Dip Phenomenon in High-Curved Turbulent Flows and Application of Entropy Theory

The estimation of velocity profile in turbulent open channels is a difficult task due to the significant effects of the secondary flow. The present paper investigates the mechanism of the velocity-dip phenomenon, whereby the location of the maximum velocity appears to be below the free surface. Previous studies conducted in straight channels relate the mechanism of the velocity-dip phenomenon to secondary flow induced by anisotropy of turbulence. This work focuses on high-curved channels where the secondary motion, which is also induced by the channel’s curvature, evolves along the bend. The width-to-depth ratio, B/h, is one of the most important parameters that are affecting the secondary motion entity. In particular, the present study aims to investigate the evolution of the velocity-dip along the bend for two values of the width-to-depth ratio and the applicability of the entropic model for the dip-phenomenon estimation. The results show that the velocity-dip is more accentuated for low values of the width-to-depth ratio, where the secondary motion plays a fundamental role in the distribution of the downstream flow velocity, although the velocity-dip is also present when the aspect ratio is higher than 10. Furthermore, the velocity profiles that were estimated by applying the entropic model are in good agreement with the measured ones, especially for B/h < 10.


Introduction
The understanding of flow characteristics in different bed and plane shape conditions is very important in hydraulic engineering practice.In fact, the identification of such characteristics is useful in understanding the nature of turbulence and consequent transport processes.
Literature shows (among others [1,2]) that the turbulent structure in open-channel flows can be divided into three sub-regions: a wall region, which is controlled by the inner variables (flow viscosity, local shear velocity, distance normal to the boundary); an intermediate region, which is controlled both by wall and free surface proprieties; a free surface region, which is controlled by outer variables (flow depth, maximum or mean velocity).
The log law is universally accepted for the velocity distribution in the inner region; experiments (see in [2]) show that in the outer region the velocity data deviate from the log law.To express the velocity distribution in both the inner and outer regions, some authors ( [3][4][5][6][7]) suggested including an additional term, called the wake term.
Some researchers ( [8,9]), analyzing the velocity field and the importance of secondary currents over three-dimensional bedforms, also highlighted a deviation of flow from the logarithm behavior.
The effect of the free surface on turbulence is particularly important ( [10,11]).Literature widely reports on the velocity-dip phenomenon due to the position of the maximum velocity below the This behavior may influence the shape of the velocity profiles, and thus the entity of the velocity-dip phenomenon, along the bend.
Thus, the specific purpose of this paper is twofold: (1) to analyze the velocity-dip phenomenon along the bend for two values of the width-to-depth ratio; and, (2) to explore the applicability of the entropic model ( [24]) to estimate the velocity profile and the velocity-dip phenomenon, starting from the acknowledgment of the surface velocity.

Experimental Dataset
The data used in the present work were collected in the ambit of previous studies that aimed to analyze the kinematic characteristics of flow along a high-amplitude meandering channel.Details of the experimental setup and measurements conditions can be found in previous works (see as an example in [33]) and only features of particular interest for the following analyses are briefly presented here.
The channel follows the sine-generated curve with a deflection angle at the inflection section θ 0 = 110 • (see Figure 1).
Water 2018, 10, x FOR PEER REVIEW 3 of 10 in a large amplitude sine-generated laboratory flume, confirmed the aforementioned behavior.This behavior may influence the shape of the velocity profiles, and thus the entity of the velocity-dip phenomenon, along the bend.Thus, the specific purpose of this paper is twofold: (1) to analyze the velocity-dip phenomenon along the bend for two values of the width-to-depth ratio; and, (2) to explore the applicability of the entropic model ( [24]) to estimate the velocity profile and the velocity-dip phenomenon, starting from the acknowledgment of the surface velocity.

Experimental Dataset
The data used in the present work were collected in the ambit of previous studies that aimed to analyze the kinematic characteristics of flow along a high-amplitude meandering channel.Details of the experimental setup and measurements conditions can be found in previous works (see as an example in [33]) and only features of particular interest for the following analyses are briefly presented here.
The channel follows the sine-generated curve with a deflection angle at the inflection section θ0= 110° (see Figure 1).The channel cross-section is rectangular with width B = 0.50 m; the fixed vertical sidewalls are of plexiglass and the bed, with longitudinal centerline slope S = 0.371%, is flat and made of quartz sand (d50 = medium sediment diameter = 0.65 mm, σg = geometric standard deviation = 1.34).The meandering channel is around 25 m long in order to accommodate two meander wavelengths; the upstream and downstream ends of the meandering channel are connected to two straight channels 3 m and 2 m long, respectively.
In the present paper, the data collected during two runs conducted with two different values of flow discharge have been used: Run 1 was conducted with a flow discharge of Q = 0.019 m 3 /s (channel-averaged flow depth h = 5.5 cm; B/h = 9.09 < 10); Run 2 was conducted with a flow discharge Q = 0.007 m 3 /s (channel-averaged flow depth h = 3.0 cm, B/h = 16.67 > 10).Herein, in accordance with other authors ( [13,34]), B/h < 10 indicates "small" width-to-depth ratio and B/h > 10 indicates "large" width-to-depth ratio.
During each run, the water surface was measured using a Profiler Indicator PV09 (Delft Hydraulics) with precision of 0.1 mm.The instantaneous local longitudinal and transverse velocity components were measured in cross-sections spaced about 50 cm or so apart, starting from the first inflection section of the meandering channel (see Figure 1), by using a two-dimensional (2D) sidelooking Acoustic Doppler Velocimeter (ADV) manufactured by SonTek Inc. (San Diego, CA, USA), in points spaced of 1-1.5 mm along the verticals of nine transverse abscissas symmetrically to the The channel cross-section is rectangular with width B = 0.50 m; the fixed vertical sidewalls are of plexiglass and the bed, with longitudinal centerline slope S = 0.371%, is flat and made of quartz sand (d 50 = medium sediment diameter = 0.65 mm, σ g = geometric standard deviation = 1.34).The meandering channel is around 25 m long in order to accommodate two meander wavelengths; the upstream and downstream ends of the meandering channel are connected to two straight channels 3 m and 2 m long, respectively.
In the present paper, the data collected during two runs conducted with two different values of flow discharge have been used: Run 1 was conducted with a flow discharge of Q = 0.019 m 3 /s (channel-averaged flow depth h = 5.5 cm; B/h = 9.09 < 10); Run 2 was conducted with a flow discharge Q = 0.007 m 3 /s (channel-averaged flow depth h = 3.0 cm, B/h = 16.67 > 10).Herein, in accordance with other authors ( [13,34]), B/h < 10 indicates "small" width-to-depth ratio and B/h > 10 indicates "large" width-to-depth ratio.
During each run, the water surface was measured using a Profiler Indicator PV09 (Delft Hydraulics) with precision of 0.1 mm.The instantaneous local longitudinal and transverse velocity components were measured in cross-sections spaced about 50 cm or so apart, starting from the first inflection section of the meandering channel (see Figure 1), by using a two-dimensional (2D) side-looking Acoustic Doppler Velocimeter (ADV) manufactured by SonTek Inc. (San Diego, CA, USA), in points spaced of 1-1.5 mm along the verticals of nine transverse abscissas symmetrically to the channel axis (see details in [33]).Bearing in mind the specific aim of the present work, our attention is focused only on the longitudinal velocity component.Furthermore, in the present work attention concerns only two consecutive inflection sections (sections A and C of Figure 1) and the apex section B of Figure 1.

Pertinent Aspects and Summary of Previous Results
Before proceeding further, it seems important to briefly summarize here some peculiar results of investigations performed in the same meandering flume of that considered in this work: -Termini ( [33]) conducted experiments under the hydraulic conditions that are considered in the present work in order to examine the effect of the continuously changing channel's curvature on flow pattern.She verified that, because of the changing channel curvature, the flow accelerates near the outer bank from the beginning of the bend to the inflection section downstream and decelerates near the inner bank.The flow accelerated zone is more evident for B/h > 10 than for B/h < 10.Such a different behaviour is caused by the secondary circulation that develops more significantly in the case of B/h < 10 than in the case of B/h > 10, and that attenuates the effect of the convective acceleration close to the outer bank (see also in [31]).- Termini and Moramarco ([29]) investigated the effect of the downstream variation of the channel's curvature on the applicability of the linear entropic relationship between the maximum velocity, u max , and the mean flow velocity, u m , through a dimensionless parameter Φ(M).As result, they observed that the ratio u m/ u max , and thus the value of the parameter Φ(M), varies along the bend.In particular, Termini and Moramarco ([29]) demonstrated that, in contrast to what observed in straight channels ( [28]) where Φ(M) assumed a value that was almost constant and on average equal to 0.65, along the meandering bend the parameter Φ(M) increases passing from the inflection section to the apex section and decreases from the apex section to the inflection section downstream, assuming a value of around 0.8 close to the apex section and of around 0.6 at the inflection section.This behavior can be observed from Figure 2, where the values of the parameter Φ(M), as obtained by Termini and Moramarco ([29]), along the bend are reported.
Water 2018, 10, x FOR PEER REVIEW 4 of 10 channel axis (see details in [33]).Bearing in mind the specific aim of the present work, our attention is focused only on the longitudinal velocity component.Furthermore, in the present work attention concerns only two consecutive inflection sections (sections A and C of Figure 1) and the apex section B of Figure 1.

Pertinent Aspects and Summary of Previous Results
Before proceeding further, it seems important to briefly summarize here some peculiar results of investigations performed in the same meandering flume of that considered in this work: - Termini ( [33]) conducted experiments under the hydraulic conditions that are considered in the present work in order to examine the effect of the continuously changing channel's curvature on flow pattern.She verified that, because of the changing channel curvature, the flow accelerates near the outer bank from the beginning of the bend to the inflection section downstream and decelerates near the inner bank.The flow accelerated zone is more evident for B/h > 10 than for B/h < 10.Such a different behaviour is caused by the secondary circulation that develops more significantly in the case of B/h < 10 than in the case of B/h > 10, and that attenuates the effect of the convective acceleration close to the outer bank (see also in [31]).-Termini and Moramarco ( [29]) investigated the effect of the downstream variation of the channel's curvature on the applicability of the linear entropic relationship between the maximum velocity, umax, and the mean flow velocity, um, through a dimensionless parameter Φ(M).As result, they observed that the ratio um/umax, and thus the value of the parameter Φ(M), varies along the bend.In particular, Termini and Moramarco ([29]) demonstrated that, in contrast to what observed in straight channels ( [28]) where Φ(M) assumed a value that was almost constant and on average equal to 0.65, along the meandering bend the parameter Φ(M) increases passing from the inflection section to the apex section and decreases from the apex section to the inflection section downstream, assuming a value of around 0.8 close to the apex section and of around 0.6 at the inflection section.This behavior can be observed from Figure 2, where the values of the parameter Φ(M), as obtained by Termini and Moramarco ([29]), along the bend are reported.

Measured Velocity Profiles and Effect of the Aspect Ratio on the Velocity-Dip
The time-averaged longitudinal velocity, u, was estimated for each measurement point by using the series of the measured instantaneous longitudinal velocity component.Thus, the u-profiles (hereafter denoted as "measured profiles") were estimated for each measurement vertical.Then, the

Measured Velocity Profiles and Effect of the Aspect Ratio on the Velocity-Dip
The time-averaged longitudinal velocity, u, was estimated for each measurement point by using the series of the measured instantaneous longitudinal velocity component.Thus, the Water 2018, 10, 306 5 of 10 u-profiles (hereafter denoted as "measured profiles") were estimated for each measurement vertical.Then, the distance δ (dip) between the maximum of flow velocity below the water surface was evaluated for each vertical measured profile.
From the analysis of all the measured profiles appeared clear that the dip-phenomenon occurs for both the width-to-depth ratios.But, the value of the dip decreases as the width-to-depth ratio increases.As an example, Figure 3 shows the velocity profiles that were measured in peculiar verticals (i.e., where the dip was evident) of the examined sections for both the aspect ratios.Furthermore, it was observed that, while in Run 1 (i.e., for B/h < 10) the velocity-dip is evident in all the measured profiles, in Run 2 (i.e., for B/h > 10), the velocity-dip is especially evident in the regions that are close to the banks.This behavior is consistent with previous literature findings obtained in straight channels (among others [2,13]).
The entity of the distance δ (dip) varies along the bend.This behavior can be observed from Table 1, where the values of the distance δ determined for each measured profile as percentage of the local flow depth are reported for both runs, and from Figure 4, which compares the δ-values that were estimated in the apex section and in the inflection sections for both the width-to-depth ratios.From Table 1 and Figure 4, it can be noted that the velocity-dip is more evident when the aspect ratio B/h is less than 10, but it is also present when the aspect ratio B/h is higher than 10.In particular, as Table 1 shows, in average, the dip assumes a value of around 53% of the flow depth in Run 1 (i.e., for B/h < 10) and of around 34% of the flow depth in Run 2 (i.e., for B/h > 10).
Water 2018, 10, x FOR PEER REVIEW 5 of 10 distance δ (dip) between the maximum of flow velocity below the water surface was evaluated for each vertical measured profile.
From the analysis of all the measured profiles appeared clear that the dip-phenomenon occurs for both the width-to-depth ratios.But, the value of the dip decreases as the width-to-depth ratio increases.As an example, Figure 3 shows the velocity profiles that were measured in peculiar verticals (i.e., where the dip was evident) of the examined sections for both the aspect ratios.Furthermore, it was observed that, while in Run 1 (i.e., for B/h < 10) the velocity-dip is evident in all the measured profiles, in Run 2 (i.e., for B/h > 10), the velocity-dip is especially evident in the regions that are close to the banks.This behavior is consistent with previous literature findings obtained in straight channels (among others [2,13]).
The entity of the distance δ (dip) varies along the bend.This behavior can be observed from Table 1, where the values of the distance δ determined for each measured profile as percentage of the local flow depth are reported for both runs, and from Figure 4, which compares the δ-values that were estimated in the apex section and in the inflection sections for both the width-to-depth ratios.From Table 1 and Figure 4, it can be noted that the velocity-dip is more evident when the aspect ratio B/h is less than 10, but it is also present when the aspect ratio B/h is higher than 10.In particular, as Table 1 shows, in average, the dip assumes a value of around 53% of the flow depth in Run 1 (i.e., for B/h < 10) and of around 34% of the flow depth in Run 2 (i.e., for B/h > 10).This is probably due to the fact that, as literature demonstrates ( [32,35]), the entity of the secondary circulation increases as the ratio B/h decreases.Thus, the observed behavior also indicates that the evolution of the velocity-dip phenomenon along the bend is strongly related to the evolution of the cross-sectional secondary motion along the bend itself.
Figure 4 also indicates that, for B/h < 10, the velocity-dip always assumes significant values: at the inflection section upstream, the velocity-dip has the higher values that are close to the left (outer) bank; at the apex section high values of the velocity-dip occur almost in the whole cross section until that at the inflection section downstream δ assumes the higher values close to the right (inner) bank.For B/h > 10, the velocity-dip assumes always very small values and the higher values of the velocitydip can be found close to the left (outer) bank at the apex section.This is probably due to the fact that, as literature demonstrates ( [32,35]), the entity of the secondary circulation increases as the ratio B/h decreases.Thus, the observed behavior also indicates that the evolution of the velocity-dip phenomenon along the bend is strongly related to the evolution of the cross-sectional secondary motion along the bend itself.
Figure 4 also indicates that, for B/h < 10, the velocity-dip always assumes significant values: at the inflection section upstream, the velocity-dip has the higher values that are close to the left (outer) bank; at the apex section high values of the velocity-dip occur almost in the whole cross section until that at the inflection section downstream δ assumes the higher values close to the right (inner) Water 2018, 10, 306 6 of 10 bank.For B/h > 10, the velocity-dip assumes always very small values and the higher values of the velocity-dip can be found close to the left (outer) bank at the apex section.) developed an iterative procedure to estimate the velocity profile and the velocity-dip based on the entropy profile, which can be written as:

Application of the Entropic Model to Estimate the Velocity-Dip Phenomenon
Recently, Moramarco et al. ( [24]) developed an iterative procedure to estimate the velocity profile and the velocity-dip based on the entropy profile, which can be written as: Water 2018, 10, 306 7 of 10 where u max,v is the maximum velocity along the each measurement vertical, D is the water depth, z is the distance the velocity point from the bed, x is the distance of the considered vertical from the left bank, and M is the entropic parameter, which can be estimated by using the entropic relation ( [25]): where u max = max[u max,v ] which is located at z-axis.The velocity-dip is estimated by using the modified expression proposed by Yang et al. ( [16]), written as ( [24]): where a p is an iterative parameter representing the dip at z-axis, and which is updated at each iteration, p, by adding a constant value (0.001 m), and starting from the initial value a 1 = 0.0001.By inspecting Equations ( 1)-( 4), it is clear that, in contrast to other formulations that are proposed in literature to estimate the velocity profile and the velocity-dip (among others [4,12,13]), the aforementioned procedure only depends on parameters that have a clear physical meaning and are easily measurable.
Starting from the observed surface velocity, u surf , and the knowledge of Φ(M), the maximum velocity u max,v and the velocity-dip are computed, respectively, through Equations ( 2) and ( 4) by initializing a p = a 1 .Then, the velocity profiles are computed through Equation ( 1) and the maximum velocity u max = max[u max,v ] is assessed.Once the velocity profiles are identified, the mean flow velocity, u m , can be assessed by applying the velocity area method ( [38]) at the estimated velocity profiles; therefore, the ratio u max (a 1 ) can be computed by Equation ( 3).The iterative procedure ends when the optimal value of Φ com M p is achieved in accordance with: Thus, for each measurement vertical and for each run, the velocity profiles (hereon called as "estimated velocity profiles") have been determined by applying the aforementioned iteration procedure and starting from the acknowledgment of the measured surface velocity.Figure 5 shows, as an example, the comparison between the measured and the estimated velocity profiles in the selected measurement verticals.From this figure, it can be observed that the estimated velocity profiles are in good agreement with the measured ones.This is especially evident in the case of B/h < 10.
Water 2018, 10, x FOR PEER REVIEW 7 of 10 where umax = max [umax,v] which is located at z-axis.The velocity-dip is estimated by using the modified expression proposed by Yang et al. ([16), written as ( [24]): where is an iterative parameter representing the dip at z-axis, and which is updated at each iteration, p, by adding a constant value (0.001 m), and starting from the initial value = 0.0001.
By inspecting Equations ( 1)-( 4), it is clear that, in contrast to other formulations that are proposed in literature to estimate the velocity profile and the velocity-dip (among others [4,12,13]), the aforementioned procedure only depends on parameters that have a clear physical meaning and are easily measurable.
Starting from the observed surface velocity, usurf, and the knowledge of Φ(M), the maximum velocity umax,v and the velocity-dip are computed, respectively, through Equations ( 2) and ( 4) by initializing ap = a1.Then, the velocity profiles are computed through Equation ( 1) and the maximum velocity umax = max[umax,v] is assessed.Once the velocity profiles are identified, the mean flow velocity, um, can be assessed by applying the velocity area method ( [38]) at the estimated velocity profiles; therefore, the ratio can be computed by Equation (3).The iterative procedure ends when the optimal value of is achieved in accordance with: Thus, for each measurement vertical and for each run, the velocity profiles (hereon called as "estimated velocity profiles") have been determined by applying the aforementioned iteration procedure and starting from the acknowledgment of the measured surface velocity.Figure 5 shows, as an example, the comparison between the measured and the estimated velocity profiles in the selected measurement verticals.From this figure, it can be observed that the estimated velocity profiles are in good agreement with the measured ones.This is especially evident in the case of B/h < 10.The good fit between all of the δ-values determined from the estimated velocity profiles and those that were obtained the measured velocity profiles has been verified by using the root mean squared error (σ) as indicator.It is: where δ i,m and δ i,e are the δ-values determined, respectively, from the measured and from the estimated profiles, N is the total number of measurements that are considered for each profile.Figure 6 reports the pair of values (δ i,m , δ i,e ) and the line of perfect agreement (bisector line).The error bar is defined by the value of σ.As shown in Figure 6, with a few exceptions, the points arrange around the bisector line and the σ is quite low in comparison to the magnitude of the measured values.
Water 2018, 10, x FOR PEER REVIEW 8 of 10 The good fit between all of the δ-values determined from the estimated velocity profiles and those that were obtained from the measured velocity profiles has been verified by using the root mean squared error (σ) as indicator.It is: where δi,m and δi,e are the δ-values determined, respectively, from the measured and from the estimated profiles, N is the total number of measurements that are considered for each profile.Figure 6 reports the pair of values (δi,m, δi,e) and the line of perfect agreement (bisector line).The error bar is defined by the value of σ.As shown in Figure 6, with a few exceptions, the points arrange around the bisector line and the σ is quite low in comparison to the magnitude of the measured values.

Discussion and Concluding Remarks
The mechanism of the dip phenomenon is investigated in a high-amplitude meandering channel, which is characterized by radii of curvature continuously variable in stream-wise direction, and, as previous findings indicate (see as an example [31,32]), convective flow accelerations occur and the entity of secondary motion changes along the bend modifying the turbulent activity at the channel's boundaries.
As it should be clear from the "Introduction", previous researches that were conducted in straight channels relate the velocity-dip to the secondary currents occurring especially in narrow channels where the side-walls effect is strong.Experiments (see among others in [2,13]) show that in narrow channels the side-walls produce anisotropy of turbulence but the velocity-dip is also determined by the free-surface effect.In wide channels, there is a central area where the side-wall effect disappears and only the free-surface effect occurs.
The results obtained in this work have shown that the velocity-dip phenomenon is more accentuated for low aspect ratio (B/h < 10), but it is also present for large aspect ratio (B/h > 10).In particular, in the first case, the velocity-dip, δ, assumes always significant values, but it varies along the bend.In particular, at the inflection section upstream it is more accentuated close to the left (outer) bank; then at the apex section high values of the velocity-dip occur almost in the whole cross section until that at the inflection section downstream δ assumes the higher values close to the right (inner) bank.For high aspect ratio, the velocity-dip assumes always very small values and, in accordance with results obtained in straight channels, the dip-phenomenon assumes the lower values especially in the central region of the cross section.Furthermore, the variation of the δ-values along the bend that were observed for high aspect ratio is less evident than that observed for low aspect ratio.

Discussion and Concluding Remarks
The mechanism of the dip phenomenon is investigated in a high-amplitude meandering channel, which is characterized by radii of curvature continuously variable in stream-wise direction, and, as previous findings indicate (see as an example [31,32]), convective flow accelerations occur and the entity of secondary motion changes along the bend modifying the turbulent activity at the channel's boundaries.
As it should be clear from the "Introduction", previous researches that were conducted in straight channels relate the velocity-dip to the secondary currents occurring especially in narrow channels where the side-walls effect is strong.Experiments (see among others in [2,13]) show that in narrow channels the side-walls produce anisotropy of turbulence but the velocity-dip is also determined by the free-surface effect.In wide channels, there is a central area where the side-wall effect disappears and only the free-surface effect occurs.
The results obtained in this work have shown that the velocity-dip phenomenon is more accentuated for low aspect ratio (B/h < 10), but it is also present for large aspect ratio (B/h > 10).In particular, in the first case, the velocity-dip, δ, assumes always significant values, but it varies along the bend.In particular, at the inflection section upstream it is more accentuated close to the left (outer) bank; then at the apex section high values of the velocity-dip occur almost in the whole cross section until that at the inflection section downstream δ assumes the higher values close to the right (inner) bank.For high aspect ratio, the velocity-dip assumes always very small values and, in accordance with results obtained in straight channels, the dip-phenomenon assumes the lower values especially in the

Table 1 .
δ-values estimated for each measured profile as percentage of the local flow depth.

Table 1 .
δ-values estimated for each measured profile as percentage of the local flow depth.