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DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2010.3.451DYNAMICAL SYSTEMS SERIES SVolume 3, Number 3, September 2010 pp. 451–471
NEW STATISTICAL SYMMETRIES OF THE MULTI-POINT
EQUATIONS AND ITS IMPORTANCE FOR TURBULENT
SCALING LAWS
Martin Oberlack1,2,3 and Andreas Rosteck1
1Chair of Fluid Dynamics, Department of Mechanical Engineering, TU DarmstadtPetersenstr. 30, 64287 Darmstadt, Germany2Center of Smart Interfaces, TU DarmstadtPetersenstr. 32, 64287 Darmstadt, Germany
3GS Computational Engineering, TU DarmstadtDolivostr. 15, 64293 Darmstadt, Germany
Abstract. We presently show that the infinite set of multi-point correlationequations, which are direct statistical consequences of the Navier-Stokes equa-tions, admit a rather large set of Lie symmetry groups. This set is considerableextended compared to the set of groups which are implied from the originalset of equations of fluid mechanics. Specifically a new scaling group and trans-lational groups of the correlation vectors and all independent variables havebeen discovered. These new statistical groups have important consequences onour understanding of turbulent scaling laws to be exemplarily revealed by two
examples. Firstly, one of the key foundations of statistical turbulence theoryis the universal law of the wall with its essential ingredient is the logarithmiclaw. We demonstrate that the log-law fundamentally relies on one of the newtranslational groups. Second, we demonstrate that the recently discovered ex-ponential decay law of isotropic turbulence generated by fractal grids is onlyadmissible with the new statistical scaling symmetry. It may not be bornefrom the two classical scaling groups implied by the fundamental equations offluid motion and which has dictated our understanding of turbulence decaysince the early thirties implicated by the von-Karman-Howarth equation.
1. Introduction. Although there has been considerable progress in theory as wellas in experiments, turbulence still is one of the most important and most challengingunresolved problems in classic mechanics. It has indirectly been chosen as one ofthe seven Millennium problems of Mathematics (with six still unresolved) sinceexistence and smoothness of Navier-Stokes equations have been posed as one ofthem.
The special importance of turbulence is determined by its presence in numerousnatural and technical systems. Examples for natural turbulent flows are atmo-spheric flow and oceanic current which to calculate is a crucial point in climateresearch. Only with the advent of super computers it became apparent that theNavier-Stokes equations provide a very good continuum mechanical model for tur-bulent flows. Still, the application of Navier-Stokes equations to practical flow
2000 Mathematics Subject Classification. Primary: 76F02, 76F40, 76F05; Secondary: 76M60.Key words and phrases. Turbulent scaling laws, multi-point correlations, Lie symmetry group
theory.
451
452 MARTIN OBERLACK AND ANDREAS ROSTECK
problems at high Reynolds number without invoking any extra assumptions is stillseveral decades away.
However, in most applications it is not at all necessary to know all the detailedfluctuating of velocity and pressure present in a turbulent flows but for the mostpart statistical measures would be sufficient.
This was in fact the key idea of O. Reynolds who was the first to suggest a sta-tistical description of turbulence. The Navier-Stokes equations, however, constitutea non-linear and, due to the pressure Poisson equation, a non-local set of equations.As an immediate consequence of this the equations for the mean values leads to aninfinite set of statistical equations, or, if truncated at some level of statistics, anun-closed system is generated.
In order to obtain at least some insight into the statistical behavior of turbulencewe presently apply Lie group theory to the full infinite set of statistical equationsinvestigating two canonical turbulent flow situations.
2. Equations of statistical turbulence theory.
2.1. Navier-Stokes equations. The initial point of the entire analysis to follow isbased on the three dimensional Navier-Stokes equations for an incompressible fluidunder the assumption of a Newtonian material with constant density and viscosity.In Cartesian tensor notation we have the continuity equation
∂Uk
∂xk= 0 (1)
and the momentum equation writes
DUi
Dt= −1
ρ
∂P
∂xi+ ν
∂2Ui
∂xk∂xk, i = 1, 2, 3 , (2)
whereD
Dt=
∂
∂t+ Uk
∂
∂xk(3)
denotes the material derivative. t ∈ R+, x ∈ R
3, U = U(x, t) and P = P (x, t)represent time, position vector, instantaneous velocity vector and pressure. Thedensity ρ and the viscosity ν are positive constants. Furthermore pressure can benormalized with the constant density. The new pressure term reduces to
P ∗ =P
ρ, (4)
which, inserted into (2), leads to the modified momentum equation and the asteriskis omitted from here on
Mi(x) =DUi
Dt+
∂P
∂xi− ν
∂2Ui
∂xk∂xk= 0 , i = 1, 2, 3 , (5)
where all terms have been collected on one side to further down deal with the Mnotation.
2.2. Statistical averaging. In the following we define the classical Reynolds aver-aging. Though definitely mathematical more sound it is not intended to define thenecessary multi-point probability density function to derive the multi-point equa-tions in the subsequent section.
The quantity Z represents an arbitrary statistical variable, i.e. U and P , whichin the following we also denote as instantaneous value. According to the classic
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 453
definition by Reynolds all instantaneous quantities are decomposed into their meanand their fluctuation value
Z = Z + z . (6)
The overbar always marks a statistically averaged quantity whereas the lower-casez denotes the fluctuation value of Z. The most general definition of statisticallyaveraged quantity is given by an ensemble average operator K
Z = Z(x, t) = K [Z(x, t)] = limN→∞
(
1
N
N∑
n=1
Zn(x, t)
)
. (7)
The definition of the mean value leads, according to the Reynolds decomposition,to the fluctuation value of Z
z = Z − Z . (8)
From the averaging operator above, several calculation rules may be derived e.g.
z = 0 ,
Z = Z ,
Z1 + Z2 = Z1 + Z2 ,
∂Z
∂s=
∂Z
∂s,
Z(1)Z(2) . . . Z(m)z(n) = 0 ,
z(1)z(2) . . . z(k) 6= 0 with k > 1 , (9)
which in the following sub-sections will be employed to derive the multi-point equa-tions.
2.3. Reynolds averaged transport equations. After U and P are decomposedaccording to (6), we may apply K to the continuity equation (1) and the momentumequation (5) leading to its averaged versions i.e. continuity equation
∂Uk
∂xk= 0 , (10)
and momentum equation
DUi
Dt= − ∂P
∂xi+ ν
∂2Ui
∂xk∂xk− ∂uiuk
∂xk, i = 1, 2, 3, (11)
whereD
Dt=
∂
∂t+ Uk
∂
∂xk(12)
denotes the mean material derivative.At this point we observe the well-known closure problem of turbulence since,
compared to the original set of equations, the unknown Reynolds stress tensor uiuk
has appeared. However, rather different from the classical approach we will notproceed with deriving the Reynolds stress tensor transport equation which containsadditional four unclosed tensors. Instead the multi-point correlation approach isput forward the reason being twofold.
454 MARTIN OBERLACK AND ANDREAS ROSTECK
First, if the infinite set of correlation equations is considered the closure problemis somewhat bypassed. Second, the multi-point correlation delivers additional in-formation on the turbulence statistics such as length scale information which maynot be gained from the Reynolds stress tensor, which is a single-point approach.
For this we need the equations for the fluctuating quantities u and p whichare derived by taking the differences between the averaged and the non-averagedequations. The resulting fluctuation equations read
∂uk
∂xk= 0 , (13)
and
Ni(x) =Dui
Dt+ uk
∂Ui
∂xk− ∂uiuk
∂xk+
∂uiuk
∂xk+
∂p
∂xi− ν
∂2ui
∂xk∂xki = 1, 2, 3 . (14)
2.4. Multi-point correlation equations. The idea of two- and multi-point cor-relation equations in turbulence was presumably first established by Friedmann andKeller ([8]). In the beginning it was assumed that all correlation equations of ordershigher than two may be neglected. Theoretical considerations led to the result thatall higher correlations have to be taken into account. Consequently, all multi-pointcorrelation equations have to be considered in the symmetry analysis to follow.
Two different sets of multi-point correlation (MPC) equations will be derivedbelow. The first is based on the instantaneous values of U and P while the secondone is in accordance with the classical notation based on the fluctuating quantitiesu and p.
2.4.1. MPC equations: instantaneous approach. In order to write the MPC equa-tions in a very compact form, we introduce the following notation. The multi-pointvelocity correlation tensor of order n + 1 is defined as follows:
Hin+1= Hi(0)i(1)...i(n)
= Ui(0)(x(0)) · . . . · Ui(n)(x(n)) , (15)
where the first index of the H tensor defines the tensor character of the term andthe second index in braces denotes the order of the tensor. The curly brackets pointout that not an index of a tensor but an enumeration is meant. It is importantto mention that the indices start with 0 which is an advantage when introducing anew coordinate system based on the Euclidean distance of two space points. Thevalue in curly brackets is the actual order of the tensor and takes into account thatcounting starts at zero. Apparently we have the connection to the mean velocityaccording to Hi1
= Hi(0) = Ui.In some cases the list of indices is interrupted by one or more other indices which
is pointed out by attaching the replaced value in brackets to the index
Hin+1[i(l) 7→k(l)] =
Ui(0)(x(0)) · . . . · Ui(l−1)(x(l−1))Uk(l)
(x(l))Ui(l+1)(x(l+1)) · . . . · Ui(n)
(x(n)) . (16)
This is further extended by
Hin+2[i(n+1) 7→k(l)][x(n+1) 7→ x(l)] = Ui(0)(x(0)) · . . . · Ui(n)(x(n))Uk(l)
(x(l)) , (17)
where not only that index i(n+1) is replaced by k(l), but also that the independentvariable x(n+1) is replaced by x(l). If indices are missing e.g. between i(l−1) andi(l+1) we define
Hin[i(l) 7→∅] = Ui(0)(x(0)) · . . . · Ui(l−1)(x(l−1))Ui(l+1)
(x(l+1)) · . . . · Ui(n)(x(n)) . (18)
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 455
Finally, if the pressure is involved we write
Iin[l] = Ui(0)(x(0)) · . . . · Ui(l−1)(x(l−1))P (x(l))Ui(l+1)
(x(l+1)) · . . . · Ui(n)(x(n)) ,
(19)which is, considering all the above definitions, sufficient to derive the MPC equationsfrom the equations of instantaneous velocity and pressure i.e. equation (1) and (5).
Applying the Reynolds averaging operator (7) according to the sum below
Sin+1(x(0), . . . , x(n)) = Mi(0)(x(0))Ui(1)(x(1)) · . . . · Ui(n)
(x(n))
+ Ui(0)(x(0))Mi(1)(x(1))Ui(2)(x(2)) · . . . · Ui(n)(x(n))
+ . . .
+ Ui(0)(x(0)) · . . . · Ui(n−2)(x(n−2))Mi(n−1)
(x(n−1))Ui(n)(x(n))
+ Ui(0)(x(0)) · . . . · Ui(n−1)(x(n−1))Mi(n)
(x(n)) , (20)
we obtain the S-equation which writes
Sin+1=
∂Hin+1
∂t+
n∑
l=0
[
∂Hin+2[i(n+1) 7→k(l)][x(n+1) 7→ x(l)]
∂xk(l)
+∂Iin[l]
∂xi(l)
− ν∂2Hin+1
∂xk(l)∂xk(l)
]
= 0
for n = 1, . . . ,∞ .
(21)
Loosely speaking equation (21) implies the statistical information of the Navier-Stokes equations at the expense to deal with an infinite dimensional chain of differ-ential equations starting with order 2 i.e. n = 1. The rather remarkable consequenceof the derivation is that (21) is a linear equation which considerably simplifies thefinding of Lie symmetries to be pointed out below.
From equation (1) a continuity equation for Hin+1and Iin[l] can be derived.
This leads to∂Hin+1[i(l) 7→k(l)]
∂xk(l)
= 0 for l = 0, . . . , n (22)
and∂Iin[k][i(l) 7→m(l)]
∂xm(l)
= 0 for k, l = 0, . . . , n and k 6= l . (23)
At this point we adopt the classic notation of distance vectors. Accordingly theusual position vector x is employed and the remaining independent spatial variablesare expressed as the difference of two position vectors x(l) and x(0). The coordinatetransformation and the corresponding rules of derivation are
x = x(0) , r(l) = x(l) − x(0) with l = 1, . . . , n (24)
and∂
∂xk(0)
=∂
∂xk−
n∑
l=1
∂
∂rk(l)
,∂
∂xk(l)
=∂
∂rk(l)
for l ≥ 1 . (25)
For consistency reasons the first index i(0) is replaced by i. Thus the indices ofthe tensor Hn+1 are Hii(1)i(2)...i(n)
. Using the rules of transformation (24) and
(25) the S-equation leads to
Sin+1=
∂Hin+1
∂t+
∂Hin+2[i(n+1) 7→k][x(n+1) 7→ x]
∂xk
456 MARTIN OBERLACK AND ANDREAS ROSTECK
−n∑
l=1
[
∂Hin+2[i(n+1)7→k(l)][x(n+1)7→x]
∂rk(l)
−∂Hin+2[i(n+1)7→k(l)][x(n+1)7→r(l)]
∂rk(l)
]
+∂Iin[0]
∂xi+
n∑
l=1
−∂Iin[0]
∂rm(l)
∣
∣
∣
∣
∣
[m(l) 7→i]
+∂Iin[l]
∂ri(l)
− ν
[
∂2Hin+1
∂xk∂xk
+
n∑
l=1
(
−2∂2Hin+1
∂xk∂rk(l)
+
n∑
m=1
∂2Hin+1
∂rk(m)∂rk(l)
+∂2Hin+1
∂rk(l)∂rk(l)
)]
= 0
for n = 1, . . . ,∞ , (26)
and the two continuity equations become
∂Hin+1[i(0) 7→k]
∂xk−
n∑
j=1
∂Hin+1[i(0) 7→k(j)]
∂rk(j)
= 0 ,
∂Hin+1[i(l) 7→k(l)]
∂rk(l)
= 0 for l = 1, . . . , n
(27)
and
∂Iin[k][i7→m]
∂xm−
n∑
j=1
∂Iin[k][i7→m(j)]
∂rm(j)
= 0 for k = 1, . . . , n ,
∂Iin[k][i(l) 7→m(l)]
∂rm(l)
= 0 for k = 0, . . . , n , l = 1, . . . , n , k 6= l .
(28)
2.4.2. MPC equations: fluctuation approach. In the present subsection we adoptthe classical approach i.e. all correlation functions are based on the fluctuatingquantities u and p as introduced by Reynolds and not on the full instantaneousquantities as in the previous sub-section. Hence, similar to (15) we have the multi-point correlation for the fluctuation velocity
Rin+1= Ri(0)i(1)...i(n)
= ui(0)(x(0)) · . . . · ui(n)(x(n)) . (29)
Further, all other correlations defined in sub-section 2.4.1 are defined accordinglyi.e. equivalent to the definitions (16)-(19) we respectively define Rin+1[i(l) 7→k(l)],
Rin+2[i(n+1) 7→k(l)][x(n+1) 7→ x(l)], Rin[i(l) 7→∅] and Pin[l].
Finally, we define the correlation equation in analogy to (20) where Mi is replacedby the equation for the fluctuations (14) denoted by Ni and Ui and P are substitutedby ui and p. The resulting equation is denoted by Tin+1
Tin+1=
∂Rin+1
∂t+
n∑
l=0
[
Uk(l)(x(l))
∂Rin+1
∂xk(l)
+ Rin+1[i(l) 7→k(l)]
∂Ui(l)(x(l))
∂xk(l)
+∂Pin[l]
∂xi(l)
− ν∂2Rin+1
∂xk(l)∂xk(l)
− Rin[i(l) 7→∅]
∂ui(l)uk(l)(x(l))
∂xk(l)
+∂Rin+2[i(n+1) 7→k(l)][x(n+1) 7→ x(l)]
∂xk(l)
]
= 0
for n = 1, . . . ,∞ . (30)
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 457
The first tensor equation of this infinite chain propagates Ri2which has a close
link to the Reynolds stress tensor
limx(k)→x(l)
Ri2= lim
x(k)→x(l)
Ri(0)i(1) = ui(0)ui(1)(x(l)) mit k 6= l , (31)
which is the key unclosed quantity in the Reynolds stress transport equation (11).Here x(k) and x(l) can be arbitrary vectors out of x(0), . . . , x(n).
Also equation (30) implies all statistical information of the Navier-Stokes equa-tions. However, apart from the latter simple relation to the Reynolds stress tensorit possesses the key disadvantage of being a non-linear infinite dimensional systemof differential equations which make the extraction of Lie symmetries from thisequation rather cumbersome. There are two essential sources of non-linearity inthese equations. One is the known convection non-linearity which links the meanvelocity to all correlation equations. The second source of non-linearity originatesfrom the second row of equation (30). It is based on the fact that the gradientof the Reynolds stress tensor is contained in the equations of fluctuation. Hence,considering the following identity, this term is not equal to zero for turbulent flowsand for multi-point correlation tensors of order higher than two
Ri1[i(l) 7→∅] = 0 . (32)
As a direct consequence all multi-point correlation equations of order n > 1 arecoupled to the two-point correlation equation.
From equation (13) a continuity equation for Rin+1and Pin[l] can be derived.
They have identical form to (22) and (23) for Hin+1and Iin
and hence will beomitted for brevity.
As above in 2.4.1 the classic distance vector notation is adopted which leads to
Tin+1=
DRin+1
Dt+
n∑
l=1
[
Uk(l)(x + r(l)) − Uk(l)
(x)] ∂Rin+1
∂rk(l)
+ Rin+1[i7→k]∂Ui(x)
∂xk+
n∑
l=1
Rin+1[i(l) 7→k(l)]
∂Ui(l)(x(l))
∂xk(l)
+∂Pin[0]
∂xi+
n∑
l=1
−∂Pin[0]
∂rm(l)
∣
∣
∣
∣
∣
[m(l) 7→i]
+∂Pin[l]
∂ri(l)
− ν
[
∂2Rin+1
∂xk∂xk+
n∑
l=1
(
−2∂2Rin+1
∂xk∂rk(l)
+n∑
m=1
∂2Rin+1
∂rk(m)∂rk(l)
+∂2Rin+1
∂rk(l)∂rk(l)
)]
− Rin[i7→∅]∂uiuk(x)
∂xk−
n∑
l=1
Rin[i(l) 7→∅]
∂ui(l)uk(l)(x(l))
∂xk(l)
[x(l) 7→ x + r(l)]
+∂Rin+2[i(n+1) 7→k][x(n+1) 7→ x]
∂xk−
n∑
l=1
∂Rin+2[i(n+1) 7→k(l)][x(n+1) 7→ x]
∂rk(l)
+
n∑
l=1
∂Rin+2[i(n+1) 7→k(l)][x(n+1) 7→ r(l)]
∂rk(l)
= 0 , (33)
for n = 1, . . . ,∞, and the two continuity equations transform alike.From (6)-(9) it is apparent that there is a unique relation between the instanta-
neous and the fluctuation approach though the actual crossover is somewhat cum-bersome in particular with increasing tensor order because they may only be given
458 MARTIN OBERLACK AND ANDREAS ROSTECK
in recursive form. Since needed later we give the first relations
Hi(0) = Ui(0) (34)
Hi(0)i(1) = Ui(0) Ui(1) + Ri(0)i(1) (35)
Hi(0)i(1)i(2) = Ui(0) Ui(1) Ui(2)
+ Ri(0)i(1) Ui(2) + Ri(0)i(2) Ui(1) + Ri(1)i(2) Ui(0) + Ri(0)i(1)i(2) (36)
......
where the indices also refer to the spatial points as indicated.As a special case of the equations (26) we consider n = 1, thus we derive the equa-
tion for the two-point correlation tensor. To abbreviate the notation we introducethe following nomenclature:
Ri2= Rii(1) = Rij . (37)
In this case equation (26) reduces to
Ti2=
DRij
Dt+ Rkj
∂Ui(x, t)
∂xk+ Rik
∂Uj(x, t)
∂xk
∣
∣
∣
∣
x+r
+[
Uk (x + r, t) − Uk (x, t)] ∂Rij
∂rk+
∂puj
∂xi− ∂puj
∂ri+
∂uip
∂rj
− ν
[
∂2Rij
∂xk∂xk− 2
∂2Rij
∂xk∂rk+ 2
∂2Rij
∂rk∂rk
]
+∂R(ik)j
∂xk− ∂
∂rk
[
R(ik)j − Ri(jk)
]
= 0 . (38)
The vectors puj and uip are special cases of Pin[k] and defined as
puj(x, r, t) = p(x(0), t)uj(x(1), t) and uip(x, r, t) = ui(x(0), t) p(x(1), t) . (39)
For the two-point case the continuity equations take the form
∂Rij
∂xi− ∂Rij
∂ri= 0 ,
∂Rij
∂rj= 0 (40)
and∂pui
∂ri= 0 ,
∂ujp
∂xj− ∂ujp
∂rj= 0. (41)
The non-locality of the two- and multi-point correlation equations is most obvi-ous when we use the commutation of the two-point correlation tensor. Given
ui(x(0))uj(x(1)) = uj(x(1))ui(x(0)) with equation (24) leads to the functional re-lations
Rij(x, r; t) = Rji(x + r,−r; t) (42)
and
puj(x, r; t) = ujp(x + r,−r; t) . (43)
Analogous identities can be derived for all other two- and multi-point correlationtensors.
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 459
2.4.3. Small scale asymptotics. The following singular asymptotic behavior in thecorrelation space r is fundamental for understanding the laws of turbulence de-veloped in section 3. They are based on Kolmogorov’s idea of a cascade process[10, 11] in which the energy of the turbulent eddies is transferred to the eddy next insize. The smallest characteristic length scale in the flow is given by the Kolmogorovlength
ηK =
(
ν3
ǫ
)14
, (44)
where ǫ denotes the rate of turbulent energy dissipation. Besides this measure twoother length scales may be defined and their magnitudes may be distinguish againsteach other as follows:
ηK ≪ λ ≪ ℓt . (45)
λ denotes the Taylor length and is defined as
λ =
√
νk
ǫ(46)
and further we have the integral length scale ℓt and it characterizes the magnitudeof most energetic turbulent eddies
ℓt =1
umum
∫
RkkdΩdr , (47)
where Ω denotes the angle in space. Both are related by the turbulent Reynoldsnumber
ηK
ℓt= Re
− 34
t ,λ
ℓt= Re
− 12
t where Ret =ℓt
√k
ν, (48)
and k represents the turbulent energy k = 12ukuk. Mean velocity as well as cor-
relation, e.g. Rij , are mostly determined by the high-energy eddies of magnitudeℓt and k. Eddies of magnitude ηK merely induce the necessary turbulent energydissipation.
In the language of two- and multi-point correlation this refers to the fact thatthere has to be made a distinction between the cases |r| ≫ ηK and |r| = O(ηK).
For the first case i.e. the large scale limit we make all variables dimensionlesswith ℓt and k. Implemented into the two-point correlation equations and taking thehigh Reynolds number limit i.e. Ret → ∞ this leads to a reduced form of equation(38):
T >
i2=
DR>
ij
Dt+ R>
kj
∂u>
i (x, t)
∂x>
k
+ R>
ik
∂u>
j (x, t)
∂x>
k
∣
∣
∣
∣
x+r
+ [u>
k (x + r, t) − u>
k (x, t)]∂R>
ij
∂r>
k
+∂pu>
j
∂x>
i
−∂pu>
j
∂r>
i
+∂uip
>
∂r>
j
+∂R>
(ik)j
∂x>
k
− ∂
∂r>
k
[
R>
(ik)j − R>
i(jk)
]
= 0 , (49)
where “>” stands for |r| ≫ ηK . Analogously, all multi-point correlation equationswithout viscosity terms can be derived. Obviously, for |r| ≫ ηK mean velocityand correlation functions are determined by the frictionless part of the correlationequations which is a long standing folk wisdom in the turbulence community.
460 MARTIN OBERLACK AND ANDREAS ROSTECK
In the second case of r being in the order of the Kolmogorov length scale, thedissipation term, which is the last term in the fourth row in equation (38), mustnot be omitted.
The limit demands r to be made dimensionless with λ and the odd moments ofvelocity and the pressure-velocity correlation with k
32 λ/ℓt. All remaining variables
are still rescalled by ℓt and k. In the leading order for Ret → ∞ we obtain
T <
i2=
DR<
ij
Dt+ R<
kj
∂u<
i (x, t)
∂x<
k
+ R<
ik
∂u<
j (x, t)
∂x<
k
+∂u<
k (x, t)
∂x<
l
r<
l
∂R<
ij
∂r<
k
−∂pu<
j
∂r<
i
(50)
+∂uip
<
∂r<
j
− 2∂2R<
ij
∂r<
k ∂r<
k
− ∂
∂r<
k
[
R<
(ik)j− R<
i(jk)
]
= 0 .
Analogous equations can be obtained for higher correlations. The essential differ-ence to equation (49) is that the second derivative of Rij with respect to r remainspresent. In the limit of r = 0 the latter expression represents the rate of dissipation.
The above splitting of the characteristic length scales allows to separate largescales from energy dissipation in their influence on the velocity.
The remaining of the paper deals with the large-scale quantities i.e. especiallythe mean velocity and the correlation tensors.
Due to length and complexity of the full multi-point tensor equations (21) and(30) mathematical details are primarily focused on the two-point correlations andthe related equations though can be extended straight forward for all higher ordercorrelations.
3. Symmetries of statistical transport equations. In the present section wefirst revisit the Lie symmetries of the Euler and Navier-Stokes equations. In turnthey will all be transferred to its corresponding ones for the MPC equations. Inthe second part we show that the MPC equations admit even more Lie symmetrieswhich are not reflected in the original Euler and Navier-Stokes equations.
Both sets of symmetries will finally be employed in section 4 to show that classicaland new scaling laws may not be from the classical symmetries alone but essentiallyrely on the new symmetries which we will call statistical symmetries.
The Lie groups will either be presented in global or in infinitesimal form [3].
3.1. Symmetries of the Euler and Navier-Stokes equations. The Euler equa-tions, i.e. equation (1) and (5) with ν = 0 admit a ten-parameter symmetry group,here in infinitesimal from,
X1 =∂
∂t,
X2 = xi∂
∂xi+ Uj
∂
∂Uj+ 2P
∂
∂P,
X3 = t∂
∂t− Ui
∂
∂Ui− 2P
∂
∂P,
X4 = −x2∂
∂x1+ x1
∂
∂x2− U2
∂
∂U1+ U1
∂
∂U2
X5 = −x3∂
∂x2+ x2
∂
∂x3− U3
∂
∂U2+ U2
∂
∂U3
X6 = −x3∂
∂x1+ x1
∂
∂x3− U3
∂
∂U1+ U1
∂
∂U3
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 461
X7 = f1(t)∂
∂x1+
df1(t)
dt
∂
∂U1− x1
d2f1(t)
dt2∂
∂P,
X8 = f2(t)∂
∂x2+
df2(t)
dt
∂
∂U2− x2
d2f2(t)
dt2∂
∂P,
X9 = f3(t)∂
∂x3+
df3(t)
dt
∂
∂U3− x3
d2f3(t)
dt2∂
∂P,
X10 = f4(t)∂
∂P, (51)
with f1(t)-f3(t) being at least differentiable twice and f4(t) may have arbitrarytime dependence.
The entire set of Lie symmetries span a linear vector space or may be interpretedas a multi-parameter group of the form
X =
10∑
i=1
ciXi (52)
with ci being arbitrary constants.The symmetries X7-X9 comprise translational invariance in space for constant
f1-f3 as well as the classical Galilei group if f1-f3 are linear in time. In its rathergeneral form X7-X9 and X10 are direct consequences of an incompressible flow anddo not have a counterpart in the case of compressible flows. The complete recordof all point-symmetries (51) was first published by Pukhnachev [18].
Making the formal transfer from Euler to the Navier-Stokes equations symmetryproperties change and a recombination of the two scaling symmetries X2 and X3 isobserved
XScaleNS = 2t∂
∂t+ xi
∂
∂xi− Uj
∂
∂Uj− 2P
∂
∂P, (53)
while the remaining groups stay unaltered.It should be noted that further symmetries exist for dimensional restricted cases
such as plane or axisymmetric flows (see e.g. [1, 4]).Lie’s key theorem (see e.. [3]) states the full equivalence between the infinitesimal
and the global form of a Lie group and hence we may give the symmetries (51) alsoin global form
T1 : t∗ = t + a1 , x∗ = x , U∗ = U , P ∗ = P ,
T2 : t∗ = t , x∗ = ea2x , U∗ = ea2U , P ∗ = e2a2P ,
T3 : t∗ = ea3t , x∗ = x , U∗ = e−a3U , P ∗ = e−2a3P ,
T4 − T6 : t∗ = t , x∗ = a · x , U∗ = a · U , P ∗ = P ,
T7 − T9 : t∗ = t , x∗ = x + f (t) , U∗ = U +df
dt, P ∗ = P − x · d2f
dt2,
T10 : t∗ = t , x∗ = x , U∗ = U , P ∗ = P + f4(t) , (54)
where a1-a3 are independent group-parameters, a denotes a constant rotation ma-trix with the properties a·aT = aT·a = I and |a| = 1, and f(t) = (f1(t), f2(t), f3(t))
T
respectively f4(t) satisfies the conditions mentioned after (51).
3.2. Symmetries of the MPC implied by Euler and Navier-Stokes sym-
metries. For brevity and ease of reading the analysis to follow in this and the nextsubsection 3.3 we primarily limit ourselves to the global form of the symmetries.
462 MARTIN OBERLACK AND ANDREAS ROSTECK
Adopting the classical Reynolds notation first where the instantaneous quantitiesare split into mean and fluctuating values we may directly derive from (6)-(9) and(54)
T1 : t∗ = t + a1, x∗ = x, r∗(l) = r(l), U
∗= U , P ∗ = P ,
R∗n = Rn, P
∗n = Pn,
T2 : t∗ = t, x∗ = ea2x, r∗(l) = ea2r(l), U
∗= ea2U , P ∗ = e2a2 P ,
R∗n = ena2Rn,P
∗n = e(n+2)a2Pn,
T3 : t∗ = ea3t, x∗ = x, r∗(l) = r(l), U
∗= e−a3U , P ∗ = e−2a3P ,
R∗n = e−na3Rn,P
∗n = e−(n+2)a3Pn,
T4−T6 : t∗ = t, x∗ = a · x, r∗(l) = r(l), U
∗= a · U , P ∗ = P ,
R∗n = An ⊗ Rn, P
∗n = An ⊗ Pn,
T7−T9 : t∗ = t , x∗ = x + f(t), r∗(l) = r(l), U
∗= U +
df
dt, P ∗ = P − x · d2f
dt2,
R∗n = Rn, P
∗n = Pn,
T10 : t∗ = t , x∗ = x, r∗(l) = r(l), U
∗= U , P ∗ = P + f4(t),
R∗n = Rn, P
∗n = Pn, (55)
where all function and parameter definitions are adopted from 3.1 and A is a con-catenation of rotation matrices as Ai(0)j(0)i(1)j(1)...i(n)j(n)
= ai(0)j(0)ai(1)j(1) . . . ai(n)j(n).
The latter symmetries may also be transformed into the H-notation. This willbe omitted for briefness and also because scaling laws have never been consideredin this notation.
3.3. Statistical symmetries of the MPC equations. First hints towards aconsiderably extended set of symmetries for the MPC equation may e.g. be takenfrom [14, 9]. Its importance were not observed therein - rather it was stated thatthey may be mathematical artifacts of the averaging process and probably physicallyirrelevant.
The actual finding of symmetries of the non-rotating MPC is rather difficultsince an infinite system of equations has to be analyzed. For this task, however,it is considerably easier to investigate the linear H-I-system (26)-(28) rather thanthe non-linear R-P-system (33) extended by its corresponding continuity equations.Since the latter formulation, however, is more common symmetries will finally bere-written in this notation.
Overall three sets of new symmetries have been identified. The first one may bewritten as
T ′1 : t∗ = t, x∗ = x, r∗
(l) = r(l) + a(l), H∗n = Hn, I
∗n = In, (56)
where a(l) represents the related set of group parameters. Note that this groupis not related to the classical translation group in usual x-space (here T7 − T9 inequation (54) with f = const.). In classical notation we have
T ′1 : t∗ = t, x∗ = x, r∗
(l) = r(l) + a(l), U∗
= U , P ∗ = P ,
R∗n = Rn, P
∗n = Pn. (57)
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 463
The second set of statistical symmetries was in fact already partially identifiedin [15] however, falsely taken for the Galilean group. In its general form it reads
T ′2n
: t∗ = t, x∗ = x, r∗(l) = r(l), H
∗n = Hn + Cn, I
∗n = In + Dn,
(58)where Cn and Dn refer to group parameters independent for each of the accord-ing tensor orders. To clarify this we single out the groups for the first two tensororder i.e.
T ′21
: t∗ = t, x∗ = x, r∗(l) = r(l), H∗
i(0)= Hi(0) + Ci(0) , H∗
i(0)i(1)= Hi(0)i(1) , · · ·
(59)and
T ′22
: t∗= t, x∗= x, r∗(l) = r(l), H∗
i(0)= Hi(0) , H∗
i(0)i(1)= Hi(0)i(1) + Ci(0)i(1) , · · · .
(60)Note, that even (59) and (60) refer to multi-parameter Lie groups since the constantsCi(0) and Ci(0)i(1) denote three and nine independent group parameters respectively.This may be further extended by also varying the parameter for Dn. Apparentlythe above notation implies a very large number of groups since each tensor has itsown independent corresponding group parameter.
Though each of these groups appear to be almost trivial since they are simpletranslational groups in the dependent coordinates they exhibit an increasingly com-plexity with increasing tensor order if written in the U -R formulation. Here, we usethe above two examples and rewrite them using (34) and (35) yielding
T ′21
: t∗ = t, x∗ = x, r∗(l) = r(l), U∗
i(0)= Ui(0) + Ci(0) ,
R∗i(0)i(1)
= Ri(0)i(1) + Ui(0)Ui(1) −(
Ui(0) + Ci(0)
) (
Ui(1) + Ci(1)
)
, · · · (61)
and
T ′22
: t∗ = t, x∗ = x, r∗(l) = r(l), U∗
i(0)= Ui(0) ,
R∗i(0)i(1)
= Ri(0)i(1) + Ci(0)i(1) , · · · . (62)
Finally, the third statistical group that has been identified denotes simple scalingof all MPC tensors as may be directly read off from equation (21)
T ′s : t∗ = t, x∗ = x, r∗
(l) = r(l), H∗n = easHn, I
∗n = easIn, . (63)
Again, employing (34) and (35) we obtain this group in classical notation thoughonly a recursive form may be given
T ′s : t∗ = t, x∗ = x, r∗
(l) = r(l),
U∗i(0)
= easUi(0) , R∗i(0)i(1)
= eas[
Ri(0)i(1) + (1 − eas) Ui(0) Ui(1)
]
, · · · . (64)
It should be finally added that due to the linearity of the MPC equation (21)another rather generic symmetry is admitted. This is in fact featured by all lineardifferential equations (see e.g. [3]). It merely reflects the super-position propertyof linear differential equations though usually cannot directly be adopted for thepractical derivation of group invariant solutions.
464 MARTIN OBERLACK AND ANDREAS ROSTECK
4. Turbulent scaling laws. The rather old idea of a turbulent scaling law (seee.g. [17]) usually refers to two distinct facts:
• Introducing a certain set of parameters to non-dimensionalize statistical tur-bulence variables such as the mean velocity leads to a collapse of data if oneexternal parameter is varied such as the Reynolds number.
• An explicit mathematical function is given for statistical turbulence variablessuch as the mean velocity, Reynolds stresses, etc..
Presently we only contemplate with the second definition. In order to rigorouslyderive such laws directly from the MPC equations we need to introduce the definitionof a group invariant solution (see e.g. [3]).
Suppose F = 0 is the set of differential equations under investigation, X denotesthe multi-parameter Lie symmetry group taken from the condition [XF ]|
F=0 =0 and, generically as above, x and y refer to the entire set of independent anddependent variables respectively. We define the group invariant solution y = Θ(x)comprising the two conditions
(i) y − Θ(x) is invariant under X and(ii) y = Θ(x) is a solution of the differential equation F=0 .
With this we obtain
X [y − Θ(x)] = 0 where y = Θ(x) . (65)
Differentiating this out we obtain the hyperbolic system
ξk(x,Θ)∂Θl
∂xk= ηl(x,Θ) (66)
to be solved by the method of characteristics which finally ends up with the char-acteristic condition
dx1
ξ1(x, y)=
dx2
ξ2(x, y)= · · · =
dxm
ξm(x, y)
=dy1
η1(x, y)=
dy2
η2(x, y)= · · · =
dyn
ηn(x, y), (67)
where Θ has been replaced by y. The latter is usually referred to invariant surfacecondition.
In the remaining two subsections we adopt the latter condition for the derivationof the accordant invariant solution alternatively later also named scaling laws, whichis the usual phrase in the turbulence literature.
4.1. Stationary wall-bounded turbulent shear flows. Due to its eminent prac-tical importance wall-bounded shear flow is by far the most intensively investigatedturbulent flows thereby employing a vast number of numerical, experimental andmodeling approaches and this, in fact, for more than a century.
From all the theoretical approaches the universal law of the wall is the mostwidely cited and also accepted approach with its essential ingredient the logarith-mic law. Though a variety of different approaches have been put forward for itsderivation neither of them employ the multi-point equation, which is the basis forstatistical turbulence, nor do they solve an equation that is related to the Navier-Stokes equations.
In the following we demonstrate that the log-law is an invariant solution of theinfinite set of multi-point equations and further it is shown that it essentially relieson one of the new translational groups.
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 465
Already in [13] it was observed that in the limit of high Reynolds numbers and|r| ≫ ηK the logarithmic wall law allows for a self-similar solution of the two-pointcorrelation equation (38). This is rather remarkable since for inhomogeneous flowsequation (38) is not a partial differential equation in the classic sense but a non-local differential equation. Non-locality is denoted by the fact that for a given pointx and r not only the dependent variables and derivatives are connected but alsoterms “at the point” x+r contribute to the equation. In equation (38) this is givenby the last term in the first row and the first term in the second row.
Indeed the terms mentioned above were the major cause for the limitation ofthe two-point correlation equation to homogeneous flows i.e. flows where the meanvelocity gradient is a constant, or even more simple the mean velocity itself is aconstant. It is important to note that this limitation is not needed for the presentgroup theoretical approach.
Within this subsection we exclusively examine wall-parallel turbulent flows onlydepending on the wall-normal coordinate x2. Further, we only explicitly writethe two-point correlation Rij though all results are also valid for all higher ordercorrelations. This finally yields
U1 = U1(x2), Rij = Rij(x2, r), . . . . (68)
With these geometrical assumptions we identify a reduced set of groups. Thetwo scaling groups in (55) may in infinitesimal form be written as
X2 = x2∂
∂x2+ U1
∂
∂U1+ ri
∂
∂ri+ 2Rij
∂
∂Rij+ . . . (69)
and
X3 = −U1∂
∂U1− 2Rij
∂
∂Rij+ . . . , (70)
where in the latter the term ∂/∂t has been omitted since only statistically steadyflows according to (68) are considered. The translation invariance in x2-directionhas the form
Xx2 =∂
∂x2(71)
and all previous three symmetries are implied by the Navier-Stokes equations. Fur-ther we have the statistical groups i.e. the translational group in correlation space(57)
Xri=
∂
∂ri(72)
and the translational group (61) for U1
XU1=
∂
∂U1− (U1(x2) + U1(x2 + r2))
∂
∂R11+ . . . (73)
and (62) for Rij
XRij=
∂
∂Rij+ . . . (74)
and finally the scaling group (64)
Xs = U1∂
∂U1+(
Rij − U1(x2)U1(x2 + r2)δi1δj1
) ∂
∂Rij+ . . . . (75)
In some earlier work on group invariance of two-point correlations [14, 15] (73)was not complete and was falsely interpreted as Galilean group. The symmetries(69)-(75) are mutually independent transformations of the two- and, if fully written
466 MARTIN OBERLACK AND ANDREAS ROSTECK
out, of the MPC equations under the restriction (68). Hence the linear combinationof the above symmetries form a multi-parameter group
X = k2X2 + k3X3 + kx2Xx2 + kriXri
+ kU1XU1
+ kRijXRij
+ ksXs (76)
and hence constitute a new symmetry of the entire set of correlation equations.Using the latter multi-parameter group in the equation for invariant solutions
(67) we obtain the invariance condition
dx2
k2x2 + kx2
=dr[k]
k2r[k] + kr[k]
=dU1
(k2 − k3 + ks)U1 + kU1
=dR[ij]
ξR[ij]
= · · · with
ξRij= (2k2 − 2k3 + ks)Rij
−(
ksU1(x2)U1(x2 + r2) + kU1(U1(x2) + U1(x2 + r2))
)
δi1δj1 + kRij(77)
where no summation is implied by the indices in square brackets and instead aconcatenation is implied where the indices are consecutively assigned its values.For brevity explicit dependencies on the independent variables are only given wherethere is an unambiguity.
In particular with a distinct combinations of parameters k2, k3 and ks a multitudeof flows may be described where here we only focus on the log-law. We may keepin mind that U1 exclusively depends on x2 and not on r.
The meaning of the scale symmetries (69), (70) and (75) can be discussed mostefficiently by investigating its concatenated global transformations T2 and T3 in (55)and (64) for the mean velocity and the space coordinate
x∗2 = ek2x2 and U∗
1 = ek2−k3+ks U1 . (78)
For the present case to examine the classic case of the logarithmic wall law itsbasic idea of a symmetry breaking in (78) can be best seen by revisiting the keyidea of von Karman. He assumed that close to the wall the wall-friction velocityuτ is the only parameter determining the flow. He defined the wall-friction velocityby investigating the balance of momentum (11) in x1-direction close to the wallassuming a parallel shear flow without any pressure gradient. Only the viscous andturbulent shear stresses remain and the resulting equation may be integrated oncewith respect to x2 leading to
ν∂U1
∂x2− u1u2 =
τw
ρ= u2
τ , (79)
where τw is the wall shear stress.Since uτ is a given external parameter specifying the velocity scale, scaling of
U1 and thereby an arbitrary choice of the parameters k2, k3 and ks in (78) is notadmissible. Thus we find
k2 − k3 + ks = 0 , (80)
since uτ causes a symmetry breaking in the scalability of U1.Under this assumption (77) leads to the classical functional form of the mean
velocity
U1 =kU1
k2ln
(
x2 +kx2
k2
)
+ Clog (81)
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 467
and the invariant correlations read
rk =rk
x2 +kx2
k2
, Rij =
(
x2 +kx2
k2
)− ksk2
Rij(r) +kRij
ksfor ij 6= 11
R11 = R11(r)
(
x2 +kx2
k2
)−ks/k2
−(
Clog kU1
k2ln(r2 + 1) + C2
log − kR11
ks
)
−(
2kU1
k2Clog +
k2U1
k22
ln(r2 + 1)
)
ln
(
x2 +kx2
k2
)
−k2
U1
k22
ln2
(
x2 +kx2
k2
)
,
. . . ,
(82)
where Clog in (81) and the variables marked with “˜” in (82) are constants ofintegration. These are the new invariant coordinates solely depending on r. Clog isan exception since according to (68) U1 depends on x2 alone.
Obviously equation (81) is a slightly generalized form of the classic logarithmic
wall law since by the termkx2
k2a displacement of the origin is admitted. In its
classical dimensionless form it reads
u+ =1
κln(x+
2 + A+) + C . (83)
As betoken above the non-locality of the two- and multi-point correlation equa-tions appears not to hinder the analysis and, of course, a reduced form of theequation may be given for any multi-point correlation tensor equation.
An experimental validation of the logarithmic wall law is not given here since ithas been verified in numerous publications.
The invariant variable r in (82) for the logarithmic wall law was already givenby Hunt et al. [5] though in a less general form and has been numerically verifiedby DNS data of a turbulent channel flow. Especially for the two-point correlationtensor R22 this data matches very well. This is very remarkable since the DNS datais based on a low Reynolds number. The considerations taken by Hunt et al. arenot based on an examination of the two-point correlation equations but merely ona dimensional analysis of the independent coordinate.
4.2. Turbulent decay scaling laws. Decaying turbulence is, if also homogeneityand isotropy is invoked, the most simple turbulent flow one can think of. In this casethe two-point correlation equations reduce to the von Karman-Howarth equation- a scalar transport equation for the correlation function [7]. Physically it refersto the fact that fluid in an infinite domain is stirred to generate a statisticallyhomogeneous and isotropic flow field and then the stirring force is interrupted andturbulence decays freely.
Classical theories on decaying turbulence in the limit of large Reynolds numbersuch as Birkhoff’s [2] and Loitsyansky’s [12] integral entirely rely on the groups T2
and T3 and, if also a time shift is considered, also on T1. These two scaling groupsgive rise to a one-parameter family of invariant solutions where e.g. the turbulentkinetic energy decays and the integral length scale increases according to a powerlaw
k ∼ (t + t0)−m , ℓt ∼ (t + t0)
n (84)
where the exponent is usually settled by one of the above proposed conserved inte-grals (see e.g. [16]).
468 MARTIN OBERLACK AND ANDREAS ROSTECK
Presently using the above extended set of symmetry groups (56)-(64) restrictedto the present flow case a much wider class of invariant solutions or turbulent scalinglaws is admissible. The combined infinitesimals read
X = k1X1 + k2X2 + k3X3 + kriXri
+ kRijXRij
+ ksXs . (85)
Employing the latter in (67) we obtain the invariant surface condition
dt
k3t + k1=
dr[k]
k2r[k] + kr[k]
=dR[ij]
(2k2 − 2k3 + ks)R[ij] + kR[ij]
= · · · (86)
with the notation identical to the one explained below (77).It is important to note that any solution for an arbitrary set of group parameters
of the latter invariant surface condition allows for an invariant solution of (33).Three fundamentally different cases may be distinguished depending on the choiceof the scaling group parameters k2, k3 and ks.
The classical cases may easily be extended if we assume k2, k3 and ks are distinctfrom each other and non-zero. With this we find the following invariants of thesystem (86)
rk =rk +
krk
k2
(t + t0)n, Rij(r, t) = (t + t0)
−mRij (r) , . . . . (87)
Here m and n are indeed independent numbers simply related to the scaling groupparameters. If, however, we limit ourselves to the classical case i.e. ks = 0 weobtain that n = k2/k3 and m = 2(1−k2/k3). The variables r and Rij are again theconstants of integration of the invariant surface condition (86) or in other words theinvariants of the system. They are to be taken as new independent and dependentvariables of the system (33) leading to a similarity reduction depicting the classicalpower law behavior. Therein m = 6/5, n = 2/5 and m = 10/7, n = 2/7 respectivelycorrespond to Birkhoff’s and Loitsiansky’s integrals. An overview on the classicalcases and its variations is given in [16].
The most remarkable new scaling behavior of decaying turbulence, however, wasobserved by [6, 19]. Therein they reported on an experiment where they generatedturbulence with a grid of fractal structure that is rather different to the classicalgrids which are usually manufactured from simple bars of a given thickness assem-bled in a rectangular manner at a given constant distant from each other. Thefractal grids have different base forms e.g. one set is installed by squares made ofbars. These squares have different sizes composing a geometrical series. On all ofthe four corners of the squares the next smaller square is mounted and this is iter-ated up to the fourth or fifth level. In the actual experiment air is blown throughthe fractal grid generating turbulence on very different length scales which is ratherdifferent from the classical square grids of a single size.
In order to interpret this flow from a symmetry point of view we rewrite thescaling groups in global form to obtain
t∗ = ek3t and R∗ij = e2(k2−k3)+ksRij , (88)
where we should recall from above that k2 refers to the scaling group of space.Interpreting the length of the different squares as symmetry breaking we imme-
diately observe that k2 = 0. If we further take the ratios of the flow velocity atwhich the grid is flown at and the different square grids we obtain a multitude oftime scales acting on the flows and hence we have a symmetry breaking of timemeaning k3 = 0.
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 469
Employing the latter information into equation (86) we observe two importantconclusions. First, due to k3 = 0 and combining the first and the third term in (86)we have an exponential scaling of the two- and MPC with time. The second casei.e. the first two terms in (86) for t and r is somewhat more delicate because onemight expect some traveling wave type of behavior due to the only non-zero groupparameters k1 and kr[k]
. This however is physically not feasible and hence we haveto further take kr[k]
= 0. With this we may deduce that r itself is an invariant since
all infinitesimals in the denominator of the second term in (86) vanish.Following the methodology above this leads to a invariant solution for the infinite
set of MPC tensors (33) where the first term in the row, i.e. the two-point tensor,has the following form
r = r , Rij(r, t) = e−t/t0Rij (r) , . . . , (89)
where Rij is the variable of the reduced set of MPC equations only depending onthe similarity r.
In order to compare (89) to experimentally observable one-point quantities werecall the relation between two-point correlation and Reynolds stress tensor (31)which leads to the turbulent kinetic energy k = 1
2ukuk and the integral length ℓt
scale defined in (47).If we finally employ (89) into the latter definitions we obtain the new turbulent
scaling law
k ∼ e−t/t0 , ℓt ∼ const. (90)
which is very different from the classical algebraic decay law.It was in fact exactly this behavior that was first reported in [6] and fully consol-
idated in [19]. For certain cases they also find a constant integral length scale (47)(and also Taylor length scale) downstream of the fractal grid. A variety of differentfractal grids were employed for the experiment such as cross-grids, square-grids andI-grids.
From the different sets of measurements they conducted it was noticed that thestronger the stretching of the fractal grid and bar spacing was the better the abovescaling behavior (90) was observed.
The physical interpretation of the latter results is that a broad bandwidth ofexternal scales have been imposed on the flow which are symmetry breaking.
Beside the classical algebraic scaling laws and the latter new exponential scalinglaw for decaying turbulence, we report another new turbulent scaling law which,to the best or the authors knowledge, has never been observed experimentally orreported from theoretical considerations.
For this we consider that there is a constant time scale, say τ0, acting on theflow i.e. the scaling of time is broken and hence k3 = 0. Employing the latter intothe invariant surface condition (86) and integrate it leads to the following invariantsolution for the correlation tensors
r = re−t/τ0 , Rij(r, t) = e−γt/τ0Rij (r) , . . . , (91)
where essentially only the scaling groups of space, the statistical scaling group andthe translation group in time respectively corresponding to the parameters k2, ks
and k1 have been employed.Translated into the language of one-point quantities we obtain
k ∼ e−γt/τ0 , ℓt ∼ et/τ0 , (92)
470 MARTIN OBERLACK AND ANDREAS ROSTECK
where in the latter two equations γ emerged from the scaling group parametersk2 and ks. This result, however, is to be be validated in future experiments ornumerical simulations.
5. Summary and outlook. Within the present paper it was shown that the ad-mitted symmetry groups of the infinite set of multi-point correlation equations areconsiderable extended by three classes of groups compared to those originally stem-ming from the Euler and the Navier-Stokes equations. In fact, it was demonstratedthat it is exactly these symmetries which are essentially needed to validate certainclassical laws from first principles and to derive new scaling laws.
Still, even with the new symmetry groups at hand which give a much deeperunderstanding on turbulence statistics there are still some key open questions to beanswered:
• From turbulence data it is apparent that the appearing group parameters dohave certain decisive values which are to be determined. In some very rarecases such as the classical decaying turbulence case values such as the de-cay exponent can be determined. In others, such as the logarithmic law, thenumerical value of the parameters may presently only be taken from experi-ments.
• Solutions such as the logarithmic law only explain certain regions of a turbu-lent flow and are usually embedded within other layers of turbulence. It is anopen questions how turbulent scaling laws may be matched to each other.
REFERENCES
[1] V. K. Andreev and A. A. Rodionov, Group analysis of the equations for plane flows of an
ideal fluid in the Lagrange variables, Dokl. Akad. Nauk SSSR, 298 (1988), 1358–1361.[2] G. Birkhoff, Fourier synthesis of homogeneous turbulence, Commun. Pure Appl. Math., 7
(1954), 19–44.[3] G. Bluman, A. Cheviakov and S. Anco, “Applications of Symmetry Methods to Partial Dif-
ferential Equations,” Springer-Verlag, New York, 2009.[4] B. J. Cantwell, Similarity transformations for the two-dimensional, unsteady, stream-function
equation, J. Fluid Mech., 85 (1978), 257–271.[5] J. C. R. Hunt, P. Moin, R. D. Moser and P. R. Spalart, Self-similarity of two-point correlation
in wall bounded turbulent flows, Proceedings of the Center for Turbulence Summer School,(1987), 25–36.
[6] D. Hurst and J. C. Vassilicos, Scalings and decay of fractal-generated turbulence, Phys. Fluids,19 (2007), 035103.
[7] Th. von Karman and L. Howarth, On the statistical theory of isotropic turbulence, Proc. Roy.Soc. A, 164 (1938), 192–215.
[8] L. Keller and A. Friedmann, Differentialgleichungen fur die turbulente Bewegung einer kom-
pressiblen Flussigkeit, Proc. First. Int. Congr. Appl. Mech., (1924), 395–405.[9] J. Khujadze and M. Oberlack, DNS and scaling laws from new symmetry groups of ZPG
turbulent boundary layer flow, Theoret. Comput. Fluid Dyn., 18 (2004), 391–411.[10] A. N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, C.R. Acad. Sci.
USSR, 32 (1941), 16–18.[11] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluids for very
large Reynolds numbers, Dokl. Akad.Nauk SSSR, 30 (1941).[12] L. G. Loitsyansky, Some basic laws of isotropic turbulent flow, Centr. Aero. Hydrodyn. Inst.
Moscow, 440 (1939).[13] M. Oberlack, Analysis of the two-point velocity correlations in turbulent boundary layer flows,
Annual Research Briefs, Center for Turbulence Research, Stanford University/NASA Ames,CA, USA, (1995), 209–220.
[14] M. Oberlack, “Symmetrie, Invarianz und Selbstahnlichkeit in der Turbulenz,” Habilitationthesis, RWTH Aachen, 2000.
NEW STATISTICAL SYMMETRIES AND TURBULENT SCALING LAWS 471
[15] M. Oberlack, A unified approach for symmetries in plane parallel turbulent shear flows, J.Fluid Mech., 427 (2001), 299–328.
[16] M. Oberlack, On the decay exponent of isotropic turbulence, Proc. Appl. Math. Mech., 1
(2002), 294–297.[17] St. B. Pope, “Turbulent Flows,” Cambridge University Press, Cambridge, 2000.[18] V. V. Pukhnachev, Invariant solutions of Navier-Stokes equations describing motions with
free boundary, Dokl. Akad. Nauk, 202 (1972), 302.[19] R. E. Seoud and J. C. Vassilicos, Dissipation and decay of fractal-generated turbulence, Phys.
Fluids, 19 (2007), 105108.
Received March 2010; revised May 2010.
E-mail address: [email protected]
E-mail address: [email protected]
