Low-order model reveals wall turbulence dynamics

A comprehensible low-order model for wall turbulence dynamics
Maher Lagha
Citation: Physics of Fluids (1994-present) 26, 085111 (2014); doi: 10.1063/1.4893872
View online: http://dx.doi.org/10.1063/1.4893872
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199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

PHYSICS OF FLUIDS 26, 085111 (2014)
A comprehensible low-order model for wall turbulence
dynamics
Maher Laghaa)
General Electric Aviation, 1 Neumann Way, Cincinnati, Ohio 45246, USA
(Received 3 June 2014; accepted 11 August 2014; published online 28 August 2014)
Streamwise vortices play an important role in the sustainment of wall turbulence.
They are associated with regions with strong Reynolds shear stress production. In
turbulent plane Couette flow at low Reynolds numbers, these streamwise vortices
fill the whole gap between the plates. Using a low-order model obtained from the
Navier-Stokes equations through a two-step Galerkin projection, the dynamics of
these streamwise vortices is shown to be similar to the dynamics observed in the
near-wall region of turbulent boundary layers [M. Lagha, J. Kim, J. D. Eldredge, and
X. Zhong, “Near-wall dynamics of compressible boundary layer,” Phys. Fluids 23,
065109 (2011)]. A spanwise vortex filling the whole gap between the plates and with
vorticity opposite in sign to that of the base flow, is tilted in the streamwise direction
by the spanwise shear of the streaks. The resultant vortex has a crescent shape and
its legs are two streamwise vortices. They regenerate the streaks by the lift-up effect.
Through its ability to generate periodic spatio-temporal flow patterns, this model is
shown to provide an ideal tool for studying the different mechanisms at work. C 2014
AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4893872]
I. INTRODUCTION
The near-wall region in a turbulent flow has a complex dynamic involving the interactions of
various spatio-temporal flow patterns, called coherent structures. Most of the turbulence production
occurs in that region. Spatio-temporal events where the Reynolds shear stress −u v (u is the
streamwise (x) and v the wall-normal (y) perturbation velocity components) is positive indicate the
spatial locations where energy is being extracted from the mean flow and fed into the turbulent state.
Streaks and streamwise vortices are two kinds of coherent structures having the dominant role in
turbulence production. It is well known that the generation of Reynolds stress −u v occurs on each
side of a streamwise vortex. The upwelling (v ≥ 0) and downwelling (v ≤ 0) motions associated
with this vortex generate, through the lift-up mechanism, the streamwise perturbation (i.e., streaks) u
such that the product −u v is positive on each side of the vortex. The self-generation of these coherent
structures ensures the production of Reynolds stresses and thereby maintaining the turbulent state.
While the lift-up mechanism is well understood, the generation mechanism of streamwise vortices
is still unclear.1,2
In this paper, we investigate the generation mechanism of streamwise vortices found in two
different flow configurations: turbulent boundary layer (TBL) and plane Couette flow (pCf).
A. Turbulent boundary layers
Lagha et al.3
showed that the near-wall region of turbulent boundary layers, for y+
≤ 60 (see
definition of wall units below), was populated by crescent-shaped vortices. Their cross sections
extend across the near-wall region. Their legs are streamwise vortices which induce streaks by the
lift-up mechanism. A conceptual model for the generation mechanisms of these streamwise vortices
a)Electronic mail: maher.lagha@ge.com
1070-6631/2014/26(8)/085111/23/$30.00 C 2014 AIP Publishing LLC26, 085111-1
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

085111-2 Maher Lagha Phys. Fluids 26, 085111 (2014)
was presented in Ref. 3. In that model, spanwise vortices are tilted by the spanwise shear of streaks
to form two counter-rotating streamwise vortices. This generation mechanism is further developed
in this paper.
Although the study in Ref. 3 is related to compressible turbulent boundary layers, the results
are also applicable for the incompressible case. In fact, it was shown in Ref. 4 that compressible
boundary layers at zero pressure gradient exhibit close similarities to incompressible boundary layers
and that the main turbulence statistics can be correctly described as variable-density extensions of
incompressible results. When scaled using wall quantities, the statistics for different Mach numbers
collapse with their incompressible part. Therefore, there is likely no difference in the structure and
mechanisms of compressible and incompressible turbulent boundary layer flows.4
B. Plane Couette flow
The second flow configuration that we are interested in is the turbulent plane Couette flow
at low Reynolds numbers. As shown in Fig. 1, pCf is the shear flow between two parallel plates
moving in opposite directions with velocities ±Up. The viscosity of the incompressible fluid is ν.
This flow is stable for all Reynolds numbers R = hUp/ν, defined based on the half-gap h and half
the velocity difference between the two plates, but experiences a transition to turbulence when R
exceeds the stability threshold R
exp
g ∼ 325. Below this stability threshold, the turbulent state cannot
be sustained.5
Note that the flow case involving two plates moving in opposite directions with velocity ±Up
and the flow case involving a fixed bottom wall and an upper wall moving with velocity Up = 2Up,
have the same Reynolds number R as defined above (and the same friction Reynolds number defined
below).
C. Wall units
The use of wall units allows us to establish a link between the turbulence structure in the near-
wall region of turbulent boundary layer and the turbulence structure in plane Couette flow at low
Reynolds numbers.
The near-wall region of turbulent boundary layers is defined from y+
= 0 up to y+
≈ 60.
Here, y+
= yuτ /νw and the superscript “+” denotes normalization by viscous units. The kinematic
viscosity is νw, uτ =
√
τw/ρw is the friction velocity, τw is the wall shear stress, and ρw is the density
at the wall. The friction Reynolds number is Rτ = δ+
99 = δ99uτ /νw, where δ99 is the boundary
layer thickness. For Rτ ≈ 320, y+
≈ 60 corresponds to y/δ99 ≈ 0.2.3
This region is populated
by streamwise vortices that generate streaks through the lift-up mechanism. The spanwise streaks
spacing, proportional to the diameter of these vortices, is about 100 wall units.3
For plane Couette flow, the friction velocity is uτ =
√
τw/ρ, where ρ is the density of the fluid.
The friction Reynolds number defined based on the whole gap (2h) is given by Rτ = 2huτ /ν. The
y+ = δ+
99
y+ ≈ 60
y+ = 0
(a)
−Up
Up
h
y+ = Rτ
y+ = 0
(b)
2Up
2h
y+ = Rτ
y+ = 0
(c)
FIG. 1. The considered flows with cross section of a streamwise vortex. (a) The near-wall region in TBL. (b) Plane Couette
flow. (c) Plane Couette flow with bottom wall fixed. For simplicity, the cross section of the vortex is sketched in the xy-plane
instead of the yz-plane.
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width of the channel (2h) in viscous units is hence equal to Rτ . For high Reynolds numbers, the
region of interest 0 ≤ y+
≤ 60 is close to the wall. If the lower wall corresponds to y+
= 0, then
the upper wall corresponds to Rτ , and the physical location of y+
= 60 is somewhere in between.
For example, for R = Uph/ν = 1300, Rτ ≈ 164 and y+
≈ 60 correspond to y/h ≈ 0.75.6
When
R decreases, the extent of this region increases. For R slightly above the threshold, this region is
expected to fill all the gap, and in fact, for R = 400, we have Rτ ≈ 68 and the upper wall corresponds
to y+
≈ 68.7–9
Since the “near-wall” region is now filling the whole gap between the plates, it is legitimate
to expect the presence of streamwise vortices with cross-stream section extending through all the
gap, as sketched in Fig. 1. In fact, several studies already pointed out this important feature of
these streamwise vortices in pCf for low-Reynolds numbers near R
exp
g . A short literature review
follows.
D. Streamwise vortices in low-Reynolds turbulent pCf
1. Direct numerical simulations (DNS) and experiments
In the DNS of Hamilton, Kim, and Waleffe,7
with R = 400 and Rτ ≈ 68, streamwise vortices
spanning all the gap between the driven plates were observed and studied. A self-sustained process
for these vortices was later on proposed by Waleffe.8
Similar streamwise vortices, but for slightly
higher Reynolds numbers, were also observed in the literature. Numerical simulations investigating
the effect of the domain size on the dynamics of these structures were carried out by Komminaho
et al.10
who considered a large computational box (88h × 25h). They obtained similar streamwise
vortices but their dynamics was less constrained by the size of the box and they can meander in
the flow field (Figure 8 in Ref. 10). The Reynolds number was R = 750, i.e., roughly twice the
value of the stability threshold R
exp
g ∼ 325, and Rτ ≈ 104. Finally, the studies of Bottin et al.5
(for
Reynolds numbers near R
exp
g ) and, more recently, of Kitoh et al.11
(for R = 750), give experimental
evidence for these streamwise vortices and hence confirm the previous numerical results. However,
despite this body of studies, a simple physical generation mechanism of these vortices is still
missing.
2. Low-order modeling
Simplifying the complex dynamic of the coherent structures can shed some light on the mech-
anisms at work. One of several methods consists of deriving low-dimensional models in terms of
ordinary differential equations (ODEs) taking into account some features of the turbulent flow.12
Well designed ODEs models involve some linear terms, which extract the energy from the base
flow, coupled with conservative quadratic nonlinear terms, which redistribute this energy among the
modes and lead to a sustained turbulent state. A comparative study of such models was given by
Baggett and Trefethen.13
More related to Navier-Stokes equations, some ODEs models were pioneered by Waleffe.
He projected the three velocity components onto some Fourier modes. In contrast with former
models,13
the modes have physical meanings such as streaks, mean flow, streamwise vortices, and
streaks instability. His work was followed by, among others, the Marburg group14,15
where additional
modes were included, motivated by physical considerations. For example, in addition to the eight
modes retained by Waleffe, a mode representing the modification of the profile of the base flow
was considered in Ref. 15. Some features of shear flow turbulence, such as the distributions of the
transient lifetimes, have been captured by this 9-mode model, with much fewer modes than the
extended 17-mode model suggested in Ref. 8.
A survey of the work of Waleffe and the Marburg group14,15
was recently given by Smith et al.16
In this work, Navier-Stokes equations were projected onto the most energetic modes, calculated via
the proper orthogonal decomposition17–19
from numerical data of pCf. However, their models were
geared towards reproducing the turbulent statistics rather than understanding the flow dynamics.
The common denominator between most of these studies is the fact that the Navier-Stokes
equations were directly projected into a few modes, usually the first modes of a (Fourier) basis
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satisfying the boundary conditions, with no or little a priori physical meaning. As we will see in
this paper, a two-step projection method is a simple and helpful strategy for selecting the relevant
modes.
E. Goal and organization of the paper
Since for low Reynolds number, the region 0 ≤ y+
≤ 60 is found to fill all the gap be-
tween the plates in pCf, it is legitimate to wonder if the coherent structures living in that re-
gion and the associated mechanisms are similar to those found in the near-wall region of tur-
bulent boundary layer.3
Also, is it possible to derive a comprehensible model illustrating their
dynamics?
Answering these questions using the usual techniques (DNS of pCf at low Reynolds number or
direct Galerkin projection) is not the optimal approach for this problem. On one hand, the DNS results
have both a coherent part (i.e., coherent structures) and an incoherent part (background turbulence)
tangled together. This makes the study of the coherent structures difficult, and an effort to educe
these coherent vortices and to infer the mechanisms at work will be necessary. On the other hand,
since the structures of interest fill the whole gap between the plates, a low-resolution simulation
in terms of a few wall-normal modes (i.e., in the y-direction) would be sufficient to capture the
dynamics of these streamwise vortices, and will shed light on the other coherent structures (streaks
and other vortices). Knowing these coherent structures will guide us in projecting the Navier-Stokes
equations in the remaining horizontal directions (i.e., in the (x, z)-directions) and will provide the
aimed comprehensible model.
A two-step modeling strategy taking into consideration these points is used in this study.
In the first step (Sec. II), a model is derived by projecting the Navier-Stokes equations in the
wall-normal direction on a well-defined basis. The obtained model consists of a set of partial
differential equations (PDEs) and will serve to identify the relevant coherent structures. In the
second step (Secs. III and IV), this model is projected in the horizontal directions on a basis of
localized functions describing these coherent structures. The obtained model consists of a set of
ordinary differential equations. The main results of the present study are assessed in the conclusion in
Sec. V.
II. STEP-1: PDEs MODELING
First, we denote by u the streamwise (x), v the wall-normal (y), and w the spanwise (z)
dimensionless velocity components. Lengths are scaled by the half-gap between the plates h, and
velocities by Up so that U(y) = y is the base flow for y ∈ [ − 1, 1]. The control parameter is the
Reynolds number defined as R = Uph/ν, where ν is the fluid’s kinematic viscosity.
Our modeling strategy is as follows. First, the velocity field is projected over a complete basis
of y-dependent modes
u(x, y, z, t) =
n
Un(x, z, t)Rn(y),
where Rn(y), n = 0, 1, . . . are polynomials of increasing degree and satisfying the no-slip boundary
condition at the walls.20
Similar expressions hold for the velocity components v and w. The fact
that the coherent part consists of large scale structures filling all the gap between the plates allows
us to use the first few modes (low order polynomials) to represent the coherent part. The remaining
higher modes represent the incoherent part.
Then, to untangle the coherent part from the incoherent part, the latter is simply removed by
truncating the expansion to the first few modes. At the lowest consistent order, the expansions of the
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

velocity components read
u(x, z, t, y) = U(y) + u = U(y) + U0(x, z, t)R0(y) + U1(x, z, t)R1(y), (1)
v(x, z, t, y) = v = V1(x, z, t)S1(y), (2)
w(x, z, t, y) = w = W0(x, z, t)R0(y) + W1(x, z, t)R1(y), (3)
where U0, W0, U1, W1, and V1 are the amplitudes of the perturbation components. The y-dependent
polynomials satisfying the no-slip boundary conditions on the plates (y = ±1) are: R0(y) = B(1 −
y2
), R1(y) = Cy(1 − y2
), and S1(y) = A(1 − y2
)2
, where A, B, and C are normalization constants.20
The three amplitudes U1, V1, W1 represent vortical structures with cross-stream sections filling
all the gap between the plates. The polynomial S1 keeps the same sign over the gap, whereas
R1 changes its sign at y = 0. Streamwise vortices would be represented mainly by (V1, W1) with
|U1| |W1|. Spanwise vortices would be represented mainly by (V1,U1) with |W1| |U1|. Through
the lift-up mechanism, these streamwise vortices generate low- and high-speed streaks filling all the
gap. Since the sign of the polynomial R0 is constant across all the gap, the amplitude U0 will capture
these streaks.
Inserting the velocity expansions together with a similar pressure expansion into the Navier-
Stokes equations, and using the Galerkin method, give the governing equations for the five amplitudes
U0, U1,V1, W0, and W1.20
The pressure is eliminated from the equations by using streamfunctions
and velocity potentials. The obtained model consists of three PDEs governing the evolution of two
streamfunctions 0 and 1 and one velocity potential 1 (Eqs. (A1)–(A3)).
The model was shown to reproduce the globally sub-critical behavior of the transitional pCf.
Within the model, a sustained turbulent state can be obtained for R ≥ Rg ∼ 175, where Rg is its
stability threshold. Further properties of this model and details on the numerical method can be
found in Refs. 20 and 21. A sample Matlab program is provided as the supplementary material.22
Finally, of particular interest are the Reynolds stresses associated with the turbulent energy
production P ≡ − V uv d
dy
U dV, where V is the volume of the domain. Within the PDEs model, one
readily gets P = − S αU0V1 dS, where S is the surface of the domain and α is a positive integration
constant. Since the streamwise vortices generate streaks (U0) through the lift-up mechanism, regions
where the Reynolds stress −U0V1 is positive, are those with U0 > 0 and V1 < 0 and vice versa.21
A. Crescent vortices within the PDEs model
Vortices filling all the gap can be depicted by their wall-normal velocity v at the channel
centerline y = 0 which, according to Eq. (2), is proportional to V1. As shown in Fig. 2, the distribution
of V1 highlights many crescent contours. They are roughly similar in size (≈ 2h in x and 3h in z).
Together with the flow field (U1, W1), each crescent contour represents a crescent vortex, sketched
in Fig. 3. Its legs are two counter-rotating streamwise vortices. Regions with positive (negative)
V1 correspond to regions of negative (positive) U0, represented with quasi-horizontal vectors since
|U0| |W0|. Therefore, regions with positive Reynolds stress −U0V1 (Fig. 4) are found on each
side of the legs.
20
10
20
30
40
50
60
40 60 80 100 120
-0.4
-0.2
0
0.2
0.4
X
Z
FIG. 2. Wall-normal velocity V1 in gray levels in a large computational box with Lx × Lz = 128h × 64h, R = 200.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.3
−0.2
−0.1
0
0.1
0.2
23 24 25 26 27 28
20
21
22
23
24
25
26
27
X
Z
FIG. 3. Left: the amplitude V1 in gray levels and the flow field (U0, W0) with arrows. Right: the corresponding flow field
(U1, W1) at t = 17.
The ratio of the spanwise width of the crescent vortices to the gap width is ≈3h/2h = 3/2. When
compared to the crescent vortices observed in the DNS of compressible turbulent boundary layers
(Ref. 3), a semi-quantitative feature can be derived using the wall units. Since the gap height 2h
roughly corresponds to 60 viscous units, the width of these crescent vortices will be ≈(3h/2h)60
= 90 viscous units. This value is indeed similar to the width of the crescent vortices found in the
near-wall region of compressible turbulent boundary layers (Ref. 3).
B. The generation process
The generation process follows the conceptual model presented in Ref. 3. Consider on one hand
a spanwise vortex represented by the flow components V1S1 and (U1, W1)R1 and characterized by (i)
two z-elongated patches of negative and positive V1 and by (ii) a region where |U1| |W1| and U1
≤ 0 (Figs. 5(a) and 5(b)). This negative U1 represents a patch of streamwise velocity counteracting
the base flow. It is also characterized by a z-elongated patch of positive spanwise vorticity ωz =
∂x V1 − βU1 (with β =
√
3) as shown in Fig. 5(c).
Consider, on the other hand, a quadrupolar flow field (U0, W0) with an inflow towards the
patch of ωz, easily identified as regions where |U0| |W0|, and a spanwise outflow pointing out
of this patch, identified as regions where |W0| |U0|, as shown in Fig. 5(d). The (x, z)-location
y1
y0
y−1
vp
vn
z
x
23 24 25 26 27 28
20
21
22
23
24
25
26
27
X
Z
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
FIG. 4. Left: The 3D reconstruction of the flow field (U1 R1, W1 R1, V1 S1) of Fig. 3 represents a crescent vortex. Solid-red
(dashed-blue) contours indicate regions of positive (negative) values. Right: Isocontours of the Reynolds stress −U0V1.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.3
−0.2
−0.1
0
0.1
0.2
(a)
23 24 25 26 27 28
20
21
22
23
24
25
26
27
X
Z
(b)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.5
0
0.5
1
1.5
(c)
23 24 25 26 27 28
20
21
22
23
24
25
26
27
X
Z
(d)
FIG. 5. Spatial distribution of V1 in gray levels (a), the flow field (U1, W1) (b), ωz (c), and the flow field (U0, W0) (d). The
stagnation point (U0 ≈ 0 and ∂xU0 ≤ 0) is marked with a green circle. t = 6.
where U0 ≈ 0 and ∂xU0 ≤ 0 is termed a stagnation point herein. This particular flow-structure, i.e.,
a two-dimensional quadrupolar flow (U0, W0)R0 around a retrograde spanwise vortex, was called
key-structure in Ref. 3.
Then, the two-dimensional flow (U0, W0)R0 deforms this retrograde spanwise vortex into a
crescent vortex, as illustrated in Fig. 6 using the wall-normal velocity V1 and in Fig. 7 using the
streamwise vorticity ωx = βW1 − ∂z V1. In each sub-figure of Fig. 6, the flow (U0, W0) is represented
with (red) vectors. From (a) to (d), the positive V1 patch is deformed from its initial straight shape
in the z-direction to a crescent shape. The last configuration is the crescent vortex shown in Fig. 3.
While the spanwise vortex represents its head, the two counter-rotating streamwise vortices form
its legs and they regenerate the streaks by the lift-up effect, as attested to by the positive Reynolds
stress, shown in Fig. 4. Hence, these streamwise vortices extract the energy from the base flow across
the whole gap.23
III. STEP-2: ODEs MODELING
The numerical simulations of the PDEs model has highlighted several kinds of coherent struc-
tures, summarized below in Sec. III A. The derivation of the ODEs model is outlined in Sec. III B
using an appropriate basis of functions to model these localized coherent structures.
A. The identified coherent structures
The first type of coherent structure consists of four closely counter-rotating vortices (axis along
the y-direction) and represents the quadrupolar flow (U0, W0) with a stagnation point U0 ≈ 0. The
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.3
−0.2
−0.1
0
0.1
0.2
(a)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
(b)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
(c)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.3
−0.2
−0.1
0
0.1
0.2
(d)
FIG. 6. Sequence of figures showing the deformation of the spanwise vortex to a crescent vortex (shown in Fig. 3). V1 in
gray levels and the flow field (U0, W0) with red vectors. Time t = 5 (a), t = 7 (b), t = 12 (c), t = 17 (d).
second type is the spanwise vortex (axis along the z-direction) spanning all the gap with vorticity
opposite in sign to that of the base flow, termed retrograde spanwise vortex. The third type represents
the streamwise vortices. All these coherent structures are characterized by localized vorticity fields.
Attempting to model them with a Fourier basis will require a high number of modes, and interpreting
the physical interactions between the coherent structures will become a tedious task. Fortunately,
more suitable basis functions do exist where each coherent structure can be represented by no more
than one or two modes.
B. Expansions of the fields using Hermite functions
In the following, we consider the representation of the three fields by series of Hermite functions
(Appendix C). The symmetries of the problem are first used to simplify the expansions of these fields.
The equations (Eqs. (A1)–(A3)) are equivariant with respect to the three symmetries T Z, T X, and
T R = T X ◦ T Z, (Eqs. (B1)–(B3)). The fields satisfying all three symmetries have these Hermite
expansions
sym
0 (x, z, t) =
n=0,m=0
ψ2n+1,2m+1
0 (t)hσ1
2n+1(x)hσ1
2m+1(z), (4)
sym
1 (x, z, t) =
n=0,m=0
ψ2n,2m+1
1 (t)hσ3
2n(x)hσ3
2m+1(z), (5)
sym
1 (x, z, t) =
n=0,m=0
φ2n+1,2m
1 (t)hσ6
2n+1(x)hσ6
2m(z). (6)
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(a)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(b)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(c)
X
Z
23 24 25 26 27 28
20
21
22
23
24
25
26
27
−1
−0.5
0
0.5
1
(d)
FIG. 7. Sequence of figures showing the formation of streamwise vortices (shown in Fig. 3). The vorticity ωx in gray levels
at t = 5 (a), t = 7 (b), t = 12 (c), t = 17 (d). The two x-elongated patches of ±ωx form two counter-rotating vortices.
The amplitudes (ψn,m
0 , ψn,m
1 , φn,m
1 ) introduced in these expansions are all functions of time.
To this set of symmetric components (
sym
0 ,
sym
1 ,
sym
1 ), a set of asymmetric components
(
asym
0 ,
asym
1 ,
asym
1 ) will be added. There are three sets, the first set is only equivariant under
T Z, the second under T X, and the third under T R. The first set is chosen here since the vorticity
2 0 of the dipole (U0, W0) which deforms the spanwise vortex (to give streamwise vortices) is
odd in z. Such a dipole is shown in Fig. 6(d). For this set, the Hermite expansions of the fields read
asym
0 (x, z, t) =
n=0,m=0
ψ2n,2m+1
0 hσ2
2n(x)hσ2
2m+1(z), (7)
asym
1 (x, z, t) =
n=0,m=0
ψ2n+1,2m+1
1 hσ4
2n+1(x)hσ4
2m+1(z), (8)
asym
1 (x, z, t) =
n=0,m=0
φ2n,2m
1 hσ5
2n(x)hσ5
2m(z). (9)
The next step in the modeling is the truncation of these expansions. The first three expansions
(4)–(6) are truncated to the lowest order n = m = 0. Special care has to be taken regarding the
second expansions, (7)–(9), since the retained modes have to be interconnected enough to allow the
feedback loop that produces the sustained turbulent state.
Indeed, if expansions (7)–(9) were also truncated to the lowest order (n = m = 0), no sustained
turbulent state would be obtained with an ODEs model involving only these modes. The streak U0
corresponding to ψ0,1
0 (in Eq. (7)) is x-elongated and its sign changes along the z-direction. Since the
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

lift-up effect imposes a relation between the distribution of V1 and that of U0, the spatial distribution
of V1 has to be similar to the distribution of U0 and hence has to change its sign in the z-direction
too. This is accomplished by retaining the second mode φ0,2
1 hσ5
0 (x)hσ5
2 (z) in (9), corresponding to n
= 0 and m = 1.
Then we introduce the truncated expansions in Eqs. (A1)–(A3) and project over each mode
using the scalar product (Eq. (C1)). This is straightforward for the fields 0 and 1, due to
the parity properties of their functions, which automatically guarantees their orthogonality, and
hence the separation of their different contributions. For the field 1, complications arise since
the lowest consistent truncation involves, in addition to hσ6
1 (x)hσ6
0 (z), the two functions f1(x, z) ≡
hσ5
0 (x)hσ5
0 (z) and f2(x, z) ≡ hσ5
0 (x)hσ5
2 (z). The latter two functions have the same parity and hence
their contributions are not separated. Therefore, the expansion has to be orthogonalized.
C. Orthogonalization
Let 1 be expanded as
1 =
n,m
fmnφm,n
1 ,
where fmn are functions of x and z and the coefficients φm,n
1 are time dependent. The evolution
equation of 1 (Eq. (A3)) is written symbolically in the form
∂t ( 2 − 3) 2 1 = F. (10)
The presence of the spatial operator complicates the derivation of the ODEs model. Whereas taking
the scalar product is an operation that commutes with the temporal derivative, which properly
isolates the amplitudes, this is not the case for differentiation with respect to x and z, which mixes
contributions coming from several orders. In fact, injecting the truncated expansion for 1
1(x, z, t) = φ0,0
1 (t) f1(x, z) + φ0,2
1 (t) f2(x, z) (11)
in Eq. (10) and projecting over f1 and f2 yield
d
dt
φ0,0
1 ( 2 − 3) 2 f1, f1 + d
dt
φ0,2
1 ( 2 − 3) 2 f2, f1 = F, f1 ,
d
dt
φ0,0
1 ( 2 − 3) 2 f1, f2 + d
dt
φ0,2
1 ( 2 − 3) 2 f2, f2 = F, f2 .
The non-vanishing values of the quantities ( 2 − 3) 2f2, f1 and ( 2 − 3) 2f1, f2 couple the two
amplitudes φ0,0
1 and φ0,2
1 .
Using the Gram-Schmidt orthogonalization method, a family of functions fmn can be constructed
from the Hermite functions such that the amplitudes φm,n
1 are separated
fmn, ∂t ( 2 − 3) 2 1 =
dφm,n
1
dt
= fmn, F .
We take f1 = f1 and we construct a function f2 that is orthogonal to the first function f1 such that
( 2 − 3) 2 f1, f2 = 0.
We expand f2 as a linear combination of f1 and f2:
f2 ≡ f2 + θ f1 = hσ5
0 (x)(hσ5
2 (z) + θhσ5
0 (z)),
for some constant θ. Then, requiring that
( 2 − 3) 2 f2, f1 = 0, (12)
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

gives us the value of θ. Hence, the fields are expanded and rewritten for convenience as
0(x, z, t) = A1hσ1
1 (x)hσ1
1 (z) + A2hσ2
0 (x)hσ2
1 (z), (13)
1(x, z, t) = B1hσ3
0 (x)hσ3
1 (z) + B2hσ4
1 (x)hσ4
1 (z), (14)
1(x, z, t) = C1hσ6
1 (x)hσ6
0 (z) + C0hσ5
0 (x)hσ5
0 (z) + C2hσ5
0 (x)(hσ5
2 (z) + θhσ5
0 (z)), (15)
where the amplitudes A1, A2, B1, B2, C0, C1, and C2 are functions of time. For the case studied here,
we take σ1 = σ2 = σ4 = σ5 = σ6 = 1, σ3 = 1/
√
3, and θ = 7
5
√
2
(using Eq. (12)). These constants
σi are chosen to reproduce coherent structures similar to these previously observed with the PDEs
model. With these values, the streaks spacing associated with the stream-function with amplitude
A2, is roughly 2h, computed as the distance between the two zeros of U0 in the z-direction.
D. Physical interpretation of the modeling
In the expansion of the stream function 0 (Eq. (13)), the first mode A1hσ1
1 (x)hσ1
1 (z) corresponds
to a quadrupolar flow (U0, W0). If A1 is positive, we get a stagnation point with U0 ≈ 0 and ∂xU0 ≤ 0
at x=z=0, as shown in Fig. 8 (left). Then, for the stream function 1, the first mode B1hσ3
0 (x)hσ3
1 (z)
represents a localized region of U1 = −B1hσ3
0 (x)∂zhσ3
1 (z) centered at x=z=0. If B1 is positive, then
U1 is negative and it represents a localized region where the base flow is decreased. In other words,
this negative U1 is a local correction of rotational origin (since it comes from the first mode B1 of
the stream function 1) to the base flow.
Consider now the first mode in the expansion of 1 (Eq. (15)). As shown in Fig. 9 (left), the
associated wall-normal velocity V1 associated with the mode C1hσ6
1 (x)hσ6
0 (z) has two z-elongated
regions with an x-alternating sign. Together with the associated velocity field (U1, W1), the flow
(U1, W1, V1) represents a spanwise vortex. Its direction of rotation depends on the sign of C1. For
negative C1, its rotation is in the opposite direction than the base flow, as the spanwise vortices
observed within the PDEs model (called retrograde spanwise vortices). The corresponding U1 to
this vortex is negative and it represents a localized region where the base flow is decreased. Since
this U1 comes from the velocity potential C1, it represents a correction of potential origin to the base
flow.
The remaining modes in the expansion of 1, C0, and C2, represent the streamwise vortices,
as shown in Fig. 9 (right). The velocity V1 associated with these two modes has three x-elongated
patches with a z-alternating sign. The direction of rotation of these vortices depends on the values
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
X
Z
FIG. 8. Flow field (U0, W0) associated with 0 (Eq. (13)). Left: (A1, A2) = (1, 0). The quadrupolar flow (U0, W0) is
symmetric and there is a stagnation point at x=z=0 with ∂xU0 ≤ 0. Right: a non-symmetric case with (A1, A2) = (1, −1).
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
FIG. 9. Reconstruction of the velocity components (U1, W1, V1). The wall-normal velocity V1 is shown in gray-level while
the in-plane flow field (U1, W1) is represented by vectors. Left: spanwise vortex with B1=B2=C0=C2=0 and C1=−1/2.
Right: streamwise vortices with B1=B2=C1=0 and C0=−1, C2=1.
of the amplitudes C0 and C2. The mode A2hσ2
0 (x)hσ2
1 (z) corresponds to a dipole structure (U0, W0)
where the axes of the two vortices are located at (x=0,z=±1). If A2 reaches a finite value, the
symmetry of the quadrupolar flow associated with A1 is broken, as shown in Fig. 8 (right).
The physical meaning of the retained seven modes are summarized in Table I. In the following,
we outline the derivation of the 7-ODE model and we present some of its properties. This model is
further reduced in Sec. III F to a 6-ODE model by taking only two modes for 1 in Eq. (15) rather
than three and a complete study of the dynamics is given there.
E. Derivation of the 7-ODE model and numerical results
The equations of the 7-ODE model are obtained by inserting the expansions (13)–(15) in (A1)–
(A3) and using the scalar product (Eq. (C1)). The nonlinear and linear terms are integrated in time
using a second order Adams–Bashforth scheme, whereas the diagonal viscous terms are exactly
integrated. A sample Matlab program is provided as the supplementary material in Ref. 22. From a
numerical point of view, the conservative feature of the ODEs model can be illustrated by means of
numerical simulation with suppression of the dissipation (R−1
= 0) and with no energy input from
the base flow through the linear terms (a07 = a08 = a14 = a140 = a15 = a20 = a24 = a40 = a41 ≡ 0).
For a long time evolution ( t ∼ 2000), the fluctuation of kinetic energy remained less than 0.001,
hence the numerical scheme preserves the energy.
Results obtained with this model are presented below. The initial condition is B1(0) = 3 and
A2(0) = 1. Time evolutions of the amplitudes A1, B1, and C1 (left) and C2 and A2 (right) are reported
in Fig. 10. Note that the amplitude C1 remains, on average, negative during the time, while A1 and
B1 remain mostly positive. The signs of the amplitudes A2 and C2 are alternating and their temporal
evolution is irregularly periodic, yet they evolve in phase. The interaction of the amplitudes is similar
TABLE I. Physical meaning of the retained modes.
Amplitude Physical meaning
A1 Quadrupolar flow with a stagnation point at x=z=0 if A1 ≥ 0
B1 Local correction to the base flow centered at x=z=0 if B1 ≥ 0
C1 Retrograde spanwise vortex and local correction to the base flow if C1 ≤ 0
C2 (and C0) Streamwise vortices
A2 Streaks regenerated by C2
B2 In-plane flow (U1, W1) generated by the linear shearing of A2 by the base flow
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

0 100 200 300 400 500 600 700 800
−2
−1
0
1
2
3
4
5
Time
A1,B1,C1
0 100 200 300 400 500 600 700 800
−1.5
−1
−0.5
0
0.5
1
1.5
Time
A2,C2
FIG. 10. Numerical results with the 7-ODE model. Left: temporal evolution of A1 (solid-red), B1 (dashed-blue), and C1
(dashed-dotted magenta). Right: temporal evolution of: A2 (solid-red) and C2 (dashed-blue). The temporal evolutions are
quasiperiodic.
to that obtained with the following reduced model, hence a more detailed analysis of the interactions
of these modes is given below.
F. Reduction of the 7-ODE model: Towards a periodic dynamic
The previous 7-ODE model can be further reduced by combining the two vortices C0 and C2 in
one pair of streamwise vortices. This is done by directly projecting a pair of streamwise vortices on
the Hermite functions. The lift-up effect couples the spatial distributions of the streaks and the wall-
normal velocity associated with the streamwise vortices (at least in the spanwise direction) according
to the linear lift-up term (−a2UbV1 in the U0-equation, Ref. 20). Hence, for a given streak U0 =
−∂z 0 with 0 = A2hσ2
0 (x)hσ2
1 (z), the spatial distribution of the velocity V1 which will regenerate
this streak is V1 ∝ U0 ∝ hσ2
0 (x)∂zhσ2
1 (z). Then, the potential field 1 associated with this V1 has to
be found. Since both velocities V1 = 2 1/β = 2hσ5
0 (x)∂zhσ5
1 (z)/β and V1 = hσ5
0 (x)∂zhσ5
1 (z) have
similar distributions consisting of three x-elongated patches with a z-alternating sign, we will model
the pair of streamwise vortices by 1 ∝ hσ5
0 (x)∂zhσ5
1 (z) ∝ hσ5
0 (x)(hσ5
2 (z) − hσ5
0 (z)/
√
2) and set σ5
= σ2. In contrast with the previous model, the streaks and the streamwise vortices are now better
correlated. The expansions for the three fields now read
0(x, z, t) = A1hσ1
1 (x)hσ1
1 (z) + A2hσ2
0 (x)hσ2
1 (z), (16)
1(x, z, t) = B1hσ3
0 (x)hσ3
1 (z) + B2hσ4
1 (x)hσ4
1 (z), (17)
1(x, z, t) = C1hσ6
1 (x)hσ6
0 (z) + C2hσ5
0 (x)(hσ5
2 (z) − hσ5
0 (z)/
√
2). (18)
The physical meaning of the modes is the same as for the previous model, and is summarized in
Table I. The only simplification is that now the amplitude of the pair of streamwise vortices, shown
in Fig. 11, is represented by only one mode C2. We keep the same values for the constants σi,
i = 1. . . 6 as before.
IV. THE 6-ODE MODEL
The general derivation is similar to the previous model. This time, the parity properties of the
functions involved in the expansions of 1 guarantee the orthogonality of the modes and hence the
separation of the different contributions. The six equations of the model are obtained by inserting
Eqs. (16)–(18) in (A1)–(A3) and are given by Eqs. (D8)–(D13). This 6-ODE model conserves the
kinematic energy Etot = 3A2
1 + 2A2
2 + 2
3
B2
1 + 3B2
2 + 4C2
1 + 21
2
C2
2 .
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.4
−0.2
0
0.2
0.4
0.6
FIG. 11. Streamwise vortices for the 6-ODE model obtained by reconstructing the velocity components with B1=B2=C1=0
and C2=1/2. The flow field (U1, W1) is represented by vectors and the velocity V1 in gray-level.
A. Simulation results of the 6-ODE model
As an initial condition, we take B1(0) = 3 and A2(0) = 0.01. This B1(0) is chosen such that the
amplitude of U1(x, z, 0) is of order 1. Temporal evolutions of the six amplitudes A1, A2, B1, B2, C1,
and C2 are reported in Fig. 12. A sample Matlab program is provided as the supplementary material
in Ref. 22. Their apparent temporal periodicity proves that the interactions between the amplitudes,
which sustain turbulence, repeat themselves exactly in time. In the following, such interactions are
discussed.
At t = 0, the only non-zero terms in the r.h.s of the A1-equation (D8) are the linear term a07B1
and the nonlinear term −a02 B2
1 . Each term gives an opposite contribution to the time increment of A1.
With B1(0) = 3, the contribution of a07B1 is dominant and a positive A1 is generated, corresponding
to a quadrupolar flow (U0, W0). Subsequently, the decrease of positive B1 coupled with the onset of
positive A1 results in a growth of negative C1 through the nonlinear term −a25A1B1 in Eq. (D12).
As shown in Fig. 12, C1 remains negative during the whole simulation while A1 and B1 remain
positive.
Two points in relation to the previous discussion in Sec. III D and Table I are in order. First,
during the simulation, a retrograde spanwise vortex exists (C1(t) ≤ 0 for t ≥ 0). The streamwise
velocity U1 associated with this vortex represents a correction of potential origin to the base flow,
0 100 200 300 400 500 600 700 800
−2
−1
0
1
2
3
4
Time
0 100 200 300 400 500 600 700 800
−1.5
−1
−0.5
0
0.5
1
Time
FIG. 12. Temporal evolution of the amplitudes for R = 120 with the 6-ODE model. Left: A1 in solid-red line, B1 in blue-
dashed line, and C1 in dashed-dotted magenta line. Right: A2 in solid-red line, B2 in blue-dashed line, and C2 in dashed-dotted
magenta line. The time periodicity of these time series highlights the regeneration cycle of the turbulent state. The three
amplitudes A1, B1, and C1 maintain the same sign, while the three others A2, B2, and C2 are alternating in time.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

whereas the correction of rotational origin is provided by the streamwise velocity U1 associated with
the positive B1. Second, during the simulation, the generated quadrupolar flow associated with A1
has a stagnation point with negative ∂xU0 at x=z=0.
Consider now the periodic temporal evolutions of the amplitudes A2, B2, and C2, reported in the
right panel of Fig. 12. The increase of C2, characteristic of the generation of streamwise vortices, is
driven solely by the nonlinear term −a29A2C1. This induced C2 has the same sign as A2 since −a29C1
≥ 0 and in turn, it enhances A2 through the lift-up term a14C2. The term −a29A2C1 accounts for the
action of the streak A2 on the spanwise vortex C1 (i.e., deformation) and illustrates the essence of the
physical mechanism of streamwise vortices formation. Some points are in order here to corroborate
the statements above.
First, by denoting J1 = a14 + a13C1 − a11B1, the amplification of A2 by C2 holds if the quantity
J1C2 has the same sign as the lift-up term a14C2 because the last two terms in J1 are negative (a13C1
− a11B1 ≤ 0) whereas a14 ≥ 0. Hence, the quantity J1C2 would represent the extraction of the energy
by the streamwise vortices from the corrected base flow if J1 ≥ 0. This is the case, since using the
values of the coefficients and the ranges of variation of B1 and C1 inferred from Fig. 12, we verify
that J1 is positive during the temporal evolution of the system.
Second, A2 would generate a same-signed C2 if the contribution of −a29A2C1 dominates the
one of −a32A2B1, which indeed gives an opposite-signed C2. From the range of variations of the
amplitude C1 and B1 from Fig. 12, it is clear that the contribution of −a32B1 cannot dominate that
of −a29C1. If this were the case, the streaks A2 would be damped by the opposite-signed C2 (an anti
lift-up effect), and the system would be pushed towards the laminar state.
In addition to C2, an amplitude B2 is induced by A2 through the linear term a24A2 in Eq. (D11).
This B2 would have the same sign as A2 if the contribution of the term a24A2 was greater than
the opposite-signed contributions of the terms −a22A2B1 and a23A2C1, that is, if the quantity J2 ≡
−a22B1 + a23C1 + a24 were positive, which is the case.
The periodic temporal evolution of the amplitudes, given in Fig. 12, also proves that the
associated coherent structures repeat themselves in both space and time and this is the essence of
the self-sustainment of the turbulent state, as we discuss in the following by considering what is
happening in the physical space.
The total velocity V1 and the flow (U1, W1) displayed in Fig. 13 represent a retrograde spanwise
vortex and its later evolution is given in Figs. 14 and 15. As a rule, only the velocity V1 is plotted in
order to monitor the wall-normal motion associated with the vortices, either spanwise or streamwise.
In each sub-figure, the total flow field (U0, W0) is shown with vectors.
The loss of the x-symmetry of the flow (U0, W0), which is quadrupolar when |A2| |A1|, signals
the onset of a streak U0 = −A2hσ2
0 (x)∂zhσ2
1 (z) with amplitude A2 as shown in Figs. 14(a)–14(d) and
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
FIG. 13. Reconstruction of the velocity components V1, U1, W1 using the values of the six amplitudes at t = 170. The flow
field (U1, W1) is depicted with vectors and V1 in gray levels. The later evolution of this spanwise vortex is given in Figs. 14
and 15.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(a)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−1
−0.5
0
0.5
(b)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
(c)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
(d)
FIG. 14. From spanwise to streamwise vortices. The flow (U0, W0) is depicted by arrows and V1 in gray levels. (a) t = 166,
(b) t = 172, (c) t = 176, (d) t = 180 (compare with Fig. 6). To be continued in Fig. 15.
15(e)–15(h). By acting on the spanwise vortex C1, this streak A2 generates the streamwise vortices
C2 (the term −a29A2C1 in the C2-equation).
When C2 attains a finite amplitude level (t ∼ 186), the superposition of the spanwise vortex C1
with the streamwise vortices C2 gives a crescent-shaped vortex displayed with the flow (U0, W0) in
Fig. 15(e) and with the in-plane flow (U1, W1) in Fig. 16 (left). The corresponding distribution of the
streamwise vorticity ωx = βW1 − ∂z V1, given in Fig. 16 (right), consists of two elongated regions
in the streamwise direction with opposite signs in the spanwise direction. These regions are two
streamwise vortices which regenerate the streaks and produce positive Reynolds stress −U0V1. As
can be inferred from Fig. 15(e), positive (negative) regions of U0 correspond to negative (positive)
regions of V1.
Then, the amplitude A2 decreases and the symmetric quadrupolar flow reappears (Figs. 15(f)
and 15(g)). In Fig. 17, the r.h.s of the A2-equation is plotted. The temporal growth of A2 ceases
when the r.h.s changes its sign. By splitting the r.h.s into two parts, the first is related to the lift-up
mechanism and is given by JA2C2
≡ C2(a14 − a11 B1 + a13C1) whereas the second is related to the
shearing of the in-plane flow B2 by the base flow and its correction of potential and rotational origin
JA2 B2
≡ B2(−a15 + a12C1 + a10 B1). It is clear that the sign of the r.h.s changes when the latter part
dominates the former (at t ∼ 190).
As a consequence, the two parts JA2C2
and JA2 B2
keep A2 bounded. Hence, follows a decline
of A2 to zero. It simply passes through it without remaining there and it ultimately changes sign
since the r.h.s of the equation of A2 is dominated by JA2 B2
which is positive since B2 ≤ 0 (t ∼ 200,
Fig. 17). Hence, A2 becomes positive, after which the process begins anew: this A2 generates a
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
(e)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
(f)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
(g)
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(h)
FIG. 15. As in Fig. 14. (e) t = 186, (f) t = 196, (g) t = 200, and (h) t = 216.
positive C2 from C1 by the nonlinear term −a29A2C1 and in turn, this C2 amplifies A2 by the lift-up
effect. In the physical space, this leads to the generation of another crescent vortex, as shown in
Fig. 15(h).
The amplification of A2 by C2 (i.e., the quantity JA2C2
) and its damping by B2 (i.e., the quantity
JA2 B2
) interact, making A2 oscillate between positive and negative values. The resultant spatial
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
X
Z
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
FIG. 16. Left: Spatial reconstruction of the velocity components corresponding to t = 186 in Fig. 15. The flow field (U1, W1)
is depicted with arrows and V1 in gray-level. The arrows of the flow (U1, W1) are going from the two positive regions of V1
to the negative region of V1 in the center and represent two counter-rotating streamwise vortices, associated with two patches
of negative and positive ωx (right).
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

150 160 170 180 190 200 210 220 230 240
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time
A2
150 160 170 180 190 200 210 220 230 240
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time
C2
FIG. 17. Left: r.h.s of A2-equation (D8). In thin-red line A2. The r.h.s in solid-blue line, JA2C2 in dashed magenta, and JA2 B2
in dashed-dotted black line. Right: r.h.s of C2-equation (D13). In thin-red line C2. The r.h.s in solid-blue line, −a29A2C1
term in dashed magenta and the remaining terms (in the r.h.s) in dashed-dotted black line.
patterns consist of two kinds of crescent vortices which alternate in time, with reappearance of the
retrograde spanwise vortex (at t = 166 and t = 200) between the two crescents. The existence of
both kinds of crescent vortices is well supported by the numerical simulations of the PDEs model.
Furthermore, note that the decrease of A2 is followed by that of C2. In fact, the streak A2
deforms the spanwise vortex C1 to give streamwise vortices C2, so that, when this streak vanishes,
the streamwise vortices vanish too and the spanwise vortex reappears. To further corroborate this,
consider Fig. 17 where the r.h.s of C2-equation (Eq. (D13)) is plotted together with the contribution
of the term −a29A2C1 and the remaining nonlinear terms in the r.h.s. of Eq. (D13). First, it is clear that
−a29A2C1 represents the major contribution to the r.h.s of C2-equation. Second, this term vanishes
when the amplitude A2 of the streaks vanishes (at t ≤ 200 left panel) and the r.h.s of C2 changes sign
(at t ≥ 200). Hence, the r.h.s of the C2-equation is dominated by the term −a29A2C1 and its sign is
driven by the sign of A2 (since −a29C1 ≥ 0).
B. Magnitude of the velocity components and the choice of the σi
It is interesting to note that, during the temporal evolution, the amplitude of the velocity
components is of order 1, which is the case in the numerical simulations of the PDEs model. From
this observation, some thresholds for the amplitudes (A1, B1...) can be derived. For example, when A1
vanishes (Fig. 15(e)), the maximum amplitude of the streak U0 = −A2hσ2
0 (x)∂zhσ2
1 (z) does not exceed
1 if A2 ≤
√
2π
2
∼ 1.25. The maximum value of the streak associated with the quadrupolar flow does
not exceed 1 if A1 ≤
√
πe
2
∼ 1.48. In the same way, the maximum value of the correction to the base
flow of rotational origin (U1 = −B1hσ3
0 (x)∂zhσ3
1 (z)) does not exceed 1 if B1 ≤ 3π
2
∼ 3.75 whereas
the maximum value of the correction to the base flow of potential origin (U1 = C1∂x hσ6
1 (x)hσ6
0 (z))
does not exceed 1 if |C1| ≤
√
2π
2
∼ 1.25. Figure 12 shows that these thresholds are satisfied.
It is clear that these thresholds as well as the coefficients aij are related to the choice of the σi.
As we have seen, they are mainly taken to match the spatial extent of the coherent structures. For
instance, with σ2 = 1, the spanwise width of the streak is 2h, calculated as the distance between its
two zeros located at (x=0,z=±1).
The interactions between the coherent structures relate their spatial distributions and hence the
values of σi. Streamwise vortices and streaks are related by the lift-up effect as discussed earlier and
for this reason we set σ2 = σ5. Other choices for σi are imposed by some dynamical considerations.
For example, to be deformed by a streak, a spanwise vortex has to be longer in the z-direction than
the streak (see, for example, Fig. 14). This is satisfied if σ3 < σ1 = σ2. In addition to the values taken
in the present study, σ3 = 1/
√
3 and σ1 = σ2 = 1, other values can be considered, corresponding,
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

for example, to a thinner streak with σ1 =
√
2, σ2 = 1, and σ3 = 1/
√
3. The dynamic obtained with
these values is similar to the one discussed in Sec. IV A.
V. DISCUSSION AND CONCLUSION
The physical interpretation of some terms remains, in general, a difficult task, even with ODEs
models. As we have stated in the Introduction, the principal aims in using ODEs models are to
clarify the interactions between the coherent structures and to elucidate some physical mechanisms.
However, this is not as easily guaranteed as it first appears. Consider, for example, the nonlinear term
−a29A2C1 in C2-equation (Eq. (D13)) which drives the generation of streamwise vortices (C2) from
the spanwise vortex (C1) through the action of the streaks (A2). This term comes from the Galerkin
projection of the advective term U0∂x V1 + W0∂z V1 in the V1-equation (Ref. 20) but it does not tell
the whole story about the physical mechanism at work; that is the deformation of a spanwise vortex
by the flow (U0, W0) into a crescent vortex. It is clear that without the picture of this mechanism
in mind, it is difficult to give a physical interpretation to the term −a29A2C1. Therefore, despite
the ability of the ODEs model to provide a simplified picture of the dynamical interactions that
sustain turbulence, elucidating the mechanisms behind some terms necessitates the study of the
PDEs model. Hence, both modeling approaches (PDEs and ODEs) are complementary.
To the best of the author’s knowledge, the 6-ODE model derived in this paper is the first ODEs
model in the literature: (i) that demonstrates the periodic generation of the streamwise vortices, (ii)
that is derived from the 3D Navier-Stokes equations through (two-step) Galerkin projection, and
(iii) where all its six degrees of freedom have simple physical meaning.
Finally, future studies investigating the generation mechanism of spanwise vortices will further
refine our understanding of the wall turbulence cycle.
ACKNOWLEDGMENTS
The author would like to thank the referees for their invaluable insight, thorough and constructive
comments and suggestions, which ultimately improved the quality of the paper.
The author also would like to thank Professor Paul Manneville (LadHyX, Ecole Polytechnique,
Paris, France) who introduced him to turbulence modeling, and whose passion for the underlying
coherent structures and mechanisms had a lasting effect.
APPENDIX A: EQUATIONS OF THE PDEs MODEL
The three equations of the PDEs model are
(∂t − R−1
( 2 − γ0)) 2 0 = (∂z NU0
− ∂x NW0
) + a1(3
2
∂z 2 1 − ∂x 2 1) , (A1)
(∂t − R−1
( 2 − γ1)) 2 1 = (∂z NU1
− ∂x NW1
) − a1∂x 2 0 , (A2)
(∂t − R−1
( 2 − β2
))( 2 − β2
) 2 1 = β2
(∂x NU1
+ ∂z NW1
)
+45
2
R−1
2 1 − β 2 NV1
, (A3)
where R is the Reynolds number, 2 = ∂xx + ∂zz, β =
√
3, and the nonlinear terms NU0
, NW0
, NU1
,
NW1
, and NV1
can be found in Ref. 21. Note that Eq. (A3) can be written in the form: ∂t( 2 − β2
) 2 1
= F, where F = R−1
( 2 − β2
)( 2 − β2
) 2 1 + β2
(∂x NU1
+ ∂z NW1
) + 45
2
R−1
2 1 − β 2 NV1
.
The averaged streamwise vorticity over the gap is defined as
ωx ≡
1
−1
x S1(y) dy = βW1 − ∂z V1, (A4)
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

using the perturbation velocity components (1)–(3) with x = ∂yw − ∂zv and where β=
√
3 ac-
counts for a wall-normal gradient. Similarly, the averaged spanwise vorticity is defined as
ωz ≡
1
−1
z S1(y) dy = ∂x V1 − βU1, (A5)
with z = ∂x v − ∂yu .
APPENDIX B: SYMMETRIES OF THE PDEs MODEL
The equations of the PDEs model are invariant under the following transformations:
T Z : (x, z) → (x, −z) and : (B1)
0 → − 0,
1 → − 1 , 1 → 1,
U0 → U0 , W0 → −W0,
U1 → U1 , W1 → −W1, V1 → V1,
T X : (x, z) → (−x, z) and : (B2)
0 → − 0,
1 → 1 , 1 → − 1,
U0 → −U0 , W0 → W0,
U1 → U1 , W1 → −W1, V1 → −V1,
T R = T X ◦ T Z : (x, z) → (−x, −z) and : (B3)
0 → 0,
1 → − 1 , 1 → − 1,
U0 → −U0 , W0 → −W0,
U1 → U1 , W1 → W1, V1 → −V1.
APPENDIX C: HERMITE FUNCTIONS
We introduce a class of functions which is well adapted to the application of the Galerkin
method as well as to the representation of localized vorticity fields, as used by Kloosterziel24
and
followed by Satijn et al.25
We define the one-dimensional function hn of variable x and for some integer n by
hn(x) =
1
2nn!
√
π
Hn(x)e−x2
/2
,
where Hn are the Hermite polynomials defined as Hn(x) = (−1)n
ex2 dn
dxn e−x2
.26
The orthogonality
relations for Hn are given by
∞
−∞
Hn(x)Hm(x)e−x2
dx = 0 for n = m.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

They form a complete set in L2
(R) with the weight function ex2
.27
The first few Hermite polynomials
are
H0(x) = 1, H1(x) = 2x, H2(x) = 4x2
− 2.
The hn are special cases of the confluent hyper-geometric functions and are called Hermite
functions.27
A quadratic integrable function g may be expanded as
g(x) =
∞
n=0
cnhn(x), cn =
∞
−∞
g(x)hn(x) dx
since
∞
−∞
hn(x)hm(x) dx = 0 for n = m.
The set of hn with the inner scalar product:
∞
−∞ hm(x)hn(x) dx = δmn, (n, m = 0, 1, 2 . . .) is com-
plete in L2
(R).27
If the function g is an even function of x, an expansion in the subset with n = 0, 2,
4, . . . is found. If it is odd on x, the subset with n = 1, 3, 5, . . . is found. Furthermore, by applying
the stretching transformation x → σx, a more general set of hσ
n is obtained, and is defined by
hσ
n (x) =
σ
√
πn!2n
Hn(σ x)e−σ2
x2
/2
,
for some constant σ. The hσ
n , n = 0, 1, . . . satisfy the orthogonality condition
∞
−∞
hσ
m(x)hσ
n (x) dx = δmn.
The previous results can be straightforwardly generalized to higher dimensions. For two-dimensional
problems, if a function f is in L2
(R2), a double expansion is possible
f (x, z) =
n≥0,m≥0
anmhσ
n (x)hσ
m(z),
for a given σ and where the expansion coefficients are calculated according to
anm =
∞
−∞
∞
−∞
f (x, z)hσ
n (x)hσ
m(z) dxdz.
For convenience, we note by , the scalar product
f, g =
∞
−∞
∞
−∞
f (x, z)g(x, z) dxdz. (C1)
APPENDIX D: EQUATIONS OF THE ODEs MODELS
The equations of the 7-ODE model read
(
d
dt
+
6.5
R
)A1 = a07 B1 − a08C1 − a01 A2
2 − a02 B2
1 + a060 B2C0
+a05 B1C1 − a03C2
1 + a06 B2C2 − a040C0C2 − a04C2
2 , (D1)
(
d
dt
+
5.5
R
)A2 = a14C2 − a140C0 − a15 B2 + a09 A1 A2 + a10 B1 B2
+a110 B1C0 + a12 B2C1 − a11 B1C2 + a13C1C2, (D2)
(
d
dt
+
11.5
R
)B1 = −a20 A1 + a16 A1 B1 + a18 A2 B2
−a190 A2C0 − a17 A1C1 + a19 A2C2, (D3)
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

TABLE II. Values of the coefficients aij for the 7-ODE model.
a01 a02 a03 a04 a05 a06 a07 a08 a09
0.092490 0.005179 0.107906 0.0431624 0.181842 0.253206 0.050111 0.400891 0.138736
a11 a12 a13 a14 a15 a16 a17 a18 a19
0.046627 0.305734 0.343355 0.5952 0.400891 0.023307 0.46882 0.076446 0.492019
a21 a22 a23 a24 a25 a26 a27 a28 a29
0.018086 0.063098 0.083926 0.26726 0.05824 0.0809295 0.215812 0.006011 0.124694
a31 a32 a110 a140 a35 a34 a33 a40 a040
0.142786 0.041185 0.160174 0.400891 0.0221967 0.22366 0.0611572 0.593969 0.152602
a060 a300 a41 a280 a190 a210 a10 a20 a30
0.095916 0.086807 0.173675 0.093191 0.327628 0.0959165 0.0691659 0.2255016 0.022717
(
d
dt
+
14.5
R
)B2 = a24 A2 − a22 A2 B1 − a210 A1C0 + a23 A2C1 + a21 A1C2, (D4)
(
d
dt
+
9.9
R
)C0 = a40C2/R − a33 A2 B1 + a34 A2C1 − a35 A1C2, (D5)
(
d
dt
+
10.25
R
)C1 = −a25 A1 B1 − a27 A2 B2 − a280 A2C0 + a26 A1C1 + a28 A2C2, (D6)
(
d
dt
+
10.9347
R
)C2 = a41C0/R − a32 A2 B1 − a31 A1 B2
+a300 A1C0 − a29 A2C1 + a30 A1C2, (D7)
where the values of the coefficients are given in Table II.
The equations of the 6-ODE model read
(
d
dt
+
6.5
R
)A1 = −a01 A2
2 − a02 B2
1 − a03C2
1 + a04C2
2 + a05 B1C1
+a06 B2C2 + a07 B1 − a08C1, (D8)
(
d
dt
+
5.5
R
)A2 = a09 A1 A2 + a10 B1 B2 − a11 B1C2 + a12 B2C1
+a13C1C2 − a15 B2 + a14C2, (D9)
(
d
dt
+
11.5
R
)B1 = a16 A1 B1 − a17 A1C1 + a18 A2 B2 + a19 A2C2 − a20 A1, (D10)
(
d
dt
+
14.5
R
)B2 = a21 A1C2 − a22 A2 B1 + a23 A2C1 + a24 A2, (D11)
(
d
dt
+
10.25
R
)C1 = −a25 A1 B1 + a26 A1C1 − a27 A2 B2 + a28 A2C2, (D12)
(
d
dt
+
10.7856
R
)C2 = −a29 A2C1 − a30 A1C2 − a31 A1 B2 − a32 A2 B1. (D13)
The coefficients aij, i, j = 0. . . are given in Table III.
199.168.149.85 On: Thu, 11 Sep 2014 14:05:29

TABLE III. Values of the coefficients aij for the 6-ODE model.
a01 a02 a03 a04 a05 a06 a07 a08 a09 a10 a31
0.0925 0.0052 0.1079 0.2158 0.1818 0.0904 0.0501 0.4009 0.1387 0.0691 0.0775
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a32
0.3184 0.3057 0.3433 1.2756 0.4009 0.0232 0.4687 0.0765 1.0480 0.2254 0.0059
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 . . .
0.1809 0.063 0.0839 0.2673 0.0582 0.0809 0.2158 0.1642 0.1279 0.0617 ...
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