![Figure 4. Premultiplied one-dimensional spectra kxE 1D uu as functions of the streamwise wavelength λx . (a) Scaled with similarity variables uτ and y. The distance to the wall, y + > 200, increases in the sense of the arrow. Only y/h< 0.1 has been considered in the experiments to avoid contamination by the points in the outer layer, while up to y/h=0.5 has been represented for the DNS. The dotted straight line is 0.2 log(4λx/y). (b) Scaled with outer variables Uc and h, y/h=0.1. The dotted straight line is 5× 10−4 log[24h2/(λxy)]. The solid vertical lines mark the band 6< λx/h< 24 used to obtain figure 5 and (3.3). - - - - - -, L550; , L950; , S1900. The shaded areas cover the maximum scatter of experimental boundary layers from Hites (1997), Reτ =1300–7100. The circles and the hatched regions are for pipe flow from Perry & Abell (1975). Reτ =2325–4900.](/figures/figure-4-premultiplied-one-dimensional-spectra-kxe-1d-uu-as-2mqsvyi0.webp)
J. Fluid Mech. (2004), vol. 500, pp. 135–144.
c
2004 Cambridge University Press
DOI: 10.1017/S002211200300733X Printed in the United Kingdom
135
Scaling of the energy spectra
of turbulent channels
By JUAN C. DEL
´
ALAMO
1
, JAVIER JIM
´
ENEZ
1,2
,
PAULO ZANDONADE
3
AND ROBERT D. MOSER
3
1
School of Aeronautics, Universidad Polit
´
ecnica de Madrid, 28040 Madrid, Spain
2
Centre for Turbulence Research, Stanford University, Stanford, CA 94305, USA
3
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
(Received 26 June 2003, and in revised form 27 October 2003)
The spectra and correlations of the velocity fluctuations in turbulent channels,
especially above the buffer layer, are analysed using new direct numerical simulations
with friction Reynolds numbers up to Re
τ
= 1900. It is found, and explained, that
their scaling is anomalous in several respects, including a square-root behaviour of
their width with respect to their length, and a velocity scaling of the largest modes
with the centreline velocity U
c
. It is shown that this implies a logarithmic correction
to the k
−1
energy spectrum, and that it leads to a scaling of the total fluctuation
intensities away from the wall which agrees well with the mixed scaling of de Graaff
& Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling
with U
c
at very large ones.
1. Introduction
While self-similarity is always a welcome feature in physical problems, simplifying
their solution and providing insight into their behaviour, its failure in situations in
which in principle it ought to apply is perhaps even more interesting, because it forces
us to explain what went wrong. Some of the best-known similarity arguments in fluid
dynamics are those regarding the overlap layer in wall turbulence, and it has lately
become increasingly clear that, while the logarithmic law for the mean velocity profile
may be a good approximation to the experimental data (Zagarola & Smits 1997;
¨
Osterlund et al. 2000), there are serious similarity failures in the behaviour of the
velocity fluctuations, including the scaling of their intensity in wall units and their
predicted k
−1
energy spectrum.
The reason why the intensity of the streamwise velocity fluctuations does not scale
well with the friction velocity was given by Townsend (1976), who noted that wall-
parallel motions are inactive from the point of view of the Reynolds stresses, and do
not have to scale with them. The discovery by de Graaff & Eaton (2000) that they
scale well in mixed units was however unexpected, and presents a clear theoretical
challenge, because it implies that they should be describable by a well-defined model.
Even more interesting is the failure of the k
−1
energy spectrum, which can be easily
predicted from an assumed lack of length scale for fluctuations whose wavelengths
are short with respect to the flow thickness, but long with respect to the wall
distance (Townsend 1976). It has been known since Laufer (1954) that the k
−1
law
is approximately correct, but recent detailed observations by Hites (1997) and by
Morrison et al. (2002) reveal important corrections which are inconsistent with the
original argument.
136 J. C. del
´
Alamo, J. Jim
´
enez, P. Zandonade and R. D. Moser
Case Re
τ
L
x
/h L
z
/h TU
b
/L
x
x
+
z
+
y
+
c
N
x
N
z
N
y
Series 1 L550 547 8π 4π 10 8.94.56.7 1536 1536 257
L950 934 8π 3π 9.27.63.87.6 3072 2304 385
Series 2 S550 550 ππ/277 8.94.56.7 192 192 257
S950 964 ππ/227 7.83.97.8 384 384 385
S1900 1901 ππ/222 7.83.97.8 768 768 769
Tab le 1 . Parameters of the simulations. T is the time during which the statistics were collected
after discarding initial transients. U
b
is the bulk velocity. x and z are the collocation
resolutions parallel to the wall, using N
x
and N
z
points. y
c
is the wall-normal grid spacing
at the centre of the channel, and N
y
is the number of Chebychev polynomials.
The present paper looks at the reasons for those two failures, which we will find
to be related. We use data from new direct numerical simulations of plane turbulent
channels at moderate Reynolds numbers Re
τ
= u
τ
h/ν 6 1900, where u
τ
is the wall-
friction velocity and h is the channel half-width. The new data show that the original
failure of self-similarity occurs in the relation between the lengths and widths of the
structures in the overlap region, and that this failure results in logarithmic corrections
to the k
−1
energy spectrum. These are then demonstrated using experimental data,
for which h will represent the pipe radius or the 99% boundary layer thickness.
We then show, and motivate, that the velocity scale of the largest structures is not
the friction velocity, but the mean velocity at the centreline, and use the resulting
spectral model to derive a modified mixed scaling law for the fluctuations.
The new simulations are described in § 2. The results, and their relations to the
different scaling arguments, are detailed in § 3, followed by a short concluding section.
The companion paper by Jim
´
enez, del
´
Alamo & Flores (2003) extends the present
analysis to the buffer and viscous layers, and that by del
´
Alamo et al. (2004) contains
a more detailed analysis of the structures away from the wall.
2. The numerical experiments
The numerical code integrates the Navier–Stokes equations in the form of evolution
problems for the wall-normal vorticity ω
y
and for the Laplacian of the wall-normal
velocity ∇
2
v, as in Kim, Moin & Moser (1987). The spatial discretization uses dealiased
Fourier expansions in the wall-parallel planes, and Chebychev polynomials in y.The
streamwise and spanwise coordinates and velocity components are respectively x, z
and u, w. The temporal discretization is third-order semi-implicit Runge–Kutta, as in
Moser, Kim & Mansour (1999).
The present simulations, summarized in table 1, are divided into two series. The
runs in the first one use numerical boxes with very long periodicities L
x
and L
z
,to
try to account for all the energetic structures in the flow, including those in the outer
region whose size is proportional to h, and that could not be captured accurately in
previous simulations (Jim
´
enez 1998; Kim & Adrian 1999; del
´
Alamo & Jim
´
enez 2001,
2003).
The second series of experiments focuses on the overlap layer, increasing the
Reynolds number at the expense of the size of the numerical box, especially for the
case S1900. Less emphasis will be put on the other two simulations in this series,
which were performed mainly to assess the effect of small numerical boxes on the
computed statistics. That effect cannot necessarily be neglected, since the fractions
Spectra of turbulence in channels 137
Figure 1. (a) Mean velocity defect as a function of wall distance. (b)DeviationΨ of the
mean velocity from the logarithmic function. ------, L550;
, L950; 䊊, S550; 䊐, S950;
䉭, S1900. The shaded region in (b) covers the maximum scatter of experimental boundary
layers from Smith (1994), Re
θ
= 4600–13000,
¨
Osterlund et al. (2000), Re
θ
= 2500–27000, and
de Graaff & Eaton (2000), Re
θ
= 1430–31000. The hatched area covers experimental pipes
from Perry et al. (1986), Re
τ
= 1600–3800, Durst et al. (1995), Re
τ
= 271–570 and Zagarola &
Smits (1997), Re
τ
= 1700–5.3 × 10
5
.Onlyy/h < 0.1 has been included in the experimental data
to avoid contamination by points in the outer layer.
Figure 2. Premultiplied one-dimensional uv-cospectra k|E
1D
uv
|
+
as functions of wavelength λ
and wall distance. Shaded contours, case L950; line contours, case S950. The contours are
0.2(0.2)0.8 times the maximum value from case L950. The dashed lines mark the size of the
smaller box.
of the Reynolds stress uv which are contained in the simulations L550 and L950
at scales larger than the size of the smaller boxes (i.e. in wavelengths either longer
than λ
x
= πh or wider than λ
z
= πh/2), are respectively 51% and 47% when averaged
across the whole channel.
This result agrees with recent evidence that there are important large-scale contribu-
tions to uv (Jim
´
enez 1998; del
´
Alamo & Jim
´
enez 2001; Liu, Adrian & Hanratty
2001), and is probably the reason why the mean-velocity defects from cases S550, S950
and S1900 do not scale as accurately as those from cases L550 and L950 (figure 1a).
However, figure 2 suggests that the large-scale contributions to uv take place mostly
in the outer region, in agreement with Townsend’s (1976) idea that the impermeability
of the wall limits the large-scale motions with a wall-normal component. This figure
displays the premultiplied one-dimensional uv-cospectra, k
ˆ
u(k, y)
ˆ
v
∗
(k, y), where
ˆ
u
and
ˆ
v are the Fourier coefficients of u and v and k =2
π/λ are either streamwise or
138 J. C. del
´
Alamo, J. Jim
´
enez, P. Zandonade and R. D. Moser
Figure 3. (a, b) Two-dimensional spectral densities Φ
+
as functions of λ
x
/y and λ
z
/y.The
line contours are Φ
+
uu
=0.17. (a) y
+
= 150 and the symbols are Φ
+
vv
=0.06, (b) y/h=0.15 and
the symbols are Φ
+
ww
=0.08. These levels are roughly 5% of the corresponding intensities.
, 䊊, case L550; ——–, 䊐, L950; , 䉭, S1900. The solid straight line is λ
x
= λ
z
,
and the dashed one is λ
2
z
=2κλ
x
y.(c) Sketch of the organization of Φ
uu
, as discussed in text.
(d) Isocontours of correlation coefficient C
vv
=1/3 from case L950. Symbols, C
vv
(15
+
,y): 䊐,
y
+
= 100; 䉫, y
+
= 200; ×, y
+
= 300. Lines, C
vv
(y/2,y): ——–, y
+
= 200; , y
+
= 300. The
correlation increases from left to right, and the straight lines are as in (a).
spanwise wavenumbers. Figure 2 also shows that the misrepresentation of the large
scales in the outer layer of the channel for case S950 does not affect substantially
the resolved part of the uv-cospectrum, especially as we move toward the wall. This
is also true for Re
τ
= 550, and it seems reasonable to expect the same to hold for
Re
τ
= 1900, particularly for the O(y) scales of the overlap region. This assumption is
confirmed by figure 1(b), which displays the deviations of the mean-velocity profiles
from a logarithmic law Ψ = U
+
− κ
−1
log(y
+
), using κ =0.4. It suggests that the
overlap region of case S1900 has approximately reached the typical behaviour of
high-Reynolds-number experimental wall flows, which are represented in the figure
by the hatched and shaded regions. That conclusion, which will be further confirmed
throughout the paper, is not very sensitive to the precise value chosen for κ.
3. Results and discussion
Figures 3(a)and3(b) show isolines of the premultiplied two-dimensional spectrum
of the streamwise velocity, Φ
uu
(k
x
,k
z
,y)=k
x
k
z
ˆ
u
ˆ
u
∗
,aswellasofΦ
vv
and Φ
ww
.
Because of the range of Re
τ
in the numerical channels, figure 3(a) spans a factor of
more than 3 in y/h, while figure 3(b) does the same for y
+
. Both scale well with u
2
τ
and y in the range y<λ
x
< 10 y.
Spectra of turbulence in channels 139
However, the ridge of Φ
uu
lies along λ
z
∼ (y λ
x
)
1/2
, and needs some discussion
because self-similarity would seem to require that λ
z
∼ λ
x
. A similar square-root
behaviour was found for the width of the near-wall streaks by Jim
´
enez et al. (2003),
who showed that its most probable cause was the spreading, under the effect of
lateral fluctuations, of wakes formed by compact v-structures. Although the outer
flow is less coherent than the near-wall layer, and the details of the spreading are
probably somewhat different, the source of the square-root behaviour is in both cases
the long-term dispersion by background turbulence (Townsend 1976, p. 337).
Consider the random stirring of the mean velocity gradient by active eddies of size
O(y) and intensity O(u
τ
), which are represented by the spectral densities Φ
vv
and Φ
ww
in figures 3(a)and3(b). The expected lateral deviations of the fluid elements caused
by those eddies at long times is λ
z
≈ (2 ν
T
t)
1/2
,whereν
T
≈ κu
τ
y is the turbulent eddy
diffusivity in the logarithmic region. As the wakes lengthen, time is converted into
wavelength as λ
x
= Ut,whereU is the difference between the mean flow and the
advection velocity of the forcing fluctuations. There are few data on the advection
velocities of v and w in the logarithmic layer, but Kim & Hussain (1993) find that they
differ from the local mean velocity by at most a few wall units (see also del
´
Alamo
et al. 2004). This leads to the desired relation, λ
2
z
=2κλ
x
y/U
+
, which is represented
in figures 3(a)and3(b) with U
+
= 1. As the wakes widen they also grow in height,
although the y-dependence of ν
T
leads then to a linear growth.
Refer now to the diagram in figure 3(c). Besides the two square-root segments bc
and ad, Φ
uu
is bounded at short wavelengths by the segment ab along λ
z
≈ λ
x
,which
represents the linear dispersion of fluid elements for separations shorter than the
integral scale of the active eddies (see below). This scale is of the order of λ
x
=10y,
and marks both the transition between the linear and square-root behaviours of Φ
uu
,
and the long-wave cut-offs of Φ
vv
and Φ
ww
. It is also the border between wall-attached
and wall-detached eddies. Consider the isocontours in figure 3(d) of the correlation
coefficient of
ˆ
v at two heights, defined as
C
vv
= |
ˆ
v(λ
x
, λ
z
,y
0
)
ˆ
v
∗
(λ
x
, λ
z
,y)|/(|
ˆ
v(λ
x
, λ
z
,y
0
)|
2
|
ˆ
v(λ
x
, λ
z
,y)|
2
)
1/2
. (3.1)
The lines in figure 3(d) indicate that when y
0
∼ y (y
0
= y/2 for the lines shown), and
both y and y
0
are in the logarithmic layer, the λ dependence of the correlations
scales with y. This breaks down however if λ is much larger than h. Similarly, with
y
0
fixed in the viscous layer (y
+
0
= 15 for the symbols in figure 3d), the wavelength
dependence also scales with y, but the region of significant correlation is shifted to
much larger λ
x
. This result, which was also observed for C
uu
and C
ww
, suggests that
there is a range of wavelengths which are correlated within the overlap layer, but not
with the wall, and which are therefore ‘detached’ in the sense of Townsend (1976).
The detached wavelength range extends between the two sets of contours in figure
3(d), and is bounded for large wavelengths at λ
x
≈ 10 y, because for such wavelengths,
correlations reach to the wall. It also contains all the energetic scales for v as shown in
figure 3(a). This is consistent with the idea that blocking by the wall limits the size of
the active wall-normal motions, and explains the transition from linear to square-root
behaviour of Φ
uu
. At long wavelengths the u-spectrum is no longer forced by the
active eddies, and only the incoherent square-root behaviour remains. The ridge of
Φ
uu
, which does not contain Φ
vv
nor Φ
ww
, corresponds to Townsend’s (1976) attached
inactive motions. The limit that separates active and inactive eddies is represented by
the chain-dotted lines in figure 3(c).