Non Isotropic Length Scales During the C

Flow, Turbulence and Combustion 2005 74: 145–167



C

Springer 2005

Non-Isotropic Length Scales During
the Compression Stroke of a Motored Piston Engine
STEPHAN BREUER1 , MARTIN OBERLACK2 and NORBERT PETERS3
1

Delphi Automotive Systems, Technical Centre Luxembourg, Avenue de Luxembourg, L-4940
Bascharage, G. D. de Luxembourg
2
Fachgebiet f¨ur Str¨omungsmechanik, Fachbereich f¨ur Mechanik, TU Darmstadt, Petersenstraße 13,
64287 Darmstadt, Germany; E-mail: oberlack@hyhy.tu-darmstadt.de
3
Institut f¨ur Technische Mechanik, RWTH Aachen, 52056 Aachen, Germany
Received 25 July 2004; accepted 23 October 2004
Abstract. For the case of axial compression the two-point velocity correlation equations of axisymmetric homogeneous turbulence are derived. Appropriate integrations then lead to equations for

the components of the Reynolds stress tensor as well as to those for the two independent integral
length-scales characterizing axisymmetric homogeneous turbulence. These equations contain a certain number of empirical constants. Values for these constants are taken from the literature, or were
adjusted from the present data.
The resulting model is validated using data from a motored piston engine. The flow field, which
has negligible swirl and tumble, has been measured using particle image velocimetry (PIV). Since
turbulence is axisymmetric and homogeneous in the counter region, two-dimensional PIV provides
the time history of the axial and radial length-scales. The experimental data are compared with the
mathematical model.
Key words: compressed turbulence, two-point correlation, non-isotropic turbulence, length scale
equation, piston engine, compression stroke, PIV

1. Introduction
In many turbulent flows of practical interest the assumption of a single integral
length-scale or corresponding assumptions of a scalar dissipation rate is not appropriate [1]. This is particulary true for flows with one preferred direction. Under the
absence of a mean velocity gradient we may consider the theory of homogeneous
turbulence or the specific case of axisymmetric homogeneous turbulence, which
describes the evolution of a multi-point correlation tensor in axisymmetric form
or in the case of a single-point approach two independent turbulence intensities
and two independent turbulent length-scales. Axisymmetric turbulence is a natural
extension of isotropic turbulence which accounts for the anisotropic character of

the turbulence parameters.
In order to describe the turbulence dynamics during the compression phase
in a piston engine, a single length- and time-scale is not sufficient, because of the
imposed direction of the compression. Assuming isotropy in the plane normal to the

146

S. BREUER ET AL.

compression and anisotropy perpendicular to it we need at least two length-scales
and two turbulence intensities to describe the turbulent flow adequately.
In the present paper we model the turbulence dynamics of a compression
stroke by employing the simplification of the axisymmetric turbulence extended
by one-dimensional homogeneous compression using two-point correlations. The
theory of axisymmetric turbulence traces back to the work by Batchelor [2] and
Chandrasekhar [3] who laid the foundation of this theory. Lindborg [4] recasted the
theory in a different form, in which he introduced expressions based on cylindrical
coordinates that can be directly validated experimentally.
On the basis of the latter approach the theory of axisymmetric turbulence will
be extended to the case of axisymmetric compression in the present paper. Equations will be presented for the corresponding two-point velocity correlations. In

addition, following the procedure introduced in Oberlack [1] a modelling approach
is developed to derive a closed set of equations for both the turbulence intensities
and the integral length-scale in tensorial form.
The model for the compression of axisymmetric homogeneous turbulence presented in this paper will be validated by measurements. The compression of turbulence is performed in a piston engine with a pancake chamber. In the literature the
determination of turbulent length-scales and turbulence intensities in a piston engine using laser doppler velocimetry (LDV) have been detailed by H¨uppelsh¨auser
[5], Hong and Tarng [6], and Lorenz and Prescher [7] to name only a few. In order to
determine turbulent length-scales the LDV technique is applied simultaneously at
two points in space. By varying the distance of the two points and after conducting
a statistical analysis, the two-point velocity correlation may be determined along a
given line. In all the above references the theory of isotropic turbulence has been
employed to determine length-scales from the velocity correlation data. Since the
theory of isotropic turbulence was used, non-isotropic length-scales could not be
obtained.
In this paper particle image velocimetry (PIV) will be applied to determine the
flow field and other quantities to be derived thereof. PIV is a two dimensional,
nonintrusive optical measuring technique. It was developed at the beginning of
the [8, 9], and since then has been continuously improved to become a standard
measuring technique, which also has been applied to velocity field measurements in
engines (see e.g. [10, 11] 1980s). Since PIV yields a two dimensional field of data,
a two dimensional correlation function may be calculated under the assumption

of a homogeneous axisymmetric turbulence statistic. In turn this may be used to
compute statistical quantities such as turbulent length- and velocity scales to verify
turbulence model equations.
2. Two-Point Correlation Equations
In the present and all the subsequent sections we only consider week compression
i.e. we assume that the speed of sound is much faster than any flow velocity and

147

NON-ISOTROPIC LENGTH SCALES

in particular the compression velocity. Employing this asymptotic assumption we
obtain the well known zero-Mach number limit of the Navier-Stokes equations
where density ρ is only a function of time. As an immediate consequence the
temporal density variations act as a source term in the mean continuity equation
∂ u¯ k
1 ∂ρ
=−
∂ xk
ρ ∂t

(1)

In order to derive a model for the compression of axisymmetric homogeneous
turbulence the concept of two-point correlation will be introduced. Following Rotta
[12] the two-point correlation is defined as
Ri j (x, r, t) = u i′ (x, t)u ′j (x + r, t).

(2)

In the limit of zero separation r the two-point correlation Ri j converges to the
Reynolds stress tensor u i′ u ′j . In addition to the two point correlation we define the
triple correlation as
R(ik) j (x, r, t) = u i′ (x, t)u ′k (x, t)u ′k j (x + r, t),
Ri( jk) (x, r, t) = u i′ (x, t)u ′j (x + r, t)u ′k (x + r, t),

(3)

which obeys the intercorrelation
R(ik) j (x, r, t) = R j(ik) (x + r, −r, t).

(4)

Furthermore a two-point pressure-velocity correlation is introduced, which is defined as
p ′ u i′ (x, r, t) = p ′ (x, t)u i′ (x + r, t),

u i′ p ′ (x, r, t) = u i′ (x, t) p ′ (x + r, t), (5)

with the corresponding relation
p ′ u i′ (x, r, t) = u i′ p ′ (x + r, −r, t).

(6)

Using these definitions a general transport equation for the two-point correlation
Ri j may be derived. For the case of homogeneous turbulence all correlations are
independent of the location x. The transport equation for Ri j may be simplified and
yields
∂ Ri j
∂ u¯ j
∂ Ri j

∂ u¯ i
= −Rk j
− Rik
− [u¯ k (x + r, t) − u¯ k (x, t)]
∂t
∂ xk
∂ xk
∂rk



′
′

∂u ′ p ′
∂ 2 Ri j
1 ∂p uj
∂ 
+
− i
+

(7)
R(ik) j − Ri( jk) .
+ 2ν
ρ
∂ri
∂r j
∂rk ∂rk
∂rk

∂ u¯ i
is at most a function of time and the last term in the first line simplifies
∂x j
∂ u¯ k ∂ Ri j
to −rl ∂ xl ∂rk . Note that the mean convection term on the left hand side has vanished
due to homogeneity, i.e. Ri j is independent of the spatial coordinates.

where

148

S. BREUER ET AL.

Taking the divergence of Equation (7) yields a Poisson equation for u i′ p ′
′
∂ 2 R(kl) j
1 ∂ 2 p′ u j
∂ u¯ k ∂ Rl j
=2
−
.
ρ ∂rk ∂rk
∂ xl ∂rk
∂rk ∂rl

(8)

The two-point correlation, the triple correlation and the two-point pressure-velocity
correlation have to satisfy additional constraints derived from the continuity
equation
∂ Ri j

= 0,
∂ri

∂ Ri j
= 0,
∂r j
and

∂ R(ik) j
= 0,
∂r j

∂ R(ik) j
= 0,
∂r j

∂u ′j p ′
∂r j

=0

∂ p ′ u i′
= 0.
∂ri

(9)

In case of homogeneous turbulence the Equations (4) and (6) reduce to
R j(ik) (r) = −R(ik) j (r) u ′j p ′ (r) = − p ′ u ′j (r).
2.1. E QUATIONS

(10)

FOR AXISYMMETRIC TURBULENCE

Homogeneous axisymmetric turbulence is a special case of homogeneous turbulence and is defined by one preferred direction λ. All statistical properties are
invariant with regard to translation in space, finite rotation and mirroring of the
coordinate system about λ. This implies that all tensors in Equation (7) have to be
written as axisymmetric tensors. All tensors depend on a correlation separation z
in the direction of the λ-axis and a separation l perpendicular to λ.
In order to obtain a set of scalar equations for axisymmetric turbulence, all
tensors are transferred into a cylindrical coordinate system, in which the z-axis is
parallel to the significant direction λ.
This idea was first introduced in [4] since it considerably simplifies both classical
notation from [2, 3] as well as its comparison to measurable quantities. In the present
and the following Section 2.2 this extension of the classical work and the ideas of
[4] are further extended to compressed turbulence i.e. slow compression (zero Mach
number limit) either in z- or r-direction.
According to the theory of invariants a second order axisymmetric tensor may
be written as (see e.g. [2, 3])
Ri j (λ, r) = A(l, z)ri r j + B(l, z)δi j +C(l, z)λi λ j + D(l, z)(λi r j +λ j ri ).

(11)

In Equation (11) ri is the component of the correlation vector to an arbitrary
location in space while λi is the component of the unity-vector parallel to the zaxis. According to (11) the correlation tensor Ri j only depends on four independent
components. These functions depend on the axial and radial separations z and l.
Applying the equations of continuity (9), the four components of the correlation

NON-ISOTROPIC LENGTH SCALES

149

tensor Ri j may be reduced to two independent components (see [2])
∂(l Rll ) ∂(l Rzl )
+
,
∂l
∂z
∂(l Rzl ) ∂(l Rzz )
0=
+
.
∂l
∂z

Rφφ =

(12)
(13)

Following [3] the axisymmetric vector for the pressure-velocity correlation may be
written as
Pi (r) = p ′ u i′ (r) = M(l, z)ri + N (l, z)λi .

(14)

Rewriting the latter in a cylindrical coordinate system one obtains two distinct components of the pressure-velocity correlation which are interrelated by the continuity
Equation (9) according to
0=

∂(l · p ′ u l′ ) ∂(l · p ′ u ′z )
+
.
∂l
∂z

(15)

The same decomposition as in (11) and (14) may be applied to the third order tensor,
which may according to [3] be written as
R(ik) j (r) = F(l, z)ri r j rk + G(l, z)λi λ j λk
+ H1 (l, z)ri r j λk + H2 (l, z)[ri rk λ j + r j rk λi ]
+ I1 (l, z)[r j λi λk + ri λ j λk ] + I2 (l, z)rk λi λ j
+ J1 (l, z)[ri δ jk + r j δik ] + J2 (l, z)δi j rk
+ K 1 (l, z)[λi δ jk + λ j δik ] + K 2 (l, z)δi j λk

(16)

After transforming the latter into a cylindrical coordinate system we obtain ten
distinct components of the tensor. These components are interrelated to each other
by the equation of continuity (9) according to




∂ l R(zz)l
∂ l R(zz)z
0=
+
,
 ∂l 
 ∂z 
∂ l R(zl)l
∂ l R(zl)z
R(φz)φ =
+
,
∂z


∂l
1 ∂ l R(ll)z
1 ∂ l R(ll)l
R(φl)φ =
+
,
2
∂l
2
∂z
(17)








∂ l · R(φφ)l
∂ l · R(φφ)z
∂ l · R(ll)l
∂ l · R(ll)z
+
=−
−
.
∂l
∂z
∂l
∂z
As a result, six independent components of the triple correlation tensor remain.
Note that in the limit of zero separation l → 0 and z → 0 the triple correlations
become identical to zero [4].

150

S. BREUER ET AL.

2.2. D YNAMIC

EQUATIONS FOR AXISYMMETRIC COMPRESSED TURBULENCE

In order to derive equations for homogeneous axisymmetric turbulence, the vectors
and tensors in Equation (7) have to be replaced by axisymmetric tensors and the
entire equation has to be transformed into a cylindrical coordinate system. From
the continuity Equation (13) we learn that only two independent scalar equations
remain.
A special treatment for the compression term has to be taken into account. In
case of axisymmetric homogeneous turbulence two cases of compression need to
be distinguished: compression in direction of the axis of symmetry (Figure 1 left)
and compression perpendicular to the axis of symmetry (Figure 1 right). Within this
paper only the compression in the direction of the axis of symmetry is considered,
because of the direction of a compression stroke in a piston engine.
The mean velocity fields for the cases in Figure 1 are derived from the equation
of continuity
1 ∂ρ
1 dh 0 (t)
∂ u¯ z (Z , t)
=−
≡
and
∂Z
ρ ∂t
h 0 (t) dt
1 ∂ρ
2 dr0 (t)
=−
≡
,
ρ ∂t
r0 (t) dt

1 ∂ L u¯ l (L , t)
L
∂L
(18)

where h 0 (t) and r0 (t) respectively denote the time-dependent cylinder hight and
radius. Note that Z and L may not be confused with z and l. Z and L are the physical
coordinates while z and l denote correlation space variables.
Integrating (18) we respectively obtain
u¯ z (Z , t) = Z

1 dh 0 (t)
h 0 (t) dt

and

u¯ l (L , t) = L

1 dr0 (t)
.
r0 (t) dt

(19)

Using these expressions for the compression term in (7), the explicit dependence
on the physical coordinates L and Z vanishes and the entire equation only depends
on the correlation space variables l and z.

Figure 1. Compression of axisymmetrical turbulence in z-direction (left) and in l-direction
(right).

NON-ISOTROPIC LENGTH SCALES

151

For the case of compression in axial direction the transformation of Equation (7)
to cylindrical coordinates respectively yields the correlation equation in axial direction and the cross-correlation equation
∂ Rll
d ln h 0 (t) ∂ Rll
2 ∂ p ′ u l′
=−
z
+
∂t
dt
∂z
ρ ∂l

 2
2
∂ Rll
3 ∂ Rll
2 ∂ Rzl
∂ Rll
+
+
+
+ 2ν
∂z 2
∂l 2
l ∂l
l ∂l
R(ll)l
∂ R(ll)l
∂ R(ll)z
R(φφ)l
∂ R(zl)l
+
+
−
−2
+2
l
∂l
∂z
l
∂z

(20)

and


∂ Rzl
2 ∂ p ′ u l′
d ln h 0 (t)
∂ Rzl
3Rzl + z
+
=−
∂t
dt
∂z
ρ ∂z

 2
2
∂ Rzl
1 ∂ Rzl
Rzl
∂ Rzl
+
+
−
+ 2ν
∂z 2
∂l 2
l ∂l
l2


∂ R(zz)l
∂ R(zl)z ∂ l · R(zl)z
+2
−2
.
(21)
∂z
∂z
∂z
For the subsequent analysis it may be useful to employ the equation for the correlation in axial direction Rzz , which can be derived from the equation of the
cross-correlation (21) together with the equation of continuity (14) leading to


∂ Rzz
∂ Rzz
2 ∂ p ′ u ′z
d ln h 0 (t)
2Rzz + z
+
=−
∂t
dt
∂z
ρ ∂z


2
2
∂ Rzz
∂ Rzz
1 ∂ Rzz
+
+
+ 2ν
l ∂l
∂l 2
∂z 2
R(zl)z
∂ R(zl)z
∂ R(zz)z
+2
+2
(22)
+2
∂z
l
∂l
Transformation of the Poisson equation for the pressure yields one equation for the
pressure-velocity correlation for the case of axisymmetric turbulence. Interestingly
enough the resulting equation may be integrated once with respect to z by employing
continuity equations for the pressure-velocity (15) and for the triple-correlation (17)



d ln h 0 (t)
∂ p ′ u ′z
1 ∂ p ′ u l′
=2
−
Rzl
ρ
∂z
∂l
dt


 
1 ∂ l · R(ll)z
∂ R(zz)l
∂ R(zl)z
+
−l
+ 2l
− R(φφ)l .
l
∂l
∂z
∂z
Note that the compression terms in the latter equations have different pre-factors
and hence do not allow for a reduction as is briefly described in the following for
isotropic turbulence.

152

S. BREUER ET AL.

The similar procedure as above may be applied to derive a set of equations for
isotropic turbulence under spherical compression. In this case only one equation
for the longitudinal two-point correlation remains (see [13])


1 d(lnV (t)) 1 ∂(r 2 Rrr )
1 ∂
∂ Rrr
4 ∂ Rrr
r
=−
+ 2ν 4
∂t
3
dt
r
∂r
r ∂r
∂r

 4
1 ∂ r R(rr )r
+4 4
,
(23)
r
∂r
where r is the spherical correlation coordinate. The latter equation is an extended
version of the classical von-K´arm´an-Howarth equation for incompressible isotropic
turbulence (see [14]). The Poisson equation for the pressure-velocity correlation
vanishes identically in this case.
Introducing the scaling transformation
t˜ = t
1

r˜ = r V (t)− 3
2
R˜ rr (˜r , t˜) = Rrr (r, t)V (t)− 3
1
R˜ (rr )r (˜r , t˜) = R(rr )r (r, t)V (t)− 3

(24)

˜ = νV (t)− 43
ν˜ (t)
into Equation (23) the compression term can be eliminated and the classical form
of the von-K´arm´an-Howarth equation is recovered




4 ∂ r˜ 4 R˜ (rr )r (˜r , t˜)
∂ R˜ rr (˜r , t˜)
1 ∂
∂ R˜ rr (˜r , t˜)
4
+ 4
r˜
= 2˜ν (t˜) 4
,
(25)
∂ t˜
∂ r˜
∂ r˜
r˜ ∂ r˜
r˜
with the only difference that viscosity has become a time-dependent quantity.
It is important to note that for the case of homogeneous axisymmetric turbulence
no such transformation exists, which eliminates the compression terms. This is
because the prefactors of the compression terms in Equations (20)–(22) are different
as has been mentioned above. Hence compression is an essential non-trivial property
of the equations of axisymmetric turbulence.
3. One-Point Equations for Compressed Homogeneous Axisymmetric
Turbulence
The set of equations for the two-point correlations may be used to derive equations
for the Reynolds-stress tensor, the dissipation tensor or alternatively the lengthscale tensor. As in the previous section we consider homogeneous turbulence and
hence all dependencies on the location x vanish.
The two-point correlation Ri j converges to the Reynolds-stress tensor in the
limit of zero separation r
u i′ u ′j = lim Ri j (r, t)
r→0

(26)

153

NON-ISOTROPIC LENGTH SCALES

from which we define the turbulent kinetic energy according to k = 12 u ′m u ′m . Correspondingly the dissipation tensor may be derived from the two point correlation
tensor as (see [15])


∂ 2 Ri j (r, t)
εi j (t) = lim −ν
,
(27)
r→0
∂rm ∂rm
from which we obtain the scalar dissipation rate
(28)

ε = εkk .

From the two point correlation tensor we may also define the integral length-scale
tensor (see e.g. [16])

Ri j (r, t)
3 1
i j (t) =
d V (r),
(29)
8π k(t) V
r2
from which immediately leads to a scalar integral length-scale
(t) = mm (t).

(30)

Since the explicit functional form of Ri j is unknown, [15] employed the CayleyHamilton theorem in order to model a relation between the integral length-scale
tensor, the dissipation tensor and the Reynolds-stress tensor according to
2

 m n
3 k 3/2 
ϕ(m, n) bik
dk j + dikn bkmj ,
i j =
4 ε m,n=0

(31)

where the tensors bi j and di j are respectively the anisotropy part of the Reynoldsstress and the dissipation tensor defined by
u i′ u ′j

εi j
δi j
−
(32)
2k
ε
3
The coefficients ϕ(m,n) may depend on the scalar tensor invariants of bi j and di j (see
[15]). For the model to be presented here, i j is assumed to be a linear function
of bi j and di j . In the following subsection equations for the components of the
Reynolds-stress tensor and the length scale tensor will be presented for the case of
compressed homogeneous axisymmetric turbulence.
bi j =

−

δi j
3

and

di j =

3.1. M ODELLING

THE ONE - POINT EQUATIONS FOR AXISYMMETRIC
COMPRESSED TURBULENCE

Introducing the limits l → 0 and z → 0 into Equations (20)–(22) we obtain
the Reynolds stress equation for homogeneous axisymmetric compressed turbulence. Only the trace elements of the u i′ u ′j tensor remain, which for axisymmetric
′2
turbulence only constitute two independent components u ′2
z and u l to be shown

154

S. BREUER ET AL.

subsequently. Thus, two scalar equations are sufficient to describe the Reynoldsstress tensor in case of axisymmetric turbulence. The turbulence energy is given
by
1
k = u ′2
+ u l′2 .
(33)
2 z
The integration of the Equations (20)–(22) according to the definition of the integral
length-scale (29) yields scalar equations for the components of the length-scale
tensor. As in the case of the Reynolds-stress tensor, only two independent elements
in the trace of the length-scale tensor remain as well. A scalar length-scale  is
defined by the trace of the length-scale tensor
 = zz + 2ll .

(34)

The detailed derivation beginning from the two-point correlation to the
Reynolds-stress tensor and to the length-scale tensor is pointed out in [1, 15].
Applying the one-point limit to Equations (20)–(22) extended by a model for the
dissipation and redistribution leads to



du ′z2
3 u ′z2
d ln h 0 (t)
6 u ′z2
k
= −
+ 2a3 1 −
dt
5 k
2 k
dt

 ′2



3 
zz
k2
3 uz
2
4
−c
4a0 3
(35)
− 1 + cR
−1 + ,


3
2 k
3

  ′2

′
′ 
du l2
3 u l2
d ln h 0 (t)
4 ul
k
=
− 1 + 2a3 1 −
dt
5 k
2 k
dt

 ′2



3 
ll
3 ul
2
4
k2
4a0 3
(36)
− 1 + cR
−1 + .
−c


3
2 k
3
Combining Equations (35) and (36) according to Equation (33) we derive an equation for the turbulent kinetic energy
3

dk
k2
u ′z2 d ln h 0 (t)
= −k
− 2c .
dt
k
dt

The equations for the components of the length-scale tensor are





zz
d ln h 0 (t)
4
zz u ′z2
dzz

=
− 1 + a1 1 − 3
dt

k
3

dt





√ 2 ε
zz
zz 3
− c k cR 3
−1 +2
− cε2 ,
3

 2




′ 
u l2
dll
ll
8
1 d ln h 0 (t)
ll
13 − 10
− 1 − a1 1 − 3
=

dt

k
3

5
dt





√ 2 ε
ll
ll 3
− c k cR 3
−1 +2
− cε2 .
3

 2

(37)

(38)

(39)

155

NON-ISOTROPIC LENGTH SCALES

Combining Equations (38) and (39) according to Equation (34) we obtain an equation for the turbulent kinetic energy



d
u ′ 2 1 28
zz
d ln h 0 (t)
8 zz

= −
+ z + + a1 1 −
dt
5 
k
5 15

dt


3 √
+ 2c cε2 −
k.
(40)
2
The model parameters are according to [15, 17]
c = 0.1643 a0 = 0.05 c R = 4 cεR = 0.8 and

cε2 = 1.92.

As is explained in the appendix, the parameters a1 and a3 are set to zero.
Equations (35)–(40) form a complete set of one-point equations to model compressed homogeneous axisymmetric turbulence. In contrast to the case of isotropic
turbulence, no analytical solution of this set of equations may be derived.
3.2. A PPLICATION

OF THE TURBULENCE MODEL TO THE COMPRESSION
OF TURBULENCE IN A PISTON ENGINE

The compression terms in Equations (35)–(40) are expressed by the motion of a
piston in an engine according to
−sin(α) + 21 λ sin(2α)
d ln h 0 (t)
1 dh 0 (t)
,
=ω
= ω ǫ+1
dt
h 0 (t) dα
+ cos(α) + 41 (1 − cos(2α))
ǫ−1

(41)

where ω, α, ǫ and λ are respectively rotational engine speed, crane angle, compression ratio and crank shaft ratio.
Introducing this equation implies that the top dead center (TDC) before the compression stroke is equal to 0◦ cranc angle (CA) and that the TDC of the compression
stroke is equal to 180◦ CA.
The engine, which has been employed to validate the theoretical results in
Chapter 4 possesses a crank shaft ratio λ = 0.29 and a compression ratio of
ǫ = 8.5.
In order to solve the set of Equations (35)–(40), dimensionless variables have
to be introduced, which define the state of turbulence at the beginning of the compression. Combining the initial conditions for the turbulence energy, the scalar
length-scale and the engine speed, we obtain the dimensionless variable σ , which
is defined as
√
k0
σ =
.
(42)
ω0
The value σ defines the ratio of the compression time scale to a characteristic turbulence time scale. High values of σ mean that the compression is slow compared to

156

S. BREUER ET AL.

the time scale of turbulence. In addition to σ two values are required as initial conditions, which contain the degree of Reynolds-stress and length-scale anisotropy at the
beginning of the compression. As the appropriate characteristic variables we define
γ =

u l′20
u ′2
z0

and

δ=

ll0
.
zz0

(43)

Isotropy means that both ratios, i.e. the Reynolds-stress tensor anisotropy γ as well
as the length-scale stress tensor anisotropy δ, are equal to 1. For the subsequent analysis we assume the same degree of anisotropy for both the Reynolds-stress tensor
and the length-scale tensor at the beginning of the compression and hence δ = γ .
4. Experimental Model Verification
The aim of the experimental investigations is the validation of the one-point model
equations in Section 3 of the compression of homogeneous axisymmetric turbulence. Homogeneous axisymmetric turbulence is realized by the compression of air
in a combustion chamber of a four stroke piston engine at the absence of combustion. It is important to note that there were also no tumble or swirl in the flow. In
order to derive the components of the Reynolds-stress and the length-scale tensor,
the fluid velocity within the chamber has to be determined. For this purpose we applied the particle image velocimetry (PIV) technique, which is a two dimensional,
non-intrusive, optical technique for instantaneous velocity measurement. Since we
have axisymmetric turbulence, the mean velocity and the statistical quantities are
entirely described by two-dimensional information.
4.1. E XPERIMENTAL

DEVICE

Measurements were conducted in a 1.6 L four cylinder transparent engine from
Volkswagen company, which is based on the 827 series. Optical access to the
combustion chamber was given by quartz windows (size: 20 mm × 20 mm) in
the upper part of an intermediate housing between the cylinder head and the cranc
casing. For each cylinder two windows were mounted opposite each other. The two
outer cylinders contain a third window, which is arranged perpendicular to the two
other windows.
The transparent engine is characterized by the data in Table I (see [18, 19]). The
engine has been motored by a dynamometer without fuel and combustion.
4.2. PARTICLE

IMAGE VELOCIMETRY

Since the motion of the air molecules is not visible, the flow field had to be seeded
with tracer particles, assuming that the motion of the tracer particles closely follows
the motion of the fluid elements. The quality of this assumption increases as the

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NON-ISOTROPIC LENGTH SCALES

Table I. Characteristic engine parameters.
Number of cylinders
Bore
Stroke
Compression ratio
Cranc shaft ratio
Maximum engine speed
Shape of combustion chamber

4
79.5 mm
80 mm
8.5:1
0.29
4000 rpm
disc-shaped

particle size decreases. The measuring plane is defined by a thin two dimensional
light sheet located parallel to the combustion chamber axis. The particles, which
are moving in the light sheet plane, induce an optical signal, which is recorded
in perpendicular direction by a camera. By using a pulsed light source, multiple
images of the individual particles moving in the light sheet plane are recorded on
the same photographic film. With the knowledge of the time difference between
pulses the separation of the particle images on the film yields a two dimensional
velocity vector. Using a high speed camera, these multiple images can be recorded
on separated frames.
The intake flow of the engine was seeded with a mixture of Ti O2 and Si O2
particles which have a size of approximately 10 µm. A copper vapor laser ACL25
from OXFORD-LASERS was used to create the light sheet. The laser frequency
can be selected between 8 and 32 kHz. At a nominal frequency of 10 kHz
the pulse energy is approximatively 4 mJ with a pulse length between 15 and
60 ns. The wavelength of the emitted light is 510 and 577 nm. A laser controller, which was triggered externally by a camshaft signal, synchronizes laser
pulses, camera and engine. A CORDIN drum camera was used to capture the
images. With a film length of 1 m, 50 images with a size of 24 mm × 18 mm
were recorded. For the results presented below a laser frequency of 15 kHz was
adopted.
The measuring plane was located just below the cylinder head between the inlet
valve and the exhaust valve. The laser light is coupled in and out by the two quartz
windows in opposite position. The height of the measuring plane is limited by
the size of the windows, which is 20 mm × 20 mm. Since the measurements were
performed in the first cylinder, the scattered light was recorded through a third window, which was installed perpendicular to the light sheet. The experimental setup
of the camera, the light sheet and the compression volume are shown in Figure 2.
The markers, which are shown in Figure 2 are used to define a fixed coordinate
system for each image frame. They were recorded on each image and enable a
perfect alignment of subsequent images to each other.
The images were recorded on a KODAK TMAX 400 film. In order to analyze
a set of images, the images had to be digitalized. A cross-correlation operation of
two subsequent images was performed in order to extract the velocity information

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S. BREUER ET AL.

Figure 2. Set up and measuring plane in the combustion chamber.

[20]. The cross-correlation function of two subsequent images is defined by:
∞
∞
g(x, y)h(x − x, y − y)d yd x.
(44)
Corr (x, y) =
−∞

−∞

This operation is performed on small subimages of digitized PIV-images, which
yields a displacement vector for each subimage. This displacement vector is defined
by the absolute maximum of the correlation function. For the PIV-evaluation performed within this paper also the second and the third maximum was determined.
A post-processing operation selects the displacement of the three maxima, whose
vector fits best to the entire vector field with respect to the neighboring vector
field. The best choice of such a vector is calculated from the neighboring vectors
assuming solid body rotation (45)
uA = uP + ω × rAP ,

(45)

where u P , ω, r A P and u A are respectively the velocity vector, the rotation vector,
the distance vector between the given and the remote point and the velocity at the
remote point, whereto the velocity is extrapolated. An example of a post-processed
velocity field is shown in Figure 3.
4.3. A NALYSIS

OF THE VELOCITY DATA

In order to derive characteristic statistical turbulence quantities from a measured
velocity field, the turbulent fluctuations have to be extracted from the velocity
field. The mean velocity has been calculated by averaging all velocity vectors.

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NON-ISOTROPIC LENGTH SCALES

Figure 3. Example of an instantaneous velocity field from PIV measurement.

The difference between the mean velocity field and the instantaneous velocity field
yields the field of the velocity fluctuations ui′ . These fluctuations were used to
calculate the Reynolds-stress tensor and the length-scale tensor. The Reynoldsstress tensor u i u j was calculated with
1  ′ ′
uu ,
(46)
ui u j =
M·N i j i j
where i and j are the indices denoting the directions z and l of the flow field.
With the assumption of axisymmetric turbulence all components of the
Reynolds-stress tensor were calculated using Equation (46). From the latter formula
and together with the definition of the turbulent kinetic energy and (33) we obtain
1
k = vl′ vl′ + vz′ vz′ ,
(47)
2
The length-scale is obtained from the cross- or auto-correlation function of the
velocity field, which is defined by:

1
˜ z˜ )u ′j (l˜ + l · l,
˜ z˜ + z · ˜z ) (48)
Ri j (l, z) =
u ′ (l,
(m − l) · (N − z) ˜ z˜ i
l

where ˜z and l˜ denote the grid spacing in z- and l-directions respectively. Applying Equation (29) and transforming it into an axisymmetric coordinate system,

160

S. BREUER ET AL.

we obtain
i j =

31
4k



∞
z=−∞



∞

Ri j (l, z)

l=∞

l2

l
dldz,
+ z2

(49)

which in decretized form reads
i j =

3 zl
8 k

b−1


a−1


˜j=−(b−1) i=0
˜

(2i˜ + 1)l
Fi j (li˜ , li+1
˜ , z ˜j , z ˜j+1 )
(2i˜ + 1)2 l 2 + (2 ˜j + 1)2 z 2
(50)

with
Fi j (li˜ , li+1
˜ , z ˜j+1 ).
˜ , z ˜j ) + f i j (li+1
˜ , z ˜j , z ˜j+1 ) = f i j (li˜ , z ˜j ) + f i j (li˜ , z ˜j+1 ) + f i j (li+1
(51)
The length-scale tensor for homogeneous axisymmetric turbulence as well as the
Reynolds-stress tensor contain only values in the main diagonal. The trace of the
length-scale tensor yields the scalar length-scale
(52)

 = 2 · ll + zz .

Calculating the length-scale by using the approach presented here requires the
assumption of homogeneous axisymmetric turbulence. This assumption is considerably weaker than the assumption of isotropic turbulence, which is usually adopted
to calculate a turbulent length-scale from one or zero dimensional velocity fields,
obtained from LDV measurements [5, 7].
5. Results
The model of homogeneous axisymmetric compressed turbulence was validated
against the experimental data taken from a piston engine as described in Section 4.
The boundary conditions at the beginning of compression, which are required for
the mathematical model, are obtained from the experimental data (see Table II).
Table II. Initial conditions for the calculation of axisymmetric turbulence.
Engine speed
0
k0
γ
δ
σ

2000 rpm
0.9 mm
20 m/s
0.93
1
23.7

NON-ISOTROPIC LENGTH SCALES

161

Since the main interest is focused on the development of the turbulent lengthscale, only the length-scale data is considered within this paper. Since the normalized length-scale is independent of the engine speed only the results of the
length-scale at 2000 rpm are presented within this paper.
Experiment and model agree reasonably well as will be demonstrated in the
subsequent figures. The integral length-scale  in Figure 4 increases during the
compression stroke. The minimum slope is observed between 120◦ CA and 150◦
CA and not at the end of the compression phase. The model reveals that, depending
on the initial conditions and the strength of the compression the slope may change
its sign during the compression stroke. This means that for a certain time interval
during the compression stroke, the integral length-scale decreases.
A similar behaviour is observed for the two individual components of the
length-scale tensor. The length-scale in radial direction ll in Figure 5 increases
at the beginning and at the end of the compression phase, while between 90◦
CA and 150◦ CA a weak decrease of the length-scale is noticed. A continuous
increase of the length-scale is noted in the direction of the compression in Figure 6. This is in contrast to the decrease of the combustion chamber height. The
result of the model equation, which predicts that all elements beside the trace
elements of the length-scale tensor disappear, is confirmed by the experimental
data.
The varying behaviour of the length scale evolution such as an increase or
decrease may be interpreted in the light of two main effects in Equation (40). At the
absence of compression the integral length scale always increases due to the last
term which always acts as a source term with the model constants given below (40).

Figure 4. Comparison of experimental data with the calculations of the integral length-scale
 at 2000 rpm.

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S. BREUER ET AL.

Figure 5. Comparison of experimental data with the calculations of the radial component of
the integral length-scale rr at 2000 rpm.

Figure 6. Comparison of experimental data with the calculations of the radial component of
the integral length-scale zz at 2000 rpm.

Only the compression term, the first term on the right hand side of (40) may decrease
. Since the term in square brackets is usually positive, d ln(hdt0 (t)) determines the sign
of this term. For the case of compression the latter is always negative and hence the
difference between the sink and the source term determines an increase or decrease
of the integral length-scale.

NON-ISOTROPIC LENGTH SCALES

163

Figure 7. Model calculation of the Reynolds-stress and the length-scale tensor for 1000 rpm
with almost isotropic initial conditions.

Although there is a difference between the exact values of the model and those
of the experimental data, the tendencies of both agree quite well with each other.
This means that the model, which has been derived, describes the tendency of the
length-scale evolution quite well for the compression of homogeneous axisymmetric turbulence. The difference between the theoretical and experimental values are
twofold. First the experimental data still exhibits a certain amount of scatter, though
several hundred samples have been taken. Second the validity of the assumption of
homogeneous axisymmetric turbulence in the combustion chamber of the engine
during the compression stroke is not fully valid. Though flow visualizations made
it very clear that there is no swirl and tumble in the flow, inlet- and wall-effects
decrease the validity of the underlying assumption.
In addition to the latter results the model for homogeneous axisymmetric turbulence yields information on if and when turbulence becomes isotropic. For the
present model isotropic turbulence means that the two components of both the
length-scale and the Reynolds-stress tensor are equal to each other. The result
presented in Figure 7 reveals that isotropic turbulence is essentially non-existing
particularly during the late stage of compression. This result is independent of
isotropic or anisotropic initial conditions as can be taken from Figure 7 where
both Reynolds-stress and length-scale tensors have been chosen to be isotropic
at α = 0.
6. Summary
Compression of homogeneous axisymmetric turbulence was investigated theoretically and experimentally. On the theoretical side the classical two-point approaches

164

S. BREUER ET AL.

by [2, 3] were extended to include weak compression in the zero Mach number
limit. The theory was written in cylinder coordinates as has been suggested by
Lindborg [4]. In contrast to isotropic turbulence where spherical compression may
be eliminated from the dynamic equation, the von-K´arm´an-Howarth equation, by a
simple scaling transformation, the same is not true for unidirectional compression
of homogeneous axisymmetric turbulence. As a result compression is an essential
non-trivial part of the new theory which was not even implicitly included in the
classical theories.
In addition to the two-point approach the two-tensor one-point model by
Oberlack [1] was adopted and extended for the description of the compression
of axisymmetric turbulence. This model was validated by experimental data from
a four stroke piston engine at the absence of combustion. Axisymmetric turbulence
is a good approximation of the flow in the latter device. From the model equation it
was shown that exactly two independent length-scales exist, one in axial direction
and another in radial direction. Both length-scales may be combined to the usual
scalar integral length-scale.
In case of isotropic turbulence only one integral length-scale and the turbulent kinetic energy are sufficient to characterize turbulence in the one-point
limit. Homogeneous axisymmetric turbulence turns into isotropic turbulence, when
both the length-scales and Reynolds-stresses in radial and axial direction become equal. The one-point model reveals that isotropic turbulence exists in the
case of compression of axisymmetric turbulence only as a transient state at the
very beginning of the expansion phase, if the initial conditions were chosen
isotropic.
For an experimental validation of the one-point model the PIV technique was
applied to obtain velocity data from a turbulent flow field under homogeneous
axisymmetric compression. A procedure to analyze two-dimensional turbulence
data was presented, which yields two independent length-scales for homogeneous
axisymmetric turbulence.
There are essentially two mechanisms that determine the behaviour of the lengthscale evolution i.e. an increase or decrease of the integral length-scale. At the
absence of compression the integral length scale always increases regardless of
whether isotropic or axisymmetric turbulence is considered. This well known effect
is properly modelled even with simple two-equation models. Only the compression
term, which is present in both the two- and the one-point equations may decrease .
Whether an increase or a decrease is observed depends on the sign of d ln(hdto (t)) , the
temporal change of the logarithm of the cylinder hight. For the case of compression
the latter term is always negative and hence the difference between the sink and the
source term determines an increase or decrease of the integral length-scale. In case of
an expansion d ln(hdto (t)) is always positive and hence the integral length-scale always
increases.

NON-ISOTROPIC LENGTH SCALES

165

7. Discussion on the Model Parameters a1 and a3
The parameters a1 and a3 are related by the simple expression
a1 = const1

a3
,
const2 + a3

(53)

and hence we find a1 = 0 if a3 = 0 (see e.g. [1]).

Figure 8. Combination of parameter a1 and coefficient of anisotropy γ for beginning of compression, which yields a negative sign for the change of the scalar length-scale  at the start
of compression.

Figure 9. Combination of parameter a1 and coefficient of anisotropy γ for beginning of compression, which yields a negative sign for the change of the scalar length-scale zz at the start
of compression.

166

S. BREUER ET AL.

Figure 10. Combination of parameter a1 and coefficient of anisotropy γ for beginning of
compression, which yields a negative sign for the change of the scalar length-scale ll at the
start of compression.

In the following a discussion on the parameter a1 is given for the case of rapid
compression. Therefore the length-scale Equations (38)–(40) are analyzed by neglecting the dissipation of the length-scale terms. In this limit we investigate the
sign of the compression term depending on the length-scale and the parameter a1 .
For the two components of length-scale tensor ll and zz and for the scalar
length-scale  the Diagrams (8)–(10) show the combination of the parameter a1
versus the coefficient of anisotropy γ , for which the individual length-scale decreases. The borderlines between the area of a positive sign and the area of negative
sign are described by Equations (54)–(56):

equation for :

equation for zz :
equation for ll :


 
u′2
3 5 kz + 1 − 8 zz
zz < 1
>
−
a1
for
<
28
 > 3
1 − 3 zz


u ′z2 zz
 
 
zz < 1
> 3 1− k 
for
a1
< 4 1 − 3 zz
 > 3

 
 
3 15 u ′z2 1 − zz
zz < 1
>
− −
a1
for
<
8
8 k 1 − 3 zz
 > 3


(54)

(55)
(56)

A mathematical analysis of Equations (54)–(56) reveals that a stable solution
only exists for case a1 = 0. According to Equation (53) the value for the parameter
a3 has to be zero as well.

NON-ISOTROPIC LENGTH SCALES

167

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