RAPID COMMUNICATIONS
PHYSICAL REVIEW FLUIDS
1
, 032401(R) (2016)
Nonlinear interactions isolated through scale synthesis
in experimental wall turbulence
Subrahmanyam Duvvuri
*
and Beverley McKeon
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA
(Received 8 April 2016; published 15 July 2016)
An experimental investigation of nonlinear scale interactions in a forced turbulent
boundary layer is presented here. A dynamic wall perturbation mechanism was used to
externally force two distinct large-scale synthetic modes with well-defined spatial and
temporal wave numbers in a fully turbulent flow. The focus is on characterizing the nonlinear
flow response at triadically consistent wave numbers that arises from the direct interactions
of the two synthetic modes. These experimental results isolate triadic scale interactions
in wall turbulence in a unique fashion, and provide the ability to explore the dynamics of
scale coupling in a systematic and detailed manner. The ideas advanced here are intended
to contribute towards modeling efforts of high-Reynolds-number wall turbulence.
DOI:
10.1103/PhysRevFluids.1.032401
The dynamics of turbulent flows are characterized by the presence of a wide range of energetic
scales with inherent nonlinear coupling. The high-dimensional and nonlinear nature of the problem,
as viewed from a systems perspective, makes it innately challenging. Our interest is in wall-bounded
turbulent flows, in boundary layers, channels, pipes, etc., where the presence of a bounding surface
accentuates the role of fluid viscosity in its vicinity and establishes a mean shear-stress gradient.
The resulting nonhomogeneity in the wall-normal direction alters the characteristics of turbulence
at a fundamental level and introduces unique considerations that set wall turbulence apart from
other classes of turbulent flows. In addition to its scientific importance, the immense practical scope
of applications involving wall turbulence continues to drive concerted research efforts involving
experimental, theoretical, and computational approaches to the problem [
1
].
While considerable progress has been made in explaining the linear mechanisms involved in the
dynamics of wall turbulence, the nonlinear aspects are often treated indirectly through modeling
approaches. The present work is aimed at investigating the dynamics of nonlinear scale coupling
at a fundamental level. Consider a fully developed turbulent wall flow with a three-dimensional
velocity fluctuation field
u
(
x,y,z,t
), where
x,y,z
denote the streamwise, wall-normal, and spanwise
coordinates, respectively, and
t
denotes time. We restrict our attention to two distinct scales of
motion, denoted by
̃
u
1
and
̃
u
2
, that constitute the velocity field
u
together with other scales of
turbulence. They are expressed in the form of normal modes, each with positive (
̃
u
+
) and negative
(
̃
u
−
) complex oblique wave components
̃
u
1
=
ˆ
u
1
e
i
(
k
x
1
x
+
k
z
1
z
−
ω
1
t
)
+
ˆ
u
∗
1
e
−
i
(
k
x
1
x
+
k
z
1
z
−
ω
1
t
)
=
̃
u
+
1
+
̃
u
−
1
,
̃
u
2
=
ˆ
u
2
e
i
(
k
x
2
x
+
k
z
2
z
−
ω
2
t
)
+
ˆ
u
∗
2
e
−
i
(
k
x
2
x
+
k
z
2
z
−
ω
2
t
)
=
̃
u
+
2
+
̃
u
−
2
.
(1)
Here
k
1
=
(
k
x
1
,k
z
1
,ω
1
) and
k
2
=
(
k
x
2
,k
z
2
,ω
2
) denote spatial and temporal wave numbers along the
homogeneous directions (
x,z,t
), respectively.
1
The complex amplitudes
ˆ
u
j
(and conjugates
ˆ
u
∗
j
)inthe
above equation are a function of the nonhomogeneous coordinate
y
and contain phase information
for the modes. It is worth noting that complex exponentials provide the optimal basis functions
*
subrahmanyam@caltech.edu
1
The locally parallel flow approximation is implicitly invoked for a boundary layer flow by treating the
statistics of
u
to be invariant in the streamwise direction. Results from the present experimental scenario
suggest this to be a reasonable approximation for the purposes of this paper.
2469-990X/2016/1(3)/032401(7)
032401-1
©2016 American Physical Society
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SUBRAHMANYAM DUVVURI AND BEVERLEY MCKEON
FIG. 1. Schematic of the flat plate setup (not to scale). A thin spanwise-aligned straight rib is positioned at
the wall (
x
=
0) and actuated in the wall-normal direction to force the flow simultaneously at two frequencies.
Phase-locked hot-wire anemometer measurements were made at three downstream (
x>
0) stations as marked.
The BL thickness
δ
and Reynolds number Re
θ
for the flow at station 1 (
x
=
2
.
7
δ
) are indicated; only a minor
increase in these quantities was noted over the range 0
<x<
5
.
4
δ
.
(energy norm) for decomposition of the velocity field in the homogeneous directions and thus are
routinely employed in descriptions of turbulent flow fields that possess some degree of translational
invariance. For instance, the resolvent operator model [
2
] exploits the low-rank nature of linear
dynamics that results from such a decomposition to successfully capture some key features of wall
turbulence. In addition to its attractive mathematical properties, the Fourier basis provides a clear
and simple quantitative definition of scale in terms of a wave number
k
j
.
We now consider the nature of interaction between the two modes
̃
u
1
and
̃
u
2
and begin by noting
that the nonlinear convective acceleration term (
u
·
∇
)
u
in the governing Navier-Stokes equation is
quadratic in
u
. The implications of this are readily seen from Eq. (
1
): Quadratic interactions between
wave components
̃
u
+
1
,
̃
u
−
1
,
̃
u
+
2
,
̃
u
−
2
establish a direct coupling between the modes under consideration
and their sum and difference wave-number modes, denoted by
̃
u
3
and
̃
u
4
, with wave numbers
k
3
=
k
1
+
k
2
and
k
4
=
k
1
−
k
2
, respectively. Also, self-interactions of
̃
u
1
and
̃
u
2
establish a direct
coupling with their respective harmonic modes, denoted by
̃
u
5
and
̃
u
6
, with wave numbers
k
5
=
2
k
1
and
k
6
=
2
k
2
. Such sets of three wave numbers
{
k
1
,
k
2
,
k
3
}
and
{
k
1
,
k
2
,
k
4
}
are said to possess
triadic consistency, with harmonic wave-number sets like
{
k
1
,
k
1
,
k
5
}
and
{
k
2
,
k
2
,
k
6
}
being the self-
interaction cases. The coupling between triadically consistent wave numbers, referred to as triadic
interactions, is a key aspect of the flow dynamics from which the observed statistics and structure
of wall turbulence emanate. A forcing-response-type approach is used here to experimentally study
triadic interactions in a turbulent boundary layer. Two distinct spatiotemporal modes are externally
forced, or excited, and the linear and nonlinear responses of the flow are measured and analyzed.
A flat plate zero-pressure-gradient boundary layer (BL) flow was forced by a spatially impulsive
dynamic wall perturbation; the flow has a constant free-stream velocity of
U
∞
=
22
.
1m
/
s and
is fully turbulent at the perturbation location with a layer thickness
δ
≈
16
.
6 mm and momentum
thickness Reynolds number Re
θ
≈
2780 (see Fig.
1
). The wall perturbation geometry was adapted
from earlier experiments [
3
,
4
] and consists of a thin straight rib of thickness 1.5 mm aligned along
the spanwise direction. The rib is connected to a linear motor mechanism placed underneath the
plate; a plate insert allows for smooth rib reciprocation in the wall-normal direction. The rib is
actuated by the motor with a waveform consisting of two frequency components, given by
h
(
t
)
/δ
=
a
[2
+
cos(
ω
1
t
)
+
cos(
ω
2
t
)]
,
(2)
where
h
(
t
)
/δ
is the instantaneous nondimensional height (
y
location of the rib) with
h
=
0
corresponding to the wall. The following amplitude and frequency values were chosen for the
experiment:
a
=
0
.
024,
ω
1
/
2
π
=
50 Hz, and
ω
2
/
2
π
=
35 Hz. Note that the maximum instantaneous
height of the rib is around 10% of the local BL thickness;
h
max
=
0
.
096
δ
and
h
rms
≈
0
.
054
δ
from
Eq. (
2
). The entire actuation mechanism was carefully tuned to ensure that the rib displacement
closely matched Eq. (
2
); the same was confirmed by measurements of the actual rib displacement
obtained from a linear encoder. The overall quality of the flow setup was found to be satisfactory
032401-2
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NONLINEAR INTERACTIONS ISOLATED THROUGH . . .
with no vibration or noise contamination in the tunnel test section. Phase-locked measurements of
time-resolved streamwise velocity
U
were made using a hot-wire anemometer at three different
measurement stations downstream of the perturbation (marked in Fig.
1
). At each measurement
station the wall-normal flow profile was sampled at 50 logarithmically spaced locations between
y
=
0
.
008
δ
and 3
δ
, with sufficient temporal resolution to fully resolve the turbulence spectrum
at all locations. Measurements were also made of the canonical flow (smooth flat plate without
the perturbation) for reference and were found to be in satisfactory agreement with previous
literature. The BL thickness and Reynolds number grow at a very slow pace in the region of interest
(0
<x<
5
.
4
δ
) and hence are treated to be constants for the purposes of the present analysis. Also, in
comparison with the canonical flow, no significant changes in
δ
or Re
θ
were noticed in the presence
of the dynamic forcing in the region of measurements. An estimate of friction Reynolds number Re
τ
for the flow is made from Re
θ
using the Coles-Fernholz empirical relationship to be Re
τ
≈
940; the
friction length scale, or inner scale,
ν/u
τ
(
ν
is the kinematic viscosity of air and
u
τ
is the friction
velocity) thus obtained is used along with the outer scale
δ
for reference in later figures. However,
it is important to note that the estimate of Re
τ
is strictly valid only for the canonical flow. Interested
readers are referred to [
5
] for a more detailed description of the experiment.
Before proceeding to the results, we make the following remarks with regard to the experimental
design. A spanwise-constant perturbation results in nominally two-dimensional (2D) disturbances
(
x
-
y
plane) and was chosen for ease in experimentation. The choice of perturbation frequencies and
amplitudes was guided by the objective of targeting the large-scale region of the turbulence spectrum.
The significance of large-scale fluctuations, particularly the very-large-scale motions (VLSMs), on
the overall flow dynamics is increasingly clear [
6
] and hence they are a natural starting point for
attempts at understanding the complex network of coupled spatiotemporal scales of wall turbulence.
The large scales also present a practical avenue to experimentally effect external perturbations
(or forcing) on the flow, particularly at high Reynolds numbers. Previous work by Jacobi and
McKeon [
3
] and by the present authors [
4
] demonstrates the effectiveness of a spatially impulsive
wall perturbation in forcing a single synthetic large-scale mode. The 50-Hz component in the present
case was chosen to match [
4
] and a 35-Hz component was added to simultaneously force another
distinct large-scale mode. The combination of these frequencies results in triadic responses with no
duplication and sufficient separation in the turbulence spectrum (see results). Finally, the amplitude
a
was chosen to induce disturbances of sufficient strength in the downstream region while keeping
the blockage effects associated with the rib to a minimum. We now proceed to the experimental
results and model the wall-perturbation-induced disturbances as spanwise-constant (
k
z
=
0) normal
modes.
Time-resolved streamwise velocity from the anemometer is decomposed into local mean and
fluctuating components
U
(
y,t
)
=
U
(
y
)
+
u
(
y,t
). A slight deficit in the mean momentum relative
to the canonical flow, brought about by blockage effects of the rib and also possibly due to the
dynamic forcing effects, was observed in the logarithmic region (see [
5
]). Since our interest is
in the triadic interactions associated with the synthetic modes, we restrict our attention to the
fluctuations. The power spectral density (PSD)
of
u
provides the distribution of energy across
temporal scales;
is calculated using the standard Welch method and is shown in Fig.
2(a)
as a
function of frequency
f
at station 1 (
x
=
2
.
7
δ
). Linear flow response to external forcing in the
large-scale region can immediately be seen at
f
1
=
50 Hz and
f
2
=
35 Hz in the form of energetic
narrow-band vertical streaks. The two-frequency dynamic forcing excites two distinct large-scale
motions, referred to as the forcing modes,
2
with frequencies
f
1
and
f
2
. From the earlier discussion
of triadic coupling, the interactions between the two forcing modes is expected to result in direct
nonlinear response with some degree of coherence. A closer inspection of Fig.
2(a)
reveals what is
perhaps the most striking feature of this experiment: A clear energetic response is seen at the sum
2
Note that the use of terminology “forcing modes” in the present context differs from that of the resolvent
operator framework [
2
].
032401-3