawczyk et al. (2014) that these symmetries are
related to the phenomena of intermittency and of nonGaussian behaviour in turbulence - both of which illustrate well-known, central characteristics of turbulent
flows and which, with the latter, offer a unique quantification in terms of symmetries. The infinite sequenceof the multipoint moment equations is formed from theNavier-Stokes equations,

In a recent publication Oberlack et al. (2022) by thelast two authors, they showed that velocity momentsof arbitrary order can be derived as invariant solutionsof the multi-point moment equations by means of thesymmetry-based turbulence theory. They applied it to
the log-region as well as to the deficit region in the centre of a channel flow. The results were validated witha new channel flow DNS at Reτ = 104 with high accuracy and up to moment order 6. Presently we havesignificantly extended it by computing a invariant solutions for the probability density function (PDF), which
is formally the next higher-level statistical descriptionof turbulence. This new solution is believed to be thefirst exact solution to Lundgren’s PDF equation, andin the sequel we show that the above moments can begenerated from it.

INTRODUCTION
To treat the complex flow behaviour of turbulent
fluids mathematically, a statistical approach has been
the most successful so far. For a complete mathematical description, however, an infinite number of coupledequations is usually required, a fact that was alreadyknown to Reynolds. For practical applications it istherefore necessary to reduce this set of equations toa finite set by some form of closure procedure, This,
however, considerably limits the prediction accuracy ofsuch an approach. Quite early on, it was observed thatfor canonical flows, certain distinguished regions can be
identified in which a universal behaviour is exhibitedand corresponding simplified equations can be derived
heuristically. Nowadays, these results are usually subsumed under the term scaling law and are essentiallybased on similarity arguments, which in turn often exploit, at least indirectly underlying symmetries of thephysical process. Probably one of the first examples
of such a result is the well-known logarithmic law of
the wall due to von Kármán (1930). A downside ofvon Kármán’s reasoning, however, is that it is not basedon the Navier-Stokes equations, and therefore are notderived from “first principles” in this sense. This aspect∂Ui ∂P∂ 2 Ui
∂Ui
+ Uk
+
−ν
=0
∂t
∂xk ∂xi∂xk ∂xk∂Ui=0∂xi

i = 1, 2, 3

(1)
(2)

where t ∈ R+ , x ∈ R3 , U = U (x , t), P = P (x , t) and ν respectively represent time, position vector, instantaneous
velocity vector, pressure, and kinematic viscosity. Whendealing with statistical properties, typically, Reynolds’s
decomposition is used, i.e., Uk = U k + uk , where ( . ) denotes averaging and uk is the turbulent fluctuation. We
depart here from the usual approach to analyse moment
equations based on the fluctuations and use instead statistical moments that are composed of the full instan-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
referring to aSs denote a measure of intermittency andthe further statistical symmetry

taneous velocities. The main ideas and steps of thisapproach are briefly illustrated in the following section.

∗′∶ t∗ = t , x(i)= x(i) ,T̄{n}HIGH-ORDER MOMENT SCALING LAWSŪ ∗ = Ū + a , H∗{n} = H{n} + aH{n}

For the subsequent new findings on the PDF, thecentral results for the high velocity moments in turbulent shear flows close to the wall will be outlined first.
The essential ingredients for the construction of invariant solutions are so-called symmetry transformations which refer to a variable transformationx ∗ = ϕ(x , y ; a),y ∗ = ψ(x , y ; a)defines non-gaussianty (Waclawczyk et al., 2014).
For the following comparison with the DNS data, weonly consider one-point statistics, i.e. we apply x = x(1) =x(2) = . . . = x(n) and further limit ourselves to the moments of the stream-wise velocity U1 . Hence the Hi{n}reduce to one-point moments U1n .
In Oberlack et al. (2022) it is rigorously shown, andthis shall not be repeated here, how from this for thelog region arbitrarily high moment scaling laws are generated. The key parameter√ for the log-law is the wall
shear stress velocity uτ = τw /ρ, where τw is the wall
shear stress and ρ is the density. uτ uniquely determines
the only velocity scale in the problem. uτ implies that ascaling of the mean velocities according to Eq. (7) with
arbitrary aSx , aSt and aSs is no longer feasible. Hence,in terms of symmetry theory, uτ is symmetry breaking
for the mean velocity U 1 and for the group parametersin Eq. (7) this implies aSx − aSt + aSs = 0 and it follows(3)

which leaves a differential equation form invariant when
it is written in the new ∗-variables, i.e. if the followingholds
F (x , y , y (1) , y (2) , . . . , y (p) ) = 0⇔

F (x ∗ , y ∗ , y ∗(1) , y ∗(2) , . . . , y ∗(p) ) = 0(4)

where y (p) denotes the set of all pth derivatives of y .
The basis for the analysis in Oberlack et al. (2022)
are the multi-point moment equations∂Hi{n} n ⎡⎢ ∂Hi{n+1} [i(n) ↦k(l) ] [x(n) ↦ x(l) ]+∑⎢⎢∂t∂xk(l)
l=1 ⎢⎣∂Ii{n−1} [l]∂ 2 Hi{n} ⎤
⎥
⎥=0
+
−ν
∂xi(l)∂xk(l) ∂xk(l) ⎥
⎥
⎦

(8)

1
Ū1+ = ln (y + ) + B
κ
U1n =Cn (y + )+(9)ω(n−1)− Bn , for n ≥ 2

Cn = αeβn , Bn = α̃eβ̃n , for n ≥ 2

(10)
(11)

(5)
with κ and ω are related to the group parameters from
(8) and the scaling symmetries (7) as follows κ = aSx /aH
1
and ω = 1 − aSt /aSx . ω is the exponent of the second
moment and apparently scales all higher moments.
The following result on the scaling laws in the centreof the channel is also taken from Oberlack et al. (2022).
Not imposing any symmetry breaking in the first place
for the centre of the channel, the so-called deficit region,
the following scaling laws were generated

with the multi-point velocity moments defined as follows
Hi{n} = Hi(1) ...i(n) = Ui(1) (x(1) , t) ⋅ . . . ⋅ Ui(n) (x(n) , t) (6)and Ii{n−1} [l] contains the pressure, with details may betaken from Oberlack & Rosteck (2010). In the limit ofvanishing viscosity ν → 0 equation (5) admits the following three-parameter scaling groups

U1n

(0)

− U1n

un
τ
T̄Scale ∶ x∗2 = eaSx x2 , Ū1∗ = eaSx −aSt +aSs Ū1 , . . . ,∗U1n = en(aSx −aSt )+aSs U1n(7)= Cn′ (

x2 n(σ2 −σ1 )+2σ1 −σ2
)
h

(12)

′

with Cn′ = α′ eβ n(13)with the scaling exponents of the first and second moments, σ1 and σ2 , are related to the scaling symmetries (7) as follows σ1 = 1 − aSt /aSx + aSs /aSx and

The symmetries associated with aSx and aSt are scaling of time and space that come from the Navier-Stokes
equations. The parameter aSs refers to a scaling symmetry that is purely statistical in nature and only occurs
in the statistical equation, which, although infinite dimensional, are linear. This is also in clear contrast toequations that follow from the Reynolds decomposition.
These appear non-linear by expressions like the production term. It should be noted, however, that this isostensibly the case, for the equations formed from the
Reynolds decomposition as well as the H-formulation
used here as completely equivalent. Even more, in the
H-formulation in equation (5) on sees that this forms
an infinite set of conversation equations. The scaling

σ2 = 2(1 − aSt /aSx ) + aSs /aSx . Further, U1n
is the nth U1 -moment at the centre line of the channel.
Data clearly show, that σ1 ≃ σ2 ≃ 1.95 andhence 2σ1 − σ2 = aSs /aSx remains in the exponent for
all moments, which, as no longer scaling with n, implies
strong anomalous scaling. This is fully in line with thefact, that aSs is a measure of intermittency which is thedominating effect in the channel centre.
Verification of the log-region scaling (9) and (10) is
shown in figures 1, while clear evidence of the extended
deficit scaling (12) is shown in 2, where both images are
reproductions from Oberlack et al. (2022).
(0)2

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

Figure 2: Deficit scaling of U1n of order n = 1,...6(a)in the channel centre. Solid lines are according to
Eq. (12). Symbols, DNS data as in Fig. 1. Red
dashed lines and symbols: Reτ = 5200 Lee & Moser(2015). Vertical dotdashed line: Limit for the channel’s centre region defined by equation (12). Figure
is taken from Oberlack et al. (2022).

the one- and two-point PDF are defined by

f1 (v ; x , t) = ⟨δ (v − U (x , t))⟩ ,f2 (v , w(2) ; x , r (2) , t) =⟨δ (v − U (x , t)) δ (w(2) − U (x + r (2) , t) + U (x , t)) ⟩ (14)(b)
Figure 1: Symbols from DNS data n = 1: ◯, n = 2:◻, n = 3: ◇, n = 4: △, n = 5: ▽, and n = 6: ☆; (a):+moments U1n ;: Eq. (10) with coefficients fitted to DNS data for n ≥ 2. (b): Left axis: indi++
dU1ncator function Γn = n +y:
+ for (10);
U1 +Bn dy
Γn = ω(n − 1) with ω = 0.10 and n ≥ 2. Right axis:
log-indicator function Γ = y + ∂y+ Ū1+ = κ−1 according
to (9) with κ = 0.394. Figures are taken from Oberlack et al. (2022)

were w(j) denotes the sample space variable for the velocity difference between points x = x1 and x + r (j) , j =2, 3, . . ., i.e. U (x +r (j) , t)−U (x , t), and δ denotes Dirac’sdelta function. The analogue definition for the n-point
PDF fn is straight-forward.
The associated equations governing the timeevolution of the hierarchy of PDFs {fn ∶ n = 1, 2, 3 . . .}are the Lundgren-Monin-Novikov (LMN) equations,first derived by Lundgren (1967) and independently
by Monin (1967). The first equation for the one-point
PDF f1 reads

Although in the derivation of the multi-point moment equations (5) the probability density function(PDF) has been used only formally, information of the
PDF has not been incorporated. Thus, the basis of theaforementioned symmetries and resulting scaling laws
are implicitly based on the PDF, which will be examined in detail in the following chapter.

∂f1+v ⋅∇x f1 +∇v ⋅ ∫ ∇r2 G(r2 ) (w2 ⋅ ∇r2 )2 f2 dw2 dr2∂t+ ν∇v ⋅ ∫ w2 ∆r2 f2 δ(r2 ) dw2 dr2 = 0(15)

where G(r ) = (4π)−1 ∣r ∣−1 , and ∇x , ∇r(j) , etc., denotethe nabla operator w.r.t. the variable x , r (k) , etc.. Similarly, ∆r(j) denotes the Laplace operator w.r.t. r (j) .
As one can see, equation (15) also contains the twopoint PDF f2 which is tantamount to the well-known
closure problem in turbulence. In fact, the general npoint evolution equation in difference formulation, derived by Ulinich & Lyubimov (1968), is similar in struc-

PDF BASED SCALING LAWS
A similar construction as outlined in the first section can be done for the probability density functions(PDFs), subsequently written in increment form. Thus,3

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

ture and readsw∗(k) = eaSx w(k) ,Tsc,t ∶+∇w(l) ⋅ ∫ ∇r(n+1) (G(r (n+1) − r (l) ) − G(r (n+1) ))Ttr,x ∶

(δ(r (n+1) − r (l) ) − δ(r (n+1) ))w(k) = ew(k)t = t,

x = x + ax

fn∗ = e3naSt fn∗∗∗

× (w(n+1) ⋅ ∇r(n+1) ) fn+1 dw(n+1) dr (n+1)+ ν∇v ⋅ ∫ w(n+1) ∆r(n+1) fn+1
(16)

Again one observes that every n-point equation not only
depends on fn but also on the (n + 1)-point PDF fn+1 .
Not surprisingly, the hierarchy (16) admits the same
symmetries as the underlying Navier-Stokes equations,also referred to as classical symmetries, see (Waclawczyk
et al., 2014). In the same work, the authors also translated the statistical symmetries of the multi-point moment equations (5), given by aSs in (7) and by (8), to
the LMN-hierarchy (16). In the variables of the PDF
formulation the former (i.e. associated to aSs in (7)) isof the form

(17)

Tsh ∶

t = t,

x = x,

fn∗ = fn + ψ(v )δ(W )v = v,
∗

(20)

v =v

fn∗ = fn

(21)

−3(aSx −aSt )+aSs

f1 (v ; x2 ) = x̃2

aSxw = w,x̃2aStaf˜1 (x̃2 Sx−3(aSx −aSt )+aSs
aSx+rk∗ = rk ,v

This is a system of hyperbolic ordinary differential equations (ODEs) and can therefore be solved by the standard method of characteristics, see e.g. Cantwell (2002)
for details and a general introduction to symmetry analysis.
Solving the system (22) for n = 1 and aSx ≠ 0 as an
ODE in the variable x2 yields the following expression
for an invariant one-point PDF

which was connected by Waclawczyk et al. (2014) tothe phenomenon of intermittency. This can be best understood by noting that the delta distribution in (17)corresponds to the PDF of a laminar flow. The aforementioned symmetry thus combines laminar and turbulent components. The corresponding turbulent signal
thus has an intermittent form. The second symmetry
related to (8) has the form∗r (k) = r (k)−aSt

dr (2)
dr (n)
dt
dx=== ... =
aSt t aSx x + ax aSx r (2)aSx r (n)
dw(2)
dw(n)
dv
=
= ... ==(aSx − aSt )v (aSx − aSt )w(2)(aSx − aSt )w(n)dfn=−[3n(aSx − aSt ) − aSs ]fn + [ψ(v) − aSs δ(v)]δ(W )(22)t∗ = t, x ∗ = x , r ∗(j) = r (j) , v ∗ = v , w∗(j) = w(j) ,

∗

,

∗

where aSx , aSt denote two real parameters and ax = ax ,1denotes the three-dimensional parameter for the spatialtranslation group.
Using standard techniques of symmetry analysis,
the condition for a hierarchy of PDFs {fn ∶ n = 1, 2, 3, . . .}to be invariant w.r.t. to a one-parameter group T generated by a combination of TSs , Tsh , Tsc,x , Tsc,t , Ttr,x
for a given set of parameters aSs , ψ, aSx , aSt , ax is expressed by the condition to solve the following systemof so-called characteristic equations2

fn∗ = eaSs fn + (1 − eaSs )δ(v )δ(W )

v =e

∗

∗

w(k) = w(k) ,× δ(r (n+1) ) dw(n+1) dr (n+1) = 0(19)r (k) = r (k) ,

t,

−aSt

∗

∗

+ ∇v ⋅ ∫ ∇r(n+1) G(r (n+1) )fn∗ = e−3naSx fn

x = x,t =e×w(n+1) ∆r(n+1) fn+1 dw(n+1) dr (n+1) ]

TSs ∶

aSt

∗

∗

×(w(n+1) ⋅ ∇r(n+1) )2 fn+1 dw(n+1) dr (n+1)+ν∇w(l) ⋅ ∫t∗ = t, x ∗ = eaSx x , r ∗(k) = eaSx r (k) , v ∗ = eaSx v

Tsc,x ∶n∂fn
+ v ⋅ ∇x fn + ∑ [w(l) ⋅ ∇r(l) fn
∂t
l=2

1−

∫

aSx − aSt

x̃2
1

aSt
aSx −1∗× ψ (s̃x̃2

(18)

aSt
aSx

−1

2− a

s̃

v)

aSs
Sx −aSt×v ) ds̃ + f1 (v ; x2 )
(i)

(23)

where x̃2 = Cx2 + a for some constants C > 0 and a,f˜1 (v ) denotes a free PDF in one velocity variable

describing non-Gaussian behaviour of turbulence.
Here W = (w2 , . . . , wn ) and ψ(v ) has to admit the side
condition ∫ ψ(v ) dv = 0.
For the construction of invariant PDFs, in addition
to the two statistical symmetries (17) and (18) above,
also the classical symmetries of scaling in space Tsc,x ,scaling in time Tsc,t , and the three-parameter groupspatial translations Ttr,x are considered. In the PDF
formulation, the associated symmetry transformationsaand f1 (v , x ) = (1 − x̃2 Ss Sx ) δ(v ) is the solution component due to the statistical scaling (17). This component, however, is usually not visible in data due tothe delta function. Since the expressions in x̃2 appearing in (23) have arbitrary real exponents, the expression (23) is only defined for x2 > −a/C. Finally, similar
expressions can be derived from (22) for the n-point
PDFs fn by the same methods. The resulting invariant
(i)

4

/a

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
where Dn = H̃n − D̃n and D̃n is short-hand notation for

expression is−

fn (v, W ; x2 ) = x̃2

aSt
−1
a

× f˜n (x̃2 Sx

3n(aSx −aSt )+aSs
aSx
aSt
−1
a

v, x̃2 Sx

D̃n =

×

W ) + (f1 (v; x2 )Now the deficit law (12) can be obtained from (29)by setting An = −(n(aSx − aSt ) + aSs )U1n

and H̃n =
(0)
n ′
n
U1
− uτ Cn while setting C = 1/h and a = 0 in the defi(0)nition of x̃2 . Indeed, the choice for An yields D̃n = U1n′in (30) and Dn = −unτ Cn .

(24)(s)

where the short-hand notation f1 was used for theterm in (23) with the integral over ψ(v ), i.e.
−3(aSx −aSt )+aSs
aSx

(s)

x̃2

(30)

(i)

(s)
+f1 (v; x2 )) δ(W )f1 (v , x2 ) =−Ann(aSx − aSt ) + aSs

1−∫aSx − aStx̃2

In view of definition (28) the constants An are essentially the moments of the distribution function ψ(v ).

×

By definition (30) of the D̃n , the relation D̃n = U1n
used above to obtain the deficit law (12) from (29) implies that the moments of the function ψ(v ) are de-(25)

termined by the moments U1nof the PDF f1 (v ) =f1 (v ; 0) at the centre of the channel. Assuming that all

(0)

aSt
aSx

a
2− a SsSx −aSt

s̃

1

(0)

aSt
−1
a

× ψ (s̃x̃2 Sxv ) ds̃

(0)

(0)

(0)

moments U1nexist and the corresponding momentproblem can be solved positively for the An , this meansthat the function ψ(v ) can be expressed in terms of the(0)
PDF f1 (v ). That suggests the possibility of formulating a deficit law for PDFs similar to (12).
The question whether not only the derived moment
expressions (9), (10) and (12) are in good agreementwith the DNS data but also the proposed invariant PDF
scaling law (23) is an open question and subject of current research.

(i)and f1 the same as in (23).
An important observation is that by using the PDFbased definition of the moments
U1n = ∫ f1 (v1 ; x2 )v1n dv1(26)

for the invariant PDF (23), the resulting moment expressions are in complete agreement with the invariant moment scaling laws (9)-(10) and (12), using thecorresponding scaling parameters κ, ω, σ1 and σ2 .
Three cases may be distinguished. More precisely, taking n(aSx − aSt ) + aSs = 0, for a fixed n the expression derived from (23) via (26) for this n-th moment are given
by a log-function
U1n (x2 ) = H̃n +Anln (x̃2 )aSx

CONCLUSIONS AND OUTLOOKThe central result of the present contribution isthat, on the basis of classical and statistical symmetries,
we are now able to calculate invariant solutions for moments of arbitrary order as well as for the corresponding
PDF for near-wall shear flows. It is particularly important that the moments and the PDFs are based on the
instantaneous velocities and not on the fluctuation velocities resulting from the Reynolds decomposition.
The final step is the determination of the free function ψ, which characterises non-Gaussian behaviourin (18), which explicitly appears in the PDF solution (23), and needs to be determined from DNS data.

(27)

where the constants H̃n , An are defined asH̃n = ∫ ṽ1n f˜1 (ṽ) dṽ

,

An = ∫ ṽ1n ψ(ṽ) dṽ

(28)

All other moments then turn out to be power laws.
Physically relevant seem to be the following cases.
For the classical log-law, i.e. for aSx − aSt + aSs = 0,
one obtains (9) from√(27) by setting A1 = uτ aSx /κ,H̃1 = uτ B and C = 1/(ν ρ/τw ), a = 0 in the notation x̃2 =Cx2 + a introduced above, since then x̃2 = y + is the distance to the wall in wall units. Similarly, for n ≥ 2 thescaling law of higher moments (10) follows from (29) ben
low by setting An = (n − 1)aSx ωunτ Bn and H̃n = uτ (Cn −Bn ). In fact, recalling that ω = (aSx − aSt )/aSx , thischoice reduces (30) to D̃n = −unτ Bn and hence Dn =H̃n − D̃n = unτ Cn .
In the case of the deficit power-law, i.e. n(aSx −aSt ) + aSs ≠ 0, the corresponding result for the n-th moment are

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5

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Osaka, Japan (Online), July 19-22, 2022
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