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Relationship between wall shear stresses and streamwise vortices
Article in Applied Mathematics and Mechanics · January 2019
DOI: 10.1007/s10483-019-2448-8
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Appl. Math. Mech. -Engl. Ed., 40(3), 381–396 (2019)
Applied Mathematics and Mechanics (English Edition)
https://doi.org/10.1007/s10483-019-2448-8
Relationship between wall shear stresses and streamwise vortices
in turbulent flows over wavy boundaries∗
Lihao WANG1 , Weixi HUANG1,† , Chunxiao XU1 ,
Lian SHEN2 , Zhaoshun ZHANG1
1. Applied Mechanics Laboratory, Department of Engineering Mechanics,
Tsinghua University, Beijing 100084, China;
2. Department of Mechanical Engineering and Saint Anthony Falls Laboratory,
University of Minnesota, Minneapolis, MN 55455, U. S. A.
(Received Sept. 4, 2018 / Revised Nov. 25, 2018)
Abstract The relationship between wall shear stresses and near-wall streamwise vortices
is investigated via a direct numerical simulation (DNS) of turbulent flows over a wavy
boundary with traveling-wave motion. The results indicate that the wall shear stresses
are still closely related to the near-wall streamwise vortices in the presence of a wave. The
wave age and wave phase significantly affect the distribution of a two-point correlation
coefficient between the wall shear stresses and streamwise vorticity. For the slow wave case
of c/Um = 0.14, the correlation is attenuated above the leeward side while the distribution
of correlation function is more elongated and also exhibits a larger vertical extent above
the crest. With respect to the fast wave case of c/Um =1.4, the distribution of the correlation
function is recovered in a manner similar to that in the flat-wall case. In this case, the
maximum correlation coefficient exhibits only slight differences at different wave phases
while the vertical distribution of the correlation function depends on the wave phase.
Key words direct numerical simulation (DNS), wall shear stress, near-wall streamwise
vortex, two-point correlation
Chinese Library Classification O357.5
2010 Mathematics Subject Classification
1
76F40
Introduction
Near-wall flow structures play a significant role in the dynamical process of transitional
and turbulent boundary layers[1–3] . Streamwise vortices are associated with sweep and ejection motions[4] , which is adequately understood through the quadrant analysis[5]. Additionally, streamwise vortices are important elements in the self-sustaining process of near-wall
∗ Citation: WANG, L. H., HUANG, W. X., XU, C. X., SHEN, L., and ZHANG, Z. S. Relationship
between wall shear stresses and streamwise vortices in turbulent flows over wavy boundaries. Applied
Mathematics and Mechanics (English Edition), 40(3), 381–396 (2019) https://doi.org/10.1007/s10483019-2448-8
† Corresponding author, E-mail: hwx@tsinghua.edu.cn
Project supported by the the National Natural Science Foundation of China (Nos. 91752205 and
11772172) and the “13th Five-Year Plan” Equipment Development Common Technology Pre-research
(No. 41407020501)
c Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019
382
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
turbulence[6] , and are also involved in the formation of low-speed streaks. Furthermore, the
spatial and temporal characteristics of turbulent structures are closely related to turbulence
statistics[7] . Choi et al.[8] performed a direct numerical simulation (DNS) of turbulent flows
over riblets and indicated that the high wall skin-friction region is closely related to the near-wall
streamwise vortices and proposed a drag reduction mechanism by riblets with small spacings
to restrict the location of streamwise vortices. The aforementioned ability of surface-mounted
riblets to reduce drag was originally presented in experimental studies that focused on the
turbulent boundary layer[9] and also turbulent pipe flows[10–11] . The correlation between the
streamwise vorticity and wall shear stress was first quantitatively examined by Kravchenko et
al.[12] . Additionally, Kim et al.[13] explored the relationship between the wall pressure fluctuation and streamwise vorticity. Ge et al.[14] used measurable information at a wall (i.e.,
streamwise and spanwise wall shear stresses and wall pressure fluctuations) to detect near-wall
streamwise vortices. They introduced a stochastic interference and further observed that the
relation based on the streamwise wall shear stress breaks down while the correlations based on
the spanwise wall shear stress and wall pressure fluctuation are still robust[15] .
When compared with the flat-wall turbulent boundary layer, the wavy boundary can significantly affect the turbulent flow above, and this is typically encountered in wind over water
waves[16] and also the fish-like locomotion[17–18] . De Marchis et al.[19] indicated that a stationary
wavy rough surface locally modifies the coherent pattern of the flow. Sullivan et al.[20] observed
that the near-wall low-speed streaks are disrupted by the wave motion for turbulent flows over
a traveling-wave boundary. Shen et al.[21] observed that the traveling-wave boundary can reduce drag for fast-moving waves. Yang and Shen[22] identified typical vortical structures that
are significantly dependent on the wave phase speed. With respect to a slow wave, the dominant quasi-streamwise vortices are concentrated above the windward surface while the reversed
horseshoe vortices are mainly located above the wave trough. With respect to an intermediate
wave and a fast wave, the vortical structure is characterized by bent quasi-streamwise vortices.
The aim of the present study involves exploring a quantitative relationship between wall
shear stresses and near-wall streamwise vortices in turbulent flows over a wavy boundary. Given
that a wavy boundary significantly affects near-wall vortical structures and that wall shear
stresses significantly depend on the wave motion, the relationship between wall shear stresses
and near-wall streamwise vortices is systematically investigated via a phase-resolved two-point
correlation.
2
Numerical method
The sketch of turbulent flows over a wavy boundary is shown in Fig. 1. The flow motion
is defined in the Cartesian coordinates (x, y, z ), where x, y, and z denote the streamwise,
vertical, and spanwise directions, respectively. The initial lower boundary is prescribed by a
monochromatic wave, and the corresponding wave parameters include the wavelength λ, the
phase speed c, and the wave amplitude a, as shown in Fig. 1. The wave steepness is defined as
ak, where k = 2π/λ denotes the wavenumber.
-Y
Z
0 [ Y
δ
D
B
Fig. 1
-[
6F
λ
Schematic of three-dimensional turbulent flows over a wavy boundary, where Ue is the external
flow velocity
Relationship between wall shear stresses and streamwise vortices
383
The governing equations for turbulent flows include the dimensionless incompressible NavierStokes and continuity equations that are expressed as follows:
1 2
∂u
+ u · ∇u = −∇p +
∇ u,
∂t
Re
∇ · u = 0,
(1)
(2)
2
where u = (u, v, w) denotes the velocity vector, p denotes the pressure normalized by ρUm
,
Re = Um δ/ν denotes the bulk Reynolds number, Um denotes the bulk velocity, ν denotes the
kinematic viscosity, and δ denotes the height of the open channel.
The pseudo-spectral method based on a boundary-fitted mesh[23] is used to simulate the
turbulent open-channel flow, and the high-order spectral (HOS) method is used to describe the
boundary wave motion[24–25] . The lower boundary conditions correspond to no-slip conditions
provided by the wave motion, and the free-slip condition is applied at the upper boundary. The
periodic condition is adopted in the streamwise and spanwise directions. The computational
flow domain exhibits a size of (Lx , Ly , Lz ) = (2πδ, δ, πδ) with a grid number corresponding to
(Nx , Ny , Nz ) = (192, 143, 192). Uniform grid sizes are used in the streamwise and spanwise
directions with ∆x+ ≈ 9.8 and ∆z + ≈ 4.9, respectively, and the vertical grid stretches from
+
∆ymin
≈ 0.02 in the vicinity of the wavy boundary to ∆y + ≈ 3.27 at the top of the open
channel. With respect to all the simulations, the flow is driven by a mean pressure gradient to
ensure a constant bulk flow rate. The constant simulation parameters include the bulk Reynolds
number Re = 5 000 and wave steepness ak = 0.1 while the friction Reynolds number Reτ varies
with the wave age c/Um , as shown in Table 1. All the simulations are performed for a time
length of approximately 800δ/Um, and all the statistics are obtained from 150 instantaneous
flow field data.
Table 1
Simulation parameters including the wave steepness ak, the dimensionless wave amplitude
normalized by the wall viscous unit a+ , the wave age c/Um , and the Reynolds number Reτ
based on the friction velocity uτ
Case
ak
a+
c/Um
Reτ
Flat wall
Stationary wavy surface
Slow-moving wave
Medium-moving wave
Critical-moving wave
Fast-moving wave
0
0.1
0.1
0.1
0.1
0.1
0
8.2
8.6
8.8
7.5
7.2
0
0
0.14
0.5
1.0
1.4
298
329
343
353
299
287
3
Numerical results and discussion
Figure 2 shows the mean velocity profiles of cases with different wave ages. For the flat-wall
case, the velocity profile agrees well with the linear law in the viscous sub-layer and the logarithmic law in the log region. For the case of c/Um = 0, the stationary wavy surface is treated
as a type of rough wall that decelerates the flow in the near-wall region. The mean velocity
profile still satisfies the logarithmic law with the exception of a downshift when compared with
that of the flat-wall case. The increases in the wave age from zero cause the mean velocity
profile to move down further and then move up. With respect to the case of c/Um = 1.4, the
mean velocity profile even exceeds that of the flat-wall case. For the fast-moving wave case,
dp
the total drag on the wall (calculated by the mean pressure gradient dx
and not shown here)
is lower than that of the flat-wall case. The variation in the total drag with the wave age is
also reflected in the wave amplitude a+ based on the wall viscous unit. The physical wave
amplitude is maintained as a constant in different wave-age cases, and thus the difference in a+
is attributed to the differences in the friction Reynolds numbers as listed in Table 1.
384
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
The wavy boundary mainly affects the total drag by the form drag, and thus variation in
the form drag Fd with respect to the wave age is shown in Fig. 3. When c/uτ increases from
zero, the form drag initially increases and then decreases and changes its sign from positive to
negative, thereby corresponding to the change in the mean velocity profile, as shown in Fig. 2.
It should be noted that the present DNS results of turbulent flows over a wavy boundary are
consistent with the previous studies with the exception of a higher form drag in the low waveage regime, and this is due to the differences in the lower boundary conditions. In previous
studies[20,26–27] , orbital velocities are prescribed in the streamwise and vertical directions, and
the waveform remains constant. Conversely, in the present study, velocities are given based on
the surface velocity potential in the streamwise, spanwise, and vertical directions. Furthermore,
the surface elevation slightly changes during the dynamical evolution of surface wave due to the
nonlinear effect induced by the wave steepness.
6 Z
6 MOZ B B D6N
B D6N
'E
6
Fig. 2
$VSSFOU%/4
4VMMJWBOFUBM<>
,JIBSBFUBM<>
:BOH4IFO<>
B D6N
B D6N
B D6N
Z
Comparison of mean velocity profiles
for different wave ages (color online)
Fig. 3
DV
Variations in the surface form drag
with c/uτ (color online)
In addition to the form drag associated with the normal stresses, the wall shear stresses are
the main focus of the present study. The near-wall vortical structures play a key role in turbulent
transport and are closely related to high skin friction. Additionally, the wall measurable signals
are used to detect near-wall vortical structures in the turbulent boundary layer[12–15] .
In the present study, both the tangential and spanwise wall shear stresses denoted as τt and
τz , respectively, are investigated and defined as follows:

s
∂η 2

′
′
′ .

∂η
∂u
∂v
∂u


τ
=
−
2µ
+
µ
+
µ
1
+
,

x

∂x ∂x
∂x
∂y
∂x



s



∂η 2

∂η ∂u′
∂η ∂v ′
∂v ′ .



−
µ
+
2µ
1
+
,
τ
=
−
µ
 y
∂x ∂y
∂x ∂x
∂y
∂x
(3)
s

∂η 2

′
′
′
′ .

∂η
∂u
∂η
∂w
∂v
∂w


τz = − µ
−µ
+µ
+µ
1+
,


∂x ∂z
∂x ∂x
∂z
∂y
∂x



s



∂η 2

∂η .


 τt = τx +
τy
1+
,
∂x
∂x
where τx and τy denote the streamwise and vertical components of the wall viscous stress,
respectively, η denotes the surface wave elevation, and u′ , v ′ , and w′ represent the velocity
fluctuations in the three directions. The vortical structures are visualized via the iso-surface of
the local swirling strength λci , which is defined as the imaginary part of the complex eigenvalue
Relationship between wall shear stresses and streamwise vortices
385
of ∇u · x = λx[28] that corresponds to the vorticity related to the rotation and is used to
identify a vortex. Figure 4 displays the vortical structures near the wavy boundary and the
contours of wall shear stresses. As shown in Fig. 4(a), the tangential wall shear stress is still
related to the downstream streamwise vortices in the presence of the surface wave. Clockwise
(counterclockwise) rotating streamwise vortices (Position A) are observed at the upper left
(right) of the positive tangential wall shear stress (Position B) and vice versa. However, the
spanwise wall shear stress in Fig. 4(c) is related to the vortical structures immediately above
it. Clockwise (counterclockwise) rotating streamwise vortices are observed above the regions of
the negative (positive) spanwise wall shear stress.
Z
U
"
#
[
D
Fig. 4
Z
0
U
[
0
B
Z
Y
C
Y
[
[
Y
0
[
Z
0
Y
[
E
Snapshots of near-wall vortical structures and wall shear stresses for c/Um = 0.14 (left) and
c/Um = 1.4 (right), (a) and (b) the tangential wall shear stress, (c) and (d) the spanwise
wall shear stress, where the vortical structures are identified by the iso-surface of 10% of the
maximum λci and colored with respect to the streamwise vorticity with red (blue) for positive
(negative) values. In Fig. 4(a), Position A denotes a clockwise rotating streamwise vortex, and
Position B denotes a positive region of the tangential wall shear stress (color online)
Increases in the wave age to c/Um = 1.4 lead to the appearance of long vortex sheets in the
very-near region of the wave crests and troughs, and they differ from the isolated vortical structures above the wavy boundary and spread in the spanwise direction (see Figs. 4(b) and 4(d)).
These types of vortical structures can result from the periodic wave motion as mentioned by
Yang and Shen[27] . Simultaneously, we observe that the number of isolated vortices in Fig. 4(b)
is significantly reduced when compared with that of the slow wave case in Fig. 4(a). This is
because the fast-moving wave leads to a smaller friction velocity and suppresses turbulence.
Nevertheless, the aforementioned relationship between near-wall streamwise vortices and wall
shear stresses for the slow wave case is similar to that for the case of c/Um = 1.4.
Yang and Shen[22] observed that different types of vortical structures gather at different
positions on the wavy boundary. In the case of slow wave, the quasi-streamwise vortices are
dominant above the windward side while the reversed horseshoe vortices are concentrated above
the trough. Therefore, the relationship between near-wall streamwise vortices and wall shear
stresses varies at different wave phases. In order to explore the spatial relationship between
near-wall vortices and wall shear stresses at different wave phases, the phase-resolved two-point
correlation coefficient between the streamwise vorticity and wall shear stresses is computed,
which is defined as follows:
386
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
Rτ ωx (x+ , y + , ∆x+ , ∆z + ) =
hτ (x+ , z + )ωx (x+ + ∆x+ , y + , z + + ∆z + )i
,
τRMS (x+ )ωx,RMS (x+ + ∆x+ , y + )
(4)
where ∆x+ and ∆z + denote the spatial separations in the streamwise and spanwise directions,
respectively, and τ and ωx denote the wall shear stress and streamwise vorticity, respectively.
The bracket in the numerator and the root mean square (RMS) values in the denominator correspond to an average over the spanwise direction and time, and further at the same wave phase
over all the waves in the streamwise direction. Thus, the two-point correlation coefficient is a
function of the positions (x+ , y + ) and also the spacings (∆x+ , ∆z + ). Both the tangential and
spanwise wall shear stresses are examined in the present study, and the correlation coefficients
are denoted as Rτt ωx and Rτz ωx , respectively.
First, Rτt ωx and Rτz ωx are averaged in the streamwise direction and denoted as Rτt ωx and
Rτz ωx to provide an overview of the structures of two-point correlation and compare with the
flat-wall case. Figure 5 shows the iso-surfaces of Rτt ωx = ±0.15 for both the flat and wavy
boundaries. As shown in Fig. 5(a), for the flat-wall case, two pairs of streamwise elongated
structures appear in the downstream of the location where the tangential wall shear stress is
detected. It should be noted that the pair of structures at the wall is formed by the near-wall
streamwise vortices under the no-slip condition. It is the overlying pair of structures that reflect the spatial relationship between the tangential wall shear stress and near-wall streamwise
vortices. The tilted structure reaches a maximum correlation coefficient of 0.28 at the location (∆x+ , y + , ∆z + ) = (90, 15, ±20), which agrees with the results obtained by Kravchenko
et al.[12] and Ge et al.[15] . If the tangential wall shear stress τt > 0 at the detecting point,
then the positive (negative) correlation region at the upper left (right) of it corresponds to
clockwise (counterclockwise) rotating streamwise vortices. For this case, the positive tangential
shear stress region is related to the sweep motion induced by the counter-rotating vortex pair.
Conversely, the negative tangential wall shear stress region corresponds to the ejection motion.
The results of the flat-wall case (see Fig. 5(a)) are consistent with the quadrant analysis and
conditional averaged turbulent structures in previous studies[4–5] .
Z
∆[
Z
Fig. 5
∆[ ∆Y
∆[ B
Z
D
∆Y
Z
∆[ ∆Y
C
∆Y
E
Iso-surfaces of the two-point correlation coefficient between the streamwise vorticity and the
tangential wall shear stress for Rτt ωx = 0.15 (red) and Rτt ωx = −0.15 (blue), (a) flat wall,
(b) c/Um = 0, (c) c/Um = 0.14, and (d) c/Um = 1.4 (color online)
With respect to the case of stationary wavy wall, the correlation Rτt ωx in Fig. 5(b) exhibits
a pattern similar to that of the flat-wall case. Additionally, our focus is on the upper pair of
Relationship between wall shear stresses and streamwise vortices
387
structures, and this is directly related to the near-wall streamwise vortices. When compared
with the flat-wall case, this pair of structures exhibits a smaller streamwise length scale and a
larger tilting angle, thereby indicating the confinement effect of the wavy boundary geometry
on near-wall streamwise vortices. For the case of c/Um = 0.14, the maximum correlation
coefficient at y + = 15 is 0.23 and is smaller than 0.28 for the flat-wall case. The corresponding
streamwise length scale of the upper structure is further reduced (see Fig. 5(c)). Increases in the
wave age to c/Um = 1.4 cause the iso-surface of the correlation Rτt ωx in the three-dimensional
view to resemble that of the flat-wall turbulence (see Fig. 5(d)). Furthermore, the maximum
correlation coefficient above the wall increases to 0.25. The correlation Rτt ωx in Figs. 5(c) and
5(d) is consistent with the instantaneous fields shown in Figs. 4(a) and 4(b).
Figure 6 shows the iso-surfaces of Rτz ωx = ±0.15, which corresponds to a single streamwise
elongated structure with spanwise symmetry. In a manner similar to the iso-surface of Rτt ωx , the
positive Rτz ωx region at the wall is induced via the no-slip boundary condition. Furthermore,
based on the Biot-Savart law, the streamwise vortices induce counter-rotating vortex pairs that
correspond to the two symmetrical negative correlation regions on both sides of the positive
region at the wall. As shown in Fig. 6(a), for the flat-wall case, the negative correlation region
above the detecting point is related to the near-wall streamwise vortices. If the spanwise
wall shear stress τz > 0 at the detecting point, then the negative correlation region implies the
existence of counterclockwise rotating streamwise vortices and vice versa. The correlation Rτz ωx
reaches its maximum value of 0.47 at approximately y + = 15, and this significantly exceeds
that of Rτt ωx , which is consistent with the results obtained by Ge et al.[14] . This implies that
near-wall streamwise vortices exhibit stronger correlation with the spanwise wall shear stress
than the streamwise component. Specifically, the spanwise wall shear stress is determined as a
preferential choice for drag reduction control of turbulent flows in previous studies[15,29–30] .
Z
∆[
Z
∆[
Fig. 6
∆Y
B
∆[
Z
D
∆Y
Z
∆[
∆Y
C
∆Y
E
Iso-surfaces of the two-point correlation coefficient between the streamwise vorticity and the
spanwise wall shear stress for Rτz ωx = 0.15 (red) and Rτz ωx = −0.15 (blue), (a) flat wall, (b)
c/Um = 0, (c) c/Um = 0.14, and (d) c/Um = 1.4 (color online)
As shown in Fig. 6(b), for the case of the stationary wavy wall, the correlation Rτz ωx also
exhibits a similar pattern to that for the flat-wall case. The magnitude of the upper negative
correlation increases from 0.47 to 0.51 while the negative correlation on both sides of the positive
region at the wall is weakened. With respect to the case of c/Um = 0.14, the correlation Rτz ωx
reaches its maximum value of 0.54 at approximately y + = 15 (see Fig. 6(c)) in accordance with
388
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
the instantaneous field as shown in Fig. 4(c). Additionally, as shown in Fig. 6(d), the iso-surface
of Rτz ωx for the case of c/Um = 1.4 is similar to that for the flat-wall case, and the maximum
correlation coefficient is reduced to 0.42 at y + = 15. Physically, the results indicate the variation
in near-wall streamwise vortices due to the presence of the wavy boundary. As shown in Ref. [22],
unique vortical structures were observed and found to be significantly dependent on the wave
motion.
After examining the overall effect of the wave age on the relationship between wall shear
stresses and near-wall streamwise vortices, its variation with respect to the wave phase is further
examined in detail. We select four representative phase points as the detecting points that
denote the windward surface, crest, leeward surface, and trough. Figure 7 shows the iso-surfaces
of Rτt ωx = ±0.15 at different phase positions. It is observed that the streamwise extent of the
correlation structures exhibits a significant variation with the wave phase. Specifically, for
the case of c/Um = 0.14, the iso-surface of Rτt ωx = ±0.15 does not appear in the form of
Z
0
Fig. 7
Y
[
B
C
D
E
F
G
H
I
Iso-surfaces of the two-point correlation coefficient between the streamwise vorticity and the
tangential wall shear stress for Rτt ωx = 0.15 (red) and Rτt ωx = −0.15 (blue) with respect
to the cases of c/Um = 0.14 (left) and c/Um = 1.4 (right) at (a) and (b) windward, (c) and
(d) crest, (e) and (f) leeward, (g) and (h) trough, where the wall is colored by the surface
elevation, and the blue (white) color denotes the wave crest (trough) (color online)
Relationship between wall shear stresses and streamwise vortices
389
two pairs of streamwise elongated structures, as shown in Fig. 7(e). Nevertheless, the upper
pair of structures that are directly related to the near-wall streamwise vortices remains similar.
With respect to the case of c/Um = 0.14, the correlation Rτt ωx reaches a maximum value at
different streamwise locations downstream from the detecting point at different wave phases.
The maximum correlation coefficient is located at ∆x+ = 30 when the detecting point is on
the windward side (see Fig. 7(a)) while the maximum correlation is located at ∆x+ = 56 when
the detecting point is on the wave crest (see Fig. 7(c)). Conversely, with respect to the case of
c/Um = 1.4, there is no significant difference in the upper pair of structures at different wave
phases as shown in the right column of Fig. 7. As shown in Fig. 7(d), it is noted that the lower
pair of structures extends to the upstream of the detecting point and is shown more clearly in
Fig. 9(d) below.
Figure 8 shows the contours of Rτt ωx in the yz -plane across the streamwise maximum value
position. As shown in Fig. 8(c), the correlation Rτt ωx reaches a maximum value at approximately y + = 26 above the crest while the strongest correlation is located at y + = 8 when
the detecting point is on the leeward side (see Fig. 8(e)). This is consistent with the results
of Yang and Shen[27] . The near-wall streamwise vortices begin from the trough, extend to the
downstream direction, lift up above the crest, and are subsequently weakened above the leeward
side. The present results also indicate that the upper tilted structure is lifted up to a higher
position and elongated such that it is above the wave crest (see Fig. 7(c)) when compared with
that on the leeward side (see Fig. 7(e)). With respect to the case of c/Um = 1.4, almost the
same vertical maximum correlation coefficient and the peak position at different wave phases
as shown in Fig. 8 confirm the aforementioned phenomenon. Figure 9 shows the contours of
Rτt ωx on the wavy boundary. As mentioned above, the structure at the wall is due to the effect
of near-wall streamwise vortices and no-slip boundary condition. Given the wave motion, the
velocity at the wall corresponds to the surface wave orbital velocity. As shown in Fig. 9(e), the
correlation near the detecting point is opposite to that in the downstream region, and this can
be caused by the weak mean shear on the leeward side. Furthermore, the surface wave orbital
velocity varies with the wave phase, and thus the lower pair of structures exhibits significant
differences in Fig. 9. We take Fig. 9(d) as an example, and the streamwise orbital velocity is
the fastest when the detecting point is above the wave crest, and a long shear layer is formed
at the upstream of the detecting point for the case of c/Um = 1.4.
The iso-surfaces of Rτz ωx = ±0.15 at different wave phases are shown in Fig. 10. With
respect to the case of c/Um = 0.14, the upper structure is weakened above the leeward side (see
Fig. 10(e)) when compared with that at the other three wave phases. The increases in the wave
age to c/Um = 1.4 increase the size of the upper structure above the windward side and crest
compared with that above the leeward side and trough.
In contrast to Rτt ωx , the correlation Rτz ωx reaches a maximum value immediately above
the detecting point ∆x+ = 0 and ∆z + = 0. Figure 11 shows the contours of Rτz ωx in the
yz -plane across ∆x+ = 0. The maximum correlation coefficient is approximately 0.33 above
the leeward side (see Fig. 11(e)), while the correlation coefficient reaches its maximum value of
0.61 above the trough (see Fig. 11(g)). This is consistent with the upper structure as shown in
Figs. 10(e) and 10(g). With respect to the case of c/Um = 1.4, the magnitude of the maximum
correlation is almost constant across different wave phases. The vertical extent of the upper
structure above the windward side and crest exceeds that above the leeward side and trough, as
shown in Fig. 11, and this is also evident from the corresponding three-dimensional structure as
shown in Fig. 10. Although the maximum correlation coefficient is only 0.33 above the leeward
side for the case of c/Um = 0.14, the mean value of the maximum correlation exceeds that of
c/Um = 1.4, which is consistent with Rτz ωx , as shown in Fig. 6. Figure 12 shows the contours
of Rτz ωx on the wavy boundary. The correlation at the wall exhibits a strong dependence on
both the wave phase and wave age, which is also attributed to the effects of the mean shear
velocity and wave surface orbital velocity.
390
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
Z
∆[
B
Z
Z
Z
∆[
F
∆Y Z
∆Y ∆[
G
∆Y Z
∆[
E
∆Y Z
∆[
D
∆Y ∆[
C
∆Y Fig. 8
∆Y Z
∆Y ∆[
H
∆[
I
Contours of the two-point correlation coefficient between the streamwise vorticity and the tangential wall shear stress in the yz -plane across the streamwise maximum correlation positions
for the cases of c/Um = 0.14 (left) and c/Um = 1.4 (right) detecting at (a) and (b) windward,
(c) and (d) crest, (e) and (f) leeward, (g) and (h) trough, where ∆x+ in the subfigure denotes the streamwise position, in which the correlation coefficient reaches a maximum value,
solid lines represent positive values, and dashed lines represent negative values with a level
increment of 0.05
Relationship between wall shear stresses and streamwise vortices
Fig. 9
391
Contours of the two-point correlation coefficient between the streamwise vorticity and the
tangential wall shear stress at the wavy surface for the cases of c/Um = 0.14 (left) and
c/Um = 1.4 (right) at (a) and (b) windward, (c) and (d) crest, (e) and (f) leeward, (g) and (h)
trough, where solid lines represent positive values, and dashed lines represent negative values
with a level increment of 0.05
392
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
Z
0
Fig. 10
Y
[
B
C
D
E
F
G
H
I
Iso-surfaces of the two-point correlation coefficient between the streamwise vorticity and the
spanwise wall shear stress for Rτz ωx = 0.15 (red) and Rτz ωx = −0.15 (blue) with respect
to the cases of c/Um = 0.14 (left) and c/Um = 1.4 (right) at (a) and (b) windward, (c) and
(d) crest, (e) and (f) leeward, (g) and (h) trough, where the wall is colored by the surface
elevation, and the blue (white) color denotes the wave crest (trough) (color online)
Relationship between wall shear stresses and streamwise vortices
∆[
F
∆[
E
Z
Z
∆[
D
∆[
G
Z
Z
∆[
H
∆[
C
Z
Z
∆[
B
Fig. 11
Z
Z
393
∆[
I
Contours of the two-point correlation coefficient between the streamwise vorticity and the
spanwise shear stress in the yz -plane across ∆x+ = 0 for the cases of c/Um = 0.14 (left) and
c/Um = 1.4 (right) at (a) and (b) windward, (c) and (d) crest, (e) and (f) leeward, (g) and
(h) trough, where solid lines represent positive values, and dashed lines represent negative
values with a level increment of 0.05
394
Lihao WANG, Weixi HUANG, Chunxiao XU, Lian SHEN, and Zhaoshun ZHANG
∆[
∆[
∆Y
E
∆Y
F
∆Y
G
∆[
∆[
∆[
∆[
∆Y
D
Fig. 12
∆Y
C
∆[
∆[
∆Y
B
∆Y
H
∆Y
I
Contours of the two-point correlation coefficient between the streamwise vorticity and the
spanwise shear stress at the wavy surface for the cases of c/Um = 0.14 (left) and c/Um = 1.4
(right) at (a) and (b) windward, (c) and (d) crest, (e) and (f) leeward, (g) and (h) trough,
where solid lines represent positive values, and dashed lines represent negative values with a
level increment of 0.05
Relationship between wall shear stresses and streamwise vortices
4
395
Conclusions
In the study, a DNS is performed for turbulent flows over a wavy boundary. The relationship between wall shear stresses including both the tangential and spanwise components and
near-wall streamwise vortices is investigated via the phase-resolved two-point correlation. The
overview of the correlation coefficient indicates that near-wall streamwise vortices are closely
related to the wall shear stresses, and the magnitude of maximum correlation coefficient varies
with the wave age. With respect to the case of c/Um = 0.14, the correlation between the tangential wall shear stress and the streamwise vorticity (Rτt ωx ) is attenuated, and the streamwise
length scale of the upper pair of structures is also reduced. Conversely, the correlation between
the spanwise wall shear stress and the streamwise vorticity (Rτz ωx ) is enhanced. When the
wave age increases to c/Um = 1.4, the iso-surface of correlation between the tangential wall
shear stress and the near-wall vortices is similar to that for the flat-wall case. Additionally,
the correlation between the spanwise wall shear stress and streamwise vorticity is significantly
reduced when compared with that of the case of c/Um = 0.14. Furthermore, the results reveal
that the correlation is significantly dependent on the wave phase. With respect to the case of
c/Um = 0.14, the vertical position of the maximum correlation Rτt ωx above the crest exceeds
that above the leeward side, and the streamwise length scale of the upper pair of structures
significantly decreases. This is consistent with the fact that the near-wall streamwise vortices
are lifted up above the crest and weakened above the leeward side. Additionally, the maximum
correlation coefficient Rτz ωx above the leeward side is approximately half of that above the
trough. With respect to the case of c/Um = 1.4, there is no significant difference in the upper
pair of structures of Rτt ωx in different wave phases. However, the vertical extent of the upper
structure of Rτz ωx above the windward side and crest exceeds that above the leeward side and
trough. The present findings on the relationship between wall shear stresses and near-wall
streamwise vortices can be applied to the detection of near-wall streamwise vortices based on
wall information in turbulent flows with complex-geometry boundaries.
Acknowledgements
The authors would like to thank Tsinghua National Laboratory for Information Science and Technology for support in parallel computation.
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