3.1. Transverse Velocity Distribution
Figure 3 shows the transverse distribution of longitudinal velocity for different cases. The abscissa is the normalized width
y/
B, where
B (=0.4 m) is the width of the flume. The ordinate is the longitudinal velocity, and its value is the average velocity of 120 s measured by ADV. The blue vertical dotted line in the figure indicates the boundary between the vegetation area and the non-vegetation area, that is, the center line of the flume, 20 cm from the convex bank.
Case 1 was taken as an example to show the influence of vertical positions in five characteristic sections on the transverse distribution of longitudinal velocity in
Figure 3a. For each section along the curve, the longitudinal velocity is generally small in the vegetation area but large in the non-vegetation area. In addition, there is an obvious velocity gradient between vegetation and non-vegetation areas as in the case of a straight channel [
17]. The results also show that the transverse distribution of velocity at different depth is basically similar, and there is a small velocity difference only near the side wall, for the wall effect is greater than the vegetation resistance. It implies that the change in vertical depth position does not have an impact on the trend of transverse velocity distribution and the velocity in the mixed layer when a rigid round stick with vertically uniform resistance was used as simulated vegetation. Therefore, the medium depth survey line of
z = 12 cm from the bottom was taken as an example to analyze the test cases.
The results in
Figure 3b show that the velocity distribution was jointly affected by the effects of vegetation density and vegetation distribution and the curve effect. For the same distribution cases, velocity along the transverse direction for Cases 1 and 3 (
= 2.235%) is significantly larger than that for Cases 2 and 4 (
= 4.47%), indicating that the vegetation resistance is still the main factor to affect the velocity. However, for the same difference in vegetation resistance, the variation in velocity in the concave case (the difference of velocity between Cases 2 and 4) is considerably greater than that in convex case (the difference of velocity between cases 1 and 3). It can be seen that the response of velocity to the change of vegetation density in the concave bank is faster. In addition, when vegetation density is the same in the concave or convex bank, the velocity distribution for the concave and convex bank is symmetrical with respect to the vegetation boundary (i.e.,
y/
B = 0.5) for sparse vegetation (
= 2.235%). On the other hand, when the vegetation density is greater (
= 4.47%), the velocity distribution for the concave and convex cases is no longer symmetrical. The velocity
U1 in the vegetation area is still similar, but there is a significant difference in the velocity
U2 in the non-vegetation area. For example,
U2 at the 0 ° section of case 2 is 0.4534 m/s, while for Case 4 it is 0.3477 m/s. This result reveals that the change of vegetation density on convex bank and concave bank has different effects on transverse velocity distribution as [
22] mentioned.
The effect of the channel bend on velocity distribution is mainly represented by the secondary flow. The velocity
U1 has almost no significant change in the vegetation area of each section during the development along the curve, but the velocity
U2 in the non-vegetation area of the convex case gradually increase from the 0° section to180° section. The decrease along the curve of the velocity difference
ΔU (in
Table 1) also proves the maximum velocity
U2 was not completely developed. This trend mainly appears in the velocity of concave area, while the velocity of convex area is not affected by the curve. This is possibly due to the secondary flow induced by the channel bend.
For all experimental cases, the velocity ratio
λ that characterizes the mixing layer ranges between 0.45~0.81, which is greater than 0.3 [
16,
32], the threshold to initiate the K-H instability. It is confirmed that K-H instability exists in all experimental cases, and there is an inflection point of transverse velocity profiles and large velocity gradient. Therefore, this research is applicable to the mixed layer theory affected by the K-H instability [
33].
Since there is an inflection point in the shear section of the velocity distribution, the shape is similar to the standard hyperbolic tangent function of the mixed layer:
Cases 2 and 4 with large vegetation density were taken as examples to compare the measured data with the theoretical formula in Equation (8), as shown in
Figure 4. The points in
Figure 4 are the measured data of different vertical positions, and the blue line is the theoretical curve. The results in
Figure 4 show that the transverse distribution of the measured longitudinal velocity is in good agreement with the hyperbolic tangent function of the mixed layer, suggesting the similarity of the mixed layer in straight reach and curved reach (both convex and concave cases).
Compared with the fitting results of the concave case in
Figure 4b, the results of the convex case shown in
Figure 4a are located slightly above the theoretical hyperbolic tangent curve, while the values in the concave case are below the theoretical curve. The result of the convex case with the range
is better fit to the theoretical curve, especially at the 135° and 180° sections, while the fitting degree of the concave case at both ends of the abscissa (
and
) is better in the latter part of the curve. This is mainly because the centrifugal force points to the concave bank in the curve, and the velocity ratio
λ of the convex case is larger than that in the concave case.
In addition, the thickness
θ of the mixed layer with vegetation density
= 2.235% is basically larger than that with
= 4.47%, which decreases first and then increases along the curved channel, and finally reaches the minimum in the middle section of the curved channel, as shown in
Figure 5. For the Cases 2 and 4 (
= 4.47%), the thickness
θ of the convex case increases along the section, while that of the concave case has little change in each section. The phenomenon is mainly due to the difference in longitudinal velocity distribution between the convex and concave cases, as previously described.
3.2. Turbulence Intensity Analysis
In the section,
TKE (turbulence kinetic energy) is used to estimate the intensity of turbulence, which is defined by:
where
u′,
v′ and
w′ are, respectively,
x,
y and
z three-dimensional fluctuating velocity, which is
,
and
, and
,
,
is the three-dimensional time average velocity.
Figure 6 analyzes
TKE for different cases and cross sections, taking ADV measurements at
h = 12 cm as an example. For Case 5 (bare bed,
Figure 6e),
TKE increases as the flow goes into the curved part, and
TKE at
y/
B = 0.5 reaches the peak at the 135 ° section while
TKE at
y/
B = 0.75 reaches the peak at the 180° section, indicating that the strongest turbulence mainly occurs at the section. In addition, the
TKE of the curved section is greater than that of the straight section, which means that the curved section enhances the turbulent intensity by promoting the mixing of water flow. This result is consistent with the results of Termini [
34] and Yang et al. [
22].
For Cases 1 to 4, the minimum TKE of each section appears in the vegetation area (y/B < 0.4 in Cases 1 and 2 of vegetation distributed along the convex bank and y/B > 0.6 in Cases 3 and 4 of vegetation placed along the concave bank), and reaches a stable value close to the turbulent kinetic energy of Case 5, which is approximately 0.001 m2/s2 in each section. The TKE in the vegetation zone of both Case 2 and Case 4 were lower than that of Case 5. The result shows that the vegetation with 4.47% can inhibit turbulence in the vegetation zone, leading to the reduction in TKE, which is different from the vegetation with 2.235%. Compared with the vegetation arranged on convex bank, the vegetation arranged on the concave bank has a stronger inhibition effect. However, the peak values of TKE appear near the non-vegetation area at the lateral interface between vegetation and non-vegetation areas (0.5 < y/B < 0.7 in the convex case, 0.4 < y/B < 0.5 in the concave case). This phenomenon is due to the existence of large velocity gradient and inflection point of longitudinal velocity profile in this area.
By comparing the peak
TKE values for different cases, it is found that the influence of the location along a curved channel is greater than that from vegetation density and distribution on peak
TKE values. For the convex case, the ratio of the peak
TKE values between 180° and 0° sections is 1.5~2, while the ratio becomes 3~5 for the concave case, indicating that the vegetation occupied along the concave bank has a greater impact on the turbulent characteristics. This is because the combined effect of vegetation and channel curve enhances the mixing between water bodies, and the wake effect of vegetation flow is amplified by the bend, further elevating the turbulence of water bodies. Furthermore, the centrifugal force pointing at the concave bank intensifies the turbulence intensity on the concave bank. Since the peak values of
TKE of the convex case appear in the non-vegetation area near the concave bank, the peak value is not only larger and deeper into the non-vegetation area than that of the concave case, but also decays in a slower rate. Therefore, the region of peak
TKE values perform a round shape for Cases 1 and 2 and exhibits a spike shape for Cases 3 and 4, as shown in
Figure 6.
The change in vegetation density will alter vegetation resistance, subsequently affecting the TKE. When the vegetation is denser, the TKE values become smaller, and the transverse position of the TKE peak will shift to the non-vegetation area. The maximum TKE value appears at the 135° section for cases with 4.47%, while the maximum turbulent kinetic energy can be found at 45 ° section for cases with = 2.235% density, which does not affect the variation in TKE. However, the change of the transverse position where the peak TKE value appears indicates that the TKE is comprehensively affected by the vegetation density, distribution and channel curve.
3.3. Power Spectral Density Analysis
The turbulence spectrum gives the occurrence possibility F (
f) (i.e., the spectral density) of turbulent eddies with different frequency
f at a fixed point. The spectral analysis is based on the Welch method [
35] with Hamming-type windowing [
36] (using commercial software Matlab, Mathworks). The high-frequency part represents small-scale turbulence vortex and the low-frequency part denotes large-scale turbulence vortex in the turbulence spectrum. The structure of turbulence vortex, and the transport and dissipation of
TKE can be understood through spectral analysis [
37].
Figure 7 and
Figure 8 show the variation in power spectral density (PSD)
Svv of the lateral fluctuation velocities measured by 200 Hz ADV with different frequency
f. The noise in the turbulence spectrum is removed by the moving average method.
To understand the effect of depth on PSD, the data of three depths (
h = 2 cm, 12 cm, and 18 cm from the bottom) at the center line of the 45° section for Case 2 (λ = 0.79) are compared in
Figure 7. It can be seen from the figure that the PSD in the inertial subrange and dissipation range at different heights is similar but shows certain differences in the energetic range (PSD in
h = 12 cm > PSD in
h = 2 cm > PSD in
h = 18 cm). This result indicates that the turbulence vortex structure along the vertical direction is approximately the same with a certain difference only in the large-scale vortex. Therefore, only the PSD at middle position (
h = 12 cm) is analyzed afterwards.
Figure 8 displays the variation of PSD of lateral fluctuation velocity for Cases 2 and Case 4, and the velocity ratio of each section λ was provided as well. For each case, the PSD at the vegetation and non-vegetation interface area (
y/
B = 0.5) is the largest, which is due to the strongest interaction between vegetation and water in this area and the greater turbulence intensity, consistent with the
TKE results in the previous section. The spectrum in vegetation areas and non-vegetation areas has no obvious trend in the energetic range, but the PSD in vegetation areas (
y/
B = 0.25 in
Figure 8a and
y/
B = 0.75 in
Figure 8b) reaches the minimum value in the dissipation range among the three regions, suggesting that the small-scale turbulence in vegetation area is the least, while there are more small-scale turbulence eddies in the interface area and non-vegetation area. Additionally, the PSD peak in the convex case is more obvious than that in concave case.
The attenuation rate of the PSD is compared with several characteristic slopes in the classical turbulence spectral density plot, where the blue line represents
f−3, the green line denotes
f−1, and the purple line means
f−5/3. In the bare-bed case in
Figure 8, the attenuation rate of the PSD in the dissipation region of each section from 0° to 180° is consistent with Kolmogorov’s law (
k = −5/3), indicating that the curve does not affect the attenuation of energy. However, the PSD at the 0° section is different from curve sections in the convex case and concave case. At the 0° section, the attenuation rate in the vegetation area (
y/
B = 0.25 in the convex case and
y/
B = 0.75 in the concave case) is still
k = −5/3, but in the non-vegetation area the rate is changed to
k = −1, and at the interfacial area, the rate ranges between
k = −1 and
k = −5/3. In curve sections, the attenuation rate is basically close to
k = −1 and considerably different from
k = −5/3 proposed by Kolmogorov’s law, which is consistent with the results obtained by Huai et al. [
36]. The result may be due to the existence of vegetation in the curved channel, which leads to the decrease in turbulent eddies, the increase of small-scale turbulent vortex and the energy dissipates at a greater rate than that in a straight channel because of viscous friction. In addition, the attenuation rate of the curve sections varies in different regions, and it can be ranked as:
k (non-vegetation area) >
k (vegetation area) >
k (interfacial area), and at the interface area
k is more consistent with Kolmogorov’s law.
The turbulent motion is strongest at the interfacial area, and the mixing layer in the interfacial area is affected by Kelvin–Helmholtz (KH) instability, generating a K-H vortex. The peak frequency
f KH of the dominant K-H vortex at this area obtained from
Figure 7 and
Figure 8 varies between 0.2 and 0.4 Hz. The relationship between the peak frequency
f and the momentum thickness of the mixing layer
θ is described by the Strouhal number (
St) as given below.
The classical
St value in the standard mixing layer is 0.032 [
37]. The momentum thickness of the mixed layer and the frequency of K-H instability of convex and concave cases are listed in
Table 2. In the table,
f represents the frequency calculated from
St = 0.032 in the turbulence spectrum, which is obtained by Equation (10) and represented by the vertical blue dotted line in
Figure 8.
fKH is the peak frequency determined from experiments, i.e., the dominant vortex frequency of the mixing layer in the curved channel. The comparison in
Figure 8 shows that the measured peak frequency
fKH is in good agreement with the theoretical frequency
f, which proves that the large-scale vortex structure existing at the interfacial area of the curved channel is the K-H vortex. In some sections,
f <
fKH is due to the value of
Uc is rather small. The K-H vortex peak frequency
fKH in the experiment is the largest at the 0° section and the smallest at the 180° section irrespective of vegetation arrangement, and it decreases gradually as flow moves into the curved part of the channel, indicating that the curved channel promotes the development of turbulence and increases the K-H vortex in the mixed layer.
The 90° and 135° sections in the second half of the curve were taken as an example to show the influence of vegetation density on turbulence spectrum in the curve, as shown in
Figure 9. The measured points in the
Figure 9 were located at the middle depth, 12 cm away from the bottom bed. It can be seen from the figure that the peak frequency at the interfacial area is 0.6~0.7 Hz for Case 1 (
2.235%), while the peak frequency at the interfacial area becomes 0.1~0.4 Hz for Case 3 (
4.47%), which is obviously lower than the peak frequency for Case 1 (sparse vegetation). It is the same conclusion as the rigid vegetation experiment of Caroppi [
17], that is, the peak frequency gradually increases with the decrease in vegetation density. This is because the decrease of vegetation density leads to the decrease of velocity ratio λ, which changes the turbulent structure and frequency of vortex in the mixed layer.
Furthermore, there is no significant difference for the PSD values of convex and concave cases when the vegetation density is 4.47%, which reveals that the effect of dense vegetation plays a more important role on turbulent structure than the curved channel does. However, the peak value of the concave case is not evident when the vegetation density is 2.235%.
3.4. Turbulent Bursting
The quadrant analysis was used to clarify the influence of vegetation density, distribution form, and channel curve on coherence structure near the bed. In turbulence bursting, four types of contribution to Reynolds stress are classified according to the sign of instantaneous fluctuating velocity
u′ and
w′, as follows [
38]:
The first quadrant (Q1): u′ > 0, w′ > 0 is the outward interaction;
The second quadrant (Q2): u′ < 0, w′ > 0 is the ejection;
The third quadrant (Q3): u′ < 0, w′ < 0 is the inward interaction;
The fourth quadrant (Q4): u′ > 0, w′ < 0 is the sweep.
Each quadrant can represent a type of turbulence event. A threshold value
H0 was set to remove the influence of small values in quadrant analysis. Only the Reynolds stress contribution greater than this threshold in each quadrant was calculated [
39]. This threshold value is reflected in the quadrant as four hyperbolas that replace 0, that is,
, where
u′ w′ represents the Reynolds stress. The contribution of each quadrant is represented by
Sk, where
k = I, II, III, IV represents four quadrants:
In this section, the threshold value
H0 = 1.0 [
40,
41] and the occurrence frequency
fk of turbulence events in the four quadrants can be expressed as:
where
T is the recording period.
Figure 10 shows the occurrence frequency of different turbulence events at different cross section when vegetation is present on the convex bank or concave bank. Vegetation density
4.47% was taken as the example so that the result was more obvious than the case of
2.235%. The measuring points were all located at 2 cm from the bottom bed.
Figure 10b,e show that the occurrence frequency of the inward interaction and outward interaction at the interface from the 0° to 135° section is greater than the frequency of ejection and sweep, which is the same as that of the vegetation occupied in the straight channel [
42]. The reason is that the near-bed Reynolds stresses with vegetation are generally lower than that without vegetation, because the inward and outward interactions that dominant turbulence activities have negative contribution on Reynolds stress. The inward interaction and outward interaction dominate the turbulent bursting, which indicates that the vegetation in the curved channel also changes Reynolds stress contribution. At the 180° section, the frequency ratio of ejection and sweep increases, possibly due to the influence of the curve.
For
Figure 10c,d (in the non-vegetation areas), the ejection and sweep events play a dominant role in turbulent bursting, similar to the turbulent events in the non-vegetation flow of a straight channel. In the vegetation area of the convex case, the turbulence event in the first half of the bend is obviously dominated by inward and outward interactions, while the contribution of ejection and sweep to Reynolds stress increases with the development of the bend, and the final contribution from four turbulence events is similar. However, the results of the vegetation area for the concave case are different from other areas affected by vegetation, and the Reynolds stress contribution is not determined by inward and outward interactions, but by ejection and sweep.
The turbulent bursting in the non-vegetation area of convex and concave cases is consistent with the bare-bed case in Yang’s study [
22]: the occurrence frequency of sweep and ejection is higher at the 0° section, but the frequency of inward and outward interactions gradually increases under the influence of the curve, and the frequency of occurrence for the four events is similar at the 180° section. However, the frequency of the four turbulence events after the 90° section is approximately the same in the vegetation and interfacial area affected by vegetation (
4.47%), which also happens after the 45° section in Yang’s vegetation interfacial area (
2.235%). This is because the Reynolds stress at the bottom of the vegetation area is smaller, and the contribution of each type of Reynolds stress is relatively uniform. This special phenomenon reflects that the joint impact of curved and half-sided vegetation on turbulent bursting events is complicated.
The bursting phenomenon also can be assessed according to the skewness coefficients [
43], which can provide the information of the material exchange between vegetation area and non-vegetation area. Skewness coefficients are defined as:
where
N is the sampling number,
and
are the skewness coefficients of
u and
v, respectively. The four types of contribution to Reynolds stress mentioned above can be described by skewness coefficients: outward interaction (
), ejection (
), inward interaction (
) and sweep (
).
Figure 11 shows the lateral distribution of skewness coefficients at
h = 12 cm. For the Cases 2 and 4 (
4.47%), the sweep and ejection can be clearly observed: there is sweep in the vegetation area (left region (
y < 0.5) for Case 2, right region (
y > 0.5) for Case 4) and ejection in the non-vegetation area (right region (
y > 0.5) in Case 2, left region (
y < 0.5) in Case 4), which elucidates the characteristics of the mixed layer and lateral exchange of mass and momentum.
Indeed, the lateral exchange of mass and momentum between the two areas of the channel was observed to decrease with decreasing vegetation density. For Cases 1 and 3 ( 2.235%), there is still in non-vegetation area, but has no evident trend like Cases 2 and 4. In general, the dominant turbulence event is apparent at the vegetation and non-vegetation interface for denser vegetation ( 4.47%) but not in sparser vegetation ( 2.235%).