Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows

Flow, Turbulence and Combustion62: 111–135, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

111

Symmetries, Invariance and Scaling-Laws inInhomogeneous Turbulent Shear Flows

MARTIN OBERLACKInstitut für Technische Mechanik, RWTH Aachen, Templergraben 64, 52056 Aachen, Germany

Received 27 March 1998; accepted in revised form 10 May 1999

Abstract. An approach to derive turbulent scaling laws based on symmetry analysis is presented.It unifies a large set of scaling laws for the mean velocity of stationary parallel turbulent shearflows. The approach is derived from the Reynolds averaged Navier–Stokes equations, the fluctuationequations, and the velocity product equations, which are the dyad product of the velocity fluctuationswith the equations for the velocity fluctuations. For the plane case the results include the logarithmiclaw of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couetteflow and in the centre of a rotating channel flow, and a new exponential mean velocity profile thatis found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraicscaling law is confirmed in both the centre and the near wall regions in both experimental and DNSdata of turbulent channel flows. For a non-rotating and a moderately rotating pipe about its axis analgebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both lawshaving equal scaling exponents. In case of a rapidly rotating pipe, a new logarithmic scaling law forthe axial velocity is developed. The key elements of the entire analysis are two scaling symmetriesand Galilean invariance. Combining the scaling symmetries leads to the variety of different scalinglaws. Galilean invariance is crucial for all of them. It has been demonstrated that two-equation modelssuch as thek–ε model are not consistent with most of the new turbulent scaling laws.

Key words: turbulence, scaling-laws, symmetries.

1. Introduction

The logarithmic law was first derived by von Kármán [8] using empirical mod-els and dimensional arguments. Later, Millikan [13] derived the law of the wallmore formally using the so-called “velocity defect law”, which was also intro-duced by von Kármán [8]. Even though Millikan’s derivation was much morecomprehensive from a mathematical point of view, it still had no link to the Navier–Stokes equations. Recently, some doubts have been expressed as to whether theappropriate wall-layer form is logarithmic or algebraic (see, e.g., [5]). In [15, 16],it is demonstrated that the law of the wall and other plane and round scalinglaws can be obtained from Lie group or symmetry analysis of Reynolds averagedNavier–Stokes equations.

In the present approach, the work in [15, 16] will be reexamined for both planeand circular shear flows by employing a more intuitive approach. In particular, itwill be focused on the scales characterizing the flow and how they give rise to the

112 M. OBERLACK

different scaling laws to be derived in the subsequent sections. Finally, all scalinglaws will be empirically verified by analyzing data from experiments and directnumerical simulations. Specific emphasis will be put on modelling implications ofthe symmetries and the turbulent scaling laws.

In order to clarify the analysis below, a few concepts have to be introduced.This is in particular the concept of symmetries of differential equations. Supposethe system of partial differential equations under investigation is given by

F (y, z, z(1), z(2), . . .) = 0, (1)

wherey andz are the independent and the dependent variables respectively andz(n) refers to allnth-order derivatives ofz with respect toy. A transformation

y = φ(y∗, z∗) and z = ψ(y∗, z∗) (2)

is called a symmetry or symmetry transformation of Equation (1) if the followingequivalence holds

F (y, z, z(1), z(2), . . .) = 0⇔ F (y∗, z∗, z∗(1), z∗(2), . . .) = 0, (3)

i.e., the transformation (2) substituted into Equation (1) does not change the formof Equation (1) if written in the new variablesy∗ andz∗.

Once there are one or several symmetries of the partial differential (1) known,one can use them to compute self-similar solutions to be shown utilizing the ex-ample of the one-dimensional heat equation

∂u

∂t= ∂2u

∂x2 . (4)

Among others, Equation (4) admits the following symmetry transformation

t∗ = exp(2ε)t, x∗ = exp(ε)x, u∗ = u, (5)

whereε is an arbitrary parameter.The crucial element to understand similarity solutions in general and the turbu-

lent scaling laws in the subsequent sections is the concept of invariant functions.Suppose the symmetry transformations (2) of Equation (1) are known, i.e., trans-formations which satisfy Equation (3) have been derived. A functionf (y, z) iscalled an invariant under transformation (2) if the following condition is met

f (y, z) = f (y∗, z∗). (6)

The symmetry transformation of the heat equation may serve as an example. Itis obvious that the term

η = x2

t(7)

THEORY OF TURBULENT SHEAR FLOWS 113

is invariant under transformation (5) since it does not change its functional formwritten in the new variables

x2

t= x∗2

t∗. (8)

η is the classical similarity variable of the one-dimensional heat equation and en-ables to construct a variety of self-similar solutions. In the same manner these ideaswill also be adopted for the present purpose of finding scaling laws in turbulentshear flows which are invariant solutions under several symmetry transformationsof the Reynolds averaged Navier–Stokes equations.

The paper is organized as follows. In Section 2, the governing equations arederived and major differences between the present approach and the classical ap-proach using the Reynolds stress tensor equations will be pointed out. In Section 3,invariant functions will be investigated which are scaling laws of turbulent meanflow. In Section 4, the new scaling laws which are in fact self-similar solutionswill be compared with experimental and DNS data. In Section 5, the results will bediscussed in view of turbulence modelling by investigating the classicalk–ε model.

2. Governing Equations

The subsequent analysis is based on the Reynolds averaged Navier–Stokes equa-tions

N i = ∂ui

∂t+ uk ∂ui

∂xk+ ∂p

∂xi− ν ∂

2ui

∂x2k

+ ∂τik∂xk+ 2�k eikl ul = 0, (9)

the equations for the velocity fluctuations

Ni = ∂ui

∂t+ uk ∂ui

∂xk+ uk ∂ui

∂xk− ∂τik∂xk+ ∂uiuk

∂xk

+ ∂p

∂xi− ν ∂

2ui

∂x2k

+ 2�k eikl ul = 0 (10)

and the mean and fluctuation continuity equations

C = ∂uk

∂xk= 0 and C = ∂uk

∂xk= 0. (11)

In the latter equationst , x, u, p, τ , u, p and� are, respectively, time, the spatialcoordinate, the mean velocity, the mean pressure, the Reynolds stress tensor, thefluctuating velocity, the fluctuating pressure and the rotation rate of the coordinatesystem with respect to an inertial frame.

The set of Equations (9–11) is underdetermined in the sense that it containsmore dependent variables than equations. In the classical approach of finding tur-bulent scaling laws, this difficulty has motivated the introduction of second moment

114 M. OBERLACK

equations. In the next section, the above set of Equations (9–11) will be analyzedwith respect to its symmetry properties alone, without introducing higher-ordercorrelation equations which contain additional unclosed terms.

Instead, an additional equation is introduced in the subsequent analysis whichis the velocity product equation

Niuj +Nj ui = 0. (12)

Using the latter equation in the similarity analysis leads to two major differencesbetween the present and the classical similarity approach using the ensemble aver-aged Reynolds stress transport equations only. First, in the present approach, onlythe Reynolds stresses appear as unclosed terms in the equations, and no higher-order correlations need to be considered. Hence, only a finite number of variablesare present in the system to be analyzed.

Second, it is shown in [15] that any scaling law for the velocity fluctuationand the second-order velocity product equations (12) is also a scaling law for allnth-order velocity product equations. In the classical approach using correlationfunctions, it may be difficult to show that all higher-order velocity correlations areconsistent with the scaling of the Reynolds stress equations.

3. Symmetry Transformations and Shear Flow Scaling Laws

The set of variables considered in the subsequent calculation consists of

y = [t, x, ν] and z = [u, p, τ ,u, p]. (13)

The purpose of the symmetry analysis is to find all those invertible transformations

y∗ = [t∗, x∗, ν∗] = φ(y, z; ε),z∗ = [u∗, p∗, τ ∗,u∗, p∗] = ψ(y, z; ε), (14)

which preserve the functional form of Equations (9–12), written in the new variabley∗ andz∗:

Ni = 0C = 0

Ni = 0C = 0

(Niuj +Nj ui) = 0

⇔

N ∗i = 0C∗ = 0N ∗i = 0C∗ = 0

(Niuj +Njui)∗ = 0

. (15)

The superscript∗ of any quantity denotes its evaluation according to the transform-ation (14). The transformationsφ andψ are invertible mappings which depend onthe group parameterε.

In the following subsections the symmetry transformations of (15) will be ex-amined by investigating a variety of different inhomogeneous shear flows: plane


parallel shear flows in an inertial frame and in a rotating frame, and non-rotatingand rotating pipe flow.

Once the transformation properties of the flow problem under investigation aredetermined, they are utilized to obtain invariant functions. It will be demonstratedthat these functions correspond to turbulent scaling laws.

3.1. PLANE SHEAR FLOWS

In this subsection, plane parallel shear flows will be analysed. Hence, the sym-metry transformations to be presented below have been derived by employing thefollowing restrictions

∂u1

∂x1= ∂u1

∂x3= ∂u1

∂t= u2 = u3 = 0, (16)

whereu1 is the mean velocity in the mean flow direction andu2 andu3 are respect-ively the mean velocities in the wall normal and the spanwise directions. FromEquation (16) we find thatu1 is only a function of the remaining spatial coordin-ate x2. Using the latter constraints one can show that the subsequent symmetrytransformations

x∗1 = ea1x1 + a4

a1− a3(ea1 − ea3)t

+ a4a7

a3(a1− a3)(1− ea3)− a4a7 + a5(a1− a3)

a1(a1− a3)(1− ea1),

x∗2 = ea1

(x2+ a2

a1

)− a2

a1,

x∗3 = ea1

(x3+ a6

a1

)− a6

a1,

t∗ = ea3

(t + a7

a1

)− a7

a1,

ν∗ = e2a1−a3ν,

u∗i = ea1−a3ui,

p∗ = e2(a1−a3)p,

u∗1 = ea1−a3

(u1+ a4

a1− a3

)− a4

a1− a3,

p∗ = e2(a1−a3)p, (17)

preserve the functional form of Equations (15) written in the new coordinates if� = 0 anda1–a7 are arbitrary constants.

116 M. OBERLACK

Different symmetry transformations of a given differential equation can be com-bined to derive new symmetries. The latter symmetry transformations (17) are acombination of several classical invariant transformations of fluid dynamics suchas scaling, Galilean and others.

In order to see the emergence of Equations (17), consider the examples ofscaling and translation in space ofx2. Scaling in space corresponds to the trans-formation

x′2 = ec1x2 (18)

while translation in space constitutes

x′′2 = x2 + c2. (19)

Both transformations (18) and (19) and the corresponding forms for the remainingvariables leave Equations (15) form invariant. Hence one can conclude that also acombination of Equations (18) and (19) constitutes a symmetry of Equation (15).Formally, it appears that the order of combining Equations (18) and (19) matterssince two different results are obtained:

x′2 = ec1(x2+ c2) and x′′2 = ec1x2+ c2. (20)

The parameters of the symmetry transformations are arbitrary. Hence, the twotransformations (20) are fully equivalent since a particular choice for the values ofc1 andc2 leads to the same form of transformation. In a similar manner combiningall symmetries of Equation (15) leads to (17).

This procedure can be reversed and each fundamental symmetry of (17) canbe separate out. To distinguish the symmetry transformations corresponding toeach of the seven different parametersa1–a7 all but one parameter has to be setto zero. Doing this, one finds thata1 anda3 correspond to scaling of space andtime, a5, a2 and a6 conform to translation of space with respect tox1–x3, a4 isthe transformation group parameter of the Galilean invariance anda7 comports totranslation in time. In many cases, l’Hospital’s limit theorem has been employed. Inthe following sections, the limit theorem will be used very often without explicitlymentioning it.

The transformations (17) will be considered under limits such asa1 6= a3 6= 0,a1 6= a3 = 0, a3 6= a1 = 0, a3 = a1 6= 0, anda3 = a1 = 0. It can be shown that allof these limits exist. Similar limits are to be taken for the circular shear flow cases.

As was mentioned above ifa4 is considered to be the only non-zero parameterit is observed that Equation (17) constitutes the classical Galilean invariance in thex1 direction. Later it will be shown thata4 is one of the crucial parameter for allthe turbulent scaling laws. In particular, it will be observed that also the classicallogarithmic law of the wall explicitly contains the property of Galilean invariance.

It should be noted that without the limitations (16), Equations (15) admit moresymmetries such as rotation invariance and generalization of Galilean invariance.


The physical interpretation of the scaling parametersa1 anda3 is critical in orderto understand each scaling law to follow. As an example consider the parametera1.It appears in the exponent of the factor of all spatial coordinates in Equations (17).This means that all spatial coordinates can be scaled by a certain factor if also thevelocity is appropriately scaled. Suppose a given external length scalel is present inthe flow under investigation.l may be considered as a kind of boundary condition.If the transformation (17) is applied to the flow problem with a givenl, it happensthat the transformation to the new variable is only admitted ifa1 = 0. Otherwiselwould be transformed to a new length scale explicitly depending upon the arbitraryscaling parametera1. Sincel is a given fixed quantity, the scaling symmetry withrespect to the spatial coordinates is lost anda1 can only be zero. Subsequently, anyof the parameters being zero is referred to as a “broken symmetry”.

It has been demonstrated in Section 1 by utilizing the example of the heatequation that one can use symmetry transformations to derive quantities which areinvariant under the symmetry transformations. These invariants can in turn be usedto derive self-similar solutions. Since the present purpose is to investigate meanvelocities, we only focus on those invariants which contain the mean velocity andthe coordinatex2. This is the only coordinate the mean velocity depends on bydefinition given due to Equation (16).

Each of the invariant functions, subsequently called turbulent scaling laws, andits associated symmetries, may be interpreted in terms of a given external length,time or velocity scale breaking some of the scaling symmetries. Subsequently, allCi ’s are constants. In all transformations to be considered below it is assumed thata2 anda4 are non-zero.

3.1.1. Algebraic mean velocity profile(a1 6= a3 6= 0)

The present case is the most general one. No scaling symmetry is broken. As aresult, the mean velocityu1 has the following form

u1 = C1

(x2 + a2

a1

)1−(a3/a1)

− a4

a1 − a3. (21)

Invariance of Equation (21) can be verified by employing the transformation forx2

andu1 in (17) into Equation (21). As a result, the functional form of the equation ispreserved written in the new variablesu∗1 andx∗2. The key property of the algebraicfunction is that the scaling of space and time is admitted because both scalingparametersa1 anda3 are assumed to be not equal zero and not equal each other.

3.1.2. Logarithmic mean velocity profile(a1 = a3 6= 0)

For the present combination of parameters an external velocity scale is symmetrybreaking. The scaling factor cancels from the velocity transformation as can beconfirmed in Equations (17). However, the scaling of space and time is still admit-

118 M. OBERLACK

ted but with the same scaling factors. Considering the restrictiona1 = a3 in thesymmetry transformation (17) leads to

x∗2 = ea1

(x2+ a2

a1

)− a2

a1, u∗1 = u1+ a4. (22)

The corresponding invariant mean velocity profile is given by

u1 = a4

a1log

(x2 + a2

a1

)+ C2, (23)

which is an extension of the classical logarithmic law of the wall. The structure ofEquation (23) is indeed unaltered by Equation (22) if written in the new variablesx∗2 andu∗1.

Since scaling with respect to velocity has been excluded. There is an externalfixed velocity scale acting on the flow. The inability to scale velocity is analogousto von Kármán’s classical derivation of the logarithmic law of the wall. There itwas assumed that in the near-wall region any external geometrical influence isnegligible and the only determining parameter is the friction velocityuτ . Sincethe friction velocity is a given external quantity, velocity scaling is precluded. Thescaling symmetry with respect to the spatial variables (a1 6= 0) is still retained andthe length scale varies linearly with the distance to the wall. This is an assumptionin the classical derivation of the log law of the wall but is a result of the presentanalysis.

The present derivation of the logarithmic law of the wall can also be consideredin the light of what is sometimes referred to as the “constant stress layer assump-tion”. Therein it is inferred that close to the wall there is a thin region of constantturbulent stresses. Because scaling of velocity is excluded for the present case,there can also be no scaling for the turbulent stresses. However, the reverse cannotbe concluded namely that the Reynolds stresses are constant. In fact, the log regioncan be found in a variety of different flows such as channel flow, Couette flow,boundary layer flows and others. However, there are not two flows which have thesame stress profiles. In particular, the shear stresses are very different from eachother. Still, in all of these flows a logarithmic region can be identified.

3.1.3. Exponential mean velocity profile(a1 = 0 anda3 6= 0)

Sincea1 is zero in the present case, there exists an external length scale whichis symmetry breaking. In a similar line of analysis as for the previous cases thiscorresponds to a flow which is characterized by an external length scale. It can bedemonstrated by employinga1 = 0 into the symmetry transformation (17) that thetransformations forx2 andu1 reduce to

x∗2 = x2 + a2, u∗1 = e−a3

(u1− a4

a3

)+ a4

a3. (24)


The mean velocity profile

u1 = a4

a3+ exp

(−a3

a2x2

)C3, (25)

is invariant under the transformation (24). In Section 4, it will be demonstrated thatthe latter profile corresponds to the mid wake region of a flat plate boundary layerflow. It appears that the boundary layer thickness is the external length scale whichis symmetry breaking.

3.1.4. Linear mean velocity profile(a1 = a3 = 0)

This plane shear flow case is the one where scaling of time as well as scaling ofspace is excluded. The corresponding symmetry transformation is given by

x∗2 = x2 + a2, u∗1 = u1+ a4. (26)

In terms of the admitted invariant mean velocity profile, this flow is similar to thesubsequent rotating channel flow since the mean velocity is also linear given by

u1 = a4

a2x2 + C4. (27)

Invariance of the latter mean velocity profile can be confirmed by introducingEquation (26) into Equation (27). However, in contrast to Equation (29), whereonly scaling with respect to time has been excluded, both scaling of time andspace is inhibited in the present case. Hence, both an external time and lengthscale is dominating the flow. In Section 4 it will be shown that this corresponds tothe turbulent Couette flow where the channel width and the wall velocity are thecontrolling flow parameters.

3.2. PLANE SHEAR FLOWS IN A ROTATING FRAME

Considering� 6= 0 in Equations (9–11), the transformations (17) are only admittedby Equations (15) ifa3 = 0. Hence, from the transformation of time in (17) it isapparent that a scaling of time is not admitted any more. The reason is that theinverse of the rotation rate|�| is a time scale which is externally imposed on theflow. As an immediate consequence the symmetry transformations forx2 andu1 inEquations (17) reduce to

x∗2 = ea1

(x2+ a2

a1

)− a2

a1, u∗1 = ea1

(u1+ a4

a1

)− a4

a1. (28)

The corresponding scaling law is a linear mean velocity profile

u1 = �3

(x2 + a2

a1

)C5− a4

a1. (29)

120 M. OBERLACK

The rotation rate appears naturally in the latter scaling law since in Equations (9–11)�3 can be absorbed intox if t is also normalized by the rotation rate. Once thescaling law has been established it can be rewritten in the original variables whichleads to the emergence of�3 in Equation (29).

The present case is distinguished from the previous linear mean velocity pro-files, since scaling of the spatial coordinates still holds (a1 6= 0).

3.3. NON-ROTATING AND ROTATING PIPE FLOW

In case of a circular shear flow, Equations (9–11) and the symmetry condition (15)need to be transformed into an axisymmetric coordinate system wherer, z, andφare respectively the radial, axial and azimuthal coordinates.ur , uz, anduφ are thecorresponding mean velocities. Since the mean flow is assumed to be axisymmetricand straight, the following restrictions for the velocities need to be applied

∂uz

∂z= ∂uz

∂φ= ∂uz

∂t= ∂uφ

∂z= ∂uφ

∂φ= ∂uφ

∂t= ur = 0. (30)

Only rotation about thez axis has been considered

�φ = �r = 0. (31)

As a result, both non-zero mean velocitiesuz and uφ depend only on the radialcoordinater.

Utilizing the latter restrictions it can be shown that the following symmetrytransformations

z∗ = eb1z + b4

b1− b3(eb1 − eb3)t

+ b4b7

b3(b1− b3)(1− eb3)− b4b7 + b5(b1− b3)

b1(b1− b3)(1− eb1),

r∗ = eb1r,

φ∗ = φ + (1− eb1)

(t + b7

b3

)�z + b7�z + b6,

t∗ = eb3

(t + b7

b1

)− b7

b1,

ν∗ = e2b1−b3ν,

u∗z = eb1−b3uz,

u∗r = eb1−b3ur,

u∗φ = eb1−b3uφ,


p∗ = e2(b1−b3)p + 1

2�2zr

2e2b1(e−2b3 − 1),

u∗z = eb1−b3

(uz + b4

b1− b3

)− b4

b1 − b3,

u∗φ = eb1−b3uφ +�zreb1(e−b3 − 1),

p∗ = e2(b1−b3)p, (32)

conserve the functional form of Equations (15) written in the new coordinates.It should be noted at this point that this is true regardless whether�z is zero ornon-zero. The group parameterb1–b7 have the same meaning as the correspondinga1–a7 interpreted below Equations (17).

The transformations (32) have similar features as the transformation (17) for theplane case. However, there are also some very distinct differences one of which isthe lack of a translation invariance in ther direction. This property comes due tothe fact that in a cylindrical coordinate system the Navier–Stokes equations are notautonomous with respect tor, i.e., the equations explicitly contain the radial co-ordinate. As a result no constant can be added tor without changing the functionalform of Equations (15). A second very distinct difference of the circular case to theplane case is the effect of system rotation. In the plane case system rotation leadsto a reduced set of admitted symmetries sincea3 has to be set to zero if� 6= 0. Inthe circular case, the rotation rate�z about the centre line does not introduce anylimitations on the admitted symmetry parameters.

Since there are less symmetry parameters due to the absence of the translationinvariance inr compared to the plane case there are also less distinct scaling laws.

3.3.1. Algebraic mean velocity profiles(b1 6= b3 6= 0)

As for Equation (21), there is no symmetry breaking scale imposed on the flow.Employing Equation (32) in the invariant function for the axial and azimuthal meanvelocity

uz = − b4

b1 − b3+ C6r

1−(b3/b1) (33)

and

uφ = C7r1−(b3/b1) −�zr, (34)

respectively, we find their functional form unaltered.It will be shown later that Equation (33) corresponds to the centre region of the

turbulent pipe flow both for the non-rotating and the rotating case. The�z-term inEquation (34) has opposite sign to frame rotation. Hence, in an inertial frame ofreference only the algebraic term persists. Without loss of generality,�z can be setto zero and rotation effects are to be accounted for due to the wall rotation.

122 M. OBERLACK

3.3.2. Logarithmic axial mean velocity profile(b1 = b3 6= 0)

This combination of parameters applies if an external velocity scale acts on theflow. Similar to the plane parallel shear flow case, the transformations forr, uz anduφ reduce to

r∗ = eb1r, u∗z = uz + b4, u∗φ = uφ +�zr(1− eb1), (35)

which results in a logarithmic mean velocity profile

uz = b4

b1ln(r)+ C8. (36)

It is important to note that Equation (36) does not correspond to the classical lawof the wall, since the singularity appears on the pipe axis. Subsequently, it willbe referred to as “circular log-law”. It appears that Equation (36) applies on somesection of the radius for rapidly rotating pipes, in which the wall velocity is thesymmetry breaking velocity scale. The corresponding azimuthal velocity is givenby

uφ = −�zr + C9, (37)

which is also singular at the centreline. For the same reason as described above�zcan be set to zero.

3.3.3. Potential vortex mean velocity profiles(b3 = 2b1 6= 0)

For this parameter combination the axial mean velocity can be computed to be

uz = b4

b1+ C10

r(38)

and the corresponding azimuthal velocity is the potential vortex in a rotating frameas given by

uφ = C11

r−�zr. (39)

Equations (38) and (39) are special cases of Equations (33) and (34) but correspondto a completely different type of flow.

For the subsequent comparison with data and as has already been pointed outabove, the system rotation in the circular case does not need to be considered.Hence, without loss of generality, we take�z = 0.

4. Verification of the Scaling Laws

In this section, experimental and DNS data will be presented to give an empiricalverification of the scaling laws (21–29) and (33–39).


The logarithmic law of the wall and the viscous sublayer have been validated ina large number of experiments and DNS data and will not be investigated here. Inthe viscous sublayer there is a length and a velocity scale dominating the flow andhence two scale symmetries are broken (a1 = a3 = 0). The length and velocityscales are respectivelyν/uτ anduτ .

From the circular cases the potential vortex (39) will not be analyzed here sinceits flow characteristic is known from experiments (see, e.g., [2]). So far, only theaxial mean velocity profile (38) could not be assigned to a specific turbulent flow. Inorder to avoid the duplication of well documented invariant solutions, it is focusedon four plane and three circular cases.

4.1. FLAT PLATE TURBULENT BOUNDARY LAYER FLOW

Three sets of experimental data at medium to high Reynolds numbers have beenchosen for comparison with the exponential velocity profile. The data of [1, 4, 19]cover the Reynolds number range Reθ = 15000–370000, where

θ =∞∫

0

(1− u

u∞

)u

u∞dy

is the momentum thickness andu∞ is the free stream velocity.As has been pointed out above, it appears that the exponential law (25) de-

scribes the outer part of a high Reynolds number flat plate boundary layer flow.In order to match the theory and the data, the exponential mean velocity profile inEquation (25) will be written in outer scaling

u∞ − uuτ

= α exp(−β y

1

), (40)

whereα andβ are universal constants,

1 =∞∫

0

u∞ − uuτ

dy

is the Rotta–Clauser length scale anduτ is the friction velocity.In Figure 1 the turbulent boundary layer data are plotted as log[(u∞ − u)/uτ ]

vs. y/1. All the data appear to converge to a straight line in the regiony/1 ≈0.025–0.15. The data of [19] show an extended region for the exponential lawup to abouty/1 ≈ 0.23. With increasing Reynolds number the applicability ofthe exponential law appears to increase. The log region is valid withiny/1 ≈ 0–0.025 and does not follow the exponential (40). The applicability of the exponentialregion is approximately five to eight times longer than the logarithmic law.

If the exponential velocity profile (40) were to be valid over the entire boundarylayer, an integration of (40) from zero to infinity would giveα = β. A least square

124 M. OBERLACK

Figure 1. Mean velocity of the zero-pressure gradient turbulent boundary layer flow:◦,Reθ = 370000 [19];2, ♦, Reθ = 60000 [4];+, Reθ = 15000,×, Reθ = 20000 [1]; —,10.34 exp(−9.46y/1).

fit of the presented data leads to approximately the latter equivalence withα =10.34 andβ = 9.46.

4.2. TWO-DIMENSIONAL TURBULENT CHANNEL FLOW

Here the experimental data of [14, 20] and the low Reynolds number DNS data of[10] will be used for the investigation of the algebraic scaling law.

In fact, two algebraic regimes have been detected. One where the origin of theindependent coordinate is at the wall and one where it is at the centre of the channel.The region of validity of an algebraic scaling law near the centre line appears to bemore clear than for the near wall region. Since for the algebraic scaling law (21),both constantsa1 anda3 have to be non-zero and distinct, the region for whichthe algebraic scaling law applies has the highest degree of symmetry. The centreregion seems to be more suitable for that, since in the near-wall region the frictionvelocityuτ is symmetry breaking.

For the algebraic law in the centre of the channel, the appropriate outer scalingfor the channel has been proposed to be

uc − uuτ

= ϕ(yb

)γ, (41)

whereϕ andγ are constants,y originates on the channel centre line,uc is the centreline velocity andb is the channel half width.

In Figure 2, the data of [14, 20] have been plotted in log-log scaling for theReynolds number range Rec = 18000–40000, where Rec is based on the centre


Figure 2. Mean velocity of the turbulent channel flow:◦, Rec = 40000;2, Rec = 23000[20]; ♦, Rec = 18000 [14]; —, 5.83(y/b)1.69.

Figure 3. Mean velocity of the turbulent channel flow from [10]: —, Rec = 7900; – – –,Rec = 3300.

line velocity and channel half width. There is some obvious indication that thecentre region up to abouty/b ≈ 0.8 closely follows an algebraic scaling law givenby Equation (41). The unknown constants in Equation (41) have been fitted toϕ = 5.83 andγ = 1.69 the data.

An even more profound indication regarding the algebraic law may be obtainedfrom the DNS data of [10]. In Figure 3, the data are plotted with log-log scalingand an almost perfectly straight line is visible for both Rec = 3300 and 7900 from

126 M. OBERLACK

Figure 4. Mean velocity of the turbulent channel flow from [10]: — Rec = 7900: – – –Rec = 3300.

the centreline up to abouty/b = 0.75. The scaling extends slightly further out forthe Rec = 7900 case. Since both Reynolds numbers in the DNS are low, a weakReynolds number dependence of bothϕ andγ is observed.

Figure 4 shows a log-log plot of the mean velocity of the channel flow data inwall units. Up to abouty+ = 3 the linear law of the viscous sublayer is valid. In therange 50< y+ < 250 for Rec = 7900 an almost perfectly straight line is visibleand a least square fit of an algebraic law in this range results in a much highercorrelation coefficient than a least square fit of a logarithmic function.

4.3. ROTATING TWO-DIMENSIONAL TURBULENT CHANNEL FLOW

An early experimental investigation of the rotating channel flow circling about thespanwise direction has been done by Johnston et al. [7] who obtained a linearcenter region at sufficiently high rotation rates. Compared with the non-rotatingchannel flow, which exhibits the highest degree of symmetry in the centre regionof the flow (a1 6= a3 6= 0), here the external time scalet� = �−1

3 is symmetrybreaking which leads toa3 = 0. This affects in particular the center region of theflow which appears to become linear. The latter fact can also be obtained fromemployinga3 = 0 in Equation (21). However, in contrast to the turbulent Couetteflow, to be discussed in the following subsection, a scaling symmetry with respectto the spatial coordinates still exists (a1 6= 0).

Equation (29) may be rewritten as

u1 = σ1�3x2+ σ2. (42)


Figure 5. Mean velocity of the rotating two-dimensional turbulent channel flow from [11]:– – –, Ro= 0.2; —, Ro= 0.5.

A recent DNS by Kristoffersen and Andersson [11] also confirms the latter formof the mean velocity profiles. Their results are shown in Figure 5 for two differ-ent rotation rates. The data are presented employing the rotation number Ro=2|�|h/um, a non-dimensional measure of the rotation rate. Both Johnston et al. [7]and Kristoffersen and Andersson [11] found the slope coefficient in Equation (42)to beσ1 ≈ 2.

It has already been recognized in the literature that the latter value coincideswith zero absolute vorticity 2�− du1/dx2. An additional feature of this particularvalue for σ1 is that the linear region of the flow becomes neutrally stable. Anoverview on the literature regarding the latter flow characteristics is given in [11].

4.4. THE TURBULENT PLANE COUETTE FLOW

Even though a linear mean velocity profile is also obtained for the present flow, itis very distinct from the previous test case. Here, the lowest degree of symmetryis considered where a length and a velocity scale is dominating the flow and bothscaling symmetries in Equations (17) are broken (a1 = a3 = 0). Hence, compar-ing to Equation (42), also scaling with respect to the spatial coordinate is lost. Itappears that the present case applies to the turbulent plane Couette flow. The meanvelocity may be written as

u1 = ϑ1uw

hx2 + ϑ2, (43)

whereh anduw are is the channel half width and the wall velocity, respectively,andϑ1 andϑ2 are constants.

128 M. OBERLACK

Figure 6. Mean velocity in turbulent plane Couette flow:◦, El Telbany and Reynolds [3]; —,Lee and Kim [12].

Several experimental and numerical investigations of the plane Couette flow,have been reported in the literature. The experimental work by El Telbany andReynolds [3] and the DNS data by Lee and Kim [12] have been depicted in Fig-ure 6. With relatively high accuracy about 80% of the center region is fitted by thelinear mean velocity profile.

As already mentioned, a second linear mean velocity profile, which is also dom-inated by two external scales, is the viscous sublayer. Here, the scales areν/uτ anduτ , the viscous length scale and the friction velocity, respectively. The latter caseis very well known and can be verified in Figure 4.

4.5. NON-ROTATING PIPE FLOW

From the invariant solution (33), the proposed new defect law for the pipe flow isgiven by

uc − uzuτ

= χ( rR

)ψ, (44)

whereχ andψ are constants. In contrast to the usual defect law for the pipe flow,the coordinater has its origin at the pipe centre rather than at the pipe wall.

In Figure 7, the data of Zagarola [21] for medium to high Reynolds numberturbulent pipe flows are plotted in the form suggested by Equation (44). (Notethat the individual curves are shifted vertically in order to better see the individualcurves.)

It is apparent from the log-log plot of Figure 7 that in the range 0.1 ≤ r/R ≤ 0.8all data vary linearly. The deviation forr/R ≤ 0.1 may be due to the large amplific-


Figure 7. Mean velocity of the turbulent pipe flow from Zagarola [21]:•, 4, +, ×, ♦, 2, ◦,Rem = 4.2·104−−3.5·107; —, 7.5(r/R)1.77; – – –, 101.27.5(r/R)1.77. The data are shiftedvertically by the factor 100.2.

ation of errors when plotting the difference betweenuc anduz in log coordinates.The data of Zagarola [21] suggest that the constants in Equation (44) areχ = 7.5andψ = 1.77.

4.6. ROTATING PIPE FLOW

In contrast to the laminar flow in a rotating pipe, where the azimuthal velocityclosely follows solid-body rotation (see, e.g., [18]), in the turbulent flow case, analgebraic scaling law (34) with an exponent larger than one is apparent in manyexperimental and DNS data. This can be rewritten as

uφ

uw= ζ

( rR

)ψ, (45)

whereuw is the azimuthal velocity at the wall andζ andψ are constants.In Figure 8 experimental data for the azimuthal mean velocity in rotating pipes

at moderate rotation numbers are presented. This indicates that for the outer partof the pipe radius the data closely follow an algebraic scaling law, and the rangeof validity depends on Reynolds number and rotation numberN = uw/um whereum is the bulk velocity. The inner region of the rotating pipe exhibits a significantdeviation from the power law. This is due to solid-body rotation near the pipe axis.The extension towards the pipe axis is affected by the Reynolds and the rotationnumber.

One can deduce from the two invariant solutions (33) and (34) that the exponentψ of the scaling law for the axial velocity in Equation (44) is also a constant that

130 M. OBERLACK

Figure 8. Azimuthal mean velocity in a rotating pipe flow:◦, Rem = 20000,N = 0.5; 2,Rem = 20000,N = 1.0;♦, Rem = 50000,N = 0.5 [18];×, Rem = 50000,N = 1.0 [9].

has the same value asψ . The scaling law in Equation (44) has to be extended toaccount for rotation effects in the pre-factorχ . Since an additional velocity scaleuw is induced at the wall, a modified scaling law for axial mean velocity is proposed

uc − uzuτ

= χ(uw

uτ

)( rR

)ψ, (46)

whereχ is not a constant but rather a function of the velocity ratio.In Figure 9 experimental data of the axial and azimuthal mean velocity have

been plotted in log-log scaling for a moderate rotation number. The two curvesare parallel and straight for some part of the pipe radius as suggested by Equa-tions (45) and (46). No functional form has be assigned toχ(uw/uτ ) from thepresent analysis.

As the rotation number increases, the wall rotation velocityuw becomes thedominant velocity scale at the wall and the axial velocity changes to a circular loglaw (36). The suggested scaling law is

uz

uw= λ log

( rR

)+ ω. (47)

In Figure 10, the axial mean velocity data of Orlandi and Fatica [17] are plottedin semi-log scaling, corresponding to (47). A straight line matches about 30% ofthe pipe radius in the range 0.5 ≤ r/R ≤ 0.8. The region of applicability of thisnew log-region is different from the logarithmic law of the wall which is valid for0.8 ≤ r/R ≤ 1.0. Also the coefficientλ in (47) is negative and approximatelyequal to−1. The additive constantω is about 0.354.


Figure 9. Axial and azimuthal mean velocity in a rotating pipe from [9] for Rem = 50000andN = 1.0: ◦, (uc − uz)/uw: 2, uφ/uw .

Figure 10. Axial mean velocity in a rotating pipe from [17] for Rem = 4900 andN = 2: —,uz/uw from DNS: - - - -,uz/uw = − log(r/R)+ 0.354.

5. Model Implications

An important application of the present theory is turbulence modelling. Commonstatistical turbulence models, in particular two-equation models, may not be con-sistent with all the symmetries and the associated scaling laws calculated in thepresent theory. It is proposed as a necessary condition that turbulence models

132 M. OBERLACK

should have the symmetry properties computed in the present analysis as well asshould be consistent with the scaling laws.

In order to clarify the symmetry constraints for Reynolds averaged turbulencemodels the classicalk–ε model [6] will be investigated with respect to plane shearflows. Under this restriction thek–ε models reads

νt

(du1

dx2

)2

− ε + ck d

dx2

(νt

dk

dx2

)= 0,

cε1νt

(du1

dx2

)2ε

k− cε2

ε2

k+ cε d

dx2

(νt

dε

dx2

)= 0 (48)

and the momentum equation inx1-direction reduces to

d

dx2

(νt

du1

dx2

)= 0, (49)

where

νt = cµ k2

ε. (50)

A formal analysis confirms that thek–ε model is invariant under all neces-sary symmetry transformations, i.e., translation in space, Galilean invariance andscaling with respect to space and time.

In the following it will become apparent that obeying all symmetries is onlya necessarycondition. However, it is not a sufficient condition to describe allturbulent scaling laws derived in the Sections 3.1 and 3.2.

Right from the outset it is obvious that thek–ε model cannot account for systemrotation since the rotation rate� is not explicitly contained in the model equations.The k and theε-equations do not account for Coriolis effects. In principle, themomentum equation does respond to rotation. However, for plane shear flows themomentum equation contains no Coriolis terms. As an immediate consequence thelinear scaling law (29) cannot be obtained because symmetry breaking with respectto time, i.e.,a3 = 0, is not induced due to system rotation.

In contrast, the logarithmic law is indeed consistent with both Equations (48)and (49). Regarding the turbulent kinetic energy and dissipation one obtains

k = const. and ε ∼ 1

x2+ a1

a2

. (51)

This results is not too surprising since the log-law has been employed for calibrat-ing thek–ε model.

The remaining three scaling laws, namely the algebraic law, the exponential lawand the second linear law, do not comply with Equations (48–50). Interestingly


enough, the source of disagreement is different for the algebraic and the exponen-tial law and the second linear law. The algebraic and the exponential law can bemade consistent with the two model equations by choosingk andε to be

k ∼(x2+ a2

a1

)2(1−a3/a1)

, ε ∼(x2+ a2

a1

)3(1−a3/a1)

(52)

for the algebraic law and

k ∼ exp

(−2a3

a2x2

), ε ∼ exp

(−3a3

a2x2

)(53)

for the exponential law. However, implementing either Equation (52) or (53) to-gether with the corresponding turbulent scaling laws into the momentum equationleads in both cases to an incompatibly. Hence, the algebraic and the exponentiallaw cannot be a solutions of thek–ε model.

For the linear profile of the turbulent Couette flow the turbulent eddy viscosityνt needs to be constant in order to solve the momentum equation. However, thisleads to a contradiction between thek and theε equation for the common valuesof thek–ε model parameters.

From all the plane shear flow scaling laws derived in Sections 3.1 and 3.2only the logarithmic law is consistent with thek–ε model. Similar problems ofthe k–ε model are observed with respect to scaling laws for round shear flows inSection 3.3.

6. Summary

It has been demonstrated by the application of symmetry and invariant methods toparallel turbulent shear flows that a large class of invariant solutions for the meanvelocity can be derived. For the plane case, these solutions include the logarithmiclaw of the wall, an algebraic law, and a linear profile; a new exponential meanprofile has also been found. The circular scaling laws include an algebraic law forthe axial and the azimuthal velocity, the potential vortex and a circular log law.

Using DNS and experimental data the exponential and algebraic laws have beenvalidated in the outer part of boundary layer and channel flows respectively. For theturbulent boundary layer, high-quality data are available, and there is little doubtregarding the existence of an exponential region. For the turbulent channel flow,the DNS data exhibit an almost perfect algebraic centre region, but the data areat low Reynolds number and show Reynolds number dependence of the scalinglaw parameters. The experimental data also clearly show the algebraic region, butcontain more scatter.

Another algebraic scaling law in the vicinity of the wall has also been confirmedin the present investigation using the low Reynolds number DNS data of Kim et al.[10].

134 M. OBERLACK

Two, from a symmetry point of view, different linear scaling laws have beenderived. One manifests itself in the centre region of a turbulent Couette flow and hasbeen identified by DNS and experimental data [3, 12]. It is determined by a lengthscale and a velocity scale, the channel width and the velocity of the moving wall,respectively. In experiments and DNS data of a rotating channel flow another linearmean velocity profile has been found with its slope scaling with�z. This linearmean velocity is distinct from the previous case since only one scaling symmetrywith respect to time is broken.

In a non-rotating pipe, the algebraic law for the axial velocity covers about 80%of the pipe radius. This was confirmed for three decades of Reynolds number byusing the data of Zagarola [21]. In the case of the rotating pipe flow the algebraiclaw for the azimuthal velocity was confirmed for a range of Reynolds numbers androtation numbers for at least 70% of the radius. Based on experimental and DNSdata it was concluded that, except in the central region of the pipe, these scalinglaws were independent of Re and�z.

For high rotation numbers, the azimuthal velocity at the wall (rather than thefriction velocity) becomes the dominant velocity scale imposed on the flow, andin this case a new circular logarithmic law is found. The validity of this law wasdemonstrated by using the DNS data of Orlandi and Fatica [17] at the rotationnumberN = 2. The new circular log-law differs from the usual log-law of the wallbecause it scales with the radius and its location is closer to the pipe axis and has awider extent.

An important application of the present theory is turbulence modelling. It hasbeen shown by investigating thek–ε model that common two-equation models arenot consistent with the turbulent scaling laws derived in the previous sections. Thek–ε model observes all symmetry requirements. However, this is only a necessarycondition for predicting all turbulent scaling laws. A sufficient condition for con-sistency with all turbulent scaling laws is obtained if theses laws also constitutesolutions of the turbulence model under investigation. The presented symmetryproperties and scaling laws in plane turbulent shear flows may be considered as anew realizability concept.

References

1. DeGraaff, D.B., Webster, D.R. and Eaton, J.K., The effect of Reynolds number on boundarylayer turbulence.Exp. Thermal Fluid Sci.18(4) (1999) 341–346.

2. Donaldson, duP.C. and Bilanin, A.J., Vortex wakes of conventional aircraft.AGARDAG-204(1975).

3. El Telbany, M.M.M. and Reynolds, A.J., Velocity distributions in plane turbulent channel flows.J. Fluid Mech.100(1980) 1–29.

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5. George, W.K., Castillo, L. and Knecht, P., The zero pressure-gradient turbulent boundary layer.Technical Report No. TRL-153. Turbulence Research Laboratory, School of Engineering andApplied Sciences, SUNY, Buffalo, NY (1996).

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7. Johnston, J.P., Halleen, R.M. and Lazius, D.K., Effects of spanwise rotation on the structure oftwo-dimensional fully developed turbulent channel flow.J. Fluid Mech.56 (1972) 533–557.

8. von Kármán, Th., Mechanische Ähnlichkeit und Turbulenz.Nachr. Ges. Wiss. Göttingen68(1930).

9. Kikuyama, K., Murakami, M., Nishibori, K. and Maeda, K., Flow in axially rotating pipe.Bulletin JSME26(214) (1983) 506–513.

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11. Kristoffersen, R. and Andersson, H.I., Direct simulations of low-Reynolds-number turbulentflow in a rotating channel.J. Fluid Mech.256(1993) 163–197.

12. Lee, M.J. and Kim, J., The structure of turbulence in a simulated plane Couette flow. In: F.Durst, B.E. Launder, F.W. Schmidt and J.H. Whitelaw (eds),8th Symposium on Turbulent ShearFlows, Munich, Paper 5-3 (1991).

13. Millikan, C.B., A critical discussion of turbulent flows in channels and circular tubes. In:Proceedings 5th International Congress on Applied Mechanics, Cambridge, MA (1939)pp. 386–392.

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16. Oberlack, M., Similarity in non-rotating and rotating turbulent pipe flows.J. Fluid Mech.379(1999) 1–22.

17. Orlandi, P. and Fatica, M., Direct simulations of turbulent flow in a pipe rotating about its axis.J. Fluid Mech.343(1997) 43–72.

18. Reich, G., Strömung und Wärmeübertragung in einem axial rotierenden Rohr. Dissertation,Darmstadt University (1988).

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Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows