Law-of-the-wall for small-scale streamwise turbulence intensity in
high-Reynolds-number turbulent boundary layers
B. Ganapathisubramani
∗
University of Southampton, Southampton SO17 1BJ, UK
(Dated: October 13, 2018)
Abstract
Following the dimensional analysis approach carried out in previous studies, it is hypothesized
that the small-scale fluctuations should only depend on the inner scales, analogous to the Prandtl’s
law-of-the-wall for the mean flow. This allows us to examine the high frequency regime of the
streamwise energy spectra where a “law-of-the-wall” in spectra would hold. Observations in high-
Reynolds-number turbulent boundary layer data indicate that a conservative estimate for the start
of this law-of-the-wall is f
+
= 0.005 (which corresponds to 200 viscous time-units) across a range
of wall-normal positions and Reynolds numbers. This is sufficient to capture the energetic viscous-
scaled motions such as the near-wall streaks, which has a time scale of approximately 100 viscous
units. This spectral collapse is consistent with the observations in internal flows and external
flows in other studies. Furthermore, the spectral collapse leads to a universal scaling (based on
skin-friction velocity and kinematic viscosity) for the small-scale streamwise turbulence variance
(consistent with the hypothesis) across the entire boundary layer. A logarithmic variation of this
small-scale variance is observed farther away from the wall.
∗
g.bharath@soton.ac.uk
1
I. BACKGROUND
A number of previous efforts have focussed on proposing scaling laws for turbulent energy
spectra of turbulent wall-flows. Perry & Abell [15] was perhaps the first to develop scaling
laws for streamwise energy spectra using dimensional analysis approach. This work has been
extensively extended in various subsequent studies [16–18]. The primary focus of these efforts
were on the overlap scaling of streamwise energy spectra where the energy content is inversely
proportional to the wall-normal position. The choice of different velocity and length-scales
for the dimensional analysis in these previous studies were underpinned by the attached-
eddy hypothesis, a model of wall turbulence that provides critical insights. Very recently,
Zamalloa et al.[23] used dimensional analysis and proposed “model-free” scaling relations
for the turbulent-energy spectra in different regions (near-wall, log and outer wake regions)
of turbulent wall-flows. Specifically, they demonstrated the presence of law-of-the-wall in
the high wavenumber regime of energy spectra in pipe/channel flows using experimental and
DNS data over a limited range of Reynolds numbers.
In this paper, we present a re-interpretation of the dimensional analysis work carried out
by Perry & Abell [15] and Zamalloa et al. [23] as well as observations of law-of-the-wall
in small-scale turbulence intensity and in the high frequency regime of streamwise energy
spectra in high Reynolds number turbulent boundary layers. We highlight some subtle
differences (compared to those previous studies) in the choice of velocity and length scales
(used in dimensional analysis) and justify these choices based on physical reasoning and
experimental observations.
II. LAW-OF-THE-WALL FOR STREAMWISE ENERGY SPECTRA
This section is a recap of the dimensional analysis performed by [15] and [23]. However,
we highlight the choice of different scaling variables and the reasoning required for it. Let
φ
11
(k
1
, y) be the power spectral density of the streamwise velocity fluctuation for a longitu-
dinal wavenumber k
1
at location y away from the wall. Then, the integral over all k
1
of this
power spectral density is equal to the turbulent energy of the streamwise velocity component
at that wall-normal location.
2
Z
∞
0
φ
11
(k
1
, y)dk
1
= u
2
(y) (1)
Now, k
1
φ
11
(k
1
, y) is the turbulent energy contained at a given wavenumber k
1
at a certain
wall-normal location. Following [15] and [23] and their dimensional analysis approach, it is
clear that we need a velocity scale to non-dimensionalise k
1
φ
11
, and potentially two different
length scales for the wavenumber and wall-normal position.
k
1
φ
11
(k
1
, y)
˜
U
2
= F (kL
1
, y/L
2
) (2)
where, L
1
is the relevant length scale for wavenumber and L
2
is the relevant length-scale
for wall-normal position.
˜
U is the relevant velocity scale for the energy. We have to to use
physical arguments to arrive at appropriate choices for these scales.
Recall that Prandtl’s law-of-the-wall for mean flow postulates that at high Reynolds
numbers, close to the wall (y << δ, where δ is the outer length scale), there is an inner layer
in which the mean velocity is determined by the viscous scales, independent of outer length
and velocity scales. Based on this, we arrive at an equation for the mean-flow, which is the
law-of-the-wall:
U
U
τ
= f
w
yU
τ
ν
(3)
U
τ
is the skin-friction velocity (U
τ
=
p
τ
w
/ρ, where τ
w
is the wall-shear-stress and ρ is
the density of the fluid) and ν is the kinematic viscosity of the fluid.
We can follow the same reasoning as for the mean profile to determine the appropriate
values for
˜
U, L
1
and L
2
. As an observer at the wall with “no knowledge” of the outer flow,
there is only one choice for
˜
U, which is the viscous velocity scale (U
τ
). There may be two
possible candidates for L
1
: viscous length scale (ν/U
τ
) and wall-normal position (y). For
L
2
, there is only one possible candidate, which is the viscous length scale ν/U
τ
(note that
we will represent scales non-dimensionalised with the viscous scales with a superscript ‘+’).
Then, the spectral scaling becomes,
k
1
φ
11
(k
1
, y)
U
2
τ
= F (k
1
L
1
, y
+
) (4)
[15] used their insights from attached-eddy hypothesis and chose L
1
= y that highlights
distance from the wall scaling. They used U
τ
to be the velocity scale. Here, we take a
3
different approach to determine L
1
. We postulate that there exists a law-of-the-wall for
the small-scale velocity fluctuations, i.e. the integral of the spectrum over a certain high
wavenumber range. In the near-wall region, the variance of the small-scales over a range of
high wavenumbers is only influenced by inner velocity scale such that,
u
2
S
U
2
τ
= g
w
(y
+
) (5)
where, u
2
S
is the variance of the streamwise velocity fluctuations in the high wavenumber
regime that should conform to law-of-the-wall and g
w
is the function that represents law-
of-the-wall for these small-scale motions. This will result in a universal form that is only
dependent on viscous scales and is independent of outer influence. Therefore, we can choose
L
1
as ν/U
τ
. For these high wavenumbers, [15] used the classical Kolmogorov scaling of η
(length scale) and v
η
(velocity scale) that depends on the dissipation of turbulent kinetic
energy (hi). However, given that the Kolmogorov scales changes with wall-normal direction,
especially in the near-wall region, the inner-scales are a much more stringent requirement
compared to Kolmogorov scales. Additionally, it ensures that the small-scale turbulence
intensity has the same scaling parameters as the mean flow.
This above assumption is consistent recent observations of the small-scale behaviour in
wall-bounded flows. Recently, Ref.[20] showed the universality in the small-scales across
different flows even in low Reynolds numbers. This was especially true farther away from
the wall. Ref.[2] showed that the kinetic energy dissipation (and the Kolmogrov scale) in the
near-wall region of a turbulent channel flow scales with the inner scales. They show excellent
collapse of η
+
versus y
+
with increasing Reynolds numbers. Dissipation is a weighted integral
of the energy spectrum, where the weighting is towards the higher wavenumbers. Therefore,
the collapse in small-scale turbulence intensity is similar to the collapse of integrated dissi-
pation spectra. The dissipation based length scale (i.e. the Kolmgorov scale) provide a good
scaling parameter for boundary-free flows, however, in the presence of the wall, skin-friction
velocity (U
τ
) captures the integrated effects of the large-scales and small-scales and there-
fore represents a better scaling parameter. In fact, Ref. [2] built on previous efforts[5, 22]
and showed that the probability density functions of instantaneous Kolmogorov scale can
be collapse with an outer-length scale that is derived based on Townsend’s attached Eddy
model. This was necessary to capture the effect of large-scale on the small-scales Based on
4
these arguments, we set L
1
to be ν/U
τ
. Therefore,
g
w
(y
+
) =
Z
∞
M
+
F (k
+
1
, y
+
)d[ln(k
+
1
)] (6)
is a universal function in the inner region across Reynolds numbers. At fixed values of
y
+
close enough to the wall (i.e. y << δ), the above equation indicates that there should
be a universal value of M
+
that is independent of y
+
.
Zamalloa et al. [23] use y as their length scale L
1
, however, they recognised that this
scaling can be replaced with a scaling similar to the one mentioned above. They did not
examine this in any detail. They used data and observed that the spectra collapses for high
wavenumbers, if, k
1
y ≥ 10(y
+
/Re
τ
). This translates to k
+
1
≥ 10/Re
τ
or k
1
δ ≥ 10, which
essentially means that there should be one order of magnitude difference in the wavenumber
range compared to the large-scales of the flow. These observations were limited in Reynolds
number range (only up to Re
τ
≈ 3000) and were confined to internal flows. This limit
of collapse might have an outer influence since the scale separation is not sufficiently large.
However, the choice of U
τ
as the velocity scale should ensure that there is collapse in the high
wavenumber regime. Therefore, it is important to examine this collapse at higher Reynolds
numbers to determine if there exists an appropriate wavenumber cut-off that eliminates the
dependence of distance from the wall (or Reynolds number). This is examined in the next
section.
II I. OBSERVATIONS IN TURBULENT BOUNDARY LAYERS
It is very difficult to obtain wavenumber spectra at high Reynolds number without in-
voking Taylor’s hypothesis where we have to assume or model a frequency-wavenumber
mapping function. Typically, this mapping function is assumed to just depend on the local
mean velocity. This could lead to incorrect observations on the nature of collapse of spectra.
Therefore, here we resort to using frequency spectra from hot-wire data without invoking
Taylor’s hypothesis (or other equivalent mapping). In this case, law-of-the-wall for spectra
becomes,
fφ
11
(f, y)
U
2
τ
= F (fT
1
, y
+
) (7)
5
![FIG. 3. Inner-normalised pre-multiplied spectra at y+ ≈ 15 for different levels of freestream turbulence (FST). Lines are coloured from lightest to darkest in order of increasing turbulence intensity (the range is from 7% to 13%). The Taylor microscale of FST ranges from 300 to 760. The abscissa shows inner-normalised frequency f+. Further details on the different turbulence intensities, integral length scales and boundary layer characteristics can be found in [6].](/figures/fig-3-inner-normalised-pre-multiplied-spectra-at-y-15-for-36qmjlfq.webp)
